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2500	https://www.sciencedirect.com/science/article/abs/pii/S0997754618305946	Influence of Froude number and submergence depth on wave patterns - ScienceDirect Typesetting math: 100% Skip to main contentSkip to article Journals & Books Access throughyour organization Purchase PDF Search ScienceDirect Article preview Abstract Introduction Section snippets References (38) Cited by (27) European Journal of Mechanics - B/Fluids Volume 75, May–June 2019, Pages 258-270 Influence of Froude number and submergence depth on wave patterns Author links open overlay panel Huiyu Wu a, Jiayi He a, Hui Liang b, Francis Noblesse a Show more Add to Mendeley Share Cite rights and content Highlights •Consider a monohull ship, a submerged point source, and a fully-submerged body. •Illustrate influence of Froude number and submergence depth on wave patterns. •Appearance of actual wave patterns greatly differs from Kelvin pattern in many cases. •Dominant waves can be found outside or well inside the cusps of the Kelvin wake. •The Kelvin–Havelock–Peters (KHP) farfield analytical approximation is applied. Abstract The Kelvin–Havelock–Peters farfield approximation is applied to illustrate the influence of the Froude numberF based on a ship length L, and – for a body below the free surface – the influence of the Froude numberF Δ based on the submergence depth Δ, on a ship wave pattern in deep water. The influence of the Froude numberF is illustrated for a free-surface piercing monohull ship; and the influence of the Froude number F Δ is illustrated for a point source submerged at a depth Δ below the free surface. The influences of F and of the submergence depth Δ∕L, which is related to F Δ and is used instead of F Δ, are illustrated for a fully-submerged body. These numerical illustrations show that a ship wave pattern – which does not depend on L and Δ within Kelvin’s classical analysis for a ship modeled as a 1-point wavemaker – in fact is greatly influenced by the Froude numbers F and F Δ. In particular, at high Froude numbers, e.g.at F=1.5, a ship wave pattern mostly contains divergent waves that are most apparent well inside the cusps of the Kelvin wake due to interferences between the dominant waves created by the bow and the stern of the ship, as was previously explained. A very different wave pattern is obtained at low Froude numbers, e.g.at F=0.2, for which the dominant waves are found outside the cusps of the Kelvin wake. Indeed, the wave patterns created by surface-piercing or submerged bodies can differ greatly from Kelvin’s classical pattern of transverse and divergent waves found inside a 39° wedge aft of a ship. Introduction The farfield waves and wave pattern created by a free-surface piercing ship or a fully-submerged body that travels at a constant speed V in calm water of large depth is a classical topic that has been widely considered in a broad literature; E.g., , , , , , , , , , , , , , , , . In particular, important features of a farfield ship wave pattern, commonly called Kelvin wake, have been explained by Kelvin and are basic knowledge given in every textbook on water waves. Kelvin’s classical analysis shows that ship waves are found inside a wedge with half angle ψ K≡arcsin(1∕3)≈19°2 8′and actually consist of two systems of waves, called transverse and divergent waves, as is depicted in Fig.1. Kelvin’s analysis assumes that the waves due to a ship originate from a single point, which may be taken at the centroid of the ship. Although the 1-point wavemaker model is adequate to explain the basic features of ship waves depicted in Fig.1, this crude ship model cannot account for important features that depend on the ship type (e.g.,free-surface piercing monohull ship or catamaran, fully-submerged body), the ship size (length) and the geometry of the ship hull (notably, the beam/length and draft/length ratios). Indeed, the Kelvin wave pattern depicted in Fig.1 only depends on the coordinates (X,Y)g∕V 2 and is valid for every ship, including monohull ships, catamarans and fully-submerged bodies. In particular, Kelvin’s wave pattern does not depend on the ship length L and, for a fully-submerged body, on the depth of submergence Δ; i.e.,Kelvin’s pattern is independent of the Froude numbers F≡V∕g L and F Δ≡V∕g Δ However, it is well known that the Froude numbers F and F Δ greatly influence the actual appearance of a ship wave pattern. For instance, divergent waves are negligible for a submerged body at a small Froude number F Δ, but dominate – for a free-surface piercing ship as well as for a fully-submerged body – at a high Froude number F. The influence of the Froude number F is also apparent from the fact that the highest ship waves are divergent waves found well inside the Kelvin wake at high Froude numbers F, whereas the dominant ship waves at low Froude numbers F can be found outside the cusps (even though the Kelvin wake depicted in Fig.1 shows no waves beyond the cusps). Indeed, the Froude number F, and interferences between the transverse and divergent waves that are predominantly created by the bow and the stern of a ship, have large influences on the farfield wave pattern (and the related wave drag); e.g., . These interference effects evidently cannot be analyzed via Kelvin’s 1-point wavemaker model of a ship. The influence of the Froude number F based on the ship length L is illustrated in this study for a free-surface piercing monohull ship; and the influence of the Froude number F Δ based on the submergence depth Δ is illustrated for a point source submerged below the free surface. The influences of F and F Δ are also illustrated for a fully-submerged body of length L that is submerged at a depth δ≡Δ∕L=F 2∕F Δ 2 The parameter δ, commonly used to define the submergence depth, is used instead of F Δ=F∕δ in this study of the wave pattern of a fully-submerged body. As was already noted, interferences among the divergent waves created by a free-surface piercing ship hull that travels at a high Froude number F in calm water are analyzed for monohull ships and catamarans in, , where simple ship models (a point source near the bow and a point sink near the stern) are considered. Wave interferences at high speed are further studied in, , , via more realistic hydrodynamic models based on distributions of sources and sinks over the ship hull surface. These studies of interferences among divergent waves in deep water, extended to the (significantly more complicated) case of uniform finite water depth in , , show that the highest divergent waves created by fast monohull ships or catamarans are found along ray angles ψ, measured from the path x<0 of the ship, that are smaller than Kelvin’s angle ψ K defined by (1). The analysis of wave interferences considered in , , , , , , , only considers divergent waves, i.e.ignores transverse waves, and is based on Kelvin’s classical stationary-phase approximation tofarfield ship waves. This farfield analytical approximation is only valid inside the Kelvin wake, i.e.for ray angles ψ within the range −ψ K<ψ<ψ K. Indeed, Kelvin’s stationary-phase approximation is singular at the cusps ψ=±ψ K. This singularity is inconsequential for the high-Froude-number analysis that is considered in , , , , , , because the ray angles ψ=±ψ m a x where the highest divergent waves are created as a result of constructive interferences are found well inside the cusps ψ=±ψ K of the Kelvin wake at sufficiently high Froude numbers. However, the apparent wake angle ψ m a x that corresponds to the ray angles along which the highest divergent waves are found is nearly equal to the Kelvin wake angle ψ K for moderately high Froude numbers; specifically, for F≈0.6 for monohull ships, . Thus, the analysis given in , , , , , , , , based on Kelvin’s stationary phase approximation as was already mentioned, is not valid for low or moderately high Froude numbers (for F<0.7 for monohull ships). This limitation is noted and illustrated in, , , , , , ; see, e.g.Fig.1 of and Fig.3 of. Moreover, transverse waves cannot be ignored at low or moderately high Froude numbers. An analysis of farfield ship waves that accounts for divergent as well as transverse waves, and is valid for high and low Froude numbers, is considered here. This analysis is based on the Kelvin–Havelock–Peters (KHP) approximation given in, instead of Kelvin’s classical stationary-phase approximation used in , , , , , , . The KHP approximation combines three classical analytical approximations to farfield ship waves: (i) the Kelvin approximation, which is only valid inside the Kelvin wake and is (weakly) singular at the cusps as was already noted, (ii) the Havelock approximation, which is finite at the cusps but is only valid there, and (iii) the Peters approximation, which is only valid outside Kelvin’s wake and is (weakly) singular at the cusps, like Kelvin’s approximation. The KHP approximation does not differ significantly from the Kelvin and Peters approximations inside or outside the cusps ψ=±ψ K. However, it is finite and agrees with Havelock’s approximation at the cusps, where the Kelvin and Peters approximations are singular as was already noted. The KHP approximation is slightly less accurate than the Chester–Friedman–Ursell approximation in the vicinity of the cusps, but is more realistic than Kelvin’s and Peters’ approximations at the cusps as was already noted. The KHP approximation only involves elementary functions, like the Kelvin, Havelock and Peters approximations, and indeed is a simple modification of these approximations, whereas the approximation given in involves the Airy function and its derivative. The farfield approximations given by Kelvin, Havelock, Peters and Chester–Friedman–Ursell have also been considered in, , , , , , , , . The straightforward, fully-analytical, KHP approximation is used here. This approximation is applied to illustrate the influence of the Froude number F on the wave pattern created by a monohull ship at Froude numbers F within the range 0.2≤F≤1.5. The influence of the Froude number F Δ based on the depth of submergence is illustrated for a point source below the free surface at Froude numbers F Δ within the range 0.5≤F Δ≤10. Lastly, the influences of the Froude number and the submergence depth are illustrated for a fully-submerged body at Froude numbers F and submergence depths δ within the ranges 0.2≤F≤1.5 and 0.1≤δ≤0.4. Access through your organization Check access to the full text by signing in through your organization. Access through your organization Section snippets The Kelvin–Havelock–Peters approximation Thus, the farfield waves created by a ship of length L that travels at a constant speed V along a straight path, in calm water of large depth and lateral extent, are considered. The ship waves are observed from a moving system of Cartesian coordinates (X,Y,Z) attached to the ship and thus appear steady. The X-axis is taken along the path of the ship and points toward the ship bow. The Z-axis is vertical and points upward. The origin (0,0,0) of the coordinates is taken at the point where the Surface-piercing monohull ship The farfield waves created by a simple free-surface piercing monohull ship, with a constant draft and rectangular framelines, are now considered. The sides of the mean wetted ship hull surface Σ are defined as 2 y∕b=±β(x)where−0.5≤x≤0.5 and−d≤z≤0 The beam/length and draft/length ratios b≡B∕L and d≡D∕L are chosen as b=0.15 and d=0.05, and the local beam β(x) of the waterline is taken as β=1−(x+ℓ B−0.5)2∕ℓ B 2 for 0.5−ℓ B≤x≤0.5 1 for ℓ S−0.5≤x≤0.5−ℓ B 1−(x−ℓ S+0.5)2∕ℓ S 2 for−0.5≤x≤ℓ S−0.5 Thus, the ship hull Point source below the free surface The farfield waves created by a point source submerged at a depth Δ below the free surface and located at (0,0,−Δ) is now considered. The Froude number F Δ based on the submergence depth Δ is defined by (2). The amplitude function A in the wave integrals (5), (10b) and the related farfield KHP approximation (13) is A=1+q 2 e−(1+q 2)∕F Δ 2 The free-surface elevation E g∕V 2 defined by (10a), the KHP approximation (13)–(14) and the wave-amplitude function (19) associated with a point source is considered Fully-submerged body The farfield waves created by a simple fully-submerged axisymmetric body are now considered. The body surface Σ is depicted in Fig.7 and defined as y 2+(z+δ)2 r 2=1−(x+ℓ B−0.5)2∕ℓ B 2 for 0.5−ℓ B≤x≤0.5 1 for ℓ S−0.5≤x≤0.5−ℓ B[1−(x−ℓ S+0.5)2∕ℓ S 2]2 for−0.5≤x≤ℓ S−0.5 The maximum body radius/length ratio r≡R∕L is chosen as r=0.05. The body surface Σ consists of a hemispheroidal bow region of length ℓ B, a stern region of length ℓ S obtained by rotating a parabola, and a parallel midbody region of length ℓ M=1−ℓ B− Conclusion The free-surface elevation associated with the farfield waves created by a ship can be expressed as a linear superposition of elementary waves, as in (5), i.e.E g V 2=1 π Re∫−q∞q∞A e i h φ d q where h φ≡1+q 2(x˜+q y˜)The classical analysis given by Kelvin is entirely based on the phase h φ of the trigonometric function in (21). Indeed, the amplitude function A in (21) is essentially ignored in Kelvin’s simplified analysis. Kelvin’s classical ‘phase analysis’ provides important basic information, Recommended articles References (38) Liang H. et al. Capillary–gravity ship wave patterns J. Hydrodyn. Ser. B (2017) Wu H. et al. Neumann–Michell theory of short ship waves Eur. J. Mech. B Fluids (2018) Noblesse F. et al. Why can ship wakes appear narrower than Kelvin’s angle? Eur. J. Mech. B Fluids (2014) Zhu Y. et al. Elementary ship models and farfield waves Eur. J. Mech. B Fluids (2018) Zhang C. et al. Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources Eur. J. Mech. B Fluids (2015) He J. et al. Interference effects on the Kelvin wake of a catamaran represented via a hull-surface distribution of sources Eur. J. Mech. B Fluids (2016) Noblesse F. et al. Observations and computations of narrow Kelvin ship wakes J. Ocean Eng. Sci. (2016) Zhu Y. et al. Michell and Hogner models of far-field ship waves Appl. Ocean Res. (2017) Zhu Y. et al. Basic models of farfield ship waves in shallow water J. Ocean Eng. Sci. (2018) Zhu Y. et al. Hogner model of wave interferences for farfield ship waves in shallow water Appl. Ocean Res. (2018) Wu H. et al. The Kelvin–Havelock–Peters farfield approximation to ship waves Eur. J. Mech. B Fluids (2018) Maz’ya V.G. et al. On ship waves Wave Motion (1993) Huang F. et al. Numerical implementation and validation of the Neumann–Michell theory of ship waves Eur. J. Mech. B Fluids (2013) He J. et al. Boundary integral representations of steady flow around a ship Eur. J. Mech. B Fluids (2018) Thomson W. On ship waves Proc. Inst. Mech. Eng. (1887) Ursell F. On Kelvin’s ship-wave pattern J. Fluid Mech. (1960) E.O. Tuck, D.C. Scullen, L. Lazauskas, Wave patterns and minimum wave resistance for high-speed vessels, in: The 24th... Reed A.M. et al. Ship wakes and their radar images Annu. Rev. Fluid Mech. (2002) Rabaud M. et al. Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. (2013) View more references Cited by (27) Airborne optical polarization imaging for observation of submarine Kelvin wakes on the sea surface: Imaging chain and simulation 2021, ISPRS Journal of Photogrammetry and Remote Sensing Citation Excerpt : The most distinguishable and informative submarine wake is the V-shaped Kelvin wake behind the stern (Kelvin, 1906); both theoretical (e.g., Havelock, 1934; Yim, 1963; Scullen and Tuck, 1995; Gourlay and Dawson, 2015) and computational fluid dynamics (CFD) models (Sun et al., 2018) can accurately simulate the wake’s elevation distribution. Such hydrodynamic studies prove that the formation mechanism of a submarine wake is exactly the same as that of a ship wake, and the variation of wake structure with submarine’s diving depth is still a popular direction for CFD simulation (e.g., Wu et al., 2019; Amiri et al., 2019). In terms of remote sensing observation of wakes, the same hydrodynamic formation mechanism eliminates the need for studying the imaging mechanisms of ship and submarine wakes separately. Show abstract The interaction between a moving submarine and seawater generates characteristic wakes on the sea surface, enabling indirect detection of undersea objects via airborne remote sensing. Here, we demonstrate the feasibility of using visible-light polarization imaging to observe submarine wakes. The key links in the imagine chain are considered separately. These include the polarization patterns of skylight, the elevations and slopes of submarine wakes and sea waves, and the changes in the sea surface polarized bidirectional reflectance characteristics due to modulation of gravity–capillary waves by the wake’s velocity field. A complete model of the airborne optical polarization imaging process is constructed and images are simulated via ray tracing. All theories proposed are verified by a series of terrestrial observation experiments. The results show that both the sea surface roughness modulation by the wake’s velocity field and the sea surface slope formed by wake elevation play significant roles in the imaging process. The wake features in the Stokes vector linear polarization component (Q, U) images are effectively enhanced, and the environmental adverse effect on these images is smaller than that on the intensity images. The degree of linear polarization (DoLP) and angle of polarization (AoP) images exhibit acceptable contrast under certain zenith and azimuth angles. Thus, our analysis confirms that airborne optical polarization imaging has considerable potential for observing wakes and other small- and medium-scale ocean dynamic processes. ### Kelvin–Froude wake patterns of a traveling pressure disturbance 2021, European Journal of Mechanics B Fluids Show abstract According to Kelvin, a point pressure source uniformly traveling over the surface of deep calm water leaves behind universal wake pattern confined within 39° sector and consisting of the so-called transverse and diverging wavefronts. Actual ship wakes differ in their appearance from both each other and Kelvin’s prediction. The difference can be attributed to a deviation from the point source limit and for given shape of the disturbance quantified by the Froude number F. We show that within linear theory effect of arbitrary disturbance on the wake pattern can be mimicked by an effective pressure distribution. Further, resulting wake patterns are qualitatively different depending on whether water-piercing is present or not (“sharp” vs “smooth” disturbances). For smooth pressure sources, we generalize Kelvin’s stationary phase argument to encompass finite size effects and classify resulting wake patterns. Specifically, we show that there exist two characteristic Froude numbers, F 1 and F 2>F 1, such as the wake is only present if F≳F 1. For F 1≲F≲F 2, the wake consists of the transverse wavefronts confined within a sector of an angle that may be smaller than Kelvin’s. An additional 39° wake made of both the transverse and diverging wavefronts is found for F≳F 2. If the pressure source has sharp boundary, the wake is always present and features additional interference effects. Specifically, for a constant pressure line segment source mimicking slender ship the wake pattern can be understood as due to two opposing effect wakes resembling (but not identical to) Kelvin’s and originating at segment’s ends. ### Viscous effects on the fundamental solution to ship waves 2019, Journal of Fluid Mechanics ### Surface wave characteristics of a volume source horizontally translating in a stratified fluid 2020, Physics of Fluids ### Wake features of moving submerged bodies and motion state inversion of submarines 2020, IEEE Access ### Wave profile along a ship hull, short farfield waves, and broad inner Kelvin wake sans divergent waves 2019, Physics of Fluids View all citing articles on Scopus View full text © 2018 Elsevier Masson SAS. All rights reserved. Recommended articles Taylor–Couette vortex flow of ceramic dispersions European Journal of Mechanics - B/Fluids, Volume 75, 2019, pp. 279-285 Marguerite Bienia, …, Cécile Pagnoux ### Functional distance in human gait transition Acta Psychologica, Volume 161, 2015, pp. 170-176 Mohammad Abdolvahab, Claudia Carello ### Experimental investigation of unsteady laminar mixed convection from a horizontal heated cylinder in contra-flow: Buoyancy and confinement effects on the three-dimensional heat transfer response European Journal of Mechanics - B/Fluids, Volume 75, 2019, pp. 165-179 A.I.Saldaña, …, L.Martínez-Suástegui ### Protection of dilator function of coronary arteries from homocysteine by tetramethylpyrazine: Role of ER stress in modulation of BK Ca channels Vascular Pharmacology, Volume 113, 2019, pp. 27-37 Wen-Tao Sun, …, Qin Yang ### Numerical simulation of unsteady flows with forced periodical oscillation around hydrofoils using locally power-law preconditioning method European Journal of Mechanics - B/Fluids, Volume 75, 2019, pp. 153-164 S.M.Derazgisoo, …, A. Askari Lehdarboni ### Probing the influence of confinement and wettability on droplet displacement behavior: A mesoscale analysis European Journal of Mechanics - B/Fluids, Volume 75, 2019, pp. 327-338 Pitambar Randive, …, Partha P.Mukherjee Show 3 more articles About ScienceDirect Remote access Contact and support Terms and conditions Privacy policy Cookies are used by this site.Cookie settings All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply. We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. You can manage your cookie preferences using the “Cookie Settings” link. 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2501	https://pages.cs.wisc.edu/~dsmyers/cs547/lecture_9_memoryless_property.pdf	CS 547 Lecture 9: Conditional Probabilities and the Memoryless Property Daniel Myers Joint Probabilities For two events, E and F, the joint probability, written P(EF), is the the probability that both events occur. For example, let E be “the probability that a die roll is even” and F be “the probability that a die roll is greater than 3”. We have the following sets to describe each event: E = {2, 4, 6} F = {4, 5, 6} E ∩F = {4, 6} The probability that the joint event occurs is the probability that the outcome is in E ∩F, which is 2 6. Conditional Probabilities Frequently, we are interested in analyzing the probability of an event with respect to some known information. If E and F are events, the conditional probability of E given F is P(E\|F) = P(EF) P(F) Intuitively, P(EF) is the portion of the sample space that is in both E and F. If we know that a point of the sample space is deﬁnitely in F, the probability that it is also in E is given by the portion of F that overlaps with E, which is exactly what the formula calculates. Simply rearranging the formula gives us a way to calculate joint probabilities in terms of conditional proba-bilities P(EF) = P(E\|F) P(F) Independence Two events are independent if P(E\|F) = P(E). That is, knowing that F has occurred tells us nothing about whether E also occurs. Using the formula for joint probability, we get P(EF) = P(E\|F)P(F) = P(E)P(F) The joint probability of independent events is just the product of their individual probabilities. Using this knowledge, we’re now able to interpret some earlier results. Recall the pmf of the geometric random variable, f(k) = (1 −p)k−1p We deﬁned this to be the probability that a series of independent Bernoulli trials yields its ﬁrst success on trial k. That is, we are interested in the probability of a sequence of k −1 failures, each of which occur with probability 1 −p, and one success, which occurs with probability p. Because the Bernoulli trials are independent, the probability of this sequence is simply the product of the probabilities of its individual events, which gives pmf formula. 1 Distribution of the Minimum of Exponentials Suppose we have a set of independent and identically distributed (iid) variables, X1, X2, . . . , Xk, which are exponentially distributed with the same parameter λ. What is the distribution of the random variable X = min(X1, X2, . . . , Xk)? Let’s reason about P(X > t). X is deﬁned as the minimum of all the Xi, so the only way that X can be greater than a certain value t is if all of its components are also greater than t. Therefore, P(X > t) = P(X1 > t and X2 > t and . . . and Xk > t) The individual Xi are all independent, so we can re-write the joint probability as the product of their individual probabilities. P(X > t) = P(X1 > t) P(X2 > t) . . . P(Xk > t) To ﬁnd the ﬁnal distribution, we need to know P(Xi > t) when Xi ∼exp(λ). This is given by the complementary CDF (CCDF) of the random variable, which is deﬁned in terms of the CDF, F(t), as F(t) = P(Xi > t) = 1 −P(Xi ≤t) = 1 −F(t) For the exponential distribution, F(t) = 1 −F(t) = 1 −(1 −e−λt) = e−λt Now that we have the exponential CCDF, we can ﬁnd the distribution of X. P(X > t) = P(X1 > t) P(X2 > t) . . . P(Xk > t) = e−λt e−λt . . . e−λt = e−kλt The distribution of the minimum of a set of k iid exponential random variables is also exponentially dis-tributed with parameter kλ. This result generalizes to the case where the variables are still independent, but have diﬀerent parameters. The Memoryless Property The memoryless proeprty tells us about the conditional behavior of exponential random variables. It’s one of our key results, which we’ll use in deriving the solution of queueing systems. Let X be exponentially distributed with parameter λ. Suppose we know X > t. What is the probability that X is also greater than some value s + t? That is, we want to know P(X > s + t \| X > t) This type of problem shows up frequently in queueing systems where we’re interested in the time between events. For example, suppose that jobs in our system have exponentially distributed service times. If we have a job that’s been running for one hour, what’s the probability that it will continue to run for more than two hours? 2 Using the deﬁnition of conditional probability, we have P(X > s + t \| X > t) = P(X > s + t and X > t) P(X > t) If X > s + t, then X > t is redundant, so we can simplify the numerator. P(X > s + t \| X > t) = P(X > s + t) P(X > t) Using the CCDF of the exponential distribution, P(X > s + t \| X > t) = P(X > s + t) P(X > t) = e−λ(s+t) e−λt The e−λt terms cancel, giving the surprising result P(X > s + t \| X > t) = e−λs It turns out that the conditional probability does not depend on t! The probability of an exponential random variable exceeding the value s + t given t is the same as the variable originally exceeding that value s, regardless of t. In our job example, the probability that a job runs for one additional hour is the same as the probability that it ran for one hour originally, regardless of how long it’s been running. The exponential distribution is memoryless because the past has no bearing on its future behavior. Every instant is like the beginning of a new random period, which has the same distribution regardless of how much time has already elapsed. The exponential is the only memoryless continuous random variable. Implications of the Memoryless Property The memoryless property makes it easy to reason about the average behavior of exponentially distributed items in queuing systems. Suppose we’re observing a stream of events with exponentially distributed interarrival times. Because of the memoryless property, the expected time until the next event is always 1 λ, no matter how long we’ve been waiting for a new arrival to occur. This behavior is a bit counterintuitive. We might expect that arrivals get more likely the longer we wait. For example, if the bus is supposed to come every ten minutes, and we have been waiting for nine minutes without seeing a bus, we expect that the next bus should be along very soon. If the time between bus arrivals is exponentially distributed, however, the memoryless property tells us that our waiting time – no matter how long it’s been – is of no use in predicting when the next bus will arrive1. Suppose we have a queue with exponentially distributed service times. If a new customer arrives to the queue to ﬁnd someone in service, the residual service time is the time until the currently running customer ﬁnishes service and departs the queue. Because of the memoryless property, the distribution of the residual service times does not depend on how long the customer has been in service. The probability that the current customer runs for an additional minute and then departs is the same as the probability that a new customer just entering service runs for one minute. Likewise, the average remaining service time is simply s, the expected time for a new customer just entering service. 1Of course, the time between real bus arrivals is never exponentially distributed. People want to know exactly when their buses will come, so bus schedules are nearly deterministic, which justiﬁes our intuition about longer waiting times increasing the probability of an arrival. 3 Failure Rates In the ﬁeld of reliability theory, it’s common to use a random variable to represent the lifespan of a component. One of the main problems in this area is predicting the likelihood that a component fails in the very near future given its current age. This likelihood is summarized by the failure rate (also called the hazard rate) of the component2. For most components, the failure rate changes with time. There are three possible relationships. • increasing failure rate • decreasing failure rate • constant failure rate If the failure rate is increasing, then failures becomes more likely as the component ages. Most manufactured products behave in this way – they’re built to last for a certain amount of time, then fail. Failures become more likely as the product wears out and the end of its lifespan approaches. A decreasing failure rate implies that the probability of failure decreases with the passage of time. In other words, the longer a component has worked, the more likely it is to continue working. UNIX processes have been shown to have decreasing failure rates – the longer a job runs, the more likely it is to continue running3. Human lifespans also have decreasing failure rate, as surviving further into adulthood makes it more likely that you will live to old age, at least until you reach the upper limit of your natural lifespan. The ﬁnal category corresponds to components with exponentially distributed lifespans. Because of the memoryless property, the length of time a component has functioned in the past has no bearing on its future behavior, so the probability that the component fails in the near future is always the same and doesn’t depend on its current age. 2It’s possible to formally derive a function for the failure rate in terms of the lifespan distribution. The actual function isn’t directly relevant to our work, so I’ve chosen not to present it here. 3This justiﬁes the use of schedulers that steadily decrease the priority of a job as it runs. More on this in future lectures. 4
2502	https://www.cis.upenn.edu/~cis5150/cis515-13-sl7.pdf	Chapter 9 Euclidean Spaces 9.1 Inner Products, Euclidean Spaces The framework of vector spaces allows us deal with ratios of vectors and linear combinations, but there is no way to express the notion of length of a line segment or to talk about orthogonality of vectors. A Euclidean structure will allow us to deal with metric notions such as orthogonality and length (or distance). First, we deﬁne a Euclidean structure on a vector space. 445 446 CHAPTER 9. EUCLIDEAN SPACES Deﬁnition 9.1. A real vector space E is a Euclidean space i↵it is equipped with a symmetric bilinear form ': E ⇥E ! R which is also positive deﬁnite, which means that '(u, u) > 0, for every u 6= 0. More explicitly, ': E ⇥E ! R satisﬁes the following axioms: '(u1 + u2, v) = '(u1, v) + '(u2, v), '(u, v1 + v2) = '(u, v1) + '(u, v2), '(λu, v) = λ'(u, v), '(u, λv) = λ'(u, v), '(u, v) = '(v, u), u 6= 0 implies that '(u, u) > 0. The real number '(u, v) is also called the inner product (or scalar product) of u and v. 9.1. INNER PRODUCTS, EUCLIDEAN SPACES 447 We also deﬁne the quadratic form associated with ' as the function Φ: E ! R+ such that Φ(u) = '(u, u), for all u 2 E. Since ' is bilinear, we have '(0, 0) = 0, and since it is positive deﬁnite, we have the stronger fact that '(u, u) = 0 i↵ u = 0, that is Φ(u) = 0 i↵u = 0. Given an inner product ': E ⇥E ! R on a vector space E, we also denote '(u, v) by u · v, or hu, vi, or (u\|v), and p Φ(u) by kuk. 448 CHAPTER 9. EUCLIDEAN SPACES Example 1. The standard example of a Euclidean space is Rn, under the inner product · deﬁned such that (x1, . . . , xn) · (y1, . . . , yn) = x1y1 + x2y2 + · · · + xnyn. This Euclidean space is denoted by En. Example 2. Let E be a vector space of dimension 2, and let (e1, e2) be a basis of E. If a > 0 and b2 −ac < 0, the bilinear form deﬁned such that '(x1e1+y1e2, x2e1+y2e2) = ax1x2+b(x1y2+x2y1)+cy1y2 yields a Euclidean structure on E. In this case, Φ(xe1 + ye2) = ax2 + 2bxy + cy2. 9.1. INNER PRODUCTS, EUCLIDEAN SPACES 449 Example 3. Let C[a, b] denote the set of continuous func-tions f : [a, b] ! R. It is easily checked that C[a, b] is a vector space of inﬁnite dimension. Given any two functions f, g 2 C[a, b], let hf, gi = Z b a f(t)g(t)dt. We leave as an easy exercise that h−, −i is indeed an inner product on C[a, b]. When [a, b] = [−⇡, ⇡] (or [a, b] = [0, 2⇡], this makes basically no di↵erence), one should compute hsin px, sin qxi, hsin px, cos qxi, and hcos px, cos qxi, for all natural numbers p, q ≥1. The outcome of these calculations is what makes Fourier analysis possible! 450 CHAPTER 9. EUCLIDEAN SPACES Example 4. Let E = Mn(R) be the vector space of real n ⇥n matrices. If we view a matrix A 2 Mn(R) as a “long” column vector obtained by concatenating together its columns, we can deﬁne the inner product of two matrices A, B 2 Mn(R) as hA, Bi = n X i,j=1 aijbij, which can be conveniently written as hA, Bi = tr(A>B) = tr(B>A). Since this can be viewed as the Euclidean product on Rn2, it is an inner product on Mn(R). The corresponding norm kAkF = q tr(A>A) is the Frobenius norm (see Section 6.2). 9.1. INNER PRODUCTS, EUCLIDEAN SPACES 451 Let us observe that ' can be recovered from Φ. Indeed, by bilinearity and symmetry, we have Φ(u + v) = '(u + v, u + v) = '(u, u + v) + '(v, u + v) = '(u, u) + 2'(u, v) + '(v, v) = Φ(u) + 2'(u, v) + Φ(v). Thus, we have '(u, v) = 1 2[Φ(u + v) −Φ(u) −Φ(v)]. We also say that ' is the polar form of Φ. One of the very important properties of an inner product ' is that the map u 7! p Φ(u) is a norm. 452 CHAPTER 9. EUCLIDEAN SPACES Proposition 9.1. Let E be a Euclidean space with inner product ' and quadratic form Φ. For all u, v 2 E, we have the Cauchy-Schwarz inequality: '(u, v)2 Φ(u)Φ(v), the equality holding i↵u and v are linearly dependent. We also have the Minkovski inequality: p Φ(u + v)  p Φ(u) + p Φ(v), the equality holding i↵u and v are linearly dependent, where in addition if u 6= 0 and v 6= 0, then u = λv for some λ > 0. 9.1. INNER PRODUCTS, EUCLIDEAN SPACES 453 Sketch of proof . Deﬁne the function T : R ! R, such that T(λ) = Φ(u + λv), for all λ 2 R. Using bilinearity and symmetry, we can show that Φ(u + λv) = Φ(u) + 2λ'(u, v) + λ2Φ(v). Since ' is positive deﬁnite, we have T(λ) ≥0 for all λ 2 R. If Φ(v) = 0, then v = 0, and we also have '(u, v) = 0. In this case, the Cauchy-Schwarz inequality is trivial, 454 CHAPTER 9. EUCLIDEAN SPACES If Φ(v) > 0, then λ2Φ(v) + 2λ'(u, v) + Φ(u) = 0 can’t have distinct roots, which means that its discrimi-nant ∆= 4('(u, v)2 −Φ(u)Φ(v)) is zero or negative, which is precisely the Cauchy-Schwarz inequality. The Minkovski inequality can then be shown. 9.1. INNER PRODUCTS, EUCLIDEAN SPACES 455 The Minkovski inequality p Φ(u + v)  p Φ(u) + p Φ(v) shows that the map u 7! p Φ(u) satisﬁes the triangle inequality, condition (N3) of deﬁnition 6.1, and since ' is bilinear and positive deﬁnite, it also satisﬁes conditions (N1) and (N2) of deﬁnition 6.1, and thus, it is a norm on E. The norm induced by ' is called the Euclidean norm induced by '. Note that the Cauchy-Schwarz inequality can be written as \|u · v\| kuk kvk , and the Minkovski inequality as ku + vk kuk + kvk . We now deﬁne orthogonality. 456 CHAPTER 9. EUCLIDEAN SPACES 9.2 Orthogonality, Duality, Adjoint Maps Deﬁnition 9.2. Given a Euclidean space E, any two vectors u, v 2 E are orthogonal, or perpendicular i↵ u · v = 0. Given a family (ui)i2I of vectors in E, we say that (ui)i2I is orthogonal i↵ui · uj = 0 for all i, j 2 I, where i 6= j. We say that the family (ui)i2I is orthonor-mal i↵ui · uj = 0 for all i, j 2 I, where i 6= j, and kuik = ui · ui = 1, for all i 2 I. For any subset F of E, the set F ? = {v 2 E \| u · v = 0, for all u 2 F}, of all vectors orthogonal to all vectors in F, is called the orthogonal complement of F. Since inner products are positive deﬁnite, observe that for any vector u 2 E, we have u · v = 0 for all v 2 E i↵ u = 0. It is immediately veriﬁed that the orthogonal complement F ? of F is a subspace of E. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 457 Example 5. Going back to example 3, and to the inner product hf, gi = Z ⇡ −⇡ f(t)g(t)dt on the vector space C[−⇡, ⇡], it is easily checked that hsin px, sin qxi = ⇢ ⇡ if p = q, p, q ≥1, 0 if p 6= q, p, q ≥1 hcos px, cos qxi = ⇢ ⇡ if p = q, p, q ≥1, 0 if p 6= q, p, q ≥0 and hsin px, cos qxi = 0, for all p ≥1 and q ≥0, and of course, h1, 1i = R ⇡ −⇡dx = 2⇡. As a consequence, the family (sin px)p≥1 [ (cos qx)q≥0 is orthogonal. It is not orthonormal, but becomes so if we divide every trigonometric function by p⇡, and 1 by p 2⇡. 458 CHAPTER 9. EUCLIDEAN SPACES Proposition 9.2. Given a Euclidean space E, for any family (ui)i2I of nonnull vectors in E, if (ui)i2I is or-thogonal, then it is linearly independent. Proposition 9.3. Given a Euclidean space E, any two vectors u, v 2 E are orthogonal i↵ ku + vk2 = kuk2 + kvk2 . One of the most useful features of orthonormal bases is that they a↵ord a very simple method for computing the coordinates of a vector over any basis vector. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 459 Indeed, assume that (e1, . . . , em) is an orthonormal basis. For any vector x = x1e1 + · · · + xmem, if we compute the inner product x · ei, we get x · ei = x1e1 · ei + · · · + xiei · ei + · · · + xmem · ei = xi, since ei · ej = ⇢ 1 if i = j, 0 if i 6= j, is the property characterizing an orthonormal family. Thus, xi = x · ei, which means that xiei = (x·ei)ei is the orthogonal projec-tion of x onto the subspace generated by the basis vector ei. If the basis is orthogonal but not necessarily orthonormal, then xi = x · ei ei · ei = x · ei keik2. 460 CHAPTER 9. EUCLIDEAN SPACES All this is true even for an inﬁnite orthonormal (or or-thogonal) basis (ei)i2I. ! However, remember that every vector x is expressed as a linear combination x = X i2I xiei where the family of scalars (xi)i2I has ﬁnite support, which means that xi = 0 for all i 2 I −J, where J is a ﬁnite set. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 461 Thus, even though the family (sin px)p≥1 [ (cos qx)q≥0 is orthogonal (it is not orthonormal, but becomes one if we divide every trigonometric function by p⇡, and 1 by p 2⇡; we won’t because it looks messy!), the fact that a function f 2 C0[−⇡, ⇡] can be written as a Fourier series as f(x) = a0 + 1 X k=1 (ak cos kx + bk sin kx) does not mean that (sin px)p≥1 [ (cos qx)q≥0 is a basis of this vector space of functions, because in general, the families (ak) and (bk) do not have ﬁnite support! In order for this inﬁnite linear combination to make sense, it is necessary to prove that the partial sums a0 + n X k=1 (ak cos kx + bk sin kx) of the series converge to a limit when n goes to inﬁnity. This requires a topology on the space. 462 CHAPTER 9. EUCLIDEAN SPACES A very important property of Euclidean spaces of ﬁnite dimension is that the inner product induces a canoni-cal bijection (i.e., independent of the choice of bases) between the vector space E and its dual E⇤. Given a Euclidean space E, for any vector u 2 E, let 'u: E ! R be the map deﬁned such that 'u(v) = u · v, for all v 2 E. Since the inner product is bilinear, the map 'u is a linear form in E⇤. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 463 Thus, we have a map [: E ! E⇤, deﬁned such that [(u) = 'u. Theorem 9.4. Given a Euclidean space E, the map [: E ! E⇤, deﬁned such that [(u) = 'u, is linear and injective. When E is also of ﬁnite di-mension, the map [: E ! E⇤is a canonical isomor-phism. The inverse of the isomorphism [: E ! E⇤is denoted by ]: E⇤! E. 464 CHAPTER 9. EUCLIDEAN SPACES As a consequence of Theorem 9.4, if E is a Euclidean space of ﬁnite dimension, every linear form f 2 E⇤cor-responds to a unique u 2 E, such that f(v) = u · v, for every v 2 E. In particular, if f is not the null form, the kernel of f, which is a hyperplane H, is precisely the set of vectors that are orthogonal to u. Theorem 9.4 allows us to deﬁne the adjoint of a linear map on a Euclidean space. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 465 Let E be a Euclidean space of ﬁnite dimension n, and let f : E ! E be a linear map. For every u 2 E, the map v 7! u · f(v) is clearly a linear form in E⇤, and by Theorem 9.4, there is a unique vector in E denoted as f ⇤(u), such that f ⇤(u) · v = u · f(v), for every v 2 E. Proposition 9.5. Given a Euclidean space E of ﬁnite dimension, for every linear map f : E ! E, there is a unique linear map f ⇤: E ! E, such that f ⇤(u) · v = u · f(v), for all u, v 2 E. The map f ⇤is called the adjoint of f (w.r.t. to the inner product). 466 CHAPTER 9. EUCLIDEAN SPACES Linear maps f : E ! E such that f = f ⇤are called self-adjoint maps. They play a very important role because they have real eigenvalues and because orthonormal bases arise from their eigenvectors. Furthermore, many physical problems lead to self-adjoint linear maps (in the form of symmetric matrices). Linear maps such that f −1 = f ⇤, or equivalently f ⇤◦f = f ◦f ⇤= id, also play an important role. They are isometries. Rota-tions are special kinds of isometries. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 467 Another important class of linear maps are the linear maps satisfying the property f ⇤◦f = f ◦f ⇤, called normal linear maps. We will see later on that normal maps can always be diagonalized over orthonormal bases of eigenvectors, but this will require using a Hermitian inner product (over C). Given two Euclidean spaces E and F, where the inner product on E is denoted as h−, −i1 and the inner product on F is denoted as h−, −i2, given any linear map f : E ! F, it is immediately veriﬁed that the proof of Proposition 9.5 can be adapted to show that there is a unique linear map f ⇤: F ! E such that hf(u), vi2 = hu, f ⇤(v)i1 for all u 2 E and all v 2 F. The linear map f ⇤is also called the adjoint of f. 468 CHAPTER 9. EUCLIDEAN SPACES Remark: Given any basis for E and any basis for F, it is possible to characterize the matrix of the adjoint f ⇤of f in terms of the matrix of f, and the symmetric matrices deﬁning the inner products. We will do so with respect to orthonormal bases. We can also use Theorem 9.4 to show that any Euclidean space of ﬁnite dimension has an orthonormal basis. Proposition 9.6. Given any nontrivial Euclidean space E of ﬁnite dimension n ≥1, there is an orthonormal basis (u1, . . . , un) for E. There is a more constructive way of proving Proposition 9.6, using a procedure known as the Gram–Schmidt or-thonormalization procedure. Among other things, the Gram–Schmidt orthonormal-ization procedure yields the so-called QR-decomposition for matrices, an important tool in numerical methods. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 469 Proposition 9.7. Given any nontrivial Euclidean space E of dimension n ≥1, from any basis (e1, . . . , en) for E, we can construct an orthonormal basis (u1, . . . , un) for E, with the property that for every k, 1 k n, the families (e1, . . . , ek) and (u1, . . . , uk) generate the same subspace. Proof . We proceed by induction on n. For n = 1, let u1 = e1 ke1k. For n ≥2, we deﬁne the vectors uk and u0 k as follows. u0 1 = e1, u1 = u0 1 ku0 1k, and for the inductive step u0 k+1 = ek+1 − k X i=1 (ek+1 · ui) ui, uk+1 = u0 k+1 ' 'u0 k+1 ' '. We need to show that u0 k+1 is nonzero, and we conclude by induction. 470 CHAPTER 9. EUCLIDEAN SPACES Remarks: (1) Note that u0 k+1 is obtained by subtracting from ek+1 the projection of ek+1 itself onto the orthonormal vectors u1, . . . , uk that have already been computed. Then, we normalize u0 k+1. The QR-decomposition can now be obtained very easily. We will do this in section 9.4. (2) We could compute u0 k+1 using the formula u0 k+1 = ek+1 − k X i=1 ek+1 · u0 i ku0 ik2 ! u0 i, and normalize the vectors u0 k at the end. This time, we are subtracting from ek+1 the projection of ek+1 itself onto the orthogonal vectors u0 1, . . . , u0 k. This might be preferable when writing a computer pro-gram. 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 471 (3) The proof of Proposition 9.7 also works for a count-ably inﬁnite basis for E, producing a countably inﬁnite orthonormal basis. Example 6. If we consider polynomials and the inner product hf, gi = Z 1 −1 f(t)g(t)dt, applying the Gram–Schmidt orthonormalization proce-dure to the polynomials 1, x, x2, . . . , xn, . . . , which form a basis of the polynomials in one variable with real coeﬃcients, we get a family of orthonormal polyno-mials Qn(x) related to the Legendre polynomials. The Legendre polynomials Pn(x) have many nice proper-ties. They are orthogonal, but their norm is not always 1. The Legendre polynomials Pn(x) can be deﬁned as follows: 472 CHAPTER 9. EUCLIDEAN SPACES If we let fn be the function fn(x) = (x2 −1)n, we deﬁne Pn(x) as follows: P0(x) = 1, and Pn(x) = 1 2nn! f (n) n (x), where f (n) n is the nth derivative of fn. They can also be deﬁned inductively as follows: P0(x) = 1, P1(x) = x, Pn+1(x) = 2n + 1 n + 1 xPn(x) − n n + 1 Pn−1(x). It turns out that the polynomials Qn are related to the Legendre polynomials Pn as follows: Qn(x) = r 2n + 1 2 Pn(x). 9.2. ORTHOGONALITY, DUALITY, ADJOINT MAPS 473 As a consequence of Proposition 9.6 (or Proposition 9.7), given any Euclidean space of ﬁnite dimension n, if (e1, . . . , en) is an orthonormal basis for E, then for any two vec-tors u = u1e1 +· · ·+unen and v = v1e1 +· · ·+vnen, the inner product u · v is expressed as u·v = (u1e1+· · ·+unen)·(v1e1+· · ·+vnen) = n X i=1 uivi, and the norm kuk as kuk = ku1e1 + · · · + unenk = v u u t n X i=1 u2 i. We can also prove the following proposition regarding or-thogonal spaces. Proposition 9.8. Given any nontrivial Euclidean space E of ﬁnite dimension n ≥1, for any subspace F of dimension k, the orthogonal complement F ? of F has dimension n −k, and E = F ⊕F ?. Furthermore, we have F ?? = F. 474 CHAPTER 9. EUCLIDEAN SPACES 9.3 Linear Isometries (Orthogonal Transformations) In this section, we consider linear maps between Eu-clidean spaces that preserve the Euclidean norm. Deﬁnition 9.3. Given any two nontrivial Euclidean spaces E and F of the same ﬁnite dimension n, a function f : E ! F is an orthogonal transformation, or a lin-ear isometry i↵it is linear and kf(u)k = kuk , for all u 2 E. Thus, a linear isometry is a linear map that preserves the norm. 9.3. LINEAR ISOMETRIES (ORTHOGONAL TRANSFORMATIONS) 475 Remarks: (1) A linear isometry is often deﬁned as a linear map such that kf(v) −f(u)k = kv −uk , for all u, v 2 E. Since the map f is linear, the two deﬁ-nitions are equivalent. The second deﬁnition just focuses on preserving the distance between vectors. (2) Sometimes, a linear map satisfying the condition of deﬁnition 9.3 is called a metric map, and a linear isom-etry is deﬁned as a bijective metric map. Also, an isometry (without the word linear) is sometimes deﬁned as a function f : E ! F (not necessarily linear) such that kf(v) −f(u)k = kv −uk , for all u, v 2 E, i.e., as a function that preserves the distance. 476 CHAPTER 9. EUCLIDEAN SPACES This requirement turns out to be very strong. Indeed, the next proposition shows that all these deﬁnitions are equivalent when E and F are of ﬁnite dimension, and for functions such that f(0) = 0. Proposition 9.9. Given any two nontrivial Euclidean spaces E and F of the same ﬁnite dimension n, for every function f : E ! F, the following properties are equivalent: (1) f is a linear map and kf(u)k = kuk, for all u 2 E; (2) kf(v) −f(u)k = kv −uk, for all u, v 2 E, and f(0) = 0; (3) f(u) · f(v) = u · v, for all u, v 2 E. Furthermore, such a map is bijective. 9.3. LINEAR ISOMETRIES (ORTHOGONAL TRANSFORMATIONS) 477 For (2), we shall prove a slightly stronger result. We prove that if kf(v) −f(u)k = kv −uk for all u, v 2 E, for any vector ⌧2 E, the function g: E ! F deﬁned such that g(u) = f(⌧+ u) −f(⌧) for all u 2 E is a linear map such that g(0) = 0 and (3) holds. Remarks: (i) The dimension assumption is only needed to prove that (3) implies (1) when f is not known to be linear, and to prove that f is surjective, but the proof shows that (1) implies that f is injective. (ii) The implication that (3) implies (1) holds if we also assume that f is surjective, even if E has inﬁnite dimen-sion. 478 CHAPTER 9. EUCLIDEAN SPACES In (2), when f does not satisfy the condition f(0) = 0, the proof shows that f is an aﬃne map. Indeed, taking any vector ⌧as an origin, the map g is linear, and f(⌧+ u) = f(⌧) + g(u) for all u 2 E. By Proposition 3.11, this shows that f is aﬃne with as-sociated linear map g. This fact is worth recording as the following proposition. 9.3. LINEAR ISOMETRIES (ORTHOGONAL TRANSFORMATIONS) 479 Proposition 9.10. Given any two nontrivial Euclidean spaces E and F of the same ﬁnite dimension n, for every function f : E ! F, if kf(v) −f(u)k = kv −uk for all u, v 2 E, then f is an aﬃne map, and its associated linear map g is an isometry. In view of Proposition 9.9, we will drop the word “linear” in “linear isometry”, unless we wish to emphasize that we are dealing with a map between vector spaces. 480 CHAPTER 9. EUCLIDEAN SPACES 9.4 The Orthogonal Group, Orthogonal Matrices In this section, we explore some of the fundamental prop-erties of the orthogonal group and of orthogonal matrices. As an immediate corollary of the Gram–Schmidt orthonor-malization procedure, we obtain the QR-decomposition for invertible matrices. 9.4. THE ORTHOGONAL GROUP, ORTHOGONAL MATRICES 481 Proposition 9.11. Let E be any Euclidean space of ﬁnite dimension n, and let f : E ! E be any linear map. The following properties hold: (1) The linear map f : E ! E is an isometry i↵ f ◦f ⇤= f ⇤◦f = id. (2) For every orthonormal basis (e1, . . . , en) of E, if the matrix of f is A, then the matrix of f ⇤is the transpose A> of A, and f is an isometry i↵A satisﬁes the identities A A> = A>A = In, where In denotes the identity matrix of order n, i↵ the columns of A form an orthonormal basis of E, i↵the rows of A form an orthonormal basis of E. Proposition 9.11 shows that the inverse of an isometry f is its adjoint f ⇤. Proposition 9.11 also motivates the following deﬁnition: 482 CHAPTER 9. EUCLIDEAN SPACES Deﬁnition 9.4. A real n ⇥n matrix is an orthogonal matrix i↵ A A> = A>A = In. Remarks: It is easy to show that the conditions A A> = In, A>A = In, and A−1 = A>, are equivalent. Given any two orthonormal bases (u1, . . . , un) and (v1, . . . , vn), if P is the change of basis matrix from (u1, . . . , un) to (v1, . . . , vn) since the columns of P are the coordinates of the vectors vj with respect to the basis (u1, . . . , un), and since (v1, . . . , vn) is orthonormal, the columns of P are orthonormal, and by Proposition 9.11 (2), the matrix P is orthogonal. The proof of Proposition 9.9 (3) also shows that if f is an isometry, then the image of an orthonormal basis (u1, . . . , un) is an orthonormal basis. 9.4. THE ORTHOGONAL GROUP, ORTHOGONAL MATRICES 483 Recall that the determinant det(f) of an endomorphism f : E ! E is independent of the choice of a basis in E. Also, for every matrix A 2 Mn(R), we have det(A) = det(A>), and for any two n⇥n-matrices A and B, we have det(AB) = det(A) det(B). Then, if f is an isometry, and A is its matrix with respect to any orthonormal basis, A A> = A>A = In implies that det(A)2 = 1, that is, either det(A) = 1, or det(A) = −1. It is also clear that the isometries of a Euclidean space of dimension n form a group, and that the isometries of determinant +1 form a subgroup. 484 CHAPTER 9. EUCLIDEAN SPACES Deﬁnition 9.5. Given a Euclidean space E of dimen-sion n, the set of isometries f : E ! E forms a group denoted as O(E), or O(n) when E = Rn, called the orthogonal group (of E). For every isometry, f, we have det(f) = ±1, where det(f) denotes the determinant of f. The isometries such that det(f) = 1 are called rotations, or proper isometries, or proper orthogonal transformations, and they form a subgroup of the special linear group SL(E) (and of O(E)), denoted as SO(E), or SO(n) when E = Rn, called the special orthogonal group (of E). The isometries such that det(f) = −1 are called im-proper isometries, or improper orthogonal transfor-mations, or ﬂip transformations. 9.5. QR-DECOMPOSITION FOR INVERTIBLE MATRICES 485 9.5 QR-Decomposition for Invertible Matrices Now that we have the deﬁnition of an orthogonal matrix, we can explain how the Gram–Schmidt orthonormaliza-tion procedure immediately yields the QR-decomposition for matrices. Proposition 9.12. Given any n⇥n real matrix A, if A is invertible then there is an orthogonal matrix Q and an upper triangular matrix R with positive diag-onal entries such that A = QR. Proof . We can view the columns of A as vectors A1, . . . , An in En. If A is invertible, then they are linearly independent, and we can apply Proposition 9.7 to produce an orthonormal basis using the Gram–Schmidt orthonormalization proce-dure. 486 CHAPTER 9. EUCLIDEAN SPACES Recall that we construct vectors Qk and Q 0k as follows: Q 01 = A1, Q1 = Q 01 kQ 01k, and for the inductive step Q 0k+1 = Ak+1− k X i=1 (Ak+1·Qi) Qi, Qk+1 = Q 0k+1 kQ 0k+1k, where 1 k n −1. If we express the vectors Ak in terms of the Qi and Q 0i, we get the triangular system A1 = kQ 01kQ1, . . . Aj = (Aj · Q1) Q1 + · · · + (Aj · Qi) Qi + · · · + kQ 0jkQj, . . . An = (An · Q1) Q1 + · · · + (An · Qn−1) Qn−1 + kQ 0nkQn. If we let rk k = kQ 0kk, ri j = Aj · Qi when 1 i j −1, and then Q = (Q1, . . . , Qn) and R = (ri j), then R is upper-triangular, Q is orthogonal, and A = QR. 9.5. QR-DECOMPOSITION FOR INVERTIBLE MATRICES 487 Remarks: (1) Because the diagonal entries of R are pos-itive, it can be shown that Q and R are unique. (2) The QR-decomposition holds even when A is not in-vertible. In this case, R has some zero on the diagonal. However, a di↵erent proof is needed. We will give a nice proof using Householder matrices (see also Strang ). Example 7. Consider the matrix A = 0 @ 0 0 5 0 4 1 1 1 1 1 A We leave as an exercise to show that A = QR with Q = 0 @ 0 0 1 0 1 0 1 0 0 1 A and R = 0 @ 1 1 1 0 4 1 0 0 5 1 A 488 CHAPTER 9. EUCLIDEAN SPACES Another example of QR-decomposition is A = 0 @ 1 1 2 0 0 1 1 0 0 1 A where Q = 0 @ 1/ p 2 1/ p 2 0 0 0 1 1/ p 2 −1/ p 2 0 1 A and R = 0 @ p 2 1/ p 2 p 2 0 1/ p 2 p 2 0 0 1 1 A The QR-decomposition yields a rather eﬃcient and nu-merically stable method for solving systems of linear equa-tions. 9.5. QR-DECOMPOSITION FOR INVERTIBLE MATRICES 489 Indeed, given a system Ax = b, where A is an n ⇥n invertible matrix, writing A = QR, since Q is orthogonal, we get Rx = Q>b, and since R is upper triangular, we can solve it by Gaus-sian elimination, by solving for the last variable xn ﬁrst, substituting its value into the system, then solving for xn−1, etc. The QR-decomposition is also very useful in solving least squares problems (we will come back to this later on), and for ﬁnding eigenvalues. It can be easily adapted to the case where A is a rect-angular m ⇥n matrix with independent columns (thus, n m). In this case, Q is not quite orthogonal. It is an m ⇥ n matrix whose columns are orthogonal, and R is an invertible n ⇥n upper triangular matrix with positive diagonal entries. For more on QR, see Strang . 490 CHAPTER 9. EUCLIDEAN SPACES It should also be said that the Gram–Schmidt orthonor-malization procedure that we have presented is not very stable numerically, and instead, one should use the mod-iﬁed Gram–Schmidt method. To compute Q 0k+1, instead of projecting Ak+1 onto Q1, . . . , Qk in a single step, it is better to perform k pro-jections. We compute Qk+1 1 , Qk+1 2 , . . . , Qk+1 k as follows: Qk+1 1 = Ak+1 −(Ak+1 · Q1) Q1, Qk+1 i+1 = Qk+1 i −(Qk+1 i · Qi+1) Qi+1, where 1 i k −1. It is easily shown that Q 0k+1 = Qk+1 k . The reader is urged to code this method.
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Skip to lesson content Get ready for AP® Calculus Course: Get ready for AP® Calculus>Unit 1 Lesson 17: Trigonometric identities on the unit circle Sine & cosine identities: symmetry Tangent identities: symmetry Sine & cosine identities: periodicity Tangent identities: periodicity Trig identities from reflections and rotations Trig values of special angles Math> Get ready for AP® Calculus> Get ready for limits and continuity> Trigonometric identities on the unit circle © 2025 Khan Academy Terms of usePrivacy PolicyCookie NoticeAccessibility Statement Sine & cosine identities: symmetry AL.Math: PC.34 Google Classroom Microsoft Teams About About this video Transcript Sal finds several trigonometric identities for sine and cosine by considering horizontal and vertical symmetries of the unit circle.Created by Sal Khan. Skip to end of discussions Questions Tips & Thanks Want to join the conversation? Log in Sort by: Top Voted sarahh 11 years ago Posted 11 years ago. Direct link to sarahh's post “Do I have to memorize all...” more Do I have to memorize all of these formulas? Is it important enough to know? Answer Button navigates to signup page •2 comments Comment on sarahh's post “Do I have to memorize all...” (33 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer doctorfoxphd 11 years ago Posted 11 years ago. Direct link to doctorfoxphd's post “No, you can get through a...” more No, you can get through a lot of math without memorizing, but it just takes a lot longer to do the problems. Sometimes it is just plain easier to memorize a couple of formulas than to try to dig back to the basics and reconstruct the formulas. In the case of the symmetry relationships, it is a great time-saver to know these. There are ways of reconstructing the information if you forget. One way is to memorize the signs for the different trig functions in the four quadrants. The way I remind myself of these formulas is to think of a point in the first quadrant (both x and y will be positive, so all sine and cosine values will be positive, as will tangent). Then I think of a point in the second quadrant (x will be negative, since all the values for x will be less than zero, and y will be positive. As a result, sine will be positive, but cosine will be negative, and all tangent values will be negative.) In the third quadrant, all x and y values will be negative, so all sine and cosine values will be negative. Tangent will be positive because a negative divided by a negative is positive.) The final quadrant is the fourth quadrant, and there, all x values are positive, but all y values are negative, so sine will be negative, cosine will be positive and tangent values will be negative. So, you CAN recreate the information by logic. In the meantime, others can use the symmetries and be done with the problem and maybe with the next problem as well. Also, there are some ways that the questions can be asked that make it difficult to use this method, and if you are not very conversant with unit circle and translating points to sine and cosine, then you may have some tough slogging ahead. Knowing this set of symmetries becomes handy in Physics and many other applications. 1 comment Comment on doctorfoxphd's post “No, you can get through a...” (170 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... armystorms 12 years ago Posted 12 years ago. Direct link to armystorms's post “What is a radian? I thin...” more What is a radian? I think I have some vague description of a ratio and something with PI but I don't honestly remember. Answer Button navigates to signup page •1 comment Comment on armystorms's post “What is a radian? I thin...” (28 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Alaska Test Prep 9 years ago Posted 9 years ago. Direct link to Alaska Test Prep's post “My take on radians: Imagi...” more My take on radians: Imagine a unit circle (centered at (0,0) & radius=1). Now, imagine taking a string and laying it down all the way around this circle. What is the length of the string? In other words, what is the circumference of this circle? Elementary school tells us that C = πD = 2πr, so C = 2π. In other words, 2π of string gets us one complete revolution around this circle. But how far does π get us? If we only had π string, where on the circle would we be? We'd be half-way, to point (-1,0). And, π/2 would get us 1/4 of the way around the circle, to (0,1), and 3π/2 would get us 3/4 around the circle, to point (0,-1). So, a radian is a measure of "How many radii" do we need to get us to this point in the circle.... Comment Button navigates to signup page (24 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Neal 11 years ago Posted 11 years ago. Direct link to Neal's post “I must say that was a lot...” more I must say that was a lot to take in for one video, usually there's the odd "There's this reason" or "There's that" , not much made alot of sense to me with without examples or references . Answer Button navigates to signup page •Comment Button navigates to signup page (13 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer doctorfoxphd 11 years ago Posted 11 years ago. Direct link to doctorfoxphd's post “I thought it was mostly r...” more I thought it was mostly review, but just pulled together as a different way of looking at the symmetries around a unit circle. So, what exactly was confusing? There are a number of more basic videos on trigonometry that may help, but I am not sure where to recommend that you start. Then after you are familiar with the basics, it would be easier to see how the concepts fit together. The key was showing how to name the reflected angles, and then how to construct the cosine and sine statements for each angle, and finally how to relate the pairs. And it all starts with the unit circle, so if you are hazy on that, it would be a great place to start your review. For example, let's say that we are looking at an angle of π/3 on the unit circle. The value of sin (π/3) is ½√3 while cos (π/3) has a value of ½ The value of sin (-π/3) is -½√3 while cos (-π/3) has a value of ½ Already we can see that cos theta = cos -theta with this example. And look at that: sin -theta = -sin theta just like Sal said. If we go through all the other reflected angles, with this specific example of an angle, we will get the same relationships that Sal just walked us through. One other example is π - theta This looks mysterious until you realize that in our situation, theta equals π/3 and π - π/3 gives us an angle of ⅔π (In degrees, that is 120 degrees) cos ⅔π is going to be a negative value because it is in the second quadrant, and from working with unit circles, we should remember that it is evaluated as cos π/3 except for the negative. So, cos (π - π/3) = - cos π/3 and cos π/3 = - cos (π - π/3) Basically, if you have these symmetries, you have a multitude of sine and cosine values as long as you know what sine of theta is and cosine of theta is. It may help you to continue around the circle with common angles like π/6 and π/4 (not to mention the rest of the π/3 gang). That may help you to see how the symmetrical relationships are great time savers. 2 comments Comment on doctorfoxphd's post “I thought it was mostly r...” (18 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... я̿€̿ρ̿Ł̿ɪ̿т̿Ǻ̿ƶ̿ 11 years ago Posted 11 years ago. Direct link to я̿€̿ρ̿Ł̿ɪ̿т̿Ǻ̿ƶ̿'s post “What is SOH CAH TOA?” more What is SOH CAH TOA? Answer Button navigates to signup page •Comment Button navigates to signup page (0 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Barrett Southworth 11 years ago Posted 11 years ago. Direct link to Barrett Southworth's post “(S)ine (O)pposite (H)ypot...” more (S)ine (O)pposite (H)ypotenuse (C)osine (A)djacent (H)ypotenuse (T)angent (O)pposite (A)djacent (the non-hypotenuse adjacent) The first letter in each of the letter triplets names the ratio the second letter of each triplet is the numerator, the third letter of each triplet is the denominator. Comment Button navigates to signup page (30 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... abc123 2 years ago Posted 2 years ago. Direct link to abc123's post “I know it's advised that ...” more I know it's advised that ultimately it's easier to try and memorize these identities and formulas but a big issue i've been having moving onto higher math is that I just can't memorize everything without an understanding of why. Why do we use these sine and cosine identities, what types of problems are they useful for and how did those problems form in math history? Answer Button navigates to signup page •1 comment Comment on abc123's post “I know it's advised that ...” (8 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Nathan Reyes 7 months ago Posted 7 months ago. Direct link to Nathan Reyes's post “1. Sal tries to tell the ...” more 1. Sal tries to tell the identities of sine and cosine. While you are right that memorizing them is easier, it's fine to just remember just a few of them so that you can just build up from what you know in case you forget. The sine and cosine identities are useful in lots of different stuff. Like mechanical engineering, sound waves, and etc. Almost every single problem in Math have been built from the ground up slowly in history. People long ago just kept adding more to Math until today. Ultimately, you don't really need to learn it at a young age, but it's important to know for future use. Hope this helps! Comment Button navigates to signup page (4 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Isaacskilling 11 years ago Posted 11 years ago. Direct link to Isaacskilling's post “what is the diffrence bet...” more what is the diffrence between -data and +data Answer Button navigates to signup page •1 comment Comment on Isaacskilling's post “what is the diffrence bet...” (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer 𝒘𝒂𝒌𝒆 𝒖𝒑. 2 years ago Posted 2 years ago. Direct link to 𝒘𝒂𝒌𝒆 𝒖𝒑.'s post “At 7:04, why does Sal wri...” more At 7:04 , why does Sal write -sin(θ) = sin(θ+π) instead of sin(-θ) = sin(π+θ)? Answer Button navigates to signup page •1 comment Comment on 𝒘𝒂𝒌𝒆 𝒖𝒑.'s post “At 7:04, why does Sal wri...” (5 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Benjamin Leng 2 years ago Posted 2 years ago. Direct link to Benjamin Leng's post “Look at the circle carefu...” more Look at the circle carefully: sin(θ) is the opposite of -sin(θ). Move horizontally to the left and see that -sin(θ)=sin(θ+π). In conclusion. Sal's way is just an alternative to your way. It still makes sense either way. Sal also uses the same analogy applied to cos(θ+π)=-cos(θ) in 7:29 . Comment Button navigates to signup page (5 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... sahithigorthi 2 years ago Posted 2 years ago. Direct link to sahithigorthi's post “At around 5:35, the video...” more At around 5:35 , the video says sin(-theta) = -sin(theta). I don't understand why we are putting a negative in front of the sin(theta) that is written in green, especially because sin is positive in the first quadrant. Is it to make the two equal? Also, if it is to make the two equal, can't we just write it as -sin(-theta) = sin(theta)? Answer Button navigates to signup page •Comment Button navigates to signup page (4 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer ELAINE MAUA 2 years ago Posted 2 years ago. Direct link to ELAINE MAUA's post “The sine function is an o...” more The sine function is an odd function and a function is considered an odd function if it satisfies the function: f(-x) = -f(x). For an odd function, if you replace x with its negative value, then the function’s value becomes the negative of the original value. Something else I feel is important to explain is that sin -x represents the sine of the angle -x (Remember that if you move clockwise on the unit circle, the values are negative). An example would be sin -30. On the other hand, -sin x implies that we negate the value of sin x. To put this into perspective, let me give an example: -sin 30 = sin -30. To confirm this, just use a calculator, both functions will give you -0.5. The negative is not put before sine to equate the two functions, rather it is put because sine satisfies the odd function. 1 comment Comment on ELAINE MAUA's post “The sine function is an o...” (6 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Kamran Rashidov 2 years ago Posted 2 years ago. Direct link to Kamran Rashidov's post “PLEASE HELP. 3:57 how com...” more PLEASE HELP. 3:57 how come it can be (cos(pi-t), sin(pi-t))? The yellow marked angle is not 90 degree angle. and it means we cannot apply Pythagorean theorem to pi-t. Then how we did (cos(pi-t), sin(pi-t))? Answer Button navigates to signup page •1 comment Comment on Kamran Rashidov's post “PLEASE HELP. 3:57 how com...” (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer theglasscat 2 years ago Posted 2 years ago. Direct link to theglasscat's post “Still working on understa...” more Still working on understanding all this myself so correct me if I'm wrong, but I think (cos(pi-t), sin(pi-t)) is just representing the point of the circle in relation to the graph that the circle is on, and you wouldn't apply the Pythagorean theorem to the created triangle in this case, since, like you said, it is not a right triangle. 1 comment Comment on theglasscat's post “Still working on understa...” (6 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Mətin 7 years ago Posted 7 years ago. Direct link to Mətin's post “I cannot understand that ...” more I cannot understand that how do we find the cos sin tan etc. of an angle which is more than pi/2. There are no hypetenuse or adjacent. So how it is possible that cos(theta) equals to -cos(pi - theta)? Isn't pi- theta larger than theta? Answer Button navigates to signup page •1 comment Comment on Mətin's post “I cannot understand that ...” (3 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer VVCephei 7 years ago Posted 7 years ago. Direct link to VVCephei's post “When you start working wi...” more When you start working with the unit circle, you kinda have to abstract from thinking of trig functions in terms of "real triangles" that you can look at or hold in your hands. Or if you do feel the need to visualize it, i guess you can imagine a triangle that gets turned inside out, making one of it's side's length negative... or something like that. Comment Button navigates to signup page (5 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Show more... Video transcript Voiceover:Let's explore the unit circle a little bit more in depth. Let's just start with some angle theta, and for the sake of this video, we'll assume everything is in radians. This angle right over here, we would call this theta. Now let's flip this, I guess we could say, the terminal ray of this angle. Let's flip it over the X and Y-axis. Let's just make sure we have labeled our axes. Let's flip it over the positive X-axis. If you flip it over the positive X-axis, you just go straight down, and then you go the same distance on the other side. You get to that point right over there, and so you would get this ray. You would get this ray that I'm attempting to draw in blue. You would get that ray right over there. Now what is the angle between this ray and the positive X-axis if you start at the positive X-axis? Well, just using our conventions that counterclockwise from the X-axis is a positive angle, this is clockwise. Instead of going theta above the X-axis, we're going theta below, so we would call this, by our convention, an angle of negative theta. Now let's flip our original green ray. Let's flip it over the positive Y-axis. If you flip it over the positive Y-axis, we're going to go from there all the way to right over there then we can draw ourselves a ray. My best attempt at that is right over there. What would be the measure of this angle right over here? What was the measure of that angle in radians? We know if we were to go all the way from the positive X-axis to the negative X-axis, that would be pi radians because that's halfway around the circle. This angle, since we know that that's theta, this is theta right over here, the angle that we want to figure out, this is going to be all the way around. It's going to be pi minus, it's going to be pi minus theta. Notice, pi minus theta plus theta, these two are supplementary, and they add up to pi radians or 180 degrees. Now let's flip this one over the negative X-axis. If we flip this one over the negative X-axis, you're going to get right over there, and so you're going to get an angle that looks like this, that looks like this. Now what is going to be the measure of this angle? If we go all the way around like that, what is the measure of that angle? To go this far is pi, and then you're going another theta. This angle right over here is theta, so you're going pi plus another theta. This whole angle right over here, this whole thing, this whole thing is pi plus theta radians. Pi plus theta, let me just write that down. This is pi plus theta. Now that we've figured out these have different symmetries about them, let's think about how the sines and cosines of these different angles relate to each other. We already know that this coordinate right over here, that is sine of theta, sorry, the X-coordinate is cosine of theta. The X-coordinate is cosine of theta, and the Y-coordinate is sine of theta. Or another way of thinking about it is this value on the X-axis is cosine of theta, and this value right over here on the Y-axis is sine of theta. Now let's think about this one down over here. By the same convention, this point, this is really the unit circle definition of our trig functions. This point, since our angle is negative theta now, this point would be cosine of negative theta, comma, sine of negative theta. And we can apply the same thing over here. This point right over here, the X-coordinate is cosine of pi minus theta. That's what this angle is when we go from the positive X-axis. This is cosine of pi minus theta. And the Y-coordinate is the sine of pi minus theta. Then we could go all the way around to this point. I think you see where this is going. This is cosine of, I guess we could say theta plus pi or pi plus theta. Let's write pi plus data and sine of pi plus theta. Now how do these all relate to each other? Notice, over here, out here on the right-hand side, our X-coordinates are the exact same value. It's this value right over here. So we know that cosine of theta must be equal to the cosine of negative theta. That's pretty interesting. Let's write that down. Cosine of theta is equal to ... let me do it in this blue color, is equal to the cosine of negative theta. That's a pretty interesting result. But what about their sines? Well, here, the sine of theta is this distance above the X-axis, and here, the sine of negative theta is the same distance below the X-axis, so they're going to be the negatives of each other. We could say that sine of negative theta, sine of negative theta is equal to, is equal to the negative sine of theta, equal to the negative sine of theta. It's the opposite. If you go the same amount above or below the X-axis, you're going to get the negative value for the sine. We could do the same thing over here. How does this one relate to that? These two are going to have the same sine values. The sine of this, the Y-coordinate, is the same as the sine of that. We see that this must be equal to that. Let's write that down. We get sine of theta is equal to sine of pi minus theta. Now let's think about how do the cosines relate. The same argument, they're going to be the opposites of each other, where the X-coordinates are the same distance but on opposite sides of the origin. We get cosine of theta is equal to the negative of the cosine of ... let me do that in same color. Actually, let me make sure my colors are right. We get cosine of theta is equal to the negative of the cosine of pi minus theta. Now finally, let's think about how this one relates. Here, our cosine value, our X-coordinate is the negative, and our sine value is also the negative. We've flipped over both axes. Let's write that down. Over here, we have sine of theta plus pi, which is the same thing as pi plus theta, is equal to the negative of the sine of theta, and we see that this is sine of theta, this is sine of pi plus theta, or sine of theta plus pi, and we get the cosine of theta plus pi. Cosine of theta plus pi is going to be the negative of cosine of theta, is equal to the negative of cosine of theta. Even here, and you could see, you could keep going. You could try to relate this one to that one or that one to that one. You can get all sorts of interesting results. I encourage you to really try to think this through on your own and think about how all of these are related to each other based on essentially symmetries or reflections around the X or Y-axis. Creative Commons Attribution/Non-Commercial/Share-AlikeVideo on YouTube Up next: video Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Accept All Cookies Strictly Necessary Only Cookies Settings Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. 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2504	https://www.youtube.com/watch?v=dNSk4coSpUc	Maximum volume of a cone shaped cup (KristaKingMath) Krista King 273000 subscribers 938 likes Description 79801 views Posted: 1 Sep 2012 ► My Applications of Derivatives course: Learn how find the largest possible volume of a cone-shaped up made from a circular piece of paper with radius R, where a sector has been removed and sides CA and CB are joined together. To complete this optimization problem, you'll need to draw a picture of the problem and write down what you know. Then you'll imagine that you'll take a vertical slice of the cone shaped cup so that you can use the pythagorean theorem to relate the radius of the paper to the radius and height of the cone. Solve the pythagorean theorem, the constraint equation, for one of the variables so that you can plug it into the optimization equation, which will be the equation for the volume of the cone. Then simplify the equation for the volume, take the derivative and set it equal to 0 to solve for the height. Plug the height back into the pythagorean theorem to find the radius, then plug both values back into the volume equation to find the volume. ● ● ● GET EXTRA HELP ● ● ● If you could use some extra help with your math class, then check out Krista’s website // ● ● ● CONNECT WITH KRISTA ● ● ● Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;) Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!” So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: FACEBOOK // TWITTER // INSTAGRAM // PINTEREST // GOOGLE+ // QUORA // 110 comments Transcript: Intro hi everyone today we're going to talk about how to find the largest possible volume of a cone-shaped cup to complete this problem we'll draw a picture of the problem and write what we know identify optimization and constraint equations and then use the derivative of the optimization equation to find the volume let's take a look in this particular The problem problem we've been told that a cone-shaped drinking cup has been made from a circular piece of paper of radius R where a sector of the piece of paper has been removed and the edges here diagrammed in this figure CA and CB have been joined together and we've been asked to find the maximum volume of this cone-shaped cup so as with any optimization problem the first thing we want to do is draw a picture of the problem and write down everything that we know so in this case this picture here has already been given to us of the circular piece of paper but what we need to imagine is the cone-shaped cup that's created from it so we know that the sector that's been cut out here has this Center C these points A and B the angle a and we know that the radius of the circular piece of paper is r and we'll denote that with capital r what we need to realize is that if we join CA and CB if we pull those two sides together we'll get a cone-shaped cup like this that has a side here of r a height down the center of the cone of h and a base radius of lowercase R since we're being asked for the maximum volume of this cone-shaped cup we know that we're going to need the equation for the volume of a cone so we've gone ahead and written volume equal 1/3 piun r^2 H which is the equation for the volume of a cone now our next step is to realize that at the end of this problem we're going to need to get this volume equation completely in terms of capital r right now we have it in terms of lowercase R and H but because the problem as asks us to maximize the volume of this cup that has a radius right of capital r we know we're going to have to get the volume equation in terms of capital r only so we're going to have to do a couple things in order to get to that point the first of which is that we're going to have to find some way to relate capital r with lowercase R and H the way that we're going to do that is by imagining that we cut a cross-section of this cone if we took like a knife and we cut this cone straight down right we started from the top and we sawed straight through the middle of it we would get a triangle that looks like this that has the hypotenuse here on one side of R this height down the middle here of H and the base just of this half here as lowercase The solution R the base radius of the cone we know from the Pythagorean theorem that we can use this triangle to relate these three values so we have the the two sides here R and lowercase H lowercase R and H and the hypotenuse of capital r when we have two sides and hypotenuse like that of a right triangle we know that lowercase R 2+ h^2 is equal to the hypotenuse capital r squared so there's our equation that relates the three variables so now we need to get to work solving this equation here what we'll call the constraint equation for one of these other variables the goal here is going to be to solve for one of the other variables and then plug the value back into the volume equation so what we should do is realize that we have r s here in our volume or optimization equation and we have R squ here in our constraint equation since those two are equal it would be awfully convenient to be able to solve for lowercase R 2 so what we'll do is we'll subtract H squar from both sides and we'll get lowercase r s is equal to capital r 2 - h^2 now we can take the value we have on the right hand side r^ 2us h^2 and plug that in for lowercase R 2 in our volume equation so in our volume equation we'll get 1/3 Pi time capital r 2us h^ 2 H now we really have our volume equation in terms of one variable it looks like it's in ter in terms of two variables but it's actually only in terms of one variable because capital r is a constant because we've been told that the radius is equal to r r will never change and they could have just as easily asked us to find the volume of the cup made out of this piece of paper with radius four but instead they asked us for R it's representing the same thing R here is a constant so the only variable in this equation that's left is H so what we're going to do is simplify the volume equation then take its derivative with respect ECT to H treating capital r as a constant set it equal to zero and solve for H so let's go ahead and simplify first what we'll get here is 1/3 pi r 2 H when we multiply this coefficient here 1/3 piun r^ 2 H then we'll get minus 1/3 Pi H cubed when we multiply the coefficient here by h^2 by H so now that we have this for our volume equation we want to go ahead and take the derivative of the volume equation so we'll get V prime or the derivative of the volume equation remember we're treating r as a constant so this first term here 1/3 Pi r^ 2 H is the same thing right as basically saying 3 H 1/3 is a constant Pi is a constant and R is a constant so if if we were taking the derivative of 3H what we would get is 3 we'd basically take the constant coefficient in front of the H and that would be our derivative in this case we can take the constant coefficient 1/3 pi r 2 and that's our derivative so 13 pi r 2 when we take the derivative of - 13 Pi H cubed we'll use the power rule and we'll multiply this three out in front and then subtract one from the exponent so 3 1/3 will just give us 1 that'll cancel so we'll be left with just Pi out in front and then we'll subtract one from the exponent to get H squared and now that we have our derivative we can go ahead and set this equal to zero and solve for H so we will add Pi H squar to both sides we'll get 1/3 piun R 2 is equal to Pi h^ 2 we'll go ahead and divide both sides by pi and we'll get pi to cancel that's just a constant and you can see that what we're left with is just h^2 is equal to 1/3 R 2 to solve for H we'll go ahead and take the square root of both sides and we'll see that H is equal to 1 over the square OT of 3 R remember that normally when you take the square root you have to pay attention to both the positive and negative value that could result over here on the right hand side because we're dealing with a three-dimensional figure here in actual space if we took the negative value -1 over the < TK 3 R this whole value over here on the right hand side would be negative and that can't be true because if H were negative this cone wouldn't exist at all so we only have to pay attention to the positive value now that we've solved for H remember that our goal here is to get get back to the volume equation we want to make substitutions to replace r s lowercase r squ with something in terms of capital r 2 we also want to replace H with something in terms of capital r so we need to find an equation that relates lowercase R 2 to R and another equation that relates H to capital r we already have an equation solved for H in terms of r capital r it's right here so we'll go ahead and plug 1/ < tk3 R in for H but we need to find a value for lowercase R 2 in terms of capital r so the way that we'll do that is by using this equation here for lowercase R 2 so we'll take lowercase R2 is equal to capital r 2 we've solved for H2 it's right here so we'll go ahead and plug the right hand side here in for H2 so we'll get minus 1/3 capital r squared so if you can imagine now if we find a common denominator if we multiply this first term by 3 over3 to get that common denominator now we have 3 r^ 2us 1 R 2 all over 3 which will just give us R 2 = 2/3 capital r 2ar now we have an equation for H in terms of capital r and we have an equation for R 2 in terms of capital r so we'll plug both of these into our volume equation and then simplify to find our final answer now plugging into this volume equation we'll get volume equals 1/3 pi for lowercase R 2 here we'll plug in 2/3 capital r 2 so we'll get 2/3 capital r squar and then we've also solved for H so we'll plug in 1 1/ < tk3 R in for H 1/ < tk3 capital r and now we'll simplify this as much as we can you can see here that in the numerator we'll get two that'll take care of the one the two and the two we'll get Pi because we have that in our numerator here and then we'll get R to the 3 which will take care of this r s and this R here so we'll get r cubed then in our denominator you can see we have 3 3 gives us 9 < tk3 so essentially we have 9 the square < TK of 3 and that's it that is our final answer for the maximum volume of this cone-shaped cup this way now that we have a volume equation that's in terms of capital r in the future we could take any value for capital r any value for the radius of this circular piece of paper plug it in for capital r and we would immediately know the maximum volume since we now have a formula that relates the two so I hope you found that video helpful if you did like this video down below and subscribe to be notified of future videos
2505	https://byjus.com/physics/ideal-gas-law-and-absolute-zero/	Table of Contents What is Ideal Gas Law? Ideal Gas Law Units Derivation of the Ideal Gas Law Ideal Gas Solved Problems Relationship between Pressure and Temperature Frequently Asked Questions -FAQs What is Ideal Gas Law? The ideal gas law, also known as the general gas equation, is an equation of the state of a hypothetical ideal gas. Although the ideal gas law has several limitations, it is a good approximation of the behaviour of many gases under many conditions. Benoit Paul Émile Clapeyron stated the ideal gas law in 1834 as a combination of the empirical Charles’s law, Boyle’s Law, Avogadro’s law, and Gay-Lussac’s law. The ideal gas law states that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant. The empirical form of ideal gas law is given by: PV=nRT where, P is the pressure. V is the volume. n is the amount of substance. R is the ideal gas constant. Ideal Gas Law Units When we use the gas constant R = 8.31 J/K.mol, then we have to plug in the pressure P in the units of pascals Pa, volume in the units of m 3 and the temperature T in the units of kelvin K. When we use the gas constant R = 0.082 L.atm/K.mol then pressure should be in the units of atmospheres atm, volume in the units of litres L and the temperature T in the units of kelvin K. For easy reference, the above information is summarised in the table as follows: Ideal Gas equation Units R=8.31 J K.m o l R=0.082 L.a t m K.m o l Pressure in pascals Pa Pressure in atmospheres atm Volume in m 3 Volume in litres L The temperature in Kelvin K The temperature in Kelvin K Derivation of the Ideal Gas Law The ideal gas law is derived from the observational work of Robert Boyle, Gay-Lussac and Amedeo Avogadro. Combining their observations into a single expression, we arrive at the Ideal gas equation, which describes all the relationships simultaneously. The three individual expressions are as follows: Boyle’s Law V∝1 P Charles’s Law V∝T Avagadro’s Law V∝n Combining these three expressions, we get V∝n T P The above equation shows that volume is proportional to the number of moles and the temperature while inversely proportional to the pressure. This expression can be rewritten as follows: V=R n T P=n R T P Multiplying both sides of the equation by P to clear off the fraction, we get P V=n R T The above equation is known as the ideal gas equation. Ideal Gas Solved Problems 1. What is the volume occupied by 2.34 grams of carbon dioxide gas at STP? Solution: To determine the volume, rearrange the ideal gas law as follows: V=n R T/P Substituting the values as follows, we get V=[(2.34 g/44 g.m o l−1)(0.08206 L a t m m o l−1 K−1)(273.0 K)]/1.00 a t m V=1.19 L 2. Calculate the temperature at which 0.654 moles of neon gas occupies 12.30 litres at 1.95 atmospheres. Solution: To determine the temperature, rearrange the ideal gas equation as follows: T=P V/n R Substituting the value, we get T=(1.95 a t m))(12.30 L)/(0.654 m o l)(0.08206 L a t m m o l−1 K−1) T=447 K Relationship between Pressure and Temperature The graph here shows the relationship between temperature and pressure for different gases. On extrapolating this graph, we see that the graphs always intercept the x-axis at a point we now call the absolute zero, irrespective of which gas we are graphing. This point represents the beginning of the Kelvin scale, i.e. Zero K. On the Celsius scale, 0 K is equivalent to -273.5 o C. This is the coldest temperature possible. As a molecule gets colder, its energy and, consequently, its movement and vibrations decrease in amplitude. As we keep cooling it, at a point, the atom will reach a state of minimum internal energy where the atom has lost almost all its energy and is stationary. Frequently Asked Questions -FAQs Q1 What is the ideal gas state? The pressure, temperature and volume of a gas are related to each other. Q2 Who derived the ideal gas? Benoît Paul Émile Clapeyron derived the ideal gas. Q3 Why is the ideal gas inaccurate? Only under perfect conditions the theory of the ideal gas is valid. The ideal gas law effectively fails at high pressure and low temperature because molecule size and intermolecular forces become important factors to take into account and are no longer negligible. Q4 Does the ideal gas law apply to liquids? No, the ideal gas law doesn’t apply to liquids. Q5 How is the ideal gas law used in everyday life? Assume an engineer wishes to store 500 g of oxygen in a container at 1 atm and 125 degrees Fahrenheit. The Ideal Gas Law is used to determine the capacity of a container that must be constructed for the same. The video about the expansion of gases 35,714 If you liked the article, download BYJU’S– the learning app to learn more fascinating intricacies of science. Come learn from us and fall in love with learning. Test your knowledge on Ideal gas law and absolute zero Q 5 Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin! Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz Start Quiz Congrats! Visit BYJU’S for all Physics related queries and study materials Your result is as below 0 out of 0 arewrong 0 out of 0 are correct 0 out of 0 are Unattempted View Quiz Answers and Analysis X Login To View Results Mobile Number Send OTP Did not receive OTP? 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2506	https://www.youtube.com/watch?v=KpcV0djb5Nk	Counting Possible INJECTIVE (one-to-one) Functions Given Constraints Gresty Academy 57300 subscribers 15 likes Description 310 views Posted: 3 Mar 2024 Occasionally Maths Olympiads and College Entrance Tests ask a question along the lines of 'how many possible INJECTIVE (one-to-one) functions are there from set A to set B which satisfy certain constraints?'. Frankly there is no better way to understand how to do these types of questions than simply to attempt as many of them as possible for familiarity and practice. In this video we answer two such questions, both from recent JEE Main exams, and as we see during the video there is no 'simple formula' which covers them. Each question is different in its own constraint, though there are some similarities which we can take advantage of. Counting is not easy - it is quite difficult not to double count or omit possibilities, so care really needs to be taken. There are certainly overlaps with these types of questions and other types of 'counting'. As such, it may be beneficial to try out some of our other 'Counting' videos in our playlist 'Counting' For more videos on injective (one-to-one) functions, see our playlist 'Injective (one-to-one), Surjective (onto) and Bijective Functions' For more videos on JEE Main see our playlist 'JEE Aspirant' For more videos on functions generally see our playlist 'Functions' For more videos on Maths Olympiads, see our playlist 'Maths Olympiads' For more videos on College Entrance Tests see our playlist 'UPCAT and Other CETs' 5 comments Transcript: in the previous video we were doing counting functions and in this video we're going to do counting injective or one: one functions okay so let's just have a quick look at the difference uh just as a reminder a function every element in the input which is here has to match with exactly one element in the output or the codomain here with an injective or one: one function every element in the input has to match with a distinct element in the output and that is the difference so um let's just take an example uh for example uh 1 2 3 uh to 1 2 3 4 five um so basically if it was a function we could have this that would be a perfectly acceptable function where they all map to one that is not an injected function we can't have that because they have to map to distinct elements so for example an injected function would have to be something along the lines of that now how many injective functions are there well if there's no restriction whatsoever and this here has n elements and this here has M elements well then the number of injective functions is n factorial over nus M factorial um and just taking this one example here well there are three elements here and there are five elements here uh so basically this one can choose any of those five this one therefore can choose any of four and this one can choose three so that would be 5 4 3 which equals 60 or another way of writing it is n p m okay now unfortunately in most college entrance tests there are restrictions or constraints which make the complication uh the calculations more complicated so let's have a look at a couple okay here's the first one um pause the video If you so desire okay right so the number of one: one or injected functions from here the domain f a b c and d to the co- domain 0 1 2 3 4 to 10 such that we have this constraint here 2 F A minus FB add 3 FC add FD equals z okay well the first thing we want to do is just amend this uh I like to rewrite it so that the biggest number is at the beginning and so there are no negatives so basically we have that all I've done is I've just rearranged it um so that we have three FC add two F A FD equals FB okay now what we're going to do is we're going to draw a table we literally do count these so uh let's imagine that c maps to here and I'll write in a different color then this will be three times F of c and then let's see what a can possibly map to and then we have two times F of a and then we have F of D and F of B and then here we can have the number of possibles possible injective functions um which we will then uh add up okay so let's have a look um let's try and keep that in set three of see right so basically F of C let's write it in a different color this is what it will map to uh what color should I write in let's write it in uh blue okay so F of C well of course F of C could map to zero uh F of C could also map to one F of C could map to two and F of C could map to three it can't map Penny higher than that because if F of C mapped to four for example then three times F of C would already be 12 uh and F of B can only possibly go between 0 and 10 because that's what the co- domain is so we know that it can't be any higher than that and for each of these numbers so let's just draw a a little bit of a table here uh and there's one let's separate that and here's two let's separate that and then there's three down below so where FC equals zero obviously uh let's just change the color again 3 FC here would be zero 3 3 FC here would be 3 3 FC here would be 6 3 FC here would be 9 uh I'm only writing in different color so we don't get confused uh now a um could be mapped to well it could be mapped to 1 2 three or four note it cannot be mapped to zero because it's an injective function and C is mapped to zero so a can't also be mapped to zero that's the difference between an injective and uh not an injective function so it be 1 2 3 or 4 it can't be greater because as we see when we have 2 F of a that would be 2 4 6 and eight well if we had here five well that would be 10 well obviously clearly we can't have this equation satisfied if that bit alone is 10 okay so that's um if it equals one so let's have a look um uh so basically if F of C is zero and F of a is 1 well then 3 F of C add 2 F of a is z add two well that means that F of D could could be anything of uh it could be three it could be uh 2 3 4 5 6 7 or8 and therefore F of B could be 4 5 6 7 8 9 or 10 and that is basically seven possibilities so let's put the number of possibilities in red there's seven possibilities there okay now let's go to the second line where F of C uh is z so 3 F of C is z and now let's say F of a be 2 so 2 F of a is four well therefore we know that F of d d could map to one it can't map to two because a is mapping to two and it's an injected function so it can't map to two 3 4 5 or 6 which means that b would map to 5 7 8 9 or 10 which is five functions so let's put in here five more possible okay let's have a look when a f of C is zero when c maps to zero and a maps to three well that means that three F of C add two F of a would be six okay so now the f d could map to either 1 2 it can't map to three because three is there and by the way it can't map to zero either neither of these can map to zero because C is mapping to zero so it can't map to three uh or it could map to four and therefore B would map to seven 8 or 10 well that's three more so we add three more there and then finally uh F of c equal 0 and F of a equal 4 so 2 F of a equal 8 therefore D could be 1 or two again it can't be 0o because zero is already being used by C uh and it can't be anything higher than two because otherwise the equation won't work remember B can only be between n and 10 uh and that would be B would be 9 or 10 and that gives us another two okay okay so that's all of the possibilities for when C is zero okay so let's uh move on to when C is one uh so if C is 1 therefore 3 F of C is uh is three so basically a f of a could be zero it can't be one because uh C is mapped to one uh it could be two or it could be three okay so that means that uh two times that would be 0 4 or six and so therefore on our equation that means that um this one here this is a d isn't it uh that's F of D and this here is f of B so that means D could be mapped to um 3 4 5 6 or seven um or seven two sorry or two so it could be mapped to 2 3 4 5 6 or seven and that means that b could be mapped to 5 6 7 8 9 or 10 which means there is another six possibles there uh let's have a look at the next one so therefore we have uh this is sorry that the uh the headings uh have gone off the top of the screen apologies for that so therefore we have C is is one so that means 3 F of C is three and that is two so that would be seven which means that therefore F of D could D could map to zero or three which means that uh leag would map to s or 10 so that's two more and and basically all you're doing here is just going through one by one um and and seeing what maps and what doesn't map so the last one here would be if that is three and that is six well clearly that means that the only thing uh that uh D could map to is zero note it can't map to one because one is already being used by three and it's an injective function which means that this would be nine which is one more uh and then finally uh sorry not finally then we have uh two let's try and keep those titles in so if c maps to two here uh then a could map to Zer or one it can't map to two obviously because uh c maps to two um and so therefore two times it would be uh zero and oh sorry that should be in black 0er and two okay so therefore uh we could have D could map that's two time that's six at Zero D could map to 1 three or four it can't map to zero and it can't map to two because they're both taken uh which would mean that uh B would map to 7 9 or 10 which is another three uh and then finally um down here if that if C is two and a is one that means that um the only thing that uh D could be is zero it can't be one and it can't be two because they're already taken which would mean that uh B would be eight and so there's one more there and then finally if C is equal to three then a can only be zero it can't be greater than zero and so therefore 2 a would be zero and therefore the only thing that uh D could be is one which means only thing B could be is 10 so there's one more there and all we have to do now is just add those up so that is s add five at three add two at six add two add one add three add one add one gives us 31 so the answer is 31 that is the number of injective functions uh that satisfy this constraint here it is fiddly um and the problem is more whether or not you can do it in the in the time given rather than whether or not you can do it uh it's just a question of being very careful okay well let's have a look at the second question here's the second question here uh pause the video If you so desire okay so we have the probability that a randomly chosen onet to one injective function from ABCD to 1 2 3 4 5 satisfies this constraint here okay uh and these are the four probabilities that we've got all right so basically let's imagine that um uh there was no constraint then basically we have here uh the Set uh is four elements and here the set has five elements and so therefore the number of uh possible injective functions given no constraint is 5 P4 which equals 5 factorial over 1 factorial which equals 120 so that's how many possible bjective functions uh there are and in order to find out the probability um that it satisfies that we have to find out how many satisfy this constraint and then divide it by 120 okay so let's have a look um again let's put the uh the numbers in in blue so let's let's imagine that sorry let's go back to back let's imagine here that we have well first of all we got to rearrange this so 2 F of B remember put the biggest one at the beginning um add F of a uh and basically never have a minus sign it's always miles more difficult if you got minus sign so always make sure that everything uh is a plus so that is our constraint just Rewritten okay so let's have a look uh if we have F of B here and here we have 2 F of B and here we have F of a and then here we will have F of D it's going to be the same table as before F of c and then we'll have the number of possibles okay so uh let's imagine that F of B is one so that's what B would map to okay so then 2 F of B would be two um okay well F of a could be uh well it can't be one um but it could be 2 3 4 or five so let's have a look if F of B is 1 then 2 F of B is 2 and F of a is three well we can't find an F of a f of D plus F of C where their sum is three given that they're not allowed to be one and two because it's an injective function so there is zero there that work on there what about um for it equal to three well 2 F of B would be 2 add three would be five so we need to find F of d add F of c um which equals five given that it can't be one and three uh and there isn't one so we can't have it there um what about uh four uh again so that would have to be F ofd add F of C would have to equal um six well clearly that can't happen um and then the final one we would have two add five well two add five is seven and we can make seven could be three and four so F of D could be three and four or four and three so there we have two posses okay so let's have a look what about um f equal 2 F of B is 2 well therefore 2 F of B is four um and let's have a look at what um f of a could be F of a could be sorry excuse me 1 3 4 or five it can't be two obviously because it's an injective function um and we will find that go by going through exactly the same process uh that we just have uh basically we could only have it when um two of FB is four and F of a is four 4 + 4 is 8 and that would allow these to be five and three or three and five for these two numbers and this number here we cannot have any because there are zero possibilities that satisfy this constraint so that is another two and then F uh when uh equals 3 F of b equals 3 we have 2 F of b equals H sorry wrong color we have 2 F of b equals 6 remember the blue is the uh is the numbers it can map to uh and the black is the um is the numbers to put into the formula um so basically a could be one um two four or five and the only one of those uh which which maps to anything is if it equals one which will be five and two and two and five and so there will be another two there uh and finally um for if it equals if uh F of b equals 4 or F of b equal 5 then we will get here 2 4 8 and 2 5 is 10 well we cannot find any uh a d or C which satisfy this equation where um B is four or five so the total number that we can find is equal to the sum of these which is six and so therefore the probability which is what it asks us the probability that are randomly chosen is equal to 6 over the total number of injective functions which is 120 which is equal to 1 over uh um 20 which equals option D they are quite fiddly these they're not difficult but you just have to basically find a way uh where you can make a table in a way like this um so that you can count all of the different possible injective functions um and you have to find a way to do that quickly well I hope you found this useful uh what we're going to do in the next video is counting bjective functions and the video after that surjective or onto functions uh and then both have their own idio sync casties so I hope you enjoyed this please like it if you did and subscribe to the gresy YouTube channel thank you
2507	https://pmc.ncbi.nlm.nih.gov/articles/PMC8812575/	Sperm bauplan and function and underlying processes of sperm formation and selection - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Physiol Rev . 2021 Apr 21;102(1):7–60. doi: 10.1152/physrev.00009.2020 Search in PMC Search in PubMed View in NLM Catalog Add to search Sperm bauplan and function and underlying processes of sperm formation and selection Maria Eugenia Teves Maria Eugenia Teves 1 Department of Obstetrics and Gynecology, Virginia Commonwealth University, Richmond, Virginia Find articles by Maria Eugenia Teves 1,✉, Eduardo R S Roldan Eduardo R S Roldan 2 Department of Biodiversity and Evolutionary Biology, Museo Nacional de Ciencias Naturales (CSIC), Madrid, Spain Find articles by Eduardo R S Roldan 2,✉ Author information Article notes Copyright and License information 1 Department of Obstetrics and Gynecology, Virginia Commonwealth University, Richmond, Virginia 2 Department of Biodiversity and Evolutionary Biology, Museo Nacional de Ciencias Naturales (CSIC), Madrid, Spain ✉ Correspondence: M. E. Teves (maria.teves@vcuhealth.org); E. R. S. Roldan (roldane@mncn.csic.es). ✉ Corresponding author. Received 2020 Apr 6; Revised 2021 Apr 14; Accepted 2021 Apr 19; Issue date 2022 Jan 1. Copyright © 2022 The Authors Licensed under Creative Commons Attribution CC-BY 4.0. Published by the American Physiological Society. PMC Copyright notice PMCID: PMC8812575 PMID: 33880962 Open in a new tab Keywords: male fertility, spermatogenesis, spermatozoa, sperm form, sperm function Abstract The spermatozoon is a highly differentiated and polarized cell, with two main structures: the head, containing a haploid nucleus and the acrosomal exocytotic granule, and the flagellum, which generates energy and propels the cell; both structures are connected by the neck. The sperm’s main aim is to participate in fertilization, thus activating development. Despite this common bauplan and function, there is an enormous diversity in structure and performance of sperm cells. For example, mammalian spermatozoa may exhibit several head patterns and overall sperm lengths ranging from ∼30 to 350 µm. Mechanisms of transport in the female tract, preparation for fertilization, and recognition of and interaction with the oocyte also show considerable variation. There has been much interest in understanding the origin of this diversity, both in evolutionary terms and in relation to mechanisms underlying sperm differentiation in the testis. Here, relationships between sperm bauplan and function are examined at two levels: first, by analyzing the selective forces that drive changes in sperm structure and physiology to understand the adaptive values of this variation and impact on male reproductive success and second, by examining cellular and molecular mechanisms of sperm formation in the testis that may explain how differentiation can give rise to such a wide array of sperm forms and functions. 1. INTRODUCTION The sperm cell is a highly polarized and differentiated cell, whose main goal is to reach the oocyte and participate in fertilization (1). To do so, it must embark on a long journey, and success in this endeavor relies on structurally and functionally sound cells. Despite this common task, sperm cells from different species vary in structure and mechanisms underlying their functions. To understand how and why such variation has originated, it is possible to inquire about the processes of sperm formation (2–4) and about the evolutionary forces that may operate in selecting spermatozoa (5–8). There has been a long-standing interest in both, but progress has been slow, particularly when trying to integrate them. In this review we summarize aspects of sperm structure and function and attempt to relate them to processes of spermatogenesis and postcopulatory sexual selection. Our general aim is to present this outline with a view toward a better understanding of sperm biology and, in addition, to provide background for assessments of sperm fertilizing capacity and for the development of improved methods of assisted reproduction. CLINICAL HIGHLIGHTS Sperm morphology is closely associated to sperm function. In humans, sperm morphology varies considerably even within the same individual. This pleomorphic characteristic of spermatozoa has led to problems when devising methods to assessing them in a clinical context or when evaluating the effect of genetic or environmental stresses. Very strict criteria for the evaluation of human sperm morphology, such as those recommended by the World Health Organization, result in a threshold for normality of only ∼4% of spermatozoa. Sperm variation raises several questions, such as what normal sperm cells are, how we can identify them, how we can overcome the difficulty of guaranteeing standards across laboratories, and whether we can design new approaches for morphology assessments in a clinical context. Because sperm form is closely related to its function, abnormal sperm may not swim adequately and reach the site of fertilization or fertilize in vitro. This not only relates to hydrodynamic efficiency of swimming but is also indicative of problems during sperm formation that may influence function at other levels (signaling, bioenergetics) or be a sign of sperm DNA damage. The latter is relevant because mutations in the father could lead to genetic diseases in offspring. Altogether, a better knowledge of processes of sperm formation, and selection methods that rely on form and function, may lead to better diagnosis and improved techniques to assist (or control) human reproduction. Overall, sperm structure is highly conserved throughout animals, although exceptions do exist. In general terms, the sperm bauplan (i.e., its design) consists of a head and a flagellum (or tail) (9–11). The sperm head has the nucleus and an exocytotic granule called the acrosome. A neck, or connecting piece, attaches the sperm head to the flagellum and contains typical or atypical centrioles (12). The flagellum consists of the midpiece and the principal and end pieces and usually has the machinery to produce energy in the form of ATP. It also provides the propulsive force for swimming (9–11). Efficiency in sperm function relates to several aspects. First, the sperm head carrying the nucleus is streamlined for hydrodynamic efficiency. Second, the nucleus is reduced in size by compaction of chromatin, which occurs via histone replacement and protamine binding to DNA. Third, chromatin compaction also protects DNA integrity, thus minimizing damage from intra- and extracellular stress. Fourth, because genes are silenced by protamine action and chromatin compaction, the sperm has limited repair capacity. Fifth, remodeling of the sperm head takes place together with the development of a flagellum for cell motion, which requires integration of formation of both components. Sixth, sperm movement relies on propulsive forces generated by the flagellum and energy production. Finally, once the sperm cells are released or transferred to the female tract, a series of additional structural and functional changes take place in preparation for fertilization. The study of model organisms from a few taxa to understand reproductive physiology is advantageous for identification of common themes in structure and function. However, this occasionally has led us to forget that these models may represent just some limited examples in comparison to the diversity that exists in nature. Paying attention to such diversity may be informative as to the role played by structural components and for a better understanding of molecular mechanisms that control cell formation and function. To understand the diversity of sperm cells, with regard to both structure and their physiology, it is important to bear in mind that sperm cells are manufactured under delicate genetic control but that this process is also susceptible to environmental factors. After sperm production in the testes, sperm cells undergo further processing in the male and female tracts, which are also under genetic control and may be modified by environmental conditions. Mechanisms of sperm formation in the testis are generally conserved in vertebrates, with a finely orchestrated interaction between somatic and germ cells in the seminiferous tubules and a series of phases that involve proliferation, meiosis, and differentiation (13–15), but, given that there is considerable diversity in the structure of the sperm cell, it is predictable that such diversity could be explained by differences in both the architecture of the testis and the kinetics of sperm proliferation or differentiation. This will be particularly relevant in the final stages of spermatogenesis, when the sperm cell is formed and when considerable remodeling of the nucleus and the generation of sperm-specific organelles and structures take place. It is thus not surprising that much variation exists in the underlying cellular and molecular mechanisms regulating this last step of spermatogenesis, and this could be one source of heterogeneity in sperm cells. Several selective forces may be responsible for the evolution of the known diversity in sperm structure, together with variation in cell functions (e.g., the provision of energy, propulsion, exocytosis of a single acrosomal granule, interaction with the female gamete, and penetration of the oocyte vestments). In vertebrates, fertilization could be either external or internal, and this would affect the sources of energy that the sperm may use to sustain propulsion. In addition, fertilization mode defines different environments (regarding both space and time) in which the sperm needs to survive in order to encounter an oocyte and engage in fertilization. The evolution of internal fertilization may have heavily influenced the structure and the function of the male gamete (16), considering that the sperm cell has to interact with the female tract and, in addition, undergo additional steps to be able to interact with the oocyte. The female tract may present a series of formidable barriers that the sperm has to overcome, and this could serve as the basis for female selection. Furthermore, females may mate with more than one male, and under these circumstances sperm from rival males will have to compete in order to achieve fertilizations. The latter scenarios are the basis for postcopulatory sexual selection, which may have heavily influenced the evolution of sperm shape, size, and function. There have been several recent reviews covering some of the issues mentioned above; they are referred to in the relevant sections. We have also recently reviewed some information on sperm function or sperm formation and on evolutionary processes in spermatozoa. Knowledge on protein/vesicle transport during spermatogenesis has received particular attention, and we have presented an in silico analysis that identified, for the first time, the molecular interactome of proteins involved in protein trafficking during spermiogenesis (17). This overview has identified gaps in knowledge and challenges for future research. We have recently addressed several levels of proximate and evolutionary explanations that can be used as framework to gain integrated knowledge of sperm biology, particularly structure and function of sperm cells attending fertilization (18), so these issues are not covered in detail here. To assess diversity in sperm morphology and function, it is necessary to resort to a series of methods for image analysis and cellular and molecular functional tests that can inform on sperm performance; various approaches for such sperm assessments have been reviewed recently (19) and the implications for male fertility considered in this context (20). Finally, the principles of sperm competition have been examined in connection to the development of new tests for the analysis of sperm morphology or sperm performance (21). In the present overview, we focus on two particular levels of analysis. On the one hand, we address evolutionary forces (postcopulatory sexual selection) in relation to sperm structure and spermatogenesis. On the other hand, we deal with proximate (mechanistic) explanations of molecular changes underlying spermiogenesis, with a particular interest in structural diversity and the origin of biological novelties in sperm. The rationale for this review is thus to integrate various aspects of sperm biology in such a way that it could be appealing to students and specialists interested in cellular and molecular aspects of sperm formation, function, and fertility and also those interested in understanding evolutionary issues of sperm biology. In addition, we anticipate that it may provide background for those focusing on clinical aspects of male reproduction or in areas connected to animal production. This review therefore covers the following. First, it gives an overview of the sperm bauplan, that is, the structure of the sperm cell and its main components. The attention is mostly on the shape and the size of the sperm head, because considerable recent research exists on these topics. We also present an overview of sperm head diversity in mammals, with special attention to rodents, a group of species in which a series of novel structures have evolved. Second, we examine the possible role of selective forces on changes in the sperm bauplan and function. Third, we review a series of cellular and molecular mechanisms that remodel and reshape the male germ cell in the stages after meiosis. In this phase (spermiogenesis), the round spermatid, which has a structure not very different from a round somatic cell (except for the fact that it is a haploid cell), changes through different steps into a highly differentiated and compartmentalized cell. From a shape that is somewhat similar among species, it develops into an array of very different shapes and sizes. Our goal therefore is to illustrate evolution of form and function in a highly specialized cell and summarize mechanisms that can explain the onset of biological novelty and diversity in relation to selective forces and processes of cell formation. Finally, we aim to raise awareness of areas that need further research and that could benefit from using a comparative and evolutionary approach to understand diversity in form and function and the underlying molecular mechanisms. 2. SPERM STRUCTURE AND FUNCTION 2.1. Sperm Shape and Size 2.1.1. Overall structure. The general morphology of an organism or a cell constitutes its bauplan. A bauplan (which can also be regarded as a design or blueprint) is the general configuration, the structure and organization of a body plan. However, although the term makes reference to a design, it does not mean that it is entirely determined by a genetic code, because there are a number of factors that can influence its development, both within the organism and in the environment. Bauplan thus refers to the morphology that, in turn, corresponds to the shape (form) and size of an organism or a cell. Size, in turn, relates to both absolute and relative dimensions of its different parts. Finally, ultrastructure and subcellular components relating to the organization and distribution of cell organelles are also relevant. In general, the bauplan of sperm cells is conserved in animals. What follows is a summary of the general structure of sperm cells. Several comprehensive reviews with detailed information on sperm structure can be found in the recent literature (9–12, 22, 23). Spermatozoa have two major compartments, the head and the flagellum (or tail) (FIGURE 1). The sperm head contains the haploid nucleus, with highly compacted chromatin that carries hereditary information in the DNA, and the acrosome, an exocytotic granule with enzymes that upon release help the sperm cells penetrate the oocyte vestments during fertilization. Extranuclear and extra-acrosomal regions can be identified in the sperm head. Furthermore, there are cytoplasmic compartments defined by acrosomal membranes, the nuclear envelope, and plasma membrane. The perinuclear theca is a sheath that encases the nucleus. The perforatorium is a specialized structure found in the apical region of the subacrosomal perinuclear theca that serves to anchor the acrosome to the anterior region of the nucleus. FIGURE 1. Open in a new tab Sperm structure and compartments. General structure of spermatozoa, which consists of a head and a flagellum. The head contains the nucleus and the acrosomal granule. The flagellum, which is connected to the head by the neck, is divided into 3 components: the midpiece with the mitochondria, the principal piece, and the end piece. The axial filament contains the 9 + 2 array of the microtubules. This sperm pattern shows much variation across vertebrate taxa. The flagellum attaches to the sperm head via the neck (or connecting piece), which includes typical or atypical proximal and distal centrioles. The flagellum has three recognizable regions: the midpiece, the principal piece, and the terminal piece. The midpiece has the neck as the anterior limit and the annulus, which is a transverse ring of dense material, as the posterior limit. The midpiece contains mitochondria that generate ATP via oxidative phosphorylation, whereas the principal piece can generate ATP via glycolysis. The energy from the ATP is used by the principal piece to provide the propulsive force through a structure called the axial filament or axoneme. The axoneme extends usually from the neck to the end of the flagellum and has a distinctive 9 + 2 microtubule structure in most spermatozoa. Proteins associated with the axonemal microtubules hydrolyze the ATP and convert the chemical energy released from the ATP into mechanical forces that slide the microtubules and generate the flagellar motility. This propulsion and pattern of flagellar movement is typical for the different species (24–27). Overall, spermatozoa are highly polarized and extremely differentiated cells. Importantly, their repair capacity is very limited, thus being susceptible to various factors that can generate irreversible damage. The sperm bauplan, with many of its peculiarities, thus relates to the need to protect the cell from this potential damage and carry the genetic load to the oocyte. Despite their common structure, spermatozoa from some species depart from this general bauplan. In some species sperm lack a flagellum or may bear two or more. In others, the acrosome may be absent, as in many fish species. And yet, in others, spermatozoa may bear no genome and may act to carry other sperm cells to the site of fertilization. 2.1.2. Diversity. It has been known for a long time that spermatozoa have an extraordinary diversity in morphology, with interspecific differences being greater than those seen for any other cell type (28) (FIGURE 2). This diversity has puzzled scientists for a long time, and the reasons for this variation have been difficult to grasp, to the extreme that it has, at some point, been considered to have no adaptive value (31). This diversity in spermatozoa occurs in agreement with diversity in reproductive patterns, and the possibility exists that changes in reproductive strategies may have influenced sperm structure and function. In vertebrates, for example, taxa vary in the mode of fertilization (which could be external or internal), specialized oocyte structures (32), social structures and mating behaviors, and processes of postcopulatory sexual selection (5), and these characteristics could have influenced sperm evolution. Below, a summary is presented of the main aspects regarding sperm morphology in vertebrates. This is necessarily a general overview because information is lacking for many taxa. In any case, what we already know about sperm morphology in this diverse group of animals underscores the huge diversity in shape and size (FIGURE 2). Information on sperm attributes for the main vertebrate taxa is summarized in TABLE 1. FIGURE 2. Open in a new tab Sperm diversity in vertebrates. Illustrations are drawings from Gustav Retzius (29), and they are presumably all at the same magnification and scale (30). For each species, information regarding the source of each drawing is given in parentheses, and it includes volume number (NF), plate number (Roman numbers), and figure number within each plate. Names of species follow those given by Retzius, and current names are given in parentheses when appropriate. Fishes, Chondrichthyes: (1) Chimaera sp. (NF14 XXVII-1); (2) Squalus acanthias (NF14 XXVIII-16); Osteichthyes: (3) Esox sp., freshwater species (NF12 XIX-9); (4) Salmo sp., fresh and seawater species (NF12 XX-1); (5) Pleuronectes sp., seawater (NF12 XX-16). Amphibians, Anurans: (6) Hyla arborea (NF13 XXVII-1); (7) Bufo vulgaris (= Bufo bufo) (NF13 XXIV-1); Urodeles: (8) Salamandra maculosa (= Salamandra salamandra) (NF13 XX-1). Reptiles, Chelonia: (9) Testudo sp. (NF13 XXVIII-15); Squamata: (10) Platydactylus sp. (= Rhacodactylus sp.) (NF13 XXVIII-6). Birds, nonpasserines: (11) Struthio sp. (NF16 XXV-20); (12) Gallus sp. (NF14 XXXI-1); passerines: (13) Passer passer (NF14 XXXV-1); (14)Sturnus sp. (NF14 XXXVI-17). Mammals, monotremes: (15) Tachyglossus sp. (NF13 XXIX-1); marsupials: (16) Petrogale sp. (NF13 XXXI-9); (17) Didelphys sp. showing paired spermatozoa (NF14 XXXVIII-1); Eutheria, Afrotheria: (18) Elephas sp. (NF16 XXV-1); Xenarthra: (19) Dasypus villosus (= Chaetophractus villosus) (NF13 XXXII-1); Insectivora: (20) Talpa sp. (NF14 XXXIX-1); Chiroptera: (21) Vesperugo pipistrellus (= Pipistrellus pipistrellus) (NF13 XXIX-3); Cetartiodactyla: (22) Equus sp. (NF14 L-1); (23) Dicotyles sp. (NF14 XLIX-1); (24) Cervus elaphus (NF14 LI-1); (25) Globicephalus sp. (= Globicephala sp.) (NF14 LIV-1); Carnivora: (26) Canis familiaris (NF14 LVI-1); (27) Felis catus (NF14 LVI-20); (28) Meles sp. (NF14 LVII-13); (29) Halichaerus sp. (NF14 LVII-28); Primates, prosimians: (30) Lemur sp. (NF17 XVI-1); New World monkeys: (31) Chrysothrix sciurea (= Saimiri sciureus) (NF19 XXI-1); Old World mondeys: (32) Innuus sp. (= Macaca sylvanu s) (NF17 XVI-1); great apes: (33) Gorilla sp. (NF17 XVI-27); (34) Homo sp. (NF17 XVI-35); Rodentia, Histrocoidea: (35) Hystrix sp. (NF14 XL-28); caviomorphs: (36) Cavia sp. (NF14 XLV-1); Sciuromorpha: (37) Sciurus vulgaris (NF14 XLI-1); Myomorpha: (38) Mus musculus (NF14 XLVI-1); (39) Mus norvegicus (= Rattus norvegicus) (NF14 XLVIII-1); (40) Microtus terrestris (= Microtus arvalis) (NF14 XLVI-1); (41) Lemmus sp. (NF14 XLV-15). Table 1. Diversity of sperm traits in vertebrates \| Class \| Sub-/Infraclasses \| Sperm Traits \| :---: \| Fishes \| Chondrichthyes (sharks, skates, rays, chimeras) \| Head is helical and long (>30 µm). Acrosome is conical and moderately elongated. Axoneme with 9 + 2 or 9 + 0 microtubular arrangement. Flagellum contains longitudinal columns and a midpiece with mitochondria. Sperm length: 75–230 µm. \| \| \| Osteichthyes (bony fish) \| Elongated nucleus. Midpiece is short with 2 centrioles, mitochondria, and dense body. 9 + 0 or 9 + 2 axoneme. Flagellum has membrane with 2 lateral extensions (side fins) arranged in line with central axonemal tubules. Teleost sperm have elongated midpiece and no acrosome. Sperm length: 13–272 µm, with 1 outlier (N. forsteri). \| \| Amphibians \| Anurans (frogs and toads) \| Primitive anurans: Nucleus with poorly compacted chromatin; endonuclear canal; perforatorium not well defined; acrosome is conical. Short axial rod surrounded by mitochondria. Flagellum insertion is postnuclear in most primitive taxa. Higher anurans: No perforatorium, endonuclear canal and axial rod. Sperm cell length: 19–240 µm. \| \| \| Urodeles (salamanders) \| Highly compacted chromatin, nuclear ridge, discrete endonuclear canal, perforatorium, clover-shaped acrosome with prominent acrosomal barb and subacrosomal rod. Long and stiff axial rod with abundant encircling mitochondria. Sperm length: ∼170–880 µm. \| \| \| Caecilians (Gymnophiona) \| Sperm resemble those of urodeles. Acrosome exhibits 3 zones and a base plate. Axoneme and axial rod are in proximity in the midpiece. Tail has undulating membrane. Sperm length: ∼60–255 µm. \| \| Reptiles \| Chelonia (turtles, tortoises) \| Compact nucleus with deeply penetrating intranuclear tubes and perforatorial cap. Peculiar distal centriole with central microtubules along entire caudal length, surrounded by 9 peripheral triplets. Laminated mitochondria. Sperm length: ∼50–55 µm. \| \| \| Rhynchocephalia (tuatara) \| Long helical nucleus with a tip constriction and paired endonuclear canals; conical acrosome. Spheroidal mitochondria with concentric cristae and dense lateral body. Annulus; long principal piece with dense fibrous sheath; short end piece. Sperm length: ∼150 µm. \| \| \| Squamata (lizards, snakes) \| Snakes: Prominent neck cylinder separating nucleus from midpiece; dense body lateral to the centriole. Lizards: Single perforatorium from apex of subacrosomal coneElectrolucent region around nucleusFibrous sheath extends into the midpiece. Abundant mitochondria arranged concentrically in large midpiece; irregularly distributed intermitochondrial dense plaques. Sperm length: 80–160 µm. \| \| \| Crocodylia (crocodiles, alligators) \| Inconspicuous neck. Dense body lateral to centriole. Both linear and concentric mitochondrial cristae, different from lizards and snakes. Sperm length: 80–100 µm. \| \| Birds \| Nonpasserines (ratites and some Neognathae) \| Filiform sperm, similar to those of reptiles. Perforatorium, proximal and distal centriole. Short midpiece with prominent annulus. Sperm length: 60–230 µm. \| \| \| Passerines \| Head is twisted spirally, similar to corkscrew; short nucleus. No perforatorium, proximal centriole, and annulus; mitochondria fuse forming single helical structure around axoneme (mitochondrial helix). Dense outer fibrous sheath is prominent and uniform. Sperm length: 40–290 μm. \| \| Mammals \| Protheria (monotremes: platypus, echidna) \| Sperm similar to typical reptiles and some birds. Long, elongated, filiform, helical head, with 3–5 turns; small acrosome. Tail inserts into caudal end of head via small neck. Short midpiece with mitochondria. Axoneme (9 + 2) with 9 small dense fibers surrounding it. Sperm length: ∼110–120 µm. \| \| \| Metatheria (marsupials) \| Head shape reflects nuclear shape, generally. Australian marsupials have heads with bilateral symmetry. Head resembles wedge, tapering rostrally, with prominent midventral gutter accommodating neck and part of anterior midpiece (except Phascolarctidae). Tail inserts in center or ∼2/3 of ventral surface. Bandicoots and vombatoids have hook-shaped head. American marsupials have asymmetric sperm (except for the Chilean monito del monte, Dromiciops gliroides, which has structure similar to many Australian marsupials). Small button-shaped acrosome on anterior-dorsal surface of head in some species; acrosome resembles thin layer over dorsal surface in others. Wombats and koalas have bent connecting piece and distal insertion to head. Evidence of equatorial segment and posterior ring in marsupials; no equivalent to eutherian postacrosomal sheath. Flagellum connects via long articulated connecting piece to indentation or implantation fossa centrally located in head (except in shrew opossums, koala, 3 wombat species). Sperm length: ∼80–350 µm. \| \| \| Eutheria \| Head shape reflects nuclear shape, generally. Head predominantly oval or ellipsoid and symmetric Acrosome generally covers 2/3 of head. Distinct equatorial segment, forming a stable cuff around middle of head, and postacrosomal region. In rodents, some species bear round, symmetrical head, but most have falciform, asymmetric heads. Acrosomes could be very big and extend well beyond the nucleus. Many rodents have apical head appendices (muroid rodents) or basal ones (some caviomorphs).Sperm length: 30–250 µm. \| Open in a new tab Based on information from the literature (23, 30, 33–56). 2.1.2.1. fishes. In sharks, skates, rays, and chimeras (Chondrichthyes), spermatozoa are simple (23). The sperm head has a helical shape. In addition to the central axoneme, the flagellum has longitudinal columns. To generate motion, the central axoneme rotates along the flagellum while the longitudinal columns are fixed at positions number 3 and 8 of the doublet. In bony fish (Osteichthyes), the sperm nucleus is elongated and the flagellum contains a flagellar membrane with two lateral extensions resembling two side fins of uncertain function that are thought to improve swimming capacity (36, 50). In teleost fishes, the main group of Osteichthyes and the largest group of vertebrates, spermatozoa do not have an acrosome. This may be linked to the occurrence of internal fertilization and penetration of the oocyte via a micropyle (23). 2.1.2.2. amphibians. In this group, sperm cells are more filiform and have a perforatorium and a tapering acrosome. The flagellum has a rigid “axial rod” that forms the main axis, displacing the axoneme laterally, and also an undulating plasma membrane traversing the axoneme length, generating propulsion (23). Differences found in spermatozoa of amphibians could relate to differences in the mode of fertilization. Whereas, with a few exceptions, anurans predominantly have external fertilization, the urodeles and caecilians engage in internal fertilization. 2.1.2.3. reptiles. The reptiles include turtles and tortoises (Chelonia), tuatara (Rhynchocephalia), lizards and snakes (Squamata), and crocodiles and alligators (Crocodylia). They all have internal fertilization and experience decoupling between insemination and fertilization, which leads to long sperm storage times. The sperm cell in reptilians is curved and filiform. There are many differences between reptile species, within and among orders, in relation to the presence of a compartmentalized acrosome complex or the structure of the midpiece (57–60). The midpiece has abundant mitochondria, and the principal piece contains a 9 + 2 axoneme, which is surrounded by electron-dense fibrous sheaths that resemble those in higher-order mammals. Sperm cells vary in length among the four orders (23, 45, 60). 2.1.2.4. birds. Birds are peculiar in that they control body temperature (as do mammals) and that their body temperature is high. In birds, testes are located inside the abdomen, and therefore spermatogenesis takes place at high temperatures in comparison to the situation in other vertebrates. In general terms, extragonadal storage is less prevalent in comparison to other taxa such as mammals, and sperm cells are released from the testes during ejaculation. After ejaculation, sperm cells find their way to the female sperm storage tubules and remain there for a period of up to 2 wk before fertilization. Sperm morphology shows diversity among birds. Two major bird groups can be recognized on the basis of sperm morphology: nonpasserines and passerines (songbirds). The nonpasserine birds include the ratites and some members of the Neognathae. The sperm cells in these species are filiform and are similar to those of reptiles. On the other hand, passerine birds have a sperm head with a short nucleus, and the head is twisted spirally, bearing resemblance to a corkscrew. Interestingly, a spirally twisted sperm head, as seen in passerine birds, is also seen in other taxa, such as land snails, but, curiously, they differ in the direction in which the head is twisted. Rotation of the sperm head in birds is clockwise, whereas in land snails it is counterclockwise (61). It is not clear how or why these sperm head forms have evolved and why they are so conserved within taxa. Furthermore, the question arises as to whether the chirality in spermatozoa relates or not to the asymmetry in the body of the animals that produce them (61). 2.1.2.5. mammals. The three infraclasses of mammals (Prototheria, Metatheria, and Eutheria) have sperm structures that are characteristic of each group and specific to each of them (23) (FIGURE 3). FIGURE 3. Open in a new tab Sperm head structure in mammals. A: diagram showing a longitudinal section of equidna (Protheria) sperm head with a very elongated form. Below the acrosome there is a postacrosomal ring. B: generalized structure of the marsupial sperm head. C: generalized structure of the eutherian sperm head. Modified from the literature (23, 33, 54). 2.1.2.5.1. Monotremes. Monotremes (platypus and echidnas) derived from therapsid reptiles and are the earliest branching of the mammalian lineage. Platypus and echidna sperm cells are very similar to typical reptiles (and also some bird spermatozoa) (23, 34, 54). 2.1.2.5.2. Marsupials. In marsupials, the structural organization of the sperm cells is similar to that of eutherian mammals, but spermatozoa have some characteristics that clearly separate them from the latter (54). Different sperm head shapes can be identified in each marsupial family. It is symmetric, resembling a wedge, in most marsupials from Australia. It is asymmetric in most marsupials from the Americas (54). With the exception of shrew opossums (American marsupials), koala, and the three wombat species, the flagellum of all species connects via a long curved or straight articulated connecting piece to an indentation or implantation fossa located centrally in the sperm head. This articulated piece allows the head to rotate on the flagellum, a process that takes place during sperm maturation in the epididymis and that is reversed later, at the time of capacitation in the female tract (23, 54). Marsupial spermatozoa are the longest among mammals, ranging in total sperm length from ∼80 to 350 µm (30). 2.1.2.5.3. Eutherians. The eutherians are the most abundant group of mammals, with >4,000 species. Rodents are the most diverse and consist of >2,200 species. The general structure of the eutherian sperm cell is conserved, and thus it follows the same basic bauplan, although considerable variations in size, shape, and structural specializations have been reported. It has a sperm head with a nucleus and an acrosomal vesicle, a connecting piece, and a flagellum that is divided into midpiece, principal piece, and end piece, but several structural differences distinguish eutherian spermatozoa from those of other vertebrates and also from monotremes and marsupials. The predominant sperm head shape in eutherians is oval or ellipsoid and symmetric. In rodents, although some species bear this head shape, the majority show a departure from this pattern, exhibiting shapes that are generally falciform and asymmetric. Many rodent species exhibit sperm head appendices that further enhance the asymmetric nature of the sperm head. The apical extension of the head in muroid rodents is usually known as the hook (or hooks if there are several). Muroid lineages vary in the shape and size of this appendix. Eutherian spermatozoa vary widely in dimensions. The Chinese hamster sperm is currently regarded as the longest sperm in eutherians, with a total length of ∼250 µm and midpiece and principal piece lengths of ∼100 µm and ∼140 µm, respectively (30). In any case, it is shorter than the longest marsupial sperm and far smaller than the longest animal sperm that is found in the fruit fly Drosophila bifurca, in which spermatozoa could be 5.8 cm long, that is ∼20 times the size of the male that produces them (50). 2.2. Sperm Functions Function refers here to how structures are linked to survival and reproduction of cells and, ultimately, of organisms; thus, they relate to mechanisms and processes. In this context, study of function should try to avoid reduction to physics and chemistry and also avoid teleological explanations, although without discarding teleonomic concepts (18, 62, 63). For spermatozoa, functions include survival and the processes related to the ability to reach the site of fertilization and interact with the oocytes. In general terms, spermatozoa are motile cells that swim actively once released to an external medium, or transferred to the female tract, in order to reach the oocytes. In mammals, sperm cells that leave the testis do not yet have the ability to interact with oocytes. They need to experience a series of maturational changes in the excurrent ducts of the male tract that serve to develop the capacity to express active motility. There is an additional series of changes in the female tract that involve modifications at the molecular and cellular levels and that confer the capacity to interact with the oocytes, traverse the cumulus oophorus, experience acrosomal exocytosis, penetrate the zona pellucida, and fuse with the oolemma. There are great losses of spermatozoa in the female tract (or after spawning in external fertilization), and it has been speculated at length as to why are there so many spermatozoa. In mammals, this may relate to the fact that males cannot predict the timing of ovulation and thus it is to their advantage to produce a large, and diverse, number of sperm cells that can respond to signals or be ready to fertilize at different times. 2.2.1. Sperm transport and preparation for fertilization. Upon release from the male, either at spawning or after transfer to the female tract, spermatozoa begin to move in a process generally known as “activation.” In mammals, spermatozoa are stored in a quiescent (repressed) state in the cauda epididymis after their long transit in the epididymis. Upon contact with fluids that are secreted by the male’s accessory glands, the sperm cells initiate forward progression. There are differences between species because some males ejaculate in the vagina, others in the uterine cervix, and yet others in the uterus (64). This means that there are differences in the fluids they come in contact with and also the barriers they need to negotiate. Furthermore, the timing between mating and ovulation varies widely, and in some species such as bats sperm need to survive in the female tract for a very long time (65). In the uterus, sperm are moved passively by contractions of this organ, but they need to swim actively through the utero-tubal junction and reach the lower oviductal isthmus, that is, the lower portion of the oviduct (66). It has been experimentally shown that, at this stage, sperm usually associate with the epithelial cells of the oviduct and become quiescent (64). The binding is mediated by species‐specific carbohydrate moieties present on the cilia localized at the apical portion of epithelial cells and the surface of the sperm head (64, 65). Only the sperm that establish this association remain viable (67, 68). The shape of the sperm cell seems to be important in the type of association that is established with the oviductal wall, and, moreover, the sperm has to bear an intact acrosome (i.e., has not undergone the exocytotic process); otherwise it will not be able to attach (69). This period of residence in the lower isthmus varies among species and is believed to be associated to the time between the onset of estrus (and mating) and ovulation. During this time, spermatozoa experience a series of additional changes that would render them capable of detaching from the oviductal wall by expressing a different pattern of motion and start swimming toward the site of fertilization in the oviductal ampulla. This new pattern of motility is known as hyperactivation because sperm now has a much more active beating pattern when examined under in vitro conditions. Because the oviductal lumen contains fluids that enhance the viscosity of the medium, the role of hyperactivation, which endows spermatozoa with a stronger propulsive capacity of the flagellum, seems to be to allow cells to move forward under these conditions, whereas those that do not develop hyperactivation cannot advance in this viscous fluid (64, 69–71). This hyperactivated motility is also important later on when sperm penetrate the oocyte vestments (72, 73). The development of this type of motion has been the center of much attention, with many studies dealing with its underlying molecular changes (27, 74, 75). Hyperactivation is triggered by an influx of Ca 2+ through the cation channels CatSper (76). The process may also involve other ion channels and increase in bicarbonate, cAMP, and phosphorylation of specific proteins (74). It is important to note that most of the molecular signaling studies were reported for mouse and human spermatozoa. There is little information for other species, and this is an area of research that deserves further attention. Remarkably, it is also still poorly understood how hyperactivated motility relates to sperm shape in different species. During their residence in the female tract, spermatozoa develop the ability to interact with the oocyte (77–79), a process known as “capacitation,” although it most likely occurs when the sperm are attached to the oviductal isthmus. There is still controversy regarding the use of the term for the entire period of residence in the female tract (19, 21, 80, 81). Several authors reserve the term just for the changes that prepare the sperm for hyperactivation and acrosomal exocytosis, excluding these latter processes from capacitation (82–87). Capacitation is thought to occur only in mammals, and many efforts to understand the underlying molecular mechanisms have been carried out, with several recent reviews representing a good source of information on this process (1, 81, 88–90). It is well accepted that during capacitation there are biophysical modifications in the plasma membrane as a result of increased concentration of and removal of cholesterol by albumin. The initial influx of activates adenylate cyclase (ADCY10), leading to increase in cAMP, PKA activation, and subsequent protein tyrosine phosphorylation. Several regulatory and cross talk mechanisms interact during this process, and they are not yet fully elucidated. It has been shown that PKA is essential for CFTR activity and for sustained increase in . Additionally, the membrane depolarizes and ion transporters are activated to regulate influx and efflux of ions. Ca 2+, K+, Na+, and H+ are the main cations transported, and Cl− and are the main anions (90). It is remarkable that since the discovery of capacitation by Austin and Chang 70 years ago there are still many unknowns around its essential molecular signaling. Although some of the ion channels and transporters have been identified in the sperm, there are still more unknown steps as well as participating players. Moreover, our knowledge about the sequential activation of events is still rather rudimentary. Because capacitation is an essential event for fertilization, further efforts, resources, and cutting-edge technology should be invested to advance in the understanding of this molecular mechanism. Furthermore, it may also be rewarding to ask how spermatozoa are programmed in the testis, or modified in the epididymis, to be able to undergo the process of capacitation. Additional mechanisms have been described to occur before fertilization. For instance, in species with internal fertilization, several studies have reported the participation of sperm-guiding mechanisms that move the spermatozoa toward (rheotaxis, thermotaxis, and chemotaxis) or away from (chemorepulsion) the egg surface (66). The first are important for the gametes to meet, and the latter may function as a way to avoid polyspermy. Unfortunately, the molecular signaling for these mechanisms has not yet been extensively studied, and there are only a few reports dealing with their molecular signaling in mammals (91). Regarding the chemotactic signaling, it has been shown that human spermatozoa respond chemotactically to a picomolar gradient of progesterone after binding of progesterone to its cell surface receptor. This initiates the activation of a transmembrane adenylate cyclase (tmAC)/cAMP/PKA pathway followed by protein tyrosine phosphorylation and Ca 2+ mobilization [through inositol (1,4,5)-trisphosphate receptor (IP 3 R) and store-operated Ca 2+ (SOC) channels]. The soluble guanylate cyclase (sGC)/cGMP/PKG cascade is activated later, and a possible further Ca 2+ influx through another plasma membrane calcium channel occurs (92). The chemotactic signaling was also investigated using bourgeonal, an aromatic compound, as a chemoattractant (93). These authors also reported the participation of the tmAC/cAMP/PKA pathway and subsequent tyrosine phosphorylation of proteins. Additionally, the sGC/cGMP/PKG pathway has been proposed to underlie human sperm chemotaxis in response to nitric oxide donors (94). Despite many recent advances, there are several questions that wait to be answered. For instance, among others, What is the identity of the progesterone receptor? What other ions in addition to Ca 2+ participate in this signaling? What other channels and/or transporters participate in the signaling? What are the proteins phosphorylated, and where do they localize in the sperm cell? What other molecules are involved? What is the sequential activation of the events? And most importantly, is this signaling conserved among species? No major structural changes appear to take place in the sperm during capacitation, On the other hand, the subsequent step of exocytosis of the acrosome involves a major structural reorganization, because this organelle occupies a considerable part of the head (2). Fusion at multiple points between plasma membrane and underlying outer membrane of the acrosome results in massive release of enzymes through the pores and, more importantly, the final detachment of the acrosomal cap. This results in an overall change in the structure of the cell. Importantly, the release of the acrosomal cap in many species, with significant modification in sperm head shape, could have implications for sperm hydrodynamic efficiency in the final steps of penetration of the cumulus oophorus and the zona pellucida. The acrosome reaction signaling has been intensively studied over the last two decades. Excellent articles reviewing the vast bibliography in this area have recently been published (89, 95, 96). The working model for this signaling involves Ca 2+ as the key cation transported, activation of adenylate cyclase, EPAC, RAB family proteins (Rab3, Rab27), PLC, and fusion of membranes through the soluble N-ethylmaleimide-sensitive factor (NSF)-attachment protein receptor (SNARE) complex (97). There are also recent summaries of the sequential activation of the signaling cascade (95). 2.2.2. Bioenergetics. Spermatozoa require energy to maintain homeostasis and cell integrity, signaling, and, above all, to sustain motility, a function that can consume >75% of all energy produced (98, 99). In species with external fertilization, spermatozoa do not draw energy substrates from the medium and rely on endogenous reserves (100). In many species, sperm survive for a short period of time and thus must encounter the oocytes quickly. Interestingly, sperm from some animal species that live in habitats with very low oxygen levels lack mitochondria (101). In species with internal fertilization, spermatozoa can draw from energy substrates present in seminal plasma or in various regions of the female tract (102, 103). Species differ in metabolism and can produce ATP via glycolysis, oxidative phosphorylation (OXPHOS), or both pathways (104–106). Among mammals, bovids appear to use both pathways, cats generate the majority of ATP via OXPHOS, and human spermatozoa employ glycolysis for energy production (104–106). Stallion spermatozoa require ATP generated by mitochondria to maintain membrane integrity, whereas they need both glycolysis and OXPHOS for motility (107). In rodents, there are differences between taxa. In the guinea pig (which belongs to the caviomorph group), spermatozoa produce ATP via both glycolysis and OXPHOS. In the mouse (which is a myomorph rodent), studies in laboratory strains have concluded that ATP production takes place mainly by glycolysis, but more recent studies on wild-derived mice revealed that the situation is more complex. Thus, among closely related mouse species, production of ATP may vary, with energy being generated mainly via glycolysis in some but in others OXPHOS seems to make significant contributions (108). The levels of ATP are crucial for sperm swimming performance. This is underscored by the finding that, across species, there is a significant positive relationship between the amount of ATP in sperm and sperm velocity (109, 110). It is also relevant to consider how sperm consume ATP. Studies in rodents showed that there are differences in how sperm from different species use ATP, independent of its production. The pattern of ATP utilization was found to have an effect on sperm performance, including swimming efficiency (111). Sperm energy requirements may also vary during the life of the sperm cell, but little information exists on this matter. Furthermore, sperm cells may resort to different metabolic pathways in different physiological states (102, 112). Thus, it could be speculated that spermatozoa that have undergone capacitation and show hyperactivated motility could have different energy demands. It is tempting to propose that with a more vigorous movement, such as that observed during hyperactivation, sperm may require more ATP. Whether spermatozoa produce more ATP or use it more efficiently, and what the pathways to generate it are, remain a matter for future research. 2.3. Heterogeneity in Form and Function Despite a general pattern of shape and size that can be recognized for spermatozoa of each species, there is heterogeneity in both traits even for a given male. Thus, there is variation both between and within a species and, in addition, within an individual male, which is reflected in the heterogeneity of sperm released or ejaculated. This heterogeneity in shape and size will impact on sperm function. Variation at these different levels may result from a number of factors and may have various adaptive advantages. There may be differences during the process of sperm production, in posttesticular maturation and storage, and also in how (and when) sperm are released or transferred to the female tract. For example, spermatogonial stem cells form syncytia, and in rodents 1,000 or more spermatogonia can be interconnected and undergo meiotic divisions synchronously, resulting in ∼4,000 connected spermatids undergoing differentiation; the magnitude of mitotic amplification varies between species, being 128- or 16-long syncytia in monkeys and humans (13). There is heterogeneity in the transcriptional states (gene expression) of spermatogonia that relates, in part, to the length of the syncytia (113). Studies of behavior of individual cells have revealed that individual stem cells have variable fates with regard to the number of self-renewing and differentiating cells in a cohort derived from a single spermatogonial stem cell (114). Fate variance among clones would occur over time (clonal drifts) and may result in the eventual domination by a single clone, thus resulting in “population asymmetry” (113, 114). It follows that heterogeneity of an ejaculate would have to take into account that it could be the product of competing germ cell clones. Another example relates to epigenetic heterogeneity. Epigenetic modifications influence the function of male germ cells, and alterations in levels of DNA methylation are known to be associated to sperm abnormalities, decreases in fertilization rates, and deficient embryo development (115). In human sperm, evidence for heterogeneity in DNA methylation of one maternally methylated gene (KCNQ1OT1) has been found in samples from men with abnormal sperm parameters (116). Sperm from normozoospermic samples had a homogeneous pattern of DNA methylation, but in samples with low sperm motility (oligoasthenozoospermic men) there are discrete groups of spermatozoa with either normal or abnormal patterns of methylation (116). On the basis of these findings, and additional information, possible scenarios in which spermatogonial stem cell variability can lead to sperm epigenetic heterogeneity have been presented (115). The relative importance of internal factors or those from the environment, and their impact on fertility, for this heterogeneity deserves additional attention. Heterogeneity may also arise as result of postmeiotic gene expression, and this may lead to sperm within a single ejaculate being subject to selection. The idea that postmeiotic expression of genes may result in unequal sperm has generated controversy (117–121) because gene transcripts and proteins may be fully shared among cells during sperm formation because of cytoplasmic bridges among haploid spermatids (122–126). There is, however, evidence for unequal sharing of some gene products in the mouse (127–129) and a link between sperm genotype and sperm phenotype in zebrafish and human (130, 131). The extent of specific haploid transcription is unclear (129, 132), and it may even take place in a mitochondrial-type ribosome (133). Because haploid gene expression may bear on the sperm phenotype (e.g., form or function), this may be the basis of haploid selection in a single ejaculate, as revealed by differences in zebrafish short- versus long-lived sperm that were related to differential offspring fitness (126, 130) This is to be distinguished from selection by competition between ejaculates from different males (see below). A special case of sperm diversity based on haploid gene expression relates to possible differences in phenotype due to expression of X or Y chromosome-borne genes. In mammals, there is heterogeneity with regard to sperm bearing either an X or a Y chromosome, and it is thought that about half of sperm in an ejaculate bear each type of chromosome. As shown in the mouse, sex chromosomes have a very high level of copy number amplification of postmeiotically expressed genes (134, 135). This situation is different from that in birds, in which all the sperm carry a Z chromosome, since males are the homogametic sex in these taxa. Biases in sex ratios exist in mammalian offspring, and it is thought that either females may chose one type of sperm over another or males are capable of producing, or releasing, more sperm carrying one or the other type of chromosome (136, 137). Evidence is still controversial as to the possible underlying mechanisms mediating biases in sex ratios. One example has been identified in mouse sperm in which Yq-deleted mouse males (MF1-XYRIIIqdel) produce equal numbers of X- and Y-bearing sperm, but such sperm show variant morphologies and are functionally different from each other, exhibiting motility differences that would explain offspring sex ratio due to differential transport in the female tract (138). Thus, postmeiotic differential gene expression between X- and Y-bearing sperm may influence sperm head size or shape, swimming performance, surface proteins, other aspects of sperm function, or a combination of these traits (e.g., relations between head shape and hydrodynamic efficiency) (138–141), making them more susceptible to external factors or of being differentially selected in the female tract. Heterogeneity in spermatozoa has been categorized and quantified both within and between males in order to understand sperm subpopulation structure and possible relationships with fertilizing ability (142, 143). Differences in subpopulation structure between males, based on sperm size and shape, have been identified for several species of domestic animals (144–147). Such distinctions are relevant for male fertility because a clear relationship between the proportion of sperm forms in a given subpopulation and fertility of the males has been found (142). Similarly, subpopulation structure based on sperm motility (kinematic parameters) has been characterized in domestic species (148–151) and relationships between proportion of sperm in a population and male fertility uncovered (143). Very few studies exist in which shape and motility parameters have been combined to characterize sperm subpopulations (152), and to the best of our knowledge there are no reports yet on links between a combined analysis of shape and motility subpopulations and sperm fertility. As a cautionary note, it is perhaps important to bear in mind that diversity in sperm morphology and kinetic traits (or lack of them) may be related to methodologies employed and there is a risk of generating artifacts during analyses (reviewed in Refs. 19, 153–155). In any case, the question that arises is why males produce not only vast numbers of spermatozoa but also these heterogeneous populations if only a few sperm cells would be successful at fertilization. 3. EVOLUTION OF MAMMALIAN SPERMATOZOA Diversity in spermatozoa has generated considerable interest. Many efforts have been directed toward understanding patterns of evolution of sperm cell morphology and function. Significant interest has also been placed in trying to understand the selective pressures underlying such diversity. This section therefore deals first with the patterns of evolution of spermatozoa and second with evolutionary forces that may explain sperm diversity. 3.1. Evolutionary Patterns in Spermatozoa From a biological point of view, males are defined as the sex that produces the smaller gametes (i.e., spermatozoa), as opposed to females, which are those that produce the larger gametes (i.e., oocytes) (156, 157). It is thought that, ancestrally, only small monomorphic gametes were produced (158). Therefore, an important question is what pressures have caused gamete dimorphism (anisogamy) to evolve in such a crucial transition (159). Divergence into two sexes seems to be an (almost) inevitable consequence of sexual reproduction, and it seems to be driven by gamete competition (160). The subject of early divergence of two gametes of different sizes and morphologies is beyond the scope of this overview. The reader is referred to recent reviews that cover several aspects, including modeling, of the evolution of anisogamy (160–163) and how it integrates in the so-called sexual cascade (164). Early studies focusing on evolution of sperm attempted to draw general patterns for the entire animal kingdom or concentrated on species with external fertilization (165–169). Species with external fertilization, which release gametes into the water, are believed to have sperm cells that are generally simple in their morphology, with a round head and a flagellum of ∼50 µm in length with a 9 + 2 axoneme (Refs. 170, 171, but see Ref. 172). On the other hand, in species with internal fertilization, spermatozoa are more complex and have longer flagella (170). Additional changes related to internal fertilization may be a reorganization of mitochondria and, in mammals, the appearance of accessory fibers in the flagellum that probably help when swimming in the more viscous fluids of the female tract (173–175). Studies of evolutionary patterns of mammalian sperm have focused on two main aspects, one addressing changes in sperm size in relation to body size (that is, possible allometry), together with scaling of the different sperm components, and the other trying to understand patterns of sperm head change in different animal lineages. In addition, the significance of changes in sperm length or variation of sperm head shape in relation to sperm swimming ability has also received consideration. These issues are considered next. In mammals, sperm size varies widely. Although relatively small differences exist in sperm head length, differences in flagellum length are considerable between species. Overall, marsupial sperm are much longer (range: 80–350 µm) than eutherian sperm (30–250 µm) (30, 176). Among eutherians, rodents are the species with the widest range (35–250 µm) (30, 176). Proportions of the different sperm components also vary; for example, comparison between marsupials and rodents shows differences in the sperm head (6% vs. 7.5% of total length, respectively), midpiece (11% vs. 23%), and principal piece (83% vs. 70%) (30). Although early studies in mammals proposed that sperm size (and size of its components) was inversely related to body size (30), subsequent studies revealed that the relationship is built on extreme values, with large mammals having small sperm and small ones having, in general, longer spermatozoa, and that there is no significant relationship between body size and sperm length (177). In fact, sperm size is rather uniform across many mammalian orders, and it is only in some groups (marsupials, rodents) in which a wide variation in sizes is observed (30, 178). With regard to different sperm components, it seems that there is a positive association between lengths of various sperm parts, with a proportional increase of different components (177). Sperm midpiece and principal piece lengths are positively associated with both head length and area, and principal piece length is positively associated with midpiece length (177). Interestingly, sperm head dimensions are not related to either genome mass or chromosome number (177). Diversity in sperm head shape has been examined in rodents in order to understand the direction of evolution, using a combination of phylogenetic trees based on genetic, chromosome, morphological, and biogeographical data together with sperm morphological information. These analyses revealed that in several rodent lineages, including caviomorphs (the group that includes guinea pigs, among others) and myomorphs (the group with mice, voles, hamsters, South and North American cricetids, and African nesomyids), there are spermatozoa with a simple, round-oval head shape and short flagella. Each lineage also shows species with a peculiar sperm head that becomes more complex (and in some cases includes more apical appendices) as sperm becomes longer. This pattern led to the suggestion (176) that, in rodents, ancestors for each lineage had a simple sperm head, with a short flagellum, and that an increase in complexity and length has occurred in different lineages, with each lineage developing different head morphs. Thus, increases in sperm complexity involved the repeated appearance of an asymmetric, falciform head, with the development of hooks or the displacement of the site of flagellum insertion in the base of the tail. Such a view envisioned that the diversity in sperm forms was related to adaptive advantages for spermatozoa (176). An alternative interpretation of rodent sperm evolutionary patterns considered that the ancestral sperm morphology for myomorph rodents was an asymmetric, falciform one, with a hook in the apical end of the sperm head, from which more complex heads had evolved. In this view, the round-oval, paddlelike sperm shapes seen in myomorph rodents were interpreted as being derived morphs, rather than the ancestral ones (179, 180). The controversy was resolved when the range of myomorph rodents examined was expanded (181). Spalacid species, a family of the superfamily Muroidea from eastern Asia, the Horn of Africa, the Middle East, and south-east Europe, were found to have simple, symmetric, ellipsoid sperm heads without a hook. They, together with the Dipodidae, seem to represent the oldest split in the muroid superfamily, and thus the common ancestor with the rest of Muroidea probably had this type of simple, symmetric sperm head (181). The reorganization of the sperm head, with the development of asymmetry and an apical hook, probably arose in a common ancestor of mice, cricetids, and nesomyines (FIGURE 4). Furthermore, there seems to be coevolution of head shape and sperm size, because the more complex or elongated rodent sperm heads are seen in species with the longest spermatozoa, and this is true for different rodent lineages (176). In any case, there remains the issue of the repeated reversal of the complex sperm head trait to a simpler head pattern with regard to both mechanisms and evolutionary pressure (or lack thereof). FIGURE 4. Open in a new tab Evolution of sperm head shape in muroid rodents. A model for the evolution of sperm head shape in muroid rodents. Spermatozoa in different lineages show specific patterns, with an increase in complexity, which is accompanied by an increase in length (not shown). A simple, symmetric, round or oval sperm head was probably present at the base of the muroid evolution at the time when divergence of these lineages occurred 25–30 million yr ago. An elongated sperm head with an apical hook appears to have evolved in a common ancestor of the Nesomyidae, Cricetidae, and Muridae, with this sperm form being common throughout the nesomyid, cricetid, and murid rodents. Based on results in Roldan et al. (176) and Breed et al. (181). The evolution of longer sperm could be adaptive, because longer sperm may swim faster (182). Empirical evidence for this idea was first presented in a comparative, interspecific study of mammals (182, 183) and then expanded to other taxa (reviewed in Ref. 184). In rodents, sperm swimming velocity is associated with the size of all sperm components (185, 186). In addition, sperm head shape also impacts on swimming velocity. In a study across mammals, sperm with more elongated heads were found to have higher straightline velocity, i.e., they swim faster (183), whereas in rodents changes in the sperm head, including the appearance of the hook, are associated with increased overall sperm velocity (186, 187). Changes in sperm morphology (following the appearance of a hook in the sperm head) may be linked to the development of associations between spermatozoa (that is, the formation of the so-called “sperm trains”) and display of faster swimming velocity. The phenomenon of association of sperm cells and swimming in trains, with accompanying faster sperm, was described in rodent species Apodemus sylvaticus and the genus Peromyscus, and detailed characterization of such swimming behavior has been presented (188–190). The associations observed in rodent sperm resemble somehow those observed in sperm from American, but not Australian, marsupials (191, 192), although in these species pairing of sperm occurs in the epididymis (193), in contrast to the associations in rodent spermatozoa, which seem to occur after ejaculation. Marsupial sperm also swim in pairs when migrating in the reproductive tract. It has been postulated that this behavior may serve to protect the acrosomes (because sperm heads are joined together over their flat acrosomal surface), but most likely it results in paired spermatozoa swimming faster in the viscous environment of the female tract (194). This phenomenon should also be distinguished from sperm associations that occur in the epididymis, during sperm maturation, and that result in ejaculation of sperm groups (e.g., guinea pig rouleaux) (195). Because forward-moving trains result mainly when sperm cells associate by the heads, it has been speculated that head hooks could have an important role in this phenomenon and, furthermore, that they have evolved in connection to this swimming pattern (Ref. 196, see Ref. 197 for additional references). However, an analysis of several rodent species without or with a head hook has not shown a consistent pattern of train formation and presence of an apical appendix (197). Therefore, it seems that train formation is an exceptional strategy for faster sperm swimming that appears to have evolved in parallel in a limited set of species. A thorough study is needed to characterize sperm structures involved in these sperm associations and to understand the swimming patterns of these associations as well as the kinetics of train formation and sperm release from such trains in relation to different stages of sperm transport in the female tract. At the intraspecific level, analyses of red deer spermatozoa showed that cells with elongated heads swam faster, and a similar association was found for the proportion of the principal piece in relation to the length of the flagellum; in contrast, a negative relation was observed between midpiece length and swimming velocity (198). In mice (199), an intraspecific comparison revealed a relationship between sperm form and sperm function, and in this case midpiece length was found to predict swimming velocity in lines bred under different selective pressures. Explanations for this apparent discrepancy invoked differences in sperm head shape between these species, but, since swimming is also influenced by bioenergetics, it is possible that sperm from deer and mouse are obtaining ATP via different pathways, which may influence sperm performance together with cell morphology (109). In addition, differences in sperm head shapes between these species may have different impacts on drag (200). In birds, it was predicted that increased sperm velocity may be associated with an enlarged midpiece (because of its energetic role) or overall flagellum length (because of the propulsive force it generates) or the proportion between sperm components. A comparative analysis in passerine birds revealed that sperm velocity significantly associated with sperm dimensions in the predicted direction (201). Similar results were obtained in intraspecific studies in zebra finches, in which sperm velocity was positively correlated with sperm length; because both were found to be heritable, selection for faster sperm will simultaneously lead to the evolution of longer sperm, and vice versa (202). The effect of length on sperm velocity is also suggested by the observation that spermatozoa with longer flagellum lengths preferentially reach the site of fertilization in zebra finches (203). In this species, a trade-off between length and thickness of the sperm midpiece was reported, with longer midpieces being proportionally thinner with, interestingly, similar mitochondrial material, which can have implications for energy production (204). In intraspecific comparisons in the house sparrow, sperm velocity was found to be correlated with head-to-flagellum length ratio: sperm with small heads relative to their flagellum showed higher swimming velocity, which is consistent with the idea that less drag (because of a smaller head) and more propulsion (because of a longer flagellum) make for more efficient swimming (205). The shape of the sperm head was revealed to be important for passerine birds. Spermatozoa with heads that have a more pronounced helical form (with long acrosome, short nucleus, wide helical membrane, and a more pronounced waveform along the sperm head) exhibited faster swimming (206). 3.2. Evolutionary Forces Underlying Diversity in Sperm Bauplan and Function Once the patterns of evolution of sperm cells are recognized, including direction of changes and some reversals of complex traits, it is possible to ask what selective forces may underlie them. Importantly, inquiring about selective pressures is one of several types of questions that can be asked to understand diversity in sperm cells. As recognized by Mayr (62) and expanded by Tinbergen (207), causality in biological systems could distinguish ultimate or proximate questions. Whereas evolutionary explanations address ultimate causation, that is, distant in time, proximate causation addresses immediate factors, close in time, such as physiological mechanisms. This distinction is important and allows us to ask “why” and “how” questions separately, with different toolkits, in an effort to avoid confusing levels of analyses. Among the evolutionary explanations, attention is given to either adaptive value or phylogeny (history) of traits. Included in the proximate explanations are either the mechanisms or the development (changes during one or more stages in life of individuals) of traits (208–211). Integration or complementation of levels of explanation is certainly desirable because separation of such levels may limit advances in our understanding of biological systems (212, 213). Thus, complementation of questions regarding evolution and development has given rise to the field of evolutionary developmental biology, which allows us, for example, to focus on issues of developmental constraints or reciprocal causation (208, 209). These aspects, and an initial attempt to relate them to sperm biology, have been discussed elsewhere (18). Whereas sect. 4 considers mechanisms of sperm formation to identify processes that may help us understand the origin of biological novelty and diversity in morphology and function, the rest of this section deals with evolutionary questions. Three main selective pressures are considered to understand evolution of morphology and function of spermatozoa and, in turn, processes of sperm formation in the testis. One relates to mode of fertilization (or fertilization environment), another to postcopulatory sexual selection, and yet another to common descent or phylogeny (214). 3.2.1. Mode of fertilization. This factor has probably exerted an important influence on the evolution of gametes. Vertebrate species differ in their mode of fertilization and, hence, in the way their gametes meet and interact. Internal fertilization takes place in all elasmobranchs (sharks, rays, skates, sawfish), some oisteichthyans (bony fishes), all gymnophione amphibians (caecilians), one species of anuran amphibian, most urodele amphibians (salamanders), and all amniotes (reptiles, birds, mammals) (32). In externally fertilizing species, the structure of the sperm cell is thought to be rather simple (170). Because water is a uniform environment, spermatozoa have to swim to reach the oocyte, and such simple structure is adaptive under these conditions. Internal fertilization is believed to have led to a series of modifications generating a longer and more complex sperm cell (48, 171). A recent comparative analysis of >4,000 vertebrate species has indeed found robust evidence that longer spermatozoa have evolved in species with internal fertilization (215, 216). In internal fertilizers, sperm cells have to swim from the site of deposition, usually a cloaca, urogenital sinus, or vagina, although in some species they are placed directly into the uterus or part of the oviduct (69). The distance to swim may vary from a few millimeters in small species to a meter or more in some large species such as elephants or whales. Although contractions of the female tract are important in some species, forward progressive sperm motility is a key requirement. Sperm traits have been found to relate to size of the female reproductive organs or time spent in the female tract (217, 218). Much of the analysis on mode of fertilization has focused on the impact it has on the interaction between sperm and oocyte (73). In mammals, the extracellular oocyte coats (or vestments) have increased in size. The cellular coat (the cumulus oophorus), which is the first vestment that the sperm has to penetrate, is a multilayer of cells that surrounds the oocyte and has an abundant matrix that would slow sperm in its transit. The acellular coat (the zona pellucida) has become thicker in eutherian mammals, and thus the interaction of the sperm cells with this structure has resulted in several changes in the spermatozoon. Sperm need to generate more force to penetrate the zona, and to this end the sperm head has flattened and, in addition, an apical structure with a more resistant structure has appeared: a perforatorium with a scythelike shape is present along the rostrum of the head, adjacent to the nucleus. Both the flattened head and the perforatorium may serve to cut a path through the zona pellucida when the spermatozoon tries to gain access to the oocyte (73). Because exocytosis of the sperm acrosome exposes the inner membrane of the granule, the head has also experienced modifications in the region of the sperm that will fuse with the oocyte’s plasma membrane. In addition, because penetration now requires stronger propulsive forces, a change in the pattern of movement that takes place during sperm capacitation (i.e., hyperactivation) contributes to more vigorous movements that help in the path to zona penetration and also contributes to the change in form and reorganization of subcellular structures. In species with internal fertilization there may be additional influences due to differences in their reproductive mode, that is, whether species lay eggs (oviparity) or bear young eventually leading to live births (viviparity). In a comparative study in snakes, oviparous species were found to have longer spermatozoa than viviparous species (56), a result that was confirmed in a larger sample across vertebrate species (215). The reasons for this effect are not entirely clear. It could be that, since cost of reproduction may differ between oviparous and viviparous species, postcopulatory selection may also vary. It is also possible that differences are related to considerable variation in mechanisms of sperm storage and transport between these taxa and, moreover, in the overall morphology of the female reproductive tracts and temporal differences in ovulation and fertilization. Such differences can impact on sperm physiology and, in turn, on sperm bauplan. In oviparous species, such as birds, eggs are released and fertilized in sequence, whereas in viviparous species eggs are released and fertilized simultaneously. This would most likely generate completely different scenarios for males and their sperm (219). 3.2.2. Postcopulatory sexual selection. A set of selective pressures, collectively known as postcopulatory sexual selection, has been invoked to understand variations in sperm shape, size, and performance. These pressures have been named in correspondence to the sexual selection forces that take place before mating (220) and that are thought to explain certain adaptations that are not linked to survival but to reproductive success (164, 221). Sexual selection involves modes of selection in which members of one sex choose members of the other sex for matings (intersexual selection) or members of one sex compete among themselves to gain access to the opposite sex (intrasexual selection) (21, 222). It has been found that these pressures lead to the development of certain sexual characters that represent advantages in competition, such as body size, the size or shape of antlers or horns, or coloration. When females choose, they can do it on the basis of their preference for a given male trait. When these selective forces operate after mating, they are known as postcopulatory sexual selection (21). Selection by females manifests as a process of cryptic female choice in which they may select one type of sperm cell over another. The competition between males to gain access to females takes the form of competition between spermatozoa from rival males (sperm competition), with the aim of achieving fertilization of oocytes. Altogether, these phenomena may be interpreted as either females choosing particular males or sperm with particular attributes or relatively passive females accepting the winner of fights among males or their spermatozoa. However, on a more general level, and taking into account that in some species females are more brightly colored or aggressive than males, it matters, for a more general formulation of the principle of sexual selection, which is the sex with the higher or lower potential reproductive rate instead of male or female (222). Cryptic female choice arises from female-driven mechanisms during or following mating that bias sperm use and have an impact on the share of male paternity (73, 223). Several aspects of female biology (behavior, morphology, physiology) exhibit potential for cryptic female choice during mating, sperm transit or storage, through to sperm-oocyte interaction, biasing fertilization toward the sperm of specific male or males (224). Nevertheless, until now it has not been clearly demonstrated, probably because it requires distinction of components of male and female variance that contribute to sperm retention and paternity; thus, much remains as theoretical speculation (224). With regard to its adaptive value, under certain conditions, cryptic female choice could have implications for female fitness or sexual conflict (225, 226). To gain evidence for cryptic female choice it may be necessary, for instance, to identify a female trait that influences sperm retention or utilization after mating and that the female response is nonrandom in a way that results in sperm of certain males being differentially favored based on genotype or phenotype (224). There are examples of sperm ejection by contractions of the female tract, which thus limit transit from lower sections of the female tract (227). Females may also influence sperm storage based on the degree of relatedness to males (228) or affect sperm swimming performance via reproductive fluids, including processes of chemoattraction (225, 229). Finally, there is also the possibility of choice at the time of oocyte-sperm interaction either in relation to inbreeding avoidance (230) or promoting a given major histocompatibility complex haplotype (231). In mouse species, adaptation to the risk of polyspermy may influence oocyte defensiveness, which may serve to filter and select spermatozoa on the basis of compatibility or quality (232). Sperm competition happens when females are promiscuous and mate with two or more males during their period of sexual receptivity (16, 233). In the case of external fertilization, sperm competition takes place when spermatozoa from several males are released more or less simultaneously and they attempt to gain fertilization of oocytes that are released concomitantly (234). Although many studies have focused on the evolutionary framework of sperm competition (5–7, 48, 235), there is still little knowledge regarding the underlying mechanisms that influence sperm competitive ability (221, 236). In recent decades there has been much interest and effort in trying to elucidate the impact of sperm competition on evolution of spermatozoa, both with regard to their numbers, shape, and size and also in relation to function. The conclusions obtained from much work suggest that an increase in levels of sperm competition results in an increase in relative number of spermatozoa, that is, the number of spermatozoa corrected by body mass (because larger males will produce more sperm to compensate for larger female tracts); sperm numbers evolve together with increases in relative testis mass (222). There are also changes in the shape of the sperm head in response to sperm competition, because this selective force seems to promote elongation of the sperm head in mammals (183). The adaptive significance of modifications in shape relates to the realization that sperm with more elongated heads may swim more efficiently (183). Other modifications in the sperm head are difficult to associate with sperm swimming, among other reasons because of the difficulties of quantifying shape (19, 21, 237). The evolution of appendices such as hooks modifies head shape and renders them more asymmetric, so inference of shape from head dimensions, as done in studies with simple, symmetric sperm heads, is not reliable. Analyses of the curvature of hook insertion in the main part of the head, or the length of the hook, have revealed that both are positively associated to levels of sperm competition in rodents (187, 196). Changes in hook angle or length probably influence the hydrodynamic efficiency of sperm head shape. The association between sperm competition and sperm size (or proportions between the lengths of sperm components) has been observed in many taxa in both inter- and intraspecific comparative analyses (21, 182, 184). There are some exceptions to this trend, which can perhaps be explained by other selective forces (fertilization mode or other reproductive patterns). In any case, a vast majority of comparative analyses, in many animal taxa, have revealed a positive association between sperm competition and length of spermatozoa (238). The influence of sperm competition on sperm numbers (see above) has led to the proposal that in order to produce more sperm males would reduce sperm size (239). However, it has been shown that this may not be the case, at least in mammals, because higher levels of sperm competition also result in longer spermatozoa and both numbers and dimensions may coevolve positively (240). Sperm competition influence on sperm size reflects positively on sperm velocity (184, 201). There is also a direct association between sperm competition and sperm swimming velocity (186, 201). It is possible that sperm competition influences sperm velocity via processes other than (in addition to) the effect of sperm length. For instance, changes in sperm energy production and consumption may also impact on sperm performance. Although relatively little is known about how sperm from many species produce their ATP, some studies have used midpiece length or volume (where mitochondria reside) to examine whether sperm competition influences sperm metabolism (241–243). Comparative analyses of actual ATP levels have revealed differences in the amount of sperm ATP in rodents from several lineages (mice, voles, hamsters) and a positive relationship with sperm competition levels (110). Considerable differences were also found in a comparison across mammalian species, and, again, a positive association with sperm competition was uncovered (109). Modulation of energy production or consumption could occur before appreciable changes in sperm bauplan, but it remains to be established whether changes in performance (function), for example fueled by enhanced metabolism, precede changes in morphology (form), as theory would predict, because the evidence is still controversial (206, 244, 245). In addition, swimming performance is a multiparametric function in which components such as velocity and trajectory can be distinguished. Different patterns of motility, such as those seen after activation or hyperactivation, should also be distinguished in mammals. Therefore, a more careful dissection and examination of sperm kinematic parameters under appropriate conditions and characterization of sperm subpopulations, under the prism of postcopulatory selection, would be most helpful in the future. Swimming patterns are relevant because they are important determinants of male fertility, as revealed by studies in several species (246, 247). Other sperm functions are also under selection by sperm competition. The processes that mammalian spermatozoa undergo in preparation for fertilization (capacitation and acrosomal exocytosis) are influenced by sperm competition: A comparative study using several mouse species revealed a positive association between the level of sperm competition and the proportion of cells that undergo changes compatible with capacitation or respond to a physiological agonist of acrosomal exocytosis (248), but little is so far known about the underlying molecular modifications in physiological processes that could take place in response to sperm competition. One important point to bear in mind is that these physiological processes are still poorly known and debate still exists as to how they should be framed in relation to the life of the sperm cell. For example, controversy still exists as to what constitutes capacitation of mammalian sperm and, therefore, its definition (19). In the original definition (78), the set of modifications in the female tract that prepare the sperm to engage in fertilization have been regarded as capacitation (80, 81). This definition has subsequently been challenged and perhaps needs to be refined to include only events before hyperactivation or acrosomal exocytosis (19). In addition, it has been suggested (48) that this process may not be different from other physiological interactions between sperm and female-derived factors that modify sperm behavior in other taxa, regardless of whether they engage in external or internal fertilization. Following this argument, it has been proposed that the current definition of capacitation should be entirely abandoned (48). Thus, whether the process is unique to mammals, because it includes some distinctive conditions or processes, or could be assimilated to a general phenomenon of sperm × female interactions should be examined further, using a variety of criteria. 3.2.3. Common descent (phylogeny). Another factor that may influence sperm evolution is phylogeny. Thus, sperm morphology may be similar among species because of common descent, that is, species that descend from a common ancestor, with similarities in shape and size due to this fact rather than to responses to selective forces (178, 183, 249). This is usually examined in comparative studies carried out in a phylogenetic framework to account for possible phylogenetic inertia. If such phylogenetic signals cannot be eliminated in analyses, then it is argued that it is not feasible to attribute evolutionary changes to a particular selective force. In any case, phylogeny represents more than just common descent, with the existence of developmental constraints on possible changes, particularly in form (250–252). This is, so far, a rather neglected area in the study of sperm evolution or mechanisms of sperm formation and differentiation, and perhaps more attention should be paid to this issue in the future. BOX 1 Spermatogonia: Spermatogonia are located near the lamina propria in the seminiferous tubules and in contact with the Sertoli cells. In primates (e.g., monkey and man), there are three morphologically distinct subpopulations of spermatogonia: Ad spermatogonia, Ap spermatogonia, and B spermatogonia. In humans, Ad and Ap spermatogonia represent the undifferentiated stem cells and type B the differentiating spermatogonia. Ad spermatogonia do not divide under normal conditions of testicular function. However, Ap spermatogonia divide by mitosis to maintain their own cellular stock and, for each of them, generate two B spermatogonia. In humans, only one generation of B spermatogonia has been characterized, whereas there are four in the macaque. There are six differentiated spermatogonia in the mouse (A1, A2, A3, A4, In, and B). Spermatocytes: Spermatocytes are the cells originating from B spermatogonia. They divide by meiosis. The process of meiosis involves two cell divisions, the first involves primary spermatocytes, and the second meiotic division, producing the haploid spermatids, involves secondary spermatocytes. The first meiotic division takes several days (24 days in humans); on the other hand, the second meiotic division progresses rapidly (lasting ∼5 h in humans). Primary spermatocytes are joined to each other by intercellular bridges similar to those found between spermatogonia. They are separated from adjacent Sertoli cells by distinct intercellular spaces that are modified in some regions by desmosome-like structures. The secondary spermatocytes have the haploid number of chromosomes, although their DNA content is still diploid so that when they complete meiosis the resulting spermatids have both a haploid chromosomal content and DNA content. Spermatids: Spermatids are the products of the second meiotic division and lack the ability to divide. Their progressive transformation takes 23 days in humans, 22 days in rats, and 14 days in mice. It involves a morphological and functional differentiation process including formation of the acrosome, chromatin condensation, and the elimination of excess cytoplasm. There are two types of cells, round and elongating spermatids. 4. SPERM FORMATION (SPERMATOGENESIS) 4.1. Phases of Spermatogenesis The formation of the male gamete is a highly organized process that takes place in the seminiferous epithelium. This process, known as spermatogenesis, is the sequence by which stem cells of the germ line develop into spermatozoa (BOX 1). It is usually divided into three well-characterized phases: 1) The proliferative phase is the first phase and involves proliferation of diploid spermatogonia by mitosis. During this phase, two types of spermatogonia are produced (undifferentiated and differentiated spermatogonia). Undifferentiated spermatogonia are normally present in all stages of the seminiferous epithelium. They are responsible for the renewal of their own stock of cells and for the production of differentiated spermatogonia. Differentiated spermatogonia are cells committed to sperm production. Studies on stem cell renewal and differentiation have dramatically increased in recent years, with new advances in spermatogonia-based approaches to preserve male fertility or restore fertility in infertility cases (253–256). 2) The meiotic phase follows the proliferative phase and involves two meiotic divisions where primary spermatocytes reduce the number of chromosomes, leading to haploid cells and genomic recombination (257). 3) The last phase (spermiogenesis) is essential for sperm differentiation (258). During spermiogenesis, the haploid round spermatids resulting from meiosis undergo substantial structural and functional changes. During this phase there are remarkable events including the formation of new organelles such as the chromatoid body composed of RNA, the acrosome originating by fusion of vesicles, and the manchette working as a scaffold for protein trafficking and nuclear remodeling. Trafficking of proteins via acrosome-acroplaxome-manchette, nuclear condensation and remodeling, and acquisition of the species-specific shape, elimination of residual cytoplasm, assembly of the flagella, and spermiation are also hallmark events of the spermiogenic phase (FIGURE 5). FIGURE 5. Open in a new tab Schematic representation of the mouse spermatogenic process. Spermatogenesis advances from the base to the lumen of the seminiferous tubule in a dynamic contact with Sertoli cells. Spermatogonia reside in the basal compartment and proliferate to primary spermatocytes (preleptotene, leptotene, zygotene, and pachytene). Secondary spermatocytes undergo second meiosis and became round spermatids. The sperm differentiation phase starts at this point with morphological and molecular transformations of round spermatids into elongating spermatids. Elongated spermatids are cells in the process of spermiation undergoing the final cellular transformations. In the last steps of spermiation, most of the cytoplasm will be removed and phagocytized by Sertoli cells to form fully developed spermatozoa. At the beginning of spermiogenesis in mammals, round spermatids have a spherical nucleus located in the center of the cell and the chromatin is decondensed (steps 1–7). Other organelles such as the acrosome and a primitive tail are developing and experiencing transformation during these steps. The acrosome is assembled from trans-Golgi and extra-Golgi vesicle pathways, and then it spreads, covering the apical part of the nucleus. Simultaneous with acrosome assembly, flagellum formation starts with docking of the centrosome to the nuclear membrane at the opposite side of the nucleus. The axoneme then extends from the proximal centriole. As the cell differentiation process advances, the nucleus moves toward one pole of the cell (step 8 in rat and mouse). Then, the cells start to mold an asymmetric nucleus supported by the manchette scaffold until step 14. Assembly of accessory structures in the tail develops after axoneme initiation. Fibrous sheath longitudinal columns begin to appear around the distal end of the axoneme and then extend “backwards” as the axoneme extends. The transversal ribs are assembled during steps 11–15 in the rat, and outer dense fibers start simultaneously with manchette formation. The mitochondrial sheath is the final structure of the flagellum to be assembled. Chromatin condensation begins around steps 12 and 13. In nonrodent species with spatulated sperm head, the nucleus lacks dorso-ventral asymmetry and the characteristic hook, but the other events remain similar. For these dramatic changes to happen, several mechanisms have been proposed. A recent excellent review (259) deals with cellular and molecular aspects of spermiogenesis that are not covered in detail here. The underlying genetic control of spermatogenesis is complex and not well understood. Increasing evidence supports the idea that transcriptional mechanisms are quite different in haploid germ cells with respect to somatic cells (260). These differences include the use of distinct promoter elements and specific transcription factors as well as mechanistically different routes for activation (261). Widespread transcription in the testis has been shown to involve >80% of all genes in humans as well as in other species. Recently, a transcriptome analytic study has characterized the dynamics of gene expression in germ cells during the course of spermatogenesis (262). This study has shown that a total of 108 genes are uniquely expressed in pachytene spermatocytes and their expression is not carried over to round spermatids or sperm. Moreover, 323 genes are exclusively expressed in round spermatids, and 178 are only present in sperm. Interestingly, six genes expressed in spermatocytes are lost in spermatids but reappear in sperm. Many of the proteins synthesized during spermatogenesis are transcribed from the haploid nucleus and/or are translated from stored mRNAs. A comparative sperm proteomic analysis in three closely related mouse species has revealed significant differences between species in proteins involved in fertilization, including those that govern axoneme components and metabolic proteins (263). In addition, proteins that may underlie the diversification of spermatozoa were more likely to experience translational repression, suggesting that composition of spermatozoa is affected by the evolution of mechanisms that control translation. Because these species vary in their levels of sperm competition, it was possible to identify classes of proteins that are functionally coherent in relation to differences in postcopulatory sexual selection. Moreover, relaxed or intensified levels of sperm competition seemed to impact on the molecular composition of sperm cells (263). More recently, it has been proposed that widespread testis transcription facilitates germline DNA repair and ultimately modulates gene evolution rates (264). It was found that genes expressed during spermatogenesis have lower rates of nucleotide mutations compared with unexpressed genes. Moreover, the increased transcription during spermatogenesis may facilitate a transcriptional scanning process that systematically detects and repairs damage in the DNA by transcription-coupled repair. Interestingly, 90% of all protein-coding genes in germ cells are expressed. In contrast, somatic cells in the testis express only 60% of the genes, of which >99% overlap with the genes expressed in germ cells (264). 4.2. Control of Spermatogenesis Although it is generally considered that spermatogenesis is species specific, with regard both to the morphology of the sperm cell generated after differentiation and to the timing required to generate the mature sperm cell, there are a series of conserved mechanisms. Moreover, it is possible that early stages of spermatogenesis involving proliferation of spermatogonia may bear a degree of similarity among species, whereas the latter stages of spermatid differentiation may be more divergent, as they involve processes that end up in species-specific sperm cell morphologies. The characteristics of spermatogonial stem cells and dynamics of spermatogonial proliferation, along with regulatory mechanisms (253, 255), as well as those corresponding to the meiotic phase (265), have received much attention and are not covered here. Instead, we focus on aspects of sperm differentiation and control of sperm remodeling during the last phase of spermatogenesis. Thus, there is the question of what controls the morphology and timing of sperm differentiation and remodeling. Evidence suggests that the germ cell, and not the Sertoli cell, is the cell that primarily defines the remodeling of spermatids. We first review the roles of Sertoli cells and then summarize work on heterologous (xenogeneic) spermatogonia transplantation that has led to the conclusion that germ cells themselves are in control of spermatid differentiation and remodeling. Sertoli cells play fundamental roles in spermatogenesis because they 1) maintain the blood-testis barrier, which creates the appropriate environment for the occurrence of spermatogenesis; 2) provide physical support for the germ cells; 3) serve as “nurse cells” because of their involvement in the nutritional support of the male germ cells; 4) control hormonal regulation; 5) carry out the phagocytosis of apoptotic germ cells and residual bodies; and 6) release the spermatids toward the tubular lumen during spermiation (266). Some studies have also shown that the number of germ cells supported by a single Sertoli cell is limited and depends on the species (267, 268). However, it is unclear how much of the process of germ cell differentiation is driven by these cells. Spermatogonia can be transferred from an infertile male mouse to a fertile one in which the donor spermatogonial stem cells establish spermatogenesis and produce spermatozoa that transmit the donor haplotype to progeny (269, 270). This observation prompted studies of spermatogonia transplantation between species to examine whether the Sertoli or germ cells control the process of spermatogenesis. Marked testis cells from transgenic rats transplanted to the testes of immunodeficient mice colonized mouse seminiferous tubules and subsequently exhibited continuous spermatogenesis generating spermatozoa with a head and flagellum morphology characteristic of the rat (269, 271). Interestingly, although rat Sertoli cells were transplanted with rat donor germ cells, they were absent in seminiferous tubules colonized by the rat germ cells (272, 273). This suggests flexibility in the supporting role of the Sertoli cell in species that diverged a long time ago. Because rat spermatogenesis has a cycle length that is 50% longer than that of mouse spermatogenesis (rat: 52 days, mouse: 35 days), the question that also emerged was whether, in the mouse, rat spermatogenesis (if influenced by mouse Sertoli cells) had a shorter cycle. It was found that the germ cell genotype is the one that controls the cell cycle during spermatogenesis. Furthermore, both rat and mouse spermatogenesis developed concomitantly in the same mouse host after transplantation. Under these conditions, two different timing regimens for germ cell development were observed in the recipient mouse testis: one of rat and one of mouse duration (272). Thus, rat germ cells that were supported by mouse Sertoli cells always differentiated with a timing that was characteristic of the rat (52 days) and generated the spermatogenic structural pattern of the rat. These results thus suggest that the cell differentiation process of spermatogenesis is regulated by germ cells alone. To examine whether sperm generated after xenogeneic spermatogonia transplantation are functional, rat spermatozoa produced in mouse testis were recovered from seminiferous tubules and microinjected into rat oocytes, which were then transferred to recipient females. Rat offspring were born from donor spermatogonial cells, and such offspring were fertile and had a normal imprinting pattern (274). The reciprocal xenogeneic transplantation was also successful (275, 276). Hamster testis cells transplanted into a testis of an immunodeficient mouse also underwent spermatogenesis (277), despite a bigger difference between hamster and mouse sperm morphology than between rat and mouse sperm cells and older divergence times in the former pair (>16 million yr) in comparison to the latter (∼11 million yr). In this case, abnormalities were observed in hamster spermatids in seminiferous tubules of recipient mice. Hamster spermatozoa were seen in the epididymis of recipient mice, but most spermatozoa lacked acrosomes, and heads and tails were separated. Defects in hamster spermatogenesis occurring in mouse seminiferous tubules could be due to lower compatibility and a limited capacity of mouse Sertoli cells to support the much larger hamster spermatozoa, which also have a different head shape (277). Transplantation of germ cells from other species (rabbit, dog, pig, bull, and stallion) into mouse testes was also examined (278–280). These combinations represent an increased phylogenetic distance in comparison to the previous rat-mouse or hamster-mouse transplantations. Germ cells from rabbits or dogs colonized the mouse testis but did not proliferate or differentiate beyond the stage of spermatogonial expansion (279). Pig donor germ cells formed chains and networks of round cells connected by intercellular bridges, but later stages of donor-derived spermatogenesis were not found (278, 280). Bull testis cells developed predominantly into fibrous tissue. Few stallion germ cells proliferated in mouse testes (280). Thus, germ cells from domestic animals do colonize the mouse testis but do not differentiate beyond the stage of spermatogonial expansion. Among primates, baboon spermatogonial stem cells readily established germ cell colonies in recipient mice and survived for a period of ∼6 mo, but differentiation into spermatozoa did not take place (281). After xenotransplantation of human testis cells to mouse seminiferous tubules, no donor germ cells were identified in recipient testes (282). In contrast, another study (283, 284) claimed that transplantation of human testis cells to mouse and rat seminiferous tubules led to the production of spermatozoa in >25% of recipients. Subsequent studies (285) found that a high proportion of mouse recipient testes were colonized by human testis cells obtained from several patients. Human spermatogonial stem cells survived in mouse testes for at least 6 mo and proliferated during the first month after transplantation. However, no differentiating human spermatogonia were identified, and meiotic differentiation did not take place in mouse testes. The fact that rabbit, baboon, human, pig, bull, or dog spermatogonial stem cells and undifferentiated spermatogonia survive in mouse seminiferous tubules for a long period after transplantation suggests that antigens, growth factors, and signaling molecules that are required for interaction of these cells and the testis environment are highly conserved (281). Because these species diverged from rodents ∼75–85 million yr ago (286), it seems that there could be a high degree of conservation in the first stages of spermatogenesis. On the other hand, because differentiation of germ cells does not take place in mouse seminiferous tubules when spermatogonia from species other than rat or hamster are transplanted, molecules necessary for stages of spermiogenesis appear to have undergone great divergence between rodents and rabbits, primates, ungulates, or carnivores. Support for this idea comes from studies of xenografting in which testicular tissue fragments were transplanted subcutaneously to immunodeficient mice (253, 287). Under these conditions communication between the endocrine system and the donor testis become functional, and this leads to full spermatogenesis and competent sperm cells in grafts from species that are distantly related to mice, such as primates (288, 289), carnivores (290, 291), and ungulates (292–296). In addition to the fundamental questions related to the regulation of spermatogenesis, spermatogonia transplantation has potential applications for animal transgenesis (297), studies of human infertility, domestic animal production, or animal conservation (253). The possibility of using cryopreserved testis cells (rats: Refs. 269, 274; hamsters: Ref. 277) represents an opportunity to store material from valuable animals and use them when suitable recipients are identified or culture techniques are developed. The question of whether xenogeneic germ cell transplantation could be achieved with other combinations of species is relevant, but the distance between host and the species of interest is still potentially a problem. In attempts to use xenogeneic germ cell transplantation for species conservation, procedures were explored using the domestic cat as a recipient for the preservation and propagation of male germ plasm from wild felids (298) Both syngeneic and xenogeneic transplants revealed that spermatogonial stem cells were able to colonize and differentiate successfully in the recipient cat testis, generating elongated spermatids several weeks after transplantation. After >3 mo of transplantation, ocelot spermatozoa were observed in the cat seminiferous tubules and in the epididymis; the morphology of ocelot spermatozoa differentiating under these conditions appeared normal, and it seems that they could be differentiated by shape and size (298). At the present time, it seems that the domestic cat may be suitable as a recipient for other wild felids because no signs of immunorejection were observed (298). It remains to be established whether these spermatozoa are functionally competent by examination of their fertilizing ability. In summary, the studies mentioned above support the hypothesis that the germ cell seems to drive its own process of morphological change during cell differentiation. Two main factors have been proposed to exist in the germ cells to influence such morphological changes: 1) external forces applied to the spermatid nucleus either by cytoplasmic microtubules or by ectoplasmic filaments from Sertoli cells (299) and 2) internal nuclear forces resulting from a controlled pattern of aggregation of DNA and proteins during the condensation of the chromatin (35). These mechanisms are discussed in detail in sect. 5. 4.3. Energetics of Sperm Formation A considerable amount of energy is directed toward reproductive functions, in searching for mates, production of spermatozoa, or both. It is well known that sperm cells are costly (300) and that germ cells have important energy demands. Evidence for substantial costs of spermatogenesis comes from Drosophila, in which increasing sperm length delays male reproductive maturity (301), increases resources required for sperm production and packaging (302, 303), and selects for prudent male sperm-production strategies (304). Sertoli cells provide nutritional support for male germ cells during the differentiation phase, particularly to fulfill their energetic needs (305). The metabolic requirements of Sertoli cells and germinal cells are quite different. Sertoli cells have been shown to produce lactate and pyruvate. Three-quarters of this production is made to satisfy germ cells, whereas β-oxidation is suggested to be used only for the Sertoli cells’ own energy needs (305). Germ cells have different metabolic requirements during their differentiation states. The spermatogonia are spatially located near blood sources and can rely on glucose as a source of ATP via glycolysis (306). Spermatocytes and spermatids reside in the luminal compartment of the testis and can be at some distance from the blood supply. These cells may use glycolysis to generate ATP and possibly also use lactate as a substrate (307–309). Spermatids have the enzymes that make up the glycolytic pathway, but glucose metabolism alone does not seem capable of maintaining ATP levels, because in the absence of other energy sustrates spermatids experience depletion of ATP (310). Thus, they need pyruvate and lactate for survival and likely rely on the supply of lactate from the Sertoli cells. Several genes encoding sperm-specific forms of enzymes of the glycolytic pathway are expressed only in spermatids (306, 307). There is also selective expression of lactate dehydrogenase in advanced germ cells in the adluminal compartment and, since lactate dehydrogenase C (LDHC) converts lactate into pyruvate and pyruvate can be converted to acetyl coenzyme A, which would fuel OXPHOS, postmeiotic cells may rely more on mitochondrial OXPHOS activity (309). The mitochondria in germ cells change during spermatogenesis (309). Spermatogonia and early spermatocytes have mitochondria that are similar to those in somatic cells, whereas late spermatocytes, spermatids, and spermatozoa have more condensed and efficient mitochondria (311, 312). During the final stages of spermiogenesis mitochondria are lost in residual bodies. For example, <100 mitochondria remain in the mouse sperm cell, and they rearrange in tubular structures that are anchored helically around the nine outer dense fibers and the axoneme in the sperm midpiece (313, 314). As a consequence, the amounts of mitochondrial proteins and DNA molecules are also reduced (315, 316). 4.4. Evolution of Spermatogenesis The overall process of spermatogenesis appears to be conserved among many taxa (255), although several characteristics, such as sperm morphology and timing of the differentiation process from spermatogonia to mature spermatozoa, are unique and species specific. In mammalian species, the time required for the completion of spermatogenesis is unique and of a fixed duration for each species. It has generally been assumed that cell-cell interaction between germ cells and Sertoli cells could be a limiting factor in the production of spermatozoa (317) and that the duration of cell cycles and cellular organization may be the result of this interaction. Males appear to have the plasticity to modify the production of spermatozoa and generate more sperm, or to adjust sperm morphology, in response to social conditions such as the perceived risk of sperm competition (318). Under conditions of sperm competition, males would benefit if they could transfer a higher number of spermatozoa and/or sperm with a morphology that gives them an advantage over rival males (319, 320). Males could potentially modify sperm output by increasing the relative size of testes, by modifying testis architecture, by making production more efficient, or by changing the speed at which sperm are produced. In birds and mammals subject to high sperm competition levels, the proportion of spermatogenic tissue contained within the testis increases (321–324). Species under strong sperm competition generate more round spermatids per spermatogonium and have Sertoli cells that support a greater number of germ cells, both of which are likely to increase the maximum sperm output (325). High levels of sperm competition lead to shorter length of spermatogenic cycles in mammals, which indicates faster rates of spermatogenesis (322, 326). In a comprehensive comparative analysis of almost 100 mammalian species, changes in testis size, testicular architecture, kinetics of spermatogenesis, and sperm reserves were examined in relation to sperm competition in an integrative manner (321). It was found that higher levels of sperm competition correlated with higher proportions of seminiferous tubules, shorter seminiferous epithelium cycle lengths (SECLs), which reduce the time required to produce sperm, and higher efficiencies of Sertoli cells involved in sperm maturation. These responses to sperm competition were found to result in higher daily sperm production, more sperm stored in the epididymides, and more sperm in the ejaculate (321). Although there was a strong relationship between SECL and the duration of spermiogenesis, the levels of sperm competition were not associated with the latter (i.e., the period of sperm differentiation) (321). A dynamic model considering all the information available for mammalian species was used to jointly analyze the data in order to understand the relative contribution of variations in testis architecture and kinetics of spermatogenesis in response to sperm competition (FIGURE 6). Three hierarchic levels of variables were identified based on the hypothetic relationships tested in the models: 1) sperm competition; 2) SECL, efficiency of Sertoli cells, and percentage of the testis occupied by seminiferous tubules (% of tubules); and 3) number of sperm in the caudae epididymides (sperm reserves). The slopes and intercepts estimated by phylogenetic generalized least squares (PGLS) models were used to predict the relative influence of one level on the successive levels. Dynamic variations of the three levels were assessed by first examining a 1% variation in the range of the predictor (level 1) on changes in the dependent variable of the next level (level 2). Then, for the relationships between levels 2 and 3, the variation introduced in the predictor variable at level 2 was that resulting from the variation of 1% in the range in the level 1 predictor. For example, an increase in sperm competition equivalent to 1% of sperm competition range produced an increase of 1.15% of the variable range of seminiferous tubules; such 1.15% increase in tubules resulted in 0.70% increase in sperm numbers in the cauda (321). Results clearly revealed a higher impact of sperm competition on sperm architecture than on sperm kinetics, in turn resulting in a higher effect of the former on sperm reserves in the cauda epididymis (FIGURE 6). The highest effect of sperm competition was seen on the efficiency of the Sertoli cells, which, altogether, had the highest effect on sperm reserves. FIGURE 6. Open in a new tab Relationships between sperm competition, or metabolic rate, and testis architecture, kinetics of sperm formation, and sperm numbers in mammals. The schematic representations show dynamic variations in testicular parameters, or sperm reserves in the epididymis, in relation to sperm competition or mass-specific metabolic rates. In the model, level 1 (top) corresponds to sperm competition (left) or metabolic rate (right), level 2 (middle) corresponds to testicular parameters [% tubules, efficiency of Sertoli cells, seminiferous epithelium cycle length (SECL)], and level 3 (bottom) corresponds to sperm reserves in caudae epididymides. Numbers at the end of each arrow between levels 1 and 2 are the relative variation in the dependent variable (level 2; testicular parameters) caused by a variation of 1% of the independent variable (level 1; sperm competition or metabolic rate). Numbers at the end of each arrow between level 2 and 3 variables indicate the relative variation in the dependent variable (level 3; sperm reserves) caused by the change in the variable (level 2) due to a 1% increment in level 1. Percentages of relative variation were calculated using the slopes and intercepts estimated by phylogenetic generalized least squares (PGLS) models (321). % tubules, % of the testicular tissue occupied by seminiferous tubules; sperm reserves in cauda, number of spermatozoa in the caudae epididymides. The effect of metabolic rate on testicular traits and sperm reserves was also examined because low mass-specific metabolic rate, as seen in large-bodied species, may limit processing energy and resources efficiently enough at both the organismic and cellular levels. In the same dynamic model explained above, changes in metabolic rate had a higher impact on sperm architecture than on sperm kinetics, with a higher effect of testicular architecture on sperm reserves (FIGURE 6). Furthermore, the two variables that require processing resources at faster rates (SECL and efficiency of Sertoli cells) only responded to sperm competition in species with high mass-specific metabolic rate. Thus, increases in sperm production with intense sperm competition take place via a complex network of mechanisms, with some being constrained by metabolic rate (321). 5. MOLECULAR MECHANISMS UNDERLYING SPERM FORMATION One of the most remarkable cellular changes accompanying sperm differentiation is the remodeling of head architecture (327). As described in sect. 3, there is a rich variety of shapes of sperm heads. Globular forms are common among fish. Amphibia have filiform, cylindrical, tapering, or long heads. Flattened ovoid heads are predominant in mammals, although murine and cricetid rodents generally have a falciform, sickle-shaped head. We review in this section the mechanisms that influence the remodeling of the sperm head (FIGURE 7). Most of the current knowledge comes from several studies performed in mammals, and little is known for other species or from an evolutionary point of view. Whenever possible, comparison of processes occurring in different species is attempted. FIGURE 7. Open in a new tab Schematic representation of the processes involved in sperm head shaping during spermiogenesis. The ectoplasmic specialization in Sertoli cells encircles >1/3 of the spermatid head and generates external pressure. The acroplaxome, also covering >1/3 of the head, modulates the external forces and anchors the acrosome to the nucleus. The manchette, a scaffolding of microtubules and actin filaments, encircles the 2/3 remaining area of the nucleus. It connects intimately with the nuclear lamina and generates internal forces by a zipperlike movement. Additionally, the manchette provides tracks for the trafficking of proteins. The nuclear membrane restructures its molecular composition, redistributing proteins to adjust the flexibility of the membrane to allow for external and internal forces. Replacement of histones by transition proteins and then protamines makes the last twists for chromatin condensation. 5.1. Role of Sertoli Cells The Sertoli cells are specialized cells that surround the germinal cells. Among their numerous functions, orchestrating germ cell development and differentiation, they are believed to also participate in the compression and remodeling of the sperm head via mechanical forces. During spermatid elongation (step 8), the Sertoli cells spread over more than one-third of the sperm head and thus contribute to nuclear and head reshaping (328). However, the underlying mechanisms are not well understood. The Sertoli cells display an ectoplasmic specialization (ES) that participates in the remodeling of the sperm head, and this structure is thought to be conserved in mammals. However, it is not clear how this structure would influence the formation of oval or falciform sperm heads. At the molecular level, the ES is formed by a layer of actin bundles packed in parallel hexagonal arrays and surrounded by cisternae of endoplasmic reticulum (295). Associated proteins including afadin and nectin 2 and 3 are also part of this structure (329). Nectin 2- and 3-deficient mice show severe disorganization in the sites where Sertoli cells contact with spermatids, resulting in aberrant spermiogenesis and sterility (330, 331). The actin-related protein (Arp) 2/3 complex has been localized to the ES and may contribute to the formation of the actin network at the site. This complex includes Arp3, N-Wasp, cortactin, and regulators such as Rac1 and Cdc42 (332). F-actin bundles present in the ES are believed to be noncontractile. It has been proposed that the mechanical forces may be driven by dynamic changes in polymerization and depolymerization of F-actin assisted by the Arp2/3 complex (333) and Fer kinase (334) (FIGURE 8 A). It is not yet clear how these proteins interact with F-actin and what the molecular mechanisms behind the role of Sertoli cells in the remodeling of the head are. Thus, this is an important area that needs further investigation. FIGURE 8. Open in a new tab Molecules involved in the spermatid head remodeling. A: a molecular model that illustrates the structural association between Sertoli cells and spermatids during head remodeling at the site of ectoplasmic specialization. B: diagram illustrating the binding of Cortactin to F-actin. 1: Arp2/3 complex promoting F-actin polymerization. 2: Arp2/3 regulating Cortactin function. 3: FerT mediating tyrosine phosphorylation of Cortactin. 4: Phosphorylation of Cortactin promotes F-actin depolymerization. C: schematic representation of the distribution of SUN domain proteins during nuclear elongation. Yellow, Sun3; blue, Sun1 and Sun1η; green, Sun4. Posterior and anterior spermiogenesis-specific linker of nucleoskeleton and cytoskeleton (LINC) complexes are shown in insets. INM, inner nuclear membrane; OAM, outer acrosomal membrane; ONM, outer nuclear membrane. 5.2. The Acroplaxome The acroplaxome was described for the first time in rat spermatids by Kierszenbaum and collaborators (335) as a cytoskeletal plate linking the inner acrosomal membrane to the nuclear envelope. At the molecular level, the acroplaxome is composed of F-actin, keratin 5 (335), and myosin Va (336). This structure also includes a specialized electron-dense material named marginal ring (336), which is made up of keratin 5-containing intermediate filaments inserted into a plaque associated with the leading edge of the inner acrosomal membrane. The array of F-actin is dynamically modulated in the acroplaxome by Arp2/3 complex (333), tyrosine kinases targeting cortactin (337), as well as FerT (338) (FIGURE 8 B). The testis-specific proteins profilin-3 (PFN3) and profilin-4 (PFN4), crucial for actin microfilament dynamics, were also reported to localize to the acroplaxome (339). It has been proposed that the acroplaxome plays a role in stabilizing and anchoring the developing acrosome to the nucleus of elongating spermatids. Additionally, it may also provide a mechanical planar scaffold modulating external clutching forces generated by the ectoplasmic specialization of Sertoli cells encircling the elongating spermatid nucleus (335). Mutant mouse models in which spermatid nuclear shaping was defective provided clues concerning the significance of the acroplaxome in the elongation of the spermatid nucleus. In the azh mutant mouse, a relatively large number of spermatids display abnormally shaped nuclei and spermatids in which the acroplaxome-containing region is particularly indented (336, 340). Hrb mutant males are infertile, and both spermatids and sperm are round headed and lack the acrosome. Although the acroplaxome plate is present in the mutant spermatids, its composition is affected, since it is deficient in keratin 5 but not F-actin (329). Additional proteins were shown to localize to the acroplaxome of rat spermatids, including the ubiquitin protein ligase Rnf19a, the component of the 26S proteasome Psmc3 (341), the outer dense fiber (ODF2) protein (342), the Golgi protein GMAP210, and the intraflagellar protein IFT88 (343). Recently, lamin A/C has been proposed to be part of the acroplaxome in the mouse (344). The decreased expression of lamin A/C following injections of siRNA against the Lmna gene causes significant alterations in nuclear elongation and acrosome formation (344). However, further studies are required to understand the role of lamin A/C in the acroplaxome (327). Another protein reported to be present in the acroplaxome is MARCH7, an E3 ubiquitin ligase important for protein degradation (345). Although confocal images show colocalization with β-actin in the head of elongating spermatids, it is difficult to conclude that MARCH7 is a component of the acroplaxome with the experiments performed by Zhao and collaborators (345). Superresolution or electron microscopy studies may provide better clues for the acroplaxome localization of this protein. Unfortunately, since the discovery of the acroplaxome (335), very few advances have been made in the characterization of this structure and the molecular mechanisms that take place here. Moreover, it is not clear whether this structure and its function are conserved among species. Therefore, this is an area of research that deserves further investigation. The combination of state-of-the-art superresolution microscopy and cryo-electron microscopy may bring about valuable discoveries in this area. 5.3. The Manchette The manchette is a transient perinuclear organelle that is also believed to participate in the elongation of the sperm head and nucleus. The timing of the development of the manchette is very precise. In the mouse and the rat, it assembles around step 8 during spermatid elongation and disappears when the elongation and condensation of the nucleus are nearly completed (step 14). This skirtlike organelle consists of bundles of microtubules connected to a perinuclear ring and filaments of actin intercalated between the microtubules (17, 346) (FIGURE 7). Endoplasmic reticulum is aligned along the cytoplasmic face of the manchette. The manchette microtubules appear to emanate from the perinuclear ring at the base of the acrosome. However, there is still controversy about whether the microtubules are nucleated in the perinuclear ring or are nucleated elsewhere and are later linked to this ring. Two specific nucleation sites for the manchette microtubules have been postulated: the perinuclear ring and the centrosome. However, detection of plus-end tracking proteins (EB3 and CLIP-170) together with the absence of γ-tubulin disagree with the idea of the perinuclear ring as the microtubule nucleation site (346–348). Instead, supporting evidence indicates that the centriolar adjunct serves as a nucleator of manchette microtubules with their plus ends reaching toward the perinuclear ring (348, 349). Another theory proposed multiple microtubule organizing centers that organize from existing cytoskeletal microtubules (328). Further experimental studies should be performed to bring light to this issue. The importance of the manchette in sperm head shaping has been a matter of debate because of discrepancies found in different species (35). In a number of species the manchette microtubules are absent during spermiogenesis (e.g., scorpion, Anthozoa, Chaetognatha, crested tinamou) (350–353) or disappear before profound changes in nuclear shape (35, 354). Species such as the ostrich display a circular and longitudinal manchette around the nucleus (280). However, with the help of murine mutant models it has become evident that the manchette plays an important role in head shaping during spermatid differentiation in mammals. In general, the spermatid nucleus assumes a parallel shape to the manchette, and when the manchette is absent the nucleus adopts a round shape as seen in models with caudally displaced manchette and in mutants that lack nucleus-manchette connection (reviewed in Ref. 348). During spermatid elongation, the marginal ring of the acroplaxome and the perinuclear ring of the manchette reduce their diameter as they gradually descend along the nucleus toward the spermatid tail (329). This zipperlike movement helps with nuclear condensation and shaping (348). In the case of rodents with hook-shaped heads, this zipperlike movement of the manchette generates forces that help to create the dorsal and ventral nuclear surfaces (355). Yet the molecular mechanisms driving this force are not well understood. Many molecules have been found to anchor or harbor around the manchette (17). They assist with nuclear condensation, spermatid differentiation, and tail formation (348). However, their molecular functions are largely uncharacterized. Recently, in silico analysis has drafted a putative interacting protein network that represents a framework for future investigations (17). Several proteins have also been shown to localize to the manchette, and their mutation in murine models resulted in perturbations in the manchette and alterations in the nucleus and head shape (TABLE 2). Bay way of illustration, SPAG17, a protein originally described as a microtubule central pair protein present in the flagellar axoneme (401), has been found to localize to the manchette and shown to be essential for nuclear shaping and chromatin condensation (390), a role that extends beyond its proposed role in flagellar function. SPAG17 is a poorly understood protein, and the molecular mechanisms behind its pleiotropic functions are unknown (401). Nevertheless, it appears to be associated with trafficking of proteins during spermatid differentiation (390). The coiled-coil domain containing 42 (CCDC42) protein has been found to be enriched in the perinuclear ring and to colocalize with acetylated tubulin in the manchette (358). A Ccdc42-mutant mouse displays defects in the nuclear morphology of elongating spermatids (357). The molecular function of CCDC42 is unknown, but it was shown to interact with ODF1 and ODF2 (358). Table 2. Knockout mouse models affecting the nuclear shape and the manchette during sperm differentiation \| Gene \| Protein \| Spermiogenesis Phenotype \| Interaction \| References \| :---: :---: \| Azi1/CEP131 \| CEP131 \| Short tail, disorganized sperm tail structures, ectopic and elongated manchette \| BBS4 \| (356) \| \| Ccdc42 \| CCDC42 \| Abnormal shaped heads, detached acrosome-acroplaxome, lack of flagella \| ODF1, ODF2 \| (357, 358) \| \| Cfap43 \| CFAP43 \| Abnormal manchette, disorganized ectoplasmic specialization, and sperm head and flagella deformities \| \| (359) \| \| Clip170 \| CLIP170 \| Abnormal formation and maintenance of the manchette and abnormally shaped sperm heads \| LIS1, UBE2 \| (17, 360) \| \| Cntrob \| CNTROB \| Ectopic and asymmetric perinuclear ring and manchette, detached centrosome, decapitated and disorganized tails \| KRT5, tubulin \| (361) \| \| Dlec1 \| DLEC1 \| Deformed head, shortened tail, and abnormal manchette organization \| α/β-Tubulin, TCP-1, BBS2, BBS4, BBS5, BBS6 \| (362) \| \| Drc7/Ccdc135 \| DRC7 \| Elongated manchette and sperm with shorter tails and abnormal head shapes \| DRC5 \| (363) \| \| E-Map-115 \| E-MAP-115 \| Ectopic manchette along regions of the nucleus that normally do not display manchette and tail appears normal \| Kinesin 1 \| (364) \| \| Fam46c \| FAM46C \| Headless spermatozoa due to defects in the connecting piece \| \| (365) \| \| Fused/Axin1 \| FU \| Periaxonemal abnormalities, manchette elongated and malformed, acroplaxome affected \| KIF27/ODF1 \| (366) \| \| Gopc \| GOPC \| Lack of the acrosome, lack of postacrosomal sheath and the posterior ectopic and misplaced manchette; impaired mitochondrial sheath assembly in the epididymal spermatozoa, coiled flagella \| Golgi-160, RAB6A, GRID2, BECN1, RHOQ, ACCN3, CFTR, CSPG5. \| (367, 368) \| \| Hook1/azh \| HOOK1 \| Elongated and disorganized manchette microtubules, knoblike shape of the head, weak head-tail connection, and bending of the tail \| RIMBP3, CCDC181 \| (335, 369–371) \| \| Ift20 \| IFT20 \| Abnormal head shape, elongated manchette, disorganized tail components \| SPEF2, GMAP210, COPS5, IFT88, IFT172 \| (17, 372) \| \| Ift81 \| IFT81 \| Abnormally shaped heads and lack of tails \| \| (373) \| \| Ift88 \| IFT88 \| No axoneme, disorganized tail components, malformed HTCA, ectopic perinuclear ring and manchette elongated \| GMAP210 \| (343) \| \| Ift172 \| IFT172 \| Abnormal nuclear and acrosome morphology, elongated manchette, abnormal axoneme accessory structures, multiple centrosomes \| IFT20, IFT27, IFT88 \| (17, 374) \| \| Iqcg \| IQCG \| Short tail and disorganized sperm tail structures, irregular nucleus and localized to the manchette \| Calmodulin \| (375) \| \| Katnal2 \| KATNAL2 \| Abnormal nuclear morphology, detached acrosome, elongated manchette, absence of sperm tail \| KATNB1, δ- and ε-tubulin \| (376) \| \| Katnb1 \| KATNB1 \| Sperm tail mobility affected, manchette elongated and knoblike \| Katanin 60 \| (377) \| \| Kifnb1 \| KIF3A \| No axoneme, disorganized tail components, manchette elongated and knoblike shape of the head \| KIF3B, KAP, MNS1, KBP. \| (378) \| \| Lrguk1 \| LRGUK1 \| Short tail, acrosome acroplaxome detached, manchette MTs unevenly distributed and elongated manchette \| HOOK2 \| (379) \| \| Meig1 \| MEIG1 \| Disorganized sperm tail structures, disrupted manchette structure reported, and round or detached heads \| PACRG, SPAG16 \| (380–382) \| \| Pacrg \| PACRG \| Disorganized sperm tail structures, disrupted manchette structure reported, and round or detached heads \| MEIG1 \| (383) \| \| Rim-bp3 \| RIMBP3 \| Abnormalities in sperm heads characterized by deformed nuclei and detached acrosomes \| HOOK1 \| (384) \| \| Sept12 \| SEPT12 \| Defective sperm heads, bent tails, premature chromosomal condensation, and nuclear damage \| CDC42 \| (385, 386) \| \| Spag6 \| SPAG6 \| Abnormal spermatid head, midpiece fragmentation, truncated flagella, decapitated sperm \| Snapin, COPS5, SINK2, TCTE3, SPAG16L, TAC1, Moesin, BBS4, DAZL, ACTR2, MGP SPAG17 \| (387–389) \| \| Spag17 \| SPAG17 \| Abnormal nuclear morphology and chromatin condensation, detached acrosome, elongated manchette, absence of mature sperm \| SPAG6, SPAG16 \| (389, 390) \| \| Spef2 \| SPEF2 \| Short tail, elongated manchette, and disorganized sperm tail structures \| IFT20 \| (391, 392) \| \| Spem1 \| SPEM1 \| Head bent back, midpiece wrapped around head and retained cytoplasm \| RANBP17, UBQLN1 \| (393, 394) \| \| Stk33 \| STK33 \| Oversized acrosomal tips, bifurcated heads, elongated manchette, and disorganized tail structures \| TUBA1B, ACTN4 \| (395) \| \| Sun3 \| SUN3 \| Abnormalities in the manchette and sperm heads characterized by deformed nuclei and missing, mislocalized, or fragmented acrosome, as well as multiple defects in sperm flagella \| SUN4 \| (396) \| \| Sun4 \| SUN4/SPAG4 \| Round-headed sperm, abnormal acrosome, severely disorganized manchette, and coiled tails \| SUN3, Nesprin1, ODF1 \| (397, 398) \| \| Ube2b \| UBE2B \| Mislocation of the longitudinal columns of the FS, head shape and MS abnormalities, acrosomal defects, and ectopic manchette \| RAD18 \| (399, 400) \| Open in a new tab FS, fibrous sheath; HTCA, head-to-tail coupling apparatus; MS, mitochondrial sheath; MT, microtubule. Depending on the species, the manchette is in intimate contact with the nucleus. It connects through a linker of nucleoskeleton and cytoskeleton (LINC) complex (402). Current sequence information from genome and transcriptome databases shows that the main proteins from this LINC complex (SUN and KASH) have undergone a remarkable diversification over the course of evolution (403). In particular, concomitant with the increase in complexity of the organism, the number of SUN proteins has significantly increased as well. Lower organisms, such as yeast and cellular slime molds, appear to cope with only one SUN domain protein. Nematodes and flies both contain two genes for SUN proteins, whereas the mammalian genome encodes five distinct members of the SUN protein family: Sun1, Sun2, Sun3, Sun4/SPAG4, and Sun5/SPAG4L (403). Some evolutionary studies have been performed in plants as well (404). In the mouse, two LINC complexes have been implicated in spermatid elongation: SUN3/Nesprin 1 complex, which connects the manchette to the nucleus, and SUN1/Nesprin 3, which shows an atypical nonnuclear localization at the anterior pole (348). SUN4 has been shown to interact with SUN3/Nesprin 1 complex. SUN4 knockout shows disconnection of the manchette with the nuclear envelope resulting in round-headed sperm (397, 405). Similarly, SUN3-knockout male mice are infertile, displaying a globozoospermia-like phenotype, where nuclei fail to elongate because of the absence of manchette microtubules and perinuclear rings (396). However, Sun5 mutant mouse spermatozoa contain tailless heads (406). Similarly, biallelic SUN5 mutations in humans cause male infertility due to autosomal-recessive acephalic spermatozoa syndrome instead of globozoospermia (407). These mutant models have proposed essential roles for the LINC complex in nuclear remodeling; however, the entire interactome of proteins that localize to the manchette and interact with the LINC complex is not well known. Further research dissecting these molecular interactions may bring light to the teratozoospermia defects. The manchette also participates in the transport of proteins between the nucleus and the cytoplasm. In this context, the motor protein KIFC1 and the nucleoporin protein NUP62 have been shown to interact and participate in this transport mechanism (348). RANBP17 is a RAN-binding protein also involved in transport in and out of the nucleus. It interacts with SPEM1 and UBQLN1 (393, 408). The exact role for SPEM1 is unknown, but its interaction with UBQLN1 may associate this complex with the degradation of unwanted proteins via the ubiquitin-proteasome system. Proteomic studies of different model organisms have provided very valuable information relevant to our understanding of the functions of the germ cell and conserved key elements. However, to the best of our knowledge, there are no proteomics studies focused on the manchette. Notwithstanding, some proteins that localize to the manchette are conserved during evolution (e.g., ODF1, ODF2, SPAG17, SPAG6) (409, 410). It would be interesting to know whether proteomic studies can find associations between the shape of the sperm head and manchette proteins among different species. In humans, several genes have been associated with teratozoospermia, including SPATA16, DPY19L2, PICK1, ZPBP1, and CCDC62, that are linked to globozoospermia. Additionally, mutations in SUN5, PMFBP1, TSGA10, DNAH6, BRDT, and CEP112 were implicated in acephalic spermatozoa syndrome (411). However, none of the proteins encoded by these genes has been reported to localize to the manchette yet (17). Therefore, it would be important to study the subcellular localization of these proteins in elongating spermatids to dissect whether the manchette is involved in the etiology of human teratozoospermia as it is in the mouse. Little is known about the role and molecular mechanisms of the manchette in spermatid differentiation in species other than the mouse. Therefore, a comparative overview of common or divergent processes is not possible at this time. Some descriptive studies of the morphology of this organelle exist in some species (35, 350–354, 412), but it is unclear how the manchette relates to nuclei of sperm with other shapes. It may be possible to predict differences between distant species with different nuclear shape, but it should also be borne in mind that there are examples of closely related rodent species in which there is a drastic switch from a falciform nucleus to an oval nucleus in species in which there appears to be a “reversal” to an ancestral head form. The question thus arises as to how this major remodeling of the sperm head is taking place in different species. Furthermore, there is very little information regarding evolution of molecules controlling manchette formation (or participating in manchette function). It is not clear, therefore, what degree of conservation exists in these molecules and whether one or some of these molecules are positively selected. If this were the case, it would be important to learn if this is influenced by postcopulatory sexual selection. 5.4. The Nuclear Membrane The nuclear envelope (NE) is a double membrane around the eukaryotic nucleus. It is structurally composed of two bilayers of lipids and associated proteins similar to the composition of the plasma membrane. These bilayers form the inner and outer nuclear membranes, which are connected by nuclear pore complexes allowing for nucleocytoplasmic trafficking. Lamin A/C and DPY19L2, both NE proteins, have been associated with the attachment of the acrosomal vesicle to the anterior NE margin of spermatids and nuclear shaping (344, 413). Downregulation of Lamin A/C expression in mouse testis leads to severe acrosome and nuclear morphology defects (344). Deletion of Dpy19l2 in a murine model results in defective spermiogenesis. Round spermatids show fragmented and partially missing nuclear dense lamina. Elongating spermatids show a detached acrosome-acroplaxome complex, dislocation in the manchette structure, and impaired nuclear elongation (413). In humans, deletion of the DPY19L2 gene leads to globozoospermia, a condition in which spermatozoa show a monomorphic rounded head. Globozoospermic sperm are unable to adhere to or penetrate the zona pellucida, leading to primary infertility (411). Several B-type lamins and intra-nuclear membrane (INM) proteins are also involved in nuclear shaping via remodeling of the spermatid chromatin. During the initiation of nuclear compaction and reshaping, Lamin B, LAP1, LAP2, and LBR display specific spatiotemporal expression patterns. At the beginning they distribute homogeneously or in one-half of the NE of round spermatids. As spermatid elongation takes place, these proteins translocate to the posterior nuclear pole. This spatial reorganization of proteins is believed to be responsible for chromatin redistribution and replacement of histones by protamines, accounting for changes in the nuclear volume (327). The NE has been also shown to influence nuclear shape by modulating the flexibility of the nuclear lamina network. In this context, Lamin B3 has been shown to have a dynamic distribution in elongating spermatids. Initially, the protein distributes around the NE and diffusely in the nucleoplasm of round spermatids, occupying one-half of the mouse nucleus. As spermiogenesis progresses, Lamin B3 relocates to the posterior nuclear pole (414, 415) and provides a localized increased flexibility in the nuclear lamina. Interestingly, the distribution of some NE proteins during spermatogenesis is different in germ cells from mouse, rat, and human (327), raising the question about whether these differences may account for their different sperm head shapes. The nucleus maintains extensive contacts with the cytoskeleton. This is accomplished largely through the LINC complex. LINC complexes are versatile structures recognized in all nucleated cells. They are composed of SUN and KASH domain proteins. SUN proteins localize to the inner nuclear membrane and provide a connection to nuclear structures while acting as a tether for outer nuclear membrane KASH proteins. On the other hand, KASH provides binding sites for diverse cytoskeletal components. In this way, LINC complexes form bridges linking the NE to the cytoskeleton (416). Recent studies identified quite striking polarization in the distribution of the LINC complex during sperm differentiation. SUN3 and Nesprin1 are exclusively found at the posterior NE region linked to the manchette microtubules. However, Sun1η/Nesprin3 and Sun5 (Spag4L) are present at the opposite, anterior spermatid pole, likely linking to the actin cytoskeleton in the acroplaxome (FIGURE 8 C) (403). Together, these findings suggest that LINC complexes connect the differentiating spermatid nucleus to the surrounding cytoskeletal structures (microtubule manchette and actin filaments in the acroplaxome) to transfer forces that help with nuclear shaping and elongation. 5.5. Chromatin Compaction and Protamines Heretofore, we have considered molecular mechanisms involved in sperm formation that would operate in the interaction of Sertoli cells and the germ cells or that may exist in the germ cells external to the nucleus. Next we consider molecular aspects of nuclear chromatin compaction and the role of protamines. In contrast to the situation in somatic cells, in which nuclear chromatin is compacted by histones, in sperm cells chromatin compaction is mainly linked to protamine binding to DNA. Replacement of somatic histones by protamines is gradual and actually takes place in three steps. Somatic histones are first replaced in part by testis-specific histones that are amino acid sequence variants of somatic histones. Then, testis-specific histones are replaced by transition nuclear proteins (TNPs). Finally, both testis-specific histones and transition proteins are replaced by highly basic protamines. In each step, replacement is not complete. Thus, a small fraction (1–15%) of histones remain bound to sperm DNA. Sperm nucleosomes are enriched at loci of developmental importance, including imprinted gene clusters, microRNA clusters, and HOX gene clusters (417, 418). However, recent reports claim that sperm nucleosomes remain predominantly within distal gene-poor regions and are depleted in promoters of genes for developmental regulators (419, 420). The first noticeable change in chromatin structure occurs when the sperm-specific histone H1t variant is deposited in spermatid chromatin, which is transformed into a more uniform and granular state (421). The synthesis and deposition of TNP2 precedes that of TNP1 (422, 423). With the appearance of TNPs, the chromatin starts to condense, with such condensation taking place in an apical to caudal direction (424, 425). When TNP deposition is completed, the chromatin appears to be uniformly condensed. With protamines, the chromatin is reorganized yet again (421). 5.5.1. Role of protamines. Protamines are nuclear proteins, and they are only expressed in haploid male germ cells (spermatids and the spermatozoon). Protamine genes are expressed in round spermatids soon after completion of meiosis. Transcripts are stored for various days until protein synthesis starts in elongating spermatids (426). Because protamines are important in DNA compaction, modifications in their expression and function have an impact on sperm morphology and performance (427). However, it is not clear whether protamines drive nuclear remodeling by way of nuclear/chromatin compaction, and in turn cell reshaping, or if there is a joint nuclear-cytoplasmic action remodeling the nucleus and, as a consequence, cell reshaping. In the mouse and rat, chromatin condensation by protamines begins around step 12, when some transformation in nuclear shape has already happened. Therefore, chromatin compaction driven by protamines might not be responsible for the initial morphological changes in the head (299), although they may have a role in the main steps of nuclear reshaping. In any case, it should be borne in mind that the sequence of events starting with histonelike proteins, and continuing with transition nuclear proteins, involves chromatin remodeling and condensation, so, rather than thinking solely of protamines, the process of nuclear remodeling should be viewed as underlain by these sequential steps, and thus modifications from within the nucleus may be as important as those driven from outside the nucleus. Chromatin compaction results in protection of DNA integrity, minimizing damage by various factors. In addition, compaction leads to gene silencing. This process is reversed after fertilization, when protamines are removed and replaced by histones, DNA is repaired within the capacity of the oocyte, and sperm activates embryo development (428). Protamines thus play crucial roles in the series of events ending in fertilization and are also important for development and the well-being of the offspring. Two types of protamines have been identified in mammals: protamine 1 (PRM1) and protamine 2 (PRM2). The former is expressed in all mammals, whereas the latter only appears to be expressed in primates and most rodents and in a few other species (426). Whereas PRM1 is synthesized as a mature protein, PRM2 is synthesized as a precursor, which is processed as it binds to DNA. As soon as the precursor binds to DNA, it is subjected to proteolytic processing, which finally results in the removal of ∼40% of the NH 2 terminus (429). Thus, two PRM2 variants are recognized: the precursor form (pre-PRM2) and the processed form (mature-PRM2). The domain that is sequentially cleaved off is collectively termed cleaved-PRM2. As a result of this processing of PRM2, although transcription starts at about the same time as for PRM1, there is a delay in the expression of mature-PRM2 protein because of such processing. PRM1 is a 50-amino acid nuclear protein with three characteristic domains: a central arginine-rich DNA-binding domain exhibiting 3–11 consecutive arginine residues, which on both sides is flanked by short serine- and threonine-containing segments containing phosphorylation sites. PRM1 in eutherian mammals also has cysteine residues, which can form disulfide bridges between protamine molecules, linking them tightly together. PRM1 is localized within the major groove (430), with one PRM1 molecule being bound per turn of DNA helix, which is equivalent to the binding of 10–11 base pairs of DNA per PRM1 molecule (427). PRM2 shares 50–70% sequence identity with PRM1. Since it is larger than PRM1, it can bind up to 15 base pairs of DNA. The fully processed form of the PRM2 precursor (i.e., mature-PRM2) is slightly larger than PRM1, with 63 amino acids in the mouse (426). Although it is well known that protamines serve to compact chromatin and prevent DNA damage, it is not yet clear how this happens at the structural-molecular level. Intra- and interprotamine disulfide bonds between cysteines are responsible for the DNA condensation, but it is not known how protamine cross-links with DNA and how compaction takes place through disulfide bonds. The absence of crystallographic data of protamine-DNA complexes has prevented modeling, but some proposals exist for mouse, human, and bull PRM1 (426, 431–433). Additional studies to unravel mechanisms of protamine-DNA interaction are warranted because of the important relation between deficient protamination and the etiology of DNA damage in humans and other species that can result in sub- or infertility or can give rise to genetic defects in offspring (421, 434–437). 5.5.2. Protamines and sperm head shape. Several lines of evidence suggest that transition nuclear proteins and, above all, protamines are important in reshaping the sperm nucleus during spermiogenesis and thus contribute to the shape of the sperm head. Genetic alterations associate with modifications in sperm head morphology, as seen in studies of disruption of these genes. There is also supporting evidence from correlational studies of sequence evolution and head morphology and from analyses of divergence in expression and sperm head shape. Furthermore, information from humans and domestic species suggests a link between abnormal protamine protein levels and sperm abnormalities. The relationship between sperm head shape and transition nuclear proteins has been examined after targeted deletion. In mice lacking TNP1, only subtle abnormalities were observed in sperm morphology (438). However, sperm motility was reduced severely, and ∼60% of Tnp1-null males were infertile (438). It thus seems that TNP1 may be not essential for histone displacement or chromatin condensation. The absence of TNP1 could be compensated by TNP2, but the dysregulation of nucleoprotein replacement nevertheless leads to a reduction in sperm function and fertility. Spermatogenesis in Tnp2-null mice was almost normal (439). There was only a slight increase of sperm retention in stage IX to XI seminiferous tubules, and epididymal sperm showed an increase in abnormal flagella but not in head morphology. Tnp2-null mice were fertile but produced small litters, which suggests that TNP2 may not be critical for nuclear shaping or fertility but is necessary for subsequent normal processing of PRM2 and the completion of chromatin condensation (439). The Tnp1- and Tnp2-null double-mutant mice revealed that nuclear shaping, transcriptional repression, histone displacement, and protamine deposition proceeded relatively normally, but chromatin condensation was irregular in all spermatids. In these mice, epididymal spermatozoa were drastically reduced in number and were highly abnormal, and the mice were sterile. Over 80% of epididymal spermatozoa were dead, and most of the live ones were immotile. Almost all had abnormal head morphology, with many exhibiting blunted apical tips. Most spermatozoa had missing mitochondria or abnormal configurations of the tail, and many were aggregated in clumps. Interestingly, microinjection of testicular or caput epididymal sperm from Tnp1−/−Tnp2−/− males into intact oocytes resulted in normal embryonic and fetal development and yields of live born equivalent to wild type, whereas, in contrast, cauda epididymal sperm from Tnp1−/−Tnp2−/− mice produced lower implantation rates and yields of live born than those from wild-type mice (440–442). These results reveal therefore that in Tnp1- and Tnp2-null mice inadequate sperm formation due to the absence of TNP action during spermiogenesis is exacerbated during epididymal transit, with a major decline in sperm functional capacity. Disruption of one copy of the Prm1 or Prm2 genes in the mouse resulted in a reduction in the amount of the respective protein (443). In these males, there was an alteration in nuclear formation, processing of PRM2, and normal sperm function, suggesting that both protamines are essential (443). There was an increase in morphological abnormalities, and the alterations seen most frequently were sperm with elongated heads having a reduced ventral flexure or sperm heads narrowed and reduced in curvature at the tip. Interestingly, another abnormality frequently observed was sperm with the flagellum tightly wrapped around the head. There were also differences in the degree of chromatin compaction in mature sperm (443). Further studies focusing on PRM2-deficient sperm confirmed the occurrence of head abnormalities and a less pronounced ventral flexure than that seen in wild-type mice. In some cases, the PRM2-deficient sperm had the head folded back onto the flagellum (444). In addition, these studies found, as expected, alterations in the organization and integrity of sperm DNA and, at the ultrastructural level, reduced compaction of the chromatin. Microinjection of spermatozoa that were PRM2 deficient resulted in activation of most oocytes, although few oocytes reached the blastocyst stage, suggesting damage to the paternal DNA (444). In another study (445), heterozygous Prm1+/− male mice produced with embryonic stem (ES) cells revealed that >70% of spermatozoa were abnormal. Most abnormalities (40%) were seen in the sperm flagellum, ∼20% of sperm had abnormal heads, and ∼10% of sperm had abnormalities in both. A proportion of these sperm were capable of fertilizing zona-free oocytes in vitro and to generate offspring despite having abnormal head shapes, destabilized DNA, and other abnormalities (445). In a third study, using CRISPR/Cas9-mediated gene editing in oocytes, a Prm2-deficient mouse was established (446). In contrast to a previous report (443), heterozygous males were fertile, with sperm displaying normal head morphology and motility. On the other hand, Prm2-deficient sperm showed impairment of DNA condensation and sperm acrosome formation. Sperm counts of Prm2−/− males were normal, and so were testis weight and architecture in relation to control animals, but males were sterile (446). Spermatozoa displayed severe membrane defects that resulted in lack of motility (446). One important conclusion from work in which transition nuclear protein or protamine genes have been disrupted is that, in addition to effects on the morphology of sperm nucleus, which these proteins remodel, there are also significant effects on other sperm structures such as the flagellum. This suggests a concerted action during sperm formation, even for proteins acting in the nucleus. 5.5.3. Protamine evolution and sperm head shape. Studies on sequence evolution suggest that PRM1 may have evolved from an H1-like histone (447). Protamine genes have gradually increased in length during vertebrate evolution. There is high heterogeneity between different species reflecting a rapid rate of Prm1 gene evolution (448). At gene level, variability in protamine is higher in exons than in its intron (449), whereas the 3′-end seems to be the most variable region (450). At the level of the protein, the relative proportion of arginine residues is rather constant (50–70%) but the position of arginine residues is highly variable (427). The conservation of a high arginine content may be driven by the function of protamine in sperm chromatin condensation, because protamines with a higher arginine content can form DNA complexes that are more stable. The analyses of PRM1 evolution in eutherian mammals have suggested that arginine content of this protamine is negatively correlated with head width (451), thus resulting in a more compact or elongated head, potentially contributing to higher swimming efficiency. The Prm2 gene seems to have duplicated from the Prm1 gene (426), but the question regarding the origin of PRM2 still remains unanswered. Comparative studies in rodents have revealed that cleaved-PRM2 is likely to play a role in producing small and elongated heads, making sperm more competitive, whereas there is no association between sperm head phenotype and mature-PRM2 (452, 453). Divergence in Prm2 promoter, presumably having an effect on gene regulation, was associated with differences in swimming speed, probably through an effect on head shape (454). Rats have reduced PRM2 levels that appear to be due to suppression at both the transcriptional and translational levels (455). Some hamster species (common European hamster, Chinese hamster) entirely lack PRM2 (456, 457). All primates express PRM2, but differences exist in the level of expression, and this may relate to chromatin remodeling and nuclear shape. In humans, apes, and Old and New World monkeys, the processed forms of PRM2 contain 57 amino acids (458). Ruminants (e.g., bull, ram, and goat) and boar have a Prm2 gene but appear to lack a functional PRM2 protein or show very low levels of expression (426, 427). The absence of PRM2 in these species seems to be caused by point mutations that result in a loss of arginine residues representing the DNA-binding domain and an accumulation of hydrophobic amino acids, both of which adversely impact the binding affinity of PRM2 to DNA (459). Sperm of perissodactyls (horse, zebra), lagomorphs, and elephants have two PRM2 variants (pre-PRM2 and mature-PRM2) (426). Information on PRM2 protein in sperm of other species is scarce. No species has been reported that only expresses PRM2, and, as a matter of fact, the content of PRM2 does not seem to exceed 80% in any species (427). This would indicate that the presence of PRM2 alone is not enough for proper DNA condensation. The considerable variation existing in the proportion of PRM1 and PRM2 proteins in different species may relate to the possibility that the two protamines interact with each other and bind to DNA in a consistent manner. Therefore, the predominant protamine function seems to relate to the neutralization of the negative charges along the DNA phosphodiester backbone and, as a result, to enable adjacent molecules of DNA to be packed close together. The process of compaction continues during the period of transit along the epididymis, as seen in the mouse (460), during which time the protamines form intramolecular (PRM1) and intermolecular (PRM1 and PRM2) disulfide bridges between the cysteine residues (433), resulting in the sperm DNA being condensed to a volume that is one-twentieth of the volume of a somatic nucleus. The proportion between PRM1 and PRM2 (the protamine ratio) seems to be tightly regulated within a species and varies considerably between species, in contrast to the total protamine to DNA mass in sperm nuclei, which is highly similar between different species (427). Variation in the proportion between PRM1 and PRM2 between rodent species is mainly due to variations in PRM2 content (457), and in these species there is much variation in the protamine ratio, even among closely related species (461). In stallions the proportion of PRM1 to PRM2 is ∼3 to 1 (427), and a significantly decreased Prm1-to-Prm2 mRNA ratio associates with low fertility (462). In sperm of men there are approximately equal amounts of both protamines, with aberrant protamine ratios (mRNA or protein) associating with elevated DNA fragmentation and male subfertility (20, 436). An overall comparison of species with only Prm1 expression versus those with Prm1 and Prm2 expression suggests that the latter group has a significantly enhanced likelihood of sperm DNA fragmentation (463). The shape of the sperm nucleus may be influenced by variations in the protamine ratio, since the latter may influence chromatin compaction. A study in mouse species showed that the variation seen in protamine ratios between species associates with diversity in sperm head shape (461) (FIGURE 9). In the horse, the proportion of morphologically abnormal sperm correlated negatively with the protamine mRNA ratio (which in this study was calculated as PRM2/PRM1 in contrast to the usual PRM1/PRM2), indicating that spermatozoa carrying head defects display a diminished protamine ratio (462). In the bull, abnormal spermatozoa showed either a lack of PRM1 or scattered localization in the apical/acrosomal region of the nuclei (464). In human patients, alterations in protamine ratios are usually associated with abnormal spermatozoa. Deviations of PRM1/PRM2 above or below average values are associated with infertility, and patients with altered PRM1/PRM2 are more likely to display a higher frequency of abnormal morphology, along with decreased sperm concentration and motility, and reduced penetration capacity, compared with subjects with a normal protamine ratio (465). FIGURE 9. Open in a new tab Protamine expression in mouse species and association with sperm head shape. Ratios of Prm1-to-Prm2 mRNA levels vary among closely related mouse species of the genus Mus with a parallel increase in sperm competition levels (as revealed by the proxy of relative testis mass). Based on data in Lüke et al. (452). Because of the role of protamines in nuclear condensation and the overall regulation of sperm head formation, these proteins have a major impact in the structure of the entire sperm cell and, by extension, in its function. Because differentiation of the sperm cell during spermiogenesis is also a concerted process, when both reorganization of the nucleus and the entire head and formation of the flagellum take place, concomitant alterations in head and flagellum do occur, in such a way that defects in protamine synthesis result in sperm with no or reduced motility. Moreover, defects in the cytoplasm-nuclear transport via the manchette can also lead to altered nuclear translocation of protamines. Some studies have evaluated the localization of protamines during spermatogenesis (reviewed in Ref. 466). Subcellular localization of protamines was also studied in isolated mouse (467–469) and human (470) spermatids. However, this is an area that needs further mechanistic studies. For instance, it is not known how protamines are transported from the cytoplasm to the nucleus. Additional relevant questions are: What is the protein that recruits protamines to the intramanchette transport? How is the complex protamine-recruiter-transporter targeted and delivered to the nucleus? Are protamines transported via vesicle or nonvesicle transport? It has been shown that Prm1 and Prm2 are translated at the same time (427), but are both protamines transported to the nucleus also at the same time? If there is a differential transport of PRM1 and PRM2, is this associated with PRM1-to-PRM2 ratios at different steps in spermiogenesis and this in turn with different nuclear shapes in different species? Because protamines have an important role in chromatin compaction, protection of DNA, and epigenetic marks, and this translates into how the sperm nucleus condenses and the sperm cell acquires its final shape, understanding the links between these various molecular and cellular processes is important for a thorough knowledge of determinants of fertility. 5.5.4. Heterologous expression of protamines and nuclear shape. Studies designed to examine cell reprogramming for cloning by means of somatic cell nuclear transplantation afforded the unexpected opportunity to analyze nuclear remodeling following protamine transfection. The rationale of this work was that, similarly to the reprogramming during differentiation of the male germ line, somatic cell nuclei could be reprogrammed to enhance the efficiency of cloning if submitted to a similar process under the influence of protamines. It was found that human PRM1 induces compaction of sheep fibroblast nuclei and gene silencing (471). The shape of these nuclei resembled those of highly compacted nuclei of spermatozoa. When such nuclei are microinjected into enucleated sheep oocytes, they undergo efficient protamine-to-maternal histone exchange and develop into normal blastocysts, demonstrating the functionality of such nuclei (471, 472). Moreover, embryos thus generated develop to the blastocyst stage with a quality similar to embryos derived from in vitro fertilization (IVF), thus underscoring the physiological competence of protaminized somatic cell nuclei. Subsequent work showed that mouse PRM1 can also remodel and condense fibroblast nuclei and that PRM1 from both human and mouse can induce compaction of either sheep or mouse fibroblast nuclei (472). Additional work revealed that efficiency of nuclear remodeling could be enhanced by inducing nuclear quiescence and histone hyperacetylation before transfection of fibroblasts (473). Such pretreatment promoted a much higher conversion of somatic nuclei into spermatid-like structures. Future work will likely benefit from using this system and will allow exploration of mechanisms compacting nuclei in a way that resembles that achieved in sperm. Furthermore, this approach would allow us to ask whether nuclei may be remodeled and shaped by intrinsic factors in the nucleus and its dependence on protamine sequence and expression. Thus, although this system represents a simplification of mechanisms underlying nuclear remodeling during spermatogenesis, it could be an outstanding method to examine DNA reorganization to infer processes taking place during gamete formation. 6. CONCLUSIONS AND FUTURE DIRECTIONS This review has attempted to integrate recent important findings regarding sperm bauplan and function, discussing aspects of sperm morphology (structure, shape, size) and performance in an evolutionary perspective and highlighting areas that deserve further investigation. There is a rich variety of sperm head shapes among vertebrates. Globular forms are common among fish, but there are also filiform or helical sperm heads. Amphibia have filiform, cylindrical, tapering, or long heads. Birds’ sperm heads are filiform or helical. Flattened ovoid heads are predominant in mammals, although murine and cricetid rodents generally have a falciform, sickle-shaped head. There is also considerable diversity in flagellum structure and dimensions. Changes in sperm shape and size influence sperm swimming patterns, but there is still a paucity of information on sperm biomechanics or hydrodynamic efficiency. Furthermore, considering that spermatozoa perform in aqueous environments or relatively viscous fluids in the female tract, such factors should be considered in attempts to understand sperm behavior. In addition to a general positive association between sperm shape or dimensions and swimming velocity, processes of ATP generation and consumption may also have an important bearing on sperm survival and performance. Unfortunately, there is still very limited knowledge of pathways for, and regulation of, generation of sperm ATP, with considerable differences probably existing between species. It is also worth considering that sperm may change environment as they approach the oocyte, and that the sperm pattern of movement changes as they reach the site of fertilization. These changes may influence the sperm’s bioenergetics, particularly ATP consumption. The interactions between sperm cells and female- or oocyte-derived factors thus require continued attention. Diversity of sperm morphology and performance could be examined in an evolutionary framework. In this context, the inter- and intraspecies diversity, and also intramale variation, could be analyzed to understand evolutionary history of sperm cells. Issues such as direction of trait evolution, origin of biological novelties (e.g., the appearance of head asymmetries or head appendices), sperm developmental constraints, and reversal of complex traits (such as the reappearance of simple heads in several rodent lineages) represent important questions that would command considerable attention in the future. Sperm cells may thus be good models to address these important biological phenomena. Sperm morphology and function have likely evolved under the influence of various selective forces, namely the mode of fertilization and postcopulatory sexual selection, in the framework of common descent. Vertebrates differ in their mode of fertilization, and therefore sperm and oocytes vary in the way they meet and interact. In externally fertilizing species, sperm structure is rather simple. Internal fertilization led to a series of modifications resulting in complex and longer spermatozoa with modifications in their bioenergetics and swimming behavior. In internal fertilizers, sperm cells have to swim actively, at least during part of their journey, to reach the site of fertilization. The distance to swim may vary from a few millimeters in small species to a meter or more in some large species; variation also exists in the time sperm need to survive in the female tract until the moment of fertilization. Therefore, sperm traits have likely evolved in response to both spatial and temporal determinants. Additional changes in sperm morphology or performance may relate to the interaction between sperm and oocytes that may vary in cellular or acellular coats. In internally fertilizing species, reproductive mode (oviparity vs. viviparity) may impact on characteristics of the female tract or the interval between sperm deposition and fertilization, which, in turn, may influence spermatozoa. Some progress has been made, by using comparative studies, in our understanding of the role of mode of fertilization on sperm evolution, but there are still many questions left unanswered with regard to interactions between sperm cells and the female tract and the sperm-egg interaction. Processes of sexual selection occurring after copulation may relate to sperm selection in the female tract (i.e., cryptic female choice) or competition between ejaculates to achieve fertilization (i.e., sperm competition). In the former, sperm cells of certain males may be differentially favored on the basis of genotype or phenotype. This could involve influences on sperm swimming performance via reproductive fluids, including processes of chemoattraction, or may relate to choice during oocyte-sperm interaction, including processes of ovum defensiveness. In the latter, sperm will compete in the female tract when females mate with two or more males during their period of sexual receptivity; in external fertilizers, it would happen when massive concomitant spawning takes place. For these selective processes, the evolutionary framework is rather well understood and theory and modeling have received considerable attention. On the other hand, much needs to be done to understand their underlying cellular and molecular mechanisms. Many comparative studies have focused on how these selective forces could influence sperm traits. The major effect of sperm competition on sperm numbers is now clear, but the impact on sperm morphology and performance needs additional attention. There seems to be an evident, positive, overall effect of sperm competition on whole sperm size, and proportions of different compartments; less has been explored with regard to shape. Information on influence on bioenergetics is now beginning to emerge. There seems to be a complex interaction between morphology (shape, size), bioenergetics, and swimming, so multifactorial analysis of trait evolution under sexual selection is needed. A more detailed dissection and examination of sperm kinematic parameters after sperm activation or hyperactivation and characterization of sperm subpopulations would be most helpful in the future. Functions such as capacitation and the acrosome reaction in response to natural ligands, and the underlying cellular and molecular mechanisms, need to be examined under the light of evolutionary forces. How these selective forces operate with regard to both sperm production (e.g., testis size and architecture or kinetics of sperm formation) and differentiation and cell shaping and remodeling need to be analyzed to understand the processes that prepare sperm to perform (in an external environment or in the female tract) and to acquire the competence to reach and interact with the oocyte at fertilization. Sperm formation is under a strict genetic control, which comes into play during the process of spermatogenesis in the testis and subsequent maturation after its release. Importantly, sperm development is also under the influence of environmental factors. There are still many unknowns regarding the regulation of mechanisms driving spermatogenesis. The question arises as to the possible rate of evolution of genes involved in proliferation versus differentiation, considering that the former may be more conserved than the latter, as suggested by experiments with spermatogonia transplantation. Moreover, it will be important to identify molecules essential for sperm formation that could be the subject of comparative-evolutionary studies. For example, proteins like SPAG17, ODF1, ODF2, and dynein were found to be positively selected during evolution and are also essential during spermatogenesis. Whether these proteins interact as a complex or in similar mechanisms is unknown. Several mechanisms may be responsible for nuclear shaping. Depending on the species, some may be more important than others. In mammals, particularly in rodents, the cytoskeleton structures seem to play crucial and coordinated roles. For instance, in early spermiogenesis steps (around step 5) the F-actin filaments and associated proteins forming the acroplaxome may be responsible for the first changes in nuclear shaping. This may be followed by forces generated by the microtubules and actin filaments in the manchette and the interaction with the NE through the LINC complex. Final twists could be supported by chromatin condensation due to replacement of histones by transition nuclear proteins and then protamines (FIGURE 10). In any case, the possibility that chromatin compaction promoted by protamines could have a more relevant role in head shaping should not be overlooked. FIGURE 10. Open in a new tab Schematic representation of the internal mechanisms supporting the remodeling of the spermatid head. In early spermiogenesis steps the F-actin filaments and associated proteins forming the acroplaxome may be responsible for the first changes in nuclear shaping followed by the forces generated by the microtubules and actin filaments in the manchette and the interaction with the nuclear envelope through the linker of nucleoskeleton and cytoskeleton (LINC) complex, and final twists may be supported by chromatin condensation due to replacement of histones by transition nuclear proteins (TNPs) and then protamines. Regarding regulation of chromatin condensation and how this leads to a sperm cell that is transcriptionally silent, and how epigenetic information is transferred from the sperm to the oocyte, there are still many gaps in knowledge. Questions that remain to be answered relate to the evolution of sperm protamines, their expression and transport during the final stages of sperm cell differentiation, the impact of protamine-mediated DNA compaction on sperm structure and function, and, in an overall integrative fashion, examination of the interaction between different factors that influence sperm formation and function. Characterizing and understanding the fundamental aspects of the sperm bauplan, in connection to both evolution and function, would lead to a better understanding of fertility and would allow for accurate diagnostic tests and better prognosis and treatment, and also reliable predictive assessments of both animal and human fertility. Furthermore, it would contribute to learning about the impact of environmental factors and lifestyle on the integrity of DNA and effects on future generations. 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2508	https://cloud.iditarod.com/wp-content/uploads/2018/01/Using-Logic-Lesson-Plan.pdf?0b7b1ab2	Lesson Plan Title: Using Logic to Learn Problem Solving Developed by: Heidi Sloan, 2018 Teacher on the Trail™ Discipline / Subject: Math , geography Topic: Logic, problem solving Grade Level: 3rd – high school Resources / References / Materials Teacher Needs: The Lion, the Witch, and the Wardrobe , quote by CS Lewis ww.exemplars.com/blog/education/mathematical -practice -and -problem -solving - preparing -your -teachers -for -common -core Logic puzzle s, copies for each student This is a plan that will take more than one math period to complete. Lesson Summary: Students learn step by step methods of creating logic puzzles, incorporating the geography and mushers of the Iditarod Standards Addressed: (Local, State, or National) Common Core – Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look fo r and express regularity in repeated reasoning. VA Standards Math 5.4 , 4.4, 3.4, 3.6 The student will create and solve single -step and multistep practical problems involving addition, subtraction, multiplication, and division with and without remainders of whole numbers. Online research CCSS.ELA -LITERACY.W.6.7 Conduct short research projects to answer a question, drawing on several sources and refocusing the inquiry when appropriate. Test Taking Skill: eliminate irrelevant choices Learning objectives: The student will create logic puzzles Working backward to create logic puzzles will help the student learn how to logically solve them Students will train their brains to think logically Assessment: Student rubric – classmates test the logic puzzles created by other classmates Rubric for student made logic puzzles Procedural Activities Discuss logic – quote from CS Lewis, Chronicles of Narnia , The Lion, the Witch, and the Wardrobe : “Logic!" said the Professor half to himself. "Why don't they teach logic at these schools? There are only three possibilities. Either your sister is telling lies, or she is mad, or she is telling the truth. You know she doesn't tell lies and it is obvious that she is not mad. For the moment then and unless any further evidence turns up, we must assume that she is telling the truth.” ―C.S. Lewis ,The Lion, the Witch, and the Wardrobe What i s logic? Compare to inferences in reading comprehension . Give example. Work on practice Iditarod logic puzzle with partners. Afterwards, discuss. What procedure did you use? Pass out Iditarod maps. Let students know they are going to read about at least five checkpoints on the Iditarod Trail . Give students website for Iditarod checkpoints . They should click on the map for each and zoom in via satellite v iew. N otes about the geograph y of the checkpoint can be made on their trail maps for future reference. Go to Race Center on Iditarod.com, musher bios. Students choose four to use in their logic puzzles. On a table template, mark which musher is going to be at which checkpoint. Emphasize that they are working backwards, so are beginning with the answers. Work on clues. At least one should be easy so it is guessed on the first read. The others ’ answe rs can depend on the other clues. Build clues so that at least two are not obvious until two other answers are known. Students work at creating their puzzles with a partner. When finished, rewrite without the answers. Turn in so the teacher may print out a copy to be completed by another set of classmates. The pa rtners complete a rubric for another pair’s puzzle. If something is incorrect, the original creators can correct their work and resubmit. Give students another pre -made logic puzzle to solve. Get their reactions. “Was it easier since you have already built one? ” Students answer on an exit ticket. Materials Students Need: Technological device Iditarod trail maps Two blank table template s for clues ; one for working on, one to submit Link progression: Iditarod.com , Race Center , Checkpoints or Musher Profiles Rubric for assessing peer work Technology Utilized to Enhance Learning: • Iditarod.com • Devices for Internet research • Satellite maps of checkpoints Other Information For older students, the entire race route of checkpoints can be available for clues. The logic puzzle can also have more than four options. For younger students, stick with a smaller section of the trail to create puzzle clues Students can make logical inferences about the locations of the checkpoints based on geography seen like rivers as they click on the satellite map of ea ch checkpoint . Modifications for special learners/ Enrichment Opportunities: For enrichment, students can create logic puzzle booklets to share with other classes. For special learners, work with fewer clues per puzzle. Create a Logic Puzzle Creators: ______ Clues: _________ _________ _________ _________ _________ _________ Fill in mushers’ names and the four checkpoints used in the clues. Mushers è Checkpoints ê Rubric for Logic Puzzle Clues and logic 3 The clues lead to a logical solution 2 Two or three clues lead to a partial solution 1 Not enough information in clues makes it confusing Challenge 3 Clues require thought and looking at maps of checkpoints. 2 Clues do not require map study, but make you think. 1 Clues are too obvious, making it too easy. Correct grammar 3 All clues have correct spelling, punctuation, and capitalization. 2 1 – 3 spelling, punctuation, or capitalization errors 1 4 + spelling, punctuation, or capitalization errors
2509	https://www.scribd.com/document/540872208/Padhle-Notes-Elasticity-of-Demand	Padhle Notes - Elasticity of Demand \| PDF \| Business \| Finance & Money Management Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 90%(10)90% found this document useful (10 votes) 75K views 13 pages Padhle Notes - Elasticity of Demand This document is from the website padhle.in and discusses elasticity of demand in economics class 11 commerce notes. 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2510	https://id.scribd.com/document/606547769/pasar-de-cm3-a-m3-Busqueda-de-Google	Pasar de cm3 A m3 - Búsqueda de Google \| PDF \| Litro \| Youtube Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 408 views 1 page Pasar de cm3 A m3 - Búsqueda de Google Este documento proporciona información sobre cómo convertir unidades de volumen de centímetros cúbicos (cm3) a metros cúbicos (m3). 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(6,4(# O(#(, '& 5&.,1# -"6/-1# ( M8 91#19.%O,1a&:VN5/3%.1#:&# [/.,1# ( 7&.,1# 2"6/-1# A[ ( 5 GbGH2I2ID HG\G[@D > 8 O/3.&,&#.:&# GQ&,-/-/1 '& 2130&,#/;3 '& -5< (8 &#:M/0&c1,S#`&&.#:-15 2130&,#/;3 '& -5<R5<&3 -5VR5V ]^ 2(M-%M(.&UM%# -(M-%M(.&:OM%# -5< ( M/.,1#J O(#( '& -&3.45&.,1# -8 -(M-%M('1,(-130&,#1,:-15 Búsquedasrelacionadas U(#1# ( #&+%/, O1, -('( a(-.1, '& -130&,#/;3 Aa,(--8 M/'&OM(9&,:& 2&3.45&.,1# 2"6/-1# ( 7&.,1# 2"6/-18 91%.%6&:-15 Z(-.1,&# '& -130&,#/13 M/'&#`(,&:3&. 2X7!ID \EITXTG .%'9M/6:& KN: [(# -130&,#/13&# $%& #/+%&3 #13 -15%3&# &3 a4#/-8 O`9#/-#>5('&&(#9:6M1+#O1.:-15 F/'&1.&-( > 7&'/'(# 5&O:+1:-, GQ&,-/-/1 '& U(#(, '& 5&.,18 &#:M/0&c1,S#`&&.#:-15 2130&,./, V:d+R-5< ( S+R5<> !,8 6,(/3M9:M(. 2130&,#/;3 '& -5<R5<&3 5/M4+,(51#RS/M;+,(51 ]^ 2(M-%M(.&8 -(M-%M(.&:OM%# F1M%5&3 W 7(.&5e./-(# 6(,.1M15&-1##/1:-15 GQ&,-/-/1 '& GM 5&.,1 -"6/-18 &#:M/0&c1,S#`&&.#:-15 2130&,#1,&# %3/'('&# '& 01M%5&3C 8 -(M-%M('1,(-130&,#1,:-15 2130&,./, -5< ( 5<> ?1%@%6& 91%.%6&:-15 XM -130&,./, fNN S+R5< ( +R-5<&M ,&#%M.('1 #&,e 8 6,(/3M9:M(. 2&3./5&.,1# 2"6/-1# ( 7&.,1# 2"6/-1# A-58 91%.%6&:-15 2Y[2[D TG 2X[ITXT TG[ XIHG &#:#M/'&#`(,&:3&. 2&3./5&.,1# 2"6/-1# ( 7&.,1# 2"68 91%.%6&:-15 2;51 -130&,./, 5&.,1# -"6/-1# '& +(# g[U &3 +(M13&#= > h%1,( &#:$%1,(:-15 -130&,#/13&# '/a/-/M&# '& -&3.45&.,1# -%6/-1# ( 5&.,1# -"6/-1# > ?1%@%6& 91%.%6&:-15 @D7/:'/+/.(M > /#.&5( /3.&,3(-/13(M '& 5&'/'(# &3 $%45/-( AIB .15/:'/+/.(M 2D7D 2X7!IXH \EITXTG &#:#M/'&#`(,&:3&. (6,4(# O(#(, '& 5&.,1# -"6/-1# ( M/.,1#=8 91#19.%O,1a&:VN5/3%.1#:&# &#/;3 213.&3/'1#C \3/'('&# '& 5&'/'( 7(+3/.%'&#8 M/'&OM(9&,:& 2130&,./, '& 2&3.45&.,1# -"6/-1# ( 8 91%.%6&:-15 Búsquedasrelacionadas ?1%@%6& ConvertirdeMe trosCúbico saCentímetr osCúbicos(m3acm3) Ver %6/'1 O1,C /G3-/-M1.(,&(#J VV Q%M VNKi VdK 5/M/#%(M/k(-/13&# l VJK_ 5/M 7& +%#.(:::[(# /5e+&3&# O%&'&3 &#.(, O,1.&+/'(# O1, '&,&-`1# '& (%.1,: !"# %&'()+,%-& ./.. pasar de cm3 a m3 Iniciarsesión adDownload to read ad-free Share this document Share on Facebook, opens a new window Share on LinkedIn, opens a new window Share with Email, opens mail client Copy link Millions of documents at your fingertips, ad-free Subscribe with a free trial You might also like Historia de Las Unidades de Medida No ratings yet Historia de Las Unidades de Medida 8 pages Matematica 24 de Agosto-Unidad Del Volumen No ratings yet Matematica 24 de Agosto-Unidad Del Volumen 3 pages Conversion Convertir m2 A m3 No ratings yet Conversion Convertir m2 A m3 1 page Convert Cubic CM in Cubic Meters (In Spanish) No ratings yet Convert Cubic CM in Cubic Meters (In Spanish) 1 page MEDIDAS DE VOLUMEN Exp No ratings yet MEDIDAS DE VOLUMEN Exp 7 pages Conversiones Unidades Volumen No ratings yet Conversiones Unidades Volumen 2 pages La Medida Fundamental para Medir Volúmenes y Caracteristicas No ratings yet La Medida Fundamental para Medir Volúmenes y Caracteristicas 9 pages Unidades de Volumen No ratings yet Unidades de Volumen 7 pages 2P 7 Guia Geometria Septimo2 No ratings yet 2P 7 Guia Geometria Septimo2 8 pages Conversión de Volumen Métrico No ratings yet Conversión de Volumen Métrico 4 pages Conversion Es No ratings yet Conversion Es 2 pages 4.05 Hoja de Trabajo de Masa, Volumen y Densidad No ratings yet 4.05 Hoja de Trabajo de Masa, Volumen y Densidad 4 pages Guía de Conversión de Unidades No ratings yet Guía de Conversión de Unidades 9 pages 17 Sistemas de Unidades No ratings yet 17 Sistemas de Unidades 5 pages Convertir Metros Cúbicos a Litros No ratings yet Convertir Metros Cúbicos a Litros 3 pages Taller Sobre Unidades de Volumen No ratings yet Taller Sobre Unidades de Volumen 3 pages Unidades de Capacidad para Quinto de Primaria 100% (1) Unidades de Capacidad para Quinto de Primaria 4 pages Clase 8 - Volumen - Mat Fis Aplic No ratings yet Clase 8 - Volumen - Mat Fis Aplic 3 pages Ciencias Clase 27 Abril Qquinto No ratings yet Ciencias Clase 27 Abril Qquinto 2 pages Medidas Cubicas No ratings yet Medidas Cubicas 3 pages Quim No ratings yet Quim 4 pages Problemas de Volumen 100% (1) Problemas de Volumen 4 pages Volumen Geometria No ratings yet Volumen Geometria 4 pages Los Metros Cúbicos y Los Litros Son Dos Unidades Métricas Comunes de Volumen No ratings yet Los Metros Cúbicos y Los Litros Son Dos Unidades Métricas Comunes de Volumen 4 pages Conversiones de Unidades de Medida No ratings yet Conversiones de Unidades de Medida 3 pages Taller Conversion de Volumenes No ratings yet Taller Conversion de Volumenes 7 pages Conversión de Volumen para 7° Grado No ratings yet Conversión de Volumen para 7° Grado 2 pages Guía de Volumen para Octavo Grado No ratings yet Guía de Volumen para Octavo Grado 1 page MATEMATICA15 No ratings yet MATEMATICA15 5 pages Conversión de Unidades No ratings yet Conversión de Unidades 9 pages Bulldozer CAT D10 No ratings yet Bulldozer CAT D10 15 pages Clasificacion Lineal No ratings yet Clasificacion Lineal 2 pages Calse 6 Fisica No ratings yet Calse 6 Fisica 2 pages El Volumen Metro Cubico No ratings yet El Volumen Metro Cubico 7 pages Unidades de Volumen No ratings yet Unidades de Volumen 15 pages Ejercicios Adicionales 2 No ratings yet Ejercicios Adicionales 2 3 pages Explicación de Volumen y Cubicaje de La Carga 0% (1) Explicación de Volumen y Cubicaje de La Carga 19 pages Trigonometría: Medidas Angulares No ratings yet Trigonometría: Medidas Angulares 15 pages Conceptos Basicos de Medicion 100% (1) Conceptos Basicos de Medicion 21 pages Medidas de Volumen y Peso Explicadas No ratings yet Medidas de Volumen y Peso Explicadas 9 pages Diseño Instalaciones Eléctricas Residenciales No ratings yet Diseño Instalaciones Eléctricas Residenciales 20 pages Ejercicios Conversiones No ratings yet Ejercicios Conversiones 7 pages Ficha de Unidad de Volumen 100% (1) Ficha de Unidad de Volumen 4 pages Guía de Conversión de Unidades No ratings yet Guía de Conversión de Unidades 5 pages Actividad 4 Manómetro Diferencial-Examen 1er Parcial No ratings yet Actividad 4 Manómetro Diferencial-Examen 1er Parcial 7 pages Matemáticas: Volumen y Conversiones No ratings yet Matemáticas: Volumen y Conversiones 5 pages Esc31ANO 6.2 Regla de Mezcla No ratings yet Esc31ANO 6.2 Regla de Mezcla 3 pages Magnitud y Unidades 0% (1) Magnitud y Unidades 42 pages Conversión de Unidades en Fisicoquímica No ratings yet Conversión de Unidades en Fisicoquímica 12 pages NTP PERUANA ROSCAS MÉTRICAS ISO PARA USOS GENERALES. 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2511	https://www.me.psu.edu/cimbala/me320web_Fall_2012/Links/flow_patterns.htm	Flow Patterns A streamline is a line everywhere tangent to the velocity vector at a given instant of time. (A streamline is an instantaneous pattern.) For example, consider simple shear flow between parallel plates. At some instant of time, a streamline can be drawn by connecting the velocity vector lines such that the streamline is everywhere parallel to the local velocity vector. In this example, streamlines are simply horizontal lines. A streakline is the locus of particles which have earlier passed through a prescribed point in space. (A streakline is an integrated pattern.) For example, consider simple shear flow between parallel plates. A streakline is formed by injecting dye into the fluid at a fixed point in space. As time marches on, the streakline gets longer and longer, and represents an integrated history of the dye streak. In this example, streaklines are simply horizontal lines. A pathline is the actual path traversed by a given (marked) fluid particle. (A pathline is an integrated pattern.) For example, consider simple shear flow between parallel plates. A pathline is the actual path traversed by a given (marked) fluid particle. A pathline represents an integrated history of where the fluid particle has been. In this example, pathlines are simply horizontal lines. A timeline is a set of fluid particles that form a line segment at a given instant of time. (A timeline is an integrated pattern.) For example, consider simple shear flow between parallel plates. A timeline follows the location of a line of fluid particles. A timeline represents an integrated flow pattern, since the time line continually distorts with time, as shown in the sketch. Notice the no-slip condition in action. The top of the time line moves with the top plate, i.e. at velocity V to the right. The bottom of the timeline, however, stays in the same location at all times, because the bottome plate is not moving. Note: For steady flow, streamlines, streaklines, and pathlines are all identical. However, for unsteady flow, these three flow patterns can be quite different.
2512	https://www.sciline.org/elections/surveys-polling/	Search Menu What are Reporting Resources? En español Encuestas y sondeos Vea todos nuestros recursos en español Contents Opinion polls vary enormously in structure, style, and credibility, and are easy to mis- or overinterpret. At their best, opinion polls can give an accurate snapshot of broad public sentiment on an issue. But even well-constructed polls are not particularly good at measuring small shifts in opinions over time, and their ability to predict future voter choices is decidedly mixed. Among other confounders for voter polls, people often hold off before settling on a candidate—and even then they often change their minds. The following primer provides some essentials for accurate reporting on polls and surveys. Survey methods and their general reliability Live telephone interviews with human pollsters, which reach both cell phones and landlines, are expensive but have historically been the most accurate. Probability-based online polls use randomly chosen postal addresses to reach a specific sample of people and recruit them to complete an online survey. (When the individuals are recruited by the pollster to contribute to numerous polls over time, they’re referred to as online panels.) These polls generally also have high statistical accuracy. Nonprobability online polls typically use individuals who, in response to advertisements or other general outreach efforts, have volunteered to answer a survey. These polls are less expensive than probability-based online polls, but they do not use random sampling from the entire public and so have a high risk of bias. Increasingly, pollsters are developing statistical methods and models to address this inherent problem. But these methods are complex and still evolving, so the quality of these polls varies widely and journalists should consider appropriate caveats. Automated polls (also called robopolls, or interactive voice response calls) are relatively inexpensive and include only landlines (though some pollsters don’t abide by this legal restriction), which seriously undermines the representativeness of the sample. The Associated Press Stylebook recommends that the media not report on automated polls. A variety of less-often-used survey methods exist, some of which can generate statistically accurate results but some of which should not be trusted. When in doubt, check with a polling expert. Check out our media briefing on Covering Opinion Polls and Surveys An introduction to the essentials of accurate reporting on polls and surveys. View the media briefing Things to look for in a poll Is the sampling representative? The sampled population should include individuals from all or nearly all subgroups of the population it is meant to represent. What was asked (and how was it asked)? Look carefully at the actual questions asked and make sure you’re precise in how you describe them. Remember that the order of questions can influence people’s answers, too, so it can be helpful to see the full questionnaire. Who conducted the poll? Several factors can help identify pollsters with reputations for trustworthiness. One is to check whether the organization is a member of the Transparency Initiative sponsored by the American Association for Public Opinion Research or a contributor to the Roper Center for Public Opinion Research’s data archive at Cornell University—keeping in mind that those two organizations focus on ensuring full disclosure of survey methods but do not certify the rigor of those methods, and remembering too that there are reputable pollsters who are not members of either. Who sponsored the poll? Due diligence demands that special scrutiny be applied to polls sponsored by political entities or advocacy groups, for evidence of bias. At the same time, while polls sponsored by academic institutions and large media organizations are generally designed to minimize such bias, they don’t have a perfectly clean record either. Bottom line, if you’re unfamiliar with the sponsor, do some reporting. Did the pollster weight their results? If so, how? Weighting is a statistical process by which a pollster adjusts poll data to ensure it represents the target population overall. It adjusts for the fact that it is impossible to survey everyone in a large population, as well as the reality that the fraction of people polled may differ in certain important ways from the overall population whose opinions are sought. Without weighting, polls typically under-represent younger, less educated, and non-white adults, since they are less likely to respond to polls than are other groups. Weighting is especially important when looking at state-based election polls, which often aren’t well-resourced enough to secure a sufficiently large and representative sample size. Many of the state polls that wrongly predicted the outcome of the 2016 U.S. presidential election failed in part because many people made up or changed their minds late in the campaign but also, importantly, because these state polls did not use weighting to correct for the fact that college graduates are generally more likely to respond to surveys than other adults. In key states that year, formal education was strongly associated with vote choice—something pollsters had not found to be especially important in the past. How many people were surveyed? The fewer respondents, the higher the statistical uncertainty in results. Generally speaking, 100 respondents is the minimum sample size necessary for a reportable result. But note that a poll result based on 100 respondents will have a margin of error of at least +/– 10 percentage points. Although reputable polls typically have sufficient overall sample size and appropriate weighting, some individual questions within a poll may have been asked of (or answered by) a subgroup categorized by particular demographic subsets, such as race or age. For this reason, it is possible that individual questions within a larger poll may have too few respondents to provide valid results or may be weighted inappropriately. Understanding margin of error The margin of sampling error (more typically known as the margin of error) isn’t an error in the sense of being a mistake. It’s the level of uncertainty, or the price that we pay in precision for not interviewing every single person in our target population. Margin of error is only one of many types of uncertainty in a poll’s results. Others might stem from how a question is worded or which questions are presented before others. Because margin of error is the one that’s easiest to pin down numerically, it gets the most attention. Some polls may not report margins of error. Nonprobability polls, for example, use sampling techniques not suitable for generating conventional margins of error so use other approaches to estimate uncertainty of their results. Some may report a “credibility interval,” which gives a range the pollsters believe is likely to contain the true value. If you don’t see a margin of error or credibility interval or are unsure how a poll’s uncertainty was assessed, contact the pollster or talk to a polling expert. Margins of error help you figure out the strongest possible conclusions you can draw from a poll’s results. But many people apply this measure incorrectly. For example, in mid-January 2020, some publications reported that a “majority” or “more than half” of Americans favored the President’s impeachment, conviction, and removal from office, citing a poll showing that 51% of surveyed U.S. adults answered that question affirmatively. Yet the survey results had a margin of sampling error of +/- 3.4 percentage points, which means that the true results for this population were plausibly anywhere within 3.4 percentage points on either side of the poll’s given results. At this level of precision we cannot actually conclude that more than half of U.S. adults shared this opinion. Once we account for the inherent uncertainty that comes from interviewing a sample of adults instead of the entire population, we must conclude that the most plausible range of values is between 47.6% and 54.4% (that is, between 51 minus 3.4, and 51 plus 3.4). Since it’s plausible that only about 48% of all U.S. adults favor an impeachment removal, we cannot conclude that the proportion is “more than half” or a “majority.” Ten things journalists should find out and report about polls they are covering Is this poll really a poll? Some unscrupulous campaigns and advocacy groups conduct “push polls,” which are not polls at all. Rather than aiming to tally people’s opinions, they actively seek to change people’s opinions about issues or individuals. One clue: these efforts typically fail to ask any demographic information. Who sponsored the poll and who conducted it? Assuming you are not an expert, ask some pros for an assessment of the sponsor’s reputation. At a minimum, include the name of the sponsor in your story, to hold them responsible for the work they are backing. Who was the target population? This gives important context for interpreting results. For example: a poll of likely voters, or a survey of U.S. teens ages 13 to 17. How many individuals were sampled, and where? Location is important for context, and larger sample sizes help ensure—but by no means guarantee—more reliable results. How were the interviews collected? Methodology can point to the representativeness of the sample. For example: a poll conducted by landline and cellular telephone, or interviews were conducted online and by telephone. When was the poll conducted? The date is important for interpreting results, especially in politics or other fast-changing landscapes. For example: Interviews were conducted September 15 to November 8, 2019. What was the margin of sampling error? Poll results aren’t complete without information about the uncertainty and range of plausible results. For example: The poll had a margin of sampling error of +/- 6.0 percentage points (which means that the true results are anywhere within six percentage points on either side of the given results). Was there weighting? If so, on what? For example: Results were weighted to ensure that responses accurately reflect the population’s characteristics in factors such as age, sex, race, education, and phone use. What language was used? This hints to the effort made in collecting a diverse sample. For example: The poll was conducted in English and Spanish. Consider also reminding readers of reasons why polls may not perfectly reflect reality. For example: There are many potential sources of error in polls, including the use of charged wording and the order in which questions appear. Advice from Pros When reporting poll results, avoid using decimal points or tenths of a percent—that is, report 28%, not 28.4%. (The margin of error will always be at least 1 percentage point, so tenths of a percent are effectively meaningless and misleading, suggesting that the results are more accurate than they actually are.) Don’t place too much weight on any one poll. It’s best to compare several similar polls. Neither should you presume that an aggregate of smaller polls necessarily adds to accuracy or precision. Some aggregators use more sophisticated methods than others, and the quality of their results can vary greatly. Don’t forget that even small differences in question order and choice of words can significantly alter results. (For example, asking survey participants about “euthanasia” versus “physician-assisted death”.) For some topics, consider directly quoting the question in full, so readers can see how it was asked. Remember that all poll estimates are inherently uncertain. Margins of error are typically calculated at a “95% confidence level,” which means that in about 5% of poll results — that is, five results out of 100 — the truth of what’s happening in the population will lie outside the margin of error’s bounds. Don’t assume that a poll with a large sample size has high statistical accuracy. The increased statistical precision achievable with large samples can be overwhelmed by the uncertainty and bias potentially introduced from such factors as poorly designed survey questions, flawed data collection, and improper statistical analysis. This is especially important when looking at online polls, where large numbers of respondents can be amassed cheaply. At very high levels—above 1,000 or so respondents—how many individuals were selected is less important than how they were selected and their responses analyzed. Note that terms like “nationally representative,” “organic sampling,” “next-generation sampling,” “representative of all U.S. adults,” and “random sample” may be defined differently by different pollsters. It’s best to ask precisely what is meant in each case. Also note that for election polling, the distinction between “likely voter” and “registered voter” may be especially important. Even well-designed Presidential election polls can lead you astray. One thing to watch for: nationwide polls of voters are usually designed to capture only the popular vote, not electoral college outcomes. And while poll aggregators often build electoral college weights into their models, they rely on state polls, which tend to be less well funded, smaller, and less precisely weighted than national polls. Late-deciding voters can also significantly swing elections away from poll predictions. A final, important point: Surveys are done on a vast range of topics other than electoral preferences. They provide essential data on economic activity, health status, drug use, consumer behavior, and countless other measures that are critical to responsible, democratic policy making and the intelligent allocation of resources. In many of these domains, surveys are quite good at predicting behaviors and needs. When reporting on polls and surveys, treat each with the same fairness you demand from them! Contents Best Practices for Survey Research, American Association for Public Opinion Research Election Polling Resources, American Association for Public Opinion Research More from Social Sciences, Elections Media Briefings #### Reporting on opinion polls and surveys Experts on Camera #### Dr. Lisa Bryant: Psychological factors influencing voter turnout and choices Reporting Resources #### Why it’s urgent that journalists understand the science behind vaccines, disinformation
2513	https://www.youtube.com/watch?v=ZNwJFd_zlGc	The Fascinating Relationship Between Triangles & Structural Engineering The Structures Guy 3630 subscribers 89 likes Description 7021 views Posted: 18 Apr 2020 In this video we explore what makes triangular shapes stable and rigid and how they are used in structural design. The video talks about the triangles you see around you every day, how nature uses triangles, some simple home experiments to show why triangles are rigid and some real life example of triangles use in structural engineering. Reference: Music: "Beauty Flow" Kevin MacLeod (incompetech.com) Licensed under Creative Commons: By Attribution 4.0 License 4 comments Transcript: Intro hi everybody this is a structure sky and today we're talking about the fascinating relationship between triangles and structural engineering if you look around you you will notice a lot of triangles everywhere in almost all structures triangles in structural elements provide strength and rigidity we all have studied or at least heard of the study of triangles in mathematics called trigonometry making triangles special in engineering and mathematics stick around to learn what makes triangles special construction engineering nature has figured out the Construction Engineering strength of triangles a long time ago even before humans ever existed you can see this mostly in trees well the woods and the base of the trees make a triangular shape which gives to his incredible stability under their own compulsive force we humans merely got inspired by natural to create structures that have triangles in them this is called biomimicry also patient Egyptians use a concept called the angle of repose to build the famous pyramids in Egypt angle of repose is defined to be the maximum angle of a depth to aggregative horizontal surface like the x-axis to which the material can be piled without slumping all sliding the pyramids in Egypt are built near the maximum angle of repose possible which is around 53 degrees from the x-axis you can try piling up sand and you will find out that at some point the sand will start sliding until it reaches equilibrium which will create a cone like all pyramid like shape aka a 3d triangle one other example you have seen before but probably did not think that much about his mountains mountains around us have a triangular shape for each face which make them very stable natural landforms so what makes triangles structurally stable triangle is the only shape when made by hinge connection which does allow rotation will retain a shape unless you apply a significant force on it at the top like you can see here a shape other than triangle like a square will not retain a shape when made with hinge connections and you need to hold it its hinges meaning applying external moments on it to keep its original shape however triangles don't change because all three sides prevail and support each other and do not allow the geometry to change the strength of a triangle is obtained exclusively by the axial rigidity all forces of the members you might have noticed that I have been saying hinge connections which is a key concept here because if you have a square with fixed or rigid connections it will not deform or collapse because rigid connections do not allow rotation however this rigid or fixed connection for buildings or bridges or other structural engineering projects is very expensive and something engineers avoid if possible let's conduct an experiment Home Experiments you can do at home all you need is wooden sticks playdough all PA clay or even duct tape you can see here if I hold the square it will deform and collapse because it is not stable we do not want that to happen to a building if we have went question acting on it now let's look at the hinge connected triangle you will notice that this triangle is very table if I try to apply you're on it and it will not deform or collapse in fact I created this 3d model that has all the triangles as you can see here and as a very stable model because the triangles are very rigid and efficient at distributing the loads the last thing we can do is to try to stabilize the unstable hinge connection square we achieve that by dividing the square to two triangles or more to prove this point I created a cube that looks like this as you can tell the cube is stable and will not collapse because it has triangles in each square to give it more rigidity now if we look at each face you can see that it is made of triangles by adding cross bracing to that face the spacing divide the original unstable square and two triangles structural engineers usually called boo spacing expressing because it looks like Terrell X also thus cube is a good example of a building that is a resisting a lateral load like a wind or earthquake I hope those demonstrations have shown you why triangles are stable and strong as a result structural engineers have been using triangles in a lot of applications one famous application you might have seen is trusses or joists boos are made of Apple called rural coat and a lot of triangles in the middle as you can see here those trusses are usually used for framing of buildings and warehouses stadiums airports and so much more it also bridges also use different kinds of classes to resist gravity and lateral loads also realistic domes are shells that are framed using triangular elements to distribute the loads applied on the structure there are many other examples of triangle use in structural engineering but I only mentioned few I hope you enjoyed learning more about the usefulness of triangles in structure design and enjoyed the home experiments I have shown please like and subscribe so that you don't miss next videos see you next time
2514	https://pmc.ncbi.nlm.nih.gov/articles/PMC4499780/	Global circulation patterns of seasonal influenza viruses vary with antigenic drift - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. PMC Search Update PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. 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Published in final edited form as: Nature. 2015 Jun 8;523(7559):217–220. doi: 10.1038/nature14460 Search in PMC Search in PubMed View in NLM Catalog Add to search Global circulation patterns of seasonal influenza viruses vary with antigenic drift Trevor Bedford Trevor Bedford 1 Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, Seattle, WA, USA Find articles by Trevor Bedford 1, Steven Riley Steven Riley 2 MRC Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, School of Public Health, Imperial College London, London, UK 3 Fogarty International Center, National Institutes of Health, Bethesda, MD, USA Find articles by Steven Riley 2,3, Ian G Barr Ian G Barr 4 World Health Organization (WHO) Collaborating Centre for Reference and Research on Influenza, Melbourne, Australia Find articles by Ian G Barr 4, Shobha Broor Shobha Broor 5 SGT Medical College, Hospital and Research Institute, Village Budhera, District Gurgaon, Haryana, India Find articles by Shobha Broor 5, Mandeep Chadha Mandeep Chadha 6 National Institute of Virology, Pune, India Find articles by Mandeep Chadha 6, Nancy J Cox Nancy J Cox 7 WHO Collaborating Center for Reference and Research on Influenza, Centers for Disease Control and Prevention, Atlanta, GA, USA Find articles by Nancy J Cox 7, Rodney S Daniels Rodney S Daniels 8 WHO Collaborating Center for Reference and Research on Influenza, Medical Research Council National Institute for Medical Research (NIMR), London, UK Find articles by Rodney S Daniels 8, C Palani Gunasekaran C Palani Gunasekaran 9 King Institute of Preventive Medicine and Research, Guindy, Chennai, India Find articles by C Palani Gunasekaran 9, Aeron C Hurt Aeron C Hurt 4 World Health Organization (WHO) Collaborating Centre for Reference and Research on Influenza, Melbourne, Australia 10 Melbourne School of Population and Global Health, University of Melbourne, Parkville VIC 3010, Australia Find articles by Aeron C Hurt 4,10, Anne Kelso Anne Kelso 4 World Health Organization (WHO) Collaborating Centre for Reference and Research on Influenza, Melbourne, Australia Find articles by Anne Kelso 4, Alexander Klimov Alexander Klimov 7 WHO Collaborating Center for Reference and Research on Influenza, Centers for Disease Control and Prevention, Atlanta, GA, USA Find articles by Alexander Klimov 7, Nicola S Lewis Nicola S Lewis 11 Department of Zoology, University of Cambridge, Cambridge, UK Find articles by Nicola S Lewis 11, Xiyan Li Xiyan Li 12 WHO Collaborating Center for Reference and Research on Influenza, National Institute for Viral Disease Control and Prevention, China CDC, Beijing, China Find articles by Xiyan Li 12, John W McCauley John W McCauley 8 WHO Collaborating Center for Reference and Research on Influenza, Medical Research Council National Institute for Medical Research (NIMR), London, UK Find articles by John W McCauley 8, Takato Odagiri Takato Odagiri 13 WHO Collaborating Center for Reference and Research on Influenza, National Institute of Infectious Diseases, Tokyo, Japan Find articles by Takato Odagiri 13, Varsha Potdar Varsha Potdar 6 National Institute of Virology, Pune, India Find articles by Varsha Potdar 6, Andrew Rambaut Andrew Rambaut 3 Fogarty International Center, National Institutes of Health, Bethesda, MD, USA 14 Institute of Evolutionary Biology, University of Edinburgh, Edinburgh, UK 15 Centre for Immunology, Infection and Evolution, University of Edinburgh, Edinburgh, UK Find articles by Andrew Rambaut 3,14,15, Yuelong Shu Yuelong Shu 12 WHO Collaborating Center for Reference and Research on Influenza, National Institute for Viral Disease Control and Prevention, China CDC, Beijing, China Find articles by Yuelong Shu 12, Eugene Skepner Eugene Skepner 11 Department of Zoology, University of Cambridge, Cambridge, UK Find articles by Eugene Skepner 11, Derek J Smith Derek J Smith 11 Department of Zoology, University of Cambridge, Cambridge, UK 16 Department of Viroscience, Erasmus Medical Center, Rotterdam, The Netherlands Find articles by Derek J Smith 11,16, Marc A Suchard Marc A Suchard 17 Department of Biostatistics, UCLA Fielding School of Public Health, University of California, Los Angeles, CA, USA 18 Department of Biomathematics David Geffen School of Medicine at UCLA, University of California, Los Angeles, CA, USA 19 Department of Human Genetics, David Geffen School of Medicine at UCLA, University of California, Los Angeles, CA, USA Find articles by Marc A Suchard 17,18,19, Masato Tashiro Masato Tashiro 13 WHO Collaborating Center for Reference and Research on Influenza, National Institute of Infectious Diseases, Tokyo, Japan Find articles by Masato Tashiro 13, Dayan Wang Dayan Wang 12 WHO Collaborating Center for Reference and Research on Influenza, National Institute for Viral Disease Control and Prevention, China CDC, Beijing, China Find articles by Dayan Wang 12, Xiyan Xu Xiyan Xu 7 WHO Collaborating Center for Reference and Research on Influenza, Centers for Disease Control and Prevention, Atlanta, GA, USA Find articles by Xiyan Xu 7, Philippe Lemey Philippe Lemey 20 Department of Microbiology and Immunology, Rega Institute, KU Leuven – University of Leuven, Leuven, Belgium Find articles by Philippe Lemey 20, Colin A Russell Colin A Russell 21 Department of Veterinary Medicine, University of Cambridge, Cambridge, UK Find articles by Colin A Russell 21 Author information Article notes Copyright and License information 1 Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, Seattle, WA, USA 2 MRC Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, School of Public Health, Imperial College London, London, UK 3 Fogarty International Center, National Institutes of Health, Bethesda, MD, USA 4 World Health Organization (WHO) Collaborating Centre for Reference and Research on Influenza, Melbourne, Australia 5 SGT Medical College, Hospital and Research Institute, Village Budhera, District Gurgaon, Haryana, India 6 National Institute of Virology, Pune, India 7 WHO Collaborating Center for Reference and Research on Influenza, Centers for Disease Control and Prevention, Atlanta, GA, USA 8 WHO Collaborating Center for Reference and Research on Influenza, Medical Research Council National Institute for Medical Research (NIMR), London, UK 9 King Institute of Preventive Medicine and Research, Guindy, Chennai, India 10 Melbourne School of Population and Global Health, University of Melbourne, Parkville VIC 3010, Australia 11 Department of Zoology, University of Cambridge, Cambridge, UK 12 WHO Collaborating Center for Reference and Research on Influenza, National Institute for Viral Disease Control and Prevention, China CDC, Beijing, China 13 WHO Collaborating Center for Reference and Research on Influenza, National Institute of Infectious Diseases, Tokyo, Japan 14 Institute of Evolutionary Biology, University of Edinburgh, Edinburgh, UK 15 Centre for Immunology, Infection and Evolution, University of Edinburgh, Edinburgh, UK 16 Department of Viroscience, Erasmus Medical Center, Rotterdam, The Netherlands 17 Department of Biostatistics, UCLA Fielding School of Public Health, University of California, Los Angeles, CA, USA 18 Department of Biomathematics David Geffen School of Medicine at UCLA, University of California, Los Angeles, CA, USA 19 Department of Human Genetics, David Geffen School of Medicine at UCLA, University of California, Los Angeles, CA, USA 20 Department of Microbiology and Immunology, Rega Institute, KU Leuven – University of Leuven, Leuven, Belgium 21 Department of Veterinary Medicine, University of Cambridge, Cambridge, UK ✉ Correspondence and requests for materials should be addressed to C.A.R. (car44@cam.ac.uk) Author Contributions C.A.R. and T.B. conceived the research. C.A.R. and T.B. drafted the manuscript with substantial support from P.L. and S.R.. I.G.B., S.B., M.C., N.J.C., R.S.D., C.P.G., A.C.H., A.K., A.K., X.L., J.W.M., T.O., V.P., Y.S., M.T., D.W., and X.X. coordinated and produced the influenza surveillance data. T.B. performed the modeling and data analyses along with C.A.R., S.R., P.L., M.A.S. and A.R.. T.B. created the figures. All authors discussed the results and contributed to the revision of the final manuscript. Issue date 2015 Jul 9. Reprints and permissions information is available at www.nature.com/reprints. Users may view, print, copy, and download text and data-mine the content in such documents, for the purposes of academic research, subject always to the full Conditions of use: PMC Copyright notice PMCID: PMC4499780 EMSID: EMS62811 PMID: 26053121 The publisher's version of this article is available at Nature Abstract Understanding the spatio-temporal patterns of emergence and circulation of new human seasonal influenza virus variants is a key scientific and public health challenge. The global circulation patterns of influenza A/H3N2 viruses are well-characterized1-7 but the patterns of A/H1N1 and B viruses have remained largely unexplored. Here, based on analyses of 9,604 hemagglutinin sequences of human seasonal influenza viruses from 2000–2012, we show that the global circulation patterns of A/H1N1 (up to 2009), B/Victoria, and B/Yamagata viruses differ substantially from those of A/H3N2 viruses. While genetic variants of A/H3N2 viruses did not persist locally between epidemics and were reseeded from East and Southeast (E-SE) Asia, genetic variants of A/H1N1 and B viruses persisted across multiple seasons and exhibited complex global dynamics with E-SE Asia playing a limited role in disseminating new variants. The less frequent global movement of influenza A/H1N1 and B viruses coincided with slower rates of antigenic evolution, lower ages of infection, and smaller less frequent epidemics compared to A/H3N2 viruses. Detailed epidemic models support differences in age of infection, combined with the less frequent travel of children, as likely drivers of the differences in the patterns of global circulation, suggesting a complex interaction between virus evolution, epidemiology and human behavior. Owing to the frequency and severity of human seasonal influenza A H3N2 virus epidemics, recent work has focused on the global circulation dynamics of H3N2 viruses1-7. Studies have shown that, each year, H3N2 epidemics worldwide result from the introduction of new genetic variants from East and Southeast (E-SE) Asia, where viruses circulate via a network of temporally overlapping epidemics1,2,4,5, rather than local persistence1,3,6,7. In addition to H3N2, H1N1 viruses and two antigenically diverged lineages of influenza B viruses, B/Victoria/2/1987-like (Vic) and B/Yamagata/16/1988-like (Yam), circulate among humans with lower but substantial disease burdens8,9. Despite their importance, the global circulation dynamics of former seasonal H1N1 viruses (preceeding the 2009 pandemic) and B viruses have been largely neglected. Given that influenza A and B viruses cause similar symptoms and evolve by similar mechanisms of immune escape, we hypothesized that each would follow similar patterns of global circulation, with new genetic variants originating in E-SE Asia that rapidly replace existing genetic variants. To test this hypothesis we compared the global circulation patterns of the hemagglutinin (HA) genes of H3N2, former seasonal H1N1, Vic, and Yam viruses. We assembled datasets of HA sequences with complete HA1 domains for each subtype from the World Health Organization Global Influenza Surveillance and Response System and the Influenza Research Database10 covering 2000–2012. To reduce the impact of surveillance biases, we subsampled these data to more equitable spatiotemporal distributions, resulting in datasets comprising 4006 H3N2, 2144 H1N1, 1999 Vic, and 1455 Yam HA sequences (Extended Data Fig. 1). Though deficient in viruses from Africa and Eastern Europe, these are the most geographically and temporally comprehensive seasonal influenza virus datasets assembled to date. By estimating temporally-resolved phylogenetic trees for each subtype, we revealed faster rates of nucleotide mutation and amino acid substitution in H3N2 and H1N1 than in the B viruses (consistent with previous work11,12), but more genealogical diversity in B viruses than A viruses (Extended Data Table 1). This inverse relationship between evolutionary rate and genealogical diversity is expected if increased mutation rate correlates with antigenic drift13 and drives increased adaptive evolution, thus purging HA genetic diversity14. By inferring geographic ancestry using Bayesian phylogeographic methods15, we found a consistent pattern for H3N2 viruses (Fig. 1a) in which viruses worldwide rapidly coalesce to the trunk of the tree (average time to trunk = 1.42 years), with trunk viruses mostly originating from E-SE Asia (Extended Data Fig. 2a). This finding is consistent with previously reported patterns1,2,4,5, with E-SE Asia acting as the source population for epidemics worldwide. Figure 1. Maximum clade credibility trees for primary datasets of 4006 H3N2 viruses (a), 2144 H1N1 viruses (b), 1999 Vic viruses (c) and 1455 Yam viruses (d). Open in a new tab Branch tips are colored by geographic region of virus collection; internal branches are colored by geographic region as inferred by Bayesian phylogeographic methods (region colors in persistence insets). In b) nodes 1-3 indicate co-circulating clades that diverged in 2004. In c), nodes 1 and 2 indicate divergent clades of viruses from Asia, colored vertical bars indicate antigenic variants shown in Extended Data Figure 5a (green: B/Malaysia/2506/2004-like, red: B/Hubei Songzi/52/2008-like, other post-2008 viruses: B/Brisbane/60/2008-like). The inset to the top left of each tree shows duration of region-specific persistence measured as the waiting time in years for a virus to leave its geographic region of origin. Circles represent mean persistence across sampled viruses, while lines show the inter-quartile range of persistence across sampled viruses. Region “China”, shows the combined persistence estimate for North China and South China together. In addition to China and Southeast Asia, India frequently contributed viruses to the trunk of the tree suggesting that the global circulation of H3N2 viruses is maintained by an E-SE Asian network that includes India. India’s role in the global dissemination of H3N2 viruses may have been similar historically, but India-wide influenza surveillance only began in 2004. There were brief periods, notably the 2007–2008 Northern Hemisphere winter, when regions outside E-SE Asia contributed to the trunk of the H3N2 tree. However, these instances were rare and trunk viruses from outside E-SE Asia descended directly from viruses within E-SE Asia (Fig. 1a). Quantifying the average ancestry of strains from each geographic region in the 3 years prior to sampling showed prominent roles for China, India, and Southeast Asia in seeding epidemics in all regions (Extended Data Fig. 3). Surprisingly, the global circulation patterns of former seasonal H1N1 viruses differed substantially from those observed for H3N2 viruses (Fig. 1). Like H3N2, most lineages of H1N1 viruses eventually coalesced with viruses from E-SE Asia and India. However, this coalescence was slower than for H3N2 viruses with prolonged co-circulation of geographically segregated H1N1 lineages (Fig. 1b, Extended Data Figs. 3 and 4). Geographic segregation of H1N1 viruses was particularly pronounced beginning in 2004/2005, with the emergence of three co-circulating genetic lineages (Fig. 1b, nodes 1-3) that each independently acquired HA mutations leading to antigenic evolution from the A/New Caledonia/20/1999-like phenotype to the A/Solomon Islands/3/2006-like phenotype. These lineages circulated in Southeast Asia (node 1), China (node 2) and India (node 3), with the Indian lineage eventually spreading worldwide prior to the emergence of H1N1pdm09 viruses. Phylogeographic analyses of B Vic and Yam viruses revealed further differences from H3N2 viruses with lineages frequently circulating outside of E-SE Asia for several years without evidence of seeding from E-SE Asia (Fig. 1c,d). Prominent examples include the seeding of the North American 2006/2007 Vic season directly from 2005/2006 North American viruses and the seeding of the North American 2001/2002 Yam season directly from 2000/2001 North American viruses (Extended Data Fig. 4). Similarly, lineages of viruses within E-SE Asia commonly circulated exclusively in E-SE Asia for more than 1 year. These long circulating E-SE Asian lineages were most apparent for Vic viruses where two lineages (Fig. 1c, nodes 1 and 2) persisted independently in China and SE Asia for over 5 years without spreading to other regions and led to the co-circulation of three distinct Vic antigenic variants in different parts of the world during 2007/2008 (Extended Data Fig. 5a). Patterns of persistence of genetic variants differed by subtype and region, with H3N2 viruses persisting regionally for an average of ~6 months, H1N1 for ~9 months, Vic for ~13 months and Yam for ~12 months. H3N2 viruses showed comparably short durations of persistence across the world (Fig. 1), with the exceptions of India and China. Patterns within China were characterized by North and South lineages contributing jointly to persistence as combining North and South phylogeny nodes resulted in substantially greater persistence estimates than from North or South lineages alone (Fig. 1). For H3N2, evidence for joint contributions to persistence by region pairs that exclude China is comparatively weak (Extended Data Fig. 6a, Supplementary Information). For Vic and Yam, the mean duration of persistence was longer than for H3N2 or H1N1 in most regions, particularly in India and China where mean durations were >2 years (Fig. 1, Extended Data Fig. 4). Duration of regional persistence correlated with the proportion of virus originating from that region (Extended Data Fig. 6b) and observed phylogeographic patterns were robust to subsampling assumptions (Supplementary Information, Extended Data Table 2). To investigate differences in the global migration patterns of H3N2, H1N1 and B viruses, we used the spatiotemporally-resolved phylogenies to estimate the amounts of virus movement between regions (Fig. 2). Rates of movement between pairs of regions were highly correlated between viruses with Spearman correlation coefficients ranging from 0.65 (H3N2 vs Yam) to 0.75 (H3N2 vs H1N1), suggesting similar global connectivity networks for all viruses. However, while the overall structure of the migration network was similar, H3N2 viruses moved between regions more frequently than H1N1 and B viruses (migration events per lineage per year H3N2 = 1.96, H1N1 = 1.27, Vic = 0.93, Yam = 0.97, Extended Data Table 1). Figure 2. Estimates of mean pairwise virus migration rate. Open in a new tab Line thickness between regions indicates average number of migration events per lineage per year. Arrowhead size indicates the strength of directionality of migration. For clarity, only arrows corresponding to migration rates greater than 0.25 events per lineage per year are shown. Circle area indicates the global proportion of ancestry deriving from each region. We hypothesized a relationship between rates of global movement and rates of antigenic drift: though rates of genetic evolution were similar for H3N2 and H1N1 viruses, both H1N1 and B viruses evolved antigenically more slowly than H3N2 viruses13 (Extended Data Table 1). We also hypothesized that lower rates of immune escape for B and H1N1 compared with H3N2 would lead to: younger average ages of infection as children increasingly comprise the largest pool of susceptible individuals; and smaller, less frequent epidemics owing to smaller populations of susceptible individuals13. These differences are consistent with results from several community-based cohort studies that found that children were more frequently infected with B viruses than were adults8,16,17. Age of infection data covering 2002–2011 from Australia show that H1N1 and B viruses infect younger individuals than H3N2 viruses (Extended Data Fig. 5b-d, median age of infection H3N2 = 30y, H1N1 = 20y, B = 16y) and epidemiological data from Australia and the United States show reduced size and frequency of H1N1 and B epidemics compared to H3N2 (Extended Data Fig. 5f-i). Differences in age of infection may explain differences in global circulation as children travel long distances much less frequently than adults (Extended Data Fig. 5e). A previous study hypothesized that age-specific patterns of infection could lead to differences in contact rates and the spread of influenza types within the United States over the course of a single season18. Here, we hypothesized that differential global air travel provides a plausible mechanism by which H1N1 and B viruses show increased genetic differentiation and reduced rates of global migration across multiple seasons, compared to H3N2 viruses. To test the impact of differences in age distribution of infection on global patterns of virus movement, we constructed a multi-patch transmission model. We modeled two scenarios for host movement: 1. age-independent mixing between patches; 2. age-stratified mixing with host movement derived from air travel passenger age data (Extended Data Fig. 5e). In the age-independent scenario, model parameters only differed in rate of antigenic mutation, leading to differences in observed rates of antigenic drift among viruses and hence epidemic size and frequency (Extended Data Fig. 7). Faster antigenic drift resulted in greater incidence and more adult infections (Fig. 3a,b), but only modest differences in virus lineage movement (Fig. 3c, B-like viruses differ from H3-like viruses by a factor of 1.2), consistent with slightly faster spread of antigenically novel strains. However, age-stratified mixing between patches intensified the effect of antigenic drift on migration rate and created differences in rates of movement between patches more consistent with those observed for H3N2 vs H1N1 and B (Fig. 3c, B-like viruses differ from H3-like by a factor of 1.6). In the scenario with faster antigenic drift, infections were more mobile due to greater frequency of adult infection, causing a knock-on effect on rates of viral movement. The model also suggests that the differences in patterns of regional persistence observed in the phylogenies might be shaped by a combination of differences in rates of antigenic evolution and variation in amplitude of epidemic seasonality, with slowly evolving viruses persisting longer than rapidly evolving viruses at low amplitudes of seasonal forcing (Extended Data Fig. 8a, Supplementary Information). Figure 3. Relationship of antigenic drift to incidence (a), proportion of childhood infections (b), and geographic migration rate (c), in a multi-strain multi-region model of influenza transmission. Open in a new tab Black points represent outcomes from a model in which children and adults travel between regions at equal rates. Red points represent outcomes from a model in which adults travel between regions at 5.26× the rate of children (Extended Data Fig. 5e). Solid black and red lines represent LOESS fits to the data. With 2 travel scenarios, 7 mutation rates and 8 replicates, there are 112 individual stochastic simulations (Extended Data Fig. 7). Antigenic drift was measured in cartographic units13 per year (see Methods). In a) attack rate was measured as proportion of the total population infected yearly. In c) migration rate was measured in terms of migration events per lineage per year. In the model, varying transmission rate rather than antigenic mutation rate also resulted in differences in the observed rate of antigenic drift, with higher transmission resulting in faster drift (Extended Data Fig. 8b). The relationship between antigenic drift rate and migration rate is similar regardless of whether drift is modulated by mutation rate or transmission rate (Extended Data Fig. 8b). This finding is in line with theoretical work showing that epidemiological processes can influence influenza virus evolution19,20. However, there are important virological differences between influenza viruses that are likely to impact the efficiency and tempo at which antigenic variation is generated and fixed, which could in turn affect epidemiology21-24 (Supplementary Information). Regardless of the underlying drivers, there is a remarkable correspondence in model behavior, quantified as a stable relationship between observable rate of antigenic drift and global circulation patterns. The patterns of epidemic spread observed here suggest that differences in ages of infection could explain patterns of global circulation across a variety of human viruses. Methods Sequence data Hemagglutinin (HA) coding sequences for influenza A H3N2 viruses, former seasonal H1N1 viruses (preceding the 2009 pandemic), and influenza B virus lineages Victoria (Vic) and Yamagata (Yam) collected by the World Health Organization (WHO) Global Influenza Surveillance and Response Network including the National Institute of Virology, Pune, India between 2000 and 2012 were combined with human seasonal influenza virus sequences (minimum length = 984 base pairs) covering 2000 to 2012 from the Influenza Research Database10. After removing duplicate strains and strains overly divergent based on root-to-tip distances, the data set contained 9139 H3N2 sequences, 3789 H1N1 sequences, 2577 Vic sequences and 1821 Yam sequences. Sampling locations for these sequences were parsed from strain names. Sequences were grouped into 9 geographic regions: USA/Canada, South America, Europe, India, North China, South China, Japan/Korea, Southeast Asia and Oceania. Specifics of this partitioning are shown in Extended Data Figure 1. Groups were chosen to maximize available sequences within each region while still providing enough geographic diversity to ensure nearly global coverage. Sequences from Africa, Central America, the Middle East and Russia were excluded because of a lack of sufficient numbers of sequences to provide comparable estimates to other regions. In the raw sequence data, some regions, such as the USA, were over-represented. Additionally, more recent years were over-represented compared to years at the start of the study period. In order to control for these sampling biases, we subsampled the raw data randomly by location and time to create a more equitable spatiotemporal distribution. The USA had consistently more sequences available every year from 2000 to 2012, thus in order to maintain similar total numbers of sequences for each region across the entire study period it was necessary to sample fewer sequences per year from the USA. We selected 50 sequences per region per year (40 for USA/Canada) for H3N2 and 80 sequences per region per year (45 for USA/Canada) for H1N1, Vic and Yam. This subsampling resulted in largely similar sequence counts across years and across regions for each virus, but overall more H3N2 sequences than H1N1 or B sequences, with 4006 H3N2 sequences, 2144 H1N1 sequences, 1999 Vic sequences and 1455 Yam sequences (Extended Data Fig. 1). When selecting subsampled sequences we first selected sequences with full day-month-year collection dates and then longer sequences over sequences with less precise dates or shorter sequences. HA sequence data for 1630 H3N2 isolates, 1600 H1N1 isolates, 1394 Vic isolates and 881 Yam isolates have been deposited in the Influenza Research Database10 and accession numbers for all sequences used provided as Supplementary Information. Phylogeographic inference Time-resolved phylogenetic trees were estimated for H3N2, H1N1, Vic and Yam using BEAST v1.8.125 and incorporated the SRD06 nucleotide substitution model26, a coalescent demographic model with constant effective population size and a strict molecular clock across branches. A strict molecular clock was chosen based on finding strong correlations between date of sampling and evolutionary distance in all datasets, as estimated by Path-O-Gen v1.427. Using a strict clock also reduced the risk of model over-parameterization (for example, for the complete H3N2 data set with a relaxed clock, there would be 2 × 4006 – 2 = 8010 branch-specific rates). Samples with imprecise dates (known only to the month or to the year) had their dates of sampling estimated assuming a uniform prior within the known temporal bounds28. Markov Chain Monte Carlo (MCMC) was run for 600 million steps and trees were sampled every 5 million steps after allowing a burn-in of 100 million steps, yielding a total sample of 100 trees for H1N1, Vic and Yam. With significantly more samples, H3N2 required a longer chain to converge. Here, MCMC was run in parallel for 2 chains, each with 650 million steps sampled every 3 million steps with a burn-in of 500 million steps and samples across chains combined, yielding a total of 100 sampled trees. These trees were treated as independent draws from the posterior space of trees when subsequently used in the robust counting and phylogeographic analyses29. Evolutionary rates in Extended Data Table 1 were estimated using the ‘renaissance’ counting methods of Lemey et al.30. Phylogeographic patterns were estimated using a discrete-state continuous time Markov chain (CTMC) model, in which transition rates were estimated between each pair of regions15. We assumed a non-reversible transition model31 consisting of 72 separate rate parameters, each with a Bayesian stochastic search variable selection (BSSVS) indicator variable, and a separate overall rate of geographic transition. We assumed an exponential prior with mean of 1 for each transition rate, a negative binomial prior with mean of 9 and standard deviation of 9 for the total number of non-zero rates and an exponential prior with mean of 1 migration event per lineage per year for the overall geographic transition rate. MCMC was run for 12 million steps with a burn-in of 2 million steps, and parameters sampled every 10,000 steps and trees sampled every 100,000 steps, yielding a total sample of 1000 parameter states and 100 trees on which estimates were based. Pairwise migration rate estimates had an effective sample size (ESS) of 350 at the minimum and most had ESS greater than 500. This procedure yielded posterior trees with the geographic states of internal nodes resolved. We analyzed these posterior trees using the program PACT v0.9.532 to compute the following summary statistics: a) genealogical diversity14, measuring the average time it takes for two randomly chosen contemporaneous lineages to coalesce, b) time to the most recent common ancestor (TMRCA)14, measuring the average time it takes for all contemporaneous lineages to find a common ancestor, c) genealogical F ST, measuring the degree of population structure in contemporaneous lineages calculated as F ST = (π b – π w)/π b, where π w is genealogical diversity between randomly sampled lineages from the same geographic region and π b is genealogical diversity between randomly sampled lineages from different geographic regions, d) persistence, measuring the average number of years for a tip to leave its sampled location, walking backwards up the phylogeny, e) migration rate, measuring the average number of migration events over the phylogeny divided by total tree length to give migration events per lineage per year, f) trunk location through time4, measuring the posterior distribution across sampled phylogenies of the trunk geographic state, where the trunk is defined as all branches ancestral to viruses sampled within 1 year of the most recent sample, g) region-specific ancestral geographic history, measuring the distribution of geographic locations of tips belonging to a particular region traced backwards in time through the phylogeny averaged across sampled phylogenies. Statistics (a), (b), (c), (f), and (g) were calculated across 0.1 year genealogical windows. These procedures gave an estimate of credible intervals for inferred ancestral locations across posterior phylogeographic reconstructions. Code and data availability Sequence data has been deposited with the Influenza Research Database10 and accession numbers provided as Source Data. The entire bioinformatic pipeline, including data subsampling, preparing XML files for BEAST, setting up PACT analyses and rendering figures is available at Analysis and data files are archived on the Dryad Digital Repository under DOI 10.5061/dryad.pc641. Surveillance, travel and age-structure data We investigated epidemic size and frequency using virological isolation data between 2000 and 2012 collected by the WHO Collaborating Centre for Reference and Research on Influenza at the Victorian Infectious Diseases Reference Laboratory (VIDRL), Melbourne, Australia and the Centers for Disease Control and Prevention, Atlanta, USA (Extended Data Fig. 5f–i). These isolations were categorized by date of sampling and by virus type: H3N2, H1N1, Vic, or Yam. The data from VIDRL also contained information on patient age. The age structure of incidence was estimated by constructing a distribution of age of infection from individuals >5 y (owing to the overrepresentation of <5 year old patients for all subtypes) (Extended Data Fig. 5b–d). Median age of infection was 30 y (H3N2), 20 y (H1N1) and 16 y (B) and mean age of infection was 33.9 y (H3N2), 23.1 y (H1N1) and 23.2 y (B). Median age of infection was significantly different for H3N2 vs H1N1 (P = 4.6 × 10−29, Mann-Whitney U test), H3N2 vs B (P = 1.2 × 10−62) and H1N1 vs B (P = 0.041). The patient age data from VIDRL were potentially biased by testing strategy and the generally higher severity of H3N2 virus infections. Children and working age adults were more likely to be tested than the elderly but the greater severity of H3N2 virus infections might spread and flatten the patient age distribution. For this reason we additionally tested excluding individuals >65 y and recalculating summary statistics, finding median ages of infection of 27 y (H3N2), 19 y (H1N1) and 15 y (B) and mean age of infection as 28.0 y (H3N2), 22.2 y (H1N1) and 20.3 y (B). We classified children as 0-15 years and adults as 16 years and older, and estimated proportion of childhood infections as 30% (H3N2), 52% (H1N1) and 60% (B). There are potentially other biases specific to individual sentinel physicians and hospitals that could affect sample collection. However, the estimate derived from the VIDRL data that ~60% of influenza B virus infections occur in children is consistent with other estimates (reviewed in Glezen et al.8). Other studies similarly corroborate the estimates of lower age of infection for H1N1 viruses as compared to H3N233,34. Additionally, we analyzed the distribution of ages of ~102.5 million air passengers traveling through London Heathrow and London Gatwick airports in 2011 (Extended Data Fig. 5E) reported by Civil Aviation Authority of the UK35. Assuming that children of ages 0 to 15 make up 17% of the UK population (Office of National Statistics), this distribution suggests that children engage in air travel at 19% the rate of adults. For the modeling described below, we estimated age-structured contact rates following the empirical mixing data provided by Mossong et al.36. These contact matrices were previously validated in modeling pertussis epidemiology37. We simplified the Mossong et al. mixing matrices to record child-to-child contacts, child-to-adult contacts, adult-to-child contacts and adult-to-adult contacts, where children were defined to be 0 to 15 and adults to be 16 or over. This resulted in the following mixing matrix where rates are relative to child-to-child contact rates. Epidemiological modeling An individual-based model of influenza evolution and epidemiology was constructed following methods presented in Bedford et al.38. The model used here is identical to Bedford et al. except where specified below. The present implementation used a linear-strain space39,40, in which virus phenotype is represented by a continuous variable and cross-immunity between viruses is a function of distance between viruses in this space. We parameterized the model to compare scenarios of age-structured mixing between regions and to compare viruses with different rates of antigenic drift. The model was simulated for 120 years with daily time steps and the first 100 years discarded to allow equilibrium to be reached. We modeled a metapopulation with individuals equally divided into three regions (North, Tropics, South). Individual’s ages were tracked throughout the simulation and those less than 16 years old were classified as children and those 16 or older were classified as adults. Transmission occurred by mass action, with transmission rates modified by regional compartment and by age compartment. Thus, for example, the force of infection into children in the Tropics followed where β j is the seasonally forced contact rate in region j, α ac represents adult-to-child transmission, m i represents between-region transmission in age class i, I ij represents the number of infecteds in age class i in region j, S ij represents the number of susceptibles in age class i in region j, and N j represents the total number of hosts in region j. The northern and southern regions were seasonally forced in opposite phase with a sinusoidal function following ε, while the tropics had no seasonal forcing. Each virus possessed a one-dimensional antigenic phenotype ϕ v and after recovery a host ‘remembered’ its infecting phenotype. For each contact event, the Euclidean distance from infecting phenotype ϕ v was calculated to each of the phenotypes in the host immune history ϕ h 1, …, ϕ h n. Here, one unit of antigenic distance was designed to roughly correspond to a twofold dilution of antiserum in a hemagglutination inhibition (HI) assay41. The probability ρ that infection occurred after exposure was proportional to the distance d to the closest phenotype in the host immune history, following ρ = min{d s, 1}. Each day there was a chance μ that an infection mutates to a new phenotype. This mutation rate represents a phenotypic rate, rather than genetic mutation rate, and can be thought of as arising from multiple genetic sources. When a mutation occurred, the virus’s phenotype was moved either left or right randomly and mutation size sampled from an exponential distribution with mean step size δ avg. Epidemiological parameters for the baseline epidemiological scenario with notation following Bedford et al.38 were: Base transmission rate β = 0.88 per day Duration of infection 1/ν = 5 days Birth/death rate = 1/50 years Total population size N = 45 million Seasonal forcing in north and south ε = 0.15 Antigenic scaling s = 0.07 Antigenic mutation rate μ = 0.5 to 6.5 × 10 −4 per day Average mutation size δ avg = 0.3 units Child-to-child transmission α cc = 1.00 Child-to-adult transmission α ca = 0.21 Adult-to-child transmission α ac = 0.21 Adult-to-adult transmission α aa = 0.26 Child between-region transmission m c = 0.0020 Adult between-region transmission m a = 0.0020 In the model with age-stratified mixing with host movement derived from air travel passenger age data, child between-region transmission m c was 0.0011 and adult between-region transmission m a was 0.0060. In the course of the simulation, the underlying infection history of who infects whom was recorded and output as a complete infection tree. Without ample within-host diversity owing to chronic infection, the complete infection tree also generated a fully observed phylogenetic tree. Examining geographic location across the phylogenetic tree allowed us to directly calculate migration rate as total migration events observed (transitions from one region to another) divided by total opportunity (tree length). The simulation was parameterized to model H3-like, H1-like and B-like behavior (Extended Data Fig. 7) by modulating antigenic mutation rate μ in the primary analysis (Fig. 3) or transmission rate β as a secondary analysis (Extended Data Fig. 8b). Values for μ and β were chosen based on observed attack rate, proportion of childhood infections and antigenic drift rate. Source code for the simulation is available at and parameter and results files are available at Extended Data Extended Data Figure 1. Spatial distribution of 4006 H3N2, 2144 H1N1, 1999 Vic and 1455 Yam samples. Open in a new tab Circle area is proportional to the number of sequenced viruses originating from a location. Color indicates assignment to one of 9 geographic regions. Extended Data Figure 2. Inferred location of the trunk of H3N2 tree through time in the primary dataset (a) and in a smaller secondary dataset (b). Open in a new tab Colored width at each time point indicates the posterior support for viruses from a particular geographic location comprising the trunk of the phylogenetic tree. Colors correspond to colored circles in persistence insets in Figure 1. The secondary datasets consist of 1391 H3N2 viruses, 1372 H1N1 viruses, 1394 Vic viruses and 1240 Yam viruses. Extended Data Figure 3. Average inferred geographic history of region-specific samples for H3N2, former seasonal H1N1, Vic and Yam viruses from 2000 to 2012. Open in a new tab In each panel, phylogeny tips belonging to a particular region were collected and their phylogeographic histories traced backwards in time averaging across the phylogenetic tree to combine all viruses within each region. The x-axis shows number of years backward in time from phylogeny tips from a particular region and the y-axis shows the geographic make up as stacked histogram of the ancestors of these tips, where region color-coding corresponds to the legend in Figure 1. For example, the top left panel shows the ancestry of USA and Canadian H3N2 viruses. At x = 0, all of these viruses are still in the USA or Canada and so an unbroken yellow band takes up the entire y. However, at x = 1 year, a number of different geographic regions appear on the y. This indicates that, 1 year back, ancestors of USA and Canadian viruses are primarily found in Southeast Asia, India and South China. The pattern in the top right panel shows that the ancestors of USA and Canadian Yam viruses more often remain in the USA or Canada with approximately 50% of ancestors remaining 1 year back. Each panel is constructed by averaging across region-specific tips within a tree, but also across sampled posterior trees. Extended Data Figure 4. Maximum clade credibility (MCC) trees for region-specific samples from USA/Canada, India and South China for H3N2, H1N1, Vic and Yam viruses. Open in a new tab Each tree only contains viruses from a particular geographic region and thus tips are all a single color within a tree. Branch and trunk coloring have been retained from Figure 1 to highlight the inferred geographic ancestry of each lineage. Extended Data Figure 5. Antigenic map of Vic viruses primarily collected in 2008 (a), age distribution of infections for H3N2 (b), H1N1 (c) and B (d) in Australia 2000–2011, age distribution of ~102.5 million passengers at London Heathrow and London Gatwick airports during 2011 (e), timeseries of virological characterizations from 2000 to 2012 of viruses from the USA by US CDC and from Australia by VIDRL for H3N2 (f), H1N1 (g), Vic (h) and Yam (i). Open in a new tab In (a), the positions of strains (colored circles) and antisera (uncolored squares) are fit such that the distances between strains and antisera in the map represent the corresponding hemagglutination inhibition (HI) measurements with the least error following Smith et al.41 using data on Vic viruses from the WHO Collaborating Centre for Reference and Research on Influenza at the Centers for Disease Control and Prevention, Atlanta, Georgia, USA. Strains are colored by antigenic cluster. Genetic clades corresponding to each antigenic cluster are marked with colored vertical bars in Fig 1c. The spacing between grid lines is one unit of antigenic distance corresponding to a twofold dilution of antiserum in the HI assay. In (f) to (i), virological characterizations are a surrogate for epidemiological activity that allow for accurate discrimination among H3N2, H1N1, Vic, and Yam viruses. These data generally reflect the relative magnitudes and frequencies of epidemics but in some cases will inflate magnitudes of very small epidemics due to preferential characterization of subtypes circulating at low levels. Extended Data Figure 6. Combined persistence estimates across pairs of regions for H3N2, H1N1, Vic and Yam (a) and Spearman correlation of a region’s persistence vs the region’s contribution to phylogenetic ancestry for H3N2, H1N1, Vic and Yam (b). Open in a new tab In (a) and (b), persistence is measured as the average waiting time in years for a sample to leave its origin backwards in time in the phylogeny, with waiting time averaged across tips within a tree and across sampled posterior trees. In each panel of (a), the diagonal shows persistence within each of the 9 study regions and within the combined region of ‘China’, for which nodes in North China and in South China were considered to belong to a single region. The estimates along the diagonal are equivalent to the means shown in Figure 1. Off-diagonal elements show persistence estimates for pairwise combinations of regions. For example, the off-diagonal for North and South China is exactly equivalent to the diagonal element for ‘China’ and the off diagonal for ‘China’ and India represents mean persistence when combining nodes from North China, South China and India. In (b), origin proportion is measured as the proportion of the time that a region is represented when tracing back 2 or more years from each tip in the phylogeny, averaged across tips within a tree and across sampled posterior trees. Spearman’s ρ is not significant for any individual virus. However, the probability of observing 4 instances where each virus has a ρ of at least 0.32 is significant (P = 0.0017, bootstrap resampling test). Extended Data Figure 7. Simulation results for a model parameterized for slow antigenic drift (a), moderate antigenic drift (b), and fast antigenic drift (c). Open in a new tab Colors represent geographic regions with tropics in blue, north in yellow and south in red. Region-specific incidence patterns are shown in terms of cases per 100,000 individuals per week, patterns of antigenic drift in terms of increasing antigenic distance (roughly proportional to log 2 HI units) over time and in the geographically labeled phylogeny. The parameterized antigenic mutation rate is 0.00015 antigenic mutations per infection per day in (a), 0.00035 in (b) and 0.00055 in (c), while the realized antigenic drift rate is 0.29 antigenic units per year in (a), 0.58 in (b) and 1.19 in (c). Between-region mixing is 5.26× faster in adults. Each panel shows output from a single simulation selected from the 112 shown in Figure 3, and is intended to show model behaviors over a range of parameters, not necessarily the behavior of particular viruses. Extended Data Figure 8. Simulation results showing relationship between antigenic drift and persistence as a function of seasonality (a) and simulation results showing the effects of modulating transmission rate β on model behavior (b). Open in a new tab In (a), the seasonal forcing parameter ε follows ε = 0.00 (no forcing), ε = 0.04, ε = 0.08 and ε = 0.12 (moderate seasonal forcing). Points represent outcomes from a model in which adults travel between regions at 5.26× the rate of children. Solid black lines represent linear fits to the data. With 4 seasonality scenarios, 7 mutation rates and 8 replicates, there are 224 individual simulations shown. Persistence is measured as the average time in years taken for a tip to leave its region of origin going backwards in time, up the tree. In (b), transmission rate β in contacts per day is varied and compared to its effect on observed antigenic drift (in antigenic units per year), attack rate per year, proportion of childhood infections and migration rate between regions (in events per viral lineage per year). One antigenic unit is roughly equivalent to one log 2 HI unit. Black points represent outcomes from a model in which children and adults travel between regions at equal rates. Red points represent outcomes from a model in which adults travel between regions at 5.26× the rate of children. Solid black and red lines represent LOESS fits to the data. With 2 travel scenarios, 7 transmission rates and 8 replicates, there are 112 individual simulations shown. Extended Data Table 1. Posterior mean estimates (and 95% highest posterior density intervals) across viruses for evolutionary and phylogeographic parameters. \| Statistic \| H3N2 \| H1N1 \| Vic \| Yam \| :---: :---: \| Total nucleotide rate \| 5.0 (4.8–5.2) \| 4.4 (4.2–4.6) \| 2.7 (2.6–2.9) \| 2.8 (2.6–3.0) \| \| Nonsynonymous rate \| 2.2 (2.2–2.3) \| 1.9 (1.9–2.0) \| 1.0 (0.9–1.1) \| 1.0 (0.9–1.0) \| \| Synonymous rate \| 2.8 (2.7–2.9) \| 2.6 (2.5–2.7) \| 1.8 (1.8–1.9) \| 1.8 (1.8–1.9) \| \| Antigenic drift rate† \| 1.01 (0.98–1.04) \| 0.62 (0.56–0.67) \| 0.42 (0.32–0.52) \| 0.32 (0.25–0.39) \| \| Diversity‡ \| 3.03 \| 4.59 \| 5.46 \| 6.83 \| \| TMRCA§ \| 3.89 \| 4.53 \| 5.22 \| 7.62 \| \| F ST∥ \| 0.30 \| 0.36 \| 0.37 \| 0.36 \| \| Persistence¶ \| 0.50 (0.48–0.54) \| 0.79 (0.73–0.85) \| 1.07 (0.98–1.16) \| 1.03 (0.88–1.21) \| \| Migration rate# \| 1.99 (1.85–2.10) \| 1.27 (1.18–1.37) \| 0.93 (0.86–1.02) \| 0.98 (0.83–1.14) \| Open in a new tab Evolutionary rates are measured in terms of 10−3 substitutions per site per year. † Antigenic drift rates are from Bedford et al.13table 2, and measures cartographic drift per year in terms of twofold dilution of antiserum in a hemagglutination inhibition (HI) assay. ‡ Diversity of contemporaneous lineages is measured as average time in years for two randomly sampled lineages to share a common ancestor. § Time to the most recent common ancestor (TMRCA) of contemporaneous lineages is measured as the average time in years for all lineages to find a common ancestor. ∥ F ST compares diversity within regions to diversity between regions, so that F ST = (π b – π w) / π b. ¶ Persistence is calculated as the average number of years for a tip to leave its sampled location, walking backwards up the phylogeny. Migration rate is calculated as migration events per lineage per year between any two regions. Extended Data Table 2. Posterior mean estimates across viruses and datasets of regional persistence, migration rate and geographic population structure. \| Statistic \| Dataset \| H3N2 \| H1N1 \| Vic \| Yam \| :---: :---: :---: \| \| Persistence \| Primary§ \| 0.51 \| 0.79 \| 1.07 \| 1.03 \| \| Persistence \| Secondary∥ \| 0.53 \| 0.75 \| 1.16 \| 1.11 \| \| Persistence \| Alternative¶ \| 0.50 \| 0.76 \| 1.28 \| 1.12 \| \| Migration rate† \| Primary§ \| 1.96 \| 1.27 \| 0.93 \| 0.97 \| \| Migration rate† \| Secondary∥ \| 1.89 \| 1.33 \| 0.86 \| 0.90 \| \| Migration rate† \| Alternative¶ \| 2.00 \| 1.32 \| 0.78 \| 0.89 \| \| F ST‡ \| Primary§ \| 0.30 \| 0.36 \| 0.37 \| 0.36 \| \| F ST‡ \| Secondary∥ \| 0.29 \| 0.35 \| 0.36 \| 0.37 \| \| F ST‡ \| Alternative¶ \| 0.29 \| 0.34 \| 0.36 \| 0.35 \| Open in a new tab Regional persistence is measured as the average waiting time in years for a sample to leave its origin backwards in time in the phylogeny. † Migration rate is measured as migration events per lineage per year. ‡ F ST compares diversity within regions to diversity between regions, so that F ST = (π b – π w) / π b. § The primary datasets consist of 4006 H3N2 viruses, 2144 H1N1 viruses, 1999 Vic viruses and 1455 Yam viruses. ∥ The secondary datasets consist of 1391 H3N2 viruses, 1372 H1N1 viruses, 1394 Vic viruses and 1240 Yam viruses. ¶ The alternative datasets consist of 1967 H3N2 viruses, 1439 H1N1 viruses, 1756 Vic viruses and 1223 Yam viruses divided into 10 geographic regions (USA/Canada, South America, Europe, India, Japan/Korea, Southeast Asia, Oceania, China, Central America and Africa). Supplementary Material 1 NIHMS62811-supplement-1.pdf (206.1KB, pdf) 2 NIHMS62811-supplement-2.xlsx (655.5KB, xlsx) Acknowledgments T.B. was supported by a Newton International Fellowship from the Royal Society and through NIH U54 GM111274. S.R. was supported by MRC (UK, Project MR/J008761/1), Wellcome Trust (UK, Project 093488/Z/10/Z), Fogarty International Centre (USA, R01 TW008246-01), DHS (USA, RAPIDD program), NIGMS (USA, MIDAS U01 GM110721-01) and NIHR (UK, Health Protection Research Unit funding). The Melbourne WHO Collaborating Centre for Reference and Research on Influenza was supported by the Australian Government Department of Health and thanks N. Komadina and Y.-M. Deng. The Atlanta WHO Collaborating Center for Surveillance, Epidemiology and Control of Influenza was supported by the U.S. Department of Health and Human Services. NIV thanks A.C. Mishra, M. Chawla-Sarkar, A.M. Abraham, D. Biswas, S. Shrikhande, AnuKumar B, and A. Jain. Influenza surveillance in India was expanded, in part, through US Cooperative Agreements (5U50C1024407 and U51IP000333) and by the Indian Council of Medical Research. M.A.S. was supported through NSF DMS 1264153 and NIH R01 AI 107034. Work of the WHO Collaborating Centre for Reference and Research on Influenza at the MRC National Institute for Medical Research was supported by U117512723. P.L., A.R. & M.A.S were supported by EU Seventh Framework Programme [FP7/2007-2013] under Grant Agreement no. 278433-PREDEMICS and ERC Grant agreement no. 260864. C.A.R. was supported by a University Research Fellowship from the Royal Society. Footnotes The authors declare no competing financial interests. Online Content Methods, along with Extended Data Tables and Figures, and Supplementary Information are available in the online version of the paper; References unique to these sections appear only in the online paper. 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Supplementary Materials 1 NIHMS62811-supplement-1.pdf (206.1KB, pdf) 2 NIHMS62811-supplement-2.xlsx (655.5KB, xlsx) ACTIONS View on publisher site PDF (3.0 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Methods Extended Data Supplementary Material Acknowledgments Footnotes References Associated Data Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
2515	https://gist.github.com/tsemerad/5053378	Python function for converting degrees, minutes, seconds (DMS) coordinates to decimal degrees (DD). · GitHub Skip to content Search Gists Search Gists All gistsBack to GitHubSign inSign up Sign inSign up You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert {{ message }} Instantly share code, notes, and snippets. tsemerad/dms_to_dd.py Created February 28, 2013 01:15 Show Gist options Download ZIP Star 5(5)You must be signed in to star a gist Fork 0(0)You must be signed in to fork a gist Embed Embed Embed this gist in your website. Share Copy sharable link for this gist. Clone via HTTPS Clone using the web URL. Learn more about clone URLs Clone this repository at <script src=" Save tsemerad/5053378 to your computer and use it in GitHub Desktop. CodeRevisions 1Stars 5 Embed Embed Embed this gist in your website. Share Copy sharable link for this gist. Clone via HTTPS Clone using the web URL. Learn more about clone URLs Clone this repository at <script src=" Save tsemerad/5053378 to your computer and use it in GitHub Desktop. Download ZIP Python function for converting degrees, minutes, seconds (DMS) coordinates to decimal degrees (DD). Raw dms_to_dd.py This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters Show hidden characters def dms_to_dd(d, m, s): dd=d+float(m)/60+float(s)/3600 return dd Copy link lsxinh commented Feb 25, 2021• edited Loading Uh oh! There was an error while loading. Please reload this page. thanks, just found it def dms_to_dd(d, m, s): if d=='-': dd = float(d) - float(m)/60 - float(s)/3600 else: dd = float(d) + float(m)/60 + float(s)/3600 return dd Sorry, something went wrong. Uh oh! There was an error while loading. Please reload this page. Sign up for freeto join this conversation on GitHub. Already have an account? Sign in to comment Footer © 2025 GitHub,Inc. Footer navigation Terms Privacy Security Status Community Docs Contact Manage cookies Do not share my personal information You can’t perform that action at this time.
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2517	https://study.com/academy/lesson/the-irrational-root-theorem-definition-application.html	Irrational Root Theorem Application & Examples - Lesson \| Study.com Log In Sign Up Menu Plans Courses By Subject College Courses High School Courses Middle School Courses Elementary School Courses By Subject Arts Business Computer Science Education & Teaching English (ELA) Foreign Language Health & Medicine History Humanities Math Psychology Science Social Science Subjects Art Business Computer Science Education & Teaching English Health & Medicine History Humanities Math Psychology Science Social Science Art Architecture Art History Design Performing Arts Visual Arts Business Accounting Business Administration Business Communication Business Ethics Business Intelligence Business Law Economics Finance Healthcare Administration Human Resources Information Technology International Business Operations Management Real Estate Sales & Marketing Computer Science Computer Engineering Computer Programming Cybersecurity Data Science Software Education & Teaching Education Law & Policy Pedagogy & Teaching Strategies Special & Specialized Education Student Support in Education Teaching English Language Learners English Grammar Literature Public Speaking Reading Vocabulary Writing & Composition Health & Medicine Counseling & Therapy Health Medicine Nursing Nutrition History US History World History Humanities Communication Ethics Foreign Languages Philosophy Religious Studies Math Algebra Basic Math Calculus Geometry Statistics Trigonometry Psychology Clinical & Abnormal Psychology Cognitive Science Developmental Psychology Educational Psychology Organizational Psychology Social Psychology Science Anatomy & Physiology Astronomy Biology Chemistry Earth Science Engineering Environmental Science Physics Scientific Research Social Science Anthropology Criminal Justice Geography Law Linguistics Political Science Sociology Teachers Teacher Certification Teaching Resources and Curriculum Skills Practice Lesson Plans Teacher Professional Development For schools & districts Certifications Teacher Certification Exams Nursing Exams Real Estate Exams Military Exams Finance Exams Human Resources Exams Counseling & Social Work Exams Allied Health & Medicine Exams All Test Prep Teacher Certification Exams Praxis Test Prep FTCE Test Prep TExES Test Prep CSET & CBEST Test Prep All Teacher Certification Test Prep Nursing Exams NCLEX Test Prep TEAS Test Prep HESI Test Prep All Nursing Test Prep Real Estate Exams Real Estate Sales Real Estate Brokers Real Estate Appraisals All Real Estate Test Prep Military Exams ASVAB Test Prep AFOQT Test Prep All Military Test Prep Finance Exams SIE Test Prep Series 6 Test Prep Series 65 Test Prep Series 66 Test Prep Series 7 Test Prep CPP Test Prep CMA Test Prep All Finance Test Prep Human Resources Exams SHRM Test Prep PHR Test Prep aPHR Test Prep PHRi Test Prep SPHR Test Prep All HR Test Prep Counseling & Social Work Exams NCE Test Prep NCMHCE Test Prep CPCE Test Prep ASWB Test Prep CRC Test Prep All Counseling & Social Work Test Prep Allied Health & Medicine Exams ASCP Test Prep CNA Test Prep CNS Test Prep All Medical Test Prep College Degrees College Credit Courses Partner Schools Success Stories Earn credit Sign Up Copyright Math Courses Holt McDougal Algebra 2: Online Textbook Help Irrational Root Theorem Application & Examples Contributors: Cynthia Helzner, Russell Frith Author Author: Cynthia Helzner Show more Instructor Instructor: Russell Frith Understand the irrational theorem, how it works, and its application. Learn how to solve irrational roots of the polynomials through the given examples. Updated: 11/21/2023 Table of Contents What is the Irrational Root Theorem? Applications of Irrational Root Theorem Lesson Summary Show FAQ How to solve irrational roots? To solve for any roots, whether rational or irrational, set each factor equal to zero, then solve for x. Irrational roots will come in conjugate pairs by the quadratic formula. What are irrational and rational roots? A rational root is a solution to an equation or function if the root can be expressed as a fraction of integers without using any special symbols. An irrational root is a solution to an equation or function if the root cannot be written as a fraction of integers. What is the irrational root theorem equation? The irrational root theorem states that if a polynomial has an irrational root in the form of a + sqrt(b) or a - sqrt(b), then the conjugate of that root is also a root of the polynomial. The irrational root theorem equation is thus a - sqrt(b), where a is the original root's rational part and sqrt(b) is the original root's irrational part, including its sign. Create an account LessonTranscript VideoQuizCourse An error occurred trying to load this video. Try refreshing the page, or contact customer support. You must c C reate an account to continue watching Register to view this lesson Are you a student or a teacher? I am a student I am a teacher Create Your Account To Continue Watching As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed. Get unlimited access to over 88,000 lessons. Try it now Already registered? Log in here for access Go back Resources created by teachers for teachers Over 30,000 video lessons & teaching resources—all in one place. 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Your next lesson will play in 10 seconds 0:04 Irrational Root Theorem 1:29 Examples 3:04 Lesson Summary QuizCourseView Video OnlySaveTimeline 27K views Recommended lessons and courses for you Related LessonsRelated Courses ##### Quartic Function Formula, Equation & Examples 5:30 ##### Cubic Equations \| Overview, Formula & Functions 8:33 ##### Cubic Function \| Definition, Equation & Examples 7:33 ##### Solving Cubic Equations \| Factoring & Graphing 7:34 ##### Solving Cube Root Equations 5:05 ##### Grouping to Factor Cubic Equations 6:41 ##### Using the Greatest Common Factor to Solve Cubic Equations 4:01 ##### Solving Equations on the CLEP Scientific Calculator 5:30 ##### Multiplication Property of Equality \| Overview, Example & Formula 4:05 ##### Algebraic Equation \| Definition, Formula & Examples 6:16 ##### Addition Property of Equality \| Definition, Explanation & Example 3:51 ##### Multi-Step Equations with Fractions & Decimals \| Steps & Examples 6:41 ##### Division Property of Equality \| Definition & Examples 3:51 ##### Solving Subtraction Word Problems with Two or More Variables 5:55 ##### Subtraction Property of Equality \| Overview & Examples 3:54 ##### How to Solve Equations on a Calculator 9:22 ##### Writing an Equation \| Overview & Examples 5:22 ##### Literal Equations \| Definition, Formula & Examples 5:09 ##### Algebraic Model Definition, Applications & Examples 7:02 ##### Substitution Property \| Definition & Examples 2:57 ##### Trigonometry: High School ##### Prentice Hall Algebra 2: Online Textbook Help ##### Glencoe Algebra 1: Online Textbook Help ##### McDougal Littell Algebra 2: Online Textbook Help ##### Glencoe Math Connects: Online Textbook Help ##### Glencoe Math Course: Online Textbook Help ##### Number Properties: Help & Review ##### HiSET Mathematics: Prep and Practice ##### THEA Study Guide and Test Prep ##### ACCESS Math: Online Textbook Help ##### Saxon Calculus Homeschool: Online Textbook Help ##### GED Math: Quantitative, Arithmetic & Algebraic Problem Solving ##### AP Calculus AB & BC: Exam Prep ##### ELM: CSU Math Study Guide ##### Study.com SAT Study Guide and Test Prep ##### SAT Subject Test Mathematics Level 1: Practice and Study Guide ##### SAT Subject Test Mathematics Level 2: Practice and Study Guide ##### NY Regents Exam - Integrated Algebra: Test Prep & Practice ##### CSET Math Subtest I (211) Study Guide and Test Prep ##### Supplemental Math: Study Aid What is the Irrational Root Theorem? ------------------------------------ A rational number can be written as a fraction of integers. Rational numbers do not require any special symbols like π or square root signs. Examples of rational numbers are -2, -0.1, 0, 2 3, 45.5, and 100. Note that any integer can be rational because it can be written as itself over 1. Recurring decimals are also rational (nonrepeating decimals are not). In contrast, an irrational number cannot be expressed as a fraction of integers. Examples of irrational numbers are π,e, 3, and 1−5. The last example in that list has a rational part (1), but the entire number cannot be expressed as a fraction of integers, so the number is irrational. Rational roots are rational solutions to a function, while irrational roots are irrational solutions. The irrational root theorem or irrational conjugate theorem states that if an irrational number in the form of a+b or a−b is a root of a polynomial, then that number's conjugate is also a root of the polynomial. The conjugate of an irrational number has the same integer part, if any, but the irrational part has the opposite sign as it originally did. For example, the conjugate of 2+7 is 2−7 because the rational part (2) is the same while the irrational part (7) has the opposite sign. Moreover, the conjugate of−2 is 2 (Here, the rational part of the numbers is 0). Applications of Irrational Root Theorem --------------------------------------- When learning about the irrational root theorem (i.e., the irrational conjugate theorem), doing some of its example problems is helpful. Solve these examples using the irrational root theorem when needed: Example 1 (some roots given) If two of the roots of a cubic polynomial are -2 and 4−3, what is the complete list of roots for this polynomial from least to greatest? Solution First, see if there are any irrational roots. The first root given, -2, is rational because it can be expressed as the fraction of integers (−2 1), so it does not have a conjugate. In contrast, 4−3 is irrational, so its conjugate must also be a root of the polynomial. Keep the rational part and reverse the sign of the irrational part. Thus, the third root is 4+3, which is the conjugate of 4−3. 4−3≈2.27, which is greater than -2, and 4+3, must be greater than 4−3. Thus, the complete list of roots for this polynomial from least to greatest is -2, 4−3, 4+3. Figure 1 shows the graph of the function. The three roots are the three x-intercepts because a function's real roots are its x-intercepts. Figure 1: The graph of the function in Example 1. Example 2 (function given) What are the roots of the polynomial x 3−3 x 2−2 x+6? Solution If the problem allows the use of graphing technology, then graph the function on a graphing calculator or online graphing tool to see if there are any rational x-intercepts, such as integers. The graph of the polynomial is shown in Figure 2. Figure 2: The graph of y = x^3 - 3x^2 - 2x + 6. The graph shows that there is an x-intercept at (3, 0), so x = 3 is a root. Thus, (x - 3) is a factor of the polynomial. Use polynomial long division or synthetic division to divide the original polynomial by (x - 3). The result is x 2−2. To find the remaining roots, set x 2−2 equal to 0 and solve for x: x 2−2=0 x 2=2 x=±2. Note that these two irrational roots are conjugates of each other, which agrees with the irrational root theorem. Thus, the complete list of roots is −2,2, and 3. Alternatively, if the problem does not allow graphing technology, then use factoring by grouping to get the cubic polynomial: x 3−3 x 2−2 x+6 x 2(x−3)−2(x−3)(x 2−2)(x−3). Then, set each factor equal to 0 and solve for x: x - 3 = 0 x = 3 and x 2−2=0 x 2=2 x=±2. This produces the same list of roots obtained from the first method. Example 3 (graphing) Sketch a quartic function with roots at -1, 2, and −3+2 and give its equation. Solution The third root listed is irrational, so its conjugate, −3−2, is the function's fourth root. Note that the sign of the rational part (-3) did not change when finding the conjugate. A quartic function cannot have more than four roots. Thus, there are no more roots, so the complete list of roots is -1, 2, −3−2≈−4.4, and −3+2≈−1.6. The four roots are the four x-intercepts, so plot those four points. Next, list the factors, which are (x - r), where r is a root: x−(−1)=x+1 x−2 x−(−3−2)=x+3+2 x−(−3+2)=x+3−2. So, the polynomial thus far is (x+1)(x−2)(x+3+2)(x+3−2). Using the FOIL method on the first two factors yields x 2−x−2. Conjugates can be FOILed as a difference of squares: (x+3+2)(x+3−2)((x+3)+2)((x+3)−2)(x+3)2−(2)2 x 2+6 x+9−2 x 2+6 x+7. Thus, the polynomial can be written as (x 2−x−2)(x 2+6 x+7). Distribute to get the function into polynomial form, which is x 4+5 x 3−x 2−19 x−14. Next, the y-intercept is just the constant term, -14, so plot (0, -14) as the y-intercept. A positive quartic function's end behavior is up to the left and right, so connect the intercepts in that pattern. All factors have an odd exponent (in this case, 1), so the function will cut through the x-axis at the x-intercepts rather than bounce off of the x-axis. The graph of the polynomial is shown in Figure 3. Figure 3: The graph of y = x^4 + 5x^3 - x^2 - 19x - 14. Lesson Summary -------------- A rational number can be expressed as a fraction of integers without using any special symbols, such as square root symbols. All integers and repeating decimals are rational. An irrational number cannot be expressed as a fraction of integers. Rational roots are rational solutions to a function, while irrational roots are irrational solutions. Irrational roots can be thought of as a rational part plus or minus an irrational part, such as −2+3 or 1−2 2. The rational part can be 0, which results in irrational numbers like −2 and π. According to the irrational root theorem or irrational conjugate theorem, for any irrational root in the form of a+b or a−b, the number's conjugate is also a root of the polynomial. The conjugate of an irrational number has the same rational part (if any) with an opposite sign on the irrational part. For example, the conjugate of −3−2 is −3+2. The irrational root theorem helps find the remaining roots when some roots are given. Finding the roots is needed for graphing polynomials because the real roots of a function are the function's x-intercepts (the constant term in the polynomial is the y-intercept). If the function is given, then the irrational root theorem is not needed; simply factor the polynomial, set each factor equal to 0, and solve for x to find the roots. Video Transcript Irrational Root Theorem A polynomial with integer coefficients has the following roots: Can you find at least two additional roots? The answer is yes! The irrational root theorem can be used to find additional roots for a polynomial. Let a and b be two numbers. Now, a is a rational number, meaning that the numbers to the right of the decimal point are finite or repeat in some pattern (like 0.818181...). For example, integers are rational numbers. The square root of b, on the other hand, is an irrational number. So the numbers to the right of the decimal are infinite and non-patterned (like π, for example). Adding, subtracting, multiplying, or dividing an irrational number by a rational number always gives an irrational number (the division by a rational number excludes division by zero). The irrational root theorem states that if the irrational sum of a + √b is the root of a polynomial with rational coefficients, then a - √b, which is also an irrational number, is also a root of that polynomial. Ley y = a + √b, where √b is an irrational number. The conjugate of y is a - √b. Examples Using the irrational root theorem, we can answer the question posed at the start of this lesson about that polynomial. Can you find at least two additional roots? Since the unknown polynomial has integer coefficients, then those coefficients are rational. Since 6 is not a perfect square, then √6 is an irrational number. Note that √2 is irrational, and 2 multiplied by an irrational number is also irrational. Furthermore, a rational number plus an irrational number is always irrational. Thus, two additional roots for the polynomial in question would be: As a second example, consider the roots of the polynomial: f(x) = x 3 - 4 x 2 - 11 x + 2. Note that the coefficients of f(x) are integers and thus rational numbers. Using a graph of this function, we can find that -2 is a rational root for this equation. We can use synthetic division to factor this polynomial to get: f(x) = (x + 2)(x 2 + 6 x + 1). We can factor the quadratic term using the quadratic formula to get the remaining two roots for f(x). Notice that those two roots are both irrational and conjugates. Lesson Summary The irrational root theorem is a root discovery method reserved for polynomials that have only rational coefficients and at least one irrational root. When this is the case, irrational roots of this specific type occur in conjugate pairs. That is, when given two numbers, a and b, where a is rational, b is not a perfect square, and a + √b is a root for a given polynomial, then a - √b is also root. a - √b is the conjugate of a + √b. Register to view this lesson Are you a student or a teacher? I am a student I am a teacher Unlock your education See for yourself why 30 million people use Study.com Become a Study.com member and start learning now. Become a member Already a member? Log in Go back Resources created by teachers for teachers Over 30,000 video lessons & teaching resources—all in one place. 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Try it now Holt McDougal Algebra 2: Online Textbook Help 14 chapters 233 lessons Chapter 1 Holt McDougal Algebra 2 Chapter 1: Foundations for Functions Types of Numbers & Its Classifications 6:56 min Cardinality of a Set \| Definition & Examples 4:13 min Set Builder Notation \| Definition, Symbols & Examples 6:50 min Interval Notation \| Definition, Rules & Examples 9:08 min Introduction to Groups and Sets in Algebra 13:36 min Commutative Property \| Definition, Examples & Applications 3:53 min Square Root \| Definition, Formula & Examples 7:05 min Factoring Radical Expressions 4:45 min Simplifying Square Roots \| Overview & Examples 4:49 min Addition and Subtraction Using Radical Notation 3:08 min Translating Words into Algebraic Expressions \| Phrases & Examples 6:31 min Evaluating Algebraic Expressions \| Rules & Examples 7:27 min Combining Like Terms in Algebraic Expressions 7:04 min Simplifying and Solving Exponential Expressions 7:27 min Properties of Exponents \| Formula & Examples 5:26 min How to Define a Zero and Negative Exponent 3:13 min Simplifying Expressions with Exponents \| Overview & Examples 4:52 min Scientific Notation \| Definition, Conversion & Examples 6:49 min Functions: Identification, Notation & Practice Problems 9:24 min Domain & Range of a Function \| Definition, Equation & Examples 8:32 min Function in Math \| Definition & Examples 7:57 min Transformations: How to Shift Graphs on a Plane 7:12 min Reflection Rules in Math \| Graph, Formula & Examples 6:07 min Vertical & Horizontal Compression of a Function 6:29 min Parent Function \| Graphs, Types & Examples 3:56 min Chapter 2 Holt McDougal Algebra 2 Chapter 2: Linear Functions Solving Equations Using Both Addition and Multiplication Principles 6:21 min Solving Equations Containing Parentheses 6:50 min Solutions to Systems of Equations \| Overview & Examples 4:45 min Proportion \| Definition, Formula & Types 6:05 min Ratios & Rates \| Differences & Examples 6:37 min Similar Triangles \| Definition, Application Problems & Examples 6:23 min Linear Equations \| Definition, Formula & Solution 7:28 min Forms of a Linear Equation \| Overview, Graphs & Conversion 6:38 min Undefined & Zero Slope Graph \| Definition & Examples 4:23 min Linear Equation \| Parts, Writing & Examples 8:58 min Equation of a Line Using Point-Slope Formula 9:27 min Graphing Inequalities \| Definition, Rules & Examples 7:59 min Graphing Inequalities \| Overview & Examples 12:06 min Vertical & Horizontal Shifts \| Definition & Equation 5:15 min Creating & Interpreting Scatterplots: Process & Examples 6:14 min Understanding Simple Linear Regression \| Graphing & Examples 9:52 min Problem Solving Using Linear Regression: Steps & Examples 8:38 min Solving Absolute Value Functions & Equations \| Rules & Examples 5:26 min Solve & Graph an Absolute Value Inequality \| Formula & Examples 8:02 min Absolute Value \| Overview & Practice Problems 7:09 min Absolute Value \| Graph & Transformations 8:14 min Graphing Absolute Value Functions \| Definition & Translation 6:08 min Chapter 3 Holt McDougal Algebra 2 Chapter 3: Linear Systems System of Equations in Algebra \| Overview, Methods & Examples 8:39 min Classifying Linear Systems in Math 5:20 min How Do I Use a System of Equations? 9:47 min Applying Systems of Linear Equations to Breakeven Point: Steps & Example 5:44 min Inequality Notation \| Overview & Examples 8:16 min Feasible Region Definition & Graphs 3:59 min Objective Function \| Definition & Examples 4:30 min Graphing Points & Lines in Three Dimensions 5:19 min How to Solve a Linear System in Three Variables With a Solution 5:01 min Parametric Equations \| Definition & Examples 5:44 min Graphs of Parametric Equations 5:37 min Parametric Equations in Applied Contexts 5:29 min Evaluating Parametric Equations: Process & Examples 3:43 min Chapter 4 Holt McDougal Algebra 2 Chapter 4: Matrices Matrix in Math \| Definition, Properties & Rules 5:39 min Scalars & Matrices: Properties & Application Square Matrix \| Overview & Examples 4:41 min Identity Matrix \| Definition, Properties & Examples Using Matrices to Complete Translations 4:44 min Using Matrices to Complete Rotations 3:58 min Using Matrices to Complete Reflections 3:51 min Finding the Determinant of a Matrix \| Properties, Rules & Formula 7:02 min Using Cramer's Rule with Inconsistent and Dependent Systems 4:05 min Inverse Matrix \| Definition, Types & Example 6:29 min Multiplicative Inverse of a Matrix \| Overview & Examples 4:31 min Augmented Matrix \| Definition, Form & Examples 4:21 min Row Operations & Reductions with Augmented Matrices Chapter 5 Holt McDougal Algebra 2 Chapter 5: Quadratic Functions Parabola \| Definition & Parabolic Shape Equation 4:36 min Reflection Over X & Y Axis \| Overview, Equation & Examples 3:48 min Types of Parabolas \| Overview, Graphs & Examples 6:15 min Maximum & Minimum Values of a Parabola \| Overview & Formula 9:54 min Factoring Quadratic Equations Using Reverse Foil Method 8:50 min How to Solve a Quadratic Equation by Factoring 7:53 min How to Complete the Square \| Method & Examples 8:43 min Completing the Square Practice Problems 7:31 min Imaginary Numbers \| Definition, History & Examples 8:40 min How to Equate Two Complex Numbers 5:54 min Quadratic Function \| Formula, Equations & Examples 9:20 min How to Solve Quadratics with Complex Numbers as the Solution 5:59 min How to Solve & Graph Quadratic Inequalities 6:14 min Quadratic Inequality \| Solution Sets & Examples 3:40 min Quadratic Function \| Definition, Formula & Examples 10:09 min How to Add, Subtract and Multiply Complex Numbers 5:59 min How to Graph a Complex Number on the Complex Plane 3:28 min Complex Numbers & Conjugates \| Multiplication & Division 6:40 min How to Add Complex Numbers in the Complex Plane 3:52 min Chapter 6 Holt McDougal Algebra 2 Chapter 6: Polynomial Functions What are Polynomials, Binomials, and Quadratics? 4:39 min Adding, Subtracting & Multiplying Polynomials \| Steps & Examples 6:53 min How to Evaluate a Polynomial in Function Notation 8:22 min What is the Binomial Theorem? 9:14 min Polynomial Long Division \| Overview & Examples 8:05 min Synthetic Division of Polynomials \| Method & Examples 6:51 min Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11 min Factoring by Grouping \| Definition, Steps & Examples 7:46 min Factor & Remainder Theorem \| Definition, Formula & Examples 7:00 min How to Solve Perfectly Cubed Equations 10:32 min Using the Rational Zeros Theorem to Find Rational Roots 8:45 min Viewing now Irrational Root Theorem Application & Examples 3:43 min Up next Fundamental Theorem of Algebra \| Definition, Examples & Proof 7:39 min Watch next lesson Writing a Polynomial Function With Given Zeros \| Steps & Examples 8:59 min Cubic, Quartic & Quintic Equations \| Graphs & Examples 11:14 min Basic Transformations of Polynomial Graphs 7:37 min Chapter 7 Holt McDougal Algebra 2 Chapter 7: Exponential and Logarithmic Functions Exponential Function \| Definition, Equation & Examples 7:24 min Exponential Growth & Decay \| Formula, Function & Graphs 8:41 min Inverse Function \| Graph & Examples 7:31 min Logarithms \| Overview, Process & Examples 5:23 min Evaluating Logarithms \| Properties & Examples 6:45 min Inverse of Log Functions \| Definition & Examples 7:09 min Logarithmic Properties \| Product, Power & Quotient Properties 5:11 min Practice Problems for Logarithmic Properties 6:44 min Change-of-Base Formula for Logarithms \| Rules & Examples 4:56 min Solving Logarithmic Equations \| Properties & Examples 6:50 min Exponential Equations \| Definition, Solutions & Examples 6:18 min Solving Exponential & Logarithmic Inequalities Rule of 72 Definition, Formula & Examples 3:58 min Exponentials, Logarithms & the Natural Log 8:36 min Natural Base e \| Overview & Importance 4:47 min Graphing Logarithms \| Overview, Transformations & Examples 6:53 min Graphing Logarithmic Functions \| Overview & Examples 8:08 min Stretching & Compression of Logarithmic Graphs 5:36 min Chapter 8 Holt McDougal Algebra 2 Chapter 8: Rational and Radical Functions Solving Direct Variation \| Equation, Problems & Examples 5:12 min Solving Equations of Inverse Variation 5:13 min Direct & Inverse Variation \| Equations, Relationships & Problems 6:27 min How to Multiply and Divide Rational Expressions 8:07 min Multiplying and Dividing Rational Expressions: Practice Problems 4:40 min Adding & Subtracting Rational Expressions \| Overview & Examples 8:02 min Practice Adding and Subtracting Rational Expressions 9:12 min Graphing Rational Functions That Have Polynomials of Various Degrees: Steps & Examples 8:59 min Graphing Rational Functions That Have Linear Polynomials: Steps & Examples 7:55 min Domain & Range of Rational Functions \| Definition & Graph 5:50 min Transformations of the 1/x Function Graph \| Definition & Types 7:37 min Rational Equations \| Definition, Formula & Examples 7:58 min Rational Equations: Practice Problems 13:15 min Solving Rational Inequalities \| Steps & Examples 5:27 min Rational Exponents \| Definition, Calculation & Examples 3:22 min Simplifying Algebraic Expressions with Rational Exponent 7:41 min Radical Functions \| Graph, Equation & Examples 6:47 min Transformations of Radical Functions 5:09 min Solving Radical Equations \| Overview & Examples 6:48 min Solving Radical Equations with Two Radical Terms 6:00 min Solving Radical Inequalities Chapter 9 Holt McDougal Algebra 2 Chapter 9: Properties and Attributes of Functions Graphing Basic Functions 8:01 min Piecewise Function \| Definition, Evaluation & Examples 7:22 min Piecewise Functions \| Graph & Examples 5:00 min Step Function \| Definition, Equation & Graph 4:44 min Translating Piecewise Functions 6:44 min How to Compose Functions 6:52 min How to Add, Subtract, Multiply and Divide Functions 6:43 min Applying Function Operations Practice Problems 5:17 min Domain & Range of Composite Functions \| Steps & Examples 5:58 min Inverse Functions \| Definition, Methods & Calculation 6:05 min One to One Function \| Definition, Graph & Examples 4:11 min Chapter 10 Holt McDougal Algebra 2 Chapter 10: Conic Sections Conic Sections \| Definition, Equations & Types 6:33 min How to Graph a Circle \| Equation & Examples 8:32 min Tangent of a Circle \| Definition, Formula & Examples 3:52 min Understanding Circles with Inequalities How to Write the Equation of an Ellipse in Standard Form 6:18 min Ellipse Foci Formula & Calculations 5:08 min Hyperbola \| Definition, Equation & Graphs 10:00 min Hyperbola Standard Form \| Definition, Equations & Examples 8:14 min Hyperbola Equation \| Foci Formula, Parts & Example 7:07 min Parabola \| Equation, Formula & Examples 8:17 min Parabola \| Definition, Formula & Examples 8:33 min Conic Sections \| Overview, Equations & Types 6:22 min Chapter 11 Holt McDougal Algebra 2 Chapter 11: Probability and Statistics How to Use the Fundamental Counting Principle 5:52 min Permutation Definition, Formula & Examples 6:58 min Combination in Mathematics \| Definition, Formula & Examples 7:14 min Combinations in Probability \| Equation, Formula & Calculation 11:00 min How to Calculate the Probability of Permutations 10:06 min Probability of at Least One Event \| Overview & Calculation 5:27 min Addition Rule of Probability \| Formulas & Examples 10:57 min Relative Frequency \| Formula & Examples 5:56 min Probabilities as Areas of Geometric Regions: Definition & Examples 7:06 min Independent & Dependent Events \| Overview, Probability & Examples 12:06 min Conditional Probability \| Definition, Equation & Examples 7:04 min Probability of an Event \| Simple, Compound & Complementary 6:55 min Probability of A or B \| Overlapping & Non-Overlapping Events 7:05 min Measures of Central Tendency \| Definition, Formula & Examples 8:30 min Standard Deviation Equation, Formula & Examples 13:05 min Outliers in a Data Set \| Minimums & Maximums 4:40 min Box Plot \| Definition, Uses & Examples 6:29 min Dice: Finding Expected Values of Games of Chance 13:36 min How to Use the Binomial Theorem to Expand a Binomial 8:43 min Binomial Probability & Binomial Experiments Chapter 12 Holt McDougal Algebra 2 Chapter 12: Sequences and Series Sequences in Math \| Overview, Types & Examples 4:57 min Recursive Rule \| Formulas & Examples 5:52 min Special Sequences and How They Are Generated 5:21 min How to Write a Series in Summation Notation \| Overview & Examples 4:16 min Sigma Summation Notation \| Overview & Examples 6:01 min How to Calculate an Arithmetic Series 5:45 min Arithmetic Sequences \| Formula & Examples 5:55 min How to Find and Classify an Arithmetic Sequence 9:09 min Sum of Arithmetic Sequence \| Formula & Examples 6:00 min Working with Geometric Sequences 5:26 min Finding and Classifying Geometric Sequences 9:17 min Sum of a Geometric Series \| Formula & Examples 4:57 min Sum of Infinite Geometric Series \| Formula, Sequence & Examples 4:41 min Mathematical Induction: Uses & Proofs 7:48 min Converting Repeating Decimals to Fractions \| Overview & Examples 7:06 min Chapter 13 Holt McDougal Algebra 2 Chapter 13: Trigonometric Functions Special Right Triangles \| Definition, Types & Examples 6:12 min Trigonometric Ratios \| Definition, Similar Triangles & Examples 6:49 min Practice Finding the Trigonometric Ratios 6:57 min Problem-Solving with Angles of Elevation & Depression 5:10 min Trigonometric Functions \| Cotangent, Secant & Cosecant 4:21 min Coterminal Angles \| Definition, Formula & Examples 6:37 min Trig Functions using the Unit Circle \| Formula & Examples 5:46 min Unit Circle Quadrants \| Converting, Solving & Memorizing 5:15 min Radians to Degree Formula & Examples 7:15 min Unit Circle Reference Angle \| Formula, Quadrants & Examples 3:56 min How to Solve Trigonometric Equations for X 4:57 min Properties of Inverse Trigonometric Functions 7:56 min Law of Sines Formula & Examples 6:04 min Using Sine to Find the Area of a Triangle 5:19 min The Ambiguous Case of the Law of Sines \| Definition & Examples 5:19 min Law of Cosines \| Definition & Equation 8:16 min Heron's Formula in Geometry \| Overview & Examples 5:54 min Chapter 14 Holt McDougal Algebra 2 Chapter 14: Trigonometric Graphs and Identities Sine & Cosine Waves \| Graphs, Differences & Examples 7:50 min Transforming sin & cos Graphs \| Graphing sin and cosine Functions 8:39 min Graphing Tangent Functions \| Period, Phase & Amplitude 9:42 min Sec, Cotangent & Csc Trig Functions \| Graphing & Examples 7:10 min Trig Identities \| Formula, List & Properties 7:11 min Using Graphs to Determine Trigonometric Identity 5:02 min The Negative Angle Identities in Trigonometry 5:30 min Rotation Matrix \| Properties & Examples 3:30 min Double Angle Formula \| Sin, Cos & Tan 9:44 min Half-Angle Trig Identities \| Formulas, Uses & Examples 3:48 min How to Solve Trigonometric Equations: Practice Problems Related Study Materials Irrational Root Theorem 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##### How to Solve Equations on a Calculator 9:22 ##### Writing an Equation \| Overview & Examples 5:22 ##### Literal Equations \| Definition, Formula & Examples 5:09 ##### Algebraic Model Definition, Applications & Examples 7:02 ##### Substitution Property \| Definition & Examples 2:57 ##### Trigonometry: High School ##### Prentice Hall Algebra 2: Online Textbook Help ##### Glencoe Algebra 1: Online Textbook Help ##### McDougal Littell Algebra 2: Online Textbook Help ##### Glencoe Math Connects: Online Textbook Help ##### Glencoe Math Course: Online Textbook Help ##### Number Properties: Help & Review ##### HiSET Mathematics: Prep and Practice ##### THEA Study Guide and Test Prep ##### ACCESS Math: Online Textbook Help ##### Saxon Calculus Homeschool: Online Textbook Help ##### GED Math: Quantitative, Arithmetic & Algebraic Problem Solving ##### AP Calculus AB & BC: Exam Prep ##### ELM: CSU Math Study Guide ##### Study.com SAT Study Guide and Test Prep ##### SAT Subject Test 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2518	https://www.npr.org/2016/10/30/499893072/donate-a-bit-of-your-time-to-this-superpac-puzzle	Accessibility links Keyboard shortcuts for audio player Sunday Puzzle: Donate A Bit Of Your Time To This SuperPAC (Puzzle) This week's puzzle is called "SuperPACs." Every answer is a familiar two-word phrase or name in which the first word starts with PA- and the second word starts with C. ## Sunday Puzzle The Weekly Quiz From NPR Puzzlemaster Will Shortz Donate A Bit Of Your Time To This SuperPAC (Puzzle) Heard on Weekend Edition Sunday Will Shortz Donate A Bit Of Your Time To This SuperPAC (Puzzle) Listen · 6:06 Transcript Download <iframe src=" width="100%" height="290" frameborder="0" scrolling="no" title="NPR embedded audio player"> Transcript Enlarge this image Sunday Puzzle. NPR hide caption toggle caption NPR Sunday Puzzle. NPR On-air challenge: This week's puzzle is called "SuperPACs." Every answer is a familiar two-word phrase or name in which the first word starts with PA- and the second word starts with C. For example: Official who oversees a city's green spaces --> PARKS COMMISSIONER. Last week's challenge: This is a two-week challenge. Take the digits 5, 4, 3, 2 and 1, in that order. Using those digits and the four arithmetic signs — plus, minus, times and divided by — you can get 1 with the sequence 5 - 4 + 3 - 2 - 1. You can get 2 with the sequence (5 - 4 + 3 - 2) x 1. The question is ... how many numbers from 1 to 40 can you get using the digits 5, 4, 3, 2, and 1 in that order along with the four arithmetic signs? You can group digits with parentheses, as in the example. There are no tricks to this, though. It's a straightforward puzzle. How many numbers from 1 to 40 can you get — and, specifically, what number or numbers can you not get? Answer: 39 numbers — the only number you can not get is 39. Here are the answers submitted by this week's winner. Note: Many numbers have multiple possible equations: 1=5-4+3-2-1;2=(5-4+3-2)1; 3=((5+4)/3) (2-1); 4=(5+4-3-2)1; 5=5(4-3)(2-1); 6=5-4+(32)-1; 7=5+4-3+2-1;8=(5+4-3+2)1; 9=5+4+3-2-1; 10=5+4+(3-2)1; 11=5+4+3-2+1;12=(5+4-3)(21); 13=((5+4-3)2)+1; 14=(5+4+3+2)1; 15=5+4+3+2+1; 16=5+4+(32)+1; 17=5+(43)(2-1); 18=5+(43)+2-1; 19=(54)-3+(21); 20=(54)-3+2+1; 21=(54)+3-(21); 22=(54)+3-2+1;23=(54)+3(2-1); 24=(54)+3+2-1; 25=(54)+3+(21); 26=(54)+3+2+1; 27=(54)+(32)+1; 28=5+(432)-1; 29=5+(4321); 30=5+(432)+1; 31=((543)/2)+1; 32=5(4+3)-2-1; 33=5(4+3)-(21); 34=5(4+3)-2+1; 35=5(4+3)(2-1); 36=5(4+3)+2-1; 37=5(4+3)+(21); 38=5(4+3)+2+1; (no equation for 39); 40=5(4+3+2-1) Winner: Margaret Gibbs of Littleton, Mass. Next week's challenge, from listener Peter Gordon of Great Neck, N.Y.: Think of a name in the news that has a doubled letter. It's a person's last name. Change that doubled letter to a different doubled letter, and you'll get the commercial name for a popular food. What is it? Submit Your Answer If you know the answer to next week's challenge, submit it here. Listeners who submit correct answers win a chance to play the on-air puzzle. Important: Include a phone number where we can reach you Thursday, Nov. 3, at 3 p.m. ET. Will Shortz sunday puzzle
2519	https://www.nature.com/articles/s41467-020-18737-6	Human running performance from real-world big data \| Nature Communications Your privacy, your choice We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media. By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection. See our privacy policy for more information on the use of your personal data. Manage preferences for further information and to change your choices. Accept all cookies Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). 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Advertisement View all journals Search Search Search articles by subject, keyword or author Show results from Search Advanced search Quick links Explore articles by subject Find a job Guide to authors Editorial policies Log in Explore content Explore content Research articles Reviews & Analysis News & Comment Videos Collections Subjects Follow us on Facebook Follow us on Twitter Sign up for alerts RSS feed About the journal About the journal Aims & Scope Editors Journal Information Open Access Fees and Funding Calls for Papers Editorial Values Statement Journal Metrics Editors' Highlights Contact Editorial policies Top Articles Publish with us Publish with us For authors For Reviewers Language editing services Open access funding Submit manuscript Sign up for alerts RSS feed nature nature communications articles article Human running performance from real-world big data Download PDF Download PDF Article Open access Published: 06 October 2020 Human running performance from real-world big data Thorsten EmigORCID: orcid.org/0000-0002-3272-74961& Jussi PeltonenORCID: orcid.org/0000-0003-0590-42982 Nature Communicationsvolume 11, Article number:4936 (2020) Cite this article 39k Accesses 35 Citations 195 Altmetric Metrics details Abstract Wearable exercise trackers provide data that encode information on individual running performance. These data hold great potential for enhancing our understanding of the complex interplay between training and performance. Here we demonstrate feasibility of this idea by applying a previously validated mathematical model to real-world running activities of ≈14,000 individuals with≈1.6 million exercise sessions containing duration and distance, with a total distance of≈20 million km. Our model depends on two performance parameters: an aerobic power index and an endurance index. Inclusion of endurance, which describes the decline in sustainable power over duration, offers novel insights into performance: a highly accurate race time prediction and the identification of key parameters such as the lactate threshold, commonly used in exercise physiology. Correlations between performance indices and training volume and intensity are quantified, pointing to an optimal training. Our findings hint at new ways to quantify and predict athletic performance under real-world conditions. Similar content being viewed by others Predictive athlete performance modeling with machine learning and biometric data integration Article Open access 11 May 2025 Plasma metabolomics dataset of race-walking athletes illuminating systemic metabolic reaction of exercise Article Open access 18 March 2025 Special endurance coefficients enable the evaluation of running performance Article Open access 20 June 2025 Introduction Skeletal evidence suggests that endurance running may have evolved 2 million years ago1. It probably originated as a hunting skill but has later developed to competition, dating back to ancient Olympic Games~720 BC2 and exercise form for mass population. Over the years, endurance running has undergone substantially change. Recent decades have witnessed an ever growing exercising population which uses wearable sensors to bring together astonishing volumes of data for speed, distance, heart rate, accelerations, and more3,4,5. For example, endurance athletes like runners and cyclists currently upload from GPS enabled sensors more than a billion activities per year worldwide6. In principle, these data provide an exciting opportunity to monitor human physiology noninvasively under real-world conditions outside the laboratory. Measuring the physiological response to physical activity can provide important insights for a variety of populations ranging from elite athletes to recreational exercisers to patients in rehabilitation7."),8."). However, the analysis of big data sets of large, heterogeneous groups of individuals poses a substantial challenge due to the quality of the data itself9."),10."), lack of effective theoretical models11."), and influence of environmental factors like weather conditions12."),13."). The important, robust properties of an individual’s physiology can be overshadowed by details specific to the conditions of recording. Thus, there is a demand for universal theoretical models that have been validated for noise-free exercise data and can be applied under noisy real-world conditions to derive meaningful physiological and performance information14."). To date, exercise physiologists conventionally use laboratory testing to determine parameters that measure fitness and performance potential15. A strength of laboratory testing is that it can distinguish between cardiovascular limit, maximal rate of oxygen consumption (VO 2max), neuromuscular effects, and running economy16,17. Together VO2max and running economy determine maximal aerobic speed, which is the slowest speed at which VO 2max occurs. Maximal aerobic speed correlates with race speed on shorter distances but alone cannot predict race times for longer distances such as the marathon. Exercise thresholds have been used in exercise testing to quantify metabolism. However, the determination of such thresholds, like the lactate threshold, in the laboratory is somewhat limited. Typical laboratory testing is short-lasting and does not always fully capture time and distance dependent reduction in running economy18,19. For example, only sparse results exist for the endurance limited fractional utilization of maximal aerobic power (MAP) and its dependence on exercise duration20. Moreover laboratory testing is expensive and not available to most of the population. The undeniable fact that the best test of running performance is an actual race and not laboratory tests highlights the need for models specifically constructed to extract performance indices of an athlete from their regular exercise performance. For these reasons, models that can utilize data from wearable devices and turn those into meaningful performance parameters may offer a cost effective alternative approach to laboratory testing. However, it must stressed that this type of approach does not elucidate the physiological and biomechanical mechanisms that control performance. It is an adjunct to the methods which are already used, providing additional insight into running and the potential training factors influencing performance and it does not replace the insights that we can gain from laboratory testing. Several empirical and physiological models have been put forward for explaining running world records in terms of a few physiological parameters. The noted physiologist Hill empirically proposed a hyperbola to describe the maximal power output as a function of exercise duration21. Also a purely mechanical approach, based on the runners equation of motion, has been proposed22. These approaches predict that the average racing velocity tends to be a constant value with increasing race distance which contradicts observation. While more recent approaches have combined physiology and observations to propose more realistic logarithmic relations between maximal power output and duration23, these models depend on many parameters that vary among individuals24. Recently we have developed a universal running model which builds on concepts in exercise physiology, depends only a minimal set of key performance indices that are required to predict race performance, contains no additional individual-dependent quantities and has been validated with running world-records14. Here, we show that it is also possible to obtain novel insights into individual’s running performance by applying this model to big exercise datasets. Exercise data are a valuable source of information about individual long-term training protocols. Endurance training leads to a wide spectrum of physiological responses. However, in practice, training is prescribed often only by anecdotal evidence and personal experience. This might be due to a lack of knowledge of statistically significant correlations between the relevant physiological parameters and training characteristics for large groups of individuals with different fitness status. Here, we demonstrate the feasibility to extract key performance indices from real-world running exercise data recorded with wearable exercise trackers. We apply our method to runners during their training season before a marathon race. Our universal running model characterizes a runner’s performance with two indices that measure (1) endurance (endurance index) and (2) the velocity requiring MAP output (aerobic power index). The main aim of our work is to demonstrate the feasibility of extracting performance indices from real-world racing results in a big population of runners and to use these indices to predict accurate race times and evaluate the effect and efficiency of training. Our approach represents a potentially powerful platform to enlarge dramatically the number of tested subjects in sports science by extending performance index acquisition from conventional laboratory testing to real-world conditions with the aid of mathematical modeling and wearable technology. Results Universal performance model In previous work we have developed a model that can be used to extract aerobic performance indices from race data14. To summarize, this model expresses exercise intensity on a relative power scale p, which varies between zero, corresponding to basal metabolic rate, and unity at MAP generation. MAP is expected to correspond to maximal oxygen uptake VO 2,max but this analogy needs not to be assumed in our approach. A linear relation p(v) maps running velocity v to relative power with p(v m) = 1 defining v m as an aerobic power index associated with MAP beyond which anaerobic energy supply can yield p> 1 for a short time only. Anaerobic supply contributes to maximal exercise shorter than a crossover time t c which in our model is the longest time over which MAP can be sustained. An important prediction of our model is that the maximal value of the relative power p that a runner can maintain declines logarithmically with duration, with a rate γ l, assuming that the durations are longer than t c. This finding is in agreement with a finding of A.V. Hill who observed this form of decline in running world records21. For more details on our model, see the “Methods” section. Here, we use this universal, i.e., subject independent model for human running performance, to extract aerobic performance indices from finishing times of runners worldwide by matching them with model predictions14. The analyzed data set comes from an exercise tracking platform that contains precise records of distance and duration (and hence average velocity) of running activities of≈19K individuals, who ran a total distance of 32M km over a period of 3.5 years. The data were recorded by the individuals with a GPS digital sports watch (V800, Polar Electro Oy, Oulu, Finland)25, and uploaded to the platform. Maximal performance of an individual was measured by the fastest finishing time for the four most common racing distances 5000 m, 10,000 m, half-marathon (21,097.5 m) and marathon (42,195 m) within a racing season, which is defined as the 180 days preceding the marathon race (see “Methods” section for detection of racing activities). The velocity corresponding to our parameter v m is difficult to measure in laboratory settings since VO 2,max can be achieved over a wide range of sub-maximal intensities because of an upward drift of oxygen uptake with exercise duration18,19. In general, our model can determine v m from the crossover of the race–time–distance relation at time t c, and hence is free from this complications. The simplest version of the model assumes a fixed time t c. Model predictions for sub-MAP performances do not depend on this fixed time since other choices lead only to consistently renormalized values for v m and γ l (which are then no longer associated strictly with MAP but with a slightly different power). In agreement with the application of our model to running records on both the super- and sub-MAP branches14 and laboratory testings26, we choose t c = 6 min in the following. Combining running economy and the decline of the fractional utilization of maximal power output with race duration, the fastest time T(d) over a distance d is given by the universal expression $$T(d)=-\frac{{t}{\text{c}}}{{\gamma }{\text{l}}}\frac{d}{{d}{\text{c}}}\frac{1}{{W}{-1}\left[-\frac{d}{{d}{\text{c}}}\frac{\exp (-1/{\gamma }{\text{l}})}{{\gamma }{\text{l}}}\right]}\quad {\rm{for}}\ \ d\ge {d}{\text{c}}\ ,$$ (1) where we defined d c = v m t c, and W−1 is a real branch of the Lambert W-function which is defined as the multi-valued inverse of the function (w\to w\exp (w))27. W−1(z) is real valued for −1/e ≤ z< 0 which is fulfilled for all distances d that we consider (see the “Methods” section for more detail). Note that T(d c) = t c, i.e., d c is the distance that can be maximally raced in the time t c. The condition d ≥ d c is always satisfied for the race distances considered here. We note that Eq. (1) is an exact solution of our model. It can be also obtained from earlier descriptions of the energetics of endurance running28,29,30 when the fractional utilization of MAP is described by our prediction of a slow, logarithmic decay, and a linear increase of the energy cost of running with velocity is assumed. The model parameters, called performance indices, quantify different aspects of performance and provide a unique insight into basic determinants of fitness in a large population of runners over a wide range of exercise capacities and over long time scales. The velocity v m measures combined running economy and MAP and is known to be a better predictor of performance than VO 2,max alone31. We define the endurance index as ({E}{\text{l}}=\exp (0.1/{\gamma }{\text{l}})), which encodes that 90% of v m can be maintained for an extended time E l t c>t c. The pair of performance indices v m, E l is sufficient to account for racing velocity variations for distances from d c (typically one mile in our data set) to the marathon. For example, when analyzing consistent running records of individuals, we found strong evidence that they follow the same universal scaling law of Eq. (1) as running world (or national) records do, with mean errors below 1%14. Here, our model estimates are based on an individual’s fastest times for the four fixed racing distances, 5 k, 10 k, half-marathon, and marathon. Unfortunately, we cannot determine from the available data set if performance was achieved during an actual racing event. For our approach however, it is only required that the recorded performance corresponds to the maximal effort over a given running distance achieved during the racing season. Exercise data An overview of the data analysis design is provided in Fig.1. All available subjects and activities in the data set of the exercise tracking platform were grouped by SID and marathon date, combining all individual running activities during the 180 days before the marathon, defining a season. For each season, activities with the fastest time for the four fixed race distances defined a racing season. We imposed the condition that each racing season contains at least two races. If a season contained 30 or more total running activities they were defined as training season. For consistency certain data filters were applied to all activities and races (see the “Methods” section for more detail). Two variants of racing season were defined, with the marathon included and excluded. A total of ~25,000 racing seasons with the marathon included and~10,000 racing seasons without the marathon, and~22,000 training seasons were analyzed (see Table1 for a summary of the available data and performed analyses). Fig. 1: Flowchart of the exercise data analysis. SID: subject identifier, M: marathon, M-date: date of marathon, d: total running distance, “race season”: fastest times of an athlete for at least two of the distances 5 km, 10 km, half-marathon, and marathon (±3% to account for GPS tolerance), N races: total number of races, N M: number of successful model fits, N T: number of analyzed training seasons for which physiological parameters v m, E l could be obtained and predicted actual race times within a mean error below 5%, “full training season”: at least 30 activities during the 180 days before M-date. Full size image Table 1 Summary of data sets analyzed. Full size table Accuracy of performance prediction For all individuals, we estimated their performance indices v m and γ l for each racing season by matching race events to Eq. (1) by minimizing the relative prediction error for the race times. The probability densities of these indices are shown in Fig.2. For all racing seasons with three and more races (N = 12,309), the mean error between model prediction and actual race time was only 2.0%. This suggests that our model captures correctly determinants of aerobic endurance performance. Correlations between performance indices and marathon finishing times are presented in Fig.3. To investigate the predictive power of our model in more detail, we applied our model also to the racing season with the marathon performance excluded (see Fig.4). This allowed us to estimate the marathon finishing time from the performances on shorter distances only. As a function of performance indices, in the most likely parameter range the model predicted the marathon performance with an overall accuracy of better than 10%. Only for very small (or large) endurance E l, estimated times tended to be too slow (or fast) which indicates that sub-marathon distances were raced inconsistently, leading to an under (or over) estimation of E l. Given all the possible uncertainties in marathon racing that are beyond the control of this study (e.g., weather, course profile, and motivation of the athlete), our predictions for the marathon finishing times are rather satisfying. Fig. 2: Probability density of model parameters. The crossover velocity v m which is the smallest velocity that elicits maximal aerobic power MAP and the endurance E l are obtained by applying our model to the fastest performances of a subject for the four distances 5 km, 10 km, Halfmarathon, and Marathon of a racing season. For these distributions, a total of 24,858 racing seasons have been analyzed. a The velocity v m is approximately normally distributed with a mean of 4.4 m s−1. b The probability density for the endurance E l resembles an exponential decay. c The probability density for the relative power for 1h utilization (1hU) peaks at about 82% of MAP. Full size image Fig. 3: Correlation between performance indices and marathon race time (model estimates for 24,504 racing seasons are shown here). a Visualization of the marathon race time T m in the (v m,E l) parameter plane. Performance indices are obtained from individual’s best performances during the racing season. Color changes from fast (magenta) to slower (blue) finishing times (see color legend for time in minutes). Parameter pairs (v m,E l) along the dashed curves yield the same marathon race time indicated at the top of the graph (in hh:mm format). b Color coded visualization of the number n of racing seasons analyzed as function of the parameters (v m,E l). Full size image Fig. 4: Estimate of Marathon race time from the racing season (for 9410 seasons). a Visualization of the relative difference Δ T m between actual and estimated marathon race time T m (in percent of race time) as function of crossover velocity v m and endurance E l. Magenta (blue) color indicates a faster (slower) than estimated finish. b Probability density of race time differences color coded according to groups of different race time intervals. Full size image Maximal velocity for 1 h Analysis of~25,000 racing seasons reveals a normally distributed velocity v m and an exponential decay of the probability density for the endurance E l (see Fig.2). Interestingly, VO 2,max in a study on 450 elite soccer players has also been found to obey a normal distribution32. Note that v m also measures running economy, which varies considerably among individuals and modulates performance24. In exercise physiology, the ability of a runner to maintain a certain effort is often characterized in terms of thresholds, of which a common example is lactate threshold. In our approach, however, there is a continuous relationship between power output and velocity, and the change of this relation with duration appears to be a natural measure for endurance capability. Hence, as a practical measure for endurance, we define in our model the velocity ({v}{\text{1hU}}={v}{\text{m}}[1-0.1\,\mathrm{log}\,(60\min /{t}{\text{c}})/\mathrm{log}\,({E}{\text{l}})]) that a runner can maintain for 1 h, corresponding to the maximal fractional utilization of MAP for 1 h. While any duration could be chosen here, we used 1 h in analogy to running coaches defining threshold velocity as the effort that can be maintained for about 1 h33. The 1h utilization ratio p 1hU = v 1hU/v m had been estimated previously from laboratory measurements and races for a smaller group of 18 male long distance runners to be approximately 0.82 ± 0.0534. Strikingly, our findings from the running data for~14,000 subjects corroborate this range without any invasive measurements, as demonstrated in Fig.2c. Moreover, our observation of exponentially small but finite probability for larger E l explains observed values p 1hU≈0.9 in some well trained long distance runners. We also computed the marathon race time from our model and compared it to the actual marathon time T m for all racing seasons, see Fig.3. Our model predicts theoretical curves of constant T m in the plane of performance indices (shown as dashed lines in Fig.3a). We found that the actual race times are ordered according to these curves. This shows that our selected physiological profiles, computed from sub-marathon and marathon best performances, are highly correlated with T m. It is important to understand that the position of a marathon performance in the parameter space is determined by all races and hence reflects relative importance of the indices v m and E l. This demonstrates the crucial importance of taking into account endurance in addition to MAP and running economy when assessing performance of long distance runners. Importance of endurance Our findings demonstrate the strong sensitivity of performance to endurance. For example, a runner with a velocity of v m = 5 m s−1 can improve his/her marathon time from 3 h 27 min 38 s to 2 h 53 min 8 s by doubling endurance from E l = 3 to E l = 6 (corresponding to a change in the one-hour utilization from 79 to 87% of VO 2max), without any change in VO 2,max or running economy. We also find that faster runners tend to race more consistently over all race distances than slower runners, highlighted by the dependence of the prediction error Δ T m on the marathon finishing time (see Fig.4b). For example, within our fastest group of runners with a marathon time below 160 min, the prediction error was typically less than±2.5%. This observation supports our explanation for the observed uncertainty in the endurance parameter E l. Correlation with training Finally, we compared physiological profiles to running activities within a training season. There exist a few studies of the relation between training volume and intensity, improvements of aerobic fitness and performance35. For example, it has been stated that running at velocity v m might represent an optimal stimulus for improving endurance36. There is also evidence supporting that a relatively large percentage of low-intensity training over a long period improves performance during highly intense endurance events37,38. It has been argued that running velocity at lactate threshold is the best physiological predictor for distance running performance39. To investigate the effect of training distance and speed, relative to the velocity v m, we selected consistent racing seasons defined by having a mean race time prediction error below 5%. Figure5a shows that as the total training distance d train of the training season increases, v m increases on average linearly, with a weak saturation trend at largest d train. Several studies have demonstrated an increased v m due to endurance training35. A faster velocity v m can be achieved by a better running economy and/or an increase in MAP. We hypothesize that longer training distance has generated improved running economy, in agreement with earlier observations in a group of eleven well-trained long distance runners40. Our analysis provides a statistically significant, quantitative relation between training distance and speed at MAP, v m, for~22,000 training seasons. Another explanation for this relation could be that fitter runners with a larger MAP and hence higher v m log more kilometer during their training. Unfortunately, we could not measure v m at the beginning and the end of the training season independently from two different racing seasons or time trials. We also found a linear decrease of v m with the mean relative training intensity between 50% and about 90% of v m, as shown in Fig.5b. Our findings can be interpreted as faster runners train typically at lower relative intensities which is consistent with high-intensity performance improvement due to low-intensity training. The range of training velocities increases with larger v m which reflects a wider range of accessible intensities between minimal (jogging) and maximal speed. For example, a runner with v m = 4 m s−1 typically (within one standard deviation) trains between 64 and 84% of v m or MAP, while a runner with v m = 5 m s−1 trains typically up to 66% of v m so that both runners have an almost identical upper pace~5 min km−1 for the majority of their runs. Slow runners must train at a relative high intensity if they want to avoid a transition to walking. It is important to realize that these typical ranges do not include fast, high-intensity workouts which account only for a small fraction of total training volume. However, high-intensity sessions involve also resting phases that can reduce the average velocity when timer is not stopped, potentially explaining observed intensities below~50% of v m. Fig. 5: Correlations between performance indices and training characteristics. We have measured the distance and time for each running activity during a training season for a total number of 21,605 seasons. The graphs show the observed relations between performance indices (obtained from a model fit to the racing season) and different measures of training volume and intensity. The magenta line indicates the average, the gray region one standard deviation, and the light magenta and blue shaded areas represent the standard error of the mean and standard deviation, respectively, as obtained from bootstrap resampling with replacement (see “Methods” section for more details). a Increase of crossover velocity v m with total distance d train of training runs. b Relation between crossover velocity v m and relative training intensity ({p}{{\rm{train}}}={\bar{v}}{{\rm{train}}}/{v}{\text{m}}) where ({\bar{v}}{{\rm{train}}}) is the average training velocity. c Increase and saturation of endurance E l with training impulse (TRIMP). d Exponential growth of endurance E l with relative training intensity. Full size image Optimal training impulse We found strong evidence that combined effect of training volume and intensity, known as TRaining IMPulse (TRIMP)41, enhances endurance only up to a limit. Previously, it was found in recreational long distance runners that individual TRIMP correlates with 5000 m and 10,000 m track performances42. We computed TRIMP by summing the TRIMP points of all runs of the training season. For each run, TRIMP points were assigned according to the duration of the run and its relative average velocity (\bar{v}/{v}{\text{m}}) (see “Methods” section for details). We analyzed the quantitative relation between endurance _E l and total TRIMP of a training season (see Fig.5c). We observed an initial linear increase of E l with TRIMP, a plateau around E l = 7.5 ± 2 for TRIMP~25,000, and a statistically significant final drop which may be due to over-training. This result suggests that there is an optimal TRIMP per training season, and the corresponding maximal endurance enables a close to optimal marathon race time for a given velocity v m (see Fig.3a). Finally, we probed the definition of TRIMP itself to determine if it implements the best relation between endurance and training intensity. We found a striking agreement between the exponential dependence of E l on ({\bar{v}}{{\rm{train}}}/{v}{\text{m}}) and the original definition of TRIMP based on the rise of blood lactate with intensity, as demonstrated in Fig.5d. Our findings for thousands of runners show that relations between training mode and performance indices that are usually only accessible by invasive and resource-consuming laboratory testing can be obtained reliably from running activity data. Discussion Recent advances in wearable sensor technology have enabled real-time and noninvasive measurement of physiological data during exercise. However, if we are to employ these data to better understand interplay between exercise, performance and human health, we must develop new models that are adapted to extract from the raw data quantities that are most relevant for health and performance assessment. In this work we have taken this approach for long distance running to estimate physiological model indices such as MAP and endurance, and examined their correlations with training volume and intensity by analyzing exercise data of~14,000 marathon runners worldwide. We found that our recent universal model for a logarithmic relation between fractional utilization of maximal power and exercise duration14 is crucial for going beyond previous approaches which ignored this relation, and for defining a parameter measuring endurance. This is an important complement to physiological testing in the laboratory where the required maximal effort is unpractical to achieve for distances over 20 km. Indeed, our results provide evidence of the possibility to extract precise indicators for performance and fitness status from long-duration real-world exercise tracking data. Using automated digital exercise tracking goes beyond previous outside-lab studies that relied often on frequently inaccurate self-reports of exercise. The probability distributions of the extracted performance indices show large variances, implying that studies with only a few individuals might produce misleading results, missing the large interindividual variability of response to exercise. Our work has also some limitations: For each activity, only total distance and duration was available in the data set. This could lead to biased estimates of the mean velocity, for example due to periods of rest or stopping with the device timer not stopped. For the detected correlations between performance indices and training the direction of any cause–effect relationship remained open: for example, training with a higher total TRIMP might produce better endurance, but higher endurance could also enable runners to follow training modes with a higher TRIMP. To resolve this relationship, additional data filters need to be developed to select groups of runners with similar initial performance which subsequently follow different training modes. However, the observed correlations can be of practical importance. They can be useful for estimating realistic expectations for a race for less experienced runners from their training intensity and volume. In addition, our observation that endurance peaks at a given training load (TRIMP) should help preventing over-training, i.e., unproductive increase in training that can cause injury and other health problems. It should also be stressed that real-world data always lack the controlled environment of laboratory based testing. For example, the energy cost of running has been measured very accurately in laboratory conditions43,44,45,46 and the theoretical approaches derived from these experiments have motivated the development of our model. Our work implies several directions for future research. The combination of effective models and real-world exercise data holds great potential for a change in our theoretical description and understanding of human response to physical activity over longer periods of time, optimal exercise dosing and training, early injury detection and prevention, and elite athlete performance. Approaches similar to ours could be used to develop standards for cardiorespiratory fitness based on the probability distribution of performance indices in populations with certain characteristics. More detailed, time-resolved activity data for heart rate, mechanical power output and others could be integrated in our model to improve accuracy and to extract other performance indices. Further applications of our approach include the detection of the usage of performance enhancers in professional sports, the early identification of talented athletes, and even the effect of sports equipment like new running shoe technology on performance indices47. Methods Exercise tracking platform Exercise data were obtained from Polar Flow web service48."), which is an exercise tracking platform that allows users to upload various exercise data, including running distance and velocity from GPS watches. Meta data and activity data of users are linked anonymously through user identification. Selection of subjects and activities Users of the exercise tracking platform were selected as subjects for this study under the conditions that they had completed a run over the marathon distance (42,195 m) in the period between 1 Jul 2015 and 31 Dec 2018, and used the same GPS watch (Polar V800) for activity recording to assure comparable accuracy of GPS based distance recording. We analyzed the running data of~19,000 individuals who completed~2.5M activities with a total distance of~32M km (see Table1 for details). For each individual all running activities in the 180 days before a completed marathon race were grouped together with the marathon race and the groups labeled uniquely by a subject identifier (SID) and the marathon date (M-date). Note that an individual may have have completed multiple marathons during the studied period. For each of those groups, labeled by the pair (SID, M-date), a race season was defined as the fastest runs of all activities over the four race distances 5km, 10km, half-marathon (21,097.5m) and marathon (42,195 m), if distances were available. A tolerance of±3% was allowed in the distance selection to account for GPS inaccuracy, and average race velocities were determined by assuming the actual race distances (which are more reliable than GPS recordings). We applied conditions that race velocities must increase with decreasing race distance and must be slower than current world record velocities. Inconsistent race seasons were identified by violation of these conditions and excluded from further analysis. Race seasons were defined both with and without the marathon race included. A valid race season must contain at least two different race distances. For each race season with a successful performance model fit with mean race time error below 5% (see section below) a corresponding training season was defined as all running activities with a total distance ≥1000 m in the 180 days before the marathon. Runs with apparent velocities ≥7.8 m s−1 (world record for 1000 m) were excluded. Only training seasons with 30 or more runs were considered so that runner had trained at least once per week and training seasons with longer interruptions were excluded. Performance model We mathematically describe running performance by a minimal model based on a relative power scale14. The model is formulated in terms of relative quantities to eliminate irrelevant, subject dependent quantities. The nominal power expenditure P(v) that is required to run at a constant velocity v, the so-called running economy, determines the relative power as $$p(v)=\frac{P(v)-{P}{\text{b}}}{{P}{\text{m}}-{P}{\text{b}}}=\frac{v}{{v}{\text{m}}}\ ,$$ (2) where we introduced a basal power P b that is obtained by linearly extrapolating the running economy to zero velocity and a crossover power P m that we expect to be close to the MAP associated with maximal oxygen uptake VO 2max. This power P m defines a crossover velocity v m that is close to the velocity that permits exercise with maximal time at MAP. For velocities v>v m the energy cost of running cannot be determined from oxygen uptake alone due to anaerobic energy supply. The running performance of an athlete is not only determined by p(v) (which is fixed by running economy and VO 2max) but depends crucially on the average power P max that can be maximally generated over a duration T over which it can be sustained. To run at the average velocity v max that can be maximally sustained over the time T, the nominal power P(v max) = P max(T) is required, establishing a relation between v max and T. It has been shown14 that P max(T) can be obtained from a self-consistency relation which states that the time average of the instantaneously utilized power P max(T − t) equals the sum of P max(T) and a supplemental power. This supplemental power has aerobic and anaerobic contributions and accounts for an upward shift in the power that is required to complete a run with a given average velocity, for example, due to deteriorating running economy or muscle fatigue. The existence of an upward shift has been observed experimentally and it is essential since its absence would yield a duration independent P max, which contradicts the fact that a given power cannot be sustained for an arbitrary duration. The solution of the self-consistency equation yields $${P}{\max }(T)={P}{\text{m}}-{P}{\text{l}}\mathrm{log}\,\frac{T}{{t}{\text{c}}}\quad {\rm{for}}\ \ T\ge {t}_{\text{c}}\ ,$$ (3) where P l measures the supplemental power supply and t c is a crossover time scale separating different anaerobic and aerobic forms of supplemental power. It can be shown that for T<t c, ({P}{\max }) is given by Eq. (3) with _P l replaced by another constant. By inverting ({P}{\max }(T)) and using the power–velocity relation of Eq. (2), we get the maximal time ({T}{\max }(v)={t}{\text{c}}\exp [({v}{\text{m}}-v)/({\gamma }{\text{l}}{v}{\text{m}})]) over which an average velocity v can be sustained. Here, the constant γ l = P l/(P m − P b) measures endurance ({E}{\text{l}}=\exp (0.1/{\gamma }{\text{l}})), see main text. The shortest time T(d) for covering a distance d follows from solving (T={T}{\max }(v=d/T)) for _T, yielding Eq. (1). It is important for the application to a large, inhomogeneous group of subjects that this model is universal in the sense that it only depends on three parameters v m, t c, and γ l and does not depend directly on any additional, subject-dependent parameters. Performance data analysis We tested whether or not meaningful performance indices can be deduced only from the racing performance of individuals, employing the performance model described before. For each racing season, uniquely labeled by a pair (SID, M-date), two model parameters, v m and γ l, were computed from Eq. (1) applied to all races in the racing season. In general, the time t c must be obtained from the crossover between anaerobic and aerobic regimes, and hence from races that involve both means of energy supply, i.e., events with finishing time shorter and longer than t c. Explicit comparison to racing results and laboratory testing has shown that t c = 6 min is a good approximation on average, and this estimate was used in our data analysis14. We numerically minimized the sum of the squared relative differences between the actual race time and the one predicted by Eq. (1). The nonlinear fitting was based on a Levenberg–Marquardt type algorithm with multiple starting values to minimize probability to converge only to local minimum, and with support for lower and upper parameter bounds. Parameter bounds were chosen as 2 m s−1 ≤ v m ≤ 7 m s−1, 0.039 ≤ γ l ≤ 0.135 corresponding to 2.1 ≤ E l ≤ 13.014. Fits that converged onto these bounds were excluded from further analysis. Training data analysis To quantify training of individuals during the 180-day period before a marathon, we must establish measures based on duration and distances of activities within the training season. We considered an optimal set of three variables that measure quantity, quality, and a combination of quantity and quality. Training volume was quantified by total running distance d train of a training season. To account for possibly varying physiological adaptions during different training modes, training intensity ({p}{{\rm{train}}}={\bar{v}}{{\rm{train}}}/{v}{\text{m}}) was measured by the average running velocity ({\bar{v}}{\rm{train}}) in relation to the characteristic velocity v m that was determined for each race season independently. Finally, the overall training load was evaluated by the TRIMP scale, which is frequently employed in exercise physiology and the design of training. TRIMP is a measure for both volume and intensity of exercise. We assigned to each activity of a training season a TRIMP number using the definition ({\rm{TRIMP}}={T}{{\rm{train}}}{\kappa }{1}(\bar{v}/{v}{\text{m}})\exp ({\kappa }{2}\bar{v}/{v}{\text{m}})) for activity of duration _T train and average velocity (\bar{v}) with κ 1 = 0.64, κ 2 = 1.92 for male subjects, and κ 1 = 0.86, κ 2 = 1.67 for female subjects49. The total training TRIMP number was then obtained by summing the individual TRIMP numbers of all activities within a training season. Usually TRIMP is defined in terms of the average heart rate reserve during exercise which is expected to be well approximated by the ratio (\bar{v}/{v}{\text{m}}). We are interested in the relation between physiological model parameters _v m and E l, and training variables. To measure these relations, we grouped training variables into bins of widths Δ d train = 300 km, Δ p train = 0.025 and Δ TRIMP = 2000. 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C., Maddison, R., Ali, A., Foskett, A. & Gant, N. Rapid directional change degrades GPS distance measurement validity during intermittent intensity running. PLoS ONE9, 1–6 (2014). ArticleCASGoogle Scholar Scott, M. T. U., Scott, T. J. & Kelly, V. G. The validity and reliability of global positioning systems in team sport: a brief review. J. Strength Cond. Res.30, 1470–1490 (2016). ArticlePubMedGoogle Scholar Sreedhara, V. S. M., Mocko, G. M. & Hutchison, R. E. A survey of mathematical models of human performance using power and energy. Sports Med.5, 54 (2019). Google Scholar Tatterson, A. J., Hahn, A. G., Martini, D. T. & Febbraio, M. A. Effects of heat stress on physiological responses and exercise performance in elite cyclists. J. Sci. Med. Sport3,186–193 (2000). Google Scholar Vihma, T. Effects of weather on the performance of marathon runners. Int. J. Biometeorol.54, 297–306 (2010). ArticleADSPubMedGoogle Scholar Mulligan, M., Adam, G. & Emig, T. 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Anaerobic capacity determined by maximal accumulated o 2 deficit. J. Appl. Physiol.64, 50–60 (1988). ArticlePubMedCASGoogle Scholar Hill, A. V. The physiological basis of ahletic records. Lancet206, 481–486 (1925). ArticleGoogle Scholar Keller, J. B. A theory of competitive running. Phys. Today26, 43–47 (1973). ArticleGoogle Scholar Peronnet, F. & Thibault, G. Mathematical analysis of running performance and world running records. J. Appl. Physiol.67, 453–465 (1989). ArticlePubMedCASGoogle Scholar Batliner, M. E., Kipp, S., Grabowski, A. M., Kram, R. & Byrnes, W. C. Does metabolic rate increase linearly with running speed in all distance runners? Sports Med. Int. Open2, E1–E8 (2018). PubMedGoogle Scholar Caminal, P. et al. Validity of the Polar V800 monitor for measuring heart rate variability in mountain running route conditions. Eur. J. Appl. Phys.118, 669–677 (2018). ArticleGoogle Scholar Billat, V., Binsse, V., Petit, B. & Koralsztein, J. J. High level runners are able to maintain a vo2 steady-state below vo2max in an all-out run over their critical velocity. Arch. Physiol. Biochem.106, 38–45 (1998). ArticlePubMedCASGoogle Scholar Corless, R., Gonnet, G., Hare, D., Jeffrey, D. & Knuth, D. On the lambert w function. Adv. Comp. Math.5, 329–359 (1996). ArticleMathSciNetMATHGoogle Scholar di Prampero, P. E., Atchou, G., Brueckner, J. C. & Moia, C. The energetics of endurance running. Eur. J. Appl. Phys.55, 259 (1986). Google Scholar di Prampero, P. E. et al. Energetics of best performances in middle-distance running. J. Appl. Physiol.74, 2318 (1993). ArticlePubMedGoogle Scholar Lazzer, S. et al. The energetics of ultra-endurance running. Eur. J. Appl. Phys.112, 1709 (2012). ArticleGoogle Scholar Daniels, J. T. A physiologist’s view of running economy. Med. Sci. Sports Exerc.17, 332 (1985). PubMedCASGoogle Scholar Manari, D. et al. Vo2max and vo2at: athletic performance and field role of elite soccer players. Sport Sci. Health12, 221–226 (2016). ArticleGoogle Scholar Daniels, J. Daniels’ Running Formula (Human Kinetics, 2013), 3rd edn. Farrell, P. A., Wilmore, J. H., Coyle, E. F., Billing, J. E. & Costill, D. L. Plasma lactate accumulation and distance running performance. Med. Sci. Sports11, 338–344 (1979). PubMedCASGoogle Scholar Jones, A. & Carter, H. The effect of endurance training on parameters of aerobic fitness. Sports Med.29, 373–386 (2000). ArticlePubMedCASGoogle Scholar Billat, V., Renoux, J., Pinoteau, J., Petit, B. & Koralsztein, J. Reproducibility of running time to exhaustion at vo2max in subelite runners. Med. Sci. Sports Exerc.26, 254–257 (1994). ArticlePubMedCASGoogle Scholar Esteve-Lanao, J., San Juan, A., Earnest, C., Foster, C. & Lucia, A. How do endurance runners actually train? Relationship with competition performance. Med. Sci. Sports Exerc.37, 496–504 (2005). ArticlePubMedGoogle Scholar Esteve-Lanao, J., Foster, C., Seiler, S. & Lucia, A. Impact of training intensity distribution on performance in endurance athletes. J. Strength Cond. Res.21, 943–949 (2007). PubMedGoogle Scholar Bassett, D. & Howley, E. Limiting factors for maximum oxygen uptake and determinants of endurance performance. Med. Sci. Sports Exerc.32, 70–84 (2000). ArticlePubMedGoogle Scholar Kubo, K., Tabata, T., Ikebukuro, T., Igarashi, K. & Tsunoda, N. A longitudinal assessment of running economy and tendon properties in long-distance runners. J. Strength Cond. Res.24, 1724–1731 (2010). ArticlePubMedGoogle Scholar Banister, E. W. & Calvert, T. W. Planning for future performance: implications for long term training. Can. J. Appl. Sport Sci.5, 170 (1980). PubMedCASGoogle Scholar Manzi, V., Iellamo, F., Impellizzeri, F., D’Ottavio, S. & Castagna, C. Relation between individualized training impulses and performance in distance runners. Med. Sci. Sports Exerc.41, 2090–2096 (2009). ArticlePubMedGoogle Scholar Margaria, R., Cerretelli, P., Aghemo, P. & Sassi, G. Enery cost of running. J. Appl. Physiol.18, 367–370 (1963). ArticlePubMedCASGoogle Scholar di Prampero, P. E. The energy cost of human locomotion on land and in water. Int. J. Sports Med.7, 55–72 (1986). ArticlePubMedGoogle Scholar Ferretti, G., Bringard, A. & Perini, R. An analysis of performance in human locomotion. Eur. J. Appl. Physiol.111, 391–401 (2011). ArticlePubMedGoogle Scholar Tam, E. et al. Energetics of running in top-level marathon runners from kenya. Eur. J. Appl. Physiol.112, 3797–3806 (2012). ArticlePubMedGoogle Scholar Hoogkamer, W. et al. A comparison of the energetic cost of running in marathon racing shoes. Sports Med.48, 1009–1019 (2018). ArticlePubMedGoogle Scholar Polar Flow. (2019). Borresen, J. & Lambert, M. I. The quantification of training load, the training response and the effect on performance. Sports Med.39, 779–795 (2009). ArticlePubMedGoogle Scholar Download references Acknowledgements The support by Polar Electro in obtaining the exercise data from their data base is greatly acknowledged. Author information Authors and Affiliations Université Paris-Saclay, CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, 91405, Orsay, France Thorsten Emig Polar Electro Oy, Professorintie 5, 90440, Kempele, Finland Jussi Peltonen Authors 1. Thorsten EmigView author publications Search author on:PubMedGoogle Scholar 2. Jussi PeltonenView author publications Search author on:PubMedGoogle Scholar Contributions T.E. designed the study and performed the numerical analysis. T.E. and J.P. wrote the paper. Corresponding author Correspondence to Thorsten Emig. Ethics declarations Competing interests The authors declare no competing interests. Additional information Peer review informationNature Communications thanks Guido Ferretti and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary information Reporting Summary Peer Review File Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit Reprints and permissions About this article Cite this article Emig, T., Peltonen, J. Human running performance from real-world big data. Nat Commun11, 4936 (2020). Download citation Received: 18 January 2020 Accepted: 08 September 2020 Published: 06 October 2020 DOI: Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Subjects Biological physics Computational biology and bioinformatics Physiology This article is cited by Smart Textiles for Personalized Sports and Healthcare Ziao Xu Chentian Zhang Bin Ding Nano-Micro Letters (2025) DVT: a high-throughput analysis pipeline for locomotion and social behavior in adult Drosophila melanogaster Kai Mi Yiqing Li Xingyin Liu Cell & Bioscience (2023) Recommendations for marathon runners: on the application of recommender systems and machine learning to support recreational marathon runners Barry Smyth Aonghus Lawlor Ciara Feely User Modeling and User-Adapted Interaction (2022) Learning from machine learning: prediction of age-related athletic performance decline trajectories Christoph Hoog Antink Anne K. Braczynski Bergita Ganse GeroScience (2021) Download PDF Sections Figures References Abstract Introduction Results Discussion Methods Data availability Code availability References Acknowledgements Author information Ethics declarations Additional information Supplementary information Rights and permissions About this article This article is cited by Advertisement Fig. 1: Flowchart of the exercise data analysis. View in articleFull size image Fig. 2: Probability density of model parameters. View in articleFull size image Fig. 3: Correlation between performance indices and marathon race time (model estimates for 24,504 racing seasons are shown here). View in articleFull size image Fig. 4: Estimate of Marathon race time from the racing season (for 9410 seasons). View in articleFull size image Fig. 5: Correlations between performance indices and training characteristics. View in articleFull size image Lieberman, D. E. & Bramble, D. M. The evolution of marathon running. 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ArticlePubMedPubMed CentralCASGoogle Scholar Rawstorn, J. C., Maddison, R., Ali, A., Foskett, A. & Gant, N. Rapid directional change degrades GPS distance measurement validity during intermittent intensity running. PLoS ONE9, 1–6 (2014). ArticleCASGoogle Scholar Scott, M. T. U., Scott, T. J. & Kelly, V. G. The validity and reliability of global positioning systems in team sport: a brief review. J. Strength Cond. Res.30, 1470–1490 (2016). ArticlePubMedGoogle Scholar Sreedhara, V. S. M., Mocko, G. M. & Hutchison, R. E. A survey of mathematical models of human performance using power and energy. Sports Med.5, 54 (2019). Google Scholar Tatterson, A. J., Hahn, A. G., Martini, D. T. & Febbraio, M. A. Effects of heat stress on physiological responses and exercise performance in elite cyclists. J. Sci. Med. Sport3,186–193 (2000). Google Scholar Vihma, T. Effects of weather on the performance of marathon runners. Int. J. Biometeorol.54, 297–306 (2010). ArticleADSPubMedGoogle Scholar Mulligan, M., Adam, G. & Emig, T. A minimal power model for human running performance. PLoS ONE13, 1–26 (2018). ArticleCASGoogle Scholar Hughson, R., Orok, C. & Staudt, L. A high velocity treadmill running test to assess endurance running potential. Int. J. Sports Med.5, 23–25 (1984). ArticlePubMedCASGoogle Scholar Kipp, S., Kram, R. & Hoogkamer, W. Extrapolating metabolic savings in running: Implications for performance predictions. Front. Physiol.10, 79 (2019). ArticlePubMedPubMed CentralGoogle Scholar Joyner, M. J. & Coyle, E. F. Endurance exercise performance: the physiology of champions. J. Physiol.586, 35–44 (2008). ArticlePubMedCASGoogle Scholar Sproule, J. Running economy deteriorates following 60 min of exercise at 80% vo2max. Eur. J. Appl. Physiol.77, 366–371 (1998). ArticleCASGoogle Scholar Thomas, D. Q., Fernhall, B. & Granat, H. Changes in running economy during a 5-km run in trained men and women runners. J. Strength Cond. Res.13, 162–167 (1999). Google Scholar Medbo, J. I. et al. Anaerobic capacity determined by maximal accumulated o 2 deficit. J. Appl. Physiol.64, 50–60 (1988). ArticlePubMedCASGoogle Scholar Hill, A. V. The physiological basis of ahletic records. Lancet206, 481–486 (1925). ArticleGoogle Scholar Keller, J. B. A theory of competitive running. Phys. Today26, 43–47 (1973). ArticleGoogle Scholar Peronnet, F. & Thibault, G. Mathematical analysis of running performance and world running records. J. Appl. Physiol.67, 453–465 (1989). ArticlePubMedCASGoogle Scholar Batliner, M. E., Kipp, S., Grabowski, A. M., Kram, R. & Byrnes, W. C. Does metabolic rate increase linearly with running speed in all distance runners? Sports Med. Int. Open2, E1–E8 (2018). PubMedGoogle Scholar Caminal, P. et al. Validity of the Polar V800 monitor for measuring heart rate variability in mountain running route conditions. Eur. J. Appl. Phys.118, 669–677 (2018). ArticleGoogle Scholar Billat, V., Binsse, V., Petit, B. & Koralsztein, J. J. High level runners are able to maintain a vo2 steady-state below vo2max in an all-out run over their critical velocity. Arch. Physiol. Biochem.106, 38–45 (1998). ArticlePubMedCASGoogle Scholar Corless, R., Gonnet, G., Hare, D., Jeffrey, D. & Knuth, D. On the lambert w function. Adv. Comp. Math.5, 329–359 (1996). ArticleMathSciNetMATHGoogle Scholar di Prampero, P. E., Atchou, G., Brueckner, J. C. & Moia, C. The energetics of endurance running. Eur. J. Appl. Phys.55, 259 (1986). Google Scholar di Prampero, P. E. et al. Energetics of best performances in middle-distance running. J. Appl. Physiol.74, 2318 (1993). ArticlePubMedGoogle Scholar Lazzer, S. et al. The energetics of ultra-endurance running. Eur. J. Appl. Phys.112, 1709 (2012). ArticleGoogle Scholar Daniels, J. T. A physiologist’s view of running economy. Med. Sci. Sports Exerc.17, 332 (1985). PubMedCASGoogle Scholar Manari, D. et al. Vo2max and vo2at: athletic performance and field role of elite soccer players. Sport Sci. Health12, 221–226 (2016). ArticleGoogle Scholar Daniels, J. Daniels’ Running Formula (Human Kinetics, 2013), 3rd edn. Farrell, P. A., Wilmore, J. H., Coyle, E. F., Billing, J. E. & Costill, D. L. Plasma lactate accumulation and distance running performance. Med. Sci. Sports11, 338–344 (1979). PubMedCASGoogle Scholar Jones, A. & Carter, H. The effect of endurance training on parameters of aerobic fitness. Sports Med.29, 373–386 (2000). ArticlePubMedCASGoogle Scholar Billat, V., Renoux, J., Pinoteau, J., Petit, B. & Koralsztein, J. Reproducibility of running time to exhaustion at vo2max in subelite runners. Med. Sci. Sports Exerc.26, 254–257 (1994). ArticlePubMedCASGoogle Scholar Esteve-Lanao, J., San Juan, A., Earnest, C., Foster, C. & Lucia, A. How do endurance runners actually train? Relationship with competition performance. Med. Sci. Sports Exerc.37, 496–504 (2005). ArticlePubMedGoogle Scholar Esteve-Lanao, J., Foster, C., Seiler, S. & Lucia, A. Impact of training intensity distribution on performance in endurance athletes. J. Strength Cond. Res.21, 943–949 (2007). PubMedGoogle Scholar Bassett, D. & Howley, E. Limiting factors for maximum oxygen uptake and determinants of endurance performance. Med. Sci. Sports Exerc.32, 70–84 (2000). ArticlePubMedGoogle Scholar Kubo, K., Tabata, T., Ikebukuro, T., Igarashi, K. & Tsunoda, N. A longitudinal assessment of running economy and tendon properties in long-distance runners. J. Strength Cond. Res.24, 1724–1731 (2010). ArticlePubMedGoogle Scholar Banister, E. W. & Calvert, T. W. Planning for future performance: implications for long term training. Can. J. Appl. Sport Sci.5, 170 (1980). PubMedCASGoogle Scholar Manzi, V., Iellamo, F., Impellizzeri, F., D’Ottavio, S. & Castagna, C. Relation between individualized training impulses and performance in distance runners. Med. Sci. Sports Exerc.41, 2090–2096 (2009). ArticlePubMedGoogle Scholar Margaria, R., Cerretelli, P., Aghemo, P. & Sassi, G. Enery cost of running. J. Appl. Physiol.18, 367–370 (1963). ArticlePubMedCASGoogle Scholar di Prampero, P. E. The energy cost of human locomotion on land and in water. Int. J. Sports Med.7, 55–72 (1986). ArticlePubMedGoogle Scholar Ferretti, G., Bringard, A. & Perini, R. An analysis of performance in human locomotion. Eur. J. Appl. Physiol.111, 391–401 (2011). ArticlePubMedGoogle Scholar Tam, E. et al. Energetics of running in top-level marathon runners from kenya. Eur. J. Appl. Physiol.112, 3797–3806 (2012). ArticlePubMedGoogle Scholar Hoogkamer, W. et al. A comparison of the energetic cost of running in marathon racing shoes. Sports Med.48, 1009–1019 (2018). ArticlePubMedGoogle Scholar Polar Flow. (2019). Borresen, J. & Lambert, M. I. The quantification of training load, the training response and the effect on performance. Sports Med.39, 779–795 (2009). 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2521	https://asccc-oeri.org/wp-content/uploads/2021/04/t-table.pdf	t-distribution Table t-distribution 1-tailed α df 0.05 0.025 0.01 0.005 1 6.314 12.706 31.821 63.657 2 2.920 4.303 6.965 9.925 3 2.353 3.182 4.541 5.841 4 2.132 2.776 3.747 4.604 5 2.015 2.571 3.365 4.032 6 1.943 2.447 3.143 3.707 7 1.895 2.365 2.998 3.499 8 1.860 2.306 2.896 3.355 9 1.833 2.262 2.821 3.250 10 1.812 2.228 2.764 3.169 11 1.796 2.201 2.718 3.106 12 1.782 2.179 2.681 3.055 13 1.771 2.160 2.650 3.012 14 1.761 2.145 2.624 2.977 15 1.753 2.131 2.602 2.947 16 1.746 2.120 2.583 2.921 17 1.740 2.110 2.567 2.898 18 1.734 2.101 2.552 2.878 19 1.729 2.093 2.539 2.861 20 1.725 2.086 2.528 2.845 21 1.721 2.080 2.518 2.831 22 1.717 2.074 2.508 2.819 23 1.714 2.069 2.500 2.807 24 1.711 2.064 2.492 2.797 25 1.708 2.060 2.485 2.787 t-distribution 1-tailed α df 0.05 0.025 0.01 0.005 26 1.706 2.056 2.479 2.779 27 1.703 2.052 2.473 2.771 28 1.701 2.048 2.467 2.763 29 1.699 2.045 2.462 2.756 30 1.697 2.042 2.457 2.750 31 1.696 2.040 2.453 2.744 32 1.694 2.037 2.449 2.738 33 1.692 2.035 2.445 2.733 34 1.691 2.032 2.441 2.728 35 1.690 2.030 2.438 2.724 36 1.688 2.028 2.434 2.719 37 1.687 2.026 2.431 2.715 38 1.686 2.024 2.429 2.712 39 1.685 2.023 2.426 2.708 40 1.684 2.021 2.423 2.704 50 1.676 2.009 2.403 2.678 60 1.671 2.000 2.390 2.660 70 1.667 1.994 2.381 2.648 80 1.664 1.990 2.374 2.639 90 1.662 1.987 2.368 2.632 100 1.660 1.984 2.364 2.626 z 1.645 1.960 2.326 2.576 t-distribution 2-tailed α df 0.10 0.05 0.02 0.01 1 6.314 12.706 31.821 63.657 2 2.920 4.303 6.965 9.925 3 2.353 3.182 4.541 5.841 4 2.132 2.776 3.747 4.604 5 2.015 2.571 3.365 4.032 6 1.943 2.447 3.143 3.707 7 1.895 2.365 2.998 3.499 8 1.860 2.306 2.896 3.355 9 1.833 2.262 2.821 3.250 10 1.812 2.228 2.764 3.169 11 1.796 2.201 2.718 3.106 12 1.782 2.179 2.681 3.055 13 1.771 2.160 2.650 3.012 14 1.761 2.145 2.624 2.977 15 1.753 2.131 2.602 2.947 16 1.746 2.120 2.583 2.921 17 1.740 2.110 2.567 2.898 18 1.734 2.101 2.552 2.878 19 1.729 2.093 2.539 2.861 20 1.725 2.086 2.528 2.845 21 1.721 2.080 2.518 2.831 22 1.717 2.074 2.508 2.819 23 1.714 2.069 2.500 2.807 24 1.711 2.064 2.492 2.797 25 1.708 2.060 2.485 2.787 t-distribution 2-tailed α df 0.10 0.05 0.02 0.01 26 1.706 2.056 2.479 2.779 27 1.703 2.052 2.473 2.771 28 1.701 2.048 2.467 2.763 29 1.699 2.045 2.462 2.756 30 1.697 2.042 2.457 2.750 31 1.696 2.040 2.453 2.744 32 1.694 2.037 2.449 2.738 33 1.692 2.035 2.445 2.733 34 1.691 2.032 2.441 2.728 35 1.690 2.030 2.438 2.724 36 1.688 2.028 2.434 2.719 37 1.687 2.026 2.431 2.715 38 1.686 2.024 2.429 2.712 39 1.685 2.023 2.426 2.708 40 1.684 2.021 2.423 2.704 50 1.676 2.009 2.403 2.678 60 1.671 2.000 2.390 2.660 70 1.667 1.994 2.381 2.648 80 1.664 1.990 2.374 2.639 90 1.662 1.987 2.368 2.632 100 1.660 1.984 2.364 2.626 z 1.645 1.960 2.326 2.576
2522	https://www.splashlearn.com/math-vocabulary/subtracting-polynomials	Published Time: 2023-07-13T16:25:10+00:00 Subtracting Polynomials: Definition, Methods, Examples Parents Explore by Grade Preschool (Age 2-5)KindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 Explore by Subject Math ProgramEnglish Program More Programs Homeschool ProgramSummer ProgramMonthly Mash-up Helpful Links Parenting Blog Success Stories Support Gifting Also available on Educators Teach with Us For Teachers For Schools and Districts Data Protection Addendum Impact Success Stories Resources Lesson Plans Classroom Tools Teacher Blog Help & Support More Programs SpringBoard Summer Learning Our Library All By Grade PreschoolKindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 By Subject MathEnglish By Topic CountingAdditionSubtractionMultiplicationPhonicsAlphabetVowels One stop for learning fun! Games, activities, lessons - it's all here! 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Horizontal Method of Subtracting Polynomials Vertical Method of Subtracting Polynomials Solved Examples of Subtracting Polynomials Practice Problems on Subtracting Polynomials Frequently Asked Questions about Subtracting Polynomials What Is Subtracting Polynomials? Subtracting polynomials follows a similar procedure as adding polynomials. We just have to deal with the minus sign (−) in between. Consider the expression p(x)−q(x) representing the subtraction of polynomials. Here, we are subtracting q(x) from p(x). Due to the minus sign before q(x), the signs of all the terms in the polynomial q(x) are changed (“+ changes to −” and “− changes to +”). Next, we simplify by combining the like terms and find the answer. In simple words, when subtracting polynomials, we reverse the signs of each term in the second polynomial and then add the polynomials as usual. Let us see how to subtract polynomials. Recommended Games Find the Difference by Subtracting Multiples of 10 Game Play Match by Subtracting from a Teen Number Game Play Measure Lengths in Half-inches by Subtracting Game Play Measure Lengths in Inches by Subtracting Game Play Measure Lengths on a Ruler by Subtracting Game Play Solve by Subtracting Multiples of 100 Game Play Solve Word Problems on Subtracting Money Game Play Subtracting 1-Digit Number from 3-Digit Number Game Play Subtracting 10 from 3-Digit Numbers Game Play Subtracting 10 from Teen Numbers Game Play More Games Methods of Subtracting Polynomials There are two ways to subtract polynomials: Subtracting polynomials horizontally Subtracting polynomials vertically Regardless of which method of subtracting polynomials we use, the idea and the procedure is the same. The only difference is the arrangement or placement of polynomials during the subtraction operation. Recommended Worksheets More Worksheets Horizontal Method of Subtracting Polynomials Take a look at the steps required for the horizontal method of subtracting polynomials with an example. Example: Subtract P\=2x3−3y2+3z from Q\=7y2+5x3–2z. Here, P\=2x3−3y2+3z and Q\=7y2+5x3–2z We have to find Q – P. Step 1: Arrange the two polynomials in the standard forms. Q\=7y2+5x3–2z\=5x3+7y2−2z P\=2x3−3y2+3z is already in the standard form. Step 2: Place them horizontally separated by minus sign (–) such that the second polynomial is in the parentheses. Q−P\=(5x3+7y2−2z)–(2x3−3y2+3z) Step 3: Remove the parentheses. For the second polynomial, change the sign of each term after removing the parentheses. Q−P\=5x3+7y2−2z–(2x3−3y2+3z) Q−P\=7y2+5x3–2z–2x3+3y2−3z Step 4: Arrange all like terms together and solve. Q−P\=5x3−2x3+7y2+3y2−2z−3z Q−P\=3x3+10y2−5z Vertical Method of Subtracting Polynomials In the vertical method, all the equations are arranged in columns according to the like terms. The signs are changed and the operation takes place. Take a look at the example of a vertical method of subtracting polynomials given below: Find: (7x–3y+6z)−x–4y–2z using the vertical method. Step 1: Arrange the two polynomials in the standard form. Here, both polynomials are already in the standard form. Step 2: Arrange the expressions one below the other such that the polynomial being subtracted is at the bottom. Align the like terms. There will be two rows, in which the lower row is subtracted from the upper row. Step 3: Change the signs of all the terms of the second row; positive changes to negative and negative sign changes to positive. Finally, add the like terms. Facts on Subtracting Polynomials Subtracting polynomials is the same as adding the opposite. For example, 3-5 and 3 + (-5) represent the same thing. Similarly, (7x−5y)−(3x+2y) and (7x−5y)+(−3x−2y) are the same. When subtracting polynomials, it is important to distribute the subtraction sign to each term of the polynomial being subtracted. This means changing the sign of each term before combining like terms. In algebra, the like terms are terms having the same variable raised to the same power. In other words, their variable part is the same such that the same variables have the same exponent powers. Conclusion In this article, we learned to subtract polynomial expressions, different methods to subtract polynomials, facts and examples. Now, let us practice solving problems by subtracting polynomials. Solved Examples of Subtracting Polynomials 1. Subtract 2x−5y+3z from 5x+9y−2z using the horizontal method. Solution: We have to find (5x+9y–2z)−(2x−5y+3z). Remove the parentheses by multiply the negative sign with the terms \=5x+9y−2z–2x+5y−3z Arrange the like terms together. \=5x–2x+9y+5y−2z−3z \=3x+14y−5z 2. Subtract 4x−10y+15z from 5x+8y−20z using the horizontal method. Solution: (4x−10y+15z)−(5x+8y−20z) Sign change of the second polynomial by removing the parentheses and multiplying \=4x−10y+15z−5x−8y+20z Combine the like terms. \=4x−5x−10y−8y+15z+20z \=−x−18y+35z 3. Subtract 5x–3y–4z from 7x–6y+6z using the vertical method. Solution: We have to find 7x−6y+6z−(5x–3y–4z). Arrange the expression in two rows with like terms in the same column. Change the sign of each term of the second row and proceed with the operation. 4. Subtract 3a3+5a2–7a+10 from 6a3−8a2+a+10. Solution: 6a3−8a2+a+10−(3a3+5a2–7a+10) Arrange the expression in two rows with like terms in the same column. Change the sign of each term of the second row and proceed with the operation. Thus, 6a3−8a2+a+10−(3a3+5a2–7a+10) \=3a3−13a2+8a−0 5. Subtract the polynomials horizontally 15x3+12y2−8z−7 from 20x3−6y2+31z+9. Solution: We have to find 20x3−6y2+31z+9−(15x3+12y2−8z−7). Sign change of the second polynomial through the parentheses \=20x3−6y2+31z+9−15x3−12y2+8z+7 Combine the like terms. \=20x3−15x3−6y2−12y2+31z+8z+9+7 \=5x3−18y2+39z+16 Practice Problems on Subtracting Polynomials Subtracting Polynomials: Definition, Methods, Examples, Steps, FAQs Attend this quiz & Test your knowledge. 1 Solve: (5x−2y+1)−(2x−7y+4). 3x+5y−3 7x+5y+5 3x−5y−3 3x−9y+3 CorrectIncorrect Correct answer is: 3x+5y−3 (5x−2y+1)−(2x−7y+4) \=3x+5y−3 2 (12p2+15p+3)−(2p2+17p−2)\= 10p2+2p−5 10p2−2p+5 10p2+2p−1 10p2−2p+1 CorrectIncorrect Correct answer is: 10p2−2p+5 (12p2+15p+3)−(2p2+17p−2) \=10p2−2p+5 3 (x+y)−(y−x)\= 2x 2y 2x+2y 2x−2y CorrectIncorrect Correct answer is: 2x (x+y)−(y−x)\=x+y−y+x\=2x Frequently Asked Questions about Subtracting Polynomials What property is used when subtracting polynomials? In the procedure of subtracting polynomials, the distributive property is used. This is also used while adding polynomials. Example: (3x2+2x+6)+(x2−6x−10) \=(3+1)x2+(2−6)x+(6−10) \=4x2−4x−4 What is the rule followed while subtracting polynomials? There are two important rules to follow while subtracting polynomials. They are: Arrange the like terms together while performing subtraction. Signs of each term of the subtracting polynomial should be changed. That is “+” changes to “−” and “−” changes to “+.” How can subtracting polynomials be simplified? Subtracting polynomials can be simplified by converting the problem from subtraction to addition. Change all the signs of the terms of the second polynomials through the parentheses, that is “+” changes to “−” and “-” changes to “+.” Once it is done, the normal addition of polynomials is performed. 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2523	https://en.wikipedia.org/wiki/Directory_assistance	Jump to content Directory assistance Deutsch Español Français 日本語 Norsk bokmål Suomi Svenska Edit links From Wikipedia, the free encyclopedia Finding out phone-related data: free or fee \| \| \| --- \| \| \| This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Directory assistance" – news · newspapers · books · scholar · JSTOR (September 2024) (Learn how and when to remove this message) \| In telecommunications, directory assistance or directory inquiries is a phone service used to find out a specific telephone number and/or address of a residence, business, or government entity. Technology [edit] Directory assistance systems incorporate a wide range of automation to reduce the cost of human operators. Almost all systems use custom database software to locate listings quickly. Most directory assistance systems use automated readback systems to give out the phone number. This frees the directory assistance operator to move on to another caller as soon as the correct listing is located. Some systems have "store and forward" technology which records the city and state the caller is requesting; it then plays the recording to the operator before they answer and ask the caller to further specify. Interactive voice response systems have been added to many directory assistance systems. These complex systems use speech recognition and recorded speech or speech synthesis to handle the entire call without live operator intervention. Most systems recognize location and listing. If recognition confidence is high, the best result is played to the caller. If confidence is low, the caller's request is played back to a live operator, who locates the correct listing.[citation needed] North America [edit] In the North American Numbering Plan (covering Canada, the United States, and parts of the Caribbean), directory assistance may be contacted by dialing 411 (one of the N11 codes). To get a listing in a remote or non-local area code, directory assistance is available at 1-area code-555-1212. In some cases, a 411 call from a landline will yield local, national, and sometimes international listings. Most telephone companies permit up to two listings per 411 calls. All wireless carriers offer nationwide listings with 411 except Xfinity Mobile, and some offer additional Enhanced Directory Assistance services.[citation needed] However, wireless numbers for residential customers are not available via 411. In 2021, AT&T ended directory assistance service for wireless subscribers and in 2023 ended directory assistance services for digital landline subscribers, AT&T's directory assistance listings are now inaccessible as of 2024.[original research?] Billing [edit] Historically, the tariffs for wireline telephone service allowed subscribers to place a certain number of directory assistance calls for free each month. More recently,[when?] telephone companies are charging subscribers for every directory assistance call.[citation needed] U.S. wireline telephone companies classify DA into four rate classes: 411 LDA: Local Directory Assistance. 411 is dialed, and the operator is requested to search for a listing in a group of area codes local to the caller (LATA). Example: the caller lives in area code 630 (Oak Brook, Illinois) and requests a listing for a business in area code 312 (Chicago, Illinois). In this case, AT&T Illinois bills the call. 411 NDA: National Directory Assistance. 411 is dialed and the operator is requested to search for a listing in an area code not local to the caller. For example: The caller lives in area code 630 (Oak Brook, Illinois) and requests a listing for a business in area code 213 (Los Angeles, California). In this case, AT&T Illinois bills the call. (area code) 555–1212: National Directory Assistance. This example assumes the caller is in Oak Brook, Illinois (area code 630) and uses Verizon as their long-distance carrier. Example: The caller is looking for a listing in Los Angeles, California (area code 213) and dials 213-555-1212. In this case, Verizon bills the call. 00 and ask for the international directory assistance operator. AT&T provides International Directory Assistance calls. Toll-free directory assistance [edit] In the U.S., directory assistance for companies with toll-free "800 numbers" (with area codes 800, 833, 844, 855, 866, 877, and 888) was available from toll-free directory assistance, reachable by dialing 1-800-555-1212, for many decades until it was discontinued in 2020.[citation needed] Toll-free directory assistance was provided by telecommunication providers, namely AT&T and Verizon, as mandated by the Federal Communications Commission. Companies requested to have their toll-free number listed, and paid the providers each time their phone number was released to a toll-free directory-assistance caller. In 1999, AT&T applied for permission to discontinue this service, but it remained active until the summer of 2020.[citation needed] An automated disconnection recording now plays when calling the 1-800-555-1212 number.[citation needed] Directory assistance data sources [edit] The service of 411 queries is often outsourced to a call centre that specializes in that function. Historically, when a single carrier provided most of the telephony services for a region, the data used to satisfy the search could come exclusively from that carrier's subscriber rolls. Today, when the market is fragmented amongst many carriers, the data must be aggregated by a data aggregator specializing in directory listings. The data aggregator distributes the data to the 411 services either on a "live" basis, actually servicing each query, or by periodically transferring large swaths of listings to the call center's systems for local searching. The data aggregator collects the data from the rolls of many telecommunication carriers. Some carriers, such as Vonage, do not send their customer rolls to the aggregator. Companies specializing in free directory assistance [edit] Private companies have entered the directory assistance market by offering free directory assistance. They are available in most states like Connecticut and others. Customers often must listen to an advertisement prior to receiving directory services. Australia [edit] In Australia, there are two standard directory assistance numbers that can be accessed from any phone provider; these are 1223 for national directory assistance and 1225 for international directory assistance. Other directory service numbers are carrier specific and can only be accessed by customers of that particular provider. For example, Sensis on 1234 is a premium operator-assist directory service that only certain Telstra customers are able to access. Other operator-assist directory services do exist, for example, CallConnect on 12456; however, these numbers are not guaranteed to be accessible from all phone providers within Australia. United Kingdom [edit] In the United Kingdom directory enquiries services (sometimes abbreviated as "DQ") are provided by a variety of different companies, with a variety of call charges, each company reached by dialing a six-digit number beginning with 118. These companies supply information from the Operator Services Information System (OSIS), which is run by Directory Solutions, a division of BT Wholesale. OSIS accepts updates from telecoms providers seven days a week, and supplies that information to the inquiry companies six days a week. As of 2017[update], there were over 200 providers. 118 118 (The Number) was the second most-expensive number at £11.23 for a 90-second call, but accounted for 40% of DQ calls, mostly due to heavy advertising. Until 23 August 2003 directory inquiries were available by dialing 192 for numbers in Britain, and 153 for foreign numbers, with the service supplied by the caller's telephone company. Until the 1990s, the service was free to use; then charges were introduced from 2 April 1991, although for some years directory inquiries continued to be free from payphones. Support for 192 and 153 was replaced on 24 August 2003 by competitive directory inquiries services using different numbers beginning "118". Calls to DQ services declined by an average of 38% yearly from 2014 to 2017. Pricing [edit] The pricing structure for UK directory inquiries was reformed by Ofcom on 1 July 2015. Call charges are made up of a per-minute access charge set by the caller's telephony provider, plus a per-call and/or per-minute service charge set by the provider of the directory inquiries (or other) service, which is billed by the phone provider and passed on to the service provider. The access charge for calls to 118 numbers varies from 2p to 27p per minute from landlines, or from 4p to 89p per minute from mobiles. It also applies for calls to 084, 087, and 09 numbers, and must be shown prominently in tariff lists. The service charge may be charged per call, per minute, or a combination of the two. Originally, the per-call part could range from 5p to £16, and applies as soon as the call is answered, and the per-minute part could range from 1p to £8, and may apply either from the start of the call or after the first full minute. There were 100 available service charge price points, known as SC001 to SC100. The service charge must be declared alongside the number wherever it is advertised or promoted. The applicable service charge codes are also shown in BT's pricing table, section 2, part 15. Having found the "SC" code for a particular number, it is then necessary to refer to part 19 to find the cost. Following criticism of very high and increasing charges for DQ services, Ofcom introduced a price cap to 2013 levels from 1 April 2019. Service charges, including VAT, may not exceed £3.65 per 90 seconds. The highest level service charge price points were withdrawn from use, with some of those "SC" codes re-defined to have a new, lower charge. Some other expensive price points remain, available for use only with 09 numbers but not 118 numbers. Controversies [edit] A number offering a directory inquiries service allowing people to request to be put through to a mobile phone number was established in June 2009. 118 800 proved to be controversial, however, when it was revealed that it was making available 15 million mobile numbers that it had bought from market researchers. Its website was suspended within weeks of its launch so that the company could re-engineer the site to enable the large number of ex-directory requests to be handled more efficiently. The related 118800.co.uk site was discontinued. In 2014, the 118500 service run by BT was fined £225,000 by PhonepayPlus (later renamed as the Phone-paid Services Authority) for over-charging customers and failing to clearly display call costs. BT was also ordered to refund affected customers.[citation needed] Later in 2014, a similar failure to clearly state call costs resulted in a fine for the 118118 service. In 2017, soaring call costs for directory inquiries services, including 118118 and 118500 prompted an Ofcom review of 118 services. A price cap at 2013 rates took effect on 1 April 2019. Some directory inquiries services stand accused of inappropriate methods of promoting their services, effectively scamming people into calling. Various unallocated geographic and non-geographic numbers play an announcement directing callers to call a particular directory inquiries number for help. People may hear this message when they misdial a wanted number or may appear to have missed a call from the unallocated number and hear this message if they call back.[citation needed] Charities [edit] Some services[vague] donate part of their income to charities, such as animal welfare and football clubs[failed verification]. Finland [edit] Teleoperators in Finland are legally obligated to ensure their users' names, addresses, and telephone numbers are collected and published in a telephone directory, and to, for their part, ensure their users have access to a directory inquiry service. Furthermore, teleoperators are obligated to provide this contact information to another company for the purposes of providing a directory inquiry service. In practice, teleoperators hand telephone subscriptions' contact information to Suomen Numeropalvelu Oy, which forms and relays a number database to various directory assistance private companies. Other countries [edit] Brazil In Brazil, 102 has been the number for directory assistance since 2004. China In mainland China, (area code) 114 is dialed for directory assistance in that area code. Ethiopia In Ethiopia, 8123 is dialed for directory assistance. Israel In Israel, 144 or 1344 is dialed for directory assistance. Philippines In the Philippines, 187 is dialed for PLDT and Digitel subscribers. Taiwan In Taiwan, directory assistance is available by dialing 105 from mobile phones, or by dialing 104 from landline phones. Turkey In Turkey, directory assistance is available by dialing 118 80 from mobile phones. The assistance number is also noted in its ads, usually with their jingle, 'yüz on sekiz seksen '. Egypt In Egypt, directory assistance is available by dialing 140 from mobile phones, and from landline phones. See also [edit] 555 (telephone number) Telephone directory (including Directory assistance on CD) References [edit] ^ Jump up to: a b Fred A. Bernstein (9 March 2006). "The 411 on Directory Assistance". The New York Times. ^ AT&T Inc. (16 November 1999). "Discontinuance – AT&T Toll-Free Directory Assistance Service – 1-800-555-1212". FCC Public Notices. Federal Communications Commission. Retrieved 28 December 2007. ^ "Connecticut People Search". connecticutresidentdirectory.com. ^ Jump up to: a b c d "Statement: Directory Enquiries (118) Review". Ofcom. 28 November 2018. Retrieved 8 April 2019. ^ "BT Wholesale Directory Solutions – About Us". British Telecom. Archived from the original on 30 November 2009. Retrieved 13 January 2010. ^ "Directory Enquiry Codes". Magenta Systems. 3 May 2017. Archived from the original on 6 May 2017. ^ "About BT". British Telecom. ^ "UK Calling". Ofcom. 16 September 2014. Archived from the original on 3 April 2017. ^ "UK NGN Call Charges SC001 to SC100". Uboss. 1 July 2016. Archived from the original on 9 June 2016. ^ "Subpart 6: Calls to Service Numbers (numbers starting 084, 087, 09 and 118) from 1 July 2016". Section 2:Call Charges & Exchange Line Services – Part 1: Basic Inland Call Charge. BT. 1 July 2016. Archived from the original on 30 January 2017. ^ "How does UK Calling affect businesses?". Ofcom. 16 September 2014. Archived from the original on 4 April 2017. ^ "BT Price List – Section 2, Part 15". BT.com. Retrieved 7 June 2016. ^ "BT Price List – Part 19: Calls to Directory Enquiry 118 Services". BT.com. Retrieved 7 June 2016. ^ "118 800 To Connect UK to Millions of Mobile Numbers". Real Wire. 9 June 2009. Archived from the original on 12 June 2009. Retrieved 19 July 2009. ^ Jump up to: a b Osborne, Hilary (13 July 2009). "Mobile phone directory suspended". The Guardian. London. Archived from the original on 6 September 2013. Retrieved 14 July 2009. ^ "118800 Mobile Enquiry Service Temporarily Suspended". PR Log. 29 July 2009. Archived from the original on 30 August 2012. ^ Goodman, Rob (3 April 2014). "118 118 fined £80,000 for misleading adverts". Moneywise. Archived from the original on 6 April 2014. ^ Jones, Rupert (3 April 2014). "118 118 fined for lack of clarity over pricing". The Guardian. Archived from the original on 6 April 2014. ^ "Telephone review to ensure value for callers". Ofcom. 12 May 2017. Archived from the original on 12 May 2017. ^ "Ofcom opens investigation into the cost of 118 calls". BBC News. 12 May 2017. Archived from the original on 12 May 2017. ^ "118 Numbers – 118donate". Ethcom. Archived from the original on 17 July 2015. Retrieved 7 June 2016. ^ "Finlex, 917/2014" (PDF). finlex.fi. Archived from the original (PDF) on 18 March 2015. ^ "SNOY". Suomen Numeropalvelu. ^ "Serviços de Utilidade Pública e de Emergência (SUP)" (in Brazilian Portuguese). ANATEL. Retrieved 29 June 2024. ^ "号码百事通" (in Chinese). China Telecom. Archived from the original on 16 February 2015. Retrieved 14 July 2013. ^ "Contact Us". PLDT.com. Retrieved 7 June 2016. Further reading [edit] Lawson, Mark (19 March 2005). "Dial 0 for progress". The Guardian. Wakin, Michele A.; Zimmerman, Don H. (October 1999). "Reduction and Specialization in Emergency and Directory Assistance Calls". Research on Language and Social Interaction. 32 (4): 409–437. doi:10.1207/S15327973rls3204_4. Kopardekar, Parimal; Mital, Anil (October 1994). "The effect of different work-rest schedules on fatigue and performance of a simulated directory assistance operator's task". Ergonomics. 37 (10): 1697–1707. doi:10.1080/00140139408964946. PMID 7957021. Yu, Dong; Ju, Yun-Cheng; Wang, Ye-Yi; Zweig, Geoffrey; Acero, Alex (27 August 2007). "Automated directory assistance system - from theory to practice". Interspeech 2007: 2709–2712. doi:10.21437/Interspeech.2007-65. McSweeny, A. John (1978). "Effects of response cost on the behavior of a million persons: charging for directory assistance in Cincinnati". Journal of Applied Behavior Analysis. 11 (1): 47–51. doi:10.1901/jaba.1978.11-47. PMC 1311267. PMID 16795584. Kamm, C.A.; Shamieh, C.R.; Singhal, S. (November 1995). "Speech recognition issues for directory assistance applications". Speech Communication. 17 (3–4): 303–311. doi:10.1016/0167-6393(95)00023-H. Natarajan, Premkumar; Prasad, Rohit; Schwartz, Richard M.; Makhoul, John (2002). "A scalable architecture for Directory Assistance automation". IEEE International Conference on Acoustics Speech and Signal Processing. doi:10.1109/ICASSP.2002.5743644. ISBN 978-0-7803-7402-7. S2CID 2551571. Bechet, Frederic; De Mori, Renato; Subsol, Gerard (2002). "Dynamic generation of proper name pronunciations for directory assistance". IEEE International Conference on Acoustics Speech and Signal Processing. doi:10.1109/ICASSP.2002.5743825. ISBN 978-0-7803-7402-7. S2CID 15233030. Schramm, H; Rueber, B; Kellner, A (August 2000). "Strategies for name recognition in automatic directory assistance systems". Speech Communication. 31 (4): 329–338. doi:10.1016/S0167-6393(99)00066-7. Retrieved from " Categories: Directory assistance services Telecommunication services Hidden categories: CS1 Brazilian Portuguese-language sources (pt-br) CS1 Chinese-language sources (zh) Articles with short description Short description is different from Wikidata Use dmy dates from January 2020 Articles needing additional references from September 2024 All articles needing additional references All articles with unsourced statements Articles with unsourced statements from September 2024 All articles that may contain original research Articles that may contain original research from September 2024 All articles with vague or ambiguous time Vague or ambiguous time from September 2024 Articles containing potentially dated statements from 2017 All articles containing potentially dated statements Articles with unsourced statements from January 2020 All Wikipedia articles needing clarification Wikipedia articles needing clarification from January 2025 All articles with failed verification Articles with failed verification from January 2025
2524	https://www.homeschoolmath.net/worksheets/square-roots.php	\| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| You are here: Home → Worksheets → Square rootsFree square root worksheets (PDF and html) On this page, you'll find an unlimited supply of printable worksheets for square roots, including worksheets for square roots only (grade 7) or worksheets with square roots and other operations (grades 8-10). Options include the radicand range, limiting the square roots to perfect squares only, font size, workspace, PDF or html formats, and more. If you want the answer to be a whole number, choose "perfect squares," which makes the radicand to be a perfect square (1, 4, 9, 16, 25, etc.). If you choose to allow non-perfect squares, the answer is typically an unending decimal that is rounded to a certain number of digits. The option "Only simplify, no answers as decimals" forces the answer NOT to be given as a rounded decimal, but instead the answer is simplified if possible, and the square root is left in the answer if it cannot be simplified. For example, an answer of √28 will be given in simplified form as 2√7. This option is useful for algebra 1 and 2 courses. You can also make worksheets that include one or two other operations, besides taking a square root. For some extra tips and practice problems on square roots, check out IXL's square roots lesson! Basic instructions for the worksheets Each worksheet is randomly generated and thus unique. The answer key is automatically generated and is placed on the second page of the file. You can generate the worksheets either in html or PDF format — both are easy to print. To get the PDF worksheet, simply push the button titled "Create PDF" or "Make PDF worksheet". To get the worksheet in html format, push the button "View in browser" or "Make html worksheet". This has the advantage that you can save the worksheet directly from your browser (choose File → Save) and then edit it in Word or other word processing program. Sometimes the generated worksheet is not exactly what you want. Just try again! To get a different worksheet using the same options: PDF format: come back to this page and push the button again. Html format: simply refresh the worksheet page in your browser window. Ready-made square root worksheets \| \| \| --- \| \| Solve square roots; easy; perfect squares (grades 7-8) View in browser Create PDF \| Find decimal approximations to square roots; round answers to 3 decimals View in browser Create PDF \| \| Solve square roots with addition and subtraction; perfect squares (grades 8-9) View in browser Create PDF \| Square roots and one other operation; round answers to 3 decimals (grades 9-10) View in browser Create PDF \| \| Solve square roots with other operations; perfect squares (grades 8-9) View in browser Create PDF \| Square roots, non-perfect squares allowed, answers in simplified form (grades 9-10) View in browser Create PDF \| \| \| \| --- \| \| Take square roots from decimals (grades 9-10) View in browser Create PDF \| Challenge: square roots and two other operations; perfect squares (grades 9-10) View in browser Create PDF \| Generator Use the generator to make customized worksheets for square roots. \| \|
2525	https://www.crestolympiads.com/spellbee/frugality	Frugality: Meaning, Usage, Idioms & Fun Facts Explained Toggle navigation Olympiads CREST Mathematics Olympiad (CMO) CREST Mathematics Olympiad (CMO) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST Science Olympiad (CSO) CREST Science Olympiad (CSO) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST English Olympiad (CEO) CREST English Olympiad (CEO) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST Reasoning Olympiad (CRO) CREST Reasoning Olympiad (CRO) Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST Cyber Olympiad (CCO) CREST Cyber Olympiad (CCO) Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST Mental Maths Olympiad (CMMO) CREST Mental Maths Olympiad (CMMO) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12 International Green Warrior Olympiad (IGWO) International Green Warrior Olympiad (IGWO) Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST International Drawing Olympiad (CIDO) CREST International Drawing Olympiad (CIDO) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 CREST International Spell Bee Summer (CSB) CREST International Spell Bee Summer (CSB) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 CREST International Spell Bee Winter (CSBW) CREST International Spell Bee Winter (CSBW) Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 International Teacher Olympiads International Teacher Olympiads Teacher Mathematics Olympiad Teacher Science Olympiad Teacher English Olympiad Teacher Reasoning Olympiad Class wise Class KG Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 FAQs FAQs Exam Dates Exam Syllabus Sample Papers Previous Year Papers Marking Scheme Cut-Offs & Ranking Criteria Awards & Recognition Subject Rankers Subject Cut-Off Zone Definition Contact Us Student Connect Green Warrior Initiatives Interviews Videos Rankholder's Gallery Testimonials Olympiad Exam Blog WhatsApp Channel Contact Us Preparation Olympiad Books Live Classes Free Prep Guides CREST Mathematics Olympiad (CMO) CREST Science Olympiad (CSO) CREST English Olympiad (CEO) CREST Intl Spell Bee - Summer (CSB) CREST Intl Spell Bee - Winter (CSBW) Mental Mathematics Olympiad (CMMO) Green Warrior Olympiad (IGWO) Free Downloadable Worksheets CREST Mathematics Olympiad (CMO) CREST Science Olympiad (CSO) CREST English Olympiad (CEO) Mental Mathematics Olympiad (CMMO) Free Trial School Registration Become a Coordinator Explore More Brain Yoga Blogs Country Wise Olympiads Contact Us Register now Free Trial Student Registration Teacher Registration School Registration Become a Coordinator Login here Olympiad Exam Registration Started for 2025-26 \| Check 2025-26 Olympiad Exam Dates \| Join our WhatsApp Channel for Exam Updates \| Buy 2025-26 Olympiad books here \| Download Free Maths Olympiad Printable Worksheets \| Download Free Science Olympiad Printable Worksheets \| Download Free English Olympiad Printable Worksheets\| Check Previous Years Papers CMO arrow_drop_down CSO arrow_drop_down CEO arrow_drop_down CRO arrow_drop_down CCO arrow_drop_down CMMO arrow_drop_down IGWO arrow_drop_down CIDO arrow_drop_down CSB arrow_drop_down CSBW arrow_drop_down FAQs Individual Registration Register Your School Exam Schedule Ranking Criteria Become a Co-ordinator Contact Us info assignment_ind account_balance link phone Spell Bee Winter Sample Paper Class 7 / Class 7 CSB Winter Word List / Frugality Spelling & Meaning Spell Bee Word: frugality Overview dashboard Idioms and Phrases format_quote Usage Examples menu_book Fun Facts emoji_objects Famous Quote format_quote Multiple Choice Questions help Associated Words language Participate in Spell Bee spellcheck Basic Details Word: Frugality Part of Speech: Noun Meaning: The quality of being careful with money and not wasting it. Synonyms: Thriftiness, economy, prudence Antonyms: Wastefulness, extravagance, lavishness Idioms and Phrases Penny pincher: A person who is very careful about spending money. Example: "He is such a penny pincher that he even counts the change he receives." Cut costs: To reduce expenses. Example: "To save for their holiday, they decided to cut costs on dining out." Usage Examples Example 1: Her frugality allowed her to save enough money to buy her first car. Example 2: Learning about frugality can help young people manage their money better in the future. Example 3: The family's frugality meant they could enjoy a holiday every year without going into debt. Fun Fact Did you know that the word "frugality" comes from the Latin word "frugalis," which means "fruitful" or "economical"? It originally referred to being productive in the use of resources. Famous Quote "Frugality is founded on the principal that all riches have limits." - Edmund Burke Multiple Choice Questions (MCQs) 1. What does "frugality" mean? a) Being careless with money b) The quality of being careful with money c) Spending money on luxuries d) Earning a high salary Answer: b) The quality of being careful with money 2. Which of these is a synonym of "frugality"? a) Extravagance b) Thriftiness c) Carelessness d) Splurging Answer: b) Thriftiness 3. Which of the following is an antonym of "frugality"? a) Economy b) Wastefulness c) Prudence d) Caution Answer: b) Wastefulness 4. Which sentence uses "frugality" correctly? a) His frugality led him to buy expensive clothes. b) She showed her frugality by saving money for a new bike. c) Frugality means spending all your savings. d) The frugality of the party was very loud. Answer: b) She showed her frugality by saving money for a new bike. 5. Which idiom suggests being careful with money? a) Cut costs b) Spend like there's no tomorrow c) Throw caution to the wind d) Live in the moment Answer: a) Cut costs Associated Words survivalleverappliancegenerositypimple sanitationentertainmentdrabteenagergear How to participate in CREST International Spell Bee The CREST International Spell Bee (CSB) is conducted in two editions – CREST International Spell Bee - Summer (CSB) and CREST International Spell Bee - Winter (CSBW) – giving students across the globe an opportunity to showcase and enhance their spelling proficiency. Designed for young learners from Grades 1 to 8, the competition focuses on word-building, pronunciation, and vocabulary enrichment, helping students develop strong linguistic skills. Steps to Participate: Step 1: Visit the registration page and complete the registration with your desired subject. Step 2: After Registration done, do login at www.crestolympiads.com/login. 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2526	https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_2e_(OpenStax)	Skip to main content Prealgebra 2e (OpenStax) Last updated : May 28, 2023 Save as PDF Detailed Licensing Front Matter Buy Print Copy Donate Page ID : 114850 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics. Front Matter 1: Whole Numbers 2: Introduction to the Language of Algebra 3: Integers 4: Fractions 5: Decimals 6: Percents 7: The Properties of Real Numbers 8: Solving Linear Equations 9: Math Models and Geometry 10: Polynomials 11: Graphs 12: Appendix Back Matter Contributors and Attributions Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at
2527	https://emedicine.medscape.com/article/2129484-treatment	For You News & Perspective Tools & Reference Edition English Medscape Editions About You Professional Information Newsletters & Alerts Your Watch List Formulary Plan Manager Log Out Log In Medscape Editions English Deutsch EspaÃ±ol FranÃ§ais PortuguÃªs UK Univadis from Medscape About You Professional Information Newsletters & Alerts Your Watch List Formulary Plan Manager Log Out Register Log In close Please confirm that you would like to log out of Medscape. If you log out, you will be required to enter your username and password the next time you visit. Log out Cancel Tools & Reference>Pulmonology Status Asthmaticus Treatment & Management Updated: Nov 18, 2024 Author: Wael Azzam, MD; Chief Editor: John J Oppenheimer, MD more...;) Share Print Feedback Facebook Twitter LinkedIn WhatsApp Email Status Asthmaticus Sections Status Asthmaticus Overview Practice Essentials Pathophysiology Etiology Epidemiology Prognosis Patient Education Show All Presentation History Physical Examination Complications Show All DDx Workup Approach Considerations Laboratory Studies Chest Radiography Arterial Blood Gas Analysis Pulmonary Expiratory Flow Rate Measurement Electrocardiography Impulse Oscillometry Testing Histologic Findings Show All Treatment Approach Considerations Beta2 Agonists Anticholinergics Glucocorticosteroids Other Bronchodilators Sedatives Anesthetics Extracorporeal Life Support Noninvasive Ventilation Mechanical Ventilation Heliox Complications Prevention Consultations Long-Term Monitoring Show All Guidelines Medication Medication Summary Beta2 Agonists Corticosteroids Anticholinergics, Respiratory Xanthine Derivatives Electrolytes Monoclonal Antibodies, Anti-asthmatics Show All References;) Approach Considerations Once the diagnosis has been confirmed and the severity of the asthma attack has been assesed, the primary aim of treatment is to control the bronchoconstriction and the inflammation. Beta agonists, corticosteroids, and theophylline are mainstays in the treatment of status asthmaticus. Sevoflurane, a potent inhalational agent, was successful in a case report describing its use after failure of conventional treatment in a 26-year-old woman. Fluid replacement Hydration, with intravenous (IV) infusion of normal saline at a reasonable rate, is essential. Special attention must be paid to the patient's electrolyte status. Hypokalemia may result from the use of either corticosteroids or beta agonists. Correcting hypokalemia may help to wean an intubated patient with asthma from mechanical ventilation. Hypophosphatemia may result from poor oral intake and is also an important consideration in weaning such patients. Antibiotics Routine administration of antibiotics is discouraged. Patients should receive antibiotics only when they show evidence of infection (eg, pneumonia or sinusitis). In some situations, sinus imaging with computed tomography (CT) or plain radiography [38, 39] may be essential for ruling out chronic sinusitis. Oxygen monitoring and therapy Monitoring the patient's oxygen saturation is essential during the initial treatment of status asthmaticus. Arterial blood gas (ABG) values are usually obtained to assess hypercapnia during the patient's initial assessment. Oxygen saturation is then monitored via pulse oximetry throughout the treatment protocol. It is important to note that, as mentioned previously, oxygen saturation may decrease after the use of bronchodilators secondary to an increase in ventilation/perfusion (V/Q) mismatch. Oxygen therapy is essential, with hypoxia being the leading cause of death in children with asthma. Oxygen therapy can be administered via a nasal cannula or mask, though patients with dyspnea often do not like masks. With the advent of pulse oximetry, oxygen therapy can be easily titrated to maintain the patient's oxygen saturation in the 93-95% range, as recommended by the 2024 Global Initiative for Asthma (GINA) report. In children aged 6-11 years, the goal should be a range of 94-98%; in pregnant patients or those with cardiac disease, the goal should be a value higher than 95%. In the event of significant hypoxemia, nonrebreathing masks may be used to deliver as much as 98% oxygen. Tracheal intubation and mechanical ventilation are indicated for respiratory failure. Chest tube placement Chest tube placement may be necessary in the management of pneumothorax. Nitric oxide Nitric oxide (NO) inhalers have been used in the treatment of asthma throughout history, but their use has not become a standard practice. Initial studies, such as the one by Kacmarek et al from 1996, demonstrated that inhaled NO (iNO) could act as a bronchodilator in patients with mild asthma and methacholine-induced bronchospasm, yielding a minor but significant relaxation of airway tone. Subsequent research, however, including a study by Pfeffer et al from 1996, did not find iNO to have a significant bronchodilatory effect in pediatric asthma patients. A 2000 review by Ashutosh concluded that the use of NO in asthma treatment did not withstand the test of time and should not be recommended. The American Thoracic Society (ATS) guidelines also did not endorse the use of inhaled NO for asthma treatment but recognized the utility of measuring fractional exhaled NO (FeNO) as a biomarker for monitoring airway inflammation. Leukotriene modifiers Leukotriene modifiers are useful for treating chronic asthma, but only limited data support their use in acute asthma. This treatment may be beneficial if admininstered via a nebulizer, but it remains experimental. Most studies have examined IV use. [44, 45] Montelukast can serve as an add-on treatment for asthma in general. It is mostly used for improving quality of life as an adjunct to inhaled corticosteroids, not necessarily just for status asthmaticus. One study showed minimal to no effect of using montelukast in the emergency department (ED) setting for patients with status asthmaticus. [46, 47] In general, it has not shown a significant benefit. Surgery Status asthmaticus is generally managed by means of medical therapy, with few exceptions. For example, thoracostomy is indicated in pneumothoraces. Some children may have asthma that is primarily exacerbated by gastroesophageal reflux disease (GERD). Some patients can be treated with a combination of antireflux agents (eg, proton pump inhibitors [PPIs]) and histamine 2 (H2)-receptor antagonists. Nevertheless, surgery (eg, Nissen fundoplication) occasionally proves necessary. If all other support modalities fail and extracorporeal membrane oxygenation (ECMO) is required, surgical support for cannula placement should take place at an established ECMO center. Diet Some children with asthma may have episodes triggered by food allergies. Consultation with a nutritionist may be necessary to provide appropriate dietary management. Next: Beta2 Agonists The first line of therapy is bronchodilator treatment with a beta2 agonist, typically salbutamol (albuterol). Handheld nebulizer treatments may be administered either continuously (10-15 mg/hr) or at short intervals (eg, q5-20min), depending on the severity of the bronchospasm. The nebulized, inhaled route is generally the most effective route of delivery, and the IV forms are not routinely recommended; however, IV beta2 agonists have been successful in treating some younger patients with severe refractory status asthmaticus. It is also important to note that short-acting inhaled beta2 agonists should not be withheld or underdosed during acute asthma attacks, despite a growing body of data raising concerns regarding their long-term use. These agents continue to be the preferred treatment for acute exacerbations of asthma. For intermittent dosing of albuterol, 2.5 mg (0.3-0.5 mL of a 0.5% formulation mixed with 2.5 mL of normal saline) is administered by nebulization every 20 minutes for three doses, followed by hourly treatment for the first couple of hours. Some comparative studies found metered-dose inhaler (MDI)/spacer and dry-powder inhaler (DPI) delivery of bronchodilators to be noninferior to nebulizer delivery, but they did not include severe asthma exacerbations. [48, 49] Idris et al demonstrated that even in patients with severe disease, four puffs of albuterol (0.36 mg) delivered via MDI and spacer were as effective as a 2.5 mg dose delivered via nebulization. Levalbuterol, the R isomer of albuterol, is approved by the US Food and Drug Administration (FDA) for treatment of patients with acute asthma. The clinical bronchodilator effects of this isomer are equivalent to or greater than those of racemic albuterol, but levalbuterol has fewer effects on the heart rhythm (ie, tachyarrhythmia) and is associated with fewer occurrences of tremors, The decreased prevalence of adverse effects with this single-isomer medication may allow physicians to use nebulizer therapy in patients with acute asthma more frequently and with less concern regarding the adverse effects that occur with other bronchodilators (eg, albuterol and metaproterenol). The dose of levalbuterol is either a 0.63-mg vial (for children) or a 1.26-mg vial (for adults). In a 2009 study evaluating high-dose continuous nebulized levalbuterol for pediatric status asthmaticus, the authors found that although a slight decrease might have occurred in the duration of nebulizer treatment with levalbuterol as compared with albuterol, the difference was not statistically significant. The drugs mentioned above, especially albuterol, are safe to use during pregnancy. Beta2 agonists act via stimulation of cyclic adenosine monophosphate (cAMP)-mediated bronchodilation. The airway is rich in beta receptors; the stimulation of these receptors relaxes airway smooth muscles, increases mucociliary clearance, and decreases mucus production. Beta agonists generally are most effective in the early asthma reaction phase. However, patients who present with status asthmaticus despite frequent use of beta agonists at home may have tachyphylaxis and may exhibit resistance to these agents. Similar issues may be seen in patients using long-term inhaled long-acting beta agonists. Therefore, these patients may not respond as well when these agents are given in the hospital setting. Endotracheal adrenaline in patients who are intubated has been associated with variable success in different studies; however, it has not yet been proved to confer any specific advantage. High-dosage albuterol has been studied in a limited fashion in children with status asthmaticus. The high dosage was defined as 150 mg/hr and the lower dosage as 75 mg/hr. The higher dosage resulted in a low rate of subsequent mechanical ventilation and a short length of stay (LOS) have found in the pediatric intensive care unit (ICU). There was no evidence of toxicity. Nonselective beta2 agonists Patients whose bronchoconstriction is resistant to continuous nebulizer treatments with traditional beta2 agonists may be candidates for treatment with nonselective beta2 agonists (eg, epinephrine or terbutaline). However, systemic therapy has no proven advantage over aerosol therapy with selective beta2 agents, and repetitive doses has been associated with signs of toxicity. Caution must be exercised in patients with other complicating factors (eg, congestive heart failure [CHF] or history of cardiac arrhythmia). IV isoproterenol is not recommended for the treatment of asthma, because of the risk of myocardial toxicity. Some practitioners have advocated monitoring cardiac enzyme levels in patients who receive prolonged significant infusions of IV beta agonists. Small studies in children have documented that levels of enzymes such as troponin I may be elevated during terbutaline infusion, though the levels normalize as terbutaline is discontinued. The clinical significance of such enzyme elevation remains to be determined. [54, 55] Previous Next: Anticholinergics Anticholinergic agents are believed to work centrally by suppressing conduction in vestibular cerebellar pathways. They may have an inhibitory effect on parasympathetic nervous system. They may also decrease mucus production and improve mucociliary clearance. Ipratropium bromide Ipratropium bromide, a quaternary amine that does not cross the blood-brain barrier, is the anticholinergic parasympatholytic agent of choice. The consensus dose consists of 0.5 mg of ipratropium combined with albuterol administered by nebulization. It was shown that adding ipratropium to short-acting beta2 agonists in the treatment of moderate-to-severe asthma exacerbation was associated with fewer hospitalizations and greater improvement in PEF and FEV1 in comparison with the beta2 agonists alone. [56, 57, 58] Ipratropium may also be used as an alternative bronchodilator in patients who are unable to tolerate inhaled beta2 agonists. Because children appear to have more cholinergic receptors, they are more responsive to parasympathetic stimulation than adults are. This synthetic ammonium compound is very similar structurally to atropine. Atropine, a tertiary amine, may also be used and nebulized. However, it may cross the blood-brain barrier and thuse may cause central nervous system (CNS) effects. Nebulized glycopyrrolate, another quaternary amine, is also an alternative, though it is rarely prescribed in the United States. Previous Next: Glucocorticosteroids Glucocorticosteroids are the most important treatment for status asthmaticus. These agents can decrease mucus production, improve oxygenation, reduce beta-agonist or theophylline requirements, reduce beta2-agonist tachyphylaxis, and activate properties that may prevent late bronchoconstrictive responses to allergies and provocation. In addition, corticosteroids can decrease bronchial hypersensitivity, reduce the recovery of eosinophils and mast cells in bronchioalveolar lavage fluid, decrease the number of activated lymphocytes, and help regenerate the bronchial epithelial cells. Corticosteroids may be administered either IV or orally (PO). Although most practitioners administer corticosteroids during status asthmaticusprobably because patients are ventilated or on noninvasive positive-pressure ventilation (NIPPV)studies have found early administration of oral corticosteroids to be equally effective, less invasive, less expensive and quicker. After administration, corticosteroids usually require at least 4 hours to produce a clinical effect. In status asthmaticus, data support the administration of 60-125 mg methylprednisolone IV every 6 hours for the initial 24 hours of treatment. Oral steroids are usually required for the next 10-14 days. Other corticosteroids may be used in equivalent dosages. A study of 61 pediatric patients who randomly received IV methylprednisolone, hydrocortisone, or dexamethasone in the ICU along with continuous beta2-agonist treatment reported no differences in ICU or hospital LOS, pediatric asthma severity score (PASS), need for mechanical ventilation or maximum dose of beta2-agonist. The median duration of beta2-agonist treatment was shortest with methylprednisolone (23 hr, vs 27 hr for hydrocortisone and 32 hr for dexamethasone). Corticosteroid treatment for acute asthma is necessary but has potential adverse effects. Serum glucose must be monitored. Insulin can be administered on a sliding scale if needed. Monitoring a patient's electrolyte levels, especially potassium, is essential. Hypokalemia can cause muscle weakness, which may worsen respiratory distress and cause cardiac arrhythmias. Adverse effects of pulse therapy, in some authors' experience, are minimal and are comparable to those of the traditional doses of IV corticosteroids. The adverse effects may include hyperglycemia, which is usually reversible once steroid therapy is stopped, increased blood pressure, weight gain, increased striae formation, and hypokalemia. Long-term adverse effects depend on the duration of steroid therapy after the patient leaves the hospital. Nebulized steroids The use of nebulized corticosteroids for treating status asthmaticus is controversial. Data from studies comparing nebulized budesonide with prednisone in children have suggested that the latter is more effective for treating status asthmaticus. No good scientific evidence supports using nebulized dexamethasone or triamcinolone via a handheld nebulizer. In fact, in some authors' experience, more adverse effects, including a cushingoid appearance and irritative bronchospasms, have occurred with these nebulizers. Previous Next: Other Bronchodilators Methylxanthines The role of methylxanthines (eg, theophylline and aminophylline) in the treatment of severe acute asthma has been diminished since the advent of potent selective beta agonists and their use at higher doses. At therapeutic doses, methylxanthines are weaker bronchodilators than beta agonists and have many undesirable adverse effects (eg, frequent induction of nausea and vomiting). Furthermore, most studies have failed to show additional benefit when methylxanthines are administered to patients who are already receiving frequent doses of beta agonists and corticosteroids. Nevertheless, several prospective randomized controlled trials (RCTs) in children with refractory status asthmaticus have reexamined the role of the methylxanthines theophylline and aminophylline and have demonstrated improvement in the clinical asthma scores. One study compared IV theophylline with IV terbutaline in critically ill children with refractory asthma and demonstrated that the two agents had equal therapeutic efficacy but that theophylline use was significantly less costly. Theophylline Among the effects of theophylline that are important in managing asthma are bronchodilatation, increased diaphragmatic function, and central stimulation of breathing. Usually, theophylline is given parenterally, but it can also be given orally, depending on the severity of the asthma attack and the patient's ability to take medications. This class of drugs can induce tachycardia and decrease the seizure threshold (especially in children); therefore, therapeutic monitoring is mandatory. In the past, the therapeutic levels for theophylline were typically in the range of 10-20 Î¼g/mL. However, adverse effects have been noted to occur even with those therapeutic levels. Consequently, many institutions have adopted a lower therapeutic range of 8-15 Î¼g/mL. Seizures have occurred even with levels below 10 Î¼g/mL. Theophylline also has significant drug interactions with medications such as ciprofloxacin, digoxin, and warfarin. These interactions may decrease the rate of theophylline clearance by interfering with P-450 site metabolism. On the other hand, phenytoin and cigarette smoking can increase the rate of metabolism of theophylline and thereby decrease the therapeutic level of the drug. In patients who quit smoking less than 6 months previously, the theophylline dose should be managed as if they were still smoking. Patients who smoke or those on phenytoin require higher loading and maintenance doses of theophylline. Other adverse effects can include nausea, vomiting, and palpitations. The usual loading dose of theophylline is 6 mg/kg, followed by maintenance doses of 1 mg/kg/hr in the ED setting. For patients who smoke, the maintenance dose may be higher and the loading dose may be slightly higher. Patients on phenytoin should also receive increased maintenance doses of theophylline. Patients with liver disease or elderly patients may require a maintenance dose as low as 0.25 mg/kg/hr. Aminophylline Conflicting reports on the efficacy of aminophylline therapy have made it controversial. Starting IV aminophylline may be reasonable in patients who do not respond to medical treatment with bronchodilators, oxygen, corticosteroids, and IV fluids within 24 hours. Data suggest that aminophylline may have an anti-inflammatory effect in addition to its bronchodilator properties. The loading dose is usually 5-6 mg/kg, followed by a continuous infusion of 0.5-0.9 mg/kg/hr. Magnesium sulfate IV magnesium sulfate has been a useful adjunct in patients with acute status asthmaticus that is refractory to beta2-agonist therapy. Magnesium can relax smooth muscle and hence may cause bronchodilation by competing with calcium at calcium-mediated smooth-muscle binding sites. One double-blind placebo-controlled study reported a significant increase in peak expiratory flow rate (PEFR), forced expiratory volume in 1 second (FEV1), and forced vital capacity (FVC) in children who had asthma and were treated with a single 40-mg/kg dose of magnesium sulfate infused over 20 minutes, along with steroids and inhaled bronchodilators, as compared with control subjects who received saline placebo. In addition, patients who received IV magnesium (8/16) were significantly more likely to be discharged home from the presenting ED than control subjects were (0/14). No data regarding duration of effect or efficacy with repeated doses are available, and no guidelines have described monitoring of serum magnesium levels if more than an initial magnesium dose is administered. In one small study of four children who received 40-50 mg/kg of magnesium sulfate, serum magnesium levels were all lower than 4 mg/dL; electrocardiographic (ECG) changes are generally not seen until levels exceed 4-7 mg/dL. Adverse effects may include facial warmth, flushing, tingling, nausea, and hypotension. This therapy can be tried, especially in pregnant women, as an adjunct to beta2 bronchodilator therapy. Further studies have not confirmed the effectiveness of this treatment, [66, 67] but because of its relative cheapness and harmlessness, it is still widely implemented. Inhaled magnesium sulfate has generated some interest with regard to the treatment of status asthmaticus, when combined with beta-agonist use. In a systematic review and meta-analysis of 10 RCTs that studied inhaled magnesium sulfate, the associated improvement in lung function did not translate into clinical benefit. A retrospective study by Vaiyani et al evaluated magnesium sulfate infusion for status asthmaticus in two groups of children. In one group, patients weighing less than 30 kg received a dose of 75 Î¼g/kg, whereas those weighing more than 30 kg received 50 mg/kg. In both situations, magnesium sulfate was continuously infused at 40 mg/kg/hr over 4 hours. In the other group, no loading dose was given, and 50 mg/kg/hr was infused over 5 hours. The two groups did not differ with respect to the serum magnesium concentration or the amount of bronchodilation. These and other data suggest that magnesium sulfate remains important in the treatment of status asthmaticus, with excellent tolerability. Enoximone Enoximone is an imidazole phosphodiesterase III inhibitor that has been used in patients with heart failure. In a limited report of eight patients with status asthmaticus, IV enoximone was found to be effective for treating six patients in whom maximal treatment had been exhausted. It allowed the patients to improve without the need of invasive measures such as mechanical ventilation. Further investigation will required for better definition of the therapeutic role of enoximone in this setting. Previous Next: Sedatives The 2024 GINA report recommended that sedation should be avoided during asthma exacerbations. The concern is suppression of the patients' drive to compensatory hyperventilation. The report did not mention the use of sedatives in ventilated patients. Previous Next: Anesthetics Ketamine Ketamine is a short-acting pentachlorophenol derivative that exerts bronchodilatory effects by inducing an increase in levels of endogenous catecholamine, which may bind to beta receptors and cause smooth-muscle relaxation and bronchodilation. Ketamine was used in the management of status asthmaticus in a prospective trial in patients with respiratory failure who did not respond adequately to mechanical ventilation. This agent improved airway resistance, particularly in the lower airways, and enhanced lung compliance. Significant improvement in oxygenation and hypercarbia has been reported with ketamine, even 15 minutes after administration. Case reports have also described the use of ketamine as a sedative to reduce anxiety and agitation that can exacerbate tachypnea and work of breathing and potentially avert further respiratory failure in small children with status asthmaticus. Ketamine as a continuous infusion may induce relaxation of the airways with limited anesthesia. However, its role in status asthmaticus remains limited. CNS sedation, which may require intubation, restricts its use. It has also been noted that the use of ketamine has been limited to the pediatric population and to very low dosages. [74, 75, 76] Inhaled anesthetic agents Inhaled anesthetic agents (eg, halothane, isoflurane, and enflurane) have been used with varying degrees of success in intubated patients with refractory severe asthma. Their mechanism of action is unclear, but they may have direct relaxant effects on airway smooth muscle. [77, 78] A retrospective study of 45 pediatric ICU patients receiving isoflurane with or without extracorporeal life support reported improvements in hypercarbia and acidosis within 4 hours of isoflurane administration. However, most of the evidence for the use of inhaled anesthetic agents comes from pediatric patients, and there remains a need for studies in the adult population. Other inhaled anesthetics that have been studied are propofol and sevoflurane. Prolonged propofol administration, however, may be complicated by generalized seizure, increased carbon dioxide production, and hypertriglyceridemia. Sevoflurane has been employed more commonly than halothane and isoflurane. Although this medication is relatively safe, caution must be exercised in using it because of the risk of hepatotoxicity and renal tubular injury. In children, sevoflurane has been shown in some studies to be safe and effective. In adults, careful monitoring of liver and kidney function, as well as serum fluoride concentration, is helpful for avoiding toxic levels of sevoflurane. Other anesthetics Neuromuscular blockers may be used with caution in patients who are well sedated but are exhibiting severe anxiety and dyssynchrony with the ventilator, as well as in those who are intolerant of intubation. [5, 51] Paralytics are indicated when ventilator asynchrony persists despite sedation and when the risk of generating autopositive end-expiratory pressure (auto-PEEP) or barotrauma is high. Atracurium and vecuronium are usually used. Atracurium may be associated with less risk of myopathy, but it may lead to bronchoconstriction from histamine release; vecuronium is an alternative that carries a lower risk of bronchoconstriction. In isolated case reports, NO has also been employed in the treatment of status asthmaticus and has been effective when mechanical ventilation is not adequate. [82, 83] Additionally, the use of nebulized lidocaine in combination with albuterol or levalbuterol may be effective in mitigating the vocal cord dysfunction that can accompany status asthmaticus (this is an unpublished observation by an author in clinical practice). Data have shown that lidocaine may have efficacy in asthma. [84, 85] It helps in reducing the cough component and has been shown to be an eosinophilic apoptotic agent with clinical efficacy in chronic cough. Previous Next: Extracorporeal Life Support ECMO is a highly specialized intervention requiring significant resources and support. It should be considered when hypoxia and acidosis persist despite mechanical ventilation. [86, 87] It should also be considered in patients at high risk for the development of refractory status asthmaticus. [88, 89] Such patients include, but are not limited to, those with a history of multiple intubations, respiratory failure requiring intubation within 6 hours of admission, hemodynamic instability, neurologic impairment at the time of admission, or duration of respiratory failure greater than 12 hours despite maximal medical therapy. Mikkelsen et al reported successful use of extracorporeal life support in patients with status asthmaticus and severe secondary asphyxia who otherwise were not responsive to aggressive pulmonary support. ECMO comes with its own significant risks, being associated with severe adverse effectsin particular, hemorrhage, as a primary concern, and the possible development of limb ischemia. Previous Next: Noninvasive Ventilation Noninvasive ventilation (NIV), including continuous or bilevel positive airway pressure, can be considered in patients with impending respiratory failure, in order to avoid intubationspecifically, in patients who are able to protect their away with no significant encephalopathy and in those without significant secretions. However, the impact of NIV is not as well defined and studied for asthma as it is for chronic obstructive pulmonary disease (COPD). Nevertheless, given that the pathophysiologic processes of COPD and asthma are similar, it is reasonable to expect NIV to be similarly beneficial in status asthmaticus. In contrast to patients with COPD exacerbations, asthma patients tend to require more invasive means of ventilation with mechanical ventilation when they are in status asthmaticus, probably as a consequence of the severe bronchoconstriction with secretions and the extensive inflammatory process . Ram et al demonstrated that NIPPV is beneficial in patients with severe asthma exacerbation, yielding significant improvements in hospitalization rate, number of patients discharged from the ED, percentage of predicted FEV1, FVC, PEFR, and respiratory rate. Ueda et al reported using NIPPV to wean a patient with refractory status asthmaticus who also had developed atelectasis. Leatherman et al reported that prolongation of the expiratory time can decrease dynamic inflation in patients with status asthmaticus and may have a minor positive effect on weaning in these patients. No high-power RCTs have investigated the use of NIV in asthma, but given the proven safety of NIV, it should always be given a trial early in the presentation of status asthmaticus. This has been shown to be mostly effective in the pediatric population. Previous Next: Mechanical Ventilation Mechanical ventilation should be considered as a salvage therapy in patients with status asthmaticus. It requires careful monitoring in patients with asthma because these patients have high end-expiratory pressure and thus are at high risk for pneumothorax. Indications and application Indications for intubation and mechanical ventilation include the following: Apnea or respiratory arrest Diminishing level of consciousness Impending respiratory failure marked by significantly rising carbon dioxide tension (PCO2) with fatigue, decreased air movement, and altered level of consciousness Significant hypoxemia that is poorly responsive or unresponsive to supplemental oxygen therapy alone Mechanical ventilation, when used in patients with asthma, is usually required for less than 72 hours. In occasional patients with severe bronchospasm, however, mechanical ventilation can be prolonged. In these situations, consultation with a pulmonologist or another expert in mechanical ventilatory techniques is recommended. Key principles for the use of mechanical ventilation in this setting include the following: Minimize minute ventilation - Lower the tidal volume and respiratory rate to reduce the volume of air needing exhalation, allowing more complete expiration before the next breath Optimize inspiratory-to-expiratory (I/E) ratio - Increase inspiratory flow rates (eg, to 70-100 L/min) to shorten inspiratory time and prolong expiration; this adjustment is essential for reducing dynamic hyperinflation Allow permissive hypercapnia - This controlled retention of carbon dioxide is often tolerated if pH remains above 7.2, prioritizing reduced ventilatory demands and limiting complications from forced hyperventilation Initial ventilator settings are as follows: Mode - The pressure control mode should be used to achieve low tidal volumes (6-8 mL/kg) so as to reduce lung overdistention Respiratory rate - A rate of 11-14 breaths/min helps prolong expiratory time, reducing the risk of gas trapping PEEP - PEEP should be set to 0-5 cm H2O; adding external PEEP may worsen gas trapping and should be done cautiously Inspiratory pressure limiting - The aim should be a plateau pressure (Pplat) lower than 30 cm H2O; high Pplat can indicate excessive lung stretch, risk of barotrauma, and auto-PEEP Additional considerations The decision to intubate a patient with asthma should be made with extreme caution. Positive-pressure ventilation (PPV) in a patient with asthma is complicated by the severe airway obstruction and air trapping, which result in hyperinflated lungs that may resist further inflation and place the patient at high risk for barotrauma. Therefore, mechanical ventilation should be undertaken only in the face of continued deterioration despite maximal bronchodilator therapy. In the face of high peak airway pressures, the key principle of mechanical ventilation in status asthmaticus is controlled hypoventilation with toleration of higher levels of PCO2 in order to minimize tidal volume and peak inspiratory pressure (PIP). Permissive hypercapnia can be tolerated as long as the patient remains adequately oxygenated. A lower I/E ratio (often < 1:3-4) helps allow time for optimal exhalation, facilitating ventilation and avoiding an excessive amount of further air trapping (auto-PEEP). It must be kept in mind, however, that patients may be uncomfortable and air-hungry while ventilated with low respiratory rates, prolonged exhalation times, and hypercapnia in accordance with a strategy of controlled hypoventilation. The use of PEEP is controversial. A patient with status asthmaticus who is in respiratory failure and on mechanical ventilation usually has a significant amount of air trapping that results in intrinsic PEEP (so-called auto-PEEP), which may be worsened by maintaining additional PEEP during exhalation. However, some patients may benefit from the addition of PEEP, perhaps owing to maintenance of airway patency during exhalation. Thus, in a patient who remains refractory to the initial ventilatory settings with no or very low PEEP, cautiously increasing the PEEP may prove beneficial. Traditionally, slow, controlled ventilation with heavy sedation, often with muscle relaxation, has been used to ventilate patients with status asthmaticus. Caution is warranted, however, in that the use of muscle relaxants with high-dose corticosteroids has been associated with the development of prolonged paralysis. Alternatively, some practitioners report ventilating children with status asthmaticus by means of pressure support alone. This strategy may allow the patient to set his or her own respiratory rate as determined by the physiologic time constant, while assisting ventilation and relieving the fatigue due to significantly increased work of breathing. Invasive mechanical ventilation is associated with increased hospital resource use, prolonged LOS, and even a higher risk of pneumonia. Further research into NIV should help minimize the frequency of invasive mechanical ventilation. In one case report describing status asthmaticus in a 12-year-old boy with subcutaneous emphysema and pneumomediastinum, both conditions worsened with NIV. Additional study is needed in this area. When there is doubt, invasive mechanical ventilation should be considered first in patients with significant respiratory distress who are not responding to the usual aggressive pharmacologic treatment and appropriate oxygenation. It is considered the safest approach in a patient who is in severe status asthmaticus, particularly with secondary respiratory failure. It is not, however, without adverse effects or complications. Pneumothorax and idiopathic hemothorax have both been reported. Monitoring and support Patients require supportive measures and monitoring during mechanical ventilation. Ideally, flow-volume loops should be monitored to determine whether adequate time is being provided for exhalation so as to avoid breath stacking, which occurs if the next breath is delivered before exhalation is completed. Monitoring of exhaled tidal volume and auto-PEEP is also important. Fluids and electrolytes should be monitored as well. Before arrival in the hospital, children with status asthmaticus have often had diminished oral intake and may have been vomiting because of respiratory difficulty or adverse effects from their medications. This leads to decreased intravascular volume that may be potentiated by the effects of PPV. Moreover, the medications used to treat asthma can result in significant kaliuresis, not to mention the cellular potassium shifts secondary to controlled hypoventilation's permissive hypercapnia with resultant acidemia. In addition, cardiac output may be decreased because of decreased preload that results from air trapping and auto-PEEP. This decreased cardiac output and intravascular volume may be accompanied by metabolic acidosis. Intravascular fluid expansion is needed to treat hypoperfusion, hypotension, or metabolic acidosis. On occasion, diastolic hypotension may result from high doses of beta-agonists. A vasoconstrictor (eg, norepinephrine or phenylephrine) may be considered if significant diastolic hypotension persists in the face of adequate intravascular volume. Catheter placement Placement of an indwelling arterial catheter may be considered for ABG sampling and continuous blood pressure measurement in patients with mechanical ventilation but is not generally recommended. The arterial waveform can also be used for measurement of pulsus paradoxus. Previous Next: Heliox Various other treatments have been used in patients with severe acute asthma, but none is well proven. A combination of helium and oxygen (eg, 70% helium and 30% oxygen) known as heliox has been studied, but systematic reviews comparing it with air-oxygen have suggested that heliox has no role in routine management. In any case, this treatment should be considered only in patients who are able to take deep breaths, because it is dependent on inspiratory flow. Helium is an inert gas that is less dense than nitrogen and allows better oxygenation into the small airways. Administration of heliox reduces turbulent airflow across narrowed airways, which can help reduce the work of breathing. This, in turn, can result in enhanced gas exchange and yield improvements in pH and clinical symptoms. [102, 103] It does not increase the caliber of the narrowed airway. Because of its low density, helium is more fluid under conditions of turbulence. This helps minimize airway pressure and facilitates recurrence of laminar flow. Thus, oxygenation becomes easier in the presence of increased airway resistance. There are some data to suggest that nebulized-size particles may be more uniformly distributed in the distal airways when nebulization treatments are administered with heliox rather than with a standard oxygen-nitrogen mixture. Heliox can be administered via a well-fitting face mask at flows high enough to prevent entrainment of room air. The effectiveness of heliox in reducing the density of administered gas and improving laminar airflow depends on the helium concentration of the gas: The higher the helium concentration, the more effective the result. Therefore, an 80/20 mixture (80% helium, 20% oxygen) is most effective. Heliox loses most of its clinical utility when the fraction of inspired oxygen (FiO2) exceeds 40%, reducing the percentage of helium to less than 60%. Therefore, the limitation to the use of heliox is the amount of supplemental oxygen the patient requires to maintain adequate oxygen saturation. Heliox has also been employed with mechanical ventilation to lower dynamic PIPs. Heliox should be considered in patients who are not adequately responding to conventional pharmacologic therapy, and it may aid in preventing intubation. However, limited availability, prohibitive cost, and technical issues constitute obstacles to its routine use. Previous Next: Complications In adults with status asthmaticus, the clinical presentation with overt acidemia has been significantly associated with higher rates of invasive ventilation and prolonged hospital stay with complications and mortality. Hypokalemia was noted in a minority of these patients, but, for the most part, supplementation was not required. Invasive ventilation itself is also associated with potential complications, as discussed above. All medications and potential interventions come with some risk. Previous Next: Prevention Although status asthmaticus can usually be prevented if patients are compliant with their medications and avoid triggers and stress factors, it can sometimes occur even when patients are compliant and doing well as outpatients. In such situations, it is worthwhile to search for an occult infection (eg, respiratory syncytial virus [RSV; common in children but less so in adults] or an occult sinus infection). Prevention of status asthmaticus may be aided by monitoring forced oscillation test results rather than spirometry findings. This is particularly true for children younger than 12 years. However, adults with reactive airways may be undertreated if the criterion for stability and normality is a spirometric FEV1 greater than 80% of the predicted value. Among the important preventive considerations are home medications, such as anti-inflammatory agents. Corticosteroids are now considered the mainstays of asthma maintenance therapy. Underuse of anti-inflammatory agents has been found to be related to more severe asthma, probably as a consequence of airway remodeling and persistent of inflammatory changes. Biologic agents, though not recommended for the treatment of acute status asthmaticus, are now instrumental in asthma management, offering targeted therapies that can significantly improve asthma control, reduce exacerbations, and minimize the long-term risks associated with chronic inflammation. For patients with severe asthma, these agents not only enhance quality of life but also reduce the need for systemic steroids and prevent potentially life-threatening exacerbations. An example is tezepelumab, a first-in-class human monoclonal antibody immunoglobulin G2 (IgG2) lambda that inhibits thymic stromal lymphopoietin (TSLP) and that was approved by the FDA and European Union as add-on maintenance treatment for severe uncontrolled asthma in adults and adolescents aged 12 years and older. In clinical trials, this biologic treatment significantly reduced asthma exacerbations, ED visits, and hospitalizations, regardless of asthma phenotype. [109, 110] A retrospective analysis showed that the severity of asthma at baseline and the age of the patient are the most important factors for determining the risk of recurrent status asthmaticus and predicting the severity of the attack. In other words, patients older than 60 years who are also characterized as having either moderate persistent asthma or severe persistent asthma are at high risk of developing status asthmaticus. Therefore, compliance with the National Institutes of Health (NIH) guidelines [30, 31] and the GINA recommendations for the treatment and management of patients with asthma should theoretically be an effective prophylaxis against the development of status asthmaticus. Additionally, inpatient education provided by trained volunteers to patients admitted with status asthmaticus has been associated with improvement in posthospitalization and better adherence to inhaler management. In a study by Miller et al, an asthma protocol developed by the hospital and based on the NIH guidelines resulted in improved time to treatment and better outcome in children with status asthmaticus. Previous Next: Consultations Hospital admission for asthma could be considered the result of a failure of outpatient management. Better outpatient therapy is necessary to prevent subsequent admissions. Pulmonologists or allergists should be consulted because these specialists can provide comprehensive follow-up care with the appropriate therapy, allergy testing (if indicated), and consistent follow-up testing and manipulation of medications, as required. Anesthesia support is needed if inhaled anesthetic agents are considered for refractory severe intubated status asthmaticus. Surgical teams or interventional cardiology might be helpful in the event of ECMO use. Consultation with a surgeon may also be required if a child might benefit from fundoplication (eg, if GERD is the trigger for recurrent status asthmaticus). Occsaionally, consultation with a member of social services can provide support in the complex management of a chronic illness. Adolescents who have severe, uncontrolled asthma and are nonadherent or have depression or significant behavioral issues may require the services of a psychiatrist or psychologist. Previous Next: Long-Term Monitoring Appropriate follow-up is important, as is checking the patient's peak flow meter (at home) and FEV1 (in the office). Because children with asthma commonly present with a normal FEV1, more sensitive lung function testing should be undertaken with regular impulse oscillometry system (IOS) . Medication titration may be usefully guided by IOS resistance and reactance values. Outpatient follow-up and continued care of a patient who has been hospitalized in the pediatric ICU because of severe status asthmaticus is important for optimizing long-term outcome and quality of life and for minimizing recurrent episodes of severe asthma exacerbation. Follow-up is best provided by a specialist in the treatment of asthma. The subgroup of patients who are poor perceivers of dyspnea are at increased risk for future exacerbations. An at-home peak flow meter may be valuable in this poor-perceiver population. Previous Guidelines References Cygan J, Trunsky M, Corbridge T. Inhaled heroin-induced status asthmaticus: five cases and a review of the literature. Chest. 2000 Jan. 117 (1):272-5. [QxMD MEDLINE Link]. Self TH, Shah SP, March KL, Sands CW. Asthma associated with the use of cocaine, heroin, and marijuana: A review of the evidence. J Asthma. 2017 Sep. 54 (7):714-722. [QxMD MEDLINE Link]. Rome LA, Lippmann ML, Dalsey WC, Taggart P, Pomerantz S. 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Media Gallery Antigen presentation by dendritic cell, with lymphocyte and cytokine response leading to airway inflammation and asthma symptoms. of 1 Tables Back to List Contributor Information and Disclosures Wael Azzam, MD Resident Physician, Department of Internal Medicine, MedStar Washington Hospital CenterWael Azzam, MD is a member of the following medical societies: American College of Physicians, American Society of NephrologyDisclosure: Nothing to disclose. Brian Matthew Cuneo, MD Staff Physician, Department of Pulmonology, MedStar Washington Hospital CenterDisclosure: Nothing to disclose. Specialty Editor Board Francisco Talavera, PharmD, PhD Adjunct Assistant Professor, University of Nebraska Medical Center College of Pharmacy; Editor-in-Chief, Medscape Drug ReferenceDisclosure: Received salary from Medscape for employment. Chief Editor John J Oppenheimer, MD Clinical Professor, Department of Medicine, Rutgers New Jersey Medical School; Director of Clinical Research, Pulmonary and Allergy Associates, PA John J Oppenheimer, MD is a member of the following medical societies: American Academy of Allergy Asthma and Immunology, American College of Allergy, Asthma and Immunology, New Jersey Allergy, Asthma and Immunology SocietyDisclosure: Serve(d) as a director, officer, partner, employee, advisor, consultant or trustee for: Amgen; GSK; Aquestive; Aimmune Clinical Adjudication/Data Safety Monitoring Board for AZ, GSK, Abbvie, Sanofi; Executive Editor for Annals of Allergy, Asthma & Immunology; Editor for Medscape; Reviewer for UpToDate; Honoraria from American Board of Allergy and Immunology. Michael R Bye, MD Professor of Clinical Pediatrics, Division of Pulmonary Medicine, Columbia University College of Physicians and Surgeons; Attending Physician, Pediatric Pulmonary Medicine, Morgan Stanley Children's Hospital of New York Presbyterian, Columbia University Medical Center Michael R Bye, MD is a member of the following medical societies: American Academy of Pediatrics, American College of Chest Physicians, and American Thoracic Society Disclosure: Nothing to disclose. G Patricia Cantwell, MD, FCCM Professor of Clinical Pediatrics, Chief, Division of Pediatric Critical Care Medicine, University of Miami, Leonard M Miller School of Medicine; Medical Director, Palliative Care Team, Director, Pediatric Critical Care Transport, Holtz Children's Hospital, Jackson Memorial Medical Center; Medical Manager, FEMA, Urban Search and Rescue, South Florida, Task Force 2; Pediatric Medical Director, Tilli Kids Pediatric Initiative, Division of Hospice Care Southeast Florida, Inc G Patricia Cantwell, MD, FCCM is a member of the following medical societies: American Academy of Hospice and Palliative Medicine, American Academy of Pediatrics, American Heart Association, American Trauma Society, National Association of EMS Physicians, Society of Critical Care Medicine, and Wilderness Medical Society Disclosure: Nothing to disclose. Barry J Evans, MD Assistant Professor of Pediatrics, Temple University Medical School; Director of Pediatric Critical Care and Pulmonology, Associate Chair for Pediatric Education, Temple University Children's Medical Center Barry J Evans, MD is a member of the following medical societies: American Academy of Pediatrics, American College of Chest Physicians, American Thoracic Society, and Society of Critical Care Medicine Disclosure: Nothing to disclose. Michael Goldman Professor of Internal Medicine, University of California, Los Angeles, David Geffen School of Medicine Disclosure: Nothing to disclose. Helen M Hollingsworth, MD Director, Adult Asthma and Allergy Services, Associate Professor, Department of Internal Medicine, Division of Pulmonary and Critical Care, Boston Medical Center Helen M Hollingsworth, MD is a member of the following medical societies: American Academy of Allergy Asthma and Immunology, American College of Chest Physicians, American Thoracic Society, and Massachusetts Medical Society Disclosure: Nothing to disclose. Jan Malacara, PA-C Consulting Staff, Allergy ARTS, LLP Disclosure: Nothing to disclose. Adam J Schwarz, MD Consulting Staff, Critical Care Division, Pediatric Subspecialty Faculty, Children's Hospital of Orange County Adam J Schwarz, MD is a member of the following medical societies: American Academy of Pediatrics and Phi Beta Kappa Disclosure: Nothing to disclose. Francisco Talavera, PharmD, PhD Adjunct Assistant Professor, University of Nebraska Medical Center College of Pharmacy; Editor-in-Chief, Medscape Drug Reference Disclosure: Medscape Salary Employment Mary L Windle, PharmD Adjunct Associate Professor, University of Nebraska Medical Center College of Pharmacy; Editor-in-Chief, Medscape Drug Reference Disclosure: Nothing to disclose. Close;) What would you like to print? What would you like to print? 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2528	https://flexbooks.ck12.org/cbook/cbse-science-class-7/section/9.5/primary/lesson/distance-time-graphs/	Skip to content Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign Up 9.5 Distance - Time Graphs Written by:Jean Brainard, Ph.D. Fact-checked by:The CK-12 Editorial Team Last Modified: Sep 01, 2025 Lesson Drawing line graphs can help you understand motion. In this article, you’ll learn how to draw distance-time graphs (also known as position-time graphs) and how they show velocity. Q: What’s missing from the graph being drawn in the picture above? A: The x- and y-axes are missing. Types of Graphs: Different types of graphs can be used to present information. They can help us represent data in a visual way, making it easier to understand and analyze. Bar Graphs: A bar graph is used to compare different categories or groups. It has two axes: the x-axis (horizontal) and the y-axis (vertical). The x-axis represents the categories, and the y-axis represents the values or frequencies. Below is a bar graph representing the average monthly rainfall in the capital city for the first six months of the year 2014. In the bar graph, the title is clearly shown above the graph, and both axes are labeled. The vertical bars compare the average amount of rainfall for each of the six months. Pie Charts: A pie chart is used to show the proportion of each category in a whole. It is a circle divided into segments, where each segment represents a category. 550 high school students were asked their favorite sport, and the results are shown in the pie chart below. Line Graphs: A line graph is used to show how data changes over time. It also has two axes: the x-axis (horizontal) represents time, and the y-axis (vertical) represents the values. The line graph below illustrates the absolute minimum temperature in Dubai in 2005 in Celsius. Graphing Distance and Time The motion of an object can be represented by a distance-time graph like Graph 1 in the Figurebelow. In this type of graph, the y-axis represents distance from the starting point, and the x-axis represents time. A distance-time graph shows how far an object has traveled from its starting position at any given time since it started moving. Q: In the Figureabove, what distance has the object traveled from the starting point by the time 5 seconds have elapsed? A: The object has traveled a distance of 50 meters. Slope Equals Velocity In a distance-time graph, the velocity of the moving object is represented by the slope, or steepness, of the graph line. If the graph line is horizontal, like the line after time = 5 seconds in Graph 2 in the Figurebelow, then the slope is zero, and so is the velocity. The position of the object is not changing. The steeper the line is, the greater the slope of the line is and the faster the object’s position is changing. Calculating Average Velocity from a Distance-Time Graph We can calculate the average velocity of a moving object from a distance-time graph. Average velocity equals the change in position (represented by Δd) divided by the corresponding change in time (represented by Δt): @$$\begin{align}\text{velocity}= \frac{\Delta d}{\Delta t}\\end{align}@$$ For example, in Graph 2 in the Figureabove, the average velocity between 0 seconds and 5 seconds is: @$$\begin{align}\text{velocity} & = \frac{\Delta d}{\Delta t}\ & =\frac{25 \ \text{m}-0 \ \text{m}}{5 \ \text{s}-0 \ \text{s}}\ & =\frac{25 \ \text{m}}{5 \ \text{s}}\ & =5 \ \text{m/s}\end{align}@$$ Watch this two-part video series for more distance/position vs. time graph examples. Use the following PLIX Interactive to make a position-time graph for a runner who changes speed during their run: Summary Motion can be represented by a distance-time graph, which plots distance relative to the starting point on the y-axis and time on the x-axis. The slope of a distance-time graph represents velocity. The steeper the slope is, the faster the motion is changing. Average velocity can be calculated from a distance-time graph as the change in position divided by the corresponding change in time. Review Describe how to make a distance-time graph. What is the slope of a line graph? What does the slope of a distance-time graph represent? In Graph 1 in the Figureabove, what is the object’s average velocity? Asked by Students Here are the top questions that students are asking Flexi for this concept: Overview Motion can be represented by a distance-time graph, which plots distance relative to the starting point on the y-axis and time on the x-axis. The slope of a distance-time graph represents velocity. The steeper the slope is, the faster the motion is changing. Average velocity can be calculated from a distance-time graph as the change in position divided by the corresponding change in time. 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2529	https://chem.libretexts.org/Courses/College_of_Marin/CHEM_114%3A_Introductory_Chemistry/13%3A_Solutions/13.04%3A_Solutions_of_Gases_in_Water-_How_Soda_Pop_Gets_Its_Fizz	Skip to main content 13.4: Solutions of Gases in Water- How Soda Pop Gets Its Fizz Last updated : May 20, 2018 Save as PDF 13.3: Solutions of Solids Dissolved in Water- How to Make Rock Candy 13.5: Solution Concentration- Mass Percent Page ID : 98090 ( \newcommand{\kernel}{\mathrm{null}\,}) Template:HideTOC Learning Objectives Explain how temperature and pressure affect the solubility of gases. In an earlier module of this chapter, the effect of intermolecular attractive forces on solution formation was discussed. The chemical structures of the solute and solvent dictate the types of forces possible and, consequently, are important factors in determining solubility. For example, under similar conditions, the water solubility of oxygen is approximately three times greater than that of helium, but 100 times less than the solubility of chloromethane, CHCl3. Considering the role of the solvent’s chemical structure, note that the solubility of oxygen in the liquid hydrocarbon hexane, C6H14, is approximately 20 times greater than it is in water. Other factors also affect the solubility of a given substance in a given solvent. Temperature is one such factor, with gas solubility typically decreasing as temperature increases (Figure 13.4.1). This is one of the major impacts resulting from the thermal pollution of natural bodies of water. When the temperature of a river, lake, or stream is raised abnormally high, usually due to the discharge of hot water from some industrial process, the solubility of oxygen in the water is decreased. Decreased levels of dissolved oxygen may have serious consequences for the health of the water’s ecosystems and, in severe cases, can result in large-scale fish kills (Figure 13.4.2). The solubility of a gaseous solute is also affected by the partial pressure of solute in the gas to which the solution is exposed. Gas solubility increases as the pressure of the gas increases. Carbonated beverages provide a nice illustration of this relationship. The carbonation process involves exposing the beverage to a relatively high pressure of carbon dioxide gas and then sealing the beverage container, thus saturating the beverage with CO2 at this pressure. When the beverage container is opened, a familiar hiss is heard as the carbon dioxide gas pressure is released, and some of the dissolved carbon dioxide is typically seen leaving solution in the form of small bubbles (Figure 13.4.3). At this point, the beverage is supersaturated with carbon dioxide and, with time, the dissolved carbon dioxide concentration will decrease to its equilibrium value and the beverage will become “flat.” "Fizz" The dissolution in a liquid, also known as fizz, usually involves carbon dioxide under high pressure. When the pressure is reduced, the carbon dioxide is released from the solution as small bubbles, which causes the solution to become effervescent, or fizzy. A common example is the dissolving of carbon dioxide in water, resulting in carbonated water. Carbon dioxide is weakly soluble in water, therefore it separates into a gas when the pressure is released. This process is generally represented by the following reaction, where a pressurized dilute solution of carbonic acid in water releases gaseous carbon dioxide at decompression: H2CO3(aq)→H2O(l)+CO2(g) In simple terms, it is the result of the chemical reaction occurring in the liquid which produces a gaseous product. For many gaseous solutes, the relation between solubility, Cg, and partial pressure, Pg, is a proportional one: Cg=kPg where k is a proportionality constant that depends on the identities of the gaseous solute and solvent, and on the solution temperature. This is a mathematical statement of Henry’s law: The quantity of an ideal gas that dissolves in a definite volume of liquid is directly proportional to the pressure of the gas. Example 13.4.1: Application of Henry’s Law At 20 °C, the concentration of dissolved oxygen in water exposed to gaseous oxygen at a partial pressure of 101.3 kPa (760 torr) is 1.38 × 10−3 mol L−1. Use Henry’s law to determine the solubility of oxygen when its partial pressure is 20.7 kPa (155 torr), the approximate pressure of oxygen in earth’s atmosphere. Solution According to Henry’s law, for an ideal solution the solubility, Cg, of a gas (1.38 × 10−3 mol L−1, in this case) is directly proportional to the pressure, Pg, of the undissolved gas above the solution (101.3 kPa, or 760 torr, in this case). Because we know both Cg and Pg, we can rearrange this expression to solve for k. Cgk=kPg=CgPg=1.38×10−3molL−1101.3kPa=1.36×10−5molL−1kPa−1(1.82×10−6molL−1torr−1) Now we can use k to find the solubility at the lower pressure. Cg=kPg 1.36×10−5molL−1kPa−1×20.7kPa(or1.82×10−6molL−1torr−1×155torr)=2.82×10−4molL−1 Note that various units may be used to express the quantities involved in these sorts of computations. Any combination of units that yield to the constraints of dimensional analysis are acceptable. Exercise 13.4.1 A 100.0 mL sample of water at 0 °C to an atmosphere containing a gaseous solute at 20.26 kPa (152 torr) resulted in the dissolution of 1.45 × 10−3 g of the solute. Use Henry’s law to determine the solubility of this gaseous solute when its pressure is 101.3 kPa (760 torr). Answer : 7.25 × 10−3 g Case Study: Decompression Sickness (“The Bends”) Decompression sickness (DCS), or “the bends,” is an effect of the increased pressure of the air inhaled by scuba divers when swimming underwater at considerable depths. In addition to the pressure exerted by the atmosphere, divers are subjected to additional pressure due to the water above them, experiencing an increase of approximately 1 atm for each 10 m of depth. Therefore, the air inhaled by a diver while submerged contains gases at the corresponding higher ambient pressure, and the concentrations of the gases dissolved in the diver’s blood are proportionally higher per Henry’s law. As the diver ascends to the surface of the water, the ambient pressure decreases and the dissolved gases becomes less soluble. If the ascent is too rapid, the gases escaping from the diver’s blood may form bubbles that can cause a variety of symptoms ranging from rashes and joint pain to paralysis and death. To avoid DCS, divers must ascend from depths at relatively slow speeds (10 or 20 m/min) or otherwise make several decompression stops, pausing for several minutes at given depths during the ascent. When these preventative measures are unsuccessful, divers with DCS are often provided hyperbaric oxygen therapy in pressurized vessels called decompression (or recompression) chambers (Figure 13.4.4). Deviations from Henry’s law are observed when a chemical reaction takes place between the gaseous solute and the solvent. Thus, for example, the solubility of ammonia in water does not increase as rapidly with increasing pressure as predicted by the law because ammonia, being a base, reacts to some extent with water to form ammonium ions and hydroxide ions. This reaction diagram shows three H atoms bonded to an N atom above, below, and two the left of the N. A single pair of dots is present on the right side of the N. This is followed by a plus, then two H atoms bonded to an O atom to the left and below the O. Two pairs of dots are present on the O, one above and the other to the right of the O. A double arrow, with a top arrow pointing right and a bottom arrow pointing left follows. To the right of the double arrow, four H atoms are shown bonded to a central N atom. These 5 atoms are enclosed in brackets with a superscript plus outside. A plus follows, then an O atom linked by a bond to an H atom on its right. The O atom has pairs of dots above, to the left, and below the atom. The linked O and H are enclosed in brackets with superscript minus outside. Gases can form supersaturated solutions. If a solution of a gas in a liquid is prepared either at low temperature or under pressure (or both), then as the solution warms or as the gas pressure is reduced, the solution may become supersaturated. Contributions & Attributions Wikipedia Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at 13.3: Solutions of Solids Dissolved in Water- How to Make Rock Candy 13.5: Solution Concentration- Mass Percent
2530	https://www.nagwa.com/en/explainers/350106087294/	Lesson Explainer: Rank of a Matrix \| Nagwa Lesson Explainer: Rank of a Matrix \| Nagwa Sign Up Sign In English English العربية English English العربية My Wallet Sign Up Sign In My Classes My Messages My Reports My Wallet My Classes My Messages My Reports Lesson Explainer: Rank of a Matrix Mathematics Join Nagwa Classes Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher! Check Available Classes Next Session: Thursday 25 September 2025 • 5:00pm Remaining Seats: 9 Try This In this explainer, we will learn how to find the rank and nullity of a matrix. The “rank” of a matrix is one of the most fundamental and useful properties of a matrix that can be calculated. In many senses, the rank of a matrix can be viewed as a measure of how much indispensable information is encoded by the matrix. As an example, we consider the following simple system of linear equations: Notice how the second equation is essentially identical to the first equation, other than for a multiplication by 3. Moreover, we can multiply the second equation by a constant factor of without changing any of the properties: If we were looking to solve this system of linear equations, then the second equation does not give us any extra information if we already know the first equation. In this sense, only the first equation is indispensable and the second equation is totally redundant and does not need to be written down. When working with a system of linear equations, it is entirely possible that there is one or more redundant equations. Take, for example, the linear system of equations Just like the previous example, the second equation is a copy of the first equation but, in this case, after a multiplication by . Therefore, the second equation offers no extra information and is redundant. Even worse, the third equation can be obtained from adding together the second and third equations. Therefore, the third equation is also redundant and only the first equation is indispensable. One good way of thinking about the “rank” of a matrix is that it measures the number of indispensable equations in the corresponding system of linear equations. There are other, more abstract ways of understanding the rank of a matrix without needing to refer to a system of linear equations, although these will be covered in another explainer. Before showing how to calculate the rank of a matrix, we first need to define several key concepts. Definition: Echelon Form A matrix is said to be in “echelon form” or “row echelon form” if the following two criteria are met: All zero rows are below all nonzero rows. The pivot of a nonzero row appears to the right of the pivot in all the rows above it. The echelon form of a matrix is a vital idea that was first introduced when mathematicians were looking to find general methods for solving systems of linear equations. This definition is used to subsequently define the “reduced echelon form” of a matrix, which we will not describe in this explainer but is of vital importance nonetheless. Consider the matrix We will take the first criterion of echelon form. The third row of this matrix is a zero row, which is not below the nonzero fourth row. Therefore, this matrix cannot meet the first criterion and so it is not in echelon form. However, if we had been given the matrix then the zero row would have been below all other zero rows and hence would have satisfied the first criterion. Now we consider the second criterion. The “pivot” of a row is the first nonzero entry of that row, which is also sometimes referred to as the “leading coefficient” of that row. We highlight the pivots in the matrix immediately above: The second criterion of echelon specifies that all pivots must appear to the right of any pivots in the row above. This is clearly not the case in our matrix, given that the pivot in the second row appears to the left of the pivot in the row above. If we had instead been given the matrix then every pivot is to the right of the pivots in the rows above. This new matrix satisfies both criteria of the echelon form. Definition: Rank of a Matrix When in echelon form, the “rank” of a matrix is the number of nonzero rows. Normally, this is denoted as . There are a couple of points which follow from this definition that are worth discussing. First, the rank of a matrix counts the number of nonzero rows in that matrix. Quite naturally, that means that the rank of a matrix must be less than or equal to the number of rows in that matrix. The second point arising from the above definition of the rank of a matrix is that, clearly, not every matrix is in echelon form, which means that our definition of the rank would not apply. In this case, we would ask ourselves the following question: How do we calculate the rank of a matrix if it is not already in echelon form? To answer this question, we will need one further definition and one further theorem. Definition: Elementary Row Operations For any matrix of order , the three “elementary row operations” are as follows: switching of row with row , denoted ; multiplication of a single row by a nonzero constant , denoted ; a constant multiple of a row added to , denoted . These three row operations are the key ingredients for taking a matrix and manipulating it into either echelon form or reduced echelon form, using a general and powerful technique known as Gauss–Jordan elimination. This is the approach that we will take when trying to understand the rank of a matrix, although we will not detail the full Gauss–Jordan method. We are nearly ready to begin calculating the rank for general matrices, but first we need one final theorem. Theorem: The Rank of a Matrix and Elementary Row Operations The rank of a matrix is unaffected by the elementary row operations. We will not provide a proof in this explainer, but we will instead assume that this is true and begin applying the result to aid our calculations of the matrix rank. In each of the following examples, our aim will be to take the given matrix and use elementary row operations to force this matrix into echelon form. With the knowledge that the elementary row operations do not change the rank of a matrix, we can then simply count the number of nonzero rows. Example 1: Rank of a 2 × 2 Matrix Find the rank of the matrix Answer We label the matrix as This second row of is a copy of the first row after a multiplication by 2. We, therefore, perform the row operation , which gives the matrix The second row is a zero row and appears below all nonzero rows, meeting the first criteria of echelon form. The pivot of each row is the first nonzero entry, as highlighted: The second criterion of echelon form is that all pivots appear to the right of all pivots in the rows above. Since the highlighted entry is the only pivot in this matrix, this criterion is satisfied. This matrix is in echelon form and has 1 nonzero row, which means that the rank of this matrix is 1. Since we only performed elementary row operations to move from to the matrix above, we know that shares this rank, and hence . A decent guideline for how to approach the calculation of an echelon form is to continuously highlight all pivots in the matrix of interest. This draws the eye and can provide a steer as to which elementary row operations should be performed. We will demonstrate the utility of this approach in the following examples. Unlike the reduced echelon form of a matrix, which is unique, there are infinitely many possible echelon forms. Our echelon form of is likely to be entirely different to an echelon form produced using an alternative approach. Example 2: Finding the Rank of a Matrix Find the rank of the matrix Answer We label the matrix as This matrix is certainly not in echelon form! First, we highlight all the pivots of the matrix: The pivots in the second and third rows are not to the right of the pivots in the first row, so this matrix is definitely not in echelon form, meaning that we must perform elementary row operations to achieve this. We begin by trying to secure as many zeros as possible in the first column. There is already a pivot in the top left, so we will do our best to leave this entry alone, producing zeros in the remaining entries of the first column. This can be achieved by the elementary row operations and , with the result being as follows: The pivot in the top left entry is now the only nonzero entry in the first column. The remaining pivots have been highlighted, with the pivots in the second, third, and fourth rows being to the right of the pivot in the first row. Unfortunately, the pivots in the third and fourth rows are not to the right of all pivots in the row above, which means that we must continue performing elementary row operations. Now we see that the fourth row is just a copy of the third row after a multiplication by 2. The elementary row operation gives the matrix We have a zero row at the bottom of this matrix and we will endeavor to leave this row unchanged. However, the pivot in the third row is not to the right of all pivots in the rows above, so we have not quite yet obtained an echelon form. We must find a way of removing the pivot in the third row in order to meet the second criterion of echelon form. This can be achieved by the row operation , which gives a much more agreeable structure of pivots: Now every pivot is to the right of all the pivots in the rows above it, and the only zero row is at the bottom of the matrix. Therefore, this matrix is in echelon form. Given that there are three nonzero rows, the rank of this matrix is 3. We have only used elementary row operations to change the matrix into the matrix directly above. The rank of the original matrix is equal to the rank of the matrix above, meaning that . Example 3: Finding the Rank of a Matrix Find the rank of the matrix Answer We begin by defining the following matrix and highlighting the pivots: The pivots in the second and third rows are not to the right of the pivot in the first row. Therefore, we need to remove these nonzero entries in the first column. The obvious choice of elementary row operations is and , giving the matrix where we have highlighted the pivots. The pivots in the third and fourth rows are not to the right of all pivots in the rows above, so this matrix is not in echelon form. There are many possible approaches for remedying this situation, and we choose the method that is arguably the most direct, which is by immediately removing the pivots in the third and fourth rows with the elementary row operations and . This gives the matrix There is only one remaining obstruction to the matrix above being in echelon form, and that is the pivot in the fourth column, which is not to the right of all pivots in the columns above. We perform the row operation to get The only zero row is at the bottom of the matrix and the pivots are all to the right of every pivot in the rows above. Therefore, this matrix is now in echelon form. Due to the presence of three nonzero rows, the rank of this matrix is 3. Given that we used only row elementary operations to transform from the original matrix to the matrix immediately above, we conclude that . In the final example, we will give a matrix which has a larger number of rows and columns that we will then manipulate into an echelon form. With computations such as these, it is very easy to make an arithmetic error due to the large number of calculations that are involved. The example below is convenient in that the necessary elementary row operations are all easy to determine and do not introduce any fractions into the calculations. However, this will certainly not always be the case, and we should be wary that any error in calculation is likely to proliferate through the remaining calculations, causing the final outcome to be radically different from the correct solution. Example 4: Rank of a Matrix Find the rank of the matrix Answer We define the following matrix and highlight the pivot entries: The pivots are all in the second column, which means that this matrix is not in echelon form, which requires all pivots to be to the right of every pivot in the rows above. We will eliminate the pivots in the second, third, and fourth rows using the elementary row operations , , and . The subsequent matrix is where the pivots have been highlighted. Before thinking any further about the pivots, we note that the fourth row is identical to the third row. We can quickly turn the fourth row into a zero row using the elementary row operation . This gives the matrix The third row is a copy of the second row, except for a scaling by a factor of 2. The row operation then produces the matrix There are now two zero rows which are at the bottom of the matrix, and the pivot in the second row is to the right of the pivot in the first row. This means that the matrix is in echelon form and, given that it has two nonzero rows, its rank is 2. Since we used only elementary row operations to transform the matrix into the matrix immediately above, the two matrices will have the same rank, and hence . Unless a given matrix is already in echelon form, the rank of the matrix is usually not obvious upon first inspection. This means that normally we would have to perform (at least part of) the famed Gauss–Jordan elimination, as we did above. Mastering the technique of Gauss–Jordan elimination is arguably the single most important prerequisite for being able to compute the rank of a matrix. A related topic to the rank of a matrix is the “nullity” or “null space” of a matrix, which is a quantity that counts the number of zero rows of a matrix in echelon form (rather than the number of nonzero rows, which is counted by the rank). These two concepts are used ubiquitously when studying linear algebra from a more abstract perspective, meaning that an understanding of their properties is essential. Key Points The pivots of matrix are the first nonzero entries of each row. These are sometimes referred to as the “leading coefficients.” Highlighting the pivots is normally very useful when attempting to perform Gauss–Jordan elimination or when performing elementary row operations in a general sense. A matrix is in echelon form if all zero rows are below all nonzero rows and if all pivots are to the right of any pivot in a row above. If a matrix is in echelon form, then the rank is the number of nonzero rows. The rank of a matrix is unchanged by elementary row operations. Lesson Menu Lesson Lesson Plan Lesson Explainer Lesson Playlist Join Nagwa Classes Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher! Interactive Sessions Chat & Messaging Realistic Exam Questions Nagwa is an educational technology startup aiming to help teachers teach and students learn. Company About Us Contact Us Privacy Policy Terms and Conditions Careers Tutors Content Lessons Lesson Plans Presentations Videos Explainers Playlists Copyright © 2025 Nagwa All Rights Reserved Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy Accept
2531	https://www.gauthmath.com/solution/1811806886962309/2-Find-the-coefficient-of-x3-in-the-binomial-expansion-of-a-1-3x-3-b-1-1-4-x-1-2	Solved: Find the coefficient of x^3 in the binomial expansion of: a) (1+3x)^-3 b) (1+ 1/4 x)^- 1/ [Math] Drag Image or Click Here to upload Command+to paste Upgrade Sign in Homework Homework Assignment Solver Assignment Calculator Calculator Resources Resources Blog Blog App App Gauth Unlimited answers Gauth AI Pro Start Free Trial Homework Helper Study Resources Math Counting, Permutations and Combinations Questions Question Find the coefficient of x^3 in the binomial expansion of: a) (1+3x)^-3 b) (1+ 1/4 x)^- 1/2 c) (8-2x)^- 1/3 (8-x)^ 2/3 d) Show transcript Gauth AI Solution 100%(3 rated) Answer For part (a), we use the binomial expansion formula for negative exponents: $$(1 + u)^{n} = \sum_{k=0}^{\infty} \binom{n}{k} u^k$$(1+u)n=k=0∑∞(k n)u k, where $$u = 3x$$u=3 x and $$n = -3$$n=−3 The general term is given by: $$T_k = \binom{-3}{k} (3x)^k = \frac{(-3)(-4)(-5)...(-3-k+1)}{k!} (3^k x^k)$$T k=(k−3)(3 x)k=k!(−3)(−4)(−5)...(−3−k+1)(3 k x k) We need the coefficient of $$x^{3}$$x 3, so set $$k = 3$$k=3: $$T_3 = \binom{-3}{3} (3x)^3 = \frac{-3(-4)(-5)}{3!} (27x^3)$$T 3=(3−3)(3 x)3=3!−3(−4)(−5)(27 x 3) Calculate $$\binom{-3}{3} = \frac{-3 \cdot -4 \cdot -5}{3!} = \frac{-60}{6} = -10$$(3−3)=3!−3⋅−4⋅−5=6−60=−10 Thus, the coefficient of $$x^{3}$$x 3 is: $$-10 \cdot 27 = -270.$$−10⋅27=−270. Answer: Answer for part (a): -270. For part (b), we use the same binomial expansion formula with $$u = \frac{1}{4}x$$u=4 1x and $$n = -\frac{1}{2}$$n=−2 1 The general term is: $$T_k = \binom{-\frac{1}{2}}{k} \left(\frac{1}{4}x\right)^k = \frac{(-\frac{1}{2})(-\frac{3}{2})(-\frac{5}{2})...(-\frac{1}{2}-k+1)}{k!} \left(\frac{1}{4}\right)^k x^k$$T k=(k−2 1)(4 1x)k=k!(−2 1)(−2 3)(−2 5)...(−2 1−k+1)(4 1)k x k We need the coefficient of $$x^{3}$$x 3, so set $$k = 3$$k=3: $$T_3 = \binom{-\frac{1}{2}}{3} \left(\frac{1}{4}\right)^3 x^3$$T 3=(3−2 1)(4 1)3 x 3 Calculate $$\binom{-\frac{1}{2}}{3} = \frac{-\frac{1}{2} \cdot -\frac{3}{2} \cdot -\frac{5}{2}}{3!} = \frac{-\frac{15}{8}}{6} = -\frac{15}{48} = -\frac{5}{16}$$(3−2 1)=3!−2 1⋅−2 3⋅−2 5=6−8 15=−48 15=−16 5 Thus, the coefficient of $$x^{3}$$x 3 is: $$-\frac{5}{16} \cdot \frac{1}{64} = -\frac{5}{1024}.$$−16 5⋅64 1=−1024 5. Answer: Answer for part (b): -\frac{5}{1024}. For part (c), we need to expand $$(8 - 2x)^{-\frac{1}{3}}$$(8−2 x)−3 1 and $$(8 - x)^{\frac{2}{3}}$$(8−x)3 2 For $$(8 - 2x)^{-\frac{1}{3}}$$(8−2 x)−3 1, let $$u = -\frac{2x}{8} = -\frac{x}{4}$$u=−8 2 x=−4 x and $$n = -\frac{1}{3}$$n=−3 1: $$T_k = \binom{-\frac{1}{3}}{k} \left(-\frac{x}{4}\right)^k.$$T k=(k−3 1)(−4 x)k. For $$k = 3$$k=3: $$T_3 = \binom{-\frac{1}{3}}{3} \left(-\frac{1}{4}\right)^3 x^3.$$T 3=(3−3 1)(−4 1)3 x 3. Calculate $$\binom{-\frac{1}{3}}{3} = \frac{-\frac{1}{3} \cdot -\frac{4}{3} \cdot -\frac{7}{3}}{3!} = \frac{-\frac{28}{27}}{6} = -\frac{14}{81}$$(3−3 1)=3!−3 1⋅−3 4⋅−3 7=6−27 28=−81 14 Thus, the coefficient of $$x^{3}$$x 3 from this term is: $$-\frac{14}{81} \cdot -\frac{1}{64} = \frac{14}{5184}.$$−81 14⋅−64 1=5184 14. Now for $$(8 - x)^{\frac{2}{3}}$$(8−x)3 2, let $$u = -\frac{x}{8}$$u=−8 x and $$n = \frac{2}{3}$$n=3 2: $$T_k = \binom{\frac{2}{3}}{k} \left(-\frac{x}{8}\right)^k.$$T k=(k 3 2)(−8 x)k. For $$k = 3$$k=3: $$T_3 = \binom{\frac{2}{3}}{3} \left(-\frac{1}{8}\right)^3 x^3.$$T 3=(3 3 2)(−8 1)3 x 3. Calculate $$\binom{\frac{2}{3}}{3} = \frac{\frac{2}{3} \cdot -\frac{1}{3} \cdot -\frac{4}{3}}{3!} = \frac{\frac{8}{27}}{6} = \frac{4}{81}$$(3 3 2)=3!3 2⋅−3 1⋅−3 4=6 27 8=81 4 Thus, the coefficient of $$x^{3}$$x 3 from this term is: $$\frac{4}{81} \cdot -\frac{1}{512} = -\frac{4}{41472}.$$81 4⋅−512 1=−41472 4. Combine both coefficients: $$\frac{14}{5184} - \frac{4}{41472}.$$5184 14−41472 4. Find a common denominator (41472): $$\frac{14 \cdot 8}{41472} - \frac{4}{41472} = \frac{112 - 4}{41472} = \frac{108}{41472} = \frac{9}{3456}.$$41472 14⋅8−41472 4=41472 112−4=41472 108=3456 9. Answer: Answer for part (c): \frac{9}{3456}. Helpful Not Helpful Explain Simplify this solution Gauth AI Pro Back-to-School 3 Day Free Trial Limited offer! Enjoy unlimited answers for free. Join Gauth PLUS for $0 Previous questionNext question Related Find the coefficient of x3 in the binomial expansion of: a 1+3x-3 b 1+ 1/4 x- 1/2 c 8-2x- 1/3 8-x 2/3 d 100% (3 rated) Find the coefficient of x3 in the binomial expansion of: a 1+3x-3 b 1+ 1/4 x- 1/2 c 8-2x- 1/3 8-x 2/3 d 100% (4 rated) Find the coefficient of x3 in the binomial expansion of. a 1+3x-3 b 1+ 1/4 x- 1/2 c 8-2x- 1/3 8-x 2/3 d 100% (3 rated) Find the coefficient of x3 in the binomial expansion of: a 1+3x-3 b 1+ 1/4 x- 1/2 c 8-2x- 1/3 8-x 2/3 d 3 State the range of values of x for which binomial expansions of the follov 100% (2 rated) Subtopics: Expanding 1+xn , expanding a+bxn , using partial fractions 1. Find the binomial expansion up to and including the x3 term of: b a 1+x 1/4 1-x-2 c 4+x- 1/2 -1-4x 1/3 d 2. Find the coefficient of x3 in the binomial expansion of: a 1+3x-3 b 1+ 1/4 x- 1/2 c 8-2x- 1/3 8-x 2/3 d 3. State the range of values of x for which binomial expansions of the following are valid: a 1+3x- 1/2 4+3x-4 b 4. Find the coefficient of x2 in the binomial expansion of 3+2x1-3x- 1/3 . 5. a Write frac 12x-2x+4 in the form A/x-2 + B/x+4 , where A and B are constants to be found. b Hence or otherwise, expand frac 12x-2x+4 in ascending powers of x up to and including the term in x2. . a Find the binomial expansion of square root of 1-x up to and including the term in x3. b By substituting x= 1/100 , find an approximation to square root of 99 , giving your answer to 3 significant figures. 100% (1 rated) Find the coefficient of x4 in the binomial expansion of 1/2 x2-25 100% (1 rated) Find the coefficient of x5 in the binomial expansion of 3- 1/4 x7 100% (5 rated) Find the coefficient of x4 in the binomial expansion of 3- 1/4 x5 100% (3 rated) a i How many even numbers are there from 100 to 1000 inclusive? ii I write down m consecutive odd numbers, the smallest of which is л. Find in terms of m and n, the value of the largest odd number I wrote down. b y is a positive whole number such that y/11 lies between 81/13 and 98/15 . Find all possible values of y. 80% (5 rated) The two building blocks below can be tower. Calculate the height of the tower in centimetres. ≌ ∞ 5 63% (8 rated) Gauth it, Ace it! contact@gauthmath.com Company About UsExpertsWriting Examples Legal Honor CodePrivacy PolicyTerms of Service Download App
2532	https://www.askiitians.com/forums/8-grade-maths/prove-that-a-cyclic-trapezium-is-isosceles-and-its-25_488855.htm	Prove that a cyclic trapezium is isosceles and its diagonals are equa - askIITians help@askiitians.com 1800-150-456-789 Live Courses Resources Exam Info Forum Notes Upcoming Examination Login BACK 8 grade maths>Prove that a cyclic trapezium is isoscele... Prove that a cyclic trapezium is isosceles and its diagonals are equal. Aniket Singh , 1 Month ago Grade 1 Answers Askiitians Tutor Team Last Activity: 1 Month ago A cyclic trapezium, also known as a cyclic trapezoid, is a trapezium where all vertices lie on a single circle. To show that such a trapezium is isosceles and that its diagonals are equal, we can use some properties of cyclic quadrilaterals. Properties of Cyclic Quadrilaterals In any cyclic quadrilateral, the opposite angles are supplementary. This means that the sum of the angles at one pair of opposite corners equals 180 degrees. Establishing Isosceles Properties Consider a cyclic trapezium ABCD, where AB is parallel to CD. Let the angles at vertices A and B be denoted as ∠A and ∠B, and the angles at vertices C and D as ∠C and ∠D. Since AB is parallel to CD, we have: ∠A + ∠D = 180° ∠B + ∠C = 180° From the property of cyclic quadrilaterals, we know: ∠A + ∠B = 180° ∠C + ∠D = 180° Since both pairs of angles sum to 180°, we can conclude that: ∠A = ∠B ∠C = ∠D This equality of angles indicates that the legs of the trapezium (AD and BC) are equal, confirming that ABCD is an isosceles trapezium. Proving Equal Diagonals Next, we need to show that the diagonals AC and BD are equal. In a cyclic trapezium, the diagonals can be analyzed using the Law of Cosines or by considering the triangles formed by the diagonals. Since we have established that angles ∠A = ∠B and ∠C = ∠D, triangles ABC and ABD are congruent by the Angle-Side-Angle (ASA) criterion. Therefore, we can conclude: AC = BD This equality of diagonals further supports the properties of the cyclic trapezium. Final Thoughts In summary, a cyclic trapezium is isosceles because its non-parallel sides are equal, and its diagonals are equal due to the congruence of the triangles formed by the diagonals. This demonstrates the unique properties of cyclic trapeziums effectively. Approved LIVE ONLINE CLASSES Prepraring for the competition made easy just by live online class. Full Live Access Study Material Live Doubts Solving Daily Class Assignments Start your free trial Other Related Questions on 8 grade maths Use Euclid’s division algorithm to find the HCF of the following numbers: 55 and 210. Grade 8 > 8 grade maths 1 Answer Available Last Activity: 7 Days ago Use Euclid’s division algorithm to find the HCF of: 867 and 255 Grade 8 > 8 grade maths 1 Answer Available Last Activity: 7 Days ago Two supplementary angles differ by 48°. Find the angles. Grade 8 > 8 grade maths 1 Answer Available Last Activity: 7 Days ago Two people could fit new windows in a house in 3 days. (I) One of the persons fell ill before the work started. How long would the job take now? (II) How many people would be needed to fit the windows in 1 day? Grade 8 > 8 grade maths 1 Answer Available Last Activity: 7 Days ago When two lines are perpendicular to each other, the angle is said to be ____ angle. A. Acute B. Right C. Obtuse D. Equal Grade 8 > 8 grade maths 1 Answer Available Last Activity: 7 Days ago View all Questions We help you live your dreams. Get access to our extensive online coaching courses for IIT JEE, NEET and other entrance exams with personalised online classes, lectures, study talks and test series and map your academic goals. Company About US Privacy Policy Terms and condition Course Packages Contact US +91-735-322-1155 info@askiitians.com AskiiTians.com C/O Transweb B-30, Sector-6 Noida - 201301 Tel No. +91 7353221155 info@askiitians.com , 2006-2024, All Rights reserved Type a message here... Free live chat⚡ by ·
2533	https://math.stackexchange.com/questions/2639242/class-width-vs-class-size-vs-class-interval	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Class width vs Class size vs Class interval Ask Question Asked Modified 5 years, 7 months ago Viewed 30k times 1 $\begingroup$ Please enlighten me. I'm confused with the lesson. We are now on statistics (grouped data) and I'm confused with Class width, Class size and Class interval. Can you differentiate the 3 in simple words? By the way, the $i$ stands for which of the 3? Thanks and good day statistics Share edited Feb 6, 2018 at 22:19 BruceET 52.6k88 gold badges3333 silver badges6464 bronze badges asked Feb 6, 2018 at 22:06 user484792user484792 3911 gold badge11 silver badge44 bronze badges $\endgroup$ 1 $\begingroup$ Maybe try stats.stackexchange.com for questions on terminology from statistics. $\endgroup$ GEdgar – GEdgar 2018-02-06 22:38:31 +00:00 Commented Feb 6, 2018 at 22:38 Add a comment \| 2 Answers 2 Reset to default 2 $\begingroup$ The class width is the difference between class boundaries (may or may not be the same as class limits). For example, $$10-19 \ \ \ \ 3 \ 20-29 \ \ \ \ 7 \ 30-39 \ \ \ \ 2$$ The second class limits are $20$ and $29$, while class boundaries are $19.5$ and $29.5$. Hence, the second class width is $29.5-19.5=10$. Share answered Feb 7, 2018 at 1:50 farruhotafarruhota 32.3k22 gold badges2020 silver badges5454 bronze badges $\endgroup$ Add a comment \| 0 $\begingroup$ Here are 50 test scores, sorted from smallest to largest. The numbers in brackets show the index of the first observation in each row (ten observations per row). 69 71 76 79 79 80 81 82 82 83 85 86 88 88 89 89 90 90 92 92 93 93 98 99 99 100 100 100 101 102 103 104 105 105 105 105 106 106 107 107 107 108 109 115 116 118 119 119 123 124 Here is a histogram of these data, with labels atop each of the seven bars, showing the size (number of observations) of each Class interval. The modal interval (the one with the largest count) is $(100, 110].$ The size or frequency of this interval is $15$. For this histogram, the sizes of the intervals (heights of bars) are shown on the vertical scale. Accordingly, this is called a "frequency histogram." (The sum of the frequencies must add to the size, here $n = 50,$ of the sample.) In this histogram, small tick marks at the bottom show the exact positions of these 15 scores $101, 102, \dots, 109, 109.$ (There aren't 15 distinct tick marks because some marks represent several tied observations.) The width of each interval is 10 points: ('61 up through 70', '71 up through 80', and so on.) The usual practice is to use intervals of equal width, unless there is a very good reason not to. Notice that we have to decide whether the 'round' numbers 70, 80, 90, and so on, are the smallest or largest values in an interval. The software that made this histogram puts the round numbers at the high end of each interval. It would mess up the counting if we didn't make a clear distinction whether an endpoint in included (here denoted by "$\,]$", bracket) or excluded (here denoted by "$(\,$ " prenthesis). Another method would be to label the endpoints '60.5 to 70.5', '70.5 to 80.5', and so on, because we have integer data which means that no score could ever fall at an interval endpoint. But this gives slightly messier labels along the score (bottom) axis. I don't have your book at hand, so I can't be sure what $i$ stands for. My guess is that this may be a way of numbering the class intervals: $i = 1$ for $(60,70]$; $i = 2$ for $(70, 80]$, and so on to $i = 7$ for $(120,130].$ But sometimes $i$ is used to number the observations. In that case you would have $i = 1, 2, \dots, 50$ for the observations. Share edited Feb 6, 2018 at 23:19 answered Feb 6, 2018 at 23:00 BruceETBruceET 52.6k88 gold badges3333 silver badges6464 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions statistics See similar questions with these tags. 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2534	https://www.youtube.com/watch?v=Lp2aV_4LF48	Python ANOVA using Statsmodels and Pandas Erik Marsja 2240 subscribers 218 likes Description 40195 views Posted: 12 Jun 2018 In this short tutorial we will learn how to carry out one-way ANOVA in Python. Make sure you subscribe to the channel if you haven't: In this ANOVA tutorial we are using the packages Pandas and Statsmodels. It's VERY simple and straight forward! As a bonus you will also learn how to load data from a csv file using pandas read_csv method. It is on this dataset that we do the data analysis but you can use your own data to carry out the statistical tests using Python. However, you can check my short tutorial dedicated to reading and writing csv files using Pandas here: Obviously, you will need to install Pandas and Statsmodel do ANOVA in Python: » How to install Statsmodels: » How to install Pandas: » Link to the Plantgrowth dataset used in the tutorial: » Blogpost with code and more methods to do ANOVA: Other YouTube Python data analysis tutorials: » ANOVA using Scipy and Pandas: » Repeated measures ANOVA using Statsmodels: » T-test in Python using Statsmodels or SciPy: 29 comments Transcript: Introduction hi and welcome! In this short Python tutorial we will learn how to carry out one-way ANOVA using the packages pandas and statsmodels! We use pandas because Prerequisites the dataframe object is really neat and works really great with statsmodels. To follow this tutorial you will obviously need to have these packages installed and you will find links in the description below so you will learn how to install them. As a bonus you will also learn how to load a csv file into a panda's dataframe and you will also learn how to create a box plots. Regarding box plots this first line of code is not necessary to carry out the ANOVA. However if we want to display plots in jupyter notebooks we need to have that line of code. So we if you don't want to have box plots or are not using jupyter notebooks you don't need that. So first we start by importing pandas as pd we continue by importing statsmodels api as sm and finally we import ordinary least-squares, OLS, from the statsmodels formula api. This should work now right and now that we have everything we need we Coding continue by loading the CSV file plantgrowth into a dataframe object. So you can see the link in the description below if you want to download this file and play around with it. You can also find a link to one of my blog posts that will go through four different methods to carry out one-way ANOVA using Python. Okay that said let's load the data, we use read_read and plantgrowth.csv and you have to make sure you spell everything correctly and that this file is in the same folder as your jupyter notebook or your Python script. It will not work otherwise. Now let's create a box plot, we will just use the boxplot() method of the dataframe object we created earlier data.boxplot and we want to have "weight" as the dependent variable or that's what we want to have in the y-axis right. So... "weight" and we want to group it so it's gonna be by "group" and make sure you spell everything correctly. Now this should show us a box plots and it did look box ANOVA plot group by group weight. So now we can continue with ANOVA. Okay let's type in the model "mod = ols('weight~ group', data=data)" and the data is the dataframe object and we want to fit because this is a linear regression but we want to carry out the one-way ANOVA on this model right so let's continue with that and we create an ANOVA table. And we use sm.stats.anova_lm, mod our model and fit type 2 - sorry. We want to print the results we want to print this table. Now let's see! Yeah! So here we can see we have the sum squares, degrees of freedom, our F value, and the p value. I really hope you learned something from this short Python statistics tutorial. Make sure you subscribe to my channel if you haven't already! have a nice day!
2535	https://books.google.com/books/about/Chemistry_of_the_Elements.html?hl=fr&id=EvTI-ouH3SsC	Ma bibliothÃ¨que Aide Recherche AvancÃ©e de Livres E-BOOK Ã PARTIR DE 20,29 $US Obtenir la version papier de ce livre Buy Direct from Elsevier Amazon France Amazon.com Barnes&Noble.com Books-A-Million IndieBound Trouver en bibliothÃ¨que Tous les vendeurs » \| \| \| Chemistry of the Elements N. N. Greenwood, A. Earnshaw Elsevier, 2 dÃ©c. 2012 - 1600 pages When this innovative textbook first appeared in 1984 it rapidly became a great success throughout the world and has already been translated into several European and Asian languages. Now the authors have completely revised and updated the text, including more than 2000 new literature references to work published since the first edition. No page has been left unaltered but the novel features which proved so attractive have been retained. The book presents a balanced, coherent and comprehensive account of the chemistry of the elements for both undergraduate and postgraduate students. This crucial central area of chemistry is full of ingenious experiments, intriguing compounds and exciting new discoveries. The authors specifically avoid the term `inorganic chemistry' since this evokes an outmoded view of chemistry which is no longer appropriate in the final decade of the 20th century. Accordingly, the book covers not only the 'inorganic' chemistry of the elements, but also analytical, theoretical, industrial, organometallic, bio-inorganic and other cognate areas of chemistry. The authors have broken with recent tradition in the teaching of their subject and adopted a new and highly successful approach based on descriptive chemistry. The chemistry of the elements is still discussed within the context of an underlying theoretical framework, giving cohesion and structure to the text, but at all times the chemical facts are emphasized. Students are invited to enter the exciting world of chemical phenomena with a sound knowledge and understanding of the subject, to approach experimentation with an open mind, and to assess observations reliably. This is a book that students will not only value during their formal education, but will keep and refer to throughout their careers as chemists. - Completely revised and updated - Unique approach to the subject - More comprehensive than competing titles AperÃ§u du livre » \| Pages sÃ©lectionnÃ©es Page 28 Page 62 Page de titre Table des matiÃ¨res Index Table des matiÃ¨res \| \| \| --- \| \| Chapter 1 Origin of the Elements Isotopes and Atomic Weights \| 1 \| \| \| \| Chapter 2 Chemical Periodicity and the Periodic Table \| 20 \| \| \| \| Chapter 3 Hydrogen \| 32 \| \| \| \| Chapter 4 Lithium Sodium Potassium Rubidium Caesium and Francium \| 68 \| \| \| \| Chapter 5 Beryllium Magnesium Calcium Strontium Barium and Radium \| 107 \| \| \| \| Chapter 6 Boron \| 139 \| \| \| \| Chapter 7 Aluminium Gallium Indium and Thallium \| 216 \| \| \| \| Chapter 8 Carbon \| 268 \| \| \| \| \| \| --- \| \| Chapter 21 Titanium Zirconium and Hafnium \| 954 \| \| \| \| Chapter 22 Vanadium Niobium and Tantalum \| 976 \| \| \| \| Chapter 23 Chromium Molybdenum and Tungsten \| 1002 \| \| \| \| Chapter 24 Manganese Technetium and Rhenium \| 1040 \| \| \| \| Chapter 25 Iron Ruthenium and Osmium \| 1070 \| \| \| \| Chapter 26 Cobalt Rhodium and Iridium \| 1113 \| \| \| \| Chapter 27 Nickel Palladium and Platinum \| 1144 \| \| \| \| Chapter 28 Copper Silver and Gold \| 1173 \| \| \| Plus \| \| \| --- \| \| Chapter 9 Silicon \| 328 \| \| \| \| Chapter 10 Germanium Tin and Lead \| 367 \| \| \| \| Chapter 11 Nitrogen \| 406 \| \| \| \| Chapter 12 Phosphorus \| 473 \| \| \| \| Chapter 13 Arsenic Antimony and Bismuth \| 547 \| \| \| \| Chapter 14 Oxygen \| 600 \| \| \| \| Chapter 15 Sulfur \| 645 \| \| \| \| Chapter 16 Selenium Tellurium and Polonium \| 747 \| \| \| \| Fluorine Chlorine Bromine Iodine and Astatine \| 789 \| \| \| \| Helium Neon Argon Krypton Xenon and Radon \| 888 \| \| \| \| Chapter 19 Coordination and Organometallic Compounds \| 905 \| \| \| \| Chapter 20 Scandium Yttrium Lanthanum and Actinium \| 944 \| \| \| \| \| \| --- \| \| Chapter 29 Zinc Cadmium and Mercury \| 1201 \| \| \| \| Chapter 30 The Lanthanide Elements Z5871 \| 1227 \| \| \| \| Chapter 31 The Actinide and Transactinide Elements Z90112 \| 1250 \| \| \| \| Atomic Orbitals \| 1285 \| \| \| \| Symmetry Elements Symmetry Operations and Point Groups \| 1290 \| \| \| \| Some NonSi Units \| 1293 \| \| \| \| Abundance of Elements in Crustal Rocks \| 1294 \| \| \| \| Effective Ionic Radii \| 1295 \| \| \| \| Nobel Prize for Chemistry \| 1296 \| \| \| \| Nobel Prize for Physics \| 1300 \| \| \| \| Index \| 1305 \| \| \| \| Droits d'auteur \| \| \| Moins Expressions et termes frÃ©quents 4-coordinate acid adducts alkali metals allg Angew angle anhydrous anion anorg aqueous solutions atomic weight boranes boron bridging carbon cation chains Chem chemical chemistry chlorine cluster colourless complex compounds coordination covalent crystalline crystals Dalton Trans dimer dissociation distances electrolysis electronegativity electrons elements energy Engl example fluorine fullerenes give graphite Group 15 elements H bonds halides halogen heated hydrated hydrides hydrogen hydrolysis industrial Inorg Inorganic ionic isoelectronic isotopes kJ mol-¹ known layer ligand liquid mol¯¹ molecular molecules obtained occurs octahedral orbitals organometallic oxidation oxoacids oxygen phase phosphorus planar Polyhedron polymeric prepared production properties proton reaction reactive reacts readily reducing ring room temperature salts shown in Fig solid soluble solvent species stable stoichiometry structure sulfides sulfur sulfuric acid synthesis Table tetrahedral thermal tonnes trigonal X-ray yields Informations bibliographiques \| \| \| --- \| \| Titre \| Chemistry of the Elements \| \| Auteurs \| N. N. Greenwood, A. Earnshaw \| \| Ãdition \| 2 \| \| Ãditeur \| Elsevier, 2012 \| \| ISBN \| 0080501095, 9780080501093 \| \| Longueur \| 1600 pages \| \| \| \| \| Exporter la citation \| BiBTeX EndNote RefMan \| Ã propos de Google Livres - RÃ¨gles de confidentialitÃ© - Conditions d' utilisation - Informations destinÃ©es aux Ã©diteurs - Signaler un problÃ¨me - Aide - Accueil Google
2536	https://en.wiktionary.org/wiki/dress_form	dress form - Wiktionary, the free dictionary Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main Page Community portal Requested entries Recent changes Random entry Help Glossary Contact us Special pages Feedback If you have time, leave us a note. Search Search [x] Appearance Appearance move to sidebar hide Text Small Standard Large This page always uses small font size Width Standard Wide The content is as wide as possible for your browser window. Color (beta) Automatic Light Dark This page is always in light mode. Donations Preferences Create account Log in [x] Personal tools Donations Create account Log in Pages for logged out editors learn more Contributions Talk [x] Toggle the table of contents Contents move to sidebar hide Beginning 1 EnglishToggle English subsection 1.1 Noun dress form [x] 1 language தமிழ் Entry Discussion [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Create a book Download as PDF Printable version In other projects From Wiktionary, the free dictionary English [edit] Noun [edit] dressform (pluraldress forms) A simple tailor's dummy with no head or limbs Retrieved from " Categories: English lemmas English nouns English countable nouns English multiword terms Hidden categories: Pages with entries Pages with 1 entry This page was last edited on 15 November 2016, at 17:25. Definitions and other text are available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Privacy policy About Wiktionary Disclaimers Code of Conduct Developers Statistics Cookie statement Mobile view Search Search [x] Toggle the table of contents dress form 1 languageAdd topic
2537	https://math.stackexchange.com/questions/3278052/find-the-umvue-for-prx-0-e-lambda	probability theory - Find the UMVUE for $Pr[X=0]=e^{-\lambda}$ - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Find the UMVUE for P r[X=0]=e−λ P r[X=0]=e−λ Ask Question Asked 6 years, 3 months ago Modified6 years, 3 months ago Viewed 92 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. I am working on a problem that requires me to find the UMVUE of P r[X=0]=e−λ P r[X=0]=e−λ. There are several issues that I am having so I would really appreciate your help. 1), What does it mean to be an "estimator" of a probability? I understand that estimators can estimate parameters such as the mean or variance of a distribution, but I do not quite get the intuitive meaning of this problem. . . 2), The notes that I am looking at mentions I{X 1=0}I{X 1=0} is a "natural choice" of an unbiased estimator of P r[X=0]P r[X=0]. I am thinking that because I don't understand intuitively what is going on I am not seeing why an indicator is an estimator. The notes further proceeds to simplifying E[I{X 1=0}\|∑i=1 n X i=x]E[I{X 1=0}\|∑i=1 n X i=x] which leads to (n−1 n)x(n−1 n)x I understand the rest of the algebraic portion, but because I don't know why it started off like this it does not stay in my head. Thank you for your help. probability-theory statistics parameter-estimation Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Jun 29, 2019 at 18:51 hyg17hyg17 5,281 4 4 gold badges 43 43 silver badges 90 90 bronze badges 6 Because E[I{X 1=0}]=P(X 1=0)=e−λ E[I{X 1=0}]=P(X 1=0)=e−λ. The probability is a function of λ λ which is unknown, so you are estimating it.StubbornAtom –StubbornAtom 2019-06-29 21:10:08 +00:00 Commented Jun 29, 2019 at 21:10 So, let me make sure that I understand this. I X 1=0=1 I X 1=0=1 when X 1=0 X 1=0 and =0=0 when X 1≠0 X 1≠0 so under that condition E[I X 1=0]=∑X I X 1=0∗P r[X=x]E[I X 1=0]=∑X I X 1=0∗P r[X=x]. So, if X 1=0 X 1=0 then ∵=1∗P r[X 1=0]∵=1∗P r[X 1=0] and when X 1≠0 X 1≠0 then ∵=0∗P r[X 1≠0]∵=0∗P r[X 1≠0]hyg17 –hyg17 2019-06-29 22:09:56 +00:00 Commented Jun 29, 2019 at 22:09 I see that numerically it is true. However, why would it be "natural" to think that this is the estimator? I mean, this means that the probability is either 1 or 0 right? I do not find that to be natural at all. . .hyg17 –hyg17 2019-06-29 22:12:39 +00:00 Commented Jun 29, 2019 at 22:12 Not numerically, it is exactly true. To repeat, perhaps the most trivial unbiased estimator of a parametric function like the probability P(X∈A)P(X∈A) is the indicator of the event X∈A X∈A, since E(I X∈A)=1.P(X∈A)+0.P(X∉A)E(I X∈A)=1.P(X∈A)+0.P(X∉A).StubbornAtom –StubbornAtom 2019-06-29 22:56:14 +00:00 Commented Jun 29, 2019 at 22:56 Hmm... I will just accept the algebraic part and hope that I spend enough time in stats that someday I understand.hyg17 –hyg17 2019-06-30 16:47:34 +00:00 Commented Jun 30, 2019 at 16:47 \|Show 1 more comment 0 Sorted by: Reset to default You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions probability-theory statistics parameter-estimation See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 4For what values of k∈Z∖{0}k∈Z∖{0} does there exist an unbiased estimator of e k λ e k λ? Related 1Sufficiency and UMVUE for Poisson distribution 4CRLB/UMVUE estimation of θ θ 3UMVUE of E[X 2]E[X 2] where X i X i is Poisson (λ)(λ) 4Does an UMVUE always exist? 0Using Rao-Blackwell to find UMVUE for λ 2 e−λ λ 2 e−λ for Poisson Distribution 1UMVUE for p 1−p p 1−p Hot Network Questions Fundamentally Speaking, is Western Mindfulness a Zazen or Insight Meditation Based Practice? Direct train from Rotterdam to Lille Europe Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth? In the U.S., can patients receive treatment at a hospital without being logged? 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2538	https://pubs.acs.org/doi/10.1021/jp800816a	Thermodynamic Origin of Hofmeister Ion Effects \| The Journal of Physical Chemistry B Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Manage Preferences Loading [MathJax]/extensions/MathZoom.js Recently Viewedclose modal ACS ACS Publications C&EN CAS Access through institution Log In Thermodynamic Origin of Hofmeister Ion Effects Cite Citation Citation and abstract Citation and references More citation options Share Share on Facebook X Wechat LinkedIn Reddit Email Bluesky Jump to Abstract Supporting Information Cited By Expand Collapse Back to top Close quick search form clear search J. Phys. Chem. 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Phys. Chem. B 2008, 112, 31, 9428-9436 ADVERTISEMENT Info Metrics The Journal of Physical Chemistry B Vol 112/Issue 31 Article Get e-Alerts Cite Citation Citation and abstract Citation and references More citation options Share Share on Facebook X WeChat LinkedIn Reddit Email Bluesky Jump to Abstract Supporting Information Cited By Expand Collapse Article July 16, 2008 Thermodynamic Origin of Hofmeister Ion Effects Click to copy article link Article link copied! Laurel M. Pegram† M. Thomas Record, Jr.†‡ View Author Information View Author Information Departments of Chemistry and Biochemistry, University of Wisconsin-Madison, Madison, Wisconsin 53706 E-mails: (L.M.P.) pegram@chem.wisc.edu, (M.T.R.) record@biochem.wisc.edu †Department of Chemistry. ‡Department of Biochemistry. Access Through Access is not provided via Institution Name Loading Institutional Login Options... Access Through Your Institution Add or Change Institution Explore subscriptions for institutions Other Access OptionsSupporting Information (1) The Journal of Physical Chemistry B Cite this: J. Phys. Chem. B 2008, 112, 31, 9428–9436 Click to copy citation Citation copied! Published July 16, 2008 Publication History Received 28 January 2008 Accepted 11 April 2008 Published online 16 July 2008 Published in issue 1 August 2008 research-article Copyright © 2008 American Chemical Society Request reuse permissions Article Views 3686 Altmetric - Citations 252 Learn about these metrics close Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days. Citations are the number of other articles citing this article, calculated by Crossref and updated daily.Find more information about Crossref citation counts. The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information onthe Altmetric Attention Score and how the score is calculated. Abstract Click to copy section link Section link copied! Quantitative interpretation and prediction of Hofmeister ion effects on protein processes, including folding and crystallization, have been elusive goals of a century of research. Here, a quantitative thermodynamic analysis, developed to treat noncoulombic interactions of solutes with biopolymer surface and recently extended to analyze the effects of Hofmeister salts on the surface tension of water, is applied to literature solubility data for small hydrocarbons and model peptides. This analysis allows us to obtain a minimum estimate of the hydration b 1 (H 2 O Å−2), of hydrocarbon surface and partition coefficients K p, characterizing the distribution of salts and salt ions between this hydration water and bulk water. Assuming that Na+ and SO 4 2− ions of Na 2 SO 4 (the salt giving the largest reduction in hydrocarbon solubility as well as the largest increase in surface tension) are fully excluded from the hydration water at hydrocarbon surface, we obtain the same b 1 as for air-water surface (∼0.18 H 2 O Å−2). Rank orders of cation and anion partition coefficients for nonpolar surface follow the Hofmeister series for protein processes, but are strongly offset for cations in the direction of exclusion (preferential hydration). By applying a coarse-grained decomposition of water accessible surface area (ASA) into nonpolar, polar amide, and other polar surface and the same hydration b 1 to interpret peptide solubility increments, we determine salt partition coefficients for amide surface. These partition coefficients are separated into single-ion contributions based on the observation that both Cl− and Na+ (also K+) occupy neutral positions in the middle of the anion and cation Hofmeister series for protein folding. Independent of this assignment, we find that all cations investigated are strongly accumulated at amide surface while most anions are excluded. Cation and anion effects are independent and additive, allowing successful prediction of Hofmeister salt effects on micelle formation and other processes from structural information (ASA). ACS Publications Copyright © 2008 American Chemical Society Subjects what are subjects Article subjects are automatically applied from the ACS Subject Taxonomy and describe the scientific concepts and themes of the article. Amides Hydrocarbons Ions Partition coefficient Salts Read this article To access this article, please review the available access options below. Recommended Access through Your Institution You may have access to this article through your institution. Your institution does not have access to this content. Add or change your institution or let them know you’d like them to include access. Access Through Recommend Publication Institution Name Loading Institutional Login Options... Access Through Your Institution Add or Change Institution Explore subscriptions for institutions Get instant access Purchase Access Read this article for 48 hours. Check out below using your ACS ID or as a guest. Purchase AccessRestore my guest access Recommended Log in to Access You may have access to this article with your ACS ID if you have previously purchased it or have ACS member benefits. Log in below. Login with ACS ID Purchase access Purchase this article for 48 hours $48.00 Add to cart Purchase this article for 48 hours Checkout Supporting Information Click to copy section link Section link copied! Tables containing solubility data, the composite thermodynamic quantity b 1(K p, 3 − 1) for each salt, and surface area breakdowns for the model compounds. This material is available free of charge via the Internet at jp800816a-file.001.pdf (4.93 MB) Thermodynamic Origin of Hofmeister Ion Effects 66 views 0 shares 0 downloads Skip to figshare navigation ShareDownload figshare Terms & Conditions Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: Cited By Click to copy section link Section link copied! Citation Statements beta Smart citations byscite.aiinclude citation statements extracted from the full text of the citing article. The number of the statements may be higher than the number of citations provided by ACS Publications if one paper cites another multiple times or lower if scite has not yet processed some of the citing articles. Supporting Supporting 23 Mentioning Mentioning 405 Contrasting Contrasting 2 Explore this article's citation statements onscite.ai powered by This article is cited by 252 publications. Iris B. A. Smokers, Enrico Lavagna, Rafael V. M. Freire, Matteo Paloni, Ilja K. Voets, Alessandro Barducci, Paul B. 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Effect of Anions on the Changes in the Structure and Adsorption Mechanism of Zirconium Species at the Muscovite (001)–Water Interface. The Journal of Physical Chemistry C2019, 123 (27) , 16699-16710. Ellen E. Bruce, Pho T. Bui, Bradley A. Rogers, Paul S. Cremer, Nico F. A. van der Vegt. Nonadditive Ion Effects Drive Both Collapse and Swelling of Thermoresponsive Polymers in Water. Journal of the American Chemical Society2019, 141 (16) , 6609-6616. Hemant Charaya, Xinda Li, Nathan Jen, Hyun-Joong Chung. Specific Ion Effects in Polyampholyte Hydrogels Dialyzed in Aqueous Electrolytic Solutions. Langmuir2019, 35 (5) , 1526-1533. Matthew R. Sullivan, Wei Yao, Du Tang, Henry S Ashbaugh, and Bruce C. Gibb . The Thermodynamics of Anion Complexation to Nonpolar Pockets. The Journal of Physical Chemistry B2018, 122 (5) , 1702-1713. Emiliano Brini, Christopher J. Fennell, Marivi Fernandez-Serra, Barbara Hribar-Lee, Miha Lukšič, and Ken A. Dill . 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The Journal of Physical Chemistry Letters2017, 8 (7) , 1574-1577. Halil I. Okur, Jana Hladílková, Kelvin B. Rembert, Younhee Cho, Jan Heyda, Joachim Dzubiella, Paul S. Cremer, and Pavel Jungwirth . Beyond the Hofmeister Series: Ion-Specific Effects on Proteins and Their Biological Functions. The Journal of Physical Chemistry B2017, 121 (9) , 1997-2014. Nicolas O. Johnson, Taylor P. Light, Gina MacDonald, and Yanjie Zhang . Anion–Caffeine Interactions Studied by 13C and 1H NMR and ATR–FTIR Spectroscopy. The Journal of Physical Chemistry B2017, 121 (7) , 1649-1659. Jan Heyda, Halil I. Okur, Jana Hladílková, Kelvin B. Rembert, William Hunn, Tinglu Yang, Joachim Dzubiella, Pavel Jungwirth, and Paul S. Cremer . Guanidinium can both Cause and Prevent the Hydrophobic Collapse of Biomacromolecules. Journal of the American Chemical Society2017, 139 (2) , 863-870. Bradley A. Rogers, Tye S. Thompson, and Yanjie Zhang . Hofmeister Anion Effects on Thermodynamics of Caffeine Partitioning between Aqueous and Cyclohexane Phases. The Journal of Physical Chemistry B2016, 120 (49) , 12596-12603. Yueliang Liu, Huazhou Andy Li, and Ryosuke Okuno . Measurements and Modeling of Interfacial Tension for CO2/CH4/Brine Systems under Reservoir Conditions. Industrial & Engineering Chemistry Research2016, 55 (48) , 12358-12375. Jordan W. Bye, Nicola J. Baxter, Andrea M. Hounslow, Robert J. Falconer, and Mike P. Williamson . Molecular Mechanism for the Hofmeister Effect Derived from NMR and DSC Measurements on Barnase. ACS Omega2016, 1 (4) , 669-679. Rituparna Sengupta, Adrian Pantel, Xian Cheng, Irina Shkel, Ivan Peran, Natalie Stenzoski, Daniel P. Raleigh, and M. Thomas Record, Jr. . Positioning the Intracellular Salt Potassium Glutamate in the Hofmeister Series by Chemical Unfolding Studies of NTL9. Biochemistry2016, 55 (15) , 2251-2259. Joshua D. Willott, Timothy J. Murdoch, Grant B. Webber, and Erica J. Wanless . Nature of the Specific Anion Response of a Hydrophobic Weak Polyelectrolyte Brush Revealed by AFM Force Measurements. Macromolecules2016, 49 (6) , 2327-2338. Sinem Engin Özdil, Halide Akbaş, and Mesut Boz . Synthesis and Physicochemical Properties of Double-Chain Cationic Surfactants. Journal of Chemical & Engineering Data2016, 61 (1) , 142-150. Thomas M. Scherer . Role of Cosolute–Protein Interactions in the Dissociation of Monoclonal Antibody Clusters. The Journal of Physical Chemistry B2015, 119 (41) , 13027-13038. Liel Sapir and Daniel Harries . Macromolecular Stabilization by Excluded Cosolutes: Mean Field Theory of Crowded Solutions. Journal of Chemical Theory and Computation2015, 11 (7) , 3478-3490. D. B. Knowles, Irina A. Shkel, Noel M. Phan, Matt Sternke, Emily Lingeman, Xian Cheng, Lixue Cheng, Kevin O’Connor, and M. Thomas Record . Chemical Interactions of Polyethylene Glycols (PEGs) and Glycerol with Protein Functional Groups: Applications to Effects of PEG and Glycerol on Protein Processes. Biochemistry2015, 54 (22) , 3528-3542. David Dupont, Daphne Depuydt, and Koen Binnemans . Overview of the Effect of Salts on Biphasic Ionic Liquid/Water Solvent Extraction Systems: Anion Exchange, Mutual Solubility, and Thermomorphic Properties. The Journal of Physical Chemistry B2015, 119 (22) , 6747-6757. Corinne L. D. Gibb, Estelle E. Oertling, Santhosh Velaga, and Bruce C. Gibb . Thermodynamic Profiles of Salt Effects on a Host–Guest System: New Insight into the Hofmeister Effect. The Journal of Physical Chemistry B2015, 119 (17) , 5624-5638. Jerome M. Fox, Kyungtae Kang, Woody Sherman, Annie Héroux, G. Madhavi Sastry, Mostafa Baghbanzadeh, Matthew R. Lockett, and George M. Whitesides . Interactions between Hofmeister Anions and the Binding Pocket of a Protein. Journal of the American Chemical Society2015, 137 (11) , 3859-3866. Kelvin B. Rembert, Halil I. Okur, Christian Hilty, and Paul S. Cremer . An NH Moiety Is Not Required for Anion Binding to Amides in Aqueous Solution. Langmuir2015, 31 (11) , 3459-3464. D. Roberts, R. Keeling, M. Tracka, C. F. van der Walle, S. Uddin, J. Warwicker, and R. Curtis . Specific Ion and Buffer Effects on Protein–Protein Interactions of a Monoclonal Antibody. Molecular Pharmaceutics2015, 12 (1) , 179-193. Yael Katsir and Abraham Marmur . Rate of Bubble Coalescence Following Dynamic Approach: Collectivity-Induced Specificity of Ionic Effect. Langmuir2014, 30 (46) , 13823-13830. Jan Heyda and Joachim Dzubiella . Thermodynamic Description of Hofmeister Effects on the LCST of Thermosensitive Polymers. The Journal of Physical Chemistry B2014, 118 (37) , 10979-10988. Dor Ben-Amotz and Blake M. Rankin , B. Widom . Molecular Aggregation Equilibria. Comparison of Finite Lattice and Weighted Random Mixing Predictions. The Journal of Physical Chemistry B2014, 118 (28) , 7878-7885. Eva Pluhařová, Marcel D. Baer, Christopher J. Mundy, Burkhard Schmidt, and Pavel Jungwirth . Aqueous Cation-Amide Binding: Free Energies and IR Spectral Signatures by Ab Initio Molecular Dynamics. The Journal of Physical Chemistry Letters2014, 5 (13) , 2235-2240. Josephina Werner, Erik Wernersson, Victor Ekholm, Niklas Ottosson, Gunnar Öhrwall, Jan Heyda, Ingmar Persson, Johan Söderström, Pavel Jungwirth, and Olle Björneholm . Surface Behavior of Hydrated Guanidinium and Ammonium Ions: A Comparative Study by Photoelectron Spectroscopy and Molecular Dynamics. The Journal of Physical Chemistry B2014, 118 (25) , 7119-7127. Pritam Ganguly, Timir Hajari, and Nico F. A. van der Vegt . Molecular Simulation Study on Hofmeister Cations and the Aqueous Solubility of Benzene. The Journal of Physical Chemistry B2014, 118 (20) , 5331-5339. Jan Heyda, Sebastian Soll, Jiayin Yuan, and Joachim Dzubiella . Thermodynamic Description of the LCST of Charged Thermoresponsive Copolymers. Macromolecules2014, 47 (6) , 2096-2102. Chiran Ghimire, Deepak Koirala, Malcom B. Mathis, Edgar E. Kooijman, and Hanbin Mao . Controlled Particle Collision Leads to Direct Observation of Docking and Fusion of Lipid Droplets in an Optical Trap. Langmuir2014, 30 (5) , 1370-1375. Pavel Jungwirth . Hofmeister Series of Ions: A Simple Theory of a Not So Simple Reality. The Journal of Physical Chemistry Letters2013, 4 (24) , 4258-4259. Blake M. Rankin and Dor Ben-Amotz . Analysis of Molecular Aggregation Equilibria Using Random Mixing Statistics. The Journal of Physical Chemistry B2013, 117 (49) , 15667-15674. Jana Hladílková, Jan Heyda, Kelvin B. Rembert, Halil I. Okur, Yadagiri Kurra, Wenshe R. Liu, Christian Hilty, Paul S. Cremer, and Pavel Jungwirth . Effects of End-Group Termination on Salting-Out Constants for Triglycine. The Journal of Physical Chemistry Letters2013, 4 (23) , 4069-4073. Jacob C. Lutter, Tsung-yu Wu, and Yanjie Zhang . Hydration of Cations: A Key to Understanding of Specific Cation Effects on Aggregation Behaviors of PEO-PPO-PEO Triblock Copolymers. The Journal of Physical Chemistry B2013, 117 (35) , 10132-10141. Roger C. Diehl, Emily J. Guinn, Michael W. Capp, Oleg V. Tsodikov, and M. Thomas Record, Jr. . Quantifying Additive Interactions of the Osmolyte Proline with Individual Functional Groups of Proteins: Comparisons with Urea and Glycine Betaine, Interpretation of m-Values. Biochemistry2013, 52 (35) , 5997-6010. Jana Paterová, Kelvin B. Rembert, Jan Heyda, Yadagiri Kurra, Halil I. Okur, Wenshe R. Liu, Christian Hilty, Paul S. Cremer, and Pavel Jungwirth . Reversal of the Hofmeister Series: Specific Ion Effects on Peptides. The Journal of Physical Chemistry B2013, 117 (27) , 8150-8158. Blake M. Rankin and Dor Ben-Amotz . Expulsion of Ions from Hydrophobic Hydration Shells. Journal of the American Chemical Society2013, 135 (24) , 8818-8821. Emily J. Guinn, Jeffrey J. Schwinefus, Hyo Keun Cha, Joseph L. McDevitt, Wolf E. Merker, Ryan Ritzer, Gregory W. Muth, Samuel W. Engelsgjerd, Kathryn E. Mangold, Perry J. Thompson, Michael J. Kerins, and M. Thomas Record, Jr. . Quantifying Functional Group Interactions That Determine Urea Effects on Nucleic Acid Helix Formation. Journal of the American Chemical Society2013, 135 (15) , 5828-5838. Thomas M. Scherer . Cosolute Effects on the Chemical Potential and Interactions of an IgG1 Monoclonal Antibody at High Concentrations. The Journal of Physical Chemistry B2013, 117 (8) , 2254-2266. Nadine Schwierz, Dominik Horinek, and Roland R. Netz . Anionic and Cationic Hofmeister Effects on Hydrophobic and Hydrophilic Surfaces. Langmuir2013, 29 (8) , 2602-2614. Jan Heyda, Anja Muzdalo, and Joachim Dzubiella . Rationalizing Polymer Swelling and Collapse under Attractive Cosolvent Conditions. Macromolecules2013, 46 (3) , 1231-1238. Robert C. Tyler, Jamie C. Wieting, Francis C. Peterson, and Brian F. Volkman . Electrostatic Optimization of the Conformational Energy Landscape in a Metamorphic Protein. Biochemistry2012, 51 (45) , 9067-9075. Jan Zienau and Qiang Cui . Implementation of the Solvent Macromolecule Boundary Potential and Application to Model and Realistic Enzyme Systems. The Journal of Physical Chemistry B2012, 116 (41) , 12522-12534. Rebecca J. Carlton, C. Derek Ma, Jugal K. Gupta, and Nicholas L. Abbott . Influence of Specific Anions on the Orientational Ordering of Thermotropic Liquid Crystals at Aqueous Interfaces. Langmuir2012, 28 (35) , 12796-12805. Yi Qin Gao . Simple Theory for Salt Effects on the Solubility of Amide. The Journal of Physical Chemistry B2012, 116 (33) , 9934-9943. Shuching Ou and Sandeep Patel , Brad A. Bauer . Free Energetics of Carbon Nanotube Association in Pure and Aqueous Ionic Solutions. The Journal of Physical Chemistry B2012, 116 (28) , 8154-8168. Sarah C. Flores, Jaibir Kherb, and Paul S. Cremer . Direct and Reverse Hofmeister Effects on Interfacial Water Structure. The Journal of Physical Chemistry C2012, 116 (27) , 14408-14413. Shahla Shahriari, Catarina M. S. S. Neves, Mara G. Freire, and João A. P. Coutinho . Role of the Hofmeister Series in the Formation of Ionic-Liquid-Based Aqueous Biphasic Systems. The Journal of Physical Chemistry B2012, 116 (24) , 7252-7258. Kelvin B. Rembert, Jana Paterová, Jan Heyda, Christian Hilty, Pavel Jungwirth, and Paul S. Cremer . Molecular Mechanisms of Ion-Specific Effects on Proteins. Journal of the American Chemical Society2012, 134 (24) , 10039-10046. Pierandrea Lo Nostro and Barry W. Ninham . Hofmeister Phenomena: An Update on Ion Specificity in Biology. Chemical Reviews2012, 112 (4) , 2286-2322. Elena A. Algaer and Nico F. A. van der Vegt . Hofmeister Ion Interactions with Model Amide Compounds. The Journal of Physical Chemistry B2011, 115 (46) , 13781-13787. Filippos Ioannou, Georgios Archontis, and Epameinondas Leontidis . Specific Interactions of Sodium Salts with Alanine Dipeptide and Tetrapeptide in Water: Insights from Molecular Dynamics. The Journal of Physical Chemistry B2011, 115 (45) , 13389-13400. Erik Wernersson, Jan Heyda, Mario Vazdar, Mikael Lund, Philip E. Mason, and Pavel Jungwirth . Orientational Dependence of the Affinity of Guanidinium Ions to the Water Surface. The Journal of Physical Chemistry B2011, 115 (43) , 12521-12526. Qingbo Yang and Jiang Zhao . Hofmeister Effect on the Interfacial Dynamics of Single Polymer Molecules. Langmuir2011, 27 (19) , 11757-11760. Chunhua Wang, Ying Ge, John Mortensen, and Peter Westh . Interaction Free Energies of Eight Sodium Salts and a Phosphatidylcholine Membrane. The Journal of Physical Chemistry B2011, 115 (33) , 9955-9961. Yi He, Qing Shao, Shengfu Chen, and Shaoyi Jiang . Water Mobility: A Bridge between the Hofmeister Series of Ions and the Friction of Zwitterionic Surfaces in Aqueous Environments. The Journal of Physical Chemistry C2011, 115 (31) , 15525-15531. Branden A. Deyerle and Yanjie Zhang . Effects of Hofmeister Anions on the Aggregation Behavior of PEO–PPO–PEO Triblock Copolymers. Langmuir2011, 27 (15) , 9203-9210. Mingming Ma and Dennis Bong . Determinants of Cyanuric Acid and Melamine Assembly in Water. Langmuir2011, 27 (14) , 8841-8853. Daryl K. Eggers . A Bulk Water-Dependent Desolvation Energy Model for Analyzing the Effects of Secondary Solutes on Biological Equilibria. Biochemistry2011, 50 (12) , 2004-2012. Barbara Hribar-Lee, Ken A. Dill, and Vojko Vlachy . Receptacle Model of Salting-In by Tetramethylammonium Ions. The Journal of Physical Chemistry B2010, 114 (46) , 15085-15091. Philip E. Mason, Jan Heyda, Henry E. Fischer, and Pavel Jungwirth . Specific Interactions of Ammonium Functionalities in Amino Acids with Aqueous Fluoride and Iodide. The Journal of Physical Chemistry B2010, 114 (43) , 13853-13860. Xin Chen, Sarah C. Flores, Soon-Mi Lim, Yanjie Zhang, Tinglu Yang, Jaibir Kherb, and Paul S. Cremer. Specific Anion Effects on Water Structure Adjacent to Protein Monolayers. Langmuir2010, 26 (21) , 16447-16454. Jan Heyda, Mikael Lund, Milan Ončák, Petr Slavíček and Pavel Jungwirth . Reversal of Hofmeister Ordering for Pairing of NH4+ vs Alkylated Ammonium Cations with Halide Anions in Water. The Journal of Physical Chemistry B2010, 114 (33) , 10843-10852. Anne K. Hüsecken, Florian Evers, Claus Czeslik and Metin Tolan . Effect of Urea and Glycerol on the Adsorption of Ribonuclease A at the Air−Water Interface. Langmuir2010, 26 (16) , 13429-13435. Wayne S. Kontur, Michael W. Capp, Theodore J. Gries, Ruth M. Saecker and M. Thomas Record, Jr. . Probing DNA Binding, DNA Opening, and Assembly of a Downstream Clamp/Jaw in Escherichia coli RNA Polymerase−λPR Promoter Complexes Using Salt and the Physiological Anion Glutamate. Biochemistry2010, 49 (20) , 4361-4373. Christine L. Henry and Vincent S. J. Craig. The Link between Ion Specific Bubble Coalescence and Hofmeister Effects Is the Partitioning of Ions within the Interface. Langmuir2010, 26 (9) , 6478-6483. Xiaohua Wu, Zhigang Lei, Qunsheng Li, Jiqin Zhu and Biaohua Chen. Liquid−Liquid Extraction of Low-Concentration Aniline from Aqueous Solutions with Salts. Industrial & Engineering Chemistry Research2010, 49 (6) , 2581-2588. Jeff D. Ballin, James P. Prevas, Christina R. Ross, Eric A. Toth, Gerald M. Wilson and M. Thomas Record, Jr. . Contributions of the Histidine Side Chain and the N-Terminal α-Amino Group to the Binding Thermodynamics of Oligopeptides to Nucleic Acids as a Function of pH. Biochemistry2010, 49 (9) , 2018-2030. Valerie A. Braz, Leslie A. Holladay and Mary D. Barkley . Efavirenz Binding to HIV-1 Reverse Transcriptase Monomers and Dimers. Biochemistry2010, 49 (3) , 601-610. Jan Heyda, Jordan C. Vincent, Douglas J. Tobias, Joachim Dzubiella and Pavel Jungwirth . Ion Specificity at the Peptide Bond: Molecular Dynamics Simulations of N-Methylacetamide in Aqueous Salt Solutions. The Journal of Physical Chemistry B2010, 114 (2) , 1213-1220. Michael W. Capp, Laurel M. Pegram, Ruth M. Saecker, Megan Kratz, Demian Riccardi, Timothy Wendorff, Jonathan G. Cannon and M. Thomas Record, Jr. . Interactions of the Osmolyte Glycine Betaine with Molecular Surfaces in Water: Thermodynamics, Structural Interpretation, and Prediction of m-Values. Biochemistry2009, 48 (43) , 10372-10379. Christine L. Henry and Vincent S. J. Craig. Inhibition of Bubble Coalescence by Osmolytes: Sucrose, Other Sugars, and Urea. Langmuir2009, 25 (19) , 11406-11412. Rajesh Kumar and A. Grant Mauk. Atypical Effects of Salts on the Stability and Iron Release Kinetics of Human Transferrin. The Journal of Physical Chemistry B2009, 113 (36) , 12400-12409. Gordon C. Kresheck. Isothermal Titration Calorimetry Studies of Neutral Salt Effects on the Thermodynamics of Micelle Formation. The Journal of Physical Chemistry B2009, 113 (19) , 6732-6735. Laurel M. Pegram and M. Thomas Record, Jr. . Using Surface Tension Data to Predict Differences in Surface and Bulk Concentrations of Nonelectrolytes in Water. The Journal of Physical Chemistry C2009, 113 (6) , 2171-2174. E. Leontidis and A. Aroti, L. Belloni. Liquid Expanded Monolayers of Lipids As Model Systems to Understand the Anionic Hofmeister Series: 1. A Tale of Models. The Journal of Physical Chemistry B2009, 113 (5) , 1447-1459. Jonathan Patete, John M. Petrofsky, Jeffery Stepan, Abdul Waheed and Joseph M. Serafin. Hofmeister Effect on the Interfacial Free Energy of Aliphatic and Aromatic Surfaces Studied by Chemical Force Microscopy. The Journal of Physical Chemistry B2009, 113 (2) , 583-588. Kalyani Agarwal, Nilanjan Dutta, Neelkanth Nirmalkar. Mechanistic insights into surface tension reduction: The potential contribution of nanobubbles. Journal of Molecular Liquids2025, 120, 128581. Gabriel Ortega-Quintanilla, Oscar Millet. On the Molecular Basis of the Hypersaline Adaptation of Halophilic Proteins. Journal of Molecular Biology2025, III, 169439. Bryan A. Castellanos Paez, Tao Wu, Qixin Zhong. Effects of phosphate, sodium, and potassium ions on the transacylation reaction between propylene glycol alginate and sodium caseinate. Food Chemistry2025, 481, 144117. Load more citations Get e-Alerts Get e-Alerts The Journal of Physical Chemistry B Cite this: J. Phys. Chem. B 2008, 112, 31, 9428–9436 Click to copy citation Citation copied! Published July 16, 2008 Publication History Received 28 January 2008 Accepted 11 April 2008 Published online 16 July 2008 Published in issue 1 August 2008 Copyright © 2008 American Chemical Society Request reuse permissions Article Views 3686 Altmetric - Citations 252 Learn about these metrics close Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days. Citations are the number of other articles citing this article, calculated by Crossref and updated daily.Find more information about Crossref citation counts. The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information onthe Altmetric Attention Score and how the score is calculated. Recommended Articles ### Hofmeister Series: Insights of Ion Specificity from Amphiphilic Assembly and Interface Property March 20, 2020 ACS Omega Beibei Kang , Huicheng Tang , Zengdian Zhao ,and Shasha Song ### Beyond the Hofmeister Series: Ion-Specific Effects on Proteins and Their Biological Functions February 8, 2017 The Journal of Physical Chemistry B Halil I. Okur , Jana Hladílková , Kelvin B. Rembert , Younhee Cho , Jan Heyda , Joachim Dzubiella , Paul S. Cremer ,and Pavel Jungwirth ### Hofmeister Phenomena: An Update on Ion Specificity in Biology January 17, 2012 Chemical Reviews Pierandrea Lo Nostro and Barry W. Ninham ### Specific Ion Effects on the Water Solubility of Macromolecules: PNIPAM and the Hofmeister Series September 22, 2005 Journal of the American Chemical Society Yanjie Zhang , Steven Furyk , David E. Bergbreiter ,and Paul S. 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2539	https://www.lucidchart.com/pages/flowchart-symbols-meaning-explained	PINGDOM_CANARY_STRING Flowchart Symbols and Notation What do you want to do with flowcharts? I want to learn more about flowcharts. ### I want to create my own flowchart on Lucidchart. ### I want to create a flowchart from a ready-made template. Contents Common flowchart symbols Additional flowchart symbols Standard vs. non-standard flowchart symbols Don’t be intimidated by the wide array of flowchart shapes. Each symbol has a specific meaning and context where its use is appropriate. If you get confused while drawing your flow chart, remember that most charts can be drawn with just a few common symbols, which are listed below. 4 minute read Want to create a flowchart of your own? Try Lucidchart. It's fast, easy, and totally free. Create a flowchart Common flowchart symbols These flowchart shapes and symbols are some of the most common types you'll find in most flowchart diagrams. \| Flowchart Symbol \| Name \| Description \| --- \| \| Process symbol \| Also known as an “Action Symbol,” this shape represents a process, action, or function. It’s the most widely-used symbol in flowcharting. \| \| \| Start/End symbol \| Also known as the “Terminator Symbol,” this symbol represents the start points, end points, and potential outcomes of a path. Often contains “Start” or “End” within the shape. \| \| \| Document symbol \| Represents the input or output of a document, specifically. Examples of and input are receiving a report, email, or order. Examples of an output using a document symbol include generating a presentation, memo, or letter. \| \| \| Decision symbol \| Indicates a question to be answered — usually yes/no or true/false. The flowchart path may then split off into different branches depending on the answer or consequences thereafter. \| \| \| Connector symbol \| Usually used within more complex charts, this symbol connects separate elements across one page. \| \| \| Off-Page Connector/Link symbol \| Frequently used within complex charts, this symbol connects separate elements across multiple pages with the page number usually placed on or within the shape for easy reference. \| \| \| Input/Output symbol \| Also referred to as the “Data Symbol,” this shape represents data that is available for input or output as well as representing resources used or generated. While the paper tape symbol also represents input/output, it is outdated and no longer in common use for flowchart diagramming. \| \| \| Comment/Note symbol \| Placed along with context, this symbol adds needed explanation or comments within the specified range. It may be connected by a dashed line to the relevant section of the flowchart as well. \| Want to create a flowchart of your own? Try Lucidchart. It's fast, easy, and totally free. Create a flowchart Additional flowchart symbols Many of these additional flowchart symbols are best utilized when mapping out a process flow diagram for apps, user flow, data processing, etc. \| Flowchart Symbol \| Name \| Description \| --- \| \| Database symbol \| Represents data housed on a storage service that will likely allow for searching and filtering by users. \| \| \| Paper tape symbol \| An outdated symbol rarely ever used in modern practices or process flows, but this shape could be used if you’re mapping out processes or input methods on much older computers and CNC machines. \| \| \| Summing junction symbol \| Sums the input of several converging paths. \| \| \| Predefined process symbol \| Indicates a complicated process or operation that is well-known or defined elsewhere. \| \| \| Internal storage symbol \| Commonly used to map out software designs, this shape indicates data that is stored within internal memory. \| \| \| Manual input symbol \| Represents the manual input of data into a field or step in a process, usually through a keyboard or device. Example scenario includes the step in a login process where a user is prompted to enter data manually. \| \| \| Manual operation symbol \| Indicates a step that must be done manually, not automatically. \| \| \| Merge symbol \| Combines multiple paths to become one. \| \| \| Multiple documents symbol \| Represents multiple documents or reports. \| \| \| Preparation symbol \| Differentiates between steps that prepare for work and steps that actually do work. It helps introduce the setup to another step within the same process. \| \| \| Stored data symbol \| Also known as “Data Storage” symbol, this shape represents where data gets stored within a process. \| \| \| Delay symbol \| Represents a segment of delay in a process. It can be helpful to indicate the exact length of delay within the shape. \| \| \| Or symbol \| Just as described, this shape indicates that the process flow continues two paths or more. \| \| \| Display symbol \| This shape is useful to indicate where information will get displayed within a process flow. \| \| \| Hard disk symbol \| Indicates where data is stored within a hard drive, also known as direct access storage. \| Standard vs. non-standard flowchart symbols While various standards for symbol usage and flowchart creation have been established, it’s okay to ignore the rules. Use the symbols in a way that makes sense to your audience. But if you use symbols in a non-standard fashion, be sure to do it consistently so your readers understand your meaning for that symbol each time they see it. If you're new to diagramming, you can refer to this guide on how to make a flowchart in Lucidchart with this handy symbols guide available as you construct a stunning flowchart! Additional Resources Flowchart Template for Word How to Make a Flowchart in Excel Examples and models of flowcharts What is a flowchart and how to create one How to Make a Flowchart How to Make a Flowchart in PowerPoint How to Make a Flowchart in Word What is a Flowchart If we don’t have the shape you’re looking for, Lucidchart allows you to also upload any and all shapes you want to use in addition to our own extensive shape library. Try mapping out your process flow with Lucidchart today! Want to create a flowchart of your own? Try Lucidchart. It's fast, easy, and totally free. Create a flowchart
2540	https://www.youtube.com/watch?v=FDvnEVOqdAw	Vector Geometry Proofs (1 of 3: An unexpected quadrilateral property) Eddie Woo 1940000 subscribers 254 likes Description 11344 views Posted: 17 Aug 2023 More resources available at www.misterwootube.com 14 comments Transcript: underneath this heading which is an unexpected quadrilateral property what I'd like you to do this is actually harder than it sounds I want you to draw for me a quadrilateral that is not a special quadrilateral so I don't want a parallelogram I don't want to square I don't want to kite I want something that looks really random and arbitrary which is you can see the one that I've drawn here but I want you to draw something that looks a bit unusual and the reason why I'm asking you to draw something unusual is because you probably have sitting in the back of your mind this knowledge that special quadrilaterals like let's use a rhombus for example right special quadrilaterals like a rhombus have special properties right not only are all the sides equal and rhombus like that's what makes a rhombus a rhombus but if you were to draw a rhombus the diagonals are always perpendicular with each other did you know that right um you you can't not draw a rhombus or you can't draw a rhombus where it's per which wears diagonals are at different angles you won't end up having a rhombus and every every shape has its own different things right like a parallelogram the diagonals will always bisect each other and on and on and on right so special properties okay now if you draw a random quadrilateral you kind of would expect that it doesn't have any special properties except for the fact that the angles add up to 360 degrees because you know there's a couple of triangles that you can fit in there however um we're going to explore an unexpected quadrilateral property today and we're going to prove that it's true using vectors and that's really what this idea is about in the syllabus it is can you prove things like we used to use um like prove that things are congruent two triangles are congruent right so that's a geometry proof a deductive geometry proof we're going to do two deductive geometry proofs today and they're both going to be using Vector thinking okay have you drawn a quadrilateral for me yet yeah amazing okay now what I'd like you to do is and if you've got another color here this will help but you don't have to it's just a bonus I want you to Mark in and if you've got a do you have a rule of there by any chance yeah excellent okay you can do this even more accurately than I can can you mark in the midpoint of each of the four sides okay so measure it out make it nice and accurate I'm going to eyeball it here because I don't have a ruler that I can put over the top of my iPad that I'm drawing on so I'm going to go this one's pretty easy to spot that one's going to be about there let's see about here and then that looks about right okay fantastic now those four midpoints right we can think of them as the four vertices of a new quadrilateral and in fact what I'd love you to do is to join up the four midpoints that you just created so I don't know if this is going to work let's oh I missed there we go let's see if this will snap for me nope it's not I'm just gonna have to do it like that do one more there or like so now I wonder if if you've drawn your new quadrilateral here in the middle have you noticed anything unusual about yours and mine what what have you noticed about it parallelogram it is indeed a parallelogram and it doesn't just look like a parallelogram I really like sometimes I get the chance to do this with a lot of different students at the same time like there's just you and me doing this at the moment right but no matter what quadrilateral you try and you can even draw like really weirdo ones right like say for example most people don't think of this when they think of a quadrilateral but it is it's still four-sided right and you can really quickly eyeball where the midpoints are going to be somewhere like there there uh where's this one going to be there and then maybe here even without me joining them up you can probably see oh there's a there's a parallelogram hiding there right and you've got yours you can draw as many as you like and it's just kind of like what is going on here okay there's a bit of a mystery so how would you go about proving that if you draw any quadrilateral at all you will always when you join up the midpoints or the sides you always end up with a parallelogram it's kind of wild right do you have any thoughts what would you do um like um Vector of all four vectors no matter what they are they're midpoints uh I'd probably say that there are at least like because I can see that two of the sides uh like if you split in two triangles they're like um midpoints become too parallel sides through that the I don't know maybe yeah yeah okay look Sean you're doing you're doing so well can I say you're doing like a hundred times better than I did when I first encountered this problem because um Vector thinking as you you've probably like over the last few weeks you've probably started to experience it's a bit different to like geometry thinking or other kinds of thinking and you've got to be quite comfortable it's like speaking a new language you've already got the skeleton of the proof um and so what I'm going to do is I'm going to help you to see what's going on here um so that you can you can prove and it's one of the things that's delightful about this is one of the reasons why I've given this example is that it's very little uh working there's not much to write at all in fact I'm looking at my working here there's like three lines that's it that's the whole proof there's a bit of drawing but the drawing is the fun part usually what is tiresome and frustrating about geometry proofs is you like writing for pages and pages and it's like oh man when am I done and the reasoning is very long so what I want to do with you is we're going to think of this in Vector terms and you will see how powerful Vector logic is in in constructing this proof it's very efficient okay so here's what we're going to do I want you to think about these four sides one two three four the four sides of the original quadrilateral let's just um let's get this one out I'll just Chuck them over here okay we don't need it I want you to think of the four sides of this original quadrilateral as four vectors okay now remember vectors are Direction and magnitude Direction and magnitude now with the magnitude good morning Mrs Oz by the way nice to you to join us online and with magnitude that's just the length of each side right so you can't change that but with the vectors for these sides you can actually choose which direction you want to go right so for example if you have a look at this this top one up here that's quite long right you can imagine it as starting from the left and going to the right that's one vector or the same side you can think of it as going the other way right so it's two different vectors that we can use so what I'm going to do is I'm going to choose directions for the vectors that are very deliberate which you might think why am I why am I doing it that way and you're going to see as we go through this why it is that this makes so much sense okay so let's call the top Vector a right and let's say let's go from left to right just because you know that's a bit easier to read okay so you can see I've just put it's quite small there but I've put that Arrow there at that um top right corner so that Vector a starts over here if you want you could give this coordinate a name you can call that the origin if you like but it doesn't matter start from o and then head over here let's call that Vector a okay now going from that top right corner if I put another Vector here let's call this one vector B so this is uh going here and progressing along so you said vector addition before so if I go a plus b I end up on this this end here right opposite to O now you might think a natural way to do this normally when we name things we'd go like oh A B C D and you go like clockwise or anti-clockwise or whatever I'm going to do something a little different I'm going to go back over to the origin here and I'm going to make this Vector down the bottom I'm also going to go from left to right so I'm going to call this one I've Got A and B let's call this bottom one C and then lastly to round things off I'll have Vector D and it's also sort of going from left to right as it were so I went from o over here that makes me C and then up here I'm going to have this one as d okay are you following so far so far so good okay now think about this right vectors are I keep saying it because so foundation and so important Vector is a direction and magnitude okay so when we say the midpoint of a side what does that mean in terms of a vector what are you thinking very good so what a midpoint does is it breaks let's think about say Vector a right um what a midpoint does is it breaks up a into two equal vectors right so there's this one over here and then there's this other one on the other side right now they both share the same direction as the original Vector but what's different very good so the magnitudes have changed and all we need to do is multiply by the appropriate scalar because it's the midpoint what's the scalar I should multiply by uh-huh very good so both of these vectors up the top are half a you following yeah okay fantastic I can kind of see what you're gonna do yeah yeah yeah very good okay so the logic's starting to to sort of form in your mind right so um actually what I'm going to do is I'm just going to make these orange because I realized I am um I'm starting to confuse some of the things here so make that one orange and make this one orange as well okay so you can see if we look up the top here right see how I've got half a going in this direction can you see I have another Vector beside it over here yeah you see this is the addition you're talking about before right so this Vector in here what's that yeah very good so this one here half b okay now what this tells me is if you just look up in this little this triangle up the top here right the the side of this parallelogram that you made is the same as going half a and then half B if you added those vectors together right so therefore I can just call that top side of the parallelogram half a plus half b does this make sense yeah now what's brilliant is I can use this same logic on the other side of my quadrilateral right you're spotting it so I want you to tell me where where should I be looking what what vectors should I be highlighting um you should be highlighting the second half day very good fantastic so the second half of C is down here just like I had the second half of um a and then the first half of D that's actually a very clever way to say it by the way because there's a beginning and there's an end obviously that's oops not a highlighter that's half d right there so therefore what I've got now is the half C plus half D Vector which is the other side of what I know or what I can see is a parallelogram so you've done all of the drawing this is the fun part I love this there's so much um logic that's just going to happen in a very Convent condensed place right I can say hold on a second starting from the origin when you do a plus b where do you end up over at this this spot here right um I guess I could call that P for point right so a plus b gets you to P but that's not the only way to this point right what's the other vectors I can add C plus d also get you from the origin to P so I can say hey but a plus b and c plus d they get you to the same spot so they are equal vectors so therefore this is mind-blowing how much logic is happening here when you just treat this with like arithmetic I can just multiply both sides by half and this gives me sorry I need a bracket there this gives me the half C plus half D and the half a plus half B that you could see up above and because they're vectors right what this tells you is number one sorry I keep saying it but Direction the directions equal which means in like deductive geometry language the directions being equal means they are parallel right but that's not the only thing what's the other thing that's equal between these two vectors not just Direction the magnitude so therefore they're not only parallel they are equal in length so I can say therefore uh what do we say number one uh oh sorry I I should put some names on this right um so let's call this I don't know um what letters haven't I used yet I've got o and I've got P let's call this capital a Capital Bay Capital C capital D that said it corresponds to all the little letters right so therefore a b is parallel to CD because the directions are equal and um I guess we would say you know the magnitude of c a b and c d is also equal and that's it that's that's all there is to it um a b in this case I've got the letter of the order of letters is DC is a parallelogram because opposite sides are both parallel and equal stop I just love it it's like one of the shortest geometry proofs in the world but it's this very unexpected result and vectors are really doing all the work does that make sense
2541	https://artofproblemsolving.com/wiki/index.php/Radical_axis?srsltid=AfmBOorgBohjK0dV5qFJ96na5e2OCZ4MOWJYZRnB8270_pjHYaaZ-AkZ	Art of Problem Solving Radical axis - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Radical axis Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Radical axis Contents [hide] 1 Introduction 2 Definitions 3 Results 4 Proofs 5 Exercises 6 Problems 6.1 Simple 6.2 Intermediate 6.3 Olympiad 7 See Also Introduction The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairly readily. Problems involving it can be found on many major math olympiad competitions, including the prestigious USAMO. Therefore, any aspiring math olympian should peruse this material carefully, as it may contain the keys to one's future success. Not all theorems will be fully proven in this text. The objective of this document is to introduce you to some key concepts, and then give you a chance to derive some of the beautiful results on your own. In that way, you will understand and retain the information in here much more solidly. Finally, your newfound knowledge will be tested on a few challenging problems that are exemplary examples on how radical axis theory can be used and why it pertains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. But, anyway, good luck. i Definitions The power of point with respect to circle (with radius and center ), which shall thereafter be dubbed , is defined to equal . Note that the power of a point is negative if the point is inside the circle. The radical axis of two non-concentric circles is defined as the locus of the points such that the power of with respect to and are equal. In other words, if are the center and radius of , then a point is on the radical axis if and only if (i.e., the radical axis is the line that one gets when you subtract the equations of two circles). Results Theorem 1: (Power of a Point) If a line drawn through point P intersects circle at points A and B, then . Theorem 2: (Radical Axis Theorem) a. The radical axis is a line perpendicular to the line connecting the circles' centers (line ). b. If the two circles intersect at two common points, their radical axis is the line through these two points. c. If they intersect at one point, their radical axis is the common internal tangent. d. If the circles do not intersect, and if one does not fully contain the other, their radical axis is the perpendicular to through point A, the unique point on such that . Theorem 3: (Radical Axis Concurrence Theorem) The three pairwise radical axes of three circles concur at a point, called the radical center. Theorem 4: (Radical Centre of Intersecting Circles) (EGMO Theorem 2.9) Let and be two circles with centers and . Select two points A and B on and C and D on . Then the following are equivalent lie on a circle with center not on line . Lines and intersect on the radical axis of and . Theorem 5: (EGMO Lemma 2.11) Let ABC be a triangle and consider a point in its interior. Suppose that is tangent to and , ray bisects Proofs Theorem 1 is trivial Power of a Point, and thus is left to the reader as an exercise. (Hint: Draw a line through P and the center.) Theorem 2 shall be proved here. Assume the circles are and with centers and and radii and , respectively. (It may be a good idea for you to draw some circles here.) First, we tackle part (b). Suppose the circles intersect at points and and point P lies on . Then by Theorem 1 the powers of P with respect to both circles are equal to , and hence by transitive . Thus, if point P lies on , then the powers of P with respect to both circles are equal. Now, we prove the inverse of the statement just proved; because the inverse is equivalent to the converse, the if and only if would then be proven. Suppose that P does not lie on . In particular, line does not intersect X. Then intersects circles and a second time at distinct points and , respectively. (If is tangent to , for example, we adopt the convention that ; similar conventions hold for . Power of a Point still holds in this case. Also, notice that and cannot both equal , as cannot be tangent to both circles.) Because is not equal to , so does not equal , and thus by Theorem 1 is not congruent to , as desired. This completes part (b). For the remaining parts, we employ a lemma: Lemma 1: Let be a point in the plane, and let be the foot of the perpendicular from to . Then . The proof of the lemma is an easy application of the Pythagorean Theorem and will again be left to the reader as an exercise. Lemma 2: There is an unique point P on line such that . Proof: First show that P lies between and via proof by contradiction, by using a bit of inequality theory and the fact that . Then, use the fact that (a constant) to prove the lemma. Lemma 1 shows that every point on the plane can be equivalently mapped to a line on . Lemma 2 shows that only one point in this mapping satisfies the given condition. Combining these two lemmas shows that the radical axis is a line perpendicular to , completing part (a). Parts (c) and (d) will be left to the reader as an exercise. (Also, try proving part (b) solely using the lemmas.) Now, try to prove Theorem 3 on your own! (Hint: Let P be the intersection of two of the radical axes.)The proof of this theorem along with the proof of Theorem 4are given in EGMO as Example 2.7 and Theorem 2.9. Theorem 5 Let intersect at at and let , . Clearly, is a radical axis of , . We see thatas desired. Exercises If you haven't already done so, prove the theorems and lemmas outlined in the proofs section. Note: No solutions will be provided to the following problems(laziness). If you are stuck, ask on the forum. Problem 1. Two circles and intersect at and . Point is located on such that and ~~15~~. If the radius of is , find the radius of . Note: An error in this problem previously rendered it unsolvable. Problem 2. Solve 2009 USAMO Problem 1. If you already know how to solve it. Problem 3. Two circles P and Q with radii 1 and 2, respectively, intersect at X and Y. Circle P is to the left of circle Q. Prove that point A is to the left of if and only if . Problem 4. Solve 2012 USAJMO Problem 1. Problem 5. Does Theorem 2 apply to circles in which one is contained inside the other? How about internally tangent circles? Concentric circles? Problem 6. Construct the radical axis of two circles. What happens if one circle encloses the other? Problem 7. Solve 1995 IMO Problem 1 in two different ways. Compare your solutions with the solutions provided. Problems Simple Let triangle points and be given. Denote Prove that Proof WLOG, the order of the points is as shown on diagram. Intermediate 2021 AIME Problem 13. Circles and with radii and , respectively, intersect at distinct points and . A third circle is externally tangent to both and . Suppose line intersects at two points and such that the measure of minor arc is . Find the distance between the centers of and . Olympiad 2014 USAMO Problem 5. Let be a triangle with orthocenter and let be the second intersection of the circumcircle of triangle with the internal bisector of the angle . Let be the circumcenter of triangle and the orthocenter of triangle . Prove that the length of segment is equal to the circumradius of triangle . 2017 USAMO Problem 3. Let be a scalene triangle with circumcircle and incenter Ray meets at and again at the circle with diameter cuts again at Lines and meet at and is the midpoint of The circumcircles of and intersect at points and Prove that passes through the midpoint of either or 2012 IMO Problem 5. Let be a triangle with , and let be the foot of the altitude from . Let be a point in the interior of the segment . Let be the point on the segment such that . Similarly, let be the point on the segment such that . Let . Prove that . See Also This article is a stub. Help us out by expanding it. 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2543	https://phys.libretexts.org/Bookshelves/University_Physics/Calculus-Based_Physics_(Schnick)/Volume_A%3A_Kinetics_Statics_and_Thermodynamics/10A%3A_Constant_Acceleration_Problems_in_Two_Dimensions	10A: Constant Acceleration Problems in Two Dimensions - Physics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error 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Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy 1. Home 2. Bookshelves 3. University Physics 4. Calculus-Based Physics (Schnick) 5. Volume A: Kinetics, Statics, and Thermodynamics 6. 10A: Constant Acceleration Problems in Two Dimensions Expand/collapse global location 10A: Constant Acceleration Problems in Two Dimensions Last updated Jan 16, 2023 Save as PDF 9A: One-Dimensional Motion Graphs 11A: Relative Velocity Page ID 2463 Jeffrey W. Schnick Saint Anselm College ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. In solving problems involving constant acceleration in two dimensions, the most common mistake is probably mixing the x and y motion. One should do an analysis of the x motion and a separate analysis of the y motion. The only variable common to both the x and the y motion is the time. Note that if the initial velocity is in a direction that is along neither axis, one must first break up the initial velocity into its components. 2. A horizontal square of edge length 1.20 m is situated on a Cartesian coordinate system such that one corner of the square is at the origin and the corner opposite that corner is at (1.20 m, 1.20 m). A particle is at the origin. The particle has an initial velocity of 2.20 m/s directed toward the corner of the square at (1.20 m, 1.20 m) and has a constant acceleration of 4.87⁢m/s 2 in the +x direction. Where does the particle hit the perimeter of the square? Solution Let’s start with a diagram. Now let’s make some conceptual observations on the motion of the particle. Recall that the square is horizontal so we are looking down on it from above. It is clear that the particle hits the right side of the square because: It starts out with a velocity directed toward the far right corner. That initial velocity has an x component and a y component. The y component never changes because there is no acceleration in the y direction. The x component, however, continually increases. The particle is going rightward faster and faster. Thus, it will take less time to get to the right side of the square then it would without the acceleration and the particle will get to the right side of the square before it has time to get to the far side. An important aside on the trajectory (path) of the particle: Consider an ordinary checker on a huge square checkerboard with squares of ordinary size (just a lot more of them then you find on a standard checkerboard). Suppose you start with the checker on the extreme left square of the end of the board nearest you (square 1) and every second, you move the checker right one square and forward one square. This would correspond to the checker moving toward the far right corner at constant velocity. Indeed you would be moving the checker along the diagonal. Now let’s throw in some acceleration. Return the checker to square 1 and start moving it again. This time, each time you move the checker forward, you move it rightward one more square than you did on the previous move. So first you move it forward one square and rightward one square. Then you move it forward another square but rightward two more squares. Then forward one square and rightward three squares. And so on. With each passing second, the rightward move gets bigger. (That’s what we mean when we say the rightward velocity is continually increasing.) So what would the path of the checker look like? Let’s draw a picture. Now back to the problem at hand. The way to attack these two-dimensional constant acceleration problems is to treat the x motion and the y motion separately. The difficulty with that, in the case at hand, is that the initial velocity is neither along x nor along y but is indeed a mixture of both x motion and y motion. What we have to do is to separate it out into its x and y components. Let’s proceed with that. Note that, by inspection, the angle that the velocity vector makes with the x axis is 45.0∘. Now we are ready to attack the x motion and the y motion separately. Before we do, let’s consider our plan of attack. We have established, by means of conceptual reasoning, that the particle will hit the right side of the square. This means that we already have the answer to half of the question “Where does the particle hit the perimeter of the square?” It hits it at x=1.20⁢m and y = ? . All we have to do is to find out the value of y. We have established that it is the x motion that determines the time it takes for the particle to hit the perimeter of the square. It hits the perimeter of the square at that instant in time when x achieves the value of 1.20⁢m. So our plan of attack is to use one or more of the x-motion constant acceleration equations to determine the time at which the particle hits the perimeter of the square and to plug that time into the appropriate y-motion constant acceleration equation to get the value of y at which the particle hits the side of the square. Let’s go for it. x motion We start with the equation that relates position and time: The x component of the acceleration is the total acceleration, that is a x=a. Thus, x=v 0⁢x⁢t+1 2⁢a⁢t 2 Recognizing that we are dealing with a quadratic equation we get it in the standard form of the quadratic equation. 1 2⁢a⁢t 2+v 0⁢x⁢t−x=0 Now we apply the quadratic formula: t=−v 0⁢x±v 0⁢x 2−4⁢(1 2⁢a)⁢(−x)2⁢(1 2⁢a x)t=−v 0⁢x±v 0⁢x 2+2⁢a⁢x a x substituting values with units (and, in this step, doing no evaluation) we obtain: t=−1.556⁢m s±(1.556⁢m s)2+2⁢(4.87⁢m s 2)⁢1.20⁢m 4.87⁢m s 2 Evaluating this expression yields: t=0.4518⁢s and t=−1.091⁢s. We are solving for a future time so we eliminate the negative result on the grounds that it is a time in the past. We have found that the particle arrives at the right side of the square at time t=0.4518⁢s. Now the question is, “What is the value of y at that time?” y motion Again we turn to the constant acceleration equation relating position to time, this time writing it in terms of the y variables: We note that y 0 is zero because the particle is at the origin at time 0 and ay is zero because the acceleration is in the +x direction meaning it has no y component. Rewriting this: y=V 0⁢y⁢t Substituting values with units, y=1.556⁢m s⁢(0.4518⁢s) evaluating, and rounding to three significant figures yields: y=0.703⁢m Thus, the particle hits the perimeter of the square at (1.20⁢m,0.703⁢m) 3. An important aside on the trajectory (path) of the particle: Consider an ordinary checker on a huge square checkerboard with squares of ordinary size (just a lot more of them then you find on a standard checkerboard). Suppose you start with the checker on the extreme left square of the end of the board nearest you (square 1) and every second, you move the checker right one square and forward one square. This would correspond to the checker moving toward the far right corner at constant velocity. Indeed you would be moving the checker along the diagonal. Now let’s throw in some acceleration. Return the checker to square 1 and start moving it again. This time, each time you move the checker forward, you move it rightward one more square than you did on the previous move. So first you move it forward one square and rightward one square. Then you move it forward another square but rightward two more squares. Then forward one square and rightward three squares. And so on. With each passing second, the rightward move gets bigger. (That’s what we mean when we say the rightward velocity is continually increasing.) So what would the path of the checker look like? Let’s draw a picture. 4. The positions of a particle and a thin (treat it as being as thin as a line) rocket of length 0.280⁢m are specified by means of Cartesian coordinates. At time 0 the particle is at the origin and is moving on a horizontal surface at 23.0⁢m/s at 51.0∘. It has a constant acceleration of 2.43⁢m/s 2 in the +y direction. At time 0 the rocket is at rest and it extends from (−.280⁢m, 50.0⁢m) to (0, 50.0⁢m), but it has a constant acceleration in the +x direction. What must the acceleration of the rocket be in order for the particle to hit the rocket? Solution Based on the description of the motion, the rocket travels on the horizontal surface along the line y=50.0⁢m. Let’s figure out where and when the particle crosses this line. Then we’ll calculate the acceleration that the rocket must have in order for the nose of the rocket to be at that point at that time and repeat for the tail of the rocket. Finally, we’ll quote our answer as being any acceleration in between those two values. When and where does the particle cross the line y=50.0⁢m? We need to treat the particle’s x motion and the y motion separately. Let’s start by breaking up the initial velocity of the particle into its x and y components. Now in this case, it is the y motion that determines when the particle crosses the trajectory of the rocket because it does so when y=50.0⁢m. So let’s address the y motion first. y motion of the particle Note that we can’t just assume that we can cross out yo but in this case the time zero position of the particle was given as (0, 0) meaning that y o is indeed zero for the case at hand. Now we solve for t: y=V 0⁢y⁢t+1 2⁢a y⁢t 2 1 2⁢a y⁢t 2+V 0⁢y⁢t−y=0 t=−V 0⁢y±V 0⁢y 2−4⁢(1 2⁢a y)⁢(−y)2⁢(1 2⁢a y)t=−V 0⁢y±V 0⁢y 2+2⁢a y⁢y a y t=−17.87⁢m s±(17.87⁢m s)2+2⁢(2.43⁢m s 2)⁢50.0⁢m 2.43⁢m s 2 t=2.405⁢s and t=−17.11⁢s Again, we throw out the negative solution because it represents an instant in the past and we want a future instant. Now we turn to the x motion to determine where the particle crosses the trajectory of the rocket. x motion of the particle Again we turn to the constant acceleration equation relating position to time, this time writing it in terms of the x variables: So the particle crosses the rocket’s path at (34.80⁢m, 50.0⁢m) at time t=2.450⁢s. Let’s calculate the acceleration that the rocket would have to have in order for the nose of the rocket to be there at that instant. The rocket has x motion only. It is always on the line y=50.0⁢m. Motion of the Nose of the Rocket where we use the subscript n for “nose” and a prime to indicate “rocket.” We have crossed out x o⁢n′ because the nose of the rocket is at (0, 50.0⁢m) at time zero, and we have crossed out v o⁢x⁢n′ because the rocket is at rest at time zero. x n′=1 2⁢a n′⁢t 2 Solving for a n′ yields: a n′=2⁢x n′t 2 Now we just have to evaluate this expression at t=2.405⁢s, the instant when the particle crosses the trajectory of the rocket, and at x n′=x=34.80⁢m, the value of x at which the particle crosses the trajectory of the rocket. a n′=2⁢(34.80⁢m)(2.405⁢s)2 a n′=12.0⁢m s 2 It should be emphasized that the n for “nose” is not there to imply that the nose of the rocket has a different acceleration than the tail; rather; the whole rocket must have the acceleration a′n=12.0⁢m s 2 in order for the particle to hit the rocket in the nose. Now let’s find the acceleration at′that the entire rocket must have in order for the particle to hit the rocket in the tail. Motion of the Tail of the Rocket where we use the subscript t for “tail” and a prime to indicate “rocket.” We have crossed out v′o⁢x⁢t because the rocket is at rest at time zero, but x o⁢t′ is not zero because the tail of the rocket is at (−.280, 50.0⁢m) at time zero. x t′=x o⁢t′+1 2⁢a t′⁢t 2 Solving for a t′ yields: a t′=2⁢(x t′−x o⁢t′)t 2 Evaluating at t=2.405⁢s and x t′=x=34.80⁢m yields a t′=2⁢(34.80⁢m−(−0.280⁢m))(2.405⁢s)2 a t′=12.1⁢m s 2 as the acceleration that the rocket must have in order for the particle to hit the tail of the rocket. Thus: The acceleration of the rocket must be somewhere between 12.0⁢m s 2 and 12.1⁢m s 2, inclusive, in order for the rocket to be hit by the particle.) In solving problems involving constant acceleration in two dimensions, the most common mistake is probably mixing the x and y motion. One should do an analysis of the x motion and a separate analysis of the y motion. The only variable common to both the x and the y motion is the time. Note that if the initial velocity is in a direction that is along neither axis, one must first break up the initial velocity into its components. In the last few chapters we have considered the motion of a particle that moves along a straight line with constant acceleration. In such a case, the velocity and the acceleration are always directed along one and the same line, the line on which the particle moves. Here we continue to restrict ourselves to cases involving constant acceleration (constant in both magnitude and direction) but lift the restriction that the velocity and the acceleration be directed along one and the same line. If the velocity of the particle at time zero is not collinear with the acceleration, then the velocity will never be collinear with the acceleration and the particle will move along a curved path. The curved path will be confined to the plane that contains both the initial velocity vector and the acceleration vector, and in that plane, the trajectory will be a parabola. (The trajectory is just the path of the particle.) You are going to be responsible for dealing with two classes of problems involving constant acceleration in two dimensions: Problems involving the motion of a single particle. Collision Type II problems in two dimensions We use sample problems to illustrate the concepts that you must understand in order to solve two-dimensional constant acceleration problems. A horizontal square of edge length 1.20 m is situated on a Cartesian coordinate system such that one corner of the square is at the origin and the corner opposite that corner is at (1.20 m, 1.20 m). A particle is at the origin. The particle has an initial velocity of 2.20 m/s directed toward the corner of the square at (1.20 m, 1.20 m) and has a constant acceleration of 4.87⁢m/s 2 in the +x direction. Where does the particle hit the perimeter of the square? Solution Let’s start with a diagram. Now let’s make some conceptual observations on the motion of the particle. Recall that the square is horizontal so we are looking down on it from above. It is clear that the particle hits the right side of the square because: It starts out with a velocity directed toward the far right corner. That initial velocity has an x component and a y component. The y component never changes because there is no acceleration in the y direction. The x component, however, continually increases. The particle is going rightward faster and faster. Thus, it will take less time to get to the right side of the square then it would without the acceleration and the particle will get to the right side of the square before it has time to get to the far side. An important aside on the trajectory (path) of the particle: Consider an ordinary checker on a huge square checkerboard with squares of ordinary size (just a lot more of them then you find on a standard checkerboard). Suppose you start with the checker on the extreme left square of the end of the board nearest you (square 1) and every second, you move the checker right one square and forward one square. This would correspond to the checker moving toward the far right corner at constant velocity. Indeed you would be moving the checker along the diagonal. Now let’s throw in some acceleration. Return the checker to square 1 and start moving it again. This time, each time you move the checker forward, you move it rightward one more square than you did on the previous move. So first you move it forward one square and rightward one square. Then you move it forward another square but rightward two more squares. Then forward one square and rightward three squares. And so on. With each passing second, the rightward move gets bigger. (That’s what we mean when we say the rightward velocity is continually increasing.) So what would the path of the checker look like? Let’s draw a picture. Now back to the problem at hand. The way to attack these two-dimensional constant acceleration problems is to treat the x motion and the y motion separately. The difficulty with that, in the case at hand, is that the initial velocity is neither along x nor along y but is indeed a mixture of both x motion and y motion. What we have to do is to separate it out into its x and y components. Let’s proceed with that. Note that, by inspection, the angle that the velocity vector makes with the x axis is 45.0∘. Now we are ready to attack the x motion and the y motion separately. Before we do, let’s consider our plan of attack. We have established, by means of conceptual reasoning, that the particle will hit the right side of the square. This means that we already have the answer to half of the question “Where does the particle hit the perimeter of the square?” It hits it at x=1.20⁢m and y = ? . All we have to do is to find out the value of y. We have established that it is the x motion that determines the time it takes for the particle to hit the perimeter of the square. It hits the perimeter of the square at that instant in time when x achieves the value of 1.20⁢m. So our plan of attack is to use one or more of the x-motion constant acceleration equations to determine the time at which the particle hits the perimeter of the square and to plug that time into the appropriate y-motion constant acceleration equation to get the value of y at which the particle hits the side of the square. Let’s go for it. x motion We start with the equation that relates position and time: The x component of the acceleration is the total acceleration, that is a x=a. Thus, x=v 0⁢x⁢t+1 2⁢a⁢t 2 Recognizing that we are dealing with a quadratic equation we get it in the standard form of the quadratic equation. 1 2⁢a⁢t 2+v 0⁢x⁢t−x=0 Now we apply the quadratic formula: t=−v 0⁢x±v 0⁢x 2−4⁢(1 2⁢a)⁢(−x)2⁢(1 2⁢a x) t=−v 0⁢x±v 0⁢x 2+2⁢a⁢x a x substituting values with units (and, in this step, doing no evaluation) we obtain: t=−1.556⁢m s±(1.556⁢m s)2+2⁢(4.87⁢m s 2)⁢1.20⁢m 4.87⁢m s 2 Evaluating this expression yields: t=0.4518⁢s and t=−1.091⁢s. We are solving for a future time so we eliminate the negative result on the grounds that it is a time in the past. We have found that the particle arrives at the right side of the square at time t=0.4518⁢s. Now the question is, “What is the value of y at that time?” y motion Again we turn to the constant acceleration equation relating position to time, this time writing it in terms of the y variables: We note that y 0 is zero because the particle is at the origin at time 0 and ay is zero because the acceleration is in the +x direction meaning it has no y component. Rewriting this: y=V 0⁢y⁢t Substituting values with units, y=1.556⁢m s⁢(0.4518⁢s) evaluating, and rounding to three significant figures yields: y=0.703⁢m Thus, the particle hits the perimeter of the square at (1.20⁢m,0.703⁢m) Next, let’s consider a 2-D Collision Type II problem. Solving a typical 2-D Collision Type II problem involves finding the trajectory of one of the particles, finding when the other particle crosses that trajectory, and establishing where the first particle is when the second particle crosses that trajectory. If the first particle is at the point on its own trajectory where the second particle crosses that trajectory then there is a collision. In the case of objects rather than particles, one often has to do some further reasoning to solve a 2-D Collision Type II problem. Such reasoning is illustrated in the following example involving a rocket. The positions of a particle and a thin (treat it as being as thin as a line) rocket of length 0.280⁢m are specified by means of Cartesian coordinates. At time 0 the particle is at the origin and is moving on a horizontal surface at 23.0⁢m/s at 51.0∘. It has a constant acceleration of 2.43⁢m/s 2 in the +y direction. At time 0 the rocket is at rest and it extends from (−.280⁢m, 50.0⁢m) to (0, 50.0⁢m), but it has a constant acceleration in the +x direction. What must the acceleration of the rocket be in order for the particle to hit the rocket? Solution Based on the description of the motion, the rocket travels on the horizontal surface along the line y=50.0⁢m. Let’s figure out where and when the particle crosses this line. Then we’ll calculate the acceleration that the rocket must have in order for the nose of the rocket to be at that point at that time and repeat for the tail of the rocket. Finally, we’ll quote our answer as being any acceleration in between those two values. When and where does the particle cross the line y=50.0⁢m? We need to treat the particle’s x motion and the y motion separately. Let’s start by breaking up the initial velocity of the particle into its x and y components. Now in this case, it is the y motion that determines when the particle crosses the trajectory of the rocket because it does so when y=50.0⁢m. So let’s address the y motion first. y motion of the particle Note that we can’t just assume that we can cross out yo but in this case the time zero position of the particle was given as (0, 0) meaning that y o is indeed zero for the case at hand. Now we solve for t: y=V 0⁢y⁢t+1 2⁢a y⁢t 2 1 2⁢a y⁢t 2+V 0⁢y⁢t−y=0 t=−V 0⁢y±V 0⁢y 2−4⁢(1 2⁢a y)⁢(−y)2⁢(1 2⁢a y) t=−V 0⁢y±V 0⁢y 2+2⁢a y⁢y a y t=−17.87⁢m s±(17.87⁢m s)2+2⁢(2.43⁢m s 2)⁢50.0⁢m 2.43⁢m s 2 t=2.405⁢s and t=−17.11⁢s Again, we throw out the negative solution because it represents an instant in the past and we want a future instant. Now we turn to the x motion to determine where the particle crosses the trajectory of the rocket. x motion of the particle Again we turn to the constant acceleration equation relating position to time, this time writing it in terms of the x variables: So the particle crosses the rocket’s path at (34.80⁢m, 50.0⁢m) at time t=2.450⁢s. Let’s calculate the acceleration that the rocket would have to have in order for the nose of the rocket to be there at that instant. The rocket has x motion only. It is always on the line y=50.0⁢m. Motion of the Nose of the Rocket where we use the subscript n for “nose” and a prime to indicate “rocket.” We have crossed out x o⁢n′ because the nose of the rocket is at (0, 50.0⁢m) at time zero, and we have crossed out v o⁢x⁢n′ because the rocket is at rest at time zero. x n′=1 2⁢a n′⁢t 2 Solving for a n′ yields: a n′=2⁢x n′t 2 Now we just have to evaluate this expression at t=2.405⁢s, the instant when the particle crosses the trajectory of the rocket, and at x n′=x=34.80⁢m, the value of x at which the particle crosses the trajectory of the rocket. a n′=2⁢(34.80⁢m)(2.405⁢s)2 a n′=12.0⁢m s 2 It should be emphasized that the n for “nose” is not there to imply that the nose of the rocket has a different acceleration than the tail; rather; the whole rocket must have the acceleration a′n=12.0⁢m s 2 in order for the particle to hit the rocket in the nose. Now let’s find the acceleration at′that the entire rocket must have in order for the particle to hit the rocket in the tail. Motion of the Tail of the Rocket where we use the subscript t for “tail” and a prime to indicate “rocket.” We have crossed out v′o⁢x⁢t because the rocket is at rest at time zero, but x o⁢t′ is not zero because the tail of the rocket is at (−.280, 50.0⁢m) at time zero. x t′=x o⁢t′+1 2⁢a t′⁢t 2 Solving for a t′ yields: a t′=2⁢(x t′−x o⁢t′)t 2 Evaluating at t=2.405⁢s and x t′=x=34.80⁢m yields a t′=2⁢(34.80⁢m−(−0.280⁢m))(2.405⁢s)2 a t′=12.1⁢m s 2 as the acceleration that the rocket must have in order for the particle to hit the tail of the rocket. Thus: The acceleration of the rocket must be somewhere between 12.0⁢m s 2 and 12.1⁢m s 2, inclusive, in order for the rocket to be hit by the particle. This page titled 10A: Constant Acceleration Problems in Two Dimensions is shared under a CC BY-SA 2.5 license and was authored, remixed, and/or curated by Jeffrey W. Schnick via source content that was edited to the style and standards of the LibreTexts platform. 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2544	https://www.isec.tugraz.at/wp-content/uploads/2022/09/lecture_notes_pred_logic.pdf	Lecture Notes for Logic and Computability Course Number: IND04033UF Contact Bettina Könighofer Institute for Applied Information Processing and Communications (IAIK) Graz University of Technology, Austria bettina.koenighofer@iaik.tugraz.at Table of Contents 5 Predicate Logic 3 5.1 Predicates and Quantiﬁers . . . . . . . . . . . . . . . . . . . . . . 3 5.2 Syntax of Predicate Logic . . . . . . . . . . . . . . . . . . . . . . 6 5.3 Free and Bound Variables . . . . . . . . . . . . . . . . . . . . . . 9 5.3.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.4 Semantics of Predicate Logic . . . . . . . . . . . . . . . . . . . . 10 5.4.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.4.2 Evaluate a Formula under a Model . . . . . . . . . . . . . 11 1 5 Predicate Logic The Limitations of Propositional Logic. Propositional logic has limitations when encoding declarative sentences. It is quite easy to encode logical sentence components like not, and, or, if ... then. However, we would also like to express sentence components like there exists ... and for all ... which is not possible in propositional logic. To overcome this limitation, we need a more powerful type of logic like e.g. predicate logic, also called ﬁrst-order logic. Example. Model the following sentence in propositional logic: “Every person who is 18 years or older is eligible to vote.” Solution. p with p :=“Every person who is 18 years or older is eligible to vote.” Note, that the above statement cannot be adequately expressed using only propositional logic, since propositional logic is not expressive enough to deal with quantiﬁed variables. Since the sentence is not referring to a speciﬁc person and the statement applies to all people who are 18 years or older, we are stuck. Therefore we need a richer logic like predicate logic. 5.1 Predicates and Quantiﬁers Predicate logic is an extension of propositional logic. It adds the concept of predicates and quantiﬁers to better capture the meaning of statements that cannot be adequately expressed by propositional logic. 3 4 Chapter 5. Predicate Logic Predicates A predicate is a function that takes one or more variables from a speciﬁc do-main and returns true or false depending on the values of its variables. We denote predicates with capital roman letters such as P, Q, and R. A statement involving n variables x1, x2, x3, . . . , xn can be denoted by an n-ary predicate P(x1, x2, x3, . . . , xn). Once truth value has been assigned to all the variables x1, x2, x3, . . . , xn, the statement P(x1, x2, x3, . . . , xn) becomes true or false. Example. Write a formula ϕ in predicate logic that models the following sentence: “x is smaller than 5.” Solution. ϕ := STFive(x) using the predicate STFive that returns true if x < 5 and false otherwise. Example. What are the truth values of STFive(4) and STFive(6)? Solution. STFive(4) is true, since it is the case that 4 < 5 holds. STFive(6) is false since it is not true that 6 < 5. Example. Let I(x, y) denote the predicate that compares whether x is equivalent to y + 1, i.e., it returns true if x ≡y + 1 and false otherwise. Give the truth values for I(6, 5) and I(1, 4)? Solution. I(6, 5) is true since 6 = 5 + 1. I(1, 4) is false since 1 ̸= 4 + 1. Quantiﬁers Quantiﬁers are used to express the extent to which a predicate is true over a range of elements. Using quantiﬁers to create such propositions is called quantiﬁcation. We distinguish between two types of quantiﬁcation: the universal quantiﬁcation and the existential quantiﬁcation. Universal Quantiﬁer ∀ We can express statements which assert that a property is true for all the values of a variable in a particular domain using universal quantiﬁcation. The notation ∀xP(x) denotes the universal quantiﬁcation of P(x). ∀xP(x) is read as “for all x P(x)”. The formula ∀xP(x) evaluates to true if P(x) is true for all values of x in a given domain. It is very important to explicitly specify the domain when using universal quantiﬁcation since the domain decides the possible values of x. Without the domain, the universal quantiﬁcation has no meaning. Example. Let P(x) := “x + 5 > x” with x ∈N+. What is the truth value of the statement ∀xP(x)? Solution. As x + 5 > x for any positive natural number, P(x) ≡true for all x and therefore it holds that ∀xP(x) ≡true. Existential Quantiﬁer ∃ We can express statements which assert that there is an element with a certain property by existential quantiﬁcation. The notation ∃xP(x) denotes the existential quantiﬁcation of P(x), i.e., ∃xP(x) is true if and only if P(x) is true 5.1. Predicates and Quantiﬁers 5 for at least one value of x in the domain. ∃xP(x) is read as “There is at least one such x such that P(x)” and denotes the statement. The formula ∃xP(x) is true if there exists an element x in the domain such that P(x).” Example. Let P(x) :=“x > 10” with x ∈N+. What is the truth value of the statement ∃xP(x)? Solution. P(x) is true for all natural numbers greater than 10 and false for all natural numbers less than 10. Therefore, the formula ∃xP(x) is true. Example. Model the following sentence in predicate logic: ”Every person who is 18 years or older is eligible to vote.“ Solution. Using the domain x ∈People and the predicates P(x) :=“x is 18 years”, R(x) :=“x is older than 18 years” and Q(x) :=“x is eligible to vote” we get: ∀x((P(x) ∨R(x)) ↔Q(x)). Example. Model the following sentence in predicate logic: ”Every lecturer is older than some student.“ Solution. We deﬁne the following predicates for x, y ∈People: L(x) := “x is a lecturer” S(x) := “x is a student” O(x, y) := “x is older than y” Using these predicates, we can model the sentence as follows: ∀x(L(x) →∃y(S(y) ∧O(x, y))) Example. Formalize the following reasoning using propositional logic and show whether the sequent is valid: “Every child has a mother. Maria is a child. Therefore, Maria has a mother.” Solution. We deﬁne the following atomic propositions: p := “Every child has a mother.” q := “Maria is a child.” r := “Maria has a mother.” Using propositional logic results in the following sequent: p, q ⊢r The sequent can easily disproven by the following counterexample: M := {p = ⊤, q = ⊤, r = ⊥}. Example. Formalize the reasoning from above using predicate logic. Solution. We deﬁne the following predicates: 6 Chapter 5. Predicate Logic C(x) := “x is a child” M(x) := “x has a mother” Using the domain of x, y ∈People, we can model the sequent as follows: ∀x(C(x) ↔M(x)), C(maria) ⊢M(maria) Using predicate logic, the meaning of the statements is preserved such that reasoning is possible. In the next chapter, we will extend the natural deduction calculus to predicate logic such that we are able to prove such sequents. Example. Model the following sentence in predicate logic: ”Niki and Ben have the same maternal grandmother.“ Solution. We use the binary predicate M(x, y) :=“x is the mother of y.” with x, y ∈People and get the following formula: ∀x∀y∀u∀v(M(x, y) ∧M(y, niki) ∧M(u, v) ∧M(v, ben) →x = u). The formula states that, if y and v are Niki’s and Ben’s mothers, respectively, and x and u are their mothers (i.e. Ben’s and Niki’s maternal grandmothers, respectively), then x and u are the same person. Note, that we use the inﬁx notation for the equality predicate instead of the preﬁx notation. Whenever it feels more natural you can use the inﬁx notation. You can always use the equality predicate without deﬁning it explicitly. What are functions? A function is an expression of one or more variables determined on some spe-ciﬁc domain and returns a value from that domain. We denote functions with lowercase capital roman letters such as f, g, and h. Example. Express the statement from before using the function symbol m(x) that returns the mother of x. Solution. Using the function m, the sentences from above can be encoded as follows: m(m(niki)) = m(m(ben)) 5.2 Syntax of Predicate Logic In this section we deﬁne the syntactic rules that deﬁne well-formed formulas in predicate logic. First, note that in predicate logic, we have two types of sorts. First, we have terms that talk about objects. Terms include individual objects (e..g, ben, niki), 5.2. Syntax of Predicate Logic 7 variables since they represent objects (e.g., x, y), and function symbols since they refer to objects (e.g., m(x)). Second, we have formulas that have a truth value. For example, P(x, f(y) is a formula and x, y, and f(x) are terms. Formula z }\| { P \|{z} Predicate ( x \|{z} Term , f(y) \|{z} Term ) To deﬁne the syntax of predicate logic, we use the following notation: • V : Deﬁnes the set of variable symbols, e.g., x, y, z. • F: Deﬁnes the set of function symbols, e.g., f, g, h. • P: Deﬁnes the set of predicate symbols, e.g., P, Q, R. Each predicate symbol and each function symbol comes with an arity, the number of arguments it expects. Constants are functions which don’t take any arguments, i.e., nullary functions. Terms Terms are deﬁned as follows: • Any variable is a term. • If c ∈F is a nullary function, then c is a term. • If t1, t2, ..., tn are terms and f ∈F has arity n > 0, then f(t1, t2, ..., tn) is a term. • Nothing else is a term. Note that the ﬁrst building blocks of terms are constants (nullary functions) and variables. More complex terms are built from function symbols using as many previously built terms as required by such function symbols. Formulas Formulas are deﬁned as follows: • If P ∈P is a predicate with arity n > 0 and t1, t2, ..., tn are terms over F, then P(t1, t2, ..., tn) is a formula. • If φ is a formula, then ¬φ is a formula. • If φ and ψ are formulas, then (φ ∧ψ), (φ ∨ψ), and (φ →ψ) are formulas. • If φ is a formula and x is a variable, then (∀x φ) and (∃x φ) are formulas. • Nothing else is a formula. Note, that arguments, that are given to a predicate, are always terms. 8 Chapter 5. Predicate Logic Binding Priorities We add to the binding priorities that we deﬁned for porpositional logic the convention that ∀x and ∃x bings like ¬. Therefore, the precedence rules can be given by: 1. ∀, ∃, ¬ bind most tightly; 2. then ∧; 3. then ∨; 4. then →which is right-associative. Parse Tree The parse tree is constructed in the same way as for formulas in propositional logic, but with two additional sorts of nodes: • Nodes for the quantiﬁers ∀x and ∃x have one subtree. • Predicates of the form P(t1, t2, . . . , tn) have the symbol P as a node with n subtrees, namely the parse trees of the terms t1, t2, . . . , tn. Example. Draw the syntax tree to the following formula: ∀x P(x) →¬Q(x) ∧ S(x, f(y, z)) ∨T(y) . Solution. ∀x ∧ → P x ¬ Q x ∨ S x f y z T y Figure 5.1: A syntax tree of a predicate logic formula. Red nodes are formulas, green nodes are terms. 5.3. Free and Bound Variables 9 5.3 Free and Bound Variables With the introduction of ∀and ∃, we also have to think about the scope of a these quantiﬁers and whether a variable is free or bound. Deﬁnition - Free and Bound Variables. Let φ be a formula in predicate logic. An occurrence of x in φ is free if it is a leaf node in the syntax tree of φ such that there is no path upwards from that node x to a node ∀x or ∃x. Otherwise, that occurrence of x is called bound. Deﬁnition - Scope of a Quantiﬁer. For ∀xφ, or ∃xφ, we say that φ is the scope of ∀x, respectively ∃x. In other words, if x occurs in φ, then it is bound if, and only if, it is in the scope of some ∃x or some ∀x; otherwise it is free. In terms of parse trees, the scope of a quantiﬁer is its subtree. Example. Construct a parse tree for the following formula φ and determine the scope of its quantiﬁers and which occurrences of the variables are free and which are bound: φ := ∀x (P(x) ∨Q(y, x)) ∧R(x) Solution. The syntax tree with labels denoting free and bound variables can be seen in Figure 5.2. ∧ ∀x ∨ P x Q y x R x Scope of quantiﬁer bound free bound free Figure 5.2: A syntax tree illustrating free and bound variables. The scope of ∀x is (P(x) ∨Q(y, x)). Thus, the ﬁrst two occurrences of x are bound to ∀x. Starting for the y leave node, the only quantiﬁer we run into is ∀x but that x has nothing to do with y. So y is free in this formula. The third occurrence of x is also free. 10 Chapter 5. Predicate Logic 5.3.1 Substitution Variables are place holders and can be replaced with more concrete information. Deﬁnition - Substitution. Given a variable x, a term t and a formula φ we deﬁne φ[t/x] to be the formula obtained by replacing each free occurrence of variable x in φ with t. φ[ t \|{z} Term / x \|{z} Variable ] Therefore, if all occurrences of x are bound in φ, none of them gets substituted by t. Furthermore, it is not allowed to perform substitutions such that a variable gets captured by a quantiﬁer: meaning that the variable was free before and by carrying out the substitution it gets bound by a quantiﬁer. Example. Compute φ[f(z)/x] for the following formula: φ := ∀y P(x) ∧Q(y) ∨ R(y) ∧Q(x) Solution. All occurrences of x are free and can be replayed by f(z). φ f(z)/x = ∀y P(f(z)) ∧Q(y) ∨ R(y) ∧Q(f(z)) Example. Compute ψ[f(z)/x] for the following formula: ψ = ∀x P(x) ∧Q(y) ∨ R(y) ∧Q(x) Solution. We can only substitute the x in Q(x), as the other occurrence x is bound to the ∀x quantiﬁer. ψ f(z)/x = ∀x P(x) ∧Q(y) ∨ R(y) ∧Q(f(z)) Example. Compute φ[f(y)/x] for the following formula: φ := ∀y P(x) ∧Q(y) ∨ R(y) ∧Q(x) Solution. We are not allowed to replace x with f(y), since x is free before the substitution and the term f(y) contains a y which is in the scope of the ∀y quantiﬁer. Therefore, the variable would be captured which is not allowed. φ f(y)/x := ∀y P(x) ∧Q(y) ∨ R(y) ∧Q(x) 5.4 Semantics of Predicate Logic We will extend the notion of models that we discussed for propositional logic to predicate logic. In propositional logic, a model deﬁned an assignment of truth values to all variables such that the formula evaluated to true or to false. A model in predicate logic diﬀers from a model in propositional logic in the treatment of predicates and functions. 5.4. Semantics of Predicate Logic 11 5.4.1 Models A model in predicate logic needs to deﬁne a concrete meaning to all predicate and function symbols involved. For example, the predicate P is deﬁned in the model to be the relation “greater than” on the set of real numbers. Deﬁnition - Model in Predicate Logic. A model M consists of the following set of data: • A non-empty set A, the universe/domain of concrete values; • for each nullary function symbol f ∈F, a concrete element f M ∈A; • for each nullary predicate symbol P ∈P, a truth value; • for each function symbol f ∈F with arity n > 0, a concrete function f M : An →A; • for each predicate symbol P ∈P with arity n > 0 : subset P M ⊆An. • for any free variable var: a lookup-table l : var →A. To denote a concrete instance of a function f or a predicate P in a model M, we use the notation f M and P M. We often deﬁne P M as tuples which make P true and use function tables to deﬁne f M. Example. Give a model M for the following formula: φ := ∀x∃yP(x, y) Solution. The model consists of a domain A and a concrete predicate in-stance P M. We give one possible model for φ: • A = {a, b} • P M = {(a, b), (b, a)} (meaning P(a, b) = true, P(b, a) = true, for all other cases, P evaluates to false) 5.4.2 Evaluate a Formula under a Model Given a model M, we deﬁne the satisfaction relation M ⊨φ for each logical formula φ by structural induction on φ. • P : If φ is of the form P(t1, t2, . . . , tn), then we interpret the terms t1, t2, . . . , tn in our set A by replacing all variables with their values according to the look-up table l and interpret any function symbols f ∈F by f M. In this way we compute concrete values a1, a2, ..., an of A for each of these terms. Now M ⊨P(t1, t2, . . . , tn) holds if and only if (a1, a2, ..., an) is in the set P M. • ∀x: The relation M ⊨∀xψ holds if and only if M ⊨l[x←a] ψ holds for all a ∈A. • ∃x: Dually, M ⊨∃xψ holds if and only if M ⊨l[x←a] ψ holds for some a ∈A. 12 Chapter 5. Predicate Logic Example. Given a model M : A = {a, b}, P M = {(a, b), (b, a)} and a formula φ := ∀x∃yP(x, y). Does it hold that M ⊨φ? Solution. We need to show that for any possible value for x, there exists a value for y, such that P evaluates to true. Since P(a, b) and P(b, a) are both true, this is the case and it holds that M ⊨φ. Example. Given a model M : A = {a, b}, P M = {(a, b), (b, a)} and a formula ψ = ∃x∀y P(x, y). Does it hold that M ⊨ψ? Solution. We need to show that there is value for x, such that for all possible values for y, P evaluates to true. We perform the following substitutions: • x and y substituted with a: P(a, a) = ⊥. Therefore, we try the next substitution for x. • x with b and y with a: P(b, a) = ⊤ • x with b and y with b: P(b, b) = ⊥ Therefore, there is no such x for which P evaluates to true under all possible values for y. Therefore, M ̸⊨ψ. Example. Give a formula ψ = ∃x∀y P(x, y) and a model M : A = N, P M = x ≤y \| P M = {(1, 1), (1, 2) . . . (2, 2), . . . }. Does it hold that M ⊨ψ? Solution. Let us substitute x with 1. We need to show that P evaluates to true for all values of y. • y substituted with 1: P(1, 1) = ⊤ • y substituted with 2: P(1, 2) = ⊤ • . . . • y substituted with n: P(1, n) = ⊤for any n > 1 Therefore, we can conclude that M ⊨ψ.
2545	https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_7?srsltid=AfmBOop7CfZaiD-V6NOZjzUgasXWuJSZ4YcONTFjUx06QvgprfJglPfT	Art of Problem Solving 2015 AMC 10A Problems/Problem 7 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2015 AMC 10A Problems/Problem 7 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2015 AMC 10A Problems/Problem 7 Contents [hide] 1 Problem 2 Solution 1 3 Solution 2 4 Solution 3 5 Solution 4 6 Video Solution (CREATIVE THINKING) 7 Video Solution 8 See Also Problem How many terms are in the arithmetic sequence , , , , , ? Solution 1 , so the amount of terms in the sequence , , , , , is the same as in the sequence , , , , , . In this sequence, the terms are the multiples of going up to , and there are multiples of in . However, the number 0 must also be included, adding another multiple. So, the answer is . Solution 2 Using the formula for arithmetic sequence's nth term, we see that . Solution 3 Minus each of the terms by to make the the sequence . . Solution 4 Subtract each of the terms by to make the sequence . Then divide the each term in the sequence by to get . Now it is clear to see that there are terms in the sequence. . Video Solution (CREATIVE THINKING) ~Education, the Study of Everything Video Solution ~savannahsolver See Also 2015 AMC 10A (Problems • Answer Key • Resources) Preceded by Problem 6Followed by Problem 8 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 10 Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
2546	https://ericaraylanguage.com/the-spanish-imperfect-tense-essential-for-good-storytelling/	The Spanish Imperfect Tense (essential for good storytelling!) \| Erica Ray Language Home page Classes & Courses Group Classes for All Levels Private Coaching Story Courses YouTube Blog Learn Spanish Strategies to Learn Spanish Spanish Grammar Spanish Stories Spanish Conversations Life in Mexico Contact Select Page The Spanish Imperfect Tense (essential for good storytelling!) by Erica Ray \| Feb 10, 2025 \| Spanish Grammar \| 0 comments Imagine you’re telling a story in Spanish about your childhood—describing what life was like, the things you used to do, the games you played, and how you felt. Or maybe you want to paint a picture of a place you visited, explaining how the weather was, what people were doing, and the atmosphere around you. To express all of this naturally,you need the Spanish imperfect tense. This is actuallymy favorite verb tense!Why? Because it plays a crucial role in storytelling, helping us share rich details and emotions.(And I love stories!) Because it’s one of the easiest tenses to learn!(You’ll see why in a moment!) For a full breakdown, check outmy YouTube lesson here, based on one of the original Spanish stories on the channel! What Is the Imperfect Tense? Theimperfect tenseis one of two main past tenses in Spanish (imperfect vs. preterite). But what makes it different from thepreterite(hablé, comí, viví)? Use the imperfect when describing: Habitual or repeated past actions (used to / would do) Ongoing past actions (was/were -ing) Descriptions or setting the scene (background details) Key point:The imperfect doesnotfocus on a specific start or end time—it justdescribes what things were like in the past. Use #1: Habitual or Repeated Actions Use theimperfectwhen talking about things that happenedregularly in the past, without a defined beginning or end. Examples: Cuandoeraniño,jugabaen el parque todos los días. (When I was a child, I used to play in the park every day.) Mi abuela siemprecocinabalos domingos. (My grandmother always cooked on Sundays.) Íbamosa la playa cada verano. (We used to go to the beach every summer.) A few examples of key words that often trigger the imperfect: Siempre(always),todos los días(every day),cada verano(every summer),por lo general(generally),a menudo(often). Use #2: Ongoing Actions in the Past Use theimperfectwhen an action wasin progressat some point in the past. Examples: Ellaestabaleyendo un libro cuando llamaste. (She was reading a book when you called.) Llovíamucho aquella tarde. (It was raining a lot that afternoon.) Mientrascenábamos, sonó el teléfono. (While we were having dinner, the phone rang.) Compare with the Preterite: Llovíamucho aquella tarde.(It was raining…) →Imperfect(background action) Llovió mucho ayer.(It rained a lot yesterday.) →Preterite(completed action) Use #3: Descriptions & Setting the Scene Theimperfectis often used forbackground descriptions—of people, places, weather, time, emotions, etc. Examples: Hacíamucho calor y el sol brillaba. (It was very hot, and the sun was shining.) La casaeragrande yteníamuchas ventanas. (The house was big and had many windows.) Ellaestabafeliz porqueteníaun perrito nuevo. (She was happy because she had a new puppy.) How to Form the Imperfect Tense Good news! Theimperfect is one of the easiest tenses to conjugatebecause it follows predictable patterns.Plus, there are only three irregular verbs!(That’s why it’s my favorite!) For -AR Verbs (likehablar) Yo:hablaba Tú:hablabas Él/Ella/Ud.:hablaba Nosotros:hablábamos Ellos/Ellas/Uds.:hablaban For -ER Verbs (likecomer) Yo:comía Tú:comías Él/Ella/Ud.:comía Nosotros:comíamos Ellos/Ellas/Uds.:comían For -IR Verbs (likevivir) Yo:vivía Tú:vivías Él/Ella/Ud.:vivía Nosotros:vivíamos Ellos/Ellas/Uds.:vivían Notice: All -ER and -IR verbsshare the same endings (-ía, -ías, -ía, -íamos, -ían). Only -AR verbshave a unique pattern (-aba, -abas, -aba, -ábamos, -aban). Irregular Verbs? Just Three! There are onlythree irregular verbsin the imperfect: Ser (to be) Yo:era Tú:eras Él/Ella/Ud.:era Nosotros:éramos Ellos/Ellas/Uds.:eran Ir (to go) Yo:iba Tú:ibas Él/Ella/Ud.:iba Nosotros:íbamos Ellos/Ellas/Uds.:iban Ver (to see/watch) Yo:veía Tú:veías Él/Ella/Ud.:veía Nosotros:veíamos Ellos/Ellas/Uds.:veían That’s it!Every other verb follows the regular pattern. Practice Time! Choose the correct form of the imperfect tense for each sentence. The answers are in the P.S.—but no peeking until you’re done! Cuando yo ___ (ser/irregular) niño, me encantaba jugar con mis amigos. (When I was a child, I loved playing with my friends.) Mientras ella ___ (cocinar/regular) la cena, su esposo veía la televisión. (While she was cooking dinner, her husband was watching TV.) Siempre ___ (ir/irregular) a la casa de mis abuelos los domingos. (We always went to my grandparents’ house on Sundays.) Mi hermana ___ (ver/irregular) dibujos animados todas las mañanas. (My sister used to watch cartoons every morning.) El parque ___ (estar/regular) lleno de niños jugando. (The park was full of children playing.) Mis amigos y yo ___ (vivir/regular) en una ciudad pequeña cuando éramos jóvenes. (My friends and I lived in a small town when we were young.) La maestra siempre ___ (explicar/regular) la lección con paciencia. (The teacher always explained the lesson patiently.) Nosotros ___ (tener/regular) un perro cuando éramos pequeños. (We had a dog when we were little.) Cuando hacía frío, mi abuela siempre ___ (preparar/regular) chocolate caliente. (When it was cold, my grandmother always made hot chocolate.) Antes, tú ___ (ser/irregular) más tímido en la escuela. (Before, you used to be shyer in school.) Your Challenge This week, carve out a few minutes to watch myYouTube lesson herefor a more detailed explanation about the imperfect tense and fun practice with a story! Be sure to complete all the exercises as you go along! Hasta la próxima, Erica P.S. Answers to the Practice Section: era(ser – describing past childhood) cocinaba(cocinar – past ongoing action) íbamos(ir – habitual past action) veía(ver – repeated past action) estaba(estar – setting the scene) vivíamos(vivir – describing past living situation) explicaba(explicar – habitual past action) teníamos(tener – past possession/habit) preparaba(preparar – routine past action) eras(ser – past description of personality) The following two tabs change content below. Bio Latest Posts Erica Ray Owner/Author at Erica Ray Language Hi! I'm Erica Ray. I'm a self-taught bilingual gal from the U.S., language teacher and coach, expat in Mexico and a former English/Spanish medical interpreter. I help Spanish learners go from struggling learners to confident Spanish speakers. My students learn to create and stick to a personalized and comprehensive study plan, practice their speaking regularly and establish powerful learning habits to finally progress toward conversational fluency. Latest posts by Erica Ray (see all) The Spanish Imperfect Tense (essential for good storytelling!) - February 10, 2025 Real Spanish Conversation: El Trabajo (Spanish Listening Practice for Intermediate/Advanced) - January 14, 2025 Por vs Para: The SIMPLEST way to understand and choose between them - January 14, 2025 Submit a CommentCancel reply Your email address will not be published.Required fields are marked Comment Name Email Website [x] Save my name, email, and website in this browser for the next time I comment. Δ Privacy Policy Terms and Conditions Disclaimer Refund Policy Pin It on Pinterest Share This Pinterest Facebook Twitter Gmail Optimized by Seraphinite Accelerator Turns on site high speed to be attractive for people and search engines.
2547	https://resources.wolframcloud.com/FormulaRepository/resources/36ea6bd1-8aad-4001-a92f-3c574afdaf09	Speed of Sound in an Ideal Gas by Temperature and Molar Mass \| Wolfram Formula Repository WolframAlpha.com WolframCloud.com All Sites & Public Resources... 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All Company Search Wolfram Formula Repository(Under Development) ============================================= Primary Navigation Browse by Category Astronomy Athletics Biology Chemistry Commerce Earth Science Engineering Mathematics Medical Physics Social Science All Submit New Data Contact Us Search Search Speed of Sound in an Ideal Gas by Temperature and Molar Mass The speed of sound is the distance traveled per unit time by a sound wave as it propagates through an elastic medium. The speed of sound equals the square root of the product of the molar gas constant, temperature and adiabatic index divided by the molar mass. 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2548	https://www.statsig.com/perspectives/bigger-sample-sizes-reliable-abtests	Docs Pricing Sign In Book a Live Demo Products Solutions Resources Feature Flags Product Analytics Session Replay Web Analytics Infra Analytics Marketing Experiments PLATFORM Warehouse Native Infrastructure SDKs Integrations ROLES Engineering Dev Ops Data Science Product Management INDUSTRIES Gaming B2B Saas E-Commerce Blog Customer Stories Product Updates Support Startup Program Sample Size Calculator Statsig Lite Statsig University Products Solutions Resources #### Docs #### Pricing Sign In Book a Live Demo PRODUCTS Experimentation Feature Flags Product Analytics Session Replay Web Analytics Infra Analytics Marketing Experiments PLATFORM Warehouse Native Infrastructure SDKs Integrations ROLES Engineering Dev Ops Data Science Product Management INDUSTRIES Gaming B2B Saas E-Commerce Blog Customer Stories Product Updates Support Startup Program Sample Size Calculator Statsig Lite Statsig University ###### Back to Perspectives home The Statsig Team Why bigger sample sizes lead to more reliable A/B test results Fri Sep 20 2024 Ever wondered why some A/B tests lead to groundbreaking insights while others leave you scratching your head? It's not just luck—sample size plays a critical role in the reliability of your results. Whether you're part of a startup or a seasoned data scientist, understanding how sample size impacts your testing can make all the difference. In this blog, we'll dive into the importance of sample size in A/B testing, explore the dangers of both small and overly large samples, and discuss how to find that sweet spot for your experiments. Let's get started! The critical role of sample size in A/B testing Did you know that your sample size directly impacts the reliability of A/B test results? When your sample size is too small, you're gambling with the reliability of your findings. Random errors could steer you toward false conclusions. On the flip side, having a big enough sample boosts your chances of uncovering true effects. So how do you nail down the right sample size? It all comes down to a few key factors: your baseline conversion rates, the minimum detectable effect (MDE), statistical power, and significance level. By paying attention to these elements, you can design tests that give you solid, actionable insights. Now, you might think you need a huge user base to run meaningful A/B tests. Good news—you don't! Even smaller companies can hop on the A/B testing train by aiming for larger effect sizes. By zeroing in on substantial improvements rather than tiny tweaks, startups can achieve higher statistical power without needing millions of users. At Statsig, we've seen how focusing on significant changes can help teams get valuable insights even with smaller sample sizes. Understanding statistical power is a game-changer for designing effective tests. A high statistical power (we're talking 80% or more) means your test is primed to detect significant changes when they're there. This sets you up to make data-driven decisions based on strong evidence—leading to better optimizations and, ultimately, improved business outcomes. The pitfalls of small sample sizes Let's talk about why small sample sizes can be a big problem in A/B testing. When your sample is too tiny, your test results can become unreliable, leading you astray in your decision-making. You might fail to detect significant differences between your variants, resulting in Type II errors. In other words, even if there's a real effect, your test might not have the muscle to spot it. Another headache with small samples is the risk of high p-values making true effects look insignificant. P-values help you determine whether the differences you're seeing are real or just due to chance. With a small sample, random fluctuations can skew these values, causing you to miss out on genuine insights. Relying on results from underpowered tests isn't just risky—it can be costly. You might ignore a valuable change because your test didn't pick up its significance, or worse, you might implement a change based on a fluke result. That means wasted time, effort, and resources on something that doesn't actually enhance your product or user experience. So how do you dodge these pitfalls? Make sure your A/B tests have adequate sample sizes. While some folks think you always need massive sample sizes, it's really about having enough participants to detect the effect size you're after with sufficient statistical power. By considering factors like your baseline conversion rates, minimum detectable effect, and desired significance levels, you can figure out the sweet spot for your sample size and make confident, data-driven decisions. Advantages of larger sample sizes Bigger sample sizes have some serious perks in A/B testing. They reduce the impact of random variation, leading to more accurate results. With more participants, you're more likely to detect true differences between variants—even those that are relatively small. That's because larger samples boost your statistical power, making it easier to spot genuine effects amidst the noise. One of the key benefits here is increased statistical power. With more users in your test, you're better positioned to uncover meaningful differences that might slip under the radar with a smaller group. This means you can be more confident in your findings and make solid, data-driven decisions. Another advantage is the ability to detect smaller effect sizes that can still make a significant impact on your business. While chasing big wins is exciting, those incremental improvements add up. By testing with a larger sample size, you can identify these subtle yet valuable opportunities for optimization. But remember, the ideal sample size depends on various factors like your baseline conversion rate, minimum detectable effect (MDE), and desired significance and power levels. Balancing these elements is crucial for designing effective A/B tests that give you actionable insights. Tools like sample size calculators can help you figure out the right number of participants for your specific goals. Balancing sample size: avoiding overly large samples We've talked about the benefits of larger sample sizes, but here's the twist: bigger isn't always better. While having more data sounds great, excessively large samples can make trivial differences appear statistically significant. This can lead to overpowered tests, pushing you to make changes that don't really move the needle. Overly large samples can magnify minor variations, making them seem like meaningful insights. Sure, they might be statistically significant, but do they matter in the real world? Implementing changes based on these tiny differences might not justify the cost or effort. That's why balancing statistical and practical significance is so important in effective A/B testing. So how do you find that sweet spot? Consider factors like your baseline conversion rate, minimum detectable effect (MDE), and desired statistical power. This way, your test is sensitive enough to catch meaningful changes without spending unnecessary resources on negligible differences. Remember, you don't always need massive sample sizes to run effective A/B tests. By focusing on substantial improvements and calculating the optimal sample size, even smaller companies can leverage A/B testing to make smart, data-driven decisions. At Statsig, we help teams of all sizes optimize their experiments to get the most out of their data. Closing thoughts Finding the right sample size is a balancing act that's crucial for successful A/B testing. Go too small, and you risk unreliable results; go too big, and you might chase insignificant differences. By understanding how sample size affects your tests and focusing on meaningful improvements, you can make smarter, data-driven decisions. If you're looking to dive deeper into designing effective A/B tests, check out Statsig's resources on determining sample size and understanding statistical power. We're here to help you get the most out of your experiments, no matter the size of your team or user base. Hope you found this helpful! Permalink: Recent Posts ENGINEERING ##### How we created count distinct in Statsig Cloud Aamodit Acharya Thu Aug 28 2025 How we built a fast, reliable way to measure distinct counts in Statsig Cloud—powering metrics across Experiments and Pulse. ##### Sink, swim, or scale: What startups teach us about launching AI Alexey Komissarouk, Yuzheng Sun, PhD Fri Aug 08 2025 Some startups ride the AI hyper-growth wave, while others sink under the swell. Here are four patterns and playbooks for growth challenges unique to early-stage launches. ENGINEERING ##### Optimizing cloud compute costs with GKE and compute classes Pablo Beltran Fri Jul 25 2025 How we automated spot node selection with GKE compute classes to prevent expensive drift and cut our cloud costs by 50%. ##### How Statsig lets you ship, measure, and optimize AI-generated code Sid Kumar, Brock Lumbard Thu Jul 10 2025 As more code is AI-generated, it becomes critical to have controls, experimentation, and analytics built into product development workflows. ##### Your users are your best benchmark: a guide to testing and optimizing AI products Skye Scofield Wed Jul 09 2025 Benchmarks mislead AI teams. Discover a proven framework that the world's best teams use to build safer, higher-impact AI products. ##### The more the merrier? 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2549	https://www.nature.com/articles/s41598-022-09346-y	Skip to main content Download PDF Article Open access Published: Nutrient intake differs among persons with celiac disease and gluten-related disorders in the United States Aynur Unalp-Arida1, Rui Liu2nAff4 & Constance E. Ruhl ORCID: orcid.org/0000-0002-8160-98593 Scientific Reports volume 12, Article number: 5566 (2022) Cite this article 11k Accesses 20 Citations 3 Altmetric Metrics details Abstract Persons with celiac disease (CD) may develop nutritional deficiencies, while individuals following a gluten-free diet (GFD) may lack essential nutrients. We examined nutrient intake from diet and supplements among persons with CD and GFD in the cross-sectional National Health and Nutrition Examination Survey, 2009–2014. Among 15,610 participants 20 years and older, we identified CD based on positive serology for immunoglobulin A against tissue transglutaminase, health care provider diagnosis, and adherence to a GFD. People without CD avoiding gluten (PWAG) adhered to a GFD without a diagnosis of CD. Two 24-h recalls assessed nutrient intake from diet and supplements. Compared to participants without CD or PWAG, persons with diagnosed CD had lower intake of total energy, carbohydrates, fat, and saturated and monounsaturated fatty acids. In contrast, persons with undiagnosed CD and positive serology had higher intake of those nutrients, sugar, and protein. Total carbohydrate and sugar intake was lower among PWAG. Persons with diagnosed CD had higher vitamin A and E intake, while those with undiagnosed CD had increased intake of calcium, phosphorus, magnesium, iron, zinc, copper, sodium, potassium, vitamin A, alpha-carotene, folic acid, and choline. Higher micronutrient intake with undiagnosed CD was observed more at high latitudes. PWAG had higher beta-carotene and lutein/zeaxanthin and lower folic acid intake. In the U.S. population over a 6-year period, total energy and macronutrient intake was decreased among persons with diagnosed CD, while intake of total energy, macronutrients, and multiple micronutrients was increased among persons with undiagnosed CD. Nutriomics studies of multiple analytes measured simultaneously across affected persons and populations are needed to inform screening for malabsorption and treatment strategies. Similar content being viewed by others Nutrient intakes in adult and pediatric coeliac disease patients on gluten-free diet: a systematic review and meta-analysis Article 01 March 2023 Gluten-free foods: a 'health halo' too far for oral health? Article 19 July 2022 Nutritional redundancy in the human diet and its application in phenotype association studies Article Open access 18 July 2023 Introduction Celiac disease (CD) is a chronic immune-mediated disease of the small intestine triggered by dietary gluten in genetically predisposed individuals1. In an earlier report using National Health and Nutrition Examination Survey (NHANES) 2009–2014 data, CD prevalence in the U.S. was estimated as 0.7% overall and was higher among non-Hispanic whites (1.0%) than among other race-ethnicities (0.2%)2."). The majority of celiac cases were undiagnosed in the NHANES population3, 1922–1932 (2017)."),4, 1538–1544 (2012)."). In addition, prevalence of people without CD avoiding gluten (PWAG) was 1.1% overall and did not differ by race-ethnicity3, 1922–1932 (2017)."). Clinical presentation of CD and gluten-related disorders varies across individuals and by age5, 86–93 (2016)."). Currently, the only CD treatment is lifelong strict adherence to a gluten-free diet (GFD)6, 797–809 (2012)."). Previous studies suggest that because of the restrictive diet, celiac patients may experience imbalance in nutritional intake and consume less of certain macro- and micro-nutrients (fiber, thiamin, folate, calcium, magnesium), but more of others (fat, zinc, potassium, vitamin K)5, 86–93 (2016)."),7, 349–358 (2013)."). Therefore, assessment of changes in nutritional intake and deficiencies in celiac patients is critical in preventing complications from nutrient malabsorption and in improving quality of life8, 4285–4292 (2015)."). Reduced circulating blood concentrations of six micronutrients (iron, folate, vitamin B-12, vitamin D, zinc, magnesium) are common in untreated CD9. In a recent report, newly diagnosed CD patients in the U.S. were more likely to have serum deficiencies of zinc, albumin, copper, vitamin B-12, and folate compared with population controls10. Studies conducted in various countries assessing dietary habits of adult celiac patients reported reduced intake of certain vitamins and minerals, including folate7,11, potassium7, magnesium7,12, calcium7, zinc7, phosphate7,11, fiber11, niacin11, iodine, folic acid12, iron12, zinc11, and vitamins C7, D, B-112, B-212, B-612, and B-1211, compared to age-matched study- and country-specific populations. Similar reduced micronutrient intake was reported in adult female celiac patients in Slovenia, compared to the general Central European population5, Although comparison groups in these studies were country-specific population controls, celiac patients were recruited using convenience sampling. Moreover, we are unaware of other studies of intake of a wide spectrum of dietary components in people with diagnosed and undiagnosed CD compared with PWAG and controls in a nationally representative population. The NHANES 2009–2014 measured CD serology coupled with medical condition and GFD questions. Leveraging 6 years of serology data integrated with nutrient information from food and supplements, serum nutrient concentrations, and other health measures available in NHANES, we examined nutrient intake in participants with diagnosed and undiagnosed CD and PWAG compared with nationally representative adults without these conditions. Methods Study population The NHANES is a series of nationally representative, cross-sectional surveys of the U.S. population with oversampling of non-Hispanic blacks, Hispanics, Asians (2011–2014), low income whites, and persons age 80 years or older that are conducted by the National Center for Health Statistics (NCHS) of the Centers for Disease Control and Prevention in 2-year cycles13. The survey includes in-home interviews, physical examinations, and laboratory data collected using a complex multistage, stratified, clustered probability sampling design. We analyzed survey data collected from 2009 through 2014, the years during which gluten-related disorders were assessed by interview and serology. All interviews and examinations were performed in accordance with relevant guidelines and regulations. Definitions of CD and gluten-avoidance without CD NHANES 2009–2014 participants were asked the following interview questions: (1) Has a doctor or other health professional ever told you that you have celiac disease, also called sprue? and (2) Are you on a gluten-free diet?14 Serum specimens were shipped to the Celiac Disease Research Laboratory at Mayo Clinic, Rochester, MN, for serological testing. Serum was tested for tissue transglutaminase immunoglobulin A (tTG IgA) as a screening test (sensitivity, ~ 98%) with an enzyme-linked immunosorbent assay that uses human recombinant tTG (Inova Diagnostics, San Diego, CA, USA), and results were categorized as positive (> 10 U/ml), weakly positive (4–10 U/ml), or negative (< 4.0 U/ml)4, 1538–1544 (2012)."),15. Persons were identified as having diagnosed CD if they had a clinical (provider) diagnosis and either positive (or weakly positive) serology or adherence to GFD. Similarly, persons with undiagnosed CD had positive (or weakly positively) serology without a clinical diagnosis. PWAG was defined as adherence to GFD without a diagnosis of CD and with negative serology. Persons without CD and not on a GFD are referred to herein as ‘controls’. These case definitions are diagramed in Fig. 1 for reference. In contrast to previous reports, we included all NHANES 2009–2014 participants with positive tTG IgA alone, rather than serial test positivity to tTG IgA and endomysial antibody IgA, to increase the number of cases because of the low prevalence of CD in the general population2."),3, 1922–1932 (2017)."). Nutrient intake estimates Dietary intakes were obtained from two 24-h dietary recall interviews in which respondents reported all foods and beverages consumed during the previous 24 h (midnight to midnight). The first dietary recall interview was collected in person during the Mobile Examination Center visit and the second interview was collected by telephone 3–10 days later. Detailed descriptions of the dietary interview methods and protocol are fully documented16. Daily nutrient and food energy intakes were estimated using the respective Food and Nutrient Database for Dietary Studies for participants in each NHANES cycle17. Supplement use Information on dietary supplement use was collected by both a 30-day frequency questionnaire and two 24-h dietary recall interviews. For this analysis, we used supplement data from the 24-h recalls. Participants were queried about the 24-h period prior to the interview (midnight to midnight). Persons indicating using prescription or nonprescription dietary supplements as well as non-prescription antacids that contain calcium and magnesium were asked to show the interviewer the bottles of each supplement product taken. Quantity and duration of use were further queried during the interview. Daily nutrient intakes from dietary supplements and antacids were estimated using the NHANES Dietary Supplement Database. More information is provided in NHANES documentation18. Covariates Data were collected on demographic and clinical characteristics and examined in relation to gluten-related disorders19,20: age (years), sex, race-ethnicity (non-Hispanic white, non-Hispanic black, Hispanic, other), education, income, and BMI. The highest grade of school completed was reported and categorized as less than high school graduation, a high school degree, or education beyond high school. Income was measured by the poverty income ratio (ratio of family income to poverty threshold) and categorized as tertiles (< 1.7, 1.7–<4.0, ≥ 4.0). Poverty income ratios of 1.7 and 4.0 represent family incomes of 1.7 times and 4 times, respectively, the poverty threshold for a given family size and year. BMI was categorized in kg/m2 as normal weight (< 25), overweight (25–< 30), or obese (≥ 30). Prevalence of CD and gluten-related disorders was previously found to vary by latitude in the United States; therefore, latitude was also included as a covariate to control for potential regional differences in nutrient intake3, 1922–1932 (2017)."). Latitude (°North) was geocoded by NCHS from the participant’s residential address and categorized as < 35, 35–< 40, and ≥ 40 based on earlier findings21. Geographic data were used through the NCHS Research Data Center22. . Accessed March 2022."). Serum was tested for hemoglobin, standard biochemistry, lipid profile, and selected vitamins and minerals as previously described23. Analytic sample Of 25,426 NHANES 2009–2014 participants age 20 years and older, 16,966 (67%) attended a study visit at a mobile examination center and were included. A total of 15,610 (92%) participants further completed at least one of two 24-h dietary recall interviews and 13,825 participants (81% of those examined) provided two reliable dietary recalls. We included all individuals who completed at least one reliable 24-h dietary recall in analyses (Fig. 1). Persons with missing data on CD and PWAG status were excluded. Statistical analysis Participant characteristics according to gluten-related disorders were compared using a chi-square (χ2) test and linear regression. Unadjusted means and standard errors of serum biochemical and nutritional markers were calculated according to CD and PWAG status using linear regression. We estimated the distribution of usual nutrient intakes using methods developed by the National Cancer Institute (NCI)24. The NCI method is useful to estimate the within- and between-person variances and correct for the high within-person day-to-day variation commonly observed in 24-h recalls. A two-step process is involved in the NCI method. The first step uses the MIXTRAN macro to fit a nonlinear mixed effects model using the SAS NLMIXED procedure to obtain parameter estimates of mean usual intake. The data on amount were transformed to approximate normality using Box–Cox transformation. In the second step of the NCI method, the DISTRIB macro uses parameter estimates from MIXTRAN and a Monte Carlo method to estimate the distribution of usual intake of a nutrient24. To account for the complex survey sample design, variance estimation was carried out via the Balanced Repeated Replication technique with Fay’s modification25."). Because intake of micronutrients can come from both diet and supplements, these two amounts were first summed for each participant recall day before applying the NCI method. Mean nutrient intakes were estimated unadjusted, adjusted for age, sex, and race-ethnicity, and additionally for education, income, and BMI. The NHANES-provided day one dietary recall weights were used in all nutrient intake analyses. The relationship of nutrients with gluten-related disorders was further examined stratified by latitude using linear regression analysis (SUDAAN PROC REGRESS) to calculate (least squares) mean estimates adjusted for age, sex, and race-ethnicity. For these analyses, within person mean (2-day average) from the two 24-h food and supplement recalls was utilized because it was not feasible to use the NCI methods through the NCHS Research Data Center. Multivariable-adjusted analyses excluded persons with missing values for any factor included in the model. P-values were two-sided, and a P-value of < 0.05 was considered to indicate statistical significance. All analyses utilized sample weights that accounted for unequal selection probabilities and nonresponse. Variance calculations for non-nutrient factors accounted for the design effects of the survey using Taylor series linearization26. SAS 9.4 (SAS Institute, Cary, NC, USA) and SUDAAN 11 (RTI, Research Triangle Park, NC, USA) were used for all analyses. Ethics approval and consent to participate The NCHS Research Ethics Review Board approved the survey, and all participants provided written informed consent. Results Study population characteristics Among 16,966 participants 20 years and older meeting eligibility criteria, 28 (0.2%) had diagnosed CD, 85 (0.5%) had undiagnosed CD, 184 (1.1%) were PWAG, and 15,528 (91.5%) did not have either condition and comprised the control group. We were unable to categorize 1141 (7%) participants with missing serology data. Sociodemographic characteristics by case status are summarized in Table 1. Compared to persons without CD and GFD (control group), those with diagnosed CD were older and more likely to be female and non-Hispanic white, while those with undiagnosed CD were younger, more likely to be non-Hispanic white, and did not differ by sex suggesting a differential health care seeking behavior in women or more symptomatic disease. PWAG were more likely to be female and did not differ by age or race-ethnicity. Persons with CD or PWAG had more education and income, and were less likely to be overweight. Estimated usual nutrient intake Of the survey participants included, 15,610 (92%) completed one dietary recall and 13,825 (81%) completed two recalls. Table 2 shows age-, sex- and race-ethnicity-adjusted estimated usual mean intakes of total energy and macronutrients derived from food sources by CD and PWAG status. Compared to persons without CD and GFD (control group), those with diagnosed CD had lower mean intake of total energy, total carbohydrates, total fat and saturated and monounsaturated fatty acids. In contrast, persons with undiagnosed CD had a higher mean intake of total energy, carbohydrates, sugar, protein, total fat, and saturated and monounsaturated fatty acids. Despite significantly increased macronutrient intake, persons with undiagnosed CD were less likely to be obese compared with controls. Mean total carbohydrates and sugar were lower among PWAG compared with controls. The results were similar with additional adjustment for education, income, BMI, and latitude (data not shown). We further examined macronutrient intake among persons with diagnosed and undiagnosed CD and PWAG in comparisons by latitude (Table 3). Although the number of cases is small, for macronutrients adjusted for age, sex and race-ethnicity, the lower mean intake of total energy, total carbohydrates, total fat, and saturated and monounsaturated fatty acids observed among persons with diagnosed CD was found predominantly at a latitude of less than 35°N. In contrast, the higher mean intake of total energy, total carbohydrates, sugar, protein, total fat, and saturated and monounsaturated fatty acids found among persons with undiagnosed CD tended to be observed regardless of latitude, although most differences did not reach statistical significance due to smaller sample sizes. Similarly, among PWAG, lower mean total carbohydrate and sugar intake were seen regardless of latitude, although most differences were not statistically significant. For micronutrients, unadjusted intake derived from food sources, and from food and supplements combined, is shown by disease status in Supplementary Table S1. The proportion of total intake from supplements varied by micronutrient. Table 4 shows age-, sex- and race-ethnicity-adjusted estimated usual intake of total micronutrients from food and supplements by CD and PWAG status. Persons with diagnosed CD had higher mean intake of vitamin A and vitamin E compared to persons without CD and GFD (control group). Persons with undiagnosed CD had higher mean intakes of calcium, phosphorus, magnesium, iron, zinc, copper, sodium, potassium, vitamin A, alpha-carotene, folic acid, folate and choline compared with controls. PWAG group had a higher mean intake of beta-carotene and lutein and zeaxanthin, and a lower mean intake of folic acid compared to persons without CD and GFD (control group). The results were similar with additional adjustment for education, income, BMI, and latitude (data not shown). As for macronutrients, we further examined total micronutrient intake among persons with diagnosed and undiagnosed CD and PWAG in comparisons by latitude (Table 5). For micronutrients adjusted for age, sex and race-ethnicity, the higher mean intake of calcium, phosphorus, magnesium, iron, zinc, copper, sodium, potassium, vitamin A, alpha-carotene, folic acid, folate and choline among participants with undiagnosed CD compared with controls tended to be observed regardless of latitude with the strongest relationships found at a latitude of greater than or equal to 40°N. The lower mean intake of folic acid among PWAG was also found primarily at the highest latitude. Clinical laboratory and serum nutrition values Lastly, we compared the clinical laboratory characteristics and serum concentrations of nutrition-related factors of participants with CD or PWAG to those of persons without these conditions (Table 6). Participants with diagnosed CD had lower unadjusted mean activities of alanine aminotransferase, aspartate aminotransferase, hemoglobin, creatine phosphokinase, gamma glutamyl transferase, lactate dehydrogenase, and selenium, and higher mean concentrations of HDL cholesterol. Among adults with undiagnosed CD, we observed lower mean gamma glutamyl transferase and folate concentrations. Compared with controls, PWAG had higher mean HDL cholesterol, potassium, and vitamin D concentrations. Discussion In the U.S. population, persons with undiagnosed CD had higher intake of multiple macro- and micronutrients, including total energy, carbohydrates, sugar, protein, fat, saturated and monounsaturated fatty acids, calcium, phosphorus, magnesium, iron, zinc, copper, sodium, potassium, vitamin A, alpha-carotene, folic acid, folate and choline with no evident increased obesity. Higher nutrient intake among persons with undiagnosed CD suggests compensation for potential malabsorption. In contrast, persons with diagnosed CD had lower intake of total energy, carbohydrates, fat, and saturated and monounsaturated fatty acids, and higher intake of vitamin A and vitamin E compared with controls. This could be due to imbalances in nutritional intake resulting from GFD adherence, though the small number of participants with diagnosed CD requires caution in interpreting results. Persons with diagnosed CD had lower BMI compared with controls. With healing of the intestinal mucosa on a GFD, nutrient absorption would be expected to increase and added calories could lead to weight gain. However, we did not have information on length of time on a GFD or diet adherence. There have been concerns about potential for weight gain on a GFD due to nutritional content, but that is not suggested by our findings27,28. PWAG had lower intake of carbohydrates, sugar, and folic acid, and higher intake of beta-carotene and lutein/zeaxanthin. Among available serum nutritional measures, persons with diagnosed CD had decreased hemoglobin and selenium concentrations, persons with undiagnosed CD had lower folate concentrations despite increased dietary intake, and PWAG had increased vitamin D concentrations compared with controls. Some serum markers were not measured on the entire NHANES sample, which could have limited ability to detect differences from controls (Table 6). We are unaware of other studies of a wide spectrum of macro- and micronutrients in people with diagnosed and undiagnosed CD and PWAG compared with controls in a nationally representative population. Previously, we found a lower prevalence of CD and gluten-related disorders in persons living in southern compared with northern latitudes of the United States3. In the present analysis, we examined nutrient intake by latitude of residence. Lower intake of total energy and macronutrients with diagnosed CD was found predominantly at low latitudes. Higher micronutrient intake with undiagnosed CD was observed more at high latitudes. However, these results should be interpreted with caution because of small numbers of participants at lower latitudes particularly among persons with diagnosed CD. Macro- and micronutrient deficiencies are frequently found in newly diagnosed and untreated CD, including deficiencies of iron, vitamin D, calcium, vitamin B-12, folate, zinc and vitamin B-629. This could be due to malabsorption caused by underlying etiologies in affected persons and may lead to compensatory increase in macro- and micronutrient intake possibly accounting for higher intake of multiple nutrients among persons with undiagnosed CD in the current analysis. Risk of nutritional deficiency in CD is compounded by nutritional inadequacy of the traditional GFD in which removal of gluten-containing grains can diminish nutrient content, specifically of minerals, B vitamins, and fiber29. Because the most popular gluten-free raw materials are mineral poor, deficiencies of calcium, iron, magnesium, and zinc are among the most common30."). Gluten-free processed foods have become more widely available, leading to increased concerns regarding nutritional quality of a GFD31 measure for (intervention) measures: A guide to choosing the appropriate noninvasive clinical outcome measure for intervention studies in celiac disease. Gastroenterol. Clin. N. Am. 48(1), 85–99 (2019)."). The gluten-free market is growing faster than CD prevalence, indicating increasing use of the GFD in the absence of a CD diagnosis29. This should be a source of concern given the potential for nutritional imbalances with the GFD. In a recent report on non-celiac gluten sensitivity, 65 adults with self-reported sensitivity on GFD had a high proportion of energy from fat, while intakes of vitamin D, folic acid, calcium, iodine, and iron were lower than recommended32. PWAG may include persons with non-celiac gluten sensitivity, those with CD who instituted a GFD before diagnosis, and those without CD diagnosis who follow a GFD for symptomatic relief. We did not have information on reason for GFD, so were unable to distinguish between persons with non-celiac gluten sensitivity and those following a GFD for other reasons. A limitation of 24-h dietary recall data is potential for systematic and random measurement error; however, analytic methodology used in this analysis assumes that the 24-h recall is an unbiased instrument. As previously reported, a limitation of using NHANES to study CD is inability to confirm the diagnosis by small intestinal histology3. However, upper endoscopy with small intestinal biopsy would be difficult to perform on the general U.S. population. We defined CD based on a self-reported health care provider diagnosis and adherence to a GFD or on positive serology. In contrast to previous reports, we included all participants with positive tTG IgA alone, rather than serial test positivity, to increase case number because of low prevalence of CD in the general population2."),3, 1922–1932 (2017)."). This resulted in 34 additional cases with undiagnosed CD, but may have decreased specificity. However, in a sensitivity analysis with CD defined using serial serology, nutrient relationships were similar to the main analysis. In addition, accuracy of clinical diagnosis among participants with negative serology is unknown. Serology becomes unreliable among persons who adopt a GFD prior to a formal celiac disease diagnosis33, 491–492 (2021)."). Another limitation of the survey was the small number of cases with gluten-related disorders despite 6 years of NHANES testing. Furthermore, because of the large number of comparisons, some associations could be the result of chance; therefore, results should be interpreted with caution. However, NHANES is the first nationally representative survey to collect CD serology. The limitations are balanced by the benefits of a large, national, population-based sample, particularly avoidance of ascertainment bias found in clinical studies of selected patients and ability to generalize results to the U.S. population. Conclusions Among U.S. adults, total energy and macronutrient intake tended to be lower with diagnosed CD. In contrast, persons with undiagnosed CD and positive serology had higher intake of total energy, macronutrients, and multiple micronutrients, including calcium, phosphorus, magnesium, iron, zinc, copper, sodium, potassium, vitamin A, alpha-carotene, folic acid, and choline. PWAG had lower folic acid intake. Nutriomics studies of multiple analytes measured simultaneously across affected persons and populations are needed to inform screening for malabsorption and treatment strategies. Data availability The NHANES datasets analyzed during the current study are publicly available from the National Center for Health Statistics (NCHS) ( except for geographic data (latitude) that are restricted to use through the NCHS Research Data Center ( per NCHS, Centers for Disease Control and Prevention policy. Abbreviations CD: : Celiac disease GFD: : Gluten-free diet NCI: : National Cancer Institute NCHS: : National Center for Health Statistics NHANES: : National Health and Nutrition Examination Survey PWAG: : People without CD avoiding gluten tTG IgA: : Tissue transglutaminase immunoglobulin A References Ludvigsson, J. F. et al. The Oslo definitions for coeliac disease and related terms. Gut 62(1), 43–52 (2013). Article Google Scholar 2. Choung, R. S. et al. Less hidden celiac disease but increased gluten avoidance without a diagnosis in the United States: Findings from the National Health and Nutrition Examination surveys from 2009 to 2014. Mayo Clin Proc. (2016). 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Article Google Scholar Download references Acknowledgements The authors thank the staff of the Celiac Disease Research Laboratory and the Immunodermatology Laboratory at Mayo Clinic for conducting the serological testing. The National Center for Health Statistics was the source of the National Health and Nutrition Examination Survey 2009–2014 geocoding data. All analyses, interpretations, and conclusions are those of the authors and not NCHS. The authors thank Dieudonne Nahigombeye for assistance in using the NCHS Research Data Center. Funding The work was supported by a contract from the National Institute of Diabetes and Digestive and Kidney Diseases (HHSN275201700074U). Author information Author notes Rui Liu Present address: Sacred Heart University, 5151 Park Avenue, Fairfield, CT, 06825, USA Authors and Affiliations National Institute of Diabetes and Digestive and Kidney Diseases, Democracy 2, Room 6009, 6707 Democracy Boulevard, Bethesda, MD, 20892-5458, USA Aynur Unalp-Arida 2. Social and Scientific Systems, Inc, Silver Spring, MD, USA Rui Liu 3. Social and Scientific Systems, Inc., A DLH Holdings Corp Company, 8757 Georgia Ave., Silver Spring, MD, 20910, USA Constance E. Ruhl Authors Aynur Unalp-Arida View author publications Search author on:PubMed Google Scholar 2. Rui Liu View author publications Search author on:PubMed Google Scholar 3. Constance E. Ruhl View author publications Search author on:PubMed Google Scholar Contributions A.U.A., R.L., and C.E.R. designed research; R.L. and C.E.R. performed statistical analysis; A.U.A., R.L., and C.E.R. wrote the paper; C.E.R. had primary responsibility for final content. All authors read and approved the final manuscript. Corresponding author Correspondence to Constance E. Ruhl. Ethics declarations Competing interests The authors declare no competing interests. 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To view a copy of this licence, visit Reprints and permissions About this article Cite this article Unalp-Arida, A., Liu, R. & Ruhl, C.E. Nutrient intake differs among persons with celiac disease and gluten-related disorders in the United States. Sci Rep 12, 5566 (2022). Download citation Received: Accepted: Published: DOI: Share this article Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative Subjects Gastroenterology Gastrointestinal diseases This article is cited by Characteristics of Hospitalized Patients With and Without Celiac Disease on a Gluten-Free Diet Rachel Eklund John W. Blackett Benjamin Lebwohl Digestive Diseases and Sciences (2024) ### Guidelines for best practices in monitoring established coeliac disease in adult patients Luca Elli Daniel Leffler David S. Sanders Nature Reviews Gastroenterology & Hepatology (2024) ### Application of Soy, Corn, and Bean By-products in the Gluten-free Baking Process: A Review Mariana Buranelo Egea Tainara Leal De Sousa Ailton Cesar Lemes Food and Bioprocess Technology (2023)
2550	https://atcm.mathandtech.org/EP2016/contributed/4052016_21160.pdf	Homothetic centers of three circles and their three-dimensional applications Yoichi Maeda maeda@tokai-u.jp Department of Mathematics Tokai University Japan Abstract In this paper, we recall the famous Monge’s theorem of three circles. There are several proofs for the theorem. One of the proofs is that of using three similar right cones. Inspired by the proof, we propose a three-dimensional problem of the same angles of elevation: For three similar right cones on the ground, find the places from where three angles of elevation are equal to each other. There are at most two places. With dynamic geometry software, we can simply construct the solutions. In addition, the relation between two solutions is cleared. 1. Introduction Figure 1.1 shows the famous Monge’s theorem of three circles and its three-dimensional intuitive proof: Theorem 1.1 (Monge’s theorem of three circles) For any three circles in a plane, none of which is inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. Figure 1.1 The Monge’s theorem (left) and its three-dimensional proof (right). Regarding three circles on the plane as the top view of three similar right cones as in Figure 1.1(right), we can easily know the external homothetic centers by connecting the vertices of the cones with lines. These lines are on the plane passing through three vertices of cones, hence three external homothetic centers lie on the intersection of the plane and the base plane. This is the reason why three external homothetic centers are collinear. Here is a question. When you are on the line passing through three external homothetic centers, how do the three vertices of the cones look like from your eyes? You must answer that the three vertices look like collinear. Inspired by this problem, we will propose the next problem: For three similar right cones on the ground, find the places from where three angles of elevation are equal to each other. There are infinitely many places from where three vertices look like collinear, however, there are at most two places from where three angles of elevation are equal to each other. In section 2, we review an easy construction of homothetic centers of two circles. Using the homothetic centers, we will construct the points from where the angles of elevation are equal to each other for two cones in section 3. A relation between circle of Apollonius and angle of elevation is cleared. Finally, we construct the solution of the same elevation problem of three cones in section 4. The relation between two solutions is revealed. All figures in this paper are drawn with dynamic geometry software; Cabri II plus and Cabri 3D. 2. Homothetic centers of two circles In this section, we review the simplest construction of external homothetic center and internal homothetic center of two circles shown as Figure 2.1. Construction 2.1 (Homothetic centers of two circles) 0. Input: ܥ1 and ܥ2: two circles centered at ܱ1 and ܱ2, respectively. 1. ܮ1: line passing through ܱ1 and ܱ2. 2. ܮ2: any line passing through ܱ1. 4. ܮ3: line parallel to ܮ2 passing through ܱ2. 5. ܲ1: one of intersections of ܥ1 and ܮ2. 6. ܲ2 and ܲ3: intersections of ܥ2 and ܮ3 labeled as in Figure 2.1. 7. ܮ4: line passing through ܲ1 and ܲ2. 8. ܮ5: line passing through ܲ1 and ܲ3. 9. Output: ܧ is the external homothetic center which is the intersection of ܮ4 and ܮ1. ܫ is the internal homothetic center which is the intersection of ܮ5 and ܮ1. Figure 2.1 Construction of the homothetic centers of two circles. With this construction, we can easily confirm the Monge’s theorem of three circles shown as in Figure 1.1. Now, let us regard Figure 2.1 as the top view of two similar right cones on the ground. If you are at point ܧ, then two cones look like completely overlapped. If you are at point ܫ, then the angle of elevation of one cone is equal to that of another cone. In fact, the set of points from where the angles of elevation are equal is the circle with diameter ܧܫ. 3. Angle of elevation problem of two cones In this section, we will investigate the angle of elevation problem of two cones. To do this, we have to go back to the famous Apollonius theorem shown as in Figure 3.1: Theorem 3.1 (The circle of Apollonius ( p.28) Let ܣ, ܤ be two fixed points on a line. The locus of a point ܲ which moves so that the ratio of its distances from ܣ and ܤ is constant is the circle with diameter ܥ and ܦ where ܥ is the internal division and ܦ is the external division with the ratio. Figure 3.1 Circle of Apollonius. Using Theorem 3.1, we can solve the next problem: For two circles on a plane, find the points from where the viewing angle of one circle is the same as that of another circle. Here viewing angle of the circle ܥ from a point ܲ is defined as the angle subtended by two tangent lines from ܲ to the circle ܥ shown as in Figure 3.2. Figure 3.2 Circle of the same viewing angle. Proposition 3.1 (Viewing angle of two circles) Let ܥ1 and ܥ2 be two circles in a plane, none of which is inside of the other. Let ܧ and ܫ be the external and internal homothetic centers of the two circles, respectively. Then for any point ܲ on the circle with diameter ܧܫ, the viewing angle of ܥ1 from ܲ is the same as the viewing angle of ܥ2 from ܲ. Proof. Let ܲሺܶ1ሻ and ܲሺܶ2ሻ be tangent lines from ܶ to ܥ1 and ܥ2, respectively as in Figure 3.2. It is enough to show that ∠ሺܶ1ሻܲሺܱ1ሻ ൌ ∠ሺܶ2ሻܲሺܱ2ሻ. Let ݎ1 and ݎ2 be the radii of circles ܥ1 and ܥ2, respectively. Then ܫ is the internal division of ܱ1 and ܱ2 such that ሺܱ1ሻܫ∶ ሺܱ2ሻܫ ൌ ݎ1 ∶ ݎ2. And ܧ is the external division ܱ1 and ܱ2 such that ሺܱ1ሻܧ∶ ሺܱ2ሻܧ ൌ ݎ1 ∶ ݎ2. Then, the circle with diameter ܧܫ is the Apollonius circle of ܱ1 and ܱ2 with the ratio ݎ1 ∶ ݎ2. Then, ܲሺܱ1ሻ∶ ܲሺܱ2ሻ ൌ ݎ1 ∶ ݎ2. Therefore, triangles ⊿ሺܶ1ሻܲሺܱ1ሻ and ⊿ሺܶ2ሻܲሺܱ2ሻ are similar to each other. In particular, ∠ሺܶ1ሻܲሺܱ1ሻ ൌ ∠ሺܶ2ሻܲሺܱ2ሻ. ∎ Regarding Figure 3.2 as the top view of two similar cones on the ground, we directly clear the angle of elevation of two cones, since the viewing angle is directly proportional to the angle of elevation. Figure 3.3 shows the places from where the angle of elevation of one cone is the same as that of another cone. Figure 3.3 Angle of elevation problem of two cones. 4. Angle of elevation problem of three cones With Proposition 3.1, we can find out the solution of angle of elevation problem of three cones. The solution is the intersections of three circles shown as in Figure 4.1 (left). Construction 4.1 (Angle of elevation problem of three cones) 0. Input: three cones with vertices ܣ, ܤ, and ܥ. 1. – ܤ, െܥ: reflections of ܤ and ܥ in the base plane α, respectively. 2. ܥ1: circle with diameter ܣܤ ∩ߙ and ܣሺെܤሻ ∩ߙ. 3. ܥ2: circle with diameter ܣܥ ∩ߙ and ܣሺെܥሻ ∩ߙ. 4. ܥ3: circle with diameter ܤܥ ∩ߙ and ሺെܤሻܥ ∩ߙ. 5. Output: ܵ1, ܵ2: intersections of ܥ1, ܥ2, and ܥ3. In fact, depending the configuration of the three cones, there are 2, 1, or 0 solutions as in Figure 4.1. If there is a solution, we can easily confirm that the angles of elevation are equal to each other with dynamic geometry software. To see this, construct an upside-down cone with vertex at the solution, axis perpendicular to the base plane, and passing through vertex ܣ shown as in Figure 4.1 (left). Then, we can see other vertices ܤ and ܥ are also on this upside-down cone. In this way, we can solve the angle of elevation problem of three cones. Figure 4.1 Angle of elevation problem of three cones: two solutions (left) and no solution (right). In the following argument, we assume that there are two solutions ܵ1 and ܵ2 shown as in Figure 4.1 (left). Figure 4.2 Sphere passing through three vertices of cones and two solutions ܵ1 and ܵ2. Final study is to clear the relation between two solutions ܵ1 and ܵ2. Here, let us consider the sphere ܵ݌1 passing through three vertices of cones ܣ, ܤ, and ܥ perpendicular to the base plane ߙ. Figure 4.2 shows that one of solutions ܵ1 is the inversion of another solution ܵ2 with respect to sphere ܵ݌1. Let ܧݍ be the equator which is the intersection of sphere ܵ݌1 and the base plane ߙ. With dynamic geometry software, we can check that three circles ܥ1, ܥ2, and ܥ3 in Construction 4.1 intersect orthogonally with circle ܧݍ. This means that circles ܥ1, ܥ2, and ܥ3 are invariant under the inversion with respect to circle ܧݍ on the base plane ߙ. Hence, we can show that ܵ1 is the inversion of ܵ2 with respect to sphere ܵ݌1, if we prove that ܥ1 intersects orthogonally with ܧݍ, because with the same argument, both ܥ2 and ܥ3 also intersect with ܧݍ orthogonally. Therefore, in the following discussion, let us focus on the relation between circles ܧݍ and ܥ1. Proposition 4.1 (Orthogonal intersection of ܧݍ and ܥ1 (Figure 4.3)) Let ܵܲ1 be the sphere passing through ܣ and ܤ perpendicular to the base plane ߙ. Let ܧݍ is the intersection of ܵܲ1 and the base plane ߙ. Let – ܣ be the reflection of ܣ in the base plane ߙ. Let ܧ be the intersection of line ܣܤ and the base plane ߙ. Let ܫ be the intersection of line ሺെܣሻܤ and the base plane ߙ. Let ܥ1 be the circle on the base plane ߙ with diameter ܧܫ. Then, ܧݍ and ܥ1 intersect orthogonally. Figure 4.3 Orthogonal intersection of ܧݍ and ܥ1. Before we prove Proposition 4.1, we prepare the following proposition. Proposition 4.2 (inverse relation between ܧ and ܫ (Figure 4.4)) Let ܷ be the unit circle on a complex plane. Let ܣሺߙሻ and ܤሺߚሻ be two points on ܷ. Let െܣሺα ഥሻ be the conjugate of ܣሺߙሻ. Let ܧሺݖ݁ሻ be the intersection of ܣܤ and the real axis. Let ܫሺݖ݅ሻ be the intersectin of ሺെܣሻܤ and the real axis. Then, ݖ݁∙ݖ݅ൌ1, that is, ܧ is the inversion of ܫ with respect to ܷ. Figure 4.4 Relation between ܧ and ܫ. Proof of Proposition 4.2. The equation of line ܣܤ is อ ݖ ݖ̅ 1 ߙ ߙ ത 1 ߚ ߚ̅ 1 อൌ0. Because ݖ݁ satisfies the equation and ൌݖ݁ ത ത ത , ݖ݁ൌ ఈఉ ഥିఈ ഥఉ ሺఈିఈ ഥሻିሺఉିఉ ഥሻ. In the similar way, the equation of line ሺെܣሻܤ is อ ݖ ݖ̅ 1 ߙ ത ߙ 1 ߚ ߚ̅ 1 อൌ0. Because ݖ݅ satisfies the equation and ൌݖଓ ഥ , ݖ݅ൌ ఈఉିఈ ഥఉ ഥ ሺఈିఈ ഥሻାሺఉିఉ ഥሻ. Then, ݖ݁∙ݖ݅ൌ ఈఉ ഥିఈ ഥఉ ሺఈିఈ ഥሻି൫ఉିఉ ഥ൯∙ ఈఉିఈ ഥఉ ഥ ሺఈିఈ ഥሻା൫ఉିఉ ഥ൯ൌ ఈమିఉమିఉ ഥమାఈ ഥమ ሺఈିఈ ഥሻమି൫ఉିఉ ഥ൯ మൌ1, where we use \|ߙ\| ൌ\|ߚ\| ൌ1. This completes the proof. ∎ Proof of Proposition 4.1. Let ߚ be the plane passing through ܣ, ܤ, and – ܣ. Let ܷ be the circle given as the intersection of ߚ and ܵ݌1 as in Figure 4.5. We have already seen that ܫ is the inverse of ܧ with respect to ܷ. Figure 4.5 Setting for the proof of Proposition 4.1. Let ܲ and ܳ be two intersections of ܷ and the base plane ߙ. Let ܵ݌2 be a sphere with diameter ܲܳ. Let ܰ be the top of sphere ܵ݌2. Now let us consider the stereographic projection ݂ ( p.93 , p.260, p.74) from the base plane ߙ to sphere ܵ݌2 with respect to ܰ shown as in Figure 4.6. Figure 4.6 Stereographic projection of ܥ1 and ܧݍ. Stereographic projection has two strong properties: conformal and circle-to-circle correspondence. Regarding ܲ and ܳ as the poles of sphere ܵ݌2, ݂ሺܧݍሻ is a longitude of sphere ܵ݌2, because ܲ and ܳ are on ܧݍ. On the other hand, ݂ሺܥ1ሻ is a latitude of sphere ܵ݌2, because ܫ is the inversion of ܧ with respect ܵ݌2 including ܷ. In general, latitude ݂ሺܥ1ሻ and longitude ݂ሺܧݍሻ intersect orthogonally, therefore, ܥ1 and ܧݍ which are inverse images by the stereographic projection, also intersect orthogonally. This completes the proof. ∎ In consequence, the solutions ܵ1 and ܵ2 are coincident, if and only if, ܵ1 is on the sphere passing through ܣ , ܤ, ܥ, െܣ, െܤ, and – ܥ. References Berger, M. (1987). Geometry I. Berlin Heidelberg, Germany: Springer-Verlag. Berger, M. (1987). Geometry II. Berlin Heidelberg, Germany: Springer-Verlag. Gutenmacher, V. and Vasilyev, N.B. (2004). Lines and Curves, Birkhauser, Boston, Inc. Jennings, G. (1994). Modern Geometry with Applications. Springer-Verlag New York, Inc.
2551	https://www.facebook.com/brilliantorg/posts/how-well-do-you-understand-logarithmstest-yourself-with-this-simple-algebra-prob/916540061706357/	Brilliant.org - How well do you understand logarithms?... Log In Brilliant.org's Post Brilliant.org August 30, 2014 · How well do you understand logarithms? Test yourself with this simple algebra problem: brilliant.org Understanding Logarithms Have you mastered the basics of algebra? Learn more All reactions: 254 109 comments 13 shares Like Comment Most relevant Pranay Krishna -1/2 11y 7 Ahmed Osama -1/2 11y 5 Caio It is a simple exponencial equation. No need log. 11y 3 Assem Ali -0.5 11y 2 Pedro Henrique C Navarro 4^x = 2^(-1), 2^2x = 2^(-1), 2x = -1, x= -1/2 11y 2 Manas K Pattnaik 4^x = 2^2x .... 0.5 = 1/2 = 2^-1.... 2x=-1 so x= -1/2 11y 2 Shubham Nipane WoW..!! Friends I Got 100 Rs Free Recharge Only in 5- 10 Minutes check www.Getfreetalktime.ml 11y Afnan Mahmood Take log of both sides Log 4^x = Log 0.5 x.log 4 = Log 0.5 x = Log 0.5 / Log 4 x = -1/2 Answer 11y Maaz Khan -1/2 11y Arpit Sharma -(1/2) 11y 2
2552	https://hootyshomeroom.com/teaching-decimals-to-5th-graders/	Hello there! Comparing and Ordering Decimals: Quick Tips for Teaching Decimals to 5th Graders (Without Losing Your Mind) Let’s be honest—teaching decimals to 5th graders can make even the best math teacher want to hide behind the whiteboard. One minute your students are cruising through place value, and the next they’re telling you that 0.12 is bigger than 0.9 because, “Twelve is more than nine.” Yikes. If you’re working on comparing and ordering decimals, here are a few simple (and teacher-tested) tips to help your students get it—without the groans, blank stares, or decimal drama. 1. Start with Place Value—Seriously Yes, we all say it. But before students can begin comparing and ordering decimals, they need to understand what they’re comparing. Break out the place value chart and go over tenths, hundredths, and thousandths. Have them read decimals out loud—”fifty-six hundredths” is way clearer than “point five six” when you’re trying to wrap your brain around size. And if you’ve got base-ten blocks or decimal grids, now’s the time to let those dusty manipulatives shine. Here’s how they help: Base-ten blocks: Use the flat to represent 1 whole, a rod for one-tenth, and a single cube for one-hundredth. Stack 10 rods side by side and show that they equal one whole. Then compare, for example, 0.3 (3 rods) and 0.06 (6 small cubes). Students see which amount is greater. Decimal grids: These are 10×10 grids where each square represents one-hundredth. Shade in 0.49 (49 squares) and then 0.54 (54 squares). Boom—instant visual comparison, and students get how even small differences matter when you’re working with decimals. You don’t need a full lesson—just five minutes of modeling can make a big difference. 2. Line Up Those Decimals A key strategy for comparing and ordering decimals is to line them up vertically by the decimal point. This helps students compare digits in the same place value column. Encourage students to add trailing zeros so the numbers have the same number of digits. For example: Now it’s easier to see that 0.590 is greater. 3. Use Number Lines (Your Secret Weapon) Number lines are basically cheat codes for visual learners. When students see where decimals fall, they’re more likely to understand which number is greater. Start with tenths, move to hundredths, and work your way up. You’ll be amazed how many lightbulbs go off with this one little tool. Bonus points if you can find a digital number line they can interact with! 4. Keep It Real (World) Decimals are everywhere—sports scores, measurements, time, and even snack sizes. The more your students can connect decimals to things they care about, the more likely they are to actually pay attention. Try something like this: Real-world examples not only help decimals click, but they also cut down on the classic “When are we ever gonna use this?” complaints. 5. Go Digital for Practice That Sticks If you want students to actually enjoy practicing this skill (or at least not complain about it), digital practice is your friend. My Comparing and Ordering Decimals to the Thousandths Place Google Slides activity is an easy way to reinforce the concept—without the copier jams or stacks of paper. It’s drag-and-drop, student-friendly, and can be used for: 👉 Check Comparing & Ordering Decimals Google Slides out on TPT Final Thoughts on Teaching Decimals to 5th Graders Teaching decimals to 5th graders doesn’t have to be a headache. Focus on place value, use visuals like number lines, keep it real with examples your students care about—and let digital tools do some of the heavy lifting. You’ve got this. With the right strategies, your students will catch on, and everyone will finally agree that 0.9 is greater than 0.12. Share This Post Hey there! Hi, I’m Deirdre. Thanks for dropping by. I love supporting 3rd, 4th, and 5th grade teachers with simple and engaging activities. Let me help you make teaching easier. Categories math digital games test prep folded notes seasonal Search Shop 4th Grade Decimals Review Trashketball Game Multiply and Divide Decimals Word Problems Trashketball Game 5th Grade Data Analysis Review Trashketball Game you might also like... 5 Fun Halloween Math Activities for 3rd Grade Keeping Students Focused on Math During October Chaos Digital + Printable Trashketball Template: Fun Review Games You Can Customize for Any Subject Teach Place Value to 100,000 with a Passport Project Your Students Will Love ©2024 Hooty's Homeroom. All Rights Reserved. Privacy Overview
2553	https://artofproblemsolving.com/wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality?srsltid=AfmBOorT1brMv1EDRMb_UzWrszlPj-pvLwXJ-s_wzuSw1ytwN5vn6zYh	Art of Problem Solving Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality The Root-Mean Power-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMP-AM-GM-HM) or Exponential Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (EM-AM-GM-HM) or Quadratic Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (QM-AM-GM-HM), is an inequality of the root-mean power, arithmetic mean, geometric mean, and harmonic mean of a set of positivereal numbers that says: , where , and is the . The geometric mean is the theoretical existence if the root mean power equals 0, which we couldn't calculate using radicals because the 0th root of any number is undefined when the number's absolute value is greater than or equal to 1. This creates the indeterminate form of . Then, we can say that the limit as x goes to 0 is the geometric mean of the numbers. The quadratic mean's root mean power is 2 and the arithmetic mean's root mean power is 1, as and the harmonic mean's root mean power is -1 as . Similarly, there is a root mean cube (or cubic mean), whose root mean power equals 3. When the root mean power approaches , the mean approaches the highest number. When the root mean power reaches , the mean approaches the lowest number. with equality if and only if . This inequality can be expanded to the power mean inequality, and is also known as the Mean Inequality Chain. As a consequence, we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality. This is extremely useful in problem-solving. The Root Mean Power of 2 is also known as the quadratic mean, and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality. Proof The inequality is a direct consequence of the Cauchy-Schwarz Inequality; Alternatively, the RMS-AM can be proved using Jensen's inequality: Suppose we let (We know that is convex because and therefore ). We have: Factoring out the yields: Taking the square root to both sides (remember that both are positive): The inequality is called the AM-GM inequality, and proofs can be found here. The inequality is a direct consequence of AM-GM; , so , so . Therefore, the original inequality is true. Geometric Proofs The inequality is clearly shown in this diagram for Desmos SlidersDesmos Equation NOTE: The Desmos equation will not show the line when the numbers are negative. (Note how the RMS is "sandwiched" between the minimum and the maximum) Retrieved from " Categories: Algebra Inequalities Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
2554	https://en.wikipedia.org/wiki/Magic_number_(physics)	Magic number (physics) - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Appearance move to sidebar hide Text Small Standard Large This page always uses small font size Width Standard Wide The content is as wide as possible for your browser window. Color (beta) Automatic Light Dark This page is always in light mode. Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk [x] Toggle the table of contents Contents move to sidebar hide (Top) 1 History and etymology 2 Doubly magic 3 Derivation 4 See also 5 References 6 External links Magic number (physics) [x] 34 languages Afrikaans العربية বাংলা Català Čeština Deutsch Español Esperanto فارسی Français 한국어 Հայերեն Hrvatski Bahasa Indonesia Italiano עברית Magyar Bahasa Melayu 日本語 Norsk bokmål Oʻzbekcha / ўзбекча پنجابی Plattdüütsch Română Русский Slovenčina Slovenščina Suomi Svenska தமிழ் Татарча / tatarça Українська اردو 中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Edit interlanguage links Print/export Download as PDF Printable version In other projects Wikidata item From Wikipedia, the free encyclopedia Number of protons or neutrons that make a nucleus particularly stable A graph of isotope stability, with some of the magic numbers In nuclear physics, a magic number is a number of nucleons (either protons or neutrons, separately) such that they are arranged into complete shells within the atomic nucleus. As a result, atomic nuclei with a "magic" number of protons or neutrons are much more stable than other nuclei. The seven most widely recognized magic numbers as of 2019 are 2, 8, 20, 28, 50, 82, and 126. For protons, this corresponds to the elements helium, oxygen, calcium, nickel, tin, lead, and the hypothetical unbihexium, although 126 is so far only known to be a magic number for neutrons. Atomic nuclei consisting of such a magic number of nucleons have a higher average binding energy per nucleon than one would expect based upon predictions such as the semi-empirical mass formula and are hence more stable against nuclear decay. The unusual stability of isotopes having magic numbers means that transuranium elements could theoretically be created with extremely large nuclei and yet not be subject to the extremely rapid radioactive decay normally associated with high atomic numbers. Large isotopes with magic numbers of nucleons are said to exist in an island of stability. Unlike the magic numbers 2–126, which are realized in spherical nuclei, theoretical calculations predict that nuclei in the island of stability are deformed. The difference between known binding energies of isotopes and the binding energy as predicted from the semi-empirical mass formula. Distinct sharp peaks in the contours appear only at magic numbers. Before this was realized, higher magic numbers, such as 184, 258, 350, and 462, were predicted based on simple calculations that assumed spherical shapes: these are generated by the formula 2((n 1)+(n 2)+(n 3)){\displaystyle 2({\tbinom {n}{1}}+{\tbinom {n}{2}}+{\tbinom {n}{3}})}(see Binomial coefficient). It is now believed that the sequence of spherical magic numbers cannot be extended in this way. Further predicted magic numbers are 114, 122, 124, and 164 for protons as well as 184, 196, 236, and 318 for neutrons. However, more modern calculations predict 228 and 308 for neutrons, along with 184 and 196. History and etymology [edit] Maria Goeppert Mayer While working on the Manhattan Project, the German physicist Maria Goeppert Mayer became interested in the properties of nuclear fission products, such as decay energies and half-lives. In 1948, she published a body of experimental evidence for the occurrence of closed nuclear shells for nuclei with 50 or 82 protons or 50, 82, and 126 neutrons. It had already been known that nuclei with 20 protons or neutrons were stable: that was evidenced by calculations by Hungarian-American physicist Eugene Wigner, one of her colleagues in the Manhattan Project. Two years later, in 1950, a new publication followed in which she attributed the shell closures at the magic numbers to spin-orbit coupling. According to Steven Moszkowski, a student of Goeppert Mayer, the term "magic number" was coined by Wigner: "Wigner too believed in the liquid drop model, but he recognized, from the work of Maria Mayer, the very strong evidence for the closed shells. It seemed a little like magic to him, and that is how the words 'Magic Numbers' were coined." These magic numbers were the bedrock of the nuclear shell model, which Mayer developed in the following years together with Hans Jensen and culminated in their shared 1963 Nobel Prize in Physics. Doubly magic [edit] Nuclei which have neutron numbers and proton (atomic) numbers both equal to one of the magic numbers are called "doubly magic", and are generally very stable against decay. The known doubly magic isotopes are helium-4, helium-10, oxygen-16, calcium-40, calcium-48, nickel-48, nickel-56, nickel-78, tin-100, tin-132, and lead-208. While only helium-4, oxygen-16, calcium-40, and lead-208 are completely stable, calcium-48 is extremely long-lived and therefore found naturally, disintegrating only by a very inefficient double beta minus decay process. Double beta decay in general is so rare that several nuclides exist which are predicted to decay by this mechanism but in which no such decay has yet been observed. Even in nuclides whose double beta decay has been confirmed through observations, half-lives usually exceed the age of the universe by orders of magnitude, and emitted beta or gamma radiation is for virtually all practical purposes irrelevant. On the other hand, helium-10 is extremely unstable, and has a half-life of just 260(40)yoctoseconds (2.6(4)×10−22 s). Doubly magic effects may allow the existence of stable isotopes which otherwise would not have been expected. An example is calcium-40, with 20 neutrons and 20 protons, which is the heaviest stable isotope made of the same number of protons and neutrons. Both calcium-48 and nickel-48 are doubly magic because calcium-48 has 20 protons and 28 neutrons while nickel-48 has 28 protons and 20 neutrons. Calcium-48 is very neutron-rich for such a relatively light element, but like calcium-40, it is stabilized by being doubly magic. As an exception, although oxygen-28 has 8 protons and 20 neutrons, it is unbound with respect to four-neutron decay and appears to lack closed neutron shells, so it is not regarded as doubly magic. Magic number shell effects are seen in ordinary abundances of elements: helium-4 is among the most abundant (and stable) nuclei in the universe and lead-208 is the heaviest stablenuclide (at least by known experimental observations). Alpha decay (the emission of a 4 He nucleus – also known as an alpha particle – by a heavy element undergoing radioactive decay) is common in part due to the extraordinary stability of helium-4, which makes this type of decay energetically favored in most heavy nuclei over neutron emission, proton emission or any other type of cluster decay. The stability of 4 He also leads to the absence of stable isobars of mass number 5 and 8; indeed, all nuclides of those mass numbers decay within fractions of a second to produce alpha particles. Magic effects can keep unstable nuclides from decaying as rapidly as would otherwise be expected. For example, the nuclides tin-100 and tin-132 are examples of doubly magic isotopes of tin that are unstable, and represent endpoints beyond which stability drops off rapidly. Nickel-48, discovered in 1999, is the most proton-rich doubly magic nuclide known. At the other extreme, nickel-78 is also doubly magic, with 28 protons and 50 neutrons, a ratio observed only in much heavier elements, apart from tritium with one proton and two neutrons (78 Ni: 28/50=0.56; 238 U: 92/146=0.63). In December 2006, hassium-270, with 108 protons and 162 neutrons, was discovered by an international team of scientists led by the Technical University of Munich, having a half-life of 9 seconds. Hassium-270 evidently forms part of an island of stability, and may even be doubly magic due to the deformed (American football- or rugby ball-like) shape of this nucleus. Although Z=92 and N=164 are not magic numbers, the undiscovered neutron-rich nucleus uranium-256 may be doubly magic and spherical due to the difference in size between low- and high-angular momentum orbitals, which alters the shape of the nuclear potential. Derivation [edit] Magic numbers are typically obtained by empirical studies; if the form of the nuclear potential is known, then the Schrödinger equation can be solved for the motion of nucleons and energy levels determined. Nuclear shells are said to occur when the separation between energy levels is significantly greater than the local mean separation. In the shell model for the nucleus, magic numbers are the numbers of nucleons at which a shell is filled. For instance, the magic number 8 occurs when the 1s 1/2, 1p 3/2, 1p 1/2 energy levels are filled, as there is a large energy gap between the 1p 1/2 and the next highest 1d 5/2 energy levels. The atomic analog to nuclear magic numbers are those numbers of electrons leading to discontinuities in the ionization energy. These occur for the noble gaseshelium, neon, argon, krypton, xenon, radon and oganesson. Hence, the "atomic magic numbers" are 2, 10, 18, 36, 54, 86 and 118. As with the nuclear magic numbers, these are expected to be changed in the superheavy region due to spin/orbit-coupling effects affecting subshell energy levels. Hence copernicium (112) and flerovium (114) are expected to be more inert than oganesson (118), and the next noble gas after these is expected to occur at element 172 rather than 168 (which would continue the pattern). In 2010, an alternative explanation of magic numbers was given in terms of symmetry considerations. Based on the fractional extension of the standard rotation group, the ground state properties (including the magic numbers) for metallic clusters and nuclei were simultaneously determined analytically. A specific potential term is not necessary in this model. See also [edit] Physics portal Magic number (chemistry) Superatom Superdeformation References [edit] ^ Jump up to: abKratz, J. V. (5 September 2011). The Impact of Superheavy Elements on the Chemical and Physical Sciences(PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 27 August 2013. ^"Nuclear scientists eye future landfall on a second 'island of stability'". Archived from the original on 2016-03-12. Retrieved 2014-08-10. ^Grumann, Jens; Mosel, Ulrich; Fink, Bernd; Greiner, Walter (1969). "Investigation of the stability of superheavy nuclei aroundZ=114 andZ=164". Zeitschrift für Physik. 228 (5): 371–386. Bibcode:1969ZPhy..228..371G. doi:10.1007/BF01406719. S2CID120251297. ^"Nuclear scientists eye future landfall on a second 'island of stability'". Archived from the original on 2016-03-12. Retrieved 2014-08-10. ^Grumann, Jens; Mosel, Ulrich; Fink, Bernd; Greiner, Walter (1969). "Investigation of the stability of superheavy nuclei aroundZ=114 andZ=164". Zeitschrift für Physik. 228 (5): 371–386. Bibcode:1969ZPhy..228..371G. doi:10.1007/BF01406719. S2CID120251297. ^Koura, H. (2011). Decay modes and a limit of existence of nuclei in the superheavy mass region(PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 18 November 2018. ^Out of the shadows: contributions of twentieth-century women to physics. Byers, Nina. Cambridge: Cambridge Univ. Pr. 2006. ISBN0-521-82197-5. OCLC255313795.{{cite book}}: CS1 maint: others (link) ^Mayer, Maria G. (1948-08-01). "On Closed Shells in Nuclei". Physical Review. 74 (3): 235–239. Bibcode:1948PhRv...74..235M. doi:10.1103/physrev.74.235. ISSN0031-899X. ^Wigner, E. (1937-01-15). "On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei". Physical Review. 51 (2): 106–119. Bibcode:1937PhRv...51..106W. doi:10.1103/PhysRev.51.106. ^Mayer, Maria Goeppert (1949-06-15). "On Closed Shells in Nuclei. II". Physical Review. 75 (12): 1969–1970. Bibcode:1949PhRv...75.1969M. doi:10.1103/PhysRev.75.1969. ^Audi, Georges (2006). "The history of nuclidic masses and of their evaluation". International Journal of Mass Spectrometry. 251 (2–3): 85–94. arXiv:physics/0602050. Bibcode:2006IJMSp.251...85A. doi:10.1016/j.ijms.2006.01.048. S2CID13236732. ^"The Nobel Prize in Physics 1963". NobelPrize.org. Retrieved 2020-06-27. ^"What is Stable Nuclei - Unstable Nuclei - Definition". Periodic Table. 2019-05-22. Retrieved 2019-12-22. ^Kondo, Y.; Achouri, N. L.; Falou, H. Al; Atar, L.; Aumann, T.; Baba, H.; Boretzky, K.; Caesar, C.; Calvet, D.; Chae, H.; Chiga, N.; Corsi, A.; Delaunay, F.; Delbart, A.; Deshayes, Q. (2023-08-31). "First observation of 28O". Nature. 620 (7976): 965–970. doi:10.1038/s41586-023-06352-6. ISSN0028-0836. PMC10630140. PMID37648757. ^Nave, C. R. "The Most Tightly Bound Nuclei". HyperPhysics. ^W., P. (October 23, 1999). "Twice-magic metal makes its debut - isotope of nickel". Science News. Archived from the original on May 24, 2012. Retrieved 2006-09-29. ^"Tests confirm nickel-78 is a 'doubly magic' isotope". Phys.org. September 5, 2014. Retrieved 2014-09-09. ^Audi, G.; Kondev, F. G.; Wang, M.; Huang, W. J.; Naimi, S. (2017). "The NUBASE2016 evaluation of nuclear properties"(PDF). Chinese Physics C. 41 (3) 030001. Bibcode:2017ChPhC..41c0001A. doi:10.1088/1674-1137/41/3/030001. ^Mason Inman (2006-12-14). "A Nuclear Magic Trick". Physical Review Focus. Vol.18. Retrieved 2006-12-25. ^Dvorak, J.; Brüchle, W.; Chelnokov, M.; Dressler, R.; Düllmann, Ch. E.; Eberhardt, K.; Gorshkov, V.; Jäger, E.; Krücken, R.; Kuznetsov, A.; Nagame, Y.; Nebel, F.; Novackova, Z.; Qin, Z.; Schädel, M.; Schausten, B.; Schimpf, E.; Semchenkov, A.; Thörle, P.; Türler, A.; Wegrzecki, M.; Wierczinski, B.; Yakushev, A.; Yeremin, A. (2006). "Doubly Magic Nucleus 108 270 Hs 162". Physical Review Letters. 97 (24): 242501. Bibcode:2006PhRvL..97x2501D. doi:10.1103/PhysRevLett.97.242501. PMID17280272.{{cite journal}}: CS1 maint: article number as page number (link) ^Koura, H.; Chiba, S. (2013). "Single-Particle Levels of Spherical Nuclei in the Superheavy and Extremely Superheavy Mass Region". Journal of the Physical Society of Japan. 82 (1): 014201. Bibcode:2013JPSJ...82a4201K. doi:10.7566/JPSJ.82.014201.{{cite journal}}: CS1 maint: article number as page number (link) ^Herrmann, Richard (2010). "Higher dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed Nilsson-oscillator". Physica A. 389 (4): 693–704. arXiv:0806.2300. Bibcode:2010PhyA..389..693H. doi:10.1016/j.physa.2009.11.016. ^Herrmann, Richard (2010). "Fractional phase transition in medium size metal clusters and some remarks on magic numbers in gravitationally and weakly bound clusters". Physica A. 389 (16): 3307–3315. arXiv:0907.1953. Bibcode:2010PhyA..389.3307H. doi:10.1016/j.physa.2010.03.033. S2CID50477979. External links [edit] Nave, C. R. "Shell Model of Nucleus". HyperPhysics. Scerri, Eric (2007). The Periodic Table, Its Story and Its Significance. Oxford University Press. ISBN978-0-19-530573-9. see chapter 10 especially. Moskowitz, Clara. "New magic number "inside atoms" discovered". Scientific American. Watkins, Thayer. "A Nearly Complete Explanation of the Nuclear Magic Numbers". 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2555	https://www.math.purdue.edu/academic/files/courses/2007fall/MA301/MA301Ch6.pdf	CHAPTER 6 Max, Min, Sup, Inf We would like to begin by asking for the maximum of the function f(x) = (sin x)/x. An approximate graph is indicated below. Looking at the graph, it is clear that f(x) ≤1 for all x in the domain of f. Furthermore, 1 is the smallest number which is greater than all of f’s values. o y=(sin x)/x 1 Figure 1 Loosely speaking, one might say that 1 is the ‘maximum value’ of f(x). The problem is that one is not a value of f(x) at all. There is no x in the domain of f such that f(x) = 1. In this situation, we use the word ‘supremum’ instead of the word ‘maximum’. The distinction between these two concepts is described in the following deﬁnition. Definition 1. Let S be a set of real numbers. An upper bound for S is a number B such that x ≤B for all x ∈S. The supremum, if it exists, (“sup”, “LUB,” “least upper bound”) of S is the smallest 81 82 6. MAX, MIN, SUP, INF upper bound for S. An upper bound which actually belongs to the set is called a maximum. Proving that a certain number M is the LUB of a set S is often done in two steps: (1) Prove that M is an upper bound for S–i.e. show that M ≥s for all s ∈S. (2) Prove that M is the least upper bound for S. Often this is done by assuming that there is an ǫ > 0 such that M −ǫ is also an upper bound for S. One then exhibits an element s ∈S with s > M −ǫ, showing that M −ǫ is not an upper bound. Example 1. Find the least upper bound for the following set and prove that your answer is correct. S = {1 2, 2 3, 3 4, . . . , n n + 1 . . . } Solution: We note that every element of S is less than 1 since n n + 1 < 1 We claim that the least upper bound is 1. Assume that 1 is not the least upper bound,. Then there is an ǫ > 0 such that 1 −ǫ is also an upper bound. However, we claim that there is a natural number n such that 1 −ǫ < n n + 1. This inequality is equivalent with the following sequence of inequali-ties 1 − n n + 1 < ǫ 1 n + 1 < ǫ 1 ǫ < n + 1 1 ǫ −1 < n. 6. MAX, MIN, SUP, INF 83 Reversing the above sequence of inequalities shows that if n > 1 ǫ −1, then 1 −ǫ < n n+1 showing that 1 −ǫ is not an upper bound for S. This veriﬁes our answer. If a set has a maximum, then the maximum will also be a supre-mum: Proposition 1. Suppose that B is an upper bound for a set S and that B ∈S. Then B = sup S. Proof Let ǫ > 0 be given. Then B −ǫ cannot be an upper bound for S since B ∈S and B > B −ǫ, showing that B is indeed the least upper bound. Example 2. Find the least upper bound for the following set and prove that your answer is correct. T = {1, 1 2, 2 3, 3 4, . . . , n n + 1 . . . }. Solution: From the work done in Example 1, 1 is an upper bound for S. Since 1 ∈S, 1 = sup S. Example 3. Find the max, min, sup, and inf of the following set and prove your answer. S = {2n + 1 n + 1 \| n ∈N}. Solution: We write the ﬁrst few terms of S: S = ©3 2, 5 3, 7 4, 9 5, 11 6 , . . . ª . The smallest term seems to be 3 2 and there seems to be no largest term, although all of the terms seem to be less than 2. Since limn→∞ 2n+1 n+1 = 2 we conjecture that: (a) There is no maximum, (b) sup S = 2, and (c) min S = inf S = 3 2. Proof Sup 84 6. MAX, MIN, SUP, INF We must ﬁrst show that 2 is an upper bound–i.e. 2n + 1 n + 1 < 2 2n + 1 < 2n + 2 1 < 2 which is always true. Reversing the above argument shows that 2 is an upper bound. Next we show that 2 is the least upper bound. If 2 is not the LUB, there is an ǫ > 0 such 2 −ǫ is an upper bound. However, we claim that there are n ∈N such that 2 −ǫ < 2n + 1 n + 1 showing that 2 −ǫ is not an upper bound. To prove our claim, note that the above inequality is equivalent with −ǫ < 2n + 1 n + 1 −2 ǫ > 1 n + 1 n > 1 ǫ −1 Since there exist n ∈N satisfying the above inequality, our claim is proved. Hence 2 is the LUB. Next we show that 2 is not a maximum. This means showing that there is no n such that 2 = 2n + 1 n + 1 . This however is equivalent with 2(n + 1) = 2n + 1 2n + 2 = 2n + 1 2 = 1 which is certainly false. Next we prove that min S = 3 2. We ﬁrst note that 3 2 ∈S since 3 2 = 2 · 1 + 1 1 + 1 . 6. MAX, MIN, SUP, INF 85 Hence, it suﬃces to show that 3 2 is a lower bound which we do as follows: 2n + 1 n + 1 ≥3 2 2(2n + 1) ≥3(n + 1) n ≥1 which is true for all n ∈N. Reversing the above argument shows that 3 2 is a lower bound. □ The central question in this section is “Does every non-empty set of numbers have a sup?” The simple answer is no–the set N of natural numbers does not have a sup because it is not bounded from above. O.K.–we change the question:“Does every set of numbers which is bounded from above have a sup?” The answer, it turns out, depends upon what we mean by the word “number”. If we mean “rational number” then our answer is NO!. Recall that the set of integers is the set of positive and natural numbers, together with 0. I.e. Z = {0, ±1, ±2, . . . , ±n, · · · \| n ∈N}. The set of rational numbers is the set Q = {p q \| p, q ∈Z, q ̸= 0}. Thus, for example, 2 3 and −9 7 are elements of Q. In Chapter 9 (The-orem 2) we prove that √ 2 is not rational. Now, let S be the set of all positive rational numbers r such that r2 < 2. Since the square root function is increasing on the set of positive real numbers, S = {0 < r < √ 2 \| r ∈Q}. Clearly, √ 2 is an upper bound for S. It is also a limit of values from S. In fact, we know that √ 2 = 1.414213562 + . Each of the numbers 1.4, 1.41, 1.414, 1.4142, etc. is rational and has square less than 2. Their limit is √ 2. Thus, sup S = √ 2. (See Exercise 6 below.) Since √ 2 is irrational, S is then an example of a set of rational numbers whose sup is irrational. Suppose, however, that we (like the early Greek mathematicians) only knew about rational numbers. We would be forced to say that S 86 6. MAX, MIN, SUP, INF has no sup. The fact that S does not have a sup in Q can be thought of as saying that the rational numbers do not completely ﬁll up the number line; there is a missing number “directly to the right” of S. The fact that the set R of all real numbers does ﬁll up the line is such a fundamentally important property that we take it as an axiom: the completeness axiom. (The reader may recall that in Chapter I, we mentioned that we would eventually need to add an axiom to our list. This is it.) We shall also refer to this axiom as the Least Upper Bound Axiom. (LUB Axiom for short.) Least Upper Bound Axiom: Every non-empty set of real numbers which is bounded from above has a supremum. The observation that the least upper bound axiom is false for Q tells us something important: it is not possible to prove the least upper bound axiom using only the axioms stated in Chapters 1 and 2. This is because the set of rational numbers satisfy all the axioms from Chapters 1 and 2. Thus, if the least upper bound axiom were provable from these axioms, it hold for the rational numbers. Of course, similar comments apply to minimums: Deﬁnition: Let S be a set of real numbers. A lower bound for S is a number B such that B ≤x for all x ∈S. The inﬁnum (“inf”, “GLB,” “greatest lower bound”) of S, if it exists, is the largest lower bound for S. A lower bound which actually belongs to the set is called a minimum. Fortunately, once we have the LUB Axiom, we do not need an-other axiom to guarantee the existence of inf’s. The existence of inf’s is a theorem which we will leave as an exercise. (Of course, we could have let the existence of inf’s be our completeness axiom, in which case the existence of sup’s would be a theorem.) Greatest Lower Bound Property: Every non-empty set of real numbers which is bounded from below has a inﬁmum. Proving that a certain number M is the GLB of a set S is similar to a LUB proof. It requires: 6. MAX, MIN, SUP, INF 87 (1) Proving that M is a lower bound for S–i.e. proving that M ≤s for all s ∈S. (2) Proving that M is the greatest lower bound for S. Often this is done by assuming that there is an ǫ > 0 such that M + ǫ is a lower bound for S. One then exhibits an element s of S satisfying s < M + ǫ, showing that M + ǫ is not a lower bound for S. Example 4. Prove that the inf of S = (1, 5] is 1. Solution: By deﬁnition S is the set of x satisfying 1 < x ≤5. Hence 1 is a lower bound for S. Suppose that 1 is not the GLB of S. Then there is an ǫ > 0 such that 1 + ǫ is also a lower bound for S. To contradict this, we exhibit x ∈S such that 1 < x < 1 + ǫ. Since 0 < ǫ 2 < ǫ we see that x = 1 + ǫ 2 satisﬁes 1 < x < 1 + ǫ. Since 1+ǫ is (by assumption) a lower bound for S and 5 ∈S, 1+ǫ ≤5, showing that x ∈(1, 5]. Thus, 1 + ǫ is not a lower bound, proving that 1 is the greatest lower bound. Example 5. Find upper and lower bounds for y = f(x) for x ∈ [−1, 1.5] where f(x) = −x4 + 2x2 + x Use a graphing calculator to estimate the least upper bound and the greatest lower bound for f(x). Solution: From the triangle inequality \|f(x)\| = \| −x4 + 2x2 + x\| ≤\|x4\| + \|2x2\| + \|x\| = \|x\|4 + 2\|x\|2 + \|x\| The last quantity is largest when \|x\| is largest, which occurs when \|x\| = 1.5. Hence \|f(x)\| ≤1.54 + 2(1.5)3 + 1.5 = 13.3125 88 6. MAX, MIN, SUP, INF Hence, M = 14 is an upper bound and M = −14 is a lower bound. As a check, we graph y = −x4+2x2+x with xmin= −.5, xmax= 1.5, ymin= −15 and ymax= 15, as well as the lines y = 14 and y = −14. Since the graph lies between the lines, the value of M is acceptable, although considerably larger than necessary. To estimate the least bound, we trace the curve using the trace feature of the calculator, ﬁnding that the maximum and minimum y-values are approximately 2.0559 and −.130 respectively. These values are (approximately) the least upper bound and the greatest lower bound respectively. Remark: It is important to note that in the preceding example, the values of the function at the end points of the interval are not bounds for the function because f(x) is not monotonic (i.e. it is neither increasing nor decreasing) over the stated interval. On the other hand, the largest value of \|x\|4 + 2\|x\|2 + \|x\| is at the largest endpoint because this function increases as \|x\| increases. The next example uses both the triangle inequality and the ob-servation that making the denominator of a fraction smaller increases its value. Example 6. Find a bound for y = f(x) for x ∈[−2, 2] where f(x) = x3 −3x + 1 1 + x2 Solution: We note that ¯ ¯x3 −3x + 1 1 + x2 ¯ ¯ = \|x3 −3x + 1\| 1 + x2 Since x2 > 0, we see x2 + 1 > 1 1 x2 + 1 < 1 1 = 1 Hence \|x3 −3x + 1\| 1 + x2 ≤\|x3 −3x + 1\| ≤\|x3\| + \|3x\| + 1 ≤23 + 3 · 2 + 1 = 15 Thus, our bound is M = 15. 6. MAX, MIN, SUP, INF 89 In all of the examples considered above, the least upper bound for f(x) is the maximum of f(x). This is always the case if f(x) has a maximum. Similarly, the greatest lower bound is the minimum of f(x) if f(x) has a minimum. In Chapter 4, we studied sequences which diverge because they tend to inﬁnity. Another class of sequences that have no limit are ones which might be dubbed “wishy-washy.” These are sequences that can’t make up their minds what their limit is, tending simultaneously to several numbers. An example is an = n + (−1)nn n + 1 The ﬁrst 10 values are listed below. n 1 2 3 4 5 6 7 8 9 10 an 0 1.33 0 1.60 0 1.71 0 1.78 0 1.82 It appears that some values approach (in fact equal) 0 while others approach 2. Indeed, for odd n an = n −n n + 1 = 0 which tells us that if the limit exists, it must be 0. For even n an = n + n n + 1 2n n + 1 which tends to 2. From Proposition 1, a sequence can have only one limit. Hence, there is no limit. A related type divergence is what might be referred to as “wan-dering,” where values wander in a seemingly random manner, never getting close to particular number. An example is an = cos n. The ﬁrst 10 values are listed below. n 1 2 3 4 5 6 7 8 9 10 cos n .540 −.416 −.990 −.654 .284 .960 .754 −.146 −.911 −.839 There certainly seems to be no tendency toward any one number. A sequence an is non-decreasing if a1 ≤a2 ≤a3 ≤· · · ≤an < . . . The sequence an is increasing if each of the above inequalities is strict. 90 6. MAX, MIN, SUP, INF For example, an = n2 is increasing : 1 < 22 < 32 < 42 < . . . The sequence from Example 1 is also increasing: 1 2 < 2 3 < 3 4 < 4 5 < . . . The two preceding sequences demonstrate that an increasing se-quence can either go to inﬁnity or can converge. However, an increas-ing sequence cannot wander; once it has exceeded a certain value, it can never return to that value. Hence we arrive at the following theorem: Theorem 1 (Bounded Increasing Theorem). For a non-decreasing sequence an, either limn→∞an exists or limn→∞an = ∞. Proof limn→∞an = ∞implies that for all M > 0 there is an N such that an ≥M for all n ≥N. For a non-decreasing sequence, this is equivalent with the statement that for all M there is an n such that an ≥M. Hence if limn→∞an ̸= ∞, there is an M such that an < M for all n. Thus S = {a1, a2, . . . , an, . . . }. is bounded from above. Let a = sup S. Then an ≤a for all n. Furthermore, since a is the smallest upper bound, a −ǫ is not an upper bound for any ǫ > 0. Hence, for all ǫ > 0, there is at least one aN such that a −ǫ < aN ≤a. However, since an is increasing and a is an upper bound for S, we see that for all n ≥N, aN ≤an ≤a. It follows that for all n ≥N, a −ǫ ≤an ≤a + ǫ proving that a is the limit of an, as desired. The bounded increasing theorem is one of the most important ways of proving that sequences converge. In particular, we will make good use of in Chapters 7 and 8. 6. MAX, MIN, SUP, INF 91 There is, of course, nothing special about increasing as opposed to decreasing. It is an immediate consequence of the Bounded-Increasing Theorem that a decreasing sequence which is bounded from below also has a limit. (See Exercise 10 below.) We have already used repeatedly, without comment, a conse-quence of the GLB axiom. For example, in our formal solution to Example 2 in Chapter 4, we said “. . . let ǫ > 0 be given and let n > 1/ǫ.” Certainly, we know from experience that there are natu-ral numbers n greater than 1/ǫ. However, none of the axioms from Chapters 1 and 2 tell us that such numbers exist. Their existence follows from the following theorem. We leave it as an exercise (Ex-ercise 11 below) to explain how this property follows from the GLB axiom. Theorem 2 (Archimedian Property). For all numbers M there is a natural number n such that n > M. Exercises (1) Compute the sup, inf, max and min (whenever these exist) for the following sets. 1 (a) {1 + 1/n \| n ∈N} (b) [0, 2) (c) { n2+15 n+1 \| n ∈N} (d) {x \| x ∈Q and x2 < 2} (e) {y \| y = x2 −x + 1 and x ∈R} (f) {x \| x2 −3x + 2 < 0 and x ∈R} (g) { 1 n −1 m \| n, m ∈N} (h) {1 + 1+(−1)n n \|n ∈N} (i) { 1 2, 1 3, 2 3, 1 4, 3 4, 1 5, 2 5, 3 5, 4 5, . . . }. (This is a list of the frac-tions in the interval (0, 1). The pattern is that we list fractions by increasing value of the denominator. For a given value of denominator, we go from smallest to largest, omitting fractions which are not in reduced form.) (j) {n/(1 + n2) \| n ∈N} (k) {3n2/(1 + 2n2) \| n ∈N} (l) {n/(1 −n2) \| n ∈N, n > 1} 1In set theory, the symbol ‘\|’ is read “where.” Thus, the set in part (a) is the set of numbers of the form 1 + 1/n where n is a natural number.” 92 6. MAX, MIN, SUP, INF (m) {(1 −2n2)/3n2 \| n ∈N} (n) {2n2/(3n2 −1) \| n ∈N} (o) {3n/ √ 1 + n2 \| n ∈N} (p) {3/ √ 1 + 2n2 \| n ∈N} (q) { n−1 n2+5 \| n ∈N} Hint: In the proof of the upper bound (which is re-quested in Exercise 2), you will discover that you need to prove n2 −7n + 12 ≥0. Factor this polynomial and determine when it can be negative. (r) { 3n+1 n+1 \| n ∈N} (2) Prove your answers in Exercise 1. (3) In Example 5, use a graphing calculator to estimate the sup and the inf for f(x). (4) For the following functions f(x), (i) ﬁnd a number M such that \|f(x)\| ≤M for all x in the stated interval. (ii) Use a graph to estimate the sup s and inf t for f(x). Sketch your graph on a piece of paper. (iii) As evidence for the statement that s is the LUB, ﬁnd an x such that f(x) > s −ǫ for the stated value of ǫ. (iv) As evidence for the statement that t is the GLB, ﬁnd an x such that f(x) < t + ǫ for the stated value of ǫ. (a) f(x) = x5 −x4 + x3 −x2 + x −1 x ∈[−1, 1], ǫ = .1 (b) f(x) = x3 −3x2 1 + x + x2 x ∈[0, 2], ǫ = .01 (c) f(x) = x3 −3x2 + 2x −1 + 5 sin x x ∈[−1, 2], ǫ = .05 (d) f(x) = x −4 cos x x2 + 5 x ∈[2, 4], ǫ = .02 (e) f(x) = x2 + 5 sin x x ∈[−4, 4], ǫ = .2 (f) f(x) = x4 + 1 x2 + 2 + cos x x ∈[0, π], ǫ = .03 (5) The ﬁgure below shows a rectangle inside the circle x2+y2 = 1. One side of the rectangle is formed by the x axis while two vertices lie on the circle. Find the sup and inf of the set of possible areas for the rectangle. Is the sup also a max? Is the inf also a min? 6. MAX, MIN, SUP, INF 93 (x,y) (-x,y) 2 2 1=x +y (6) Suppose that M is an upper bound for the set S. Sup-pose also that there is a sequence sn ∈S such that M = limn→∞sn. Prove that M = sup S. (7) Prove the converse of Exercise 6–i.e. suppose that M = sup S. Prove that there is a sequence an ∈S such that M = limn→∞an. Hint: For all n ∈N, M −1/n is not an upper bound. (8) Suppose that M is a lower bound for the set S. Suppose also that there is a sequence sn ∈S such that M = limn→∞sn. Prove that M = inf S. (9) Prove the converse of Exercise 8–i.e. suppose that M = inf S. Prove that there is a sequence an ∈S such that M = limn→∞an. Hint: Modify the argument from Exercise 7. (10) A sequence is said to be non-increasing if for all n, an+1 ≤an. Prove that a non-increasing sequence either tends to −∞or converges. This is the Bounded Decreasing Theorem. Hint: Consider −an. (11) Prove the Archimedian Property. Hint: Assume that the Archimedian Property is false. Explain how it follows that the set of natural numbers is bounded from above. Let s = sup N. Then there is a natural number n satisfying s −.5 < n ≤s. (Explain!) What does this say about n + 1? (12) If S is any set of numbers, we deﬁne −S = {−x\|x ∈S} Compute the sup and inf of −S for each set S in Exercise 9. How do the sup and inf of S relate to the sup and inf of −S? (13) This exercise uses the notation from Exercise 12. (a) Let S be a set of numbers that is bounded from below by a number B. Show that −S is bounded from above by −B. 94 6. MAX, MIN, SUP, INF (b) Let S be as in part (a). It follows from (a) that −S is bounded from above. The Least Upper Bound Theorem implies that −S has a least upper bound L. Prove that −L is the GLB for S. For this you must explain why (i) −L is a lower bound for S and (ii) why there is no greater lower bound. Remark It follows from (b) that S has a GLB. Hence the above sequence of arguments proves the GLB property. (14) Let a1 = 1. Let a sequence an be deﬁned by (1) an+1 = √ 2an. Thus, for example, a2 = √ 2 · 1 = 1.414 a3 = √ 2 · a2 = 1.6818 a4 = √ 2 · a3 = 1.8340 (a) Compute a5, a6, and a7 (as decimals.) You should ﬁnd that each is less than 2. (b) Prove that if an < 2 then an+1 < 2 as well. How does it follow that an < 2 for all n? (c) Prove that for all n, an+1 > an. (d) Explain how it follows from the Bounded Increasing Theorem that limn→∞an = L exists. (e) Prove that L = √ 2L. What then is the value of L? Hint: Take the limit of both sides of (1). (15) Let a1 = 1. Let a sequence an be deﬁned by (2) an+1 = √ 2 + an. (a) Compute a2, a3, and a4 (as decimals.) You should ﬁnd that each is less than 2. (b) Prove that if an < 2 then an+1 < 2 as well. How does it follow that an < 2 for all n? (c) Prove that for all n, an+1 > an. (d) Explain how it follows from the Bounded Increasing Theorem that limn→∞an = L exists. (e) Prove that L = √ 2 + L. What then is the value of a? Hint: Take the limit of both sides of (2). 6. MAX, MIN, SUP, INF 95 (16) The following exercise develops the “divide and average” method of approximating r = √ 2. From r2 = 2, we see that r = 2/r. If r1 is some approximation to r, then 2/r1 will be another. The average r2 = 1 2 ¡ r1 + 2 r1 ¢ will (hopefully) be a better approximation. We repeat this process with r2 in place of r1 to produce r3. In general, we set (3) rn+1 = 1 2 ¡ rn + 2 rn ¢ . For example, if r1 = 2, then r2 = 1 2(2 + 2/2) = 1.5 r3 = 1 2(r2 + 2/r2) = 1.416666667 r4 = 1 2(r3 + 2/r3) = 1.414215686 . (a) Compute r5, r6 and r7. (b) Prove that r2 n > 2. Hint: Write this as r2 n+1 > 2 and use formula (3) above together with some algebra. (c) Prove that for all n, 0 < rn+1 ≤rn. How does it follow from the Bounded Decreasing Property that limn→∞rn exists. (d) Show that r = limn→∞rn satisﬁes the equation r = 1 2 ¡ r + 2 r ¢ . Use this to prove that r2 = 2. Hint: Take the limit of both sides of formula (3). Remark: This exercise proves the existence of √ 2.
2556	https://zh.wikisource.org/zh-hans/%E6%99%AE%E9%80%9A%E9%AB%98%E4%B8%AD%E6%95%B0%E5%AD%A6%E8%AF%BE%E7%A8%8B%E6%A0%87%E5%87%86	跳转到内容 [关闭] 普通高中数学课程标准下载维基文库，自由的图书馆 \| \| \| \| \| \| --- --- \| \| \| 普通高中数学课程标准中华人民共和国教育部制定 \| \| \| \| \| \| 姊妹计划: 数据项 \| 前言党的十九大明确提出：“要全面贯彻党的教育方针，落实立德树人根本任务，发展素质教育，推进教育公平，培齐德智体美全面发展的社会主义建设者和接班人。基础教育课程承载着党的教育方针和教育思想，规定了教育目标和教育内容，是国家意志在教育领域的直接体现，在立德树人中发挥着关键作用。 2003年，教育部印发的普通高中课程方案和课程标准实验稿，指导了十余年来普通高中课程改革的实践，坚持了正确的改革方向和先进的教育理念，基本建立起适合我国国情、适应时代发展要求的普通高中课程体系，促进了教育观念的更新，推进了人才培养模式的变革，提升了教师队伍的整体水平，有效推动了考试评价制度的改革，为我国基础教育质量的提高作出了积极贡献。但是，面对经济、科技的迅猛发展和社会生活的深刻变化，面对新时代社会主要矛盾的转化，面对新时代对提高全体国民素质和人才培养质量的新要求，面对我国高中阶段教育基本普及的新形势，普通高中课程方案和课程标准实验稿还有一些不相适应和亟待改进之处。 2013年，教育部启动了普通高中课程修订工作。本次修订深入总结21世纪以来我国普通高中课程改革的宝贵经验，充分借鉴国际课程改革的优秀成果，努力将普通高中课程方案和课程标准修订成既符合我国实际情况，又具有国际视野的纲领性教学文件，构建具有中国特色的普通高中课程体系。一、修订工作的指导思想和基本原则（一）指导思想以马克思列宁主义、毛泽东思想、邓小平理论、“三个代表”重要思想、科学发展观、习近平新时代中国特色社会主义思想为指导，深入贯彻党的十八大、十九大精神，全面贯彻党的教育方针，落实立德树人根本任务，发展素质教育，推进教育公平，以社会主义核心价值观统领课程改革，着力提升课程思想性、科学性、时代性、系统性、指导性，推动人才培养模式的改革创新，培养德智体美全面发展的社会主义建设者和接班人。（二）基本原则 1．坚持正确的政治方向。坚持党的领导，坚持社会主义办学方向，充分体现马克思主义的指导地位和基本立场，充分反映习近平新时代中国特色社会主义思想，有机融入坚持和发展中国特色社会主义、培育和践行社会主义核心价值观的基本内容和要求，继承和弘扬中华优秀传统文化、革命文化，发展社会主义先进文化，加强法治意识、国家安全、民族团结、生态文明和海洋权益等方面的教育，培养良好政治素质、道德品质和健全人格，使学生坚定中国特色社会主义道路自信、理论自信、制度自信和文化自信，引导学生形成正确的世界观、人生观、价值观。 2．坚持反映时代要求。反映先进的教育思想和理念，关注信息化环境下的教学改革，关注学生个性化、多样化的学习和发展需求，促进人才培养模式的转变，着力发展学生的核心素养。根据经济社会发展新变化、科学技术进步新成果，及时更新教学内容和话语体系，反映新时代中国特色社会主义理论和建设新成就。 3．坚持科学论证。遵循教育教学规律和学生身心发展规律，贴近学生的思想、学习、生活实际，充分反映学生的成长需要，促进每个学生主动地、生动活泼地发展。加强调查研究和测试论证，广泛听取相关领域人员的意见建议，重大问题向权威部门、专业机构、知名专家学者咨询，求真务实，严谨认真，确保课程内容科学，表述规范。 4．坚持继承发展。对十余年普通高中课程改革实践进行系统梳理，总结提炼并继承已有经验和成功做法，确保课程改革的连续性。同时，发现并切实面对改革过程中存在的问题，有针对性地进行修订完善，在继承中前行，在改革中完善，使课程体系充满活力。二、修订的主要内容和变化（一）关于课程方案 1．进一步明确了普通高中教育的定位。我国普通高中教育是在义务教育基础上进一步提高国民素质、面向大众的基础教育，任务是促进学生全面而有个性的发展，为学生适应社会生活、高等教育和职业发展作准备，为学生的终身发展奠定基础。普通高中的培养目标是进一步提升学生综合素质，着力发展核心素养，使学生具有理想信念和社会责任感，具有科学文化素养和终身学习能力，共有自主发展能力和沟通合作能力。 2．进一步优化了课程结构。一是保留原有学习科目，调整外语规划语种，在英语、日语、俄语基础上，增加德语、法语和西班牙语。二是将课程类别调整为必修课程、选择性必修课程和选修课程。在保证共同基础的前提下，为不同发展方向的学生提供有选择的课程。三是进一步明确各类课程的功能定位，与高考综合改革相衔接：必修课程根据学生全面发展需要设置，全修全考；选择性必修课程根据学生个性发展和升学考试需要设置，选修选考；选修课程由学校根据实际情况统筹规划开设，学生自主选择修习，学而不考或学而备考，为学生就业和高校招生录取提供参考。四是合理确定各类课程学分比例，在毕业总学分不变的情况下，对原必修课程学分进行重构，由必修课程学分、选择性必修课程学分组成，适当增加选修课程学分，既保证基础性，又兼顾选择性。 3．强化了课程有效实施的制度建设。进一步明确课程实施环节的责任主体和要求，从课程标准、教材、课程规划、教学管理，以及评价、资源建设等方面，对国家、省（自治区、直辖市）、学校分别提出了要求。增设“条件保障”部分，从师资队伍建设、教学设施和经费保障等方面提出具体要求。增设“管理与监督”部分，强化各级教育行政部门和学校课程实施的责任。（二）关于学科课程标准 1．凝练了学科核心素养。中国学生发展核心素养是党的教育方针的具体化、细化。为建立核心素养与课程教学的内在联系，充分挖掘各学科课程教学对全面贯彻党的教育方针、落实立德树人的根本任务、发展素质教育的独特育人价值，各学科基于学科本质凝练了本学科的核心素养，明确了学生学习该学科课程后应达成的正确价值观念、必备品格和关键能力，对知识与技能、过程与方法、情感态度价值观三维目标进行了整合。课程标准还围绕核心素养的落实，精选、重组课程内容，明确内容要求，指导教学设计，提出考试评价和教材编写建议。 2．更新了教学内容。进一步精选了学科内容，重视以学科大概念为核心，使课程内容结构化，以主题为引领，使课程内容情境化，促进学科核心素养的落实。结合学生年龄特点和学科特征，课程内容落实习近平新时代中国特色社会主义思想，有机融入社会主义核心价值观，中华优秀传统文化、革命文化和社会主义先进文化教育内容，努力呈现经济、政治、文化、科技、社会、生态等发展的新成就、新成果，充实丰富培养学生社会责任感、创新精神、实践能力相关内容。 3．研制了学业质量标准。各学科明确学生完成本学科学习任务后，学科核心素养应该达到的水平，各水平的关键表现构成评价学业质量的标准。引导教学更加关注育人目的，更加注重培养学生核心素养，更加强调提高学生综合运用知识解决实际问题的能力，帮助教师和学生把握教与学的深度和广度，为阶段性评价、学业水平考试和升学考试命题提供重要依据，促进教、学、考有机衔接，形成育人合力。 4．增强了指导性。本着为编写教材服务、为教学服务、为考试评价服务的原则，突出课程标准的可操作性，切实加强对教材编写、教学实施、考试评价的指导。课程标准通俗易懂，逻辑更清晰，原则上每个模块或主题由“内容要求”“教学提示”“学业要求”组成，大部分学科增加了教学与评价案例，同时依据学业质量标准细化评价目标，增强了对教学和评价的指导性。本次修订是深化普通高中课程改革的重要环节，直接关系育人质量的提升。普通高中课程方案和课程标准必须在教育教学实践中接受检验，不断完善。可以预期，广大教育工作者将在过去十余年改革的基础上，在丰富而生动的教育教学实践中，不断提高课程实施水平，推动普通高中课程改革不断深化，共创普通高中教育的新辉煌，为实现国家教育现代化、建设教育强国作出新贡献。一、课程性质与基本理念（一）课程性质数学是研究数量关系和空间形式的一门科学。数学源于对现实世界的抽象，基于抽象结构，通过符号运算、形式推理、模型构建等，理解和表达现实世界中事物的本质、关系和规律。数学与人类生活和社会发展紧密关联。数学不仅是运算和推理的工具，还是表达和交流的语言。数学承载着思想和文化，是人类文明的重要组成部分。数学是自然科学的重要基础，并且在社会科学中发挥越来越大的作用，数学的应用已渗透到现代社会及人们日常生活的各个方面。随着现代科学技术特别是计算机科学、人工智能的迅猛发展，人们获取数据和处理数据的能力都得到很大的提升，伴随着大数据时代的到来，人们常常需要对网络、文本、声音、图像等反映的信息进行数字化处理，这使数学的研究领域与应用领域得到极大拓展。数学直接为社会创造价值，推动社会生产力的发展。数学在形成人的理性思维、科学精神和促进个人智力发展的过程中发挥着不可替代的作用。数学素养是现代社会每一个人应该具备的基本素养。数学教育承载着落实立德树人根本任务、发展素质教育的功能。数学教育帮助学生掌握现代生活和进一步学习所必需的数学知识、技能、思想和方法；提升学生的数学素养，引导学生会用数学眼光观察世界，会用数学思维思考世界，会用数学语言表达世界；促进学生思维能力、实践能力和创新意识的发展，探寻事物变化规律，增强社会责任感；在学生形成正确人生观、价值观、世界观等方面发挥独特作用。高中数学课程是义务教育阶段后普通高级中学的主要课程，具有基础性、选择性和发展性。必修课程面向全体学生，构建共同基础；选择性必修课程、选修课程充分考虑学生的不同成长需求，提供多样性的课程供学生自主选择；高中数学课程为学生的可持续发展和终身学习创造条件。（二）基本理念 1.学生发展为本，立德树人，提升素养高中数学课程以学生发展为本，落实立德树人根本任务，培育科学精神和创新意识，提升数学学科核心素养。高中数学课程面向全体学生，实现：人人都能获得良好的数学教育，不同的人在数学上得到不同的发展。 2.优化课程结构，突出主线，精选内容高中数学课程体现社会发展的需求、数学学科的特征和学生的认知规律，发展学生数学学科核心素养。优化课程结构，为学生发展提供共同基础和多样化选择；突出数学主线，凸显数学的内在逻辑和思想方法；精选课程内容，处理好数学学科核心素养与知识技能之间的关系，强调数学与生活以及其他学科的联系，提升学生应用数学解决实际问题的能力，同时注重数学文化的渗透。 3.把握数学本质，启发服考，改进教学高中数学教学以发展学生数学学科核心素养为导向，创设合适的教学情境，启发学生思考，引导学生把握数学内容的本质。提倡独立思考、自主学习、合作交流等多种学习方式，激发学习数学的兴趣，养成良好的学习习惯，促进学生实践能力和创新意识的发展。注重信息技术与数学课程的深度融合，提高教学的实效性。不断引导学生感悟数学的科学价值、应用价值、文化价值和审关价值。 4.重视过程评价，聚焦素养，提高质量高中数学学习评价关注学生知识技能的掌握，更关注数学学科核心素养的形成和发展，制定科学合理的学业质量要求，促进学生在不同学习阶段数学学科核心素养水平的达成。评价既要关注学生学习的结果，更要重视学生学习的过程。开发合理的评价工具，将知识技能的掌握与数学学科核心素养的达成有机结合，建立目标多元、方式多样、重视过程的评价体系。通过评价，提高学生学习兴趣，帮助学生认识自我，增强自信；帮助教师改进教学，提高质量。二、学科核心素养与课程目标（一）学科核心素养学科核心素养是育人价值的集中体现，是学生通过学科学习而逐步形成的正确价值观念、必备品格和关键能力。数学学科核心素养是数学课程目标的集中体现，是具有数学基本特征的思维品质、关键能力以及情感、态度与价值观的综合体现，是在数学学习和应用的过程中逐步形成和发展的。数学学科核心素养包括：数学抽象、逻辑推理、数学建模、直观想象、数学运算和数据分析。这些数学学科核心素养既相对独立、又相互交融，是一个有机的整体。 1.数学抽象数学抽象是指通过对数量关系与空间形式的抽象，得到数学研究对象的素养。主要包括：从数量与数量关系、图形与图形关系中抽象出数学概念及概念之间的关系，从事物的具体背景中抽象出一般规律和结构，并用数学语言予以表征。数学抽象是数学的基本思想，是形成理性思维的重要基础，反映了数学的本质特征，贯穿在数学产生、发展、应用的过程中。数学抽象使得数学成为高度概括、表达准确、结论一般、有序多级的系统。数学抽象主要表现为：获得数学概念和规则，提出数学命题和模型，形成数学方法与思想，认识数学结构与体系。通过高中数学课程的学习，学生能在情境中抽象出数学概念、命题、方法和体系，积累从具体到抽象的活动经验；养成在日常生活和实践中一般性思考问题的习惯，把握事物的本质，以简驭繁；运用数学抽象的思维方式思考并解决问题。 2.逻辑推理逻辑推理是指从一些事实和命题出发，依据规则推出其他命题的素养。主要包括两类：一类是从特殊到一般的推理，推理形式主要有归纳、类比，一类是从一般到特殊的推理，推理形式主要有演绎。逻辑推理是得到数学结论、构建数学体系的重要方式，是数学严谨性的基本保证，是人们在数学活动中进行交流的基本思维品质。逻辑推理主要表现为：掌握推理基本形式和规则，发现问题和提出命题，探索和表述论证过程，理解命题体系，有逻辑地表达与交流。通过高中数学课程的学习，学生能掌握逻辑推理的基本形式，学会有逻辑地思考问题；能够在比较复杂的情境中把握事物之间的关联，把握事物发展的脉络；形成重论据、有条理、合乎逻辑的思维品质和理性精神，增强交流能力。 3.数学建模数学建模是对现实问题进行数学抽象，用数学语言表达问题、用数学方法构建模型解决问题的素养。数学建模过程主要包括：在实际情境中从数学的视角发现问题、提出问题，分析问题、建立模型，确定参数、计算求解，检验结果、改进模型，最终解决实际问题。数学模型搭建了数学与外部世界联系的桥梁，是数学应用的重要形式。数学建模是应用数学解决实际问题的基本手段，也是推动数学发展的动力。数学建模主要表现为：发现和提出问题，建立和求解模型，检验和完善模型，分析和解决问题。通过高中数学课程的学习，学生能有意识地用数学语言表达现实世界，发现和提出问题，感悟数学与现实之间的关联；学会用数学模型解决实际问题，积累数学实践的经验；认识数学模型在科学、社会、工程技术诸多领域的作用，提升实践能力，增强创新意识和科学精神。 4.直观想象直观想象是指借助几何直观和空间想象感知事物的形态与变化，利用空间形式特别是图形，理解和解决数学问题的素养。主要包括：借助空间形式认识事物的位置关系、形态变化与运动规律；利用图形描述、分析数学问题；建立形与数的联系，构建数学问题的直观模型，探索解决问题的思路。直观想象是发现和提出问题、分析和解决问题的重要手段，是探索和形成论证思路、进行数学推理、构建抽象结构的思维基础。直观想象主要表现为：建立形与数的联系，利用几何图形描述问题，借助几何直观理解问题，运用空间想象认识事物。通过高中数学课程的学习，学生能提升数形结合的能力，发展几何直观和空间想象能力；增强运用几何直观和空间想象思考问题的意识；形成数学直观，在具体的情境中感悟事物的本质。 5.数学运算数学运算是指在明晰运算对象的基础上，依据运算法则解决数学问题的素养。主要包括：理解运算对象，掌握运算法则，探究运算思路，选择运算方法，设计运算程序，求得运算结果等。数学运算是解决数学问题的基本手段。数学运算是演绎推理，是计算机解决问题的基础。数学运算主要表现为：理解运算对象，掌握运算法则，探究运算思路，求得运算结果。通过高中数学课程的学习，学生能进一步发展数学运算能力；有效借助运算方法解决实际问题；通过运算促进数学思维发展，形成规范化思考问题的品质，养成一丝不苟、严谨求实的科学精神。 6.数据分析数据分析是指针对研究对象获取数据，运用数学方法对数据进行整理、分析和推断，形成关于研究对象知识的素养。数据分析过程主要包括：收集数据，整理数据，提取信息，构建模型，进行推断，获得结论。数据分析是研究随机现象的重要数学技术，是大数据时代数学应用的主要方法，也是“互联网+”相关领域的主要数学方法，数据分析已经深入到科学、技术、工程和现代社会生活的各个方面。数据分析主要表现为：收集和整理数据，理解和处理数据，获得和解释结论，概括和形成知识。通过高中数学课程的学习，学生能提升获取有价值信息并进行定量分析的意识和能力；适应数字化学习的需要，增强基于数据表达现实问题的意识，形成通过数据认识事物的思维品质，积累依托数据探索事物本质、关联和规律的活动经验。（二）课程目标通过高中数学课程的学习，学生能获得进一步学习以及未来发展所必需的数学基础知识、基本技能、基本思想、基本活动经验（简称“四基”）；提高从数学角度发现和提出问题的能力、分析和解决问题的能力（简称“四能”）。在学习数学和应用数学的过程中，学生能发展数学抽象、逻辑推理、数学建模、直观想象、数学运算、数据分析等数学学科核心素养。通过高中数学课程的学习，学生能提高学习数学的兴趣，增强学好数学的自信心，养成良好的数学学习习惯，发展自主学习的能力；树立敢于质疑、善于思考、严谨求实的科学精神；不断提高实践能力，提升创新意识；认识数学的科学价值、应用价值、文化价值和审美价值。三、课程结构（一）设计依据 1．依据高中数学课程理念，实现“人人都能获得良好的数学教育，不同的人在数学上得到不同的发展”，促进学生数学学科核心素养的形成和发展。 2．依据高中课程方案，借鉴国际经验，体现课程改革成果，调整课程结构，改进学业质量评价。 3．依据高中数学课程性质，体现课程的基础性、选择性和发展性，为全体学生提供共同基础，为满足学生的不同志趣和发展提供丰富多样的课程。 4．依据数学学科特点，关注数学逻辑体系、内容主线、知识之间的关联，重视数学实践和数学文化。（二）结构高中数学课程分为必修课程、选择性必修课程和选修课程。高中数学课程内容突出函数、几何与代数、概率与统计、数学建模活动与数学探究活动四条主线，它们贯穿必修、选择性必修和选修课程。数学文化融入课程内容。高中数学课程结构如下：说明：数学文化是指数学的思想、精神、语言、方法、观点，以及它们的形成和发展；还包括数学在人类生活、科学技术、社会发展中的贡献和意义，以及与数学相关的人文活动。（三）学分与选课 1．学分设置必修课程8学分，选择性必修课程6学分，选修课程6 学分。选修课程的分类、内容及学分如下。 A 类课程包括微积分、空间向量与代数、概率与统计三个专题，其中微积分2.5学分，空间向量与代数2学分，概率与统计1.5 学分。供有志于学习数理类(如数学、物理、计算机、精密仪器等)专业的学生选择。 B类课程包括微积分、空间向量与代数、应用统计、模型四个专题,其中微积分2学分，空间向量与代数1学分，应用统计2学分，模型1学分。供有志于学习经济、社会类(如数理经济、社会学等)和部分理工类(如化学、生物、机械等) 专业的学生选择。 C 类课程包括逻辑推理初步、数学模型、社会调查与数据分析三个专题，每个专题2学分。供有志于学习人文类(如语言、历史等)专业的学生选择。 D类课程包括美与数学、音乐中的数学、美术中的数学、体育运动中的数学四个专题，每个专题1学分。供有志于学习体育、艺术(包括音乐、美术) 类等专业的学生选择。 E 类课程包括拓展视野、日常生活、地方特色的数学课程，还包括大学数学先修课程等。大学数学先修课程包括三个专题:微积分、解析几何与线性代数、概率论与数理统计，每个专题6 学分。 2．课程定位必修课程为学生发展提供共同基础。是高中毕业的数学学业水平考试的内容要求，也是高考的内容要求。选择性必修课程是供学生选择的课程，也是高考的内容要求。选修课程为学生确定发展方向提供引导,为学生展示数学才能提供平台，为学生发展数学兴趣提供选择,为大学自主招生提供参考。 3．选课说明如果学生以高中毕业为目标，可以只学习必修课程，参加高中毕业的数学学业水平考试。如果学生计划通过参加高考进入高等学校学习，必须学习必修课程和选择性必修课程。参加数学高考。如果学生在上述选择的基础上，还希望多学习一些数学课程，可以在选择性必修课程或选修课程中，根据自身未来发展的需求进行选择。在选修课程中可以选择某一类课程，例如，A 类课程; 也可以选择某类课程中的某个专题，例如，E 类大学先修课程中的微积分;还可以选择某些专题的组合，例如，D 类课程中的美与数学、C类课程中的社会调查与数据分析等. 四、课程内容（一）必修课程必修课程包括五个主题，分别是预备知识、函数、几何与代数、概率与统计、数学建模活动与数学探究活动。数学文化融入课程内容。必修课程共8学分144课时，表1给出了课时分配建议，教材编写、教学实施时可以根据实际作适当调整。四、课程内容（一）必修课程必修课程包括五个主题，分别是预备知识、函数、几何与代数、概率与统计、数学建模活动与数学探究活动。数学文化融入课程内容。必修课程共8学分144课时，表1给出了课时分配建议，教材编写、教学实施时可以根据实际作适当调整。表1　必修课程课时分配建议表 \| \| \| \| --- \| 主题 \| 单元 \| 建议课时 \| \| 主题一预备知识 \| 集合 \| 18 \| \| 常用逻辑用语 \| \| 相等关系与不等关系 \| \| 从函数观点看一元二次方程和一元二次不等式 \| \| 主题二函数 \| 函数概念与性质 \| 52 \| \| 幂函数、指数函数、对数函数 \| \| 三角函数 \| \| 函数应用 \| \| 主题三几何与代数 \| 平面向量及其应用 \| 42 \| \| 复数 \| \| 立体几何初步 \| \| 主题四概率与统计 \| 概率 \| 6 \| \| 统计 \| \| 主题五数学建模活动与数学探究活动 \| 数学建模活动与数学探究活动 \| 6 \| \| 机动 \| \| 6 \| 主题一　预备知识以义务教育阶段数学课程内容为载体，结合集合、常用逻辑用语、相等关系与不等关系、从函数观点看一元二次方程和一元二次不等式等内容的学习，为高中数学课程做好学习心理、学习方式和知识技能等方面的准备，帮助学生完成初高中数学学习的过渡。【内容要求】内容包括：集合、常用逻辑用语、相等关系与不等关系、从函数观点看一元二次方程和一元二次不等式。 1．集合在高中数学课程中，集合是刻画一类事物的语言和工具。本单元的学习，可以帮助学生使用集合的语言简洁、准确地表述数学的研究对象，学会用数学的语言表达和交流，积累数学抽象的经验。内容包括：集合的概念与表示、集合的基本关系、集合的基本运算。（1）集合的概念与表示 ①通过实例，了解集合的含义，理解元素与集合的“属于”关系。 ②针对具体问题，能够在自然语言和图形语言的基础上，用符号语言刻画集合。 ③在具体情境中，了解全集与空集的含义。（2）集合的基本关系理解集合之间包含与相等的含义，能识别给定集合的子集。（3）集合的基本运算 ①理解两个集合的并集与交集的含义，能求两个集合的并集与交集。 ②理解在给定集合中一个子集的补集的含义，能求给定子集的补集。 ③能使用Venn图表达集合的基本关系与基本运算，体会图形对理解抽象概念的作用。 2．常用逻辑用语常用逻辑用语是数学语言的重要组成部分，是数学表达和交流的工具，是逻辑思维的基本语言。本单元的学习，可以帮助学生使用常用逻辑用语表达数学对象，进行数学推理，体会常用逻辑用语在表述数学内容和论证数学结论中的作用，提升交流的严谨性与准确性。内容包括：必要条件、充分条件、充要条件，全称量词、存在量词、全称量词命题与存在量词命题的否定。（1）必要条件、充分条件、充要条件 ①通过对典型数学命题的梳理，理解必要条件的意义，理解性质定理与必要条件的关系。 ②通过对典型数学命题的梳理，理解充分条件的意义，理解判定定理与充分条件的关系。 ③通过对典型数学命题的梳理，理解充要条件的意义，理解数学定义与充要条件的关系。（2）全称量词与存在量词通过已知的数学实例，理解全称量词与存在量词的意义。（3）全称量词命题与存在量词命题的否定 ①能正确使用存在量词对全称量词命题进行否定。 ②能正确使用全称量词对存在量词命题进行否定。 3．相等关系与不等关系相等关系、不等关系是数学中最基本的数量关系，是构建方程、不等式的基础。本单元的学习，可以帮助学生通过类比，理解等式和不等式的共性与差异,掌握基本不等式。内容包括：等式与不等式的性质、基本不等式。（1）等式与不等式的性质梳理等式的性质，理解不等式的概念，掌握不等式的性质。（2）基本不等式理解基本不等式。结合具体实例，能用基本不等式解决简单的求最大值或最小值的问题。 4．从函数观点看一元二次方程和一元二次不等式用函数理解方程和不等式是数学的基本思想方法。本单元的学习，可以帮助学生用一元二次函数认识一元二次方程和一元二次不等式。通过梳理初中数学的相关内容，理解函数、方程和不等式之间的联系，体会数学的整体性。内容包括：从函数观点看一元二次方程、从函数观点看一元二次不等式。（1）从函数观点看一元二次方程会结合一元二次函数的图象，判断一元二次方程实根的存在性及根的个数，了解函数的零点与方程根的关系。（2）从函数观点看一元二次不等式 ①经历从实际情境中抽象出一元二次不等式的过程，了解一元二次不等式的现实意义；能够借助一元二次函数求解一元二次不等式；并能用集合表示一元二次不等式的解集。 ②借助一元二次函数的图象，了解一元二次不等式与相应函数、方程的联系（参见案例1）。【教学提示】初中阶段数学知识相对具体，高中阶段数学知识相对抽象。教师应针对这一特征帮助学生完成从初中到高中数学学习的过渡，包括知识与技能、方法与习惯、能力与态度等方面。在集合、常用逻辑用语的教学中，教师应创设合适的教学情境，以义务教育阶段学过的数学内容为载体，引导学生用集合语言和常用逻辑用语梳理、表达学过的相应数学内容。应引导学生理解属于关系是集合的基本关系，了解元素A与元素A组成的集合{A}的差异，即,A与{A}不相同。在梳理过程中，可以针对学生的实际布置不同的任务，采用自主学习与合作学习相结合的方式组织教学活动。在相等关系与不等关系的教学中，应引导学生通过类比学过的等式与不等式的性质，进一步探索等式与不等式的共性与差异。在从函数观点看一元二次方程和一元二次不等式的教学中，可以先以讨论具体的一元二次函数变化情况为情境，引导学生发现一元二次函数与一元二次方程的关系，引出一元二次不等式概念；然后进一步引导学生探索一般的一元二次函数与一元二次方程、一元二次不等式的关系，归纳总结出用一元二次函数解一元二次不等式的程序。教学中，要根据内容的定位和教育价值，关注数学学科核心素养的培养。要让学生逐渐养成借助直观理解概念，进行逻辑推理的思维习惯，以及独立思考、合作交流的学习习惯，引导学生感悟高中阶段数学课程的特征，适应高中阶段的数学学习。学科+网【学业要求】能够在现实情境或数学情境中，概括出数学对象的一般特征，并用集合语言予以表达。初步学会用三种语言（自然语言、图形语言、符号语言）表达数学研究对象，并能进行转换。掌握集合的基本关系与基本运算。在数学表达中的作用。能够从函数的观点认识方程和不等式，感悟函数学知识之间的关联，认识函数的重要性。掌握等式与不等式的性质。重点提升数学抽象、逻辑推理和数学运算素养。主题二　函数函数是现代数学中最基本的概念，是描述客观世界中变量关系和规律的最为基本的数学语言和工具，在解决实际问题汇总发挥重要作用。函数是贯穿高中数学课程的主线。【内容要求】内容包括：函数概念与性质，幂函数、指数函数、对数函数，三角函数，函数应用。 1．函数概念与性质本单元的学习，可以帮助学生建立完整的函数概念，不仅把函数理解为刻画变量之间依赖关系的数学语言和工具，也把函数理解为实数集合之间的对应关系；能用代数运算和函数图象揭示函数的主要性质；在现实问题中，能利用函数构建模型，解决问题。内容包括：函数概念、函数性质、函数的形成与发展。（1）函数概念 ①在初中用变量之间的依赖关系描述函数的基础上，用集合语言和对应关系刻画函数，建立完整的函数概念（参见案例2），体会集合语言和对应关系在刻画函数概念中的作用。了解构成函数的要素，能求简单函数的定义域。 ②在实际情境中，会根据不同的需要选择恰当的方法（如图象法、列表法、解析法）表示函数，理解函数图象的作用。 ③通过具体实例，了解简单的分段函数，并能简单应用。（2）函数性质 ①借助函数图象，会用符号语言表达函数的单调性、最大值、最小值，理解它们的作用和实际意义。 ②结合具体函数，了解奇偶性的概念和几何意义。 ③结合三角函数，了解周期性的概念和几何意义。（3）函数的形成与发展（标有的内容为选学内容，不作为考试要求。）收集函数概念的形成与发展的历史资料，撰写论文，论述函数发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。 2．幂函数、指数函数、对数函数幂函数、指数函数与对数函数是最基本的、应用最广泛的函数，是进一步研究数学的基础。本单元的学习，可以帮助学生学会用函数图象和代数运算的方法研究这些函数的性质；理解这些函数中所蕴含的运算规律；运用这些函数建立模型，解决简单的实际问题，体会这些函数在解决实际问题中的作用。内容包括：幂函数、指数函数、对数函数。（1）幂函数通过具体实例，结合的图象，理解它们的变化规律，了解幂函数。（2）指数函数 ①通过对有理指数幂、实数指数幂（a>0,且，a≠1，x∈R）含义的认识，了解指数幂的拓展过程，掌握指数幂的运算性质。 ②通过具体实例，了解指数函数的实际意义，理解指数函数的概念。 ③能用描点法或借助计算工具画出具体指数函数的图象，探索并理解指数函数的单调性与特殊点。（3)对数函数 ①理解对数的概念和运算性质，知道用换底公式能将一般对数转化成自然对数或常用对数。 ②通过具体实例，了解对数函数的概念。能用描点法或借助计算工具画出具体对数函数的图象，探索并了解对数函数的单调性与特殊点。 ③知道对数函数与指数函数互为反函数（a＞0，且a≠1）。　 ④收集、阅读对数概念的形成与发展的历史资料，撰写小论文，论述对数发明的过程以及对数对简化运算的作用。 3．三角函数三角函数是一类最典型的周期函数。本单元的学习，可以帮助学生在用锐角三角函数刻画直角三角形中边角关系的基础上，借助单位圆建立一般三角函数的概念，体会引入弧度制的必要性；用几何直观和代数运算的方法研究三角函数的周期性、奇偶性（对称性）、单调性和最大（小）值等性质；探索和研究三角函数之间的一些恒等关系；利用三角函数构建数学模型，解决实际问题。内容包括：角与弧度、三角函数概念和性质、同角三角函数的基本关系式、三角恒等变换、三角函数应用。（1）角与弧度了解任意角的概念和弧度制，能进行弧度与角度的互化，体会引入弧度制的必要性（参见案例3）。（2）三角函数概念和性质 ①借助单位圆理解任意角三角函数（正弦、余弦、正切）的定义，能画出这些三角函数的图象，了解三角函数的周期性、奇偶性、最大（小）值。借助单位圆的对称性，利用定义推导出诱导公式（α ±，α ±π的正弦、余弦、正切）。 ②借助图象理解正弦函数在、余弦函数上、正切函数在上的性质。 ③结合具体实例，了解的实际意义；能借助图象理解参数ω，φ，A的意义，了解参数的变化对函数图象的影响。（3）同角三角函数的基本关系式理解同角三角函数的基本关系式。（4）三角恒等变换 ①经历推导两角差余弦公式的过程，知道两角差余弦公式的意义。 ②能从两角差的余弦公式推导出两角和与差的正弦、余弦、正切公式，二倍角的正弦、余弦、正切公式，了解它们的内在联系。 ③能运用上述公式进行简单的恒等变换（包括推导出积化和差、和差化积、半角公式，这三组公式不要求记忆）。（5）三角函数应用会用三角函数解决简单的实际问题，体会可以利用三角函数构建刻画事物周期变化的数学模型（参见案例4）。 4．函数应用函数应用不仅体现在用函数解决数学问题，更重要的是用函数解决实际问题。本单元的学习，可以帮助学生掌握运用函数性质求方程近似解的基本方法（二分法）；理解用函数构建数学模型的基本过程；运用模型思想发现和提出、分析和解决问题。内容包括：二分法与求方程近似解、函数与数学模型。（1）二分法与求方程近似解 ①结合学过的函数图象，了解函数的零点与方程解的关系。 ②结合具体连续函数及其图象的特点，了解函数零点存在定理，探索用二分法求方程近似解的思路并会画程序框图，能借助计算工具用二分法求方程近似解，了解用二分法求方程近似解具有一般性。（2）函数与数学模型 ①理解函数是描述客观世界中变量关系和规律的重要数学语言和工具。在实际情境中，会选择合适的函数类型刻画现实问题的变化规律。 ②结合现实情境中的具体问题，利用计算工具，比较对数函数、一元一次函数、指数函数增长速度的差异，理解“对数增长”“直线上升”“指数爆炸”等术语的现实含义。 ③收集、阅读一些现实生活、生产实际或者经济领域中的数学模型，体会人们是如何借助函数刻画实际问题的，感悟数学模型中参数的现实意义。【教学提示】教师应把本主题的内容视为一个整体，引导学生从变量之间依赖关系、实数集合之间的对应关系、函数图象的几何直观等角度整体认识函数概念；通过梳理函数的单调性、周期性、奇偶性（对称性）、最大（小）值等，认识函数的整体性质；经历运用函数解决实际问题的全过程。函数概念的引入，可以用学生熟悉的例子为背景进行抽象。例如，可以从学生已知的、基于变量关系的函数定义入手，引导学生通过生活或数学中的问题，构建函数的一般概念，体会用对应关系定义函数的必要性，感悟数学抽象的层次。函数单调性的教学，要引导学生正确地使用符号语言刻画函数最本质的性质——单调性（参见案例5）。在函数定义域、值域以及函数性质的教学过程中，应避免编制偏题、怪题，避免繁琐的技巧训练。指数函数的教学，应关注指数函数的运算法则和变化规律，引导学生经历从整数指数到有理指数幂、再到实数指数幂的拓展过程，掌握指数函数的运算法则和变化规律。对数函数的教学，应通过比较同底数的指数函数和对数函数（例如和），认识它们互为反函数。三角函数的教学，应发挥单位圆的作用，引导学生结合实际情境，借助单位圆的直观，探索三角函数的有关性质（参见案例6）。在三角恒等变换的教学中，可以采用不同的方式得到三角恒等变换基本公式；也可以在向量的学习中，引导学生利用向量的数量积推导出两角差的余弦公式。函数应用的教学，要引导学生理解如何用函数描述客观世界事物的变化规律，体会幂函数、指数函数、对数函数、三角函数等函数与现实世界的密切联系（参见案例7）。鼓励学生运用信息技术学习、探索和解决问题。例如，利用计算器、计算机画出幂函数、指数函数、对数函数、三角函数等的图象，探索、比较它们的变化规律，研究函数的性质，求方程的近似解等（参见案例8）。可以组织学生收集、阅读函数的形成与发展的历史资料，结合内容撰写报告，论述函数发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。【学业要求】能够从两个变量之间的依赖关系、实数集合之间的对应关系、函数图象的几何直观等多个角度，理解函数的意义与数学表达；理解函数符号表达与抽象定义之间的关联，知道函数抽象概念的意义。能够理解函数的单调性、最大（小）值，了解函数的奇偶性、周期性；掌握一些基本函数类（一元一次函数、反比例函数、一元二次函数、幂函数、指数函数、对数函数、三角函数等）的背景、概念和性质。能够对简单的实际问题，选择适当的函数构建数学模型，解决问题；能够从函数的观点认识方程，并运用函数的性质求方程的近似解；能够从函数观点认识不等式，并运用函数的性质解不等式。重点提升数学抽象、数学建模、数学运算、直观想象和逻辑推理素养。主题三　几何与代数几何与代数是高中数学课程的主线之一。在必修课程与选择性必修课程中，突出几何直观与代数运算之间的融合，即通过形与数的结合，感悟数学知识之间的关联，加强对数学整体性的理解。【内容标准】内容包括：平面向量及其应用、复数、立体几何初步。 1．平面向量及应用向量理论具有深刻的数学内涵、丰富的物理背景。向量既是代数研究对象，也是几何研究对象，是沟通几何与代数的桥梁。向量是描述直线、曲线、平面、曲面以及高维空间数学问题的基本工具，是进一步学习和研究其他数学领域问题的基础，在解决实际问题中发挥重要作用。本单元的学习，可以帮助学生理解平面向量的几何意义和代数意义；掌握平面向量的概念、运算、向量基本定理以及向量的应用；用向量语言、方法表述和解决现实生活、数学和物理中的问题。内容包括：向量概念、向量运算、向量基本定理及坐标表示、向量应用。（1）向量概念 ①通过对力、速度、位移等的分析，了解平面向量的实际背景，理解平面向量的意义和两个向量相等的含义。 ②理解平面向量的几何表示和基本要素。（2）向量运算 ①借助实例和平面向量的几何表示，掌握平面向量加、减运算及运算规则，理解其几何意义。 ②通过实例分析，掌握平面向量数乘运算及运算规则，理解其几何意义。理解两个平面向量共线的含义。 ③了解平面向量的线性运算性质及其几何意义。 ④通过物理中功等实例，理解平面向量数量积的概念及其物理意义，会计算平面向量的数量积。 ⑤通过几何直观，了解平面向量投影的概念以及投影向量的意义（参见案例9）。 ⑥会用数量积判断两个平面向量的垂直关系。（3）向量基本定理及坐标表示 ①理解平面向量基本定理及其意义。 ②借助平面直角坐标系，掌握平面向量的正交分解及坐标表示。 ③会用坐标表示平面向量的加、减运算与数乘运算。 ④能用坐标表示平面向量的数量积，会表示两个平面向量的夹角。 ⑤能用坐标表示平面向量共线、垂直的条件。（4）向量应用与解三角形 ①会用向量方法解决简单的平面几何问题、力学问题以及其他实际问题，体会向量在解决数学和实际问题中的作用。 ②借助向量的运算，探索三角形边长与角度的关系，掌握余弦定理、正弦定理。 ③能用余弦定理、正弦定理解决简单的实际问题。 2．复数复数是一类重要的运算对象，有广泛的应用。本单元的学习，可以帮助学生通过方程求解，理解引入复数的必要性，了解数系的扩充，掌握复数的表示、运算及其几何意义。内容包括：复数的概念、复数的运算、复数的三角表示。（1）复数的概念 ①通过方程的解，认识复数。 ②理解复数的代数表示及其几何意义，理解两个复数相等的含义。（2）复数的运算掌握复数代数表示的四则运算，了解复数加、减运算的几何意义。（3）复数的三角表示通过复数的几何意义，了解复数的三角表示，了解复数的代数形式与三角表示之间的关系，了解复数乘、除运算的三角表示及其几何意义。 3．立体几何初步立体几何研究现实世界中物体的形状、大小与位置关系。本单元的学习，可以帮助学生以长方体为载体，认识和理解空间点、直线、平面的位置关系；用数学语言表述有关平行、垂直的性质与判定，并对某些结论进行论证；了解一些简单几何体的表面积与体积的计算方法；运用直观感知、操作确认、推理论证、度量计算等认识和探索空间图形的性质，建立空间观念。内容包括：基本立体图形、基本图形位置关系、几何学的发展。（1）基本立体图形 ①利用实物、计算机软件等观察空间图形，认识柱、锥、台、球及简单组合体的结构特征，能运用这些特征描述现实生活中简单物体的结构。 ②知道球、棱柱、棱锥、棱台的表面积和体积的计算公式，能用公式解决简单的实际问题。 ③能用斜二测法画出简单空间图形（长方体、球、圆柱、圆锥、棱柱及其简单组合）的直观图。（2）基本图形位置关系 ①借助长方体，在直观认识空间点、直线、平面的位置关系的基础上，抽象出空间点、直线、平面的位置关系的定义，了解以下基本事实（基本事实1~4也称公理）和定理。基本事实1：过不在一条直线上的三个点，有且只有一个平面。基本事实2：如果一条直线上的两个点在一个平面内，那么这条直线在这个平面内。基本事实3：如果两个不重合的平面有一个公共点，那么它们有且只有一条过该点的公共直线。基本事实4：平行于同一条直线的两条直线平行。定理：如果空间中两个角的两条边分别对应平行，那么这两个角相等或互补。 ②从上述定义和基本事实出发，借助长方体，通过直观感知，了解空间中直线与直线、直线与平面、平面与平面的平行和垂直的关系，归纳出以下判定定理，并加以证明。 ◆一条直线与一个平面平行，如果过该直线的平面与此平面相交，那么该直线与交线平行。 ◆两个平面平行，若果另一个平面与这两个平面相交，那么两条交线平行。 ◆垂直于同一个平面的两条直线平行。 ◆两个平面垂直，如果一个平面内有一条直线垂直于这两个平面的交线，那么这条直线与另一个平面垂直。 ③从上述定义和基本事实出发，借助长方体，通过直观感知，了解空间中直线与直线、直线与平面、平面与平面的平行和垂直的关系，归纳出以下性质定理，并加以证明。 ◆若果平面外一条直线与此平面内的一条直线平行，那么该直线与此平面平行。 ◆如果一个平面内的两条相交直线与另一个平面平行，那么这两个平面平行。 ◆如果一条直线与一个平面内的两条相交直线垂直，那么该直线与此平面垂直。 ◆如果一个平面过另一个平面的垂线，那么这两个平面垂直。 ④能用已获得的结论证明空间基本图形位置关系的简单命题。（3）几何学的发展收集、阅读几何发展的历史资料，撰写小论文，论述几何发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。【教学提示】在平面向量及其应用的教学活动中，应从力、速度、加速度等实际情境入手，从物理、几何、代数三个角度理解向量的概念与运算法则，引导学生运用类比的方法探索实数运算与向量运算的共性与差异，可以通过力的分解引出向量基本定理，建立基底的概念和向量的坐标表示；可以引导学生运用向量解决一些物理和几何问题。例如，利用向量计算力使物体沿某方向运动所作的功，利用向量解决与平面内两条直线平行或垂直有关的问题等。对于向量的非正交分解只要求学生作一般了解，不必展开。在复数的教学中，应注重对复数的表示及几何意义的理解，避免繁琐的计算与技巧训练。对于有余力的学生，可以安排一些引申内容，如复数的三角表示等。可以适当地融入数学文化，让学生体会数系扩充过程中理性思维的作用（参见案例10）。立体几何初步的教学重点是帮助学生逐步形成空间观念，应遵循从整体到局部、从具体到抽象的原则，提供丰富的实物模型或利用计算机软件呈现空间几何体，帮助学生认识空间几何体的结构特征，进一步掌握在平面上表示空间图形的方法和技能。通过对图形的观察和操作，引导学生发现和提出描述基本图形平行、垂直关系的命题，逐步学会用准确的数学语言表达这些命题，直观解释命题的含义和表述证明的思路，并证明其中一些命题，对相应的判定定理只要求直观感知、操作确认，在选择性必修课程中将用向量方法对这些定理加以论证。可以使用信息技术展示空间图形，为理解和掌握图形几何性质（包括证明）提供直观。教师可以指导和帮助学生选择一些立体几何问题作为数学探究活动的课题（参见案例11）。可以组织学生收集、阅读几何学发展的历史资料，结合内容撰写报告，论述几何学发展过程中的重要结果、主要人物、关键事件及其对人类文明的贡献。【学业要求】能够从多种角度理解向量概念和运算法则，掌握向量基本定理；能够运用向量运算解决简单的几何和物理问题，知道数学运算与逻辑推理的关系。能够理解复数的概念，掌握复数代数表示式的四则运算。能够通过直观图理解空间图形，掌握基本空间图形及其简单组合体的概念和基本特征，解决简单的实际问题。能够运用图形的概念描述图形的基本关系和基本结果。能够证明简单的几何命题（平行、垂直的性质定理），并会进行简单应用。重点提升直观想象、逻辑推理、数学运算和教学抽象素养。主题四概率与统计概率的研究对象是随机现象，为人们从不确定性的角度认识客观世界提供重要的思维模式和解决问题的方法。统计的研究对象是数据，核心是数据分析。概率为统计的发展提供理论基础。【内容要求】内容包括：概率、统计。 1．概率本单元的学习，可以帮助学生结合具体实例，理解样本点、有限样本空间、随机事件，会计算古典概型中简单随机事件的概率，加深对随机现象的认识和理解。内容包括：随机事件与概率、随机事件的独立性。（1）随机事件与概率 ①结合具体实例，理解样本点和有限样本空间的含义，理解随机事件与样本点的关系（参见案例12）。了解随机事伴的并、交与互斥的含义，能结合实例进行随机事件的并、交运算。 ②结合具体实例，理解古典概型，能计算古典概型中简单随机事件的概率。 ③通过实例，理解概率的性质，掌握随机事件概率的运算法则。 ④结合实例，会用频率估计概率。（2）随机事件的独立性结合有限样本空间，了解两个随机事件独立性的含义。结合古典概型，利用独立性计算概率。 2．统计本单元的学习，可以帮助学生进一步学习数据收集和整理的方法、数据直观图表的表示方法、数据统计特征的刻画方法，通过具体实例，感悟在实际生活中进行科学决策的必要性和可能性；体会统计思维与确定性思维的差异、归纳推断与演绎证明的差异；通过实际操作、计算机模拟等活动，积累数据分析的经验。内容包括：获取数据的基本途径及相关概念、抽样、统计图表、用样本估计总体。（1）获取数据的基本途径及相关概念 ①知道获取数据的基本途径，包括：统计报表和年鉴、社会调查、试验设计、普查和抽样、互联网等。 ②了解总体、样本、样本量的概念，了解数据的随机性。（2）抽样 ①简单随机抽样通过实例，了解简单随机抽样的含义及其解决问题的过程，掌握两种简单随机抽样方法：抽签法和随机数法。会计算样本均值和样本方差，了解样本与总体的关系。 ②分层随机抽样通过实例，了解分层随机抽样的特点和适用范围，了解分层随机抽样的必要性，掌握各层样本量比例分配的方法。结合具体实例，掌握分层随机抽样的样本均值和样本方差（参见案例13）。 ③抽样方法的选择在简单的实际情境中，能根据实际问题的特点，设计恰当的抽样方法解决问题。（3）统计图表如根据实际问题的特点，选择恰当的统计图表对数据进行可视化描述，体会合理使用统计图表的重要性。（4）用样本估计总体 ①结合实例，能用样本估计总体的集中趋势参数（平均数、中位数、众数），理解集中趋势参数的统计含义。 ⑦结合实例，能用样本估计总体的离散程度参数（标准差、方差、极差），理解离散程度参数的统计含义。 ③结合实例，能用样本估计总体的取值规律。 ④结合实例，能用样本估计百分位数，理解百分位数的统计含义（参见案例14）。【教学提示】在概率的教学中，应引导学生通过日常生活中的实例了解随机事件与概率的意义。在随机事件和样本空间的教学中，应引导学生通过古典概型，认识样本空间，理解随机事件发生的含义；理解古典概型的特征，试验结果的有限性和每一个试验结果出现的等可能性，知道只有在这种特征下，才能定义出古典概型中随机事件发生的概率。教学中要适当介绍基本计数方法（如树状图、列表等），计算古典概率中随机事件发生的概率。在统计的教学中，应引导学生根据实际问题的需求，选择不同的抽样方法获取数据，理解数据蕴含的信息，根据数据分析的需求，选择适当的统计图表描述和表达数据，并从样本数据中提取需要的数字特征，估计总体的统计规律，解决相应的实际问题．对统计中的基本概念（如总体、样本、样本量等），应结合具体问题进行描述性说明，在此基础上适当引入严格的定义，并利用数字特征（平均值、方差等）和数据直观图表（直方图、散点图等）进行数据分析。统计学的教学活动应通过典型案例进行。教学中应通过对一些典型案例的处理，使学生经历较为系统的数据处理全过程，在此过程中学习数据分析的方法，理解数据分析的思路，运用所学知识和方法解决实际问题。可以鼓励学生尽可能运用计算器、计算机进行模拟活动，处理数据，更好地体会概率的意义和统计思想。例如，利用计算器产生随机数来模拟掷硬币试验等，利用计算机来计算样本量较大的数据的样本均值、样本方差等。【学业要求】能够掌握古典概率的基本特征，根据实际问题构建概率模型，解决简单的实际问题。能够借助古典概型初步认识有限样本空间、随机事件，以及随机事件的概率。能够根据实际问题的需求，选择恰当的抽样方法获取样本数据，并从中提取需要的数字特征推断总体，能够正确运用数据分析的方法解决简单的实际问题。能够区别统计思维与确定性思维的差异、归纳推断与演绎证明的差异。能够结合具体问题，理解统计推断结果的或然性，正确运用统计结果解释实际问题。重点提升数据分析、数学建模、逻辑推理和数学运算素养。主题五数学建模活动与数学探究活动【内容要求】数学建模活动是对现实问题进行数学抽象，用数学语言表达问题、用数学方法构建模型解决问题的过程。主要包括：在实际情境中从数学的视角发现问题、提出问题，分析问题、构建模型，确定参数、计算求解，检验结果、改进模型，最终解决实际问题。数学建摸活动是基本数学思维运用模型解决实际问题的一类综合实践活动，是高中阶段数学课程的重要内容。数学建模活动的基本过程如下：数学探究活动是围绕某个具体的数学问题，开展自主探究、合作研究并最终解决问题的过程。具体表现为：发现和提出有意义的数学问题，猜测合理的数学结论，提出解决问题的思路和方案，通过自主探索、合作研究论证数学结论。数学探究活动是运用数学知识解决数学问题的一类综合实践活动，也是高中阶段数学课程的重要内容。数学建模活动与数学探究活动以课题研究的形式开展，在必修课程中，要求学生完成其中的一个课题研究．【教学提示】课题可以由教师给定，也可以由学生与教师协商确定，课题研究的过程包括选题、开题，做题、结题四个环节。学生需要撰写开题报告，教师要组织开展开题交流活动，开题报告应包括选题意义、文献综述、解决问题思路、研究计划、预期结果等。做题是解决问题的过程，包括描述问题、教学表达、建立模型、求解模型、得到结论、反思完善等。结题包括撰写研究报告和报告研究结果，由教师组织学生开展结题答辩。根据选题的内容，报告可以采用专题作业、测量报告、算法程序、制作的实物、研究报告或小论文等多种形式（参见案例15）在数学建模活动与数学探究活动中，鼓励学生使用信息技术。【学业要求】经历数学建模活动与数学探究活动的全过程，整理资料，撰写研究报告或小论文，并进行报告、交流。对于研究报告或小论文的评价，教师应组织评价小组，可以邀请校外专家、社会人士、家长等参与评价，也可以组织学生互评。教师要引导学生遵循学术规范，坚守诚信底线。研究报告或小论文及其评价应存入学生个人学习档案，为大学招生提供参考和依据。学生可以采取独立完成或者小组合作（2~3人为宜）的方式，完成课题研究（参见案例19）。重点提升数学建模、数学抽象、数据分析、数学运算、逻辑推理和直观形象素养。（二）选择性必修课程选择性必修课程包括四个主题，分别是函数、几何与代数、概率与统计、数学建模活动与数学探究活动。数学文化融入课程内容。选择性必修课程共6学分108课时，表2给出了课时分配建议，教材编写、教学实施时可以根据实际作适当调整。表2选择性必修课程课时分配表主题单元建议课时主题一函数数列 30 一元函数导数及其应用主题二几何与代数空间向量与立体几何 44 平面解析几何主题三概率与统计计数原理 26 概率统计主题四数学建模活动与数学探究活动数学建模活动与数学探究活动 4 机动 4 主题一函数在必修课程中，学生学习了函数的概念和性质，总结了研究函数的整本方法，掌握了一些具体的基本函数类，探索了函数的应用。在本主题中，学生将学习数列和一元函数导数及其应用。数列是一类特殊的函数，是数学重要的研究对象，是研究其他类型函数的基本工具，在日常生活中也有着广泛的应用。导数是微积分的核心内容之一，是现代数学的基本概念，蕴含微积分的基本思想，导数定量地刻画了函数的局部变化，是研究函数性质的基本工具。【内容要求】内容包括：数列、一元函数导数及其应用。 1．数列本单元的学习，可以帮助学生通过对日常生活中实际问题的分析，了解数列的概念；探索并掌握等差数列和等比数列的变化规律，建立通项公式和前n项和公式：能运用等差数列、等比数列解决简单的实际问题和数学问题，感受数学模型的现实意义与应用；了解等差数列与一元一次函数、等比数列与指数函数的联系，感受数列与函数的共性与差异，体会数学的整体性。内容包括，数列概念、等差数列、等比数列、数学归纳法。（1）数列概念通过日常生活和数学中的实例，了解数列的概念和表示方法（列表、图象、通项公式），了解数列是一种特殊函数。（2）等差数列 ①通过生活中的实例，理解等差数列的概念和通项公式的意义。 ②探索并掌握等差数列的前n项和公式，理解等差数列的通项公式与前n项和公式的关系。 ③能在具体的问题情境中，发现数列的等差关系，并解决相应的问题。 ④体会等差数列与一元一次函数的关系。（3）等比数列 ①通过生活中的实例，理解等比数列的概念和通项公式的意义。 ②探索并掌握等比数列的前n项和公式，理解等比数列的通项公式与前n项和公式的关系。 ③能在具体的问题情境中，发现数列的等比关系，并解决相应的问题。 ④体会等比数列与指数函数的关系。（4）数学归纳法了解数学归纳法的原理，能用数学归纳法证明数列中的一些简单命题。 2.一元函数导数及其应用本单元的学习，可以帮助学生通过丰富的实际背景理解导数的概念，掌握导数的基本运算，运用导数研究函数的性质，并解决一些实际问题。内容包括：导数概念及其意义、导数运算、导数在研究函数中的应用、微积分的创立与发展。（1）导数概念及其意义 ①通过实例分析，经历由平均变化率过渡到瞬时变化率的过程，了解导数概念的实际背景，知道导数是关于瞬时变化率的数学表达，体会导数的内涵与思想。 ②体会极限思想。 ③通过函数图象直观理解导数的几何意义。（2）导数运算 ①能根据导数定义求函数y=c，y=x，y=x2，y=x3，y=，y=的导数。 ②能利用给出的基本初等函数的导数公式和导数的四则运算法则，求简单函数的导数；能求简单的复合函数（限于形如f（ax+b））的导数。 ③会使用导数公式表。（3）导数在研究函数中的应用 ①结合实例，借助几何直观了解函数的单调性与导数的关系，能利用导数研究函数的单调性；对于多项式函数，能求不超过三次的多项式函数的单调区间。 ②借助函数的图象，了解函数在某点取得极值的必要条件和充分条件；能利用导数求某些函数的极大值、极小值以及给定闭区间上不超过三次的多项式函数的最大值、最小值，体会导数与单调性、极值、最大（小）值的关系。（4）微积分的创立与发展收集、阅读对最积分的创立和发展起重大作用的有关资料，包括一些量要历史人物（牛顿、莱布尼茨、柯西、魏尔斯特拉斯等）和事件，采取独立完成或者小组合作的方式。完成一篇有关微积分创立与发展的研究报告。【教学提示】在数列的教学中，应引导学生通过具体实例（如购房贷款、放射性物质的衰变、人口增长等），理解等差数列、等比数列的概念、性质和应用，引导学生掌握数列中各个量之间的基本关系。应特别强调数列作为一类特殊的函数在解决实际问题中的作用，突出等差数列、等比数列的本质，引导学生通过类比的方法探索等差数列与一元一次函数、等比数列与指数函数的联系，加深对数列及函数概念的理解。在教学中可以组织学生收集、阅读数列方面的研究成果，特别是我国古代的优秀研究成果，如“杨辉三角”、《四元玉鉴》等，撰写小论文，论述数列发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献，感悟我国古代数学的辉煌成就。在一元函数导数及其应用的教学中，应通过丰富的实际背景和具体实例引入导数的概念，例如斜率、增长率、膨胀率、效率、密度、速度、加速度等，应引导学生经历由平均变化率过渡到瞬时变化率的过程，了解导数是如何刻画瞬时变化率的，感悟极限的思想；应引导学生通过具体实例感受导数在研究函数和解决实际问题中的作用，体会导数的意义。学生对导数概念的理解不可能一步到位，导数概念的学习应该贯穿在一元函数导数及其应用学习的始终。一般地，在高中阶段研究与导数有关的问题中，涉及的函数部是可导函数。在教学中可以组织学生收集、阅读微积分创立与发展的历史资料，撰写小论文，论述微积分创立与发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。【学业要求】能够结合具体实例，理解通项公式对于数列的重要性，知道通项公式是这类函数的解析表达式；通过等差数列和等比数列的研究，感悟数列是可以用来刻画现实世界中一类具有递推规律事物的数学模型。掌握通项公式与前n项和公式的关系；能够运用数列解决简单的实际问题。能够通过具体情境，直观理解导数概念，感悟极限思想，知道极限思想是人类深刻认识和表达现实世界必备的思维品质。理解导数是一种借助极限的运算，掌握导数的基本运算规则，能求简单函数和简单复合函数的导数。能够运用导数研究简单函数的性质和变化规律，能够利用导数解决简单的实际问题。知道微积分创立过程，以及微积分对数学发展的作用。重点提升数学抽象、数学运算、直观想象、数学建模和逻辑推理素养。主题二几何与代数在必修课程学习平面向量的基础上，本主题将学习空间向量，并运用空间向量研究立体几何中图形的位置关系和度量关系。解析几何是数学发展过程中的标志性成果，是微积分创立的基础。本主题将学习平面解析几何，通过建立坐标系，借助直线、圆与圆锥曲线的几何特征，导出相应方程；用代数方法研究它们的几何性质，体现形与数的结合。【内容要求】内容包括：空间向量与立体几有、平面解析几何。 1．空间向量与立体几何本单元的学习，可以帮助学生在学习平面向量的基础上，利用类比的方法理解空间向量的概念、运算、基本定理和应用，体会平面向量和空间向量的共性和差异，运用向量的方法研究空间基本图形的位置关系和度量关系，体会向量方法和综合几何方法的共性和差异，运用向量方法解决简单的数学问题和实际问题，感悟向量是研究几何问题的有效工具。内容包括：空间直角坐标系、空间向量及其运算、向量基本定理及坐标表示、空间向量的应用。（1）空间直角坐标系 ①在平面直角坐标系的基础上，了解空间直角坐标系，感受建立空间直角坐标系的必要性，会用空间直角坐标系刻画点的位置。 ②借助特殊长方体（所有被分别与坐标轴平行)顶点的坐标。探索并得出空间两点间的距离公式。（2）空间向量及其运算 ①经历由平面向量推广到空间向量的过程，了解空间向量的概念。 ②经历由平面向量的运算及其法则推广到空间向量的过程。（3）向量基本定理及坐标表示 ①了解空间向量基本定理及其意义，掌握空间向量的正交分解及其坐标表示。 ②掌握空间向量的线性运算及其坐标表示。 ③掌握空间向量的数量积及其坐标表示。 ④了解空间向量投影的概念以及投影向量的意义（参见案例9）。（4）空间向量的应用 ①能用向量语言指述直线和平面，理解直线的方向向量与平面的法向量。 ②能用向量语言表述直线与直线、直线与平面、平面与平面的夹角以及垂直与平行关系。 ③能用向量方法证明必修内容中有关直线、平面位置关系的判定定理。 ④能用向量方法解决点到直线、点到平面、相互平行的直线、相互平行的平面的距离问题（参见案例16）和简单夹角问题，并能描述解决这一类问题的程序，体会向量方法在研究几何问题中的作用。 2．平面解析几何本单元的学习，可以帮助学生在平面直角坐标系中，认识直线、围、椭圆、抛物线、双曲线的几何特征，建立它们的标准方程；运用代数方法进一步认识圆锥曲线的性质以及它们的位置关系，运用平面解析几何方法解决简单的数学问题和实际问题，感悟平面解析几何中蕴含的数学思想。内容包括：直线与方程、圆与方程、圆锥曲线与方程、平面解析几何的形成与发展。（1）直线与方程 ①在平面直角坐标系中，结合具体图形，探索确定直线位置的几何要素。 ②理解直线的倾斜角和斜率的概念，经历用代数方法刻画直线斜率的过程，掌握过两点的直线斜率的计算公式。 ③能根据斜率判定两条直线平行或垂直。 ④根据确定直线位置的几何要素，探索并掌握直线方程的几种形式（点斜式、两点式及一般式）。 ⑤能用解方程组的方法求两条直线的交点坐标。 ⑥探索并掌握平面上两点间的距离公式、点到直线的距离公式，会求两条平行直线间的距离。（2）圆与方程 ①回顾确定圆的几何要素，在平面直角坐标系中，探索并掌握圆的标准方程与一般方程。 ②能根据给定直线、圆的方程，判断直线与圆、圆与圆的位置关系。 ③能用直线和圆的方程解决一些简单的数学问题与实际问题。（3）圆锥曲线与方程 ①了解圆锥曲线的实际背景，感受圆锥曲线在刻画现实世界和解决实际问题中的作用。 ②经历从具体情境中抽象出椭圆的过程，掌握椭圆的定义、标准方程及简单几何性质。 ③了解抛物线与双曲线的定义、几何图形和标准方程，以及它们的简单几何性质。 ④通过圆锥曲线与方程的学习，进一步体会数形结合的思想。 ⑤了解椭圆、抛物线的简单应用。（4）平面解析几何的形成与发展收集、阅读平面解析几何的形成与发展的历史资料，撰写小论文、论述平面解析几何发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。【教学提示】本主题的研究对象是几何图形，所用的研究方法主要是代数方法。在空间向量与立体几何的教学中，应重视以下两方面；第一，引导学生运用类比的方法，经历向量及其运算由平面向空间的推广过程，探素空间向量与平面向量的共性和差异，引发学生思考维数增加所带来的影响：第二，鼓励学生灵活选择运用向量方法与综合几何方法，从不同角度解决立体几何问题（如距离间题），通过对比体会向量方法的优势。在上述过程中，引导学生理解向量基本定理的本质，感悟“基”的思想，并运用它解决立体几何中的问题。在平面解析几何的教学中，应引导学生经历以下过程：首先，通过实例了解几何图形的背景，例如，通过行星运行轨道、抛物运动轨迹等，使学生了解圆锥曲线的背景与应用；进而，结合情境清晰地描述图形的几何特征与问题，例如，两点决定一条直线，椭圆是到两个定点的距离之和为定长的动点的轨迹等，再结合具体问题合理地建立坐标系，用代数语言描述这些特征与问题；最后，借助几何图形的特点，形成解决问题的思路，通过直观想象和代数运算得到结果，并给出几何解释，解决问题。应充分发挥信息技术的作用，通过计算机软件向学生演示方程中参数的变化对方程所表示的曲线的影响，使学生进一步理解曲线与方程的关系。在教学中，可以组织学生收集、阅读平面解析几何的形成与发展的历史资料，撰写小论文，论述平面解析几何发展的过程、重要结果、主要人物、关键事件及其对人类文明的贡献。【学业要求】能够理解空间向量的概念、运算、背景和作用；能够依托空间向量建立空间图形及图形关系的想象力；能够掌握空间向量基本定理，体会其作用，并能简单应用；能够运用空间向量解决一些简单的实际问题，体会用向量解决一类问题的想路。能够掌握平面解析几何解决问题的基本过程：根据具体问题情境的特点，建立平面直角坐标系；根据几何问题和图形的特点，用代数语言把几何问题转化成为代数问题；根据对几何问题（图形）的分析，探索解决问题的思路，运用代数方法得到结论，给出代数结论合理的几何解释，解决几何问题。能够根据不同的情境，建立平面直线和圆的方程，建立椭圆、抛物线、双曲线的标准方程，能够运用代数的方法研究上述曲线之间的基本关系，能够运用平面解析几何的思想解决一些简单的实际问题。重点提升直观想象、数学运算、数学建模、逻辑推理和数学抽象素养。主题三　概率与统计本主题是必修课程中概率与统计内容的延续，将学习计数原理、概率、统计的相关知识。计数原理的内容包括两个基本计数原理、排列与组合、二项式定理。概率的内容包括随机事件的条件概率、离散型随机变量及其分布列、正态分布。统计的内容包括成对数据的统计相关性、一元线性回归模型、2×2列联表。【内容要求】内容包括：计数原理、概率、统计。 1．计数原理分类加法计数原理和分步乘法计数原理是解决计数问题的基础，称为基本计数原理。本单元的学习，可以帮助学生理解两个基本计数原理，运用计数原理探索排列、组合、二项式定理等问题。内容包括：两个基本计数原理、排列与组合、二项式定理。（1）两个基本计数原理通过实例，了解分类加法计数原理、分步乘法计数原理及其意义。（2）排列与组合通过实例，理解排列、组合的概念，能利用计数原理推导排列数公式、组合数公式。（3）二项式定理能用多项式运算法则和计数原理证明二项式定理（参见案例17，18），会用二项式定理解决与二项展开式有关的简单问题。 2．概率本单元的学习，可以帮助学生了解条件概率及其与独立性的关系，能进行简单计算；感悟离散型随机变量及其分布列的含义，知道可以通过随机变量更好地刻画随机现象；理解伯努利试验，掌握二项分布，了解超几何分布；感悟服从正态分布的随机变量，知道连续型随机变量；基于随机变量及其分布解决简单的实际问题。内容包括，随机事件的条件概率、离散型随机变量及其分布列、正态分布。（1）随机事件的条件概率 ①结合古典概型，了解条件概率，能计算简单随机事件的条件概率。 ②结合古典概型，了解条件概率与独立性的关系。 ③结合古典概型，会利用乘法公式计算概率。 ④结合古典概型，会利用全概率公式计算概率。了解贝叶斯公式。（2）离散型随机变量及其分布列 ①通过具体实例，了解离散型随机变量的概念，理解离散型随机变量分布及其数字特征（均值、方差）。 ②通过具体实例，了解伯努利试验，掌握二项分布及其数字特征，并能解决简单的实际问题。 ③通过具体实例，了解超几何分布及其均值，并能解决简单的实际问题。（3）正态分布 ①通过误差模型，了解服从正态分布的随机变量。通过具体实例、借助频率直方图的几何直观，了解正态分布的特征。 ②了解正态分布的均值、方差及其含义。 3．统计本单元的学习，可以帮助学生了解样本相关系数的统计含义，了解一元线性回归模型和2×2列联表，运用这些方法解决简单的实际问题。会利用统计软件进行数据分析。内容包括：成对数据的统计相关性、一元线性回归模型、2×2列联表。（1）成对数据的统计相关性 ①结合实例，了解样本相关系数的统计含义，了解样本相关系数与标准化数据向量夹角的关系。 ②结合实例，会通过相关系数比较多组成对数据的相关性。（2）一元线性回归模型 ①结合具体实例，了解一元线性回归模型的含义，了解模型参数的统计意义，了解最小二乘原理，掌握一元线性回归模型参数的最小二乘估计方法，会使用相关的统计软件。 ②针对实际问题，会用一元线性回归模型进行预测。（3）2×2列联表 ①通过实例，理解2×2列联表的统计意义。 ②通过实例，了解2×2列联表独立性检验及其应用。【教学提示】教师应通过典型案例开展教学活动，案例的情境应是丰富的、有趣的、学生熟悉的。在案例教学中要重视过程，层次清楚，从具体到抽象，从实际到理论。在计数原理的教学中，应结合具体情境，引导学生理解许多计数问题可以归结为分类和分步两类问题，引导学生根据计数原理分析问题、解决问题。在概率的教学中，应引导学生通过具体实例，理解可以用随机变量更好地刻画随机现象，感悟随机变量与随机事件的关系；理解随机事件独立性与条件概率之间的关系；通过二项分布、超几何分布、正态分布的学习，理解随机变量及其分布。在教学过程中，应在引导学生利用所学知识解决一些实际问题的基础上，适当进行严格、准确的描述。在统计的教学中，应通过具体案例，引导学生理解两个随机变量的相关性可以通过成对样本数据进行分析；理解利用一元线性回归模型可以研究变量之间的随机关系，进行预测；理解利用2×2列联表可以检验两个随机变量的独立性。在教学过程中，应通过具体案例引导学生参与数据分析的全过程，并鼓励学生使用相应的统计软件。【学业要求】能够结合具体实例，识别和理解分类加法计数原理和分步乘法计数原理及其作用，并能够运用这些原理解决简单的实际问题。能够结合具体实例，理解排列、组合、二项式定理与两个计数原理的关系，能够运用两个计数原理推导排列、组合、二项式定理的相关公式，并能够运用它们解决简单的实际问题，特别是概率中的某些问题。能够结合具体实例，理解随机事件的独立性和条件概率的关系，理解离散型随机变量在描述随机现象中的作用，掌握两个基本概率模型及其应用，了解正态分布的作用，进一步深入理解随机思想在解决实际问题中的作用。能够解决成对数据统计相关性的简单实际问题。能够结合具体实例，掌握运用一元线性回归分析的方法。掌握运用2×2列联表的方法，解决独立性检验的简单实际问题。重点提升数据分标、数学建模、逻辑推理、数学运算和数学抽象素养。主题四　数学建模活动与数学探究活动【内容要求】数学建模活动与数学探究活动以课题研究的形式开展。在选择性必修课程中，要求学生完成一个课题研究，可以是数学建模的课题研究，也可以是数学探究的课题研究。课题可以是学生在学习必修课程时已完成课题的延续，或者是新的课题。【教学提示】选题可以在教师的指导下，自主选题，也可以在必修课程中数学建模活动或数学探究活动的研究基础上继续进行深入探究。类似必修课程的要求，课题研究应经历选题、开题、做题、结题四个环节。如果选题不变，需要在研究报告中说明与必修课程中研究的差异，深入研究的新思路、新方法，得到的新结果。根据选题的内容，报告可以采用专题作业、测量报告、算法程序、制作的实物或研究论文等多种形式。【学业要求】参考必修课程的主题五。（三）选修课程选修课程是由学校根据自身情况选择设置的课程，供学生依据个人志趣自主选择，分为A,B,C,D,E五类。这些课程为学生确定发展方向提供引导，为学生展示数学才能提供平台，为学生发展数学兴趣提供选择，为大学自主招生提供参考。学生可以根据自己的志向和大学专业的要求选择学习其中的某些课程。 A类课程是供有志于学习数理类（如数学、物理、计算机、精密仪器等）学生选择的课程。 B类课程是供有志于学习经济、社会类（如数理经济、社会学等）和部分理工类（如化学、生物、机械等）学生可以选择的课程。 C类课程是供有志于学习人文类（如语言、历史等）学生选择的课程。 D类课程是供有志于学习体育、艺术（包括音乐、美术）类学生选择的课程。 E 类课程包括拓展视好、日常生活、地方特色的数学课程，还包括大学数学的先修课程等。大学数学先修课程包括: 微积分、解析几何与线性代数、概率论与数理统计。数学建模活动、数学探究活动、数学文化融入课程内容。选修课程的修习情况应列为综合素质评价的内容。不同高等院校、不同专业的招生，根据需要可以对选修课程中某些内容提出要求。国家、地方政府、社会权威机构可以组织命题考试。考试成绩应存入学生个人学习档案，供高等院校自主招生参考。 A类课程 A类课程包括微积分、空间向量与代数、概率与统计三个专题，其中微积分2.5学分，空间向量与代数2学分，概率与统计1.5学分。微积分本专题在数列极限的基础上建立函数极限和连续的概念；在具体的情境中用极限刻画导数，给出借助导数研究函数性质的一般方法；通过极限建立微分和积分的概念，阐述微分和积分的关系（微积分基本定理）及其应用。本专题要考虑高中学生的接受能力，重视课程内容的实际背景，关注数学内容的直观理解，培养学生的数学抽象、数学运算、数学建模和逻辑推理素养，为进一步学习大学数学课程奠定基础。内容包括：数列极限、函数极限、连续函数、导数与微分、定积分。 1．数列极限（1）通过典型收敛数列的极限过程（当时，，，），建立并理解数列极限的定义。（2）探索并证明基本性质：收敛数列是有界数列。（3）通过典型单调有界数列的收敛过程，理解基本事实：单调有界数列必有极限。（4）掌握数列极限的四则运算法则。（5）通过典型数列的收敛性，理解e的意义。 2．函数极限（1）通过典型函数的极限过程（当时，；当时，；当时，，且），理解函数极限的ε-δ定义。（2）掌握基本初等函数极限的四则运算。（3）掌握两个重要函数极限：，并会求其简单变形的极限。 3．连续函数（1）理解连续函数的定义。（2）了解闭区间上连续函数的有界性、介值性及其简单应用（例如，用二分法求方程近似解）。 4．导数与微分（1）借助物理背景与几何背景理解导数的意义，并能给出导数的严格数学定义。（2）通过导函数的概念，掌握二阶导数的概念，了解二阶导数的物理意义与几何意义。（3）了解复合函数的求导公式。（4）理解并证明拉格朗日中值定理，并能用其讨论函数的单调性。（5）会利用拉格朗日中值定理，证明一些函数不等式（例如，当时，有，）。（6）会利用导数讨论函数的极值问题，利用几何图形说明一个点是极值点的必要条件与充分条件（不要求数学证明）。（7）了解微分的概念及其实际意义，并会用符号表示。 5．定积分（1）通过等分区间求特殊曲边梯形面积的极限过程，理解定积分的概念及其几何意义与物理意义。（2）在单调函数定积分的计算过程中，通过微分感悟积分与导数的关系，理解并掌握牛顿-莱布尼茨公式。（3）会利用导数表和牛顿-莱布尼茨公式，求一些简单函数的定积分。（4）会利用定积分计算某些封闭图形的面积，计算球、圆锥、圆台和某些三棱锥、三棱台的体积；能利用定积分解决简单的作功问题和重心问题。空间向量与代数本专题在必修课程和选择性必修课程的基础上，通过系统学习三维空间的向量代数，表述各种运算的几何背景，实现几何与代数的融合。引入矩阵与行列式的概念，利用矩阵理论解三元一次方程组；利用向量代数，讨论三维空间中点、直线、平面的位置关系与度量；利用直观想象建立平面和空间的等距变换理论。将空间几何与线性代数融合在一起，把握问题的本质，为代数理论提供几何背景，用代数方法解决几何问题，进而解决实际问题，为大学线性代数课程的学习奠定直观基础。内容包括：空间向量代数、三阶矩阵与行列式、三元一次方程组、空间中的平面与直线、等距变换。 1．空间向量代数（1）通过几何直观，理解向量运算的几何意义。（2）探索并解释空间向量的内积与外积及其几何意义。（3）理解向量的投影与分解及其几何意义，并会应用。（4）掌握向量组的线性相关性，并能加以判断。（5）掌握向量的线性运算，理解向量空间与子空间的概念。 2．三阶矩阵与行列式（1）通过几何直观引入矩阵概念，掌握矩阵的三种基本运算及其性质。（2）了解正交矩阵及其基本性质，能用代数方法解决几何问题。（3）掌握行列式定义与性质，会计算行列式。 3．三元一次方程组（1）通过实例，探索三元一次方程组的求解过程，理解三元一次方程组的常用解法（高斯消元法），会用矩阵表示三元一次方程组。（2）掌握三元齐次线性方程组的解法，会表示一般解。（3）掌握非齐次线性方程组有解的判定，建立线性方程组的理论基础。（4）探索三元一次方程组解的结构，会表示一般解。（5）理解克拉默（Cramer）法则，会用克拉默法则求解三元一次方程组。 4．空间中的平面与直线（1）通过向量的坐标表示，建立空间平面的方程。（2）掌握空间直线方程的含义，会用方程表示空间直线。（3）理解空间点、直线、平面的位置关系，会用代数方法判断空间点、直线、平面的位置关系，会求点到直线（平面）的距离。 5．等距变换（1）了解平面变换的含义，理解平面的等距变换，特别是三种基本等距变换：直线反射、平移、旋转。（2）了解平面对称图形及变换群概念。（3）掌握常见平面等距变换及其矩阵表示。（4）了解空间变换的含义，理解空间的等距变换，特别是三种常见等距变换：平面反射、平移、旋转。（5）了解空间对称图形及变换群。（6）掌握常见空间等距变换及其矩阵表示。概率与统计本专题在必修课程和选择性必修课程的基础上展开。在概率方面，通过具体实例，进一步学习连续型随机变量及其概率分布，二维随机向量及其联合分布，并运用这些数学模型，解决一些简单的实际问题。在统计方面，结合一些具体任务，学习参数估计、假设检验，并运用这些方法解决一些简单的实际问题；在一元线性回归分析的基础上，结合具体实例，进一步学习二元线性回归分析的方法，解决一些简单的实际问题。在教学活动中，要重视课程内容的实际背景，关注学生对数学内容的直观理解；要充分考虑高中学生接受能力，更要注重学生数学学科核心素养的提升。内容包括：连续型随机变量及其分布、二维随机变量及其联合分布、参数估计、假设检验、二元线性回归模型。 1．连续型随机变量及其分布（1）借助具体实例，了解连续型随机变量及其分布，体会连续型随机变量与离散型随机变量的共性与差异。（2）结合生活中的实例，了解几个重要连续型随机变量的分布：均匀分布、正态分布、卡方分布、t-分布，理解这些分布中参数的意义，能进行简单应用。（3）了解连续型随机变量的均值和方差，知道均匀分布、正态分布、卡方分布、t-分布的均值和方差及其意义。 2．二维随机变量及其联合分布（1）在学习一维离散型随机变量的基础上，通过实例，了解二维离散型随机变量概念及其分布列、数字特征（均值、方差、协方差、相关系数），并能解决简单的实际问题。了解两个随机变量的独立性。（2）在学习一维正态随机变量的基础上，通过具体实例，了解二维正态随机变量及其联合分布，以及联合分布中参数的统计含义。 3．参数估计借助对具体实际问题的分析，知道矩估计和极大似然估计这两种参数估计方法，了解参数估计原理，能解决一些简单的实际问题。 4．假设检验（1）了解假设检验的统计思想和基本概念。（2）借助具体实例，了解正态总体均值和方差检验的方法，了解两个正态总体的均值比较的方法。（3）结合具体实例，了解总体分布的拟合优度检验。 5．二元线性回归模型（1）了解二维正态分布及其参数的意义。（2）了解二元线性回归模型，会用最小二乘原理对模型中的参数进行估计。（3）运用二元线性回归模型解决简单的实际问题。 B类课程 B类课程包括微积分、空间向量与代数、应用统计、模型四个专题，其中微积分2学分，空间向量与代数1学分，应用统计2学分，模型1学分。微积分本专题在数列极限的基础上建立函数极限的概念；在具体的情境中用极限刻画导数，给出借助导数研究函数性质的一般方法；通过极限建立微分和积分的概念，阐述微分和积分的关系（微积分基本定理）及其应用。在学习一元函数的基础上，了解二元函数及其偏导数的概念。本专题要考虑高中学生接受能力，重视课程内容的实际背景，关注数学内容的直观理解，培养学生的运算能力，为进一步学习大学相关课程奠定基础。内容包括：极限、导数与微分、定积分、二元函数。 1．极限（1）通过典型数列，了解数列的极限，掌握极限的符号，了解基本事实：单调有界数列必有极限。（2）通过具体函数犳，且，，了解函数极限和连续的概念，掌握极限的符号，了解闭区间上连续函数的性质。 2．导数与微分（1）通过导数概念，理解二阶导数的概念，了解二阶导数的物理意义与几何意义；掌握一些基本初等函数的一阶导数与二阶导数。（2）理解拉格朗日中值定理，了解它的几何解释。（3）能利用导数讨论函数的单调性，并证明某些函数不等式（例如，当时，，）。（4）会利用导数讨论函数的极值问题，利用几何图形说明一个点是极值点的必要条件与充分条件（不要求数学证明）。（5）借助导数，会求闭区间上一元一次函数、一元二次函数、一元三次函数的最大值与最小值。（6）了解微分的概念及其实际意义，会用符号表示。 3．定积分（1）了解闭区间上连续函数定积分的概念，理解其几何意义与物理意义。（2）能用等分区间方法计算特殊的黎曼和。（3）利用的单调性、等分区间的方法、拉格朗日中值定理，推导牛顿-莱布尼茨公式。（4）会利用定积分计算某些封闭平面图形的面积，计算球、圆锥、圆台和某些三棱锥、三棱台的体积；了解祖暅原理。 4．二元函数（1）通过简单实例，掌握二元函数的背景。（2）了解偏导数的定义，能计算一些简单函数的偏导数。例如，已知与分别是基本初等函数，会求，的偏导数。（3）会求一些简单二元函数的驻点，并能求相应的实际问题中的极值。（4）利用等高线法，会求一次函数在闭凸多边形区域上的最大值和最小值。（5）会求闭圆域、闭椭圆域上二元二次函数的最大值和最小值。空间向量与代数本专题在必修课程和选择性选修课程的基础上，比较系统地学习三维空间的整体结构———向量代数，感悟几何与代数的融合。引入矩阵与行列式的概念，并讨论三元一次方程组解的结构。本专题中强调几何直观，把握问题的本质，培养学生数学运算、数学抽象、逻辑推理和直观想象等素养，为大学线性代数课程的学习奠定直观基础。内容包括：空间向量代数、三阶矩阵和行列式、三元一次方程组。 1．空间向量代数（1）通过几何直观，理解向量运算的几何意义。（2）探索并解释空间向量的内积与外积及其几何意义。（3）理解向量的投影与分解及其几何意义，并会应用。（4）掌握向量组的线性相关性，并能加以判断。（5）掌握向量的线性运算，理解（低维）向量空间与子空间的概念。（6）会求点到直线、点到平面的距离，两条异面直线的距离，直线与平面的夹角。 2．三阶矩阵与行列式（1）通过几何直观引入矩阵概念，掌握矩阵的三种基本运算及其性质。（2）掌握行列式定义与性质，会计算行列式。 3．三元一次方程组（1）通过实例，探索三元一次方程组的求解过程，理解三元一次方程组的常用解法（高斯消元法），会用矩阵表示三元一次方程组。（2）掌握三元齐次线性方程组的解法，会表示一般解。（3）掌握非齐次线性方程组有解的判定，建立线性方程组的理论基础。（4）探索三元一次方程组解的结构，会表示一般解。（5）理解克拉默（Cramer）法则，会用克拉默法则求解三元一次方程组。应用统计本专题在必修课程和选择性必修课程的基础上展开。在概率课程方面，通过具体实例，进一步学习连续型随机变量及其概率分布，二维随机向量及其联合分布，并运用这些数学模型，解决一些简单的实际问题。在统计方面，结合一些具体任务，学习参数估计、假设检验和不依赖于分布的统计检验，并运用这些方法解决一些简单的实际问题；学习数据分析的两种特殊方法——聚类分析和正交设计。在教学活动中，要关注学生对数学内容的直观理解，充分考虑高学生接受能力；要重视课程内容的实际背景，更要重视课程内容的实际应用；要注重全面提升学生数学核心素养。内容包括：连续型随机变量及其分布、二维随机变量及其联合分布、参数估计、假设检验、二元线性回归模型、聚类分析、正交设计。 1.连续型随机变量及其分布（1）借助具体实例，了解连续型随机变量及其分布，体会连续型随机变量与离散型随机变量的共性与差异。（2）结合生活中的实例，了解几个重要连续型随机变量的分布：均匀分布、正态分布、卡方分布、τ-分布，理解这些分布中参数的意义，能进行简单应用。（3）了解连续型随机变量的均值和方差，知道均匀分布、正态分布、卡方分布、τ-分布的均值和方差及其意义。 2.二维随机变量及其联合分布（1）在学习一维离散型随机变量的基础上，通过实例，了解二维离散型随机变量概念及其分布列、数字特征（均值、方差、协方差、相关系数），并能解决简单的实际问题。了解两个随机变量的独立性。（2）在学习一维正态随机变量的基础上，通过具体实例，了解二维正态随机变量及其联合分布，以及联合分布中参数的统计含义。 3.参数估计借助对具体实际问题的分析，知道矩估计和极大似然估计这两种参数估计方法，了解参数估计原理，能解决一些简单的实际问题。 4.假设检验（1）了解假设检验的统计思想和基本概念。（2）借助具体实例，了解正态总体均值和方差检验的方法，了解两个正态总体的均值比较的方法。（3）结合具体实例，了解总体分布的拟合优度检验。 5.二元线性回归模型（1）了解假设检验的统计思想和基本概念。（2）借助具体实例，了解正态总体均值和方差检验的方法，了解两个正态总体的均值比较的方法。（3）结合具体实例，了解总体分布的拟合优度检验。 6.聚类分析（1）借助具体实例，了解聚类分析的意义。（2）借助具体实例，了解几种聚类分析的方法，能解决一些简单的实际问题。 7.正交设计（1）借助具体实例，了解正交设计原理。（2）借助具体实例，了解正交表，能用正交表进行实验设计。模型本专题在必修课程和选择性必修课程的基础上，通过大量的实际问题，建立一些基本的数学模型，包括线性模型、二次曲线模型、指数函数模型、三角函数模型、参变数模型。在教学中，要重视这些模型的背景、形成过程、应用范围，提升数学建模、数学抽象、数学运算和直观想象素养，提升应用能力和创新能力。内容包括：线性模型、二次函数模型、指数函数模型、三角函数模型、参变数模型。 1.线性模型（1）结合实际问题，了解一维线性模型，理解一次函数与均匀变化的关系，并能发现生活中均匀变化的实际问题。（2）结合实际问题，了解二维线性模型，探索平面上一些图形的变化，并能理解一维线性模型与二维线性模型的异同（例如，矩阵A是对角矩阵）。（3）结合实际问题，了解三维线性模型，如经济学上的投入产出模型。 2.二次函数模型借助实例（如光学模型、自由落体、边际效应），了解二次曲线模型的含义和特征，体会二次曲线模型的实际意义。 3.指数函数模型借助有关増长率的实际问题（如种群增长、放射物衰减），理解指数函数模型，感受增长率是常数的事物的单调变化。 4.三角函数模型借助具体实例，理解一类波动问题（如光波、声波、电磁波）等周期现象可以用三角函数来刻画。 5.参变数模型（1）借助具体实例，理解平面上的参变数模型，如弹道模型。（2）借助具体实例，理解空间上的参变数模型，如螺旋曲线。（3）借助一些用参变数方程描述的物理问题与几何问题，理解参变数的意义，掌握参变数变化的范围。 C类课程 C课程包括逻辑推理初步、数学模型、社会调查与数据分析三个专题，每个专题2学分。逻辑推理初步本专题内容以数学推理为主线展开，将相关逻辑知识与数学推理有机融合。通过本专题的学习，能进一步认识逻辑推理的本质，体会其在数学推理、论证中的作用；能运用相关逻辑知识正确表述自己的思想、解释社会生活中的现象，提高逻辑思维能力，发展逻辑推理素养。内容包括：数学定义、命题和推理，数学推理的前提，数学推理的类型，数学证明的主要方法，公理化思想。 1.数学定义、命题和推理通过实例，了解数学定义和数学命题，知道数学定义的基本方式，了解数学命题的表达形式，了解数学定义、数学命题和数学推理之间的关系。能理解数学命题中的条件和结论；结合实例，能对充分条件、必要条件、充要条件进行判断。 2.数学推理的前提理解同一律、矛盾律、排中律的含义，通过实例认识它们在数学推理中的作用，能在数学推理中认识推理前提的重要性。能通过实例，区分排中律与矛盾律，能在推理中正确运用排中律。 3.数学推理的类型结合学过的数学实例和生活中的实例，理解演绎推理、归纳和类比推理，在这些推理的过程中，认识数学推理的传递性。知道利用推理能够得到和验证数学的结果。通过数学和生活中的实例，认识或然性推理和必然性推理的区别。 4.数学证明的主要方法通过数学实例，认识一些常用的数学证明方法，理解这些证明方法在数学和生活中的意义。 5.公理化思想通过数学史和其他领域的典型事例，了解数学公理化的含义，了解公理体系的独立性、相容性、完备性，了解公理化思想的意义和价值。数学模型本专题在必修课程和选择性必修课程的基础上，通过具体实例，建立一些基于数学表达的经济模型和社会模型，包括存款贷款模型、投入产出模型、经济增长模型、凯恩斯模型、生产函数模型、等级评价模型、人口増长模型、信度评价模型等。在教学活动中，要让学生知道这些模型形成的背景、数学表达的道理、模型参数的意义、模型适用的范围，提升数学建模、数学抽象、数学运算和直观想象素养；知道其中的有些模型（以及模型的衍生）获得诺贝尔经济学奖的理由，理解数学的应用，提高学习数学的兴趣，提升实践能力和创新能力。内容包括：经济数学模型、社会数学模型。 1.经济数学模型（1）存款贷款模型（指数函数模型）通过对存款等实际问题的分析，抽象出复利模型；通过对住房贷款等实际问题的分析，抽象出等额本金付款模型。了解这些模型各自的特点，能用该样的模型解决简单的实际问题。（2）投入产出模型（线性方程组模型）了解投入产出模型的背景和意义，理解模型是如何通过线性方程组中的系数的解约束自变量、从而实现组合生产的计划，能用投入产出模型分析并解决简单的实际问题。（3）经济增长模型（线性回归模型）利用我国改革开放以后经济发展数据，通过实践与GDP（或者人均GDP）之间的关系建立线性回归模型（或者分段的线性回归模型），估计其中的参数，理解参数的意义。能用同样的方法分析简单的经济现象。（4）凯恩斯模型（经济理论模型）了解如何通过收入、消费和投资之间的关系建立数学模型，体会模型中系数的乘数效应，体会扩大消费与经济发展、增加国民收入之间的关系，能用模型解释简单的经济现象。（5）生产函数模型（对数线性模型）了解生产理论中柯布-道格拉斯（Cobb-Douglas）生产函数，知道如何用数学语言表达生产与劳动投入、资本投入之间的关系，知道如何把这样的表达转化为对数线性模型、如何对其中的参数进行估计，能解决简单的实际问题。 2.社会数学模型（1）等级评价模型（平均数模型）结合具体实例（如产品质量评价、热点问题筛选、跳水等技能或全能等综合性体育运动评分），了解加权平均、调和平均、稳健平均等评价模型的特点及适用范围，能用这样的模型解决简单实际问题。（2）人口增长模型（指数函数模型）结合实例（如我国人口增长数据），了解为什么可以用指数增长模型刻画人口变化的规律，知道模型中参数的意义，知道如何用模型拟合实际数据，并能判断拟合的有效性。（3）信度评价模型（Logostic回归模型）对于银行贷款用户、信用卡用户等涉及信度的问题，知道用Logostic回归模型进行信度评级的道理，知道构造两级（好、差）或者三级（好、中、差）进行评价的方法，并会简单应用。社会调查与数据分析社会调查是学生进入社会要掌握的基本能力。本专题在必修课程和选择性必修课程的基础上，结合社会调查的实际问题和在社会调查中的一些关键环节，引导学生经历社会调查的全过程，包括社会调查方案的设计、抽样设计、数据分析、报告的撰写，并结合具体社会调查案例，分析在社会调查实施过程中可能遇到的问题，以及解决这些问题的对策。本专题的基本特点是实用、具体、有效、有趣。在完成社会调查任务的过程中，要注意引导学生充分运用概率与统计知识，避免采用不科学的社会调查方法与数据分析方法，全面提升学生数学学科核心素养。内容包括：社会调查概论、社会调查方案设计、抽样设计、社会调查数据分析、社会调查数据报告、社会调查案例选讲。 1.社会调查概论（1）结合实例，了解社会调查的使用范围、分类和意义。（2）针对具体问题，了解社会调查的基本步骤：项目确定、方案设计、组织实施、数据分析、形成报告。 2.社会调查方案设计（1）结合实例，了解调查方案设计的基本内容：目的、内容、对象、项目、方式、方法及其他。（2）结合实例，探索调查方案的可行性评估。（3）结合实例，了解问卷设计的主要问题：问卷的结构与常用量表、问卷设计的程序与技巧。（4）结合实例，掌握社会调查基本方法：文案调查法、观察法、访谈法、德尔菲法、电话法等。 3.抽样设计在必修课程学习的抽样方法（简单随机抽样、分层抽样）的基础上，了解二阶与多阶抽样，能根据具体情境选择合适的抽样方法。 4.社会调查数据分析（1）结合具体实例，整理调查数据，了解常用统计图表（频数表、交叉表、直方图、茎叶图、扇形图、雷达图、箱线图）及常用统计量（均值、众数、中位数、百分位数），能够确定各种抽样方法的样本量。（3）结合具体实例，了解相关分析、回归分析、多元统计分析。 5.社会调查数据报告掌握社会调查报告的基本要求及基本内容，能够做出简单的、完整的社会调查数据报告。 6.社会调查案例选讲通过典型案例的学习，理解社会调查的意义。 D类课程 D课程包括美与数学、音乐中的数学、美术中的数学、体育运动中的数学四个专题，每个专题1学分。美与数学学会审美不仅可以陶冶情操，而且能够改善思维品质。本专题尝试从数学的角度刻画审美的共性，主要包括：简洁、对称、周期、和谐等。通过本课程的学习，学生对美的感受能够从感性走向理性，提升有志于从事艺术、体育事业学生的审美情趣和审美能力，在形象思维的基础上増强理性思维能力。内容包括：美与数学的简洁、美与数学的对称、美与数学的周期、美与数学的和谐。 1.美与数学的简洁数学可以刻画现实世界中的简洁美。例如，太阳、满月、车轮、井盖形状等美的共性与圆相关，抛物运动、行星运动轨迹等美的共性与二次曲线相关，DNA结构、向日葵花盘、海螺等美的共性与特殊曲线相关，家具、日用品、冷却塔、建筑物外形等美的共性与简单曲面相关，雪花、云彩、群山、海岸线、某些现代设计等美的共性与分形相关。 2.美与数学的对称数学可以刻画现实世界中的对称美。例如，某些动物形体、飞机造型、某些建筑物外形等美的共性与空间反射对称相关；剪纸、脸谱、风筝等传统艺术美的共性与轴对称相关；晶体等美的共性与中心对称相关，带饰、面饰等美的共性与平移对称、中心对称、轴对称相关。循环赛制、守恒定律也具有对称美。 3.美与数学的周期数学可以刻画现实世界中的周期美。例如，昼夜交替、四季循环、日月星辰运动规律、海洋波浪等美的共性与周期相关，乐曲创作、图案设计中美的共性与周期相关。 4.美与数学的和谐数学可以刻画现实世界中的和谐美。例如，人体结构、建筑物、国旗、绘画、优选法等美的共性与黄金分割相关，苗木生长、动物繁殖、向日葵种子排列规律等美的共性与斐波那契数列相关。音乐中的数学音乐的要素——音高、音响、音色、节拍、乐音、乐曲、乐器等都与数学相关，特别是音的律制与数学的关系十分密切。通过本专题的学习，学生能够更加理性地理解音乐，鉴赏音乐的美，可以提升有志于从事音乐事业学生的数学修养，増强理性思维能力。内容包括：声波与正弦函数，律制、音阶与数列，乐曲的节拍与分数，乐器中的数学，乐曲中的数学等。 1.声波与正弦函数纯音可以用正弦函数来表达，音高与正弦函数的频率相关，响度与正弦函数的振幅相关，和声、音色与正弦函数的叠加相关。 2．律制、音阶与数列音的律制用以规定音阶，三分损益律、五度相生律、纯律的音阶均与频率比、弦长比相关，十二平均律与等比数列相关。五线谱能够科学地记录乐曲。 3．乐曲的中拍与分景乐面的小节、拍、拍号与分数相关。套曲的钢琴演奏与最小公倍数相关。 4．乐器中的数学键盘乐器（如钢琴）、弦乐器（如小提琴、二胡）、管乐器（如长笛）的发声、共鸣等，都与数学相关。 5．乐曲中的数学乐曲中的高潮点、乐曲调性的转换点，常与黄金分割相关；乐曲的创作既与平移、反射、伸缩等变换相关，也与排列、组合相关。美术中的数学美术主要包括绘画、雕塑、工艺美术、建筑艺术，以及书法、篆刻艺术等。通过本专题的学习，可以帮助学生了解类术中的平移、对称、黄金分割、透视几何等数学方法，了解计算机美术的基本概念和方法，了解美术家在创作过程中所蕴含的数学思想，体会数学在美术中的作用，更加理性地鉴赏美术作品，提升直观想象和数学抽象素养。在教学过程中，应以具体实例为主线展开，将美术作品与相关的数学知识有机联系起来。内容包括：绘画与数学、其他美术作品中的数学、美术与计算机、美术家的数学思想。 1．绘画与数学名画中的数学元素，绘画中的平移与对称，绘画中的黄金分割，绘画中的透视几何。 2．其他美术作品中的数学雕塑中的黄金分割，建筑中的对称，工艺品中的对称，邮票中的数学，书法中的黄金分割。 3．美术与计算机计算机绘画的发展背景，计算机绘画所需的硬件和软件，计算机绘画实例。 4．美术家的数学思想达芬奇、毕加索、埃舍尔等的数学思想。体育运动中的数学在体育运动中，无论是运动本身还是与运动有关的事都蕴含着许多数学原理。例如，田径运动中的速度、角度、运动曲线，比赛场次安排、运动器械与运动场馆设计等。通过本专题的学习，学生能运用数学知识探索提高运动效率的途径，能运用数学方法合理安排赛事，提升有志于从事体育事业学生的数学修养，增强理性思维能力。内容包括：运动场上的数学原理、运动成绩的数据分析、运动赛事中的运筹帷幄、体育用具及设施中的数学知识。 1．运动场上的数学原理了解与田径运动、球类运动、体操运动、水上运动等相关的数学原理，探索如何提高运动效率和运动成绩。例如，根据向量分解的原理指导运动员进行跳高、跳远和投掷。 2．运动成绩的数据分析通过健康指标和运动成绩的数据，运用概率与统计知识寻求规律、探索合理方案。例如，通过日常运动和健康状况的数据，分析运动与健康的关系。 3．运动赛事中的运筹帷幄知道能借助图论、运筹等数学知识分析体育赛事的规律，进行合理安排，提升教练员的指挥策略，改善运动员赛场上的应对策略。 4．体育用具及设施中的数学知识知道在大多数体育运动用具和场馆的设计中都运用了数学知识，例如，足球、乒乓球的制作，网球拍的构造，标准跑道的规划；通过数学曲面感悟“鸟集”“水立方”等体育设施的设计原理。 E类课程 E类课程是学校根据自身的需求开发或选用的课程，包括拓展视野、日常生活、地方特色的数学课程，还包括大学数学的先修课程等。拓展视野的数学课程例如，机器人与数学、对称与群、球面上的几何、欧拉公式与闭曲面分类、数列与差分、初等数论初步。日常生活的数学课程例如，生活中的数学、家庭理财与数学。地方特色的数学课程例如，地方建筑与数学、家乡经济发展的社会调查与数据分析。大学数学的先修课程包括：微积分、解析几何与线性代数、概率论与数理统计。五、学业质量（一）学业质量内涵学业质量是学生在完成本学科课程学习后的学业成就表现。学业质量标准是以本学科核心素养及其表现水平为主要维度（参见附录1），结合课程内容，对学生学业成就表现的总体刻画。依据不同水平学业或就表现的关键特征，学业质量标准明确将学业质量划分为不同水平，并描述了不同水平学习结果的具体表现。数学学科学业质量是应该达成的数学学科核心素养的目标，是数学学科核心素养水平与课程内容的有机结合。学业质量是学生自主学习与评价、教师教学活动与评价、教材编写的指导性要求，也是相应考试命题的依据。（二）学业质量水平数学学业质量水平是六个数学学科核心素养水平的综合表现。每一个数学学科核心素养划分为三个水平（详述参见附录1)，每一个水平是通过数学学科核心素养的具体表现和体现数学学科核心素养的几个方面进行表述的。数学学科核心素养的具体表现参见“学科核心素养与课程目标”，体现数学学科核心素养的四个方面知下：情境与问题情境主要是指现实情境、数学情境、科学情境。问题是指在情境中提出的数学问题；知识与技能主要是指能够帮助学生形成相应数学学科核心素养的知识与技能；思维与表达主要是指数学活动过程中反映的思维品质、表述的严谨性和准确性；交流与反思主要是指能够用数学语言直观地解释和交流数学的概念、结论、应用和思想方法，并能进行评价、总结与拓展。水平质量描述水平一能够在熟悉的情境中，直接抽象出数学概念和规则；能够用归纳或类比的方法，发现数量或图形的性质、数量关系或图形关系，形成简单的数学命题；能够抽象出实物的几何图形，建立简单图形与实物之间的联系，体会图形与图形、图形与数量的关系；了解随机现象及简单的概率或统计问题；了解熟悉的数学模型的实际背景及其数学描述，了解数学模型中的参数、结论的实际含义，能够在熟悉的数学情境中了解运算对象，提出运算问题。能够在熟悉的数学情境中，解释数学概念和规则的含义，了解数学命题的条件与结论之间的逻辑关系，抽象出数学问题；能够通过熟悉的例子理解归纳推理、类比推理和演绎推理的基本形式，识别归纳推理、类比推理、演绎推理；掌握一些基本命题与定理的证明，并有条理地表述论证过程；能够借助图形的性质和变换（平移、对称、旋转）发现数学规律；能够推述简单图形的位置关系和度量关系及其特有性质，能够了解运算法则及其适用范围，正确进行运算，能够根据问题的特征形成合适的运算思路；能够对熟悉的概率问题，选择合适的概率模型；能够对熟悉的统计问题，选择合适的抽样方法收集数据，掌握描述、刻画、分析数据的基本统计方法；能够解决简单的数学应用问题，知道数学建模的过程包括：提出问题、建立模型、求解模型、检验给果、完善模型，能够在熟悉的实际情境中，模仿学过的数学建模过程解决问题。能够了解用数学语言表达的推理和论证；能够在解决相似的问题中感悟数学的通性通法；能够用图形描述和表达熟悉的数学问题、启迪解决这些问题的思路，体会数形结合；能够体会运算法则的意义和作用，运用运算验证简单的数学结论：能够用概率和统计的语言表达简单的随机现象；能够结合熟悉的实例，体会概率的意义，感悟统计方法的作用；对于学过的数学模型，能够举例说明数学建模的意义，体会其蕴含的数学思想。能够在交流的过程中，结合实际情境解释相关的抽象概念；能够在日常生活中利用图形直观进行交流；能够用统计图表和简单概率模型解释熟悉的随机现象：能够用运算的结果、借助或引用已有数学建模的结果说明问题；能够明确所讨论问题的内涵，有条理地表达观点。（参见案例20～35）水平二能够在关联的情境中，抽象出一般的数学概念和规则，确定运算对象和随机现象，发现问题并提出或转化为数学问题；能够想象并构建相应的几何图形，发现图形与图形、图形与数量的关系，探索图形的运动规律；能够理解归纳、类比是发现和提出数学命题的重要途径；能够将已知数学命题推广到更一般的情形；能够在新的情境中选择和运用数学方法解决问题。能够用恰当的例子解释抽象的数学概念和规则；能够理解数学命题的条件与结论，通过分析相关数学命题的条件与结论，探索论证的思路，选择合适的论证方法予以证明；能够理解和构建相关数学知识之间的联系；能够通过举反例说明某些数学结论不成立；能够掌握研究图形与图形、图形与数量之间关系的基本方法，借助图形性质探索数学规律，解决实际问题成数学问题；能够针对运算问题，合理选择运算方法、设计运算程序，运算求解；能够选择合适的数学模型表达所要解决的数学问题，理解模型中参数的意义，知道如何确定参数，建立模型，求解模型，能够根据问题的实际意义检验结果，完善模型，解决问题；能够针对具体问题，选择离散型随机变量或连续型随机变量刻画随机现象，理解抽样方法的统计意义，运用适当的概率或统计模型解决问题。能够理解用数学语言表达的概念、规则、推理和论证，理解相关概念、命题、定理之间的逻辑关系，提炼出解决一类问题的数学方法，理解其中的数学思想，初步建立网状的知识结构；能够用图形探索解决问题的思路，形成数形结合的思想；能够理解运算是一种演绎推理，在综合运用运算方法解决问题的过程中，形成规范化思考问题的品质；能够在关联的情境中，经历数学建模的过程，运用数学语言，表述数学建模过程中的问题以及解决问题的过程和结果，形成研究报告，展示研究成果；能够在运用统计方法解决问题的过程中，解释统计结果，感悟归纳推理的作用，能够用概率或统计模型表达随机现象的统计规律。在交流的过程中，能够用一般的概念解释具体现象；能够利用直观想象、数学运算探讨数学问题；能够用数据呈现的规律解释随机现象；能够用模型的思想说明问题。能够在交流的过程中，围绕主题，观点明确，论述有理有据，并能用准确的数学语言表述论证过程。（参见案例20~35）水平三能够在综合的情境中，发现其中蕴含的数学关系，用数学的眼光找到合适的研究对象，用恰当的数学语言予以表达，并运用数学思维进行分析，提出数学问题；能够借助图形探索解决问题的思路；能够在得到的数学结论基础上形成新命题。能够通过数学对象、运算或关系理解数学的抽象结构；能够掌握不同的逻辑推理方法；能够对较复杂的数学问题，通过构建过渡性命题，探索论证的途径，解决问题，能够对较复杂的运算问题，设计算法，构造运算程序，解决问题；能够综合利用图形与图形、图形与数量的关系，理解数学各分支之间的联系；能够借助直观想象建立数学与其他学科的联系，并形成理论体系的直观模型，感悟高度概括、有序多级的数学知识体系；能够在现实世界中发现问题。运用数学建模的一般方法和相关知识，创造性地建立数学模型，解决问题；能够针对不同的问题，综合或创造性地运用概率统计知识，构造相应的概率或统计模型，解决问题。在实际情境中，能够把握研究对象的数学特征，感悟通性通法的数学原理和其中蕴含的数学思想；能够运用数学语言，清晰、准确地表达数学论证和数学建模的过程和结果；能够理解建构数学体系的公理化思想；能够用程序思想理解与表达问题，理解程序思想与计算机解决问题的联系；能够通过想象对复杂的数学问题进行直观表达，抓住数学问题的本质，形成解决问题的思路，能够理解数据蕴含着信息，可以通过对信息的加工，得到数据所提供的知识和规律，理解数据分析在大数据时代的重要性。在交流的过程中，能够用数学原理解释自然现象和社会现象；能够利用直观想象探讨问题的本质及其与数学的联系；能够用程序思想理解和解释问题；能够辨明随机现象，并运用恰当的数学语言进行表述；能够通过数学建模的结论和思想阐释科学规律和社会现象；能够合理地运用数学语言和思维进行跨学科的表达与交流。（参见案例25，28，30，31，34）（三）学业质量水平与考试评价的关系数学学业质量水平一是高中毕业应当达到的要求，也是高中毕业的数学学业水平考试的命题依据；数学学业质量水平二是高考的要求，也是数学高考的命题依据；数学学业质量水平三是基于必修、选择性必修和选修课程的某些内容对数学学科核心素养的达成提出的要求，可以作为大学自主招生的参考。关于教学与评价的具体要求可参照“教学与评价建议”，关于学业水平考试与高考命题的具体要求可参照“学业水平考试与高考命题建议”，关于教材编写的具体要求可参照“教材编写建议”。六、实施建议（一）教学与评价建议在教学活动中，教师应准确把握课程目标、课程内容、学业质量的要求，合理设计教学目标，并通过相应的教学实施，在学生掌握知识技能的同时，促进数学学科核心素养的提升及水平的达成。在教学与评价中，要关注学生对具体内容的掌握情况，更要关注学生数学学科核心素养水平的表现；要关注数学学科核心素养各要素的不同特征及要求，更要关注数学学科核心素养的综合性与整体性（参见案例23）。教师应结合相应的教学内容，落实“四基”，培养“四能”，促进学生数学学科核心素养的形成和发展，达到相应水平的要求，部分学生可以达到更高水平的要求。 1．教学建议全面落实立德树人要求，深入挖掘数学学科的育人价值，树立以发展学生数学学科核心素养为导向的教学意识，将数学学科核心素养的培养贯穿于教学活动的全过程。在教学实践中，要不断探索和创新教学方式，不仅重视如何教，更要重视如何学，引导学生会学数学，养成良好的学习习惯，要努力激发学生数学学习的兴趣，促使更多的学生热爱数学。学科+网（1）教学目标制定要突出数学学科核心素养数学学科核心素养是数学课程目标的集中体现，是在数学学习的过程中逐步形成的。教师在制定教学目标时要充分关注数学学科核心素养的达成；要深入理解数学学科核心素养的内涵、价值、表现、水平及其相互联系；要结合特定教学任务，思考相应数学学科核心素养在教学中的孕育点、生长点：要注意数学学科核心素养与具体教学内容的关联；要关注数学学科核心素养目标在教学中的可实现性，研究其融入教学内容和教学过程的具体方式及载体，在此基础上确定教学目标。学生数学学科核心素养水平的达成不是一蹴而就的，具有阶段性、连续性、整合性等特点。教师应理解不同数学学科核心素养水平的具体要求，不仅关注每一节课的教学目标，更要关注主题、单元的教学目标（参见案例36），明晰这些目标对实现数学学科核心素养发展的贡献。在确定教学目标时，要把握好学生数学学科核心素养发展的各阶段目标之间的关系，合理设计各类课程的教学目标。数学学科核心素养是“四基”的继承和发展。“四基”是培养学生数学学科核心素养的沃土，是发展学生数学学科核心素养的有效载体，教学中要引导学生理解基础知识，掌握基本技能，感悟数学基本思想，积累数学基本活动经验，促进学生数学学科核心素养的不断提升。（2）情境创设和问题设计要有利于发展数学学科核心素养基于数学学科核心素养的教学活动应该把握数学的本质，创设合适的教学情境、提出合适的数学问题，引发学生思考与交流，形成和发展数学学科核心素养。教学情境和数学问题是多样的、多层次的。教学情境包括：现实情境、数学情境、科学情境，每种情境可以分为熟悉的、关联的、综合的。数学问题是指在情境中提出的问题，分为简单问题、较复杂问题、复杂问题。数学学科核心素养在学生与情绩、问题的有效互动中得到提升。在教学活动中，应结合教学任务及其蕴含的数学学科核心素养设计合适的情境和问题，引导学生用数学的眼光观察现象、发现问题，使用恰当的数学语言描述问题，用数学的思想、方法解决问题。在问题解决的过程中，理解数学内容的本质，使进学生数学学科核心素养的形成和发展。设计合适的教学情境，提出合适的数学问题是有挑战性的，也为教师的实践创新提供了平台。教师应不断学习、探索、研究、实践，提升自身的数学素养，了解数学知识之间、数学与生活、数学与其他学科的联系，开发出符合学生认知规律、有助于提升学生数学学科核心素养的优秀案例。（3）整体宝物教学内容，促进数学学科核心素养连续性和阶段性发展数学学科核心素养的发展具有连续性和阶段性。教师要以数学学科核心素养为导向，抓住函数、几何与代数、概率与统计、数学建模活动与数学探究活动等内容主线，明晰数学学科核心素养在内容体系形成中表现出的连续性和阶段性，引导学生从整体上把握课程，实现学生数学学科核心素养的形成和发展。数学建模活动与数学探究活动是综合提升数学学科核心素养的载体。教师应整体设计、分步实施数学建模活动与数学探究活动，引导学生从类比、模仿到自主创新、从局部实施到整体构想，经历“选题、开题、做题、结题”的活动过程，积累发现和提出问题、分析和解决问题的经验，养成独立思考与合作交流的习惯。应引导学生遵守学术规范，坚守诚信底线。数学文化应融入数学教学活动。在教学活动中，教师应有意识地结合相应的教学内容，将数学文化渗透在日常教学中，引导学生了解数学的发展历程，认识数学在科学技术、社会发展中的作用，感悟数学的价值，提升学生的科学精神、应用意识和人文素养，将数学文化融入教学，还有利于激发学生的数学学习兴趣，有利于学生进一步理解数学，有利于开拓学生视野、提升数学学科核心素养。（4）既要重视教，更要重视学，促进学生学会学习教师要把教学活动的重心放在促进学生学会学习上，积极探索有利于促进学生学习的多样化教学方式，不仅限于讲授与练习，也包括引导学生阅读自学、独立思考、动手实践、自主探索、合作交流等，教师要善于根据不同的内容和学习任务采用不同的教学方式，优化教学，抓住关键的教学与学习环节，增强实效。例如，丰富作业的形式，提高作业的质量，提升学生完成作业的自主性、有效性。教师要加强学习方法指导，帮助学生养成良好的数学学习习惯，敢于质疑、善于思考，理解概念、把握本质，数形结合、明晰算理，厘清知识的来龙去脉，建立知识之间的关联。教师还可以根据自身教学经验和学生学习的个性特点，引导学生总结出一些具有针对性的学习方式，因材施教。（5）重视信息技术运用，实现信息技术与数学课程的深度融合在“互联网+”时代，信息技术的广泛应用正在对数学教育产生深刻影响。在数学教学中，信息技术是学生学习和教师教学的重要辅助手段，为师生交流、生生交流、人机交流搭建了平台，为学习和教学提供了丰富的资源。因此，教师应重视信息技术的运用，优化课堂教学，转变教学与学习方式。例如，为学生理解概念创设背景，为学生探索规律启发思路，为学生解决问题提供直观，引导学生自主获取资源。在这个过程中，教师要有意识地积累数学活动案例，总结出生动、自主、有效的教学方式和学习方式。（参见案例37）教师应注重信息技术与数学课程的深度融合，实现传统教学手段难以达到的效果。例如，利用计算机展示函数图象、几何图形运动变化过程，利用计算机探究算法、进行较大规模的计算，从数据库中获得数据，绘制合适的统计图表；利用计算机的随机模拟结果，帮助学生更好地理解随机事件以及随机事件发生的概率。 2．评价建议教学评价是数学教学活动的重要组成部分。评价应以课程目标、课程内容和学业质量标准为基本依据，日常教学活动评价，要以教学目标的达成为依据。评价要关注学生数学知识技能的掌握，还要关注学生的学习态度、方法和习惯，更要关注学生数学学科核心素养水平的达成。教师要基于对学生的评价，反思教学过程，总结经验、发现问题，提出改进思路。因此，数学教学活动的评价目标，既包括对学生学习的评价，也包括对教师教学的评价。（1）评价目的评价的目的是考查学生学习的成效，进而也考查教师教学的成效。通过考查，诊断学生学习过程中的优势与不足，进而诊断教师教学过程中的优势与不足，通过诊断，改进学生的学习行为，进而改进教师的教学行为，促进学生数学学科核心素养的达成。（2）评价原则为了实现上述评价目的，教师应坚持以学生发展为本，以积极的态度促进学生不断发展，日常评价应遵循以下原则。 ①重视学生数学学科核心素养的达成教学评价要以数学学科核心素养的达成作为评价的基本要素。基于数学学科核心素养的教学要创设合适的教学情境、提出合适的数学问题。在设计教学评价工具时，应着重对设计的教学情境、提出的问题进行评价。评价内容包括：情境设计是否体现数学学科核心素养，数学问题的产生是否自然，解决问题的方法是否为通性通法，情境与问题是否有助于学生数学学科核心素养的达成。基于数学学科核心素养的教学评价具有挑战性，可以采取教研组集体研讨的方式设计评价工具和评价准则。在设计学习评价工具时，要关注知识技能的范围和难度，要有利于考查学生的思维过程、思维深度和思维广度（例如，设计好的开放题是行之有效的方法)，要关注六个数学学科核心素养的分布和水平；应聚焦数学的核心概念和通性通法，聚焦它们所承载的数学学科核心素养。 ②重视评价的整体性与阶段性基于学业质量标准和内容要求制定必修、选择性必修和选修课程的评价目标，关注评价的整体性。数学学科核心素养的达成是循序渐进的，基于内容主线对数学的理解与把握也是日积月累的。因此，应当把教学评价的总目标合理分解到日常教学评价的各个价段，关注评价的阶段性。既要关注数学知识技能的达成，更要关注相关的数学学科核心素养的提升；还应依据必修、选择性必修和选修课程内容的主线和主题，整体把握学业质量与数学学科核心素养水平。对于基于数学学科核心素养的教学评价，建立一个科学的评价体系是必要的，学校可以组织教师与有关人员，进行专门的研讨，积累经验，特别是积累通过阶段性评价不断改进教学活动的经验，最终建立适合本学校的科学评价体系。 ③重视过程评价日常评价不仅要关注学生当前的数学学科核心素养水平，更要关注学生成长和发展的过程；不仅要关注学生的学习结果，更要关注学生在学习过程中的发展和变化。学生的知识掌握、数学理解、学习自信、独立思考等是随着学习过程而变化和发展的，只有通过观察学生的学习行为和思维过程，才能发现学生思维活动的特征及教学中的问题，及时调整学与教的行为，改进学生的学习方法和思维习惯。此外，教师还要注意记录、保留和分析学生在不同时期的学习表现和学业成就，跟踪学生的学习进程，通过过程评价使学生感受成长的快乐，激发其数学学习的积极性。 ④关注学生的学习态度良好的学习态度是学生形成和发展数学学科核心素养的必要条件、也是最终形成科学精神的必要条件。在日常评价中应把学生的学习态度作为教学评价的意要目标。在对学生学习态度的评价中，应关注主动学习、认真思考、善于交流、集中精力、坚毅执着、严谨求实等。与其他目标不同，学习态度是随时表现出来的、与心理因素有关的，又是日积月累的、可以变化的。在日常教学活动中，教师要关注每一个学生的学习态度，对于特殊的学生给予重点关注。可以记录学生学习态度的变化与成长过程，从中分析问题，寻求解决问题的办法。形成良好的学习态度，需要对学生提出合适的要求，更需要教师的引导与鼓励、同学的帮助与支持，还需要良好学习氛围的激励与熏陶，需要数学教师与班主任以及其他学科教师的协同努力。（3）评价方式教学评价的主体应多元化，评价形式应多样化。评价主体的多元化是指除了教师是评价者之外，同学、家长甚至学生本人都可以作为评价者，这是为了从不同角度获取学生发展过程中的信息，特别是日常生活中关键能力、思维品质和学习态度的信息，最终给出公正客观的评价。合理利用这样的评价，可以有针对性地、有效地指导学生进一步发展。在多元评价的过程中，要重视教师与学生之间、教师与家长之间、学生与学生之间的沟通交流，努力营造良好的学习氛围。评价形式的多样化是指除了传统的书面测验外，还可以采用课堂观察、口头测验、开放式活动中的表现、课内外作业等评价的形式。这是因为一个人形成的思维品质和关键能力通常会表现在许多方面，因此需要通过多种形式的评价才能全面反映学生数学学科核心素养的达成状况。在日常评价中，可以采用形成性评价的方式。在本质上，形成性评价是与教学过程融为一体的。在教学过程中，教师既要获取学生的整体学习情况，也要关注个别学生的学习进展，在评价反思的同时调整教学活动，提高教学质量。基于数学学科核心素养的教学，在形成性评价的过程中，不仅要关注学生对知识技能掌握的程度，还要更多地关注学生的思维过程，判断学生是否会用数学的眼光观察世界，是否会用数学的思维思考世界，是否会用数学的语言表达世界。在数学建模活动与数学探究活动的教学评价中，应引导每个学生都积极参加，可以是个体活动，也可以是小组活动。教学活动包括，对于给出的问题情境，经历发现数学关联、提出数学问题、构建数学模型、完善数学模型、得到数学结论、说明结论意义的全过程；也包括根据现实情境，反复修改模型或者结论，最终提交研究报告或者小论文。无论是研究报告还是小论文，都要阐明提出问题的依据、解决问题的思路、得到结论的意义，遵循学术规范，坚守诚信底线。可以召开小型报告会，除了教师和学生之外，还可以邀请家长、有关方面的专家，对研究报告或者小论文作出评价。可以把学生完成的研究报告或者小论文以及各方评价存入学生个人档案，为大学招生提供参考。（4）评价结果的呈现与利用评价结果的呈现和利用应有利于增强学生学习数学的自信心，提高学生学习数学的兴趣，使学生养成良好的学习习惯，促进学生的全面发展。应更多地关注学生的进步，关注学生已经掌握了什么，得到了哪些提高，具备了什么能力，还有什么潜能，在哪些方面还存在不足等。要尽量避免终结性评价的“标签效应”——简单地依据评价结果对学生进行区分。评价的结果应该反映学生的个性特征和学习中的优势与不足，为改进教学的行为和方式、改进学习的行为和方法提供参考。教师要充分利用信息技术，收集、整理、分析有关反映学生学习过程和结果的数据，从而了解自己教学的成绩和问题，反思教学过程中影响学生能力发展和素养提高的原因，寻求改进教学的对策。除了考查全班学生在数学学科核心素养上的整体发展水平外，更需要根据学生个体的发展水平和特征进行个性化的反馈，特别是要以适当的方式将学生的一些积极变化及时反馈给学生。个性化的评价反馈不仅要系统、全面、客观地反映学生在数学学科核心素养发展上的成长过程和水平特征，更要为每个学生提供长期、具体、可行的指导和改进建议。（二）学业水平考试与高考命题建议对高中毕业的数学学业水平考试、数学高考的命题提出以下建议。 1．命题原则命题应依据学业质量标准和课程内容，注重对学生数学学科核心素养的考查，处理好数学学科核心素养与知识技能的关系，要充分考虑对教学的积极引导作用。在传统评分的基础上，可以根据解题情况对学生的数学学科核心素养水平的达成进行评价（参见案例20~35）。考查内容应围绕数学内容主线，聚焦学生对重要数学概念、定理、方法、思想的理解和应用，强调基础性、综合性；注重数学本质、通性通法，淡化解题技巧；融入数学文化。命题时，应有一定数量的应用问题，还应包括开放性问题和探究性同题，重点考查学生的思维过程、实践能力和创新意识，问题情境的设计应自然、合理。开放性问题和探究性问题的评分应遵循满意原则和加分原则，达到测试的基本要求视为满意，有所拓展或创新可以根据实际情况加分（参见案例20~35）。在命制应用问题、开放性问题和探究性问题时，要注意公平性和阅卷的可操作性。在高中毕业的数学学业水平考试与数学高考的考试命题中，要关注试卷的整体性。处理好考试时间和题量的关系，合理设置题量，给学生充足的思考时间；逐步减少选择题、填空题的题量；适度增加试题的思维量：关注内容与难度的分布、数学学科核心素养的比重与水平的分布；努力提高试卷的信度、效度和公平性。除了上述要求外，数学高考命题还应依据人才选拔要求，发挥数学高考的选拔功能。 2．考试命题路径基于数学学科核心素养的考试命题，应注意以下几个重要环节。（1）构建数学学科核心素养的评价框架。依据数学学科核心素养的内涵、价值和行为表现的描述，参照学业质量的三个水平，构建基于数学学科核心素养测试的评价框架。评价框架包括三个维度：第一个维度是反映数学学科核心素养的四个方面，它们分别为情境与问题、知识与技能、思维与表达、交流与反思；第二个维度是四条内容主线，它们分别为函数、几何与代数、概率与统计、数学建模活动与数学探究活动；第三个维度是数学学科核心素养的三个水平（参见附录1）。（2）依据评价框架，统筹考虑上述三个维度，编制基于数学学科核心素养的试题，每道试题都有针对性的考查重点。（3）对于每道试题，除了给出传统评分标准外，还需要给出反映相关数学学科核心素养的水平划分依据。 3．说明在命题中，选择合适的问题情境是考查数学学科核心素养的重要载体。情境包括：显示情境、数学情境、科学情境，每种情境可以分为熟悉的、关联的、综合的，数学问题是指在情境中提出的问题，从学生认识的角度分为：简单问题、较复杂问题、复杂问题。这些层次是构成数学学科核心素养水平划分的基础，也是数学学科核心素养评价等级划分的基础。对于知识与技能，要关注能够承载相应数学学科核心素养的知识、技能，层次可以分为了解、理解、掌握、运用以及经历、体验、探素。在命题中，需要突出内容主线和反应数学本质的核心概念、主要结论、通性通法、数学应用和实际应用。在命题中，应特别关注数学学习过程中思维品质的形成，关注学生会学数学的能力。（三）教材编写建议数学教材为“教”与“学”活动提供学习主题、基本线索和具体内容，是实现数学课程目标、发展学生数学学科核心素养重要的教学资源。数学教材的编写要全面落实立德树人的基本要求，充分体现数学学科特有的育人价值与功能。要贯彻高中数学课程的基本理念与要求，贯穿发展学生数学学科核心素养的主线，要体现数学内容的逻辑体系，揭示数学内容的发生、发展过程；要遵从学生认知规律，合理安排学习内容，形成教材的编排体系以及相应的特色和风格，积极探索教材的多样化。教材应有利于教师创造性教学，有利于学生自主性学习。 1．教材编写要以发展学生数学学科核心素养为宗旨（1）全面体现并落实课程标准提出的基本理念和目标要求教材编写应全面体现并落实课程标准提出的基本理念和目标要求，以学生发展为本，培养和提高学生的数学学科核心素养。为“人人都能获得良好的数学教育，不同的人在数学上得到不同的发展”提供优质的、可供学生多样选择的数学学习资源。教材编写要注重将课程标准提出的课程目标转化为实际的教学要求。应突出发展学生数学学科核心素养的目标要求，帮助学生在获得必要的基础知识和基本技能、感悟数学基本思想、不断积累数学基本活动经验的过程中，逐步提高发现和提出问题的能力、分析和解决问题的能力，发展数学实践能力及创新意识，树立科学精神，促进学生学会学习。（2）促进学生数学学科核心素养的发展发展学生数学学科核心素养是数学课程的核心目标，是教材编写的宗旨。编写者应深入理解数学学科核心素养，用以指导内容的选择和编排，应遵循学生认知规律，创设合适的问题情境，设计有效的数学学习活动，展示数学概念、结论、应用的形成发展过程。教材编写者需要以创新的精神，积极探索新的途径和方式，促进学生数学学科核心素养的发展。（3）准确把握内容要求和学业质量标准教材编写者不仅要认真研究内容要求，还要深入研究学业质量标准，准确把握学生经过学习应当达到的要求；要很好地把握学业质量标准的整体性和阶段性，统筹考虑学生的整个学习过程，设计出有利于学生达成学业目标的教材。在编写相关内容时，要把握好内容所涉及的范围，关注内容中蕴含的数学学科核心素养水平要求；也要把握好用“了解”“理解”“掌握”“运用”等行为动词所表达的内容程度要求的不同，确定教材内容的难度。 2.教材编写应体现整体性（1）凸显内容和数学学科核心素养的融合教材编写时要凸显内容和数学学科核心素养的相互融合。内容要求中没有对内容呈现的顺序提出要求，因此编写者要认真思考内容主线的逻辑结构，合理设计教材的体系。六个数学学科核心素养既相对独立、又相互交融，是一个有机的整体。编写者既要深刻理解每一个数学学科核心素养，又要把握数学学科核心素养之间的关联。特别重要的是，编写者要认真研究如何在数学内容的表述中体现数学学科核心素养，编写出数学内容与数学学科核心素养融为一体的教材。（2）注重教材的整体结构教材编写必须遵从课程标准设定的课程结构，要充分注意到必修课程是学生高中毕业的内容要求，必修和选择性必修课程是学生高考的内容要求，选修课程是供学校自主设定、学生自主选修的课程。要整体设计必修和选择性必修课程的体系，处理好数学内容的层次性与数学学科核心素养水平发展的连续性与阶段性的关系，使教材形成一个整体的结构体系。（3）体现内容之间的有机衔接高中数学内容主要分为四条主线，它们既相对独立，又相互联系。教材各个章节的设计要体现三个关注：关注同一主线内容的逻辑关系，关注不同主线内容之间的逻辑关系，关注不同数学知识所蕴含的通性通法、数学思想。数学内容的展开应循序渐进、螺旋上升，使教材成为一个有机的整体。（4）落实数学建模活动与数学探究活动数学建模活动与数学探究活动是数学内容的主线之一。这条主线不仅能够帮助学生更好地掌握知识技能，更能帮助学生学会数学地思考和实践，是学生形成和发展数学学科核心素养的有效载体。教材的编写要重视这条主线的设计，按照课程内容的要求通盘考虑、分步实施。基于这条主线的多样性和灵活性，应当在教师教学用书中提出比较详细的教学建议，使这条主线的活动能够收到实效。（5）实现内容与数学文化的融合，体现时代性教材应当把数学文化融入到学习内容中，可以适当地介绍数学和科学研究的成果，开拓学生的数学视野，激发学生的学习兴趣与好奇心，培养学生的科学精神。“课程内容”中在相应的地方给出了数学文化的提示，供编写者参考。希望教材编写者重视中国传统文化中的数学元素。（6）整体设计习题等课程资源习题是教材的重要组成部分，要提高习题的有效性，科学、准确地把握习题的容量、难度，防止“题海战术”。应开发一些具有应用性、开放性、探究性的问题，解决这样的问题有助于学生数学学科核心素养的提升。习题是课堂教学内容的巩固和深化，也应当为学生发展数学学科核心素养提供平台。要重视习题编写的针对性，也要重视习题编排的整体性。例如，练习题要关注习题的层次性、由浅入深，帮助学生在掌握知识技能的同时，进一步感悟数学的基本思想，积累数学思维的经验；思考题要关注情境和问题的创设，有利于学生理解数学知识的本质，提升数学学科核心素养；复习题要关注单元知识的系统性，帮助学生理解数学的结构，增进复习的有效性，达到相应单元的“学业要求”；复习题也要关注数学内容主线之间的关联以及六个数学学科核心素养之间的协调，有利于学生整体理解、系统掌握学过的数学内容，实现学业质量的相应要求。为了体现教材的整体特色和风格，教材的支撑性资源也应当一体设计，形成多样的课程资源。 3.教材编写应遵循“教与学”的规律（1)教材编写要有利于教师的教编写者要认真研究教学建议，教材的编写要有利于教师实现教学建议中对教师教学提出的要求。要便于教师把握知识本质，驾驭课程内容，要便于教师把握知识结构，统筹教学安排；要便于教师教学设计，创设教学情境、提出合适问题、有效组织教学；要为教师自主选择、增补和调整教学内容预留必要空间。（2）教材编写要有利于学生的学编写者要认真研究学业质量标准，教材的编写要有利于学生达成学业质量标准提出的要求。教材应具备可读性，深入浅出，易于学生理解，激发学习兴趣；应具有探索性，启发学生思考，提供思维空间；要为学生提供学习方法的指导，促进学生形成良好的学习习惯和思维习惯。（3）要处理好几个关系遵循学生数学学习规律要处理好以下几个关系。处理好数学的科学形态与教育形态之间的关系。教材的编写既要充分反映数学的本质，体现数学应有的逻辑性和严谨性，也要符合高中学生的认知规律，有利于学生自主学习、直观理解。处理好过程与结果的关系。教材不能只是数学结论的简单表述，应该体现结论产生的背景和形成发展过程，引导学生在背景和过程中主动探究、认识建构、理解结论。处理好直接经验与间接经验的关系。教材的编写要加强课程内容与学生生活以及现代社会和科技发展的联系，提高学生的学习兴趣，帮助学生积累获取知识的经验。 4.教材内容呈现方式应丰富多样内容呈现方式丰富多样可以增强教材的可读性与亲和力，更好地引导学生自主学习。多样化的设计可以体现在教材编写的各个方面，如素材选取、栏目设计、活动方式、情境类型、思路引领、习题选择、图文表达形式等。呈现方式的丰富多样，还可以通过信息技术与课程的深度融合以及课程资源开发的多样化实现。教材应具有一定的弹性，适应学生学习个性化需求，为学校、教师拓展和开发课程内容资源提供可能。例如，提供具有不同层次要求的习题供学生选用，通过特定设计的问题（非常规问题、开放性问题），引导学生展示数学理解力，满足学生自主探究的欲望，拓展学生的数学视野；也可以设定一些活动环节，让学生自己收集整理资料，形成研究成果等。 5.注重教材特色建设为提高数学教材的编写质量，应当突出所编写数学教材的特色。要认真总结课改以来数学实验教材编写的实践经验，借鉴国外优秀数学教材编写案例，广泛听取教材使用者的意见和建议，精心设计、反复修订，凝练并形成所编写教材的风格与特色。教材编写者应锐意创新、勇于实践，编写出能够经得起检验的、把数学内容与数学学科核心素养有机融合的数学教材。（四）地方与学校实施课程标准的建议 1.地方实施课程标准应注意的几个问题地方应重点关注本地区高中数学课程实施的整体推进，突出重点。通过评价，推动本地区教育的全面发展。（1）重视顶层设计，建立有效的数学教研体系逐步完善国家、省（自治区、直辖市）、地（市、县）、学校四级教研体系，重视教研顶层设计，加强与大学、研究机构等的合作，以研促教，建立合理有效的数学教研体系，由专职教研员、兼职教研员、骨干教师组成合作共同体。（2）示范引领，整体推进数学课程的实施建立一批数学课程实施的实验学校，不断探索，总结经验，引领、推动本地区整个高中数学课程的实施。（3）集中力量研究解决课程标准实施中的关键问题抓住本地区具有普遍性、全局性的关键问题，集中力量深入研究，总结经验，推广经验。例如，解决初高中过渡问题时，不仅要关注知识技能，也要关注学生学习习惯的养成，还要关注初高中学生心理的差异，等等。（4）重视过程性评价要加强对数学教学、教研、学习过程的评价，即评价数学教学经验形成的过程、数学教学研究深入的过程、数学学习规律把握的过程。日常评价与考试要根据学生的学习规律，对于重要的概念、结论和应用的评价，要循序渐进，不要一步到位。 2.学校实施课程标准应注意的几个问题（1）加强学校课程建设学校应根据自身的情况，推动国家课程的全面落实，建设有特色的校本课程，适应学生多样化发展的需求，促进学生全面发展。（2）形成有效的课程管理机制学校实施课程标准时，要形成有效的机制，处理好备课组和教研组的关系，使得备课组与教研组协同、高效工作，为数学课程的实施提供保障。学校要为课程的选择提供必要的教学条件，形成相应的管理制度，充分利用社会资源以满足学生的学习需求。（3）加强数学教师的专业发展和团队建设教师专业发展是实施课程标准的关键，学校要加强对数学教师的培训，提升教师的专业水平。学校要加强培养数学骨干教师，充分发挥骨干教师的作用，关注青年教师的成长，注重发展教师的数学教育理论、实践能力等，形成高效、专业的教师团队。（4）开展有针对性的数学教研活动教研组应定期开展教研活动，除了解决日常教学中的问题，每年还要确定需要集中研究、突破的教学难题。 3.教师实施课程标准应注意的几个问题（1）以教师专业标准的理念为指导，提升自身的专业水平《中学教师专业标准》提出了“育人为本，师德为先，能力为重，终身学习”的基本理念，从专业理念与师德、专业知识、专业能力三个维度提出了教师专业发展的基本要求。数学教师要以《中学教师专业标准》的理念为指导，以数学学科核心素养为依托，终身学习，不断实践，掌握教学所需基础知识，提升教书育人基本能力，达到《中学教师专业标准》对教师专业发展提出的基本要求。（2）数学教师要努力提升通识素养教师应主动提升自身的通识素养，包括科学素养、人文素养和信息技术素养等。应养成良好的自主学习习惯，能学习、会学习、善学习，努力成为学生主动学习、不断进取的榜样。在教学活动中，应勇于创新，包括教学方式的创新，也包括从教学实践中总结经验；包括指导学生学习方式的创新，也包括对学生认知规律的探索；包括对数学知识更为深刻的理解，也包括对数学结构的梳理。实现对自身数学教学经验的不断反思和超越。（3）数学教师要努力提升数学专业素养教学建议强调：“‘四基”是培养学生数学学科核心素养的沃土，是发展学生数学学科核心素养的有效载体。”因此，为了培养学生的数学学科核心素养，数学教师必须提升自身的“四基”水平、提升数学专业能力，自觉养成用数学的眼光发现和提出问题、用数学的思维分析和解决问题、用数学的语言表达和交流问题的习惯。可以关注以下几个方面。把握高中数学的四条主线脉络，理解知识之间的关联。把握数学核心概念的本质，明断什么是数学的通性通法。理解与高中数学关系密切的高等数学的内容，能够从更高的观点理解高中数学知识的本质。例如，通过导函数理解函数的性质，通过运算法则理解初等函数，通过矩阵变换和不变量理解几何与代数，通过样本空间和随机变量理解概率与统计。理解数学知识产生与发展过程中所蕴含的数学思想，能够通过实例理解和表述数学抽象与数学的一般性、逻辑推理与数学的严谨性、数学模型与数学应用的广泛性之间的必然联系，具有在数学教学中渗透数学基本思想的意识和能力。（4）数学教师要努力提升数学教育理论素养数学教师要有良好的数学教育理论素养，能把握数学教育的价值取向，有效落实数学教育的育人目标。可以关注以下几个方面。结合教育教学实践，阅读和理解教育与数学教育经典著作，关注前沿进展的要求。认真研读课程标准，理解和把握高中数学课程的目标，深入思考教与学的关系。基于课程标准，认真研读教材，把握“四基”与数学学科核心素养的关联。基于理论与实践，不断探索数学教学的规律，特别是学生学习高中数学的规律，探索如何把科学形态的数学转化为教育形态的数学。理解和把握评价的作用，思考如何通过评价鼓励学生学习的自觉性、如何通过评价调整自己的教学。（5）数学教师要努力提升教学实践能力数学教师应用理论指导实践，不断总结与反思自己的教学实践，不断提高教学能力，最终落实到课堂、落实到学生。可以关注以下几个方面。一是提升教学设计和实施能力。首先要把握数学知识的本质、理解其中的教育价值，把握教学中的难点，理解学生认知的特征；在此基础上，探索通过什么样的途径能够引发学生思考，让学生在掌握知识技能的同时，感悟知识的本质，实现教育价值；最后能够创设合适的情境、提出合适的问题，设计教学流程、写好教案。在实施过程中，能够有效处理预设和生成的关系，积极启发学生思考，关注每一个学生的成长。二是提升教学案例的分析能力。教学活动是不断实践的过程，实践能力的提升本质上是一种经验的积累，除自我反思之外，与同事或者教研组共同分析教学案例也是一种有效手段，同时还能促进数学教师团队的共同成长。要注意不断积累教学资源，掌握基本的教学策略。三是提升信息技术的使用能力。基于信息技术的教育资源和教学手段日新月异，正在改变着数学教与学的方式。教师要适应时代的发展，按照课程标准的要求，发挥信息技术直观便捷、资源丰富的优势，帮助学生发展数学学科核心素养。四是提升数学教育研究的能力。数学教育研究要落实到课堂，落实到学生。一方面要善于发现自己教学过程中、学生学习过程中的问题，另一方面要善于借鉴其他教师的教学经验，把这些问题或经验作为自己的研究课题，实现教学活动的理性思考，不断提升理论水平和教学能力。高中数学课程标准修订的重点是落实数学学科核心素养，这对数学教师提出了新的要求。通过校本教研、学习讨论、教学实验、展示交流等途径，数学教师要深刻认识数学学科核心素养的育人价值，把握数学学科核心素养与知识技能之间的关联，理解数学学科核心素养的内涵和水平划分，将数学学科核心素养的落实变成自己的自觉行动。要通过创设合适的学习任务、学习情境、学习活动等，把学生数学学科核心素养的养成渗透到日常教学中；要创新评价的形式和方法，把知识技能的评价与数学学科核心素养达成状况的评价有机融合，完成课程标准中提出的学业质量的要求，落实立德树人根本任务。附录1 数学学科核心素养的水平划分水平素养数学抽象水平一能够在熟悉的情境中直接抽象出数学概念和规则，能够在特例的基础上归纳并形成简单的数学命题，能够模仿学过的数学方法解决简单问题。能够解释数学概念和规则的含义，了解数学命题的条件与结论，能够在熟悉的情境中抽象出数学问题。能够了解用数学语言表达的推理和论证；能够在解决相似的问题中感悟数学的通性通法，体会其中的数学思想。在交流的过程中，结合实际情境解释相关的抽象概念。水平二能够在关联的情境中抽象出一般的数学概念和规则，能够将已知数学命题推广到更一般的情形，能够在新的情境中选择和运用数学方法解决问题。能够用恰当的例子解释抽象的数学概念和规则；理解数学命题的条件与结论；能够理解和构建相关数学知识之间的联系。能够理解用数学语言表达的概念、规则、推理和论证；能够提炼出解决一类问题的数学方法，理解其中的数学思想。在交流的过程中，能够用一般的概念解释具体现象。水平三能够在综合的情境中抽象出数学问题，并用恰当的数学语言予以表达；能够在得到的数学结论基础上形成新命题；能够针对具体问题运用或创造数学方法解决问题。能够通过数学对象、运算或关系理解数学的抽象结构，能够理解数学结论的一般性，能够感悟高度概括、有序多级的数学知识体系。在现实问题中，能够把握研究对象的数学特征，并用准确的数学语言予以表达；能够感悟通性通法的数学原理和其中蕴含的数学思想。在交流的过程中，能够用数学原理解释自然现象和社会现象。水平素养逻辑推理水平一能够在熟悉的情境中，用归纳或类比的方法，发现数量或图形的性质、数量关系或图形关系。能够在熟悉的数学内容中，识别归纳推理、类比推理、演绎推理；知道通过归纳推理、类比推理得到的结论是或然成立的，通过演绎推理得到的结论是必然成立的。能够通过熟悉的例子理解归纳推理、类比推理和演绎推理的基本形式。了解熟悉的数学命题的条件与结论之间的逻辑关系；能够证明简单的数学命题并有条理地表述论证过程。能够了解熟悉的概念、定理之间的逻辑关系。能够在交流过程中，明确所讨论问题的内涵，有条理地表达观点。水平二能够在关联的情境中，发现并提出数学问题，用数学语言予以表达；能够理解归纳、类比是发现和提出数学命题的重要途径。能够对与学过的知识有关联的数学命题，通过对条件与结果的分析，探索论证的思路，选择合适的论证方法予以证明，并能用准确的数学语言表述论证过程；能够通过举反例说明某些数学结论不成立。能够理解相关概念、命题、定理之间的逻辑关系，初步建立网状的知识结构。能够在交流的过程中，始终围绕主题，观点明确，论述有理有据。水平三能够在综合的情境中，用数学的眼光找到合适的研究对象，提出有意义的数学问题。能够掌握常用逻辑推理方法的规则，理解其中所蕴含的思想。对于新的数学问题，能够提出不同的假设前提，推断结论，形成数学命题。对于较复杂的数学问题，通过构建过渡性命题，探索论证的途径，解决问题，并会用严谨的数学语言表达论证过程。能够理解建构数学体系的公理化思想。能够合理地运用数学语言和思维进行跨学科的表达与交流。水平素养数学建模水平一了解熟悉的数学模型的实际背景及其数学描述，了解数学模型中的参数、结论的实际含义。知道数学建模的过程包括：提出问题、建立模型、求解模型、检验结果、完善模型。能够在熟悉的实际情境中，模仿学过的数学建模过程解决问题。对于学过的数学模型，能够举例说明建模的意义，体会其蕴含的数学思想；感悟数学表达对数学建模的重要性。在交流的过程中，能够借助或引用已有数学建模的结果说明问题。水平二能够在熟悉的情境中，发现问题并转化为数学问题，知道数学问题的价值与作用。能够选择合适的数学模型表达所要解决的数学问题；理解模型中参数的意义，知道如何确定参数，建立模型，求解模型；能够根据问题的实际意义检验结果，完善模型，解决问题。能够在关联的情境中，经历数学建模的过程，理解数学建模的意义；能够运用数学语言，表述数学建模过程中的问题以及解决问题的过程和结果，形成研究报告，展示研究成果。在交流的过程中，能够用模型的思想说明问题。水平三能够在综合情境中，运用数学思维进行分析，发现情境中的数学关系，提出数学问题。能够运用数学建模的一般方法和相关知识，创造性地建立数学模型，解决问题。能够理解数学建模的意义和作用；能够运用数学语言，清晰、准确地表达数学建模的过程和结果。在交流的过程中，能够通过数学建模的结论和思想阐释科学规律和社会现象。水平素养直观想象水平一能够在熟悉的情境中，建立实物的几何图形，能够建立简单图形与实物之间的联系；体会图形与图形、图形与数量的关系。能够在熟悉的数学情境中，借助图形的性质和变换（平移、对称、旋转）发现数学规律；能够描述简单图形的位置关系和度量关系及其特有性质。能够通过图形直观认识数学问题；能够用图形描述和表达熟悉的数学问题、启迪解决这些问题的思路，体会数形结合。能够在日常生活中利用图形直观进行交流。水平二能够在关联情境中，想象并构建相应的几何图形；借助图形提出数学问题，发现图形与图形、图形与数量的关系，探索图形的运动规律。能够掌握研究图形与图形、图形与数量之间关系的基本方法，能够借助图形性质探索数学规律，解决实际问题或数学问题。能够通过直观想象提出数学问题；能够用图形探索解决问题的思路；能够形成数形结合的思想，体会几何直观的作用和意义。在交流的过程中，能够利用直观想象探讨数学问题。水平三能够在综合情境中，借助图形，通过直观想象提出数学问题。能够综合利用图形与图形、图形与数量的关系，理解数学各分支之间的联系；能够借助直观想象建立数学与其他学科的联系，并形成理论体系的直观模型。能够通过想象对复杂的数学问题进行直观表达，反映数学问题的本质，形成解决问题的思路。在交流的过程中，能够利用直观想象探讨问题的本质及其与数学的联系。水平素养数学运算水平一能够在熟悉的数学情境中了解运算对象，提出运算问题。能够了解运算法则及其适用范围，正确进行运算；能够在熟悉的数学情境中，根据问题的特征建立合适的运算思路，解决问题。在运算过程中，能够体会运算法则的意义和作用，能够运用运算验证简单的数学结论。在交流的过程中，能够用运算的结果说明问题。水平二能够在关联的情境中确定运算对象，提出运算问题。能够针对运算问题，合理选择运算方法、设计运算程序，解决问题。能够理解运算是一种演绎推理；能够在综合利用运算方法解决问题的过程中，体会程序化思想的意义和作用。在交流的过程中，能够借助运算探讨问题。水平三在综合情境中，能把问题转化为运算问题，确定运算对象和运算法则，明确运算方向。能够对运算问题，构造运算程序，解决问题。能够用程序化的思想理解与表达问题，理解程序化与计算机解决问题的联系。在交流的过程中，能够用程式化思想理解和解释问题。水平素养数据分析水平一能够在熟悉的情境中了解随机现象及简单的统计或概率问题。能够对熟悉的概率问题，选择合适的概率模型，解决问题；能够对熟悉的统计问题，选择合适的抽样方法收集数据，掌握描述、刻画、分析数据的基本统计方法，解决问题。能够结合熟悉的实例，体会概率是对随机现象发生可能性大小的度量，可以通过定义的方法得到，也可以通过统计的方法进行估计；能够用统计和概率的语言表达简单的随机现象。在交流的过程中，能够用统计图表和简单概率模型解释熟悉的随机现象。水平二能够在关联情境中，识别随机现象，知道随机现象与随机变量之间的关联，发现并提出统计或概率问题。能够针对具体问题，选择离散型随机变量或连续型随机变量刻画随机现象，理解抽样方法的统计意义，能够运用适当的统计或概率模型解决问题。能够在运用统计方法解决问题的过程中，感悟归纳推理的思想，理解统计结论的意义；能够用统计或概率的思维来分析随机现象，用统计或概率模型表达随机现象的统计规律。在交流的过程中，能够用数据呈现的规律解释随机现象。水平三能够在综合情境中，发现并提出随机问题。能够针对不同的问题，综合或创造性地运用统计概率知识，构造相应的统计或概率模型，解决问题；能够分析随机现象的本质，发现随机现象的统计规律，形成新的知识。能够理解数据分析在大数据时代的重要性。能够理解数据蕴含着信息，可以通过对信息的加工，得到数据所提供的知识和规律，并用统计或概率的语言予以表达。在交流的过程中，能够辨明随机现象，并运用恰当的语言进行表述。附录2 教学案例与评价案例本附录提供了一些案例，是为了帮助教师更好地理解课程标准的要求，特别是理解数学核心素养与内容、教学、评价、考试命题的关系，为教学、评价、考试命题。案例按照标准中出现的顺序排列，有些案例是说明内容、教学、评价、考试命题中的一个问题，有些案例是说明两个或两个以上问题；有些案例体现某个数学核心素养，有些案例综合体现了几个数学学科核心素养，案例中素养表述的顺序反映了所体现素养的主次。有些案例针对在教学过程中容易出现的一些问题，是为了帮助教师答疑解惑。每一个案例前有简短说明，说明本案例针对的问题及其蕴含的数学核心素养，以及如何使用该案例。案例1　借助一元二次函数，求解一元二次不等式【目的】学习用函数统一理解初中学过的函数、方程与不等式的联系，逐渐学会利用函数解决相关的数学问题，体会数学内容之间的联系，提升直观想象与数学运算素养。【情境】基于不等式，给出相应函数图象，分析求解的程序。【分析】以下在实数范围内进行讨论。当一个问题有不同的解决方法时，需要对这些方法进行分析、比较，选择能够体现数学本质的、试用范围更广的方法。求解一元二次不等式通常有两种基本方法，一种是代数方法，先对二次三项式进行因式分解，把一元二次不等式转化为一元一次不等式组，通过求解一元一次不等式组，得到一元二次不等式的解集；另一种是函数方法，借助一元二次函数图象的直观，得到求解一元二次不等式的通性通法。后者是一种程序思想方法，具体分析如下，对于一元二次不等式，根据系数的不同，一元二次函数的图象与轴的位置关系可以分为六类，如图1所示。用函数方法求解的程序为：通过系数的符号，判定函数图象开口方向，通过一元二次方程根的判别式，判定函数图象与轴的位置关系；通过计算方程的根得到不等式的解集。图1 六类一元二次函数图象上述两种方法的共性是都与一元二次方程的根有关，差异是函数方法考虑了函数的变化规律。因此，函数方法时具有一般性的，特别是，类比上述函数方法的思维过程，还可以讨论其他类型函数的相关求解问题。案例2　函数的概念【目的】理解基于对应关系的函数概念，感悟函数概念进一步抽象的必要性。【情境】在高中函数概念的教学中，为什么要强调函数是实数集合之间的对应关系？【分析】初中学习的函数概念表述为：如果在一个变化过程中有两个变量和，对于变量的每一个值，变量都有唯一的值与它对应，那么称是的函数。它强调的是用函数描述一个变化过程。例如，在匀速直线运动中（速度为），路程随着时间的变化而变化，因此路程是时间的函数，记为。再如，在单价、数量、总价的关系中，总价随着数量的变化而变化，因此总价是数量的函数，记为，通常把这样的表述称为函数的“变量说”。但是，上述两个函数自变量的单位不同，不能进行加、减等运算。若舍去其具体背景进一步抽象，可以得到一般的正比例函数为非零常数。于是，两个正比例函数就可以进行运算了，所得结果还是一般的函数。到了高中，函数的概念表述为：给定两个非空实数集合A和B，以及对应关系f，若对于集合A中的每一个实数，集合B中有唯一实数与对应，则称为集合A上的函数，这个概念更强调实数集与实数集间的对应关系，通常把这样的表述成为函数的“对应关系说”。这样，不同的函数可以进行加、减、乘、除等运算，函数研究的内涵和应用的范围得以扩展。对应关系强调的是对应的结果，而不是对应的过程。例如，借助高中函数的表达式，可以认定函数，与函数，表示同一个函数。更一般地，可以判断两个函数是否相同：如果两个函数的定义域相同，且相同的变量值对应的函数值也相同，那么，这两个函数就是同一个函数。直观地说，如果两个函数的图象重合，这两个函数是同一个函数，此外，函数，，，，，，使用的字母不同，但它们表示的是同一个函数，因为它们的定义域和对应关系分别对应相同；反之，函数，，，，的对应关系相同，但它们是不同的函数，因为它们的定义域不同。因此，函数的表达与字母的使用无关。使用对应关系刻画函教还有更为深刻的含义，这是因为有些函数很难用解析式表示。侧如，狄利克雷函数因此，对函数概念的进一步抽象是必要的。注：1851年，德国数学家黎曼（Bernhard Riemanm，1826-1866）给出函数定义，假定x是一个变量，它可以逐次取所有可能的实数值。如果对它的每一个值，都有未知量w的唯一的一个值与之对应，则w称为x的函数。人们通常称这样的定义为函数的“对应说”，因为定义中采用了“唯一的一个值与之对应”的说法。法国布尔巴基学派（Nicolas Bourbaki）的宗旨是在集合论的基础上，用形式化的方法重新构建数学最基本的概念和法则。1939年，布尔巴基学派给出函数的定义。（Dieter Ruthing．函数概念的一些定义——从Joh．Bernoulli到N．Bourbaki[J]．数学译林，1986，3，261 Dieter Ruthing．函数概念的一些定义——从Joh．Bernoulli到N．Bourbaki[J]．数学译林，1986，3，263）设E和F是两个集合，它们可以不同，也可以相同。E中的变元x和变元y之间的一个关系称为一个函数关系，如果对于第一个x∈E，都存在唯一的y∈F，它满足与x给定的关系。称这样的运算为函数，它以上述方式将与x有给定关系的元素y∈F与每一个元素x∈E相联系。称y是函数在元素x处的值，函数值由给定的关系所确定。两个等价的函数关系确定同一个函数。人们通常称这样的定义为“关系说”，由此可以看到，高中函数定义的表述是黎曼对应说与布尔巴基学派关系说的融合，采纳了“对应”和“关系”的表述方式。后来，有些学者把布尔巴基学派的定义进一步符号化。设F是定义在集合X和Y上的一个二元关系，称这个关系为函数，如果对于每一个x∈X，都存在唯一的y∈Y，使得（x，y）∈F。这样，函数的定义九完全用数学的符号形式化了，在这个定义中，已经很难找到变量、甚至对应的影子了，进而完全摆脱了函数的物理背景。虽然这种完全形式化的定义更为一般化，却是以丧失数学直观为代价的，因此不适于基础教育阶段的数学教育。案例3 引入弧度制的必要性【目的】理解弧度制的本质是用线段长度度量角的大小，这样的度量统一了三角函数自变量和函数值的单位；进一步理解高中函数概念中为什么强调函数必须是实数集合与实数集合之间的对应，因为只有这样才能进行基本初等函数的运算（四则运算、复合、求反函数等），使函数具有更广泛的应用性。【情境】对于三角函数的教学，为什么初中数学通过直角三角形讲述，而高中数学要通过单位圆讲述？这是必要的吗？【分析】基于对应关系的函数定义，要求函数是实数与实数的对应关系，称前者的取值范围为定义域，后者的取值范围为值域。初中三角函数是对直角三角形中的边角关系的刻画，其中自变量的取值是60进位制的角度、不是10进位制的实数，不符合对应关系的函数定义。事实上，初中学习三角函数是为了解直角三角形，并不讨论三角函数的基本性质。在高中阶段，借助单位圈建立角度与对应弧长的关系，用对应弧长刻画角的大小；因为长度单位与实数单位一致，这就使得三角函数的自变量与函数值的取值都是实数，符合对应关系的函数定义。用角度作为自变量表示三角函数，还存在着一个突出的问题，就是自变量的值与函数值不能进行运算（例如，60°与 sin 60°不能相加，阻碍了三角函数通过运算法则形成其他初等函数。此外，微积分中重要极限成立，也依赖自变量x为实数。特别是，利用三角函效能够较好地描述钟摆、潮汐等周期现象，这时的自变量不一定是角度，可以是时间或其他的量。通过这样的教学，可以让学生感悟数学抽象的层次性。案例4 用三角面数刻面事物同期变化的实例【目的】通过三角函数刻画周期变化现象的实例，体会三角函数在表达和解决实际问题中的作用。【情境】用正弦函数刻画三种周期变化的现象：简谐振动（单摆、弹簧等），声波（音叉发出的纯音），交变电流。【分析】单摆、弹簧等简谐振动可以用三角函数表达为y=Asin（ωx+φ），其中x表示时间，y表示位移，A表示振幅，表示频率，φ表示初相位。图2是单摆的示意图。点O为摆球的平衡位置，如果规定摆球向右偏移的位移为正，则当摆球到达点C时，据球的位移y达到最大值A，当摆球到达点O时，摆球的位移y为0；当摆球到达点D时，摆球的位移y达到反向最大值-A；当摆球再次到达点O时，摆球的位移y又一次为0；当摆球再次到达点C时，摆球的位移y又一次达到最大值A。这样周而复始，形成周期变化。音叉发出的纯音振动可以用三角函数表达为y=Asin（ωx），其中x表示时间，y表示纯音振动时音叉的位移，表示纯音振动的频率（对应音高），A表示纯音振动的振幅（对应音强）。交变电流可以用三角函数表达为y=Asin（ωx+φ），其中x表示时间，y表示电流，A表示最大电流，表示频率，φ表示初相位。图3是交变电流产生的示意图。线圈在匀强磁场中按逆时针方向匀速旋转产生交变电流（电刷及回路等部分省略），当线圈处于图3所示的位置时，线圈中的感应电流y达到最大值A；当线圈由此位置逆时针旋转90°后到达与此平面垂直的位置时，线圈中的感应电流y为0；当线图继续逆时针旋转90°后再次到达水平位置时，线圈中的感应电流y达到反向最大值-A，当线圈继续逆时针旋转90°后再次到达垂直位置时，线圈中的感应电流y又一次为0；当线圈继续逆时针旋转90°后再次到达图示位置时，线圈中的感应电流y又一次达到最大值A。这样周而复始，形成周期变化。对于这样的案例，可以借助计算机软件做出动画，形象化地说明周期变化。案例5 函致单调性概念的抽象过程【目的】结合实例，经历从具体的直观描述到形式的符号表达的抽象过程，加深对函数单调性概念的理解，体会用符号形式化表达数学定义的必要性，知道这样的定义在讨论函数单调性问题中的作用。【情境】在初中阶段，学生已经初步了解一元一次函数、反比例函数、一元二次函数的图象具有单调性的特征。在高中阶段引入函数单调性概念时，可以从直观认识出发，提出合适的课堂讨论问题，使学生经历函数单调性概念的抽象过程。例如，可以提出如下问题。问题1 在初中阶段已经学过一元一次函数、反比例函数、一元二次函数，请根据函数图象（如图4)，分别述说x在哪个范围变化时，y随着x的增大而增大或者减小？图4 一元一次方程、反比例函数、一元二次方程示意图问题2 在日常生活中，哪些函数关系具有上述特征？问题3 如图5，f（-2）<f（2）<f（8），能否据此得出“f（x）在[-2，8]递增”的结论？为什么？问题4 依据函数单调性的定义，证明函数，x∈（2，+∞）是递增的。【分析】初中阶段，学生是经过从直观图形语言到数学自然语言的过程来认识函数的单调性的。到了高中阶段，需要在此基础上进一步用符号语言来表述函数的单调性。在使用符号语言的过程中，“任意”两字是学生遇到的一个难点，需要注意。另外，函数单调性证明过程中的运算也是一个难点。在函数单调性概念的形成中，经历由具体到抽象、由图形语言和自然语言到符号语言表达的过程，发展学生的数学抽象素养。在把握函数单调性定义时，体会全称量词、存在量词等逻辑用语的作用，发展学生的逻辑推理素养。在函数单调性证明的过程中，发展学生的数学运算素养。案例6 利用单位圆的对称性探索三角函数的诱导公式【目的】借助单位圆对称性的几何直观，探索三角函数的诱导公式，提升直观想象和逻辑推理素养。【情境】通过单位圆定义正弦、余弦函数，结合正弦、余弦函数的概念绘制正弦、余弦函数的图象。探索正弦、余弦函数的对称性，得到三角函数的诱导公式。教学片段探索正弦、余弦函数的对称性。教学过程如图6，在单位圆中，角a的终边与单位圆的交点记为P，角—α的终边与单位圆的交点记为P′，通过点P与P′关于x轴成轴对称，探索与角—α有关的诱导公式。如图7，在单位圆中，角α的终边与单位圆的交点记为P，角α+π的终边与单位圆的交点记为P′，通过点P与P关于点O成中心对称，即角a的终边绕点O旋转x以后得到角a+x的终边，探索与角α+π有关的诱导公式。引导学生通过类比，发现其他形式的对称性以及与旋转变换相关的坐标关系，尝试建立数学公式，并验证公式的正确性。引导学生自己给出记忆公式的方法，理解其中的道理。例如，理解“奇变偶不变、符号看象限”的含义。【分析】重要的数学结论往往都是“看”出来的，会“看”需要直观想象素养，诱导公式的教学内容提供了发展学生直观想象素养的平台。以上教学设计的情境，能够通过数学绪论的直观背景和数学语言的清晰表达，揭示数学结论的本质，提升学生的直观想象和逻辑推理素养。案例7 停车距离问题【目的】在数学建模活动中，经历从现实问题中确定变量、探寻关系、建立模型、计算系数、分析结论的全过程，形成和发展数学建模素养。【情境】根据现实背景，建立急刹车的停车距离数学模型，理解数学模型中系数的意义，并根据模型得到的结果，就行车安全提出建议。数学建模活动是一个科学研究的过程，可以个人单独进行，也可以组织研究小组共同开展。科学研究通常需要经历选题、开题、做题、结题四个基本步骤。选题。本案例活动的选题步骤略去。开题。结合问题，查阅相关资料，检索已有成果，用“头脑风暴”的形式集思广益，初步形成解决问题的大致思路和方案，并分析操作的可行性。尝试撰写开题报告。教师可以组织小组之间交流，请学生代表本小组介绍开题报告，交流反思后，改进并确定实施方案。做题。实施建立模型、求解模型、检验结果的过程，写出结题报告或写成小论文。结题。在班里介绍建模过程、结果和收获，由老师和其他同学给出评价。【分析】本案例中，数学建模活动大体需要经历以下几个关键环节，确定影响停车距离的主要因素。例如，停车距离与刹车前汽车行驶的速度有关；与驾驶人员的反应时间有关，因人而异，与车辆的刹车性能有关，因车而异，还与道路状况、天气状况等一些随机因素有关。构建数学模型需要确定最为关键的因素，例如，在高速公路上，如果汽车刹车性能良好，则主要考虑前两个因素。第二，建立急刹车的停车距离模型。由上面的分析，可以得到一个用生活语言表述的模型：停车距离-反应距离+制动距离。① 设d表示停车距离，d1表示反应距离，d2表示制动距离，用数学符号把上述模型表示为d=d1+d2。为了得到d1和d2的具体表达式，可以作下面的假设。关于反应距离，假设反应距离是反应时间和汽车速度的函数。反应时间是指司机意识到应当急刹车到实施刹车所需要的时间，汽车速度是指司机在实施急刹车之前汽车的速度。在一般情况下，反应距离d1与反应时间t和汽车速度v都成正比，把这个关系表示为d1=αtv，其中α为正的特定系数。在现实生活中，可以知道反应时间t>0，但很难确定具体数值。因此，最终只能确认反应距离与汽车速度成正比，即把这个关系写成d1=αv，可以认为用α替代了αt。关于制动距离，假设刹车受力大小近似等于汽车轮胎与路面的摩擦力，制动距离是刹车受力与汽车速度的函教。若F表示刹车受力，则汽车急刹车时所作的功为Fd1。根据能量守恒定律得Fd2=，其中m是汽车质量。另一方面，如果意刹车时的加速度是a，再根据牛顿第二定律得F=ma。综合上面两个式子，可以得到mad2=，即制动距离d2=。也就是说，制动距离与汽车速度平方成正比：d2=βv2，其中β是待定参数。依据①式，得 d=d1+d2=αv+βv2，② 第三，确定参数，计算求解。模型中的参数是至关重要的，一般来说不可能通过理论计算得到，因为在构建模型的过程中有许多因素没有也不可能考虑清楚。在现实模型中，参数值通常是通过统计方法得到的，是通过现实数据估计出来的。大体上有三种方法可以得到现实数据：调查、实验和试验。为了估计急刹车的停车距离模型中的参数，需要通过试验的方法得到现实数据。表1是美国公路局公布的试验数据，通过正比例关系d1=αv和d2=βv2，可以计算出表1每一行中相应的α和β的值。它们的平均数分别为α=0.21，β=0.006，这组数据可以作为对参数α，β的一种估计。于是，通过试验数据得到了停车距离模型 d=0.21v+0.006v2。③ 表1 通过试验观察到的反应距离、制动距高与停车距离（Frank R．Giordano，Maurice D．Weir，Willam P．Fox．数学建模[M]．叶其孝，姜启源，等译．3版．北京；机械工业出版社，2005：57-58．说明：原始数据的单位是英里、英尺，通过计算得到的参数α和β的值分别为1.1和0.054。为了便于理解，此处把距离单位换算为千米、米，相应的参数值也有了变化。）从③式可以看到，汽车停车距离模型是汽车速度的二次函数，因此从数学应用的角度可以认为，函数是构建数学模型的有力工具。由于模型中的参数来源于实际，在一般情况下，这个模型能够经受实践的检验。因此，急刹车的停车距离模型③普遍应用于汽车刹车设计和路面交通管理。为了便于查阅，除了构建模型、制作表格外，人们也给出直观图形。图8直观地给出了急刹车的停车距离模型。案例8 “指数爆炸”的感知和理解【目的】通过观察、计算（可利用作图、计算工具），体会幂函数、指数函数、对数函数增长速度的差异，感知和理解“指数爆炸”的含义。【情境1】使用工具进行计算,感知“指数爆炸”的含义。【分析】课程标准中提到了“指数爆炸”这个名词。如果局限于纸笔运算,在课堂上可以让学生计算整数底数的某些幂。如果借助计算工具,可以先让学生计算1.01的平方和立方,进而提出问题：猜测大概是多少?同样,可以让学生猜测的值。实际上,它们的数值分别为：这样的学习方式,不仅使学生能够直观感知“指数爆炸”的含义,还能帮助学生理解底数对幂的影响。能够帮助学生建立对指数函数变化的直观认识,底数大于1时,随着指数的增大幂变大；底数小于1时,随着指数的增大幂变小。【情境2】借助计算机进行作图,对指数函数、对数函数、幂函数的增长速度进行比较校,进一步理解“指数爆炸”的含义。【分析】可以让学生利用计算机作图工具,画出的函数图象,通过比较图象,分析这四个函数增长的快慢。特别是当x值比较大的时候,直观感知这四个函数值的差异。进一步,让学生通过与图象之间的比较、与图象之间的比较,形成更一般的猜想。例如,交点的个数、变化的差异等。借助这样的素材进行教学,可以让学生体会指数函数、幂函数、对数函数增长速度的差异,比较这三种函数的变化势；经历通过图形建立直观猜想、通过计算验证结论的思维与操作过程,提升学生的直观想象和逻辑推理素养。案例9 向量投影【目的】理解投影的作用,体会投影是构建高维空间与低维空间之间联系的桥梁,形成直观想象,了解投影与数量积运算规则的关系,体会“特殊情况”与“一般情况”的相互作用,提升逻辑推理素养。【情境】空间向量向平面投影、向直线投影,一个向量向另一个向量投影,向量投影有什么意义和作用? 【分析】向量的投影是高维空间到低维子空间的一种线性变换，得到的是低维空间向量,这里是指正交变换。在高中数学中,如图9(1),在空间中,向量向平面投影得到的是与平面x平行的向量;如图9(2),在空间中,向量向直线l投影得到的是与直线l平行的向量;如图9(3)，在空间中,向量向向量的投影,是指向量向与向量共线的向量构成的子空间的投影,得到的是与向量共线的向量。向量称为投影向量。如图9(1),不难看出,向量与向量垂直。这就意味着，当向量与向量起点相同时,终点间的距离最小。此时,三个向量,和构成一个直角三角形,借助勾股定理,可以通过几何直观更好地理解向量投影的本质。以上分析适用于高维空间到低维空间的正交投影。向量标准正交分解定理是投影作用的另一个重要体现。如图10,给定标准正交基,设向量在上的投影向量分别为,则。显然,，,是向量在标准正交基下的坐标。案例10 复数的引入【目的】了解复数概念形成的重要发展阶段,体会其中的理性思维、创新精神和数学文化。【情境】复数的产生大致经历了以下过程。在古希腊学者丢番图时代，人们已经知道一元二次方程式有两个根，但其中有一个根为虚数时，宁可认为方程不可解。直到16世纪，人们普遍认同丢番图的办法。虚数（imaginary）这个名称是法国哲学家、数学家笛卡儿给出的，写在1637年出版的《几何》（书名原文为“Ars Magne（The Great Art）”）。本文参照：M·克莱因.数学，确定性的丧失[M].李宏魁，译.长沙；湖南科学技术出版社，1999.R·笛卡尔.几何[M].袁向东，译.武汉：武汉出版社，1992.）中。欧拉第一个使用符号i表示虚数，写在1777年提交给圣彼得堡科学院的论文中，这篇论文直到1794年才发表。只有给出复数的几何表示，人们才真正感觉到了复数的存在，才心安理得地接受了复数。1797年，丹麦测量学家韦塞尔在丹麦皇家科学院宣读了一篇关于复数的论文，文中引入了虚轴、并把复数表示为平面向量。但直到100年后的1897年，韦塞尔的丹麦文的论文被翻译为法文后，复数几何表示的工作才引起数学界的广泛重视。瑞士数学家阿尔冈把复数对应的向量的长度称为模，写在1806年出版的著作《试论几何作图中虚量的表示法》中，他还进一步利用三角函数表示复数。现在，复数已经被广泛应用于流体力学、信号分析等学科，因此复数有着深厚的物理背景。在复数的基础上，英国数学家哈密顿构造了四元数，并导致了物理学中著名的麦克斯韦方程的产生。【分析】在数学史上，虚数以及复数概念的引入经历了一个曲折的过程，其中充满着数学家的想象力、创造力和不屈不饶、精益求精的精神。由此，在复数概念的教学中，可以适当介绍历史发生发展过程，一方面可以让学生感受数学的文化和精神，另一方面也有助于学生理解复数的概念和意义。如何将数学史融入中小学的数学教学是数学教育领域的一个重要课题。通过数学概念和思想方法的历史发生发展过程，一方面可以使学生感受丰富多彩的数学文化，激发数学学习的兴趣；另一方面也有助于学生对数学概念和思想方法的理解。数学史在数学课堂中的融入方式可以是多种多样的，相关的网络资源也十分丰富，教师应该根据教学的需要选择合适的资料和教学方式。案例11　正方体截面的探究【目的】结合正方体截面设计的问题串，引导学生完成探究、发现、证明新问题的过程，积累数学探究的经验。【情境】用一个平面截正方体，截面的形状将会是什么样的？启发学生提出逐渐深入的系列问题，引导学生进行逐渐深刻的思考。学生可以自主或在教师引导下提出一些问题,例如: （1）给出截面图形的分类原则,找到截得这些形状截面的方法,画出这些截面的示意图。例如。可以按照截面图形的边数进行分类（如图11）。（2）如果截面是三角形，可以截出几类不同的三角形？为什么？（3）如果截面是四边形，可以截出几类不同的四边形？为什么？（4）还能截出哪些多边形？为什么？然后进一步探讨：（5）能否截出正五边形？为什么？（6）能否截出直角三角形？为什么？（7）有没有可能截出边数超过6的多边形？为什么？（8）是否存在正六边形的截面？为什么？最后思考：（9）截面面积最大的三角形是什么形状的三角形？为什么？【分析】这是一个跨度很大的数学问题串，可以针对不同学生，设计不同的教学方式，通过多种方法实施探究。例如，可以通过切萝卜块观察，启发思路；也可以在透明的正方体盒子中注入有颜色的水，观察不同摆放位置、不同水量时的液体表面的形状；还可以借助信息技术直观快捷地展示各种可能的截面。但是，观察不能代替证明。探究的难点是分类找出所有可能的截面，并证明哪些形状的截面一定存在或一定不存在。可以鼓励学生通过操作观察，形成猜想，证明结论。经历这样逐渐深入的探究过程，有利于培养学生发现问题、分类讨论、作图表达、推理论证等能力，在具体情境中提升直观想象、数学抽象、逻辑推理等素养，积累数学探究活动经验。案例12　投掷骰子问题【目的】理解样本点、样本量、有限样本空间的概念，以及有限样本空间中随机事件的相关运算，理解随机时间的表达，体会随机思想。【情境】将一枚均匀骰子相继投掷两次，请回答以下问题：（1）写出样本点和样本空间；（2）用A表示随机事件“至少有一次掷出1点”，试用样本点表示事件A；（3）用表示随机事件“第一次掷出1点，第二次掷出点”，用B表示随机事件“第一次掷出1点”，试用随机事件表示随机事件B；（4）用C表示随机事件“点数之和为7”，并求C发生的概率。【分析】上述四个问题，依次分析如下：（1）首先确定样本点，用1，2，3，4，5，6表示掷出的点数，用（i，j）表示“第一次掷出i点，第二次掷出j点”，则相继投掷两次的所有可能结果如下：（1，1）（1，2）（1，3）（1，4）（1，5）（1，6）（2，1）（2，2）（2，3）（2，4）（2，5）（2，6）（3，1）（3，2）（3，3）（3，4）（3，5）（3，6）（4，1）（4，2）（4，3）（4，4）（4，5）（4，6）（5，1）（5，2）（5，3）（5，4）（5，5）（5，6）（6，1）（6，2）（6，3）（6，4）（6，5）（6，6）注意到（1，2）和（2，1）是不同的样本点，分别表示“第一次掷出1点，第二次掷出2点”和“第一次掷出2点，第二次掷出1点”这两个随机事件，因此样本空间共有36个样本点。把每个样本点称为基本事件。样本空间为。（2）因为随机事件A＝ “至少有一次掷出1点”，则A包括上述样本空间中所有出现1的样本点，因此。（3）。因为这些事件任何一个发生事件B就发生，所以。（4）因为，包括6个样本点，样本空间共有36个样本点，所以。通过上面的讨论可以看到，样本空间只与问题的背景有关。案例13 分层抽样【目的】理解分层随机抽样的特点,了解分层随机抽样的应用,探索快速、有效计算分层抽样数据均值和方差的方法。【情景】在大数据时代,常常需要汇总分析来自不同层次的数据。例如,基于来自不同部门或者不同时期数据的均值和方差,计算全部数据的均值和方差。请看下面的例子。某学校有高中学生有500人，其中男生320人，女生180人。希望获得全体高中学生身高的信息。按照分层抽样原则抽取了样本,通过计算得到男生身高样本均值为173.5 cm,方差为17，女生身高样本均值为163.83 cm,方差为30.03。请回答以下问题： (1)根据以上信息,能够计算出所有数据的样本均值吗?为什么? (2)应当如何计算所有数据的样本均值和方差? 【分析】按照传统的统计方法,需要把所有的数据收集到一起进行计算。但是,在大数据时代,不仅数据量非常庞大,而且要求非常迅速地提供数据结论,因此不可能把所有的数据都收集好以后再进行计算,需要创造更为简捷的方法。以上述问题为例进行分析。 (1)假设所有样本身高的均值为,根据男女生的分层方法和样本均值的定义,可以得到下面的关系式：。从上面的分析可以知道,仅仅依赖问题中提供的信息不能得到所有数据的样本均值,因为缺少男生样本量和女生样本量。因此，在提供分层样本均值的基础上,还需要知道分层的样本量,或知道男生样本量权重、女生样本量权重。 (2)假设男生样本量为32,女生样本量为18.记男生样本为,均值为,方差为;记女生样本为,势值为,方差为，所有数据样本均值为,方差为。样本总量为50。先求所有数据的样本均值。根据样本均值的定义, 。虽然数据和是未知的，但在上面的计算中，只需要样本数据之和，这可以通过样本均值和样本量的乘积得到，即所以。下面计算所有数据的样本方差，根据方差的定义，因为其中的数据是未知的，根据同样的道理，需要把上面的式子转化为各层样本方差、样本均值和样本量的函数。可以计算如下。其中同理。于是。【拓展】如果将总体分为k层,第j层抽取的样本为,第j层的样本量为,样本均值为，样本方差为，j=1,2,…,k。记,所有数据的样本均值和方差为，。注：无论总体是分为两层还是分为多居,计算方法没有实质性差异。但需要注意的是,当层次较多时数学符号的表达比较复杂,要充分考虑到学生的理解能力。这样的问题是有普遍现实意义的。例如,针对某个问题,不同网站提供了各自调查的样本均值和方差,应当如何得到所有数据的样本均值和方差?再如,针对某个问题,连续几天收集数据,得到了每天数据的样本均值和方差,应当如何得到这几天所有数据的样本均值和方差? 案例14 阶梯电价的设计【目的】通过生活中的实例,理解百分位数的统计含义及其应用,让学生体会用统计方法解决实际问题的全过程。【情境】为了实现绿色发展，避免浪费能源，某市政府计划对居民用电采用阶梯式收费的方法。为此，相关部门在该市随机调查了200户居民六月份的用电量（单位：kW·h），以了解这个城市家庭用电量的情况。数据如下： 107 101 78 99 208 127 74 223 31 131 214 135 89 66 60 115 189 135 146 127 203 97 96 62 65 111 56 151 106 8 162 91 67 93 212 159 61 63 178 194 194 216 101 98 139 78 110 192 105 96 22 50 138 251 120 112 100 201 98 84 137 203 260 134 156 61 70 100 72 164 174 131 93 100 163 80 76 95 152 182 88 247 191 70 130 49 114 110 163 202 265 18 94 146 149 147 177 339 57 109 107 182 101 148 274 289 82 213 165 224 142 61 108 137 90 254 201 83 253 113 130 82 170 110 108 63 250 237 120 84 154 288 170 123 172 319 62 133 130 127 107 71 96 140 77 106 132 106 135 132 167 82 258 542 51 107 69 98 72 48 109 134 250 42 320 113 180 144 116 530 200 174 135 160 462 139 133 304 191 283 121 132 118 134 124 178 206 626 120 274 141 80 187 88 324 136 498 169 77 57 根据以上数据，应当如何确定阶梯电价中的电量临界值，才能使得电价更为合理？【分析】选取六月份进行调查，是因为这个城市六月份的部分时间需要使用空调，因此六月份的用电量在12个月中处于中等偏上水平，如果阶梯电价临界值的确定依赖于居民月用电量的分布，例如计划实施3阶的阶梯电价，有人给出一个分布如下：75%用户在第一档（最低一档），20%用户在第二档，5%用户在第三档（最高一档）。这样需要通过样本数据估计第一档与第二档、第二档与第三档的两个电量临界值，即75%和95%这两个电量临界值。通过样本估计总体百分位数的要领是对样本数据进行排序，得到有序样本（在统计学中称之为顺序统计量）。利用电子表格软件，对上面的样本数据进行排序，可以得到下面的结果： 8 18 22 31 42 48 49 50 51 56 57 57 60 61 61 61 62 62 63 63 65 66 67 69 70 70 71 72 72 74 76 77 77 78 78 80 80 82 82 82 83 84 84 88 88 89 90 91 93 93 94 95 96 96 96 97 98 98 98 99 100 100 100 101 101 101 105 106 106 106 107 107 107 107 108 108 109 109 110 110 110 111 112 113 113 114 115 116 118 120 120 120 121 123 124 127 127 127 130 130 130 131 131 132 132 132 133 133 134 134 134 135 135 135 135 136 137 137 138 139 139 140 141 142 144 146 146 147 148 149 151 152 154 156 159 160 162 163 163 164 165 167 169 170 170 172 174 174 177 178 178 180 182 182 187 189 191 191 192 194 194 200 201 201 202 203 203 206 208 212 213 214 216 223 224 237 247 250 250 251 253 254 258 260 265 274 274 283 288 289 304 319 320 324 339 462 498 530 542 626 样本数据总共有200个，其中最小值是8，最大值是626，说明200户居民六月份的最小用电量为8 kW·h，最大用电量为626 kW·h，极差为618。初中统计内容中学过中位数，相当于50%分位数。因为数据量是200，那么这组数据的样本中位数就是有序样本第100个数和第101个数的平均数，即130，说明这个城市六月份居民用电量的中间水平在130 kW·h左右。下面确定75%和95%这两个电量临界值。类似中位数的计算，因为200×75% = 150，所以第一个临界值为有序样本中第150个数178和第151个数 178的均值，仍然是178。因为200×95% = 190，所以第二个临界值为有序样本中第190个数289和第191个数304的平均数，这个平均数为296.5（因为是对百分位数的估计，估计值可以是289和304之间任何一个数，为了便于操作可以取值为297）。依据确定了的电量临界值，阶梯电价可以规定如下：用户每月用电量不超过178kW·h（或每年用电量不超过2136kW·h），则按第一档电价标准缴费；每月用电量（单位：kW·h）在区间内（或每年用电量在区间内），其中的178kW·h按第一档电价标准收费，超过178kW·h的部分按第二档电价标准收费；每月用电量超过297kW·h（或每年用电量超过3 564kW·h），其中的178kW·h按第一档电价标准收费，（297–178=）119kW·h按第二档电价标准缴费，超过297kW·h的部分按第三档电价标准缴费。社会上对这种制定阶梯电价的原则和方法存在不同意见，教师可以引导学生讨论制定合理阶梯电价的原则和方法。案例15测量学校内、外建筑物的高度【目的】运用所学知识解决实际测量高度的问题，体验数学建模活动的完整过程。组织学生通过分组、合作等形式，完成选题、开题、做题、结题四个环节。（关于本案例的评价部分参看案例19）【情境】给出下面的测量任务：（1）测量本校的一座教学楼的高度；（2）测量本校的旗杆的高度；（3）测量学校墙外的一座不可及，但在学校操场上可以看得见的物体的高度。可以每2〜3个学生组成一个测量小组，以小组为单位完成；各人填写测量课题报告表（见表2)，一周后上交。表2测量课题报告表项目名称：___________________ 完成时间：___________________ 1.成员与分工姓名分工 2.测量对象例如，某小组选择的测量对象是：旗杆、教学楼、校外的XX大厦。 3.测量方法（请说明测量的原理、测量工具、创新点等） 4.测量数据、计算过程和结果（可以另外附图或附页） 5.研究结果（包括误差分析） 6.简述工作感受【教学过程】教师可以对学生的工作流程提出如下要求和建议。（1）成立项目小组，确定工作目标，准备测量工具。（2）小组成员查阅有关资料，进行讨论交流，寻求测量效率高的方法，设计测量方案（最好设计两套测量方案）。（3）分工合作，明确责任。例如，测量、记录数据、计算求解、撰写报告的分工等。（4）撰写报告，讨论交流。可以用照片、模型、 PPT等形式展现获得的成果。根据上述要求，每个小组要完成以下工作。（1）选题本案例活动的选题步骤略去。（2）开题可以在课堂上组织开题交流，让没一个项目小组陈述初步测量的方案，教师和其他同学可以提出质疑。例如：如果有学生提出要测量仰角来计算高度，教师可以追问：怎么测量？用什么工具测量？目的是提醒学生，事先设计出有效的测量方法和实用的测量仪器。如果有学生提出要通过测量太阳的影长计算高度，教师可以追问：几时测量比较好？如果学生提出比较测量物和参照物的影长时，教师可以追问：是同时测量好，还是先后测量好？目的是提醒学生注意测量的细节。如果有学生提出用照相机拍一张测量对象和参照物（如一个已知身高的人) 的合影，通过参照物的高度按比例计算出楼的高度。教师可以追问：参照物应该在哪里？与测量对象是什么位置关系？目的是提醒学生注意现实测量与未来计算的关联。在讨论的基础上，项目小组最终形成各自的测量方案。讨论的目的是让学生仔细想清楚测量过程中将使用的数学模型，这样可以减少实践过程中的盲目性，培养学生良好的思维习惯；同时可以让学生意识到，看似简单的问题，也有许多需要认真思考、认真对待的东西，促进科学精神的形成。（3）做题一句小组的测量方案实时测量。尽量安排各个小组在同一时间进行测量，这样有利于教师的现场观察和管理。教师需要提醒学生，要有分工、合作、责任落实到个人。在测量过程中，教师要认真巡视，记录那些态度认真、合作默契、方法恰当的测量小组和个人，供讲评时使用。特别要注意观察和发现测量中出现的问题，避免因为测量方法不合理产生较大误差，当学生出现类似的问题时，教师要把问题看做极好的教育契机，启发学生分析原因，引导他们发现出现问题的原因、寻求解决问题的办法。（4）结题在每一位学生都完成“测量报告”后，可以安排一次交流讲评活动，遴选的交流报告最好有鲜明的特点，如测量结果准确，过程完整清晰，方法有创意，误差处理得当，报告书写规范等；或者测量的结果出现明显误差，使用的方法不当。交流讲评往往是数学建模过程中最为重要的环节，可以使学生在这一过程中相互借鉴，共同提高。（有关测量评价的讨论参见案例19) 【分析】测量高度是传统的数学应用问题，这样的问题有助于培养学生分析解决问题、动手实践、误差分析等方面的能力。测量模型可以用平面几何的方法，例如，比例线段、相似形等；也可以用三角的方法，甚至可以用物理的方法，例如，考虑自由落体的时间；等等.应鼓励学生在合作学习的基础上，自主设计、自己选择测量方法解决问题。这样的教学活动，因为问题贴近学生的生活，学生比较容易上手、采用选题、开题、做题、结题四个环节实施数学建模活动，能够使学生在做中学、在学中做，从中体会数学的应用价值，并且展现个性，尝试创新。【拓展】鼓励学生提出新的问题，积累数学建模资源。例如： 1.本市的电视塔的高度是多少米？ 2.—座高度为H m的电视塔，信号传播半径是多少？信号覆盖面积有多大？ 3.找一张本市的地图，看一看本市的地域面积有多少平方千米？电视塔的位置在地图上的什么地方？按照计算得到的数据，这座电视塔发出的电视信号是否能覆盖本市？ 4.本市（外地）到北京的距离有多少千米？要用一座电视塔把信号从北京直接发送到本市，这座电视台的高度至少要多少米？ 5.如果采用多个中继站的方式，用100 m高的塔接力传输电视信号，问从北京到本地至少要建多少座100 m高的中继传递塔？ 6.考虑地球大气层和电离层对电磁波的反射作用，重新考虑问题2， 4，5。 7.如果一座电视塔（例如300 m高)不能覆盖本市，请你设计一个多塔覆盖方案。 8.至少发射几颗地球定点的通讯卫星，可以使其信号覆盖地球？ 9.如果我国要发射一颗气象监测卫星，监测我国的气象情况，请你设计一个合理的卫星定点位置或卫星轨道。 10. 在网上收集资料，了解有关“北斗卫星导航系统”的内容，在班里做一个相关内容的综述，并发表对这件事的看法。案例16 用向量方法研究距离问题【目的】针对距离问题，通过几种研究方法的比较，提炼解决问题的通行通法。在教师的指导下，学生经历梳理知识、提炼方法、感悟思想的研究过程，提升直观想象、逻辑推理和数学运算素养。这样的教学可以为空间向量与立体几何的复习课提供素材。【情境】在“几何与代数”内容的阐述中强调：“通过几何图形建立直观，通过代数公式表达规律。”正如希尔伯特所说。（摘自希尔伯特1900年在巴黎第二届国际数学家代表大会上演说《数学问题》，刊在《美国数学会通报》卷8,1902。译文参见：康斯坦丝·瑞德。希尔伯特：数学世界的亚历山大[M]。袁向东，李文林，译，上海：上海科学技术出版社，2003，116。）算术符号是文字化的图形，而几何图形则是图象化的公式。没有一个数学家能缺少这些图象化的公式，正如在数学演算中他们不能不使用加、脱括号的操作或其他的分析符号一样。距离问题是培养学生直观想象、逻辑推理和数学运算素养的很好的载体。在基础教育阶段涉及的距离问题主要有：两点间距离，点到直线距离，平行线之间距离，点到平面距离，直线到平面距离，平行平面之间距离，异面直线之间的距离 (选修)。计算距离可以用综合几何方法，也可以用解析几何方法，还可以用向量方法。教学片段1：梳理求平面上点到直线距离的几种方法。综合几何方法。给定过点A，C的直线l，B为直线l外一点，求点B 到直线l的距离。因为过点A，B，C可以得到一个平面上的三角形，因此求距离就等价于求三角形的高。基本思路是：用余弦定理确定∠A，再用正弦函数值求出AC边上的高。解析几何方法。建立平面直角坐标系，确定点B的坐标和过点A，C 的直线l的方程，然后求点B到直线l的距离。基本思路是：求与直线l垂直的直线的斜率，再求过点B的点斜式直线方程，最后求这两条相互垂直直线的交点。交点与点B的距离就是点B到直线l的距离。向量方法：建立平面直角坐标系，确定点B的坐标和过点A，C 的直线l的法向量，求点B到直线l的距离。基本思路是：求向量到法向量的投影向量，投影向量的长度就是所要求的距离。教学片段2：比较求点到平面距离和求两条异面直线距离的向量方法。点到平面距离。用向量方法求点B到平面距离基本思路：确定平面法向量，在平面内取一点A，求向量到法向量的投影向量，投影向量的长度即为所要求的距离。异面直线距离。用向量方法求异面直线距离基本思路：求出与两条直线的方向向量都垂直的法向量；在两条直线上分别取点A和B，求向量到法向量的投影向量，投影向量的长度即为所要求的距离。【分析】对于上述两个片段，可以归纳出下面的结论。片段1 通过处理距离问题三种方法的对比，可以得到垂直反映了距离的本质，垂直意味着线段长度最短，借助勾股定理可以直观、准确地揭示这个本质，两点间距离公式以及向量投影都可以看作是勾股定理的应用。可以让学生在比较的过程中分析不同方法的共性与差异，进而发现解决问题的关键。片断2 无论是对于平面还是直线，法向量都是反映垂直方向的最为直观的表达形式，法向量的方向和法向量上投影向量的长度既体现了几个图形直观，又提供了代数定量刻画。在这个过程中向量与起点无关的自由性为求距离带来很大的便利。归纳用向量研究上述距离问题的方法，可以得到通性通法，即程序思想方法：第一步，确定法向量；第二步，选择参考向量；第三步，确定参考向量到法向量的投影向量；第四步，求投影向量的长度。通过以上分析，可以体会借助几何直观的必要性。可以启发运算思路，甚至可以得到解决问题的程序。程序思想方法具有解决一类数学问题的功能，是计算（特别是运用计算机进行计算）的基本思想方法。【拓展】引导学生用向量方法给出空间所有距离的求解程序，引导学有余力的学生查阅高等数学中有关的距离问题。案例17 二项式定理【目的】根据多项式相乘的运算法则，探索二项式定理的构造性证明，体会运算法则的作用。感知运算是一种严格的逻辑推理，通过一般性运算可以发现和提出命题、掌握推理的基本形式和规则、探索和表述论证的过程，发展数学运算素养。【情境】探索二项式定理的构造性证明。【分析】首先，让学生分析得到公式的运算过程。。中间的两个步骤利用“乘法对加法的分配律”，得到的每一项都是关于a，b的二次项；最后一步利用“乘法交换律”合并同类项。在此基础上，还可以进一步分析得到公式的运算过程，中间步骤得到的每一项都是关于a，b的三次项；最后一步依然利用“乘法交换律”合并同类项。尝试让学生归纳出多项式相乘的规律，然后运用规律推出二项式定理。例如下面的过程。 1.求出每一项。因为是n个（a+b）相乘，根据多项式相乘的规律，展开式中的每一项都是一个n次项，具有形式，其中0，1，2，…，n。 2.合并同类项。需要计算形如同类项的个数。由于k个b来自不同的k个二项式（a+b），个a来自剩余的个二项式（a+b），因此同类项的个数是组合数。 3.得到展开式。根据加法原理，可以得到二项式的展开式为 =，即. 【拓展】通过类比的方法，探索概率中的二项分布。案例18　杨辉三角【目的】通过杨辉三角，了解中华优秀传统文化中的数学成就，体会其中的数学文化。【情境】图12中的表称为杨辉三角，它出现在我国南宋数学家杨辉1261年所著的《详解九章算法》一书中。这是我国数学史上的一个伟大成就。【分析】杨辉三角有许多重要性质。 1．每行两端的数都是1。 2．第n行的数字有n个。 3．第n行的第m个数可表示为，且（a+b）n的展开式中的各项系数依次对应杨辉三角的第n+1行中的各项。 4．每行数字左右对称，即第n行的第m个数与第n行的第n-m+1个数相等。 5．相邻的两行中，除1以外的每个数等于塔“肩上”两数的和，即第n+1行的第i个数等于第n行的第i-1个数与第i个数的和，可表示为，其中2≤i≤n。可用此性质写出整个杨辉三角。 6．第n行数字和为2n-1。案例19　测量学校内、外建筑物的高度项目的过程性评价【目的】本案例是案例15的深化。给出过程性评价，体现如何让学生在交流过程中展现个性、学会交流、归纳总结，发现问题、积累经验、提升素养。【评价过程】在每一个学生都完成“测量报告”后，安排交流讲评活动。安排讲评的报告应当有所侧重。例如，测量结果准确，测量过程消晰，测量方法有创意，误差处理得当，报告书写认真等；或误差明显而学生自己没有察觉，测量过程中构建的模型有待商榷等。事实表明，这种形式的交流讲评，往往是数学建模过程中学生收获最大的环节。附件：某个小组的研究报告的展示片段摘录。测量不可及“理想大厦”的方法 1．两次测角法（1）测量并记录测量工具距离地面h m；（2）用大量角器，将一遍对准大厦的顶部，计算并记录仰角α；（3）后退am，重复（2）中的操作，计算并记录仰角β；（4）楼高x的计算公式为：，其中α，β，a，h如图13所示． 2．镜面反射法（1）将镜子（平面镜）置于平地上，人后退至从镜中能够看到房顶的位置，测量人与镜子的距离；（2）将镜子后移a m，重复（1）中的操作；（3）楼高x的计算公式为，其中a1，a2是人与镜子的距离，a是两次观测时镜面之间的距离，h是人的“眼高”，如图14所示。根据光的反射原理，利用相似三角形的性质联立方程组，可以得到这个公式。实际数据测量和计算结果，测量误差简要分析：（1）两次测角法实际测量数据：第一次第二次仰角 67° 52° 后退距离为25 m，人的“眼高”为1.5 m，计算可得理想大厦的高度约为71.5 m，结果与期望值（70 m~80 m)相差不大。误差的原因是铅笔在纸板上画出度数时不够精确。减少误差的方法是几个人分别测量高度及仰角，再求平均值，误差就能更小。（2）镜面反射法实际测量数据：第一次第二次人与镜子的距离 3.84 m 3.91 m 镜子的相对距离为10 m，人的“眼高”为1.52 m。计算可得理想大厦的高度约为217 m，结果与期望值相差较大。产生误差有以下几点原因：镜面放置不能保持水平；两次放镜子的相对距离太短，容易造成误差；人眼看镜内物像时，两次不一定都看准镜面上的同一个点；人体不一定在两次测量时保证高度不变。综上所述，要做到没有误差很难，但可以通过某些方式使误差更小，我们准备用更多的测量方法找出理想的结果。对上面的测量报告，教师和同学给出评价。例如，对测量方法，教师和同学评价均为“优”，因为对不可及的测量对象选取了两种可行的测量方法；对测量结果，教师评价为“良”，同学评价为“中”，因为两种方法得到的结果相差较大。对测量结果的评价，教师和同学产生差异的原因是，教师对测量过程的部分项目实施加分，包括对自制测量仰角的工具等因素作了误差分析；同学则进一步分析产生误差的主要原因，包括：（1）测量工具问题。两次测角法的同学，自制量角工具比较粗糙，角度的刻度误差较大；镜面反射法的同学，选用的镜子尺寸太大，造成镜面间距测量有较大误差。（2）间距差的问题。这是一个普通的问题。间距差a值是测量者自己选定的，因为没有较长的卷尺测量距离，有的同学甚至选间距差a是1 m。由于间距太小，两次测量的角度差或者人与镜的距离差太小，最终导致计算结果产生巨大误差。当学生意识到了这个问题后，他们利用运动场100 m跑道的自然长度作为间距差a，使得测量精度得到较大提高。（3）不少学生用自己的审稿代替“眼高”，反映了学生没有很好地理解测量过程中的“眼高”应当是测量的高度，如照片所示。在结题交流过程中，教师通过测量的现场照片，引导学生发现问题，让学生分析测量误差产生的原因。学生们在活动中意识到，书本知识和实践能力的联系与转化是有效的学习方式。测量现场的照片和观察说明：【分析】建模活动的评价要关注结果，更要关注过程。对测量方法和结果的数学评价可以占总评价的60%，主要由教师作评价。评价依据是现场观察和学生上交的测量报告，关注的主要评价点有：（1）测量模型是否有效；（2）计算过程是否清晰准确，测量结果是否可以接受；（3）测量工具是否合理、有效；（4）有创意的测量方法（可获加分）；（5）能减少测量误差的思考和做法（可获加分）；（6）有数据处理的意识和做法（可获加分）； …… 非数学的评价可以占总评价的40%，主要评价点有：（1)每一名成员在小组测量和计算过程中的工作状态；（2）测量过程中解决困难的机智和办法；（3）讨论发言、成果汇报中的表现等。非数学的评价主要是在同学之间进行，可以要求学生给出本小组以外其他汇报小组的成绩，并写出评价的简单理由。案例20　函数图象【目的】说明数学抽象素养的表现和水平，体会评价“在熟悉的情境中直接抽象出数学概念和规则”的满意原则和加分原则。【情境】学校宿舍与办公室相距a m。某同学有重要材料要送交给老师，从宿舍出发，先匀速跑步3 min来到办公室，停留2 min，然后匀速步行10 min返回宿含。在这个过程中，这位同学行进的速度和行走的路程都是时间的函数，画出速度函数和路程函数的示意图。【分析】回顾课程标准的要求，在实际情境中能够用图象揭示图数性质，整体反映函数的基本特征。本题答案的示意图如图15所示。解答本题时，能给出速度函数或路程函数的大部分示意图，根据满意原则，可以认为达到数学抽象素养水平一的要求；能够完整画出速度函数和路程函数示意图（二者自变量一致），可以认为达到数学抽象素养水平二的要求。这个问题也可以考查直观想象等素养。案例21 传令兵问题【目的】说明数学抽象素养的表现和水平，体会评价“分析数学命题的条件与结论，在具体的情境中抽象出数学问题”的满意原则和加分原则。【情境】有一支队伍长Lm，以速度v匀速前进。排尾的传令兵因传达命令赶赴排头，到达排头后立即返回，往返速度不变。回答下列问题：（1）如果传令兵行进的速度为整个队伍行进速度的2倍，求传令兵回到排尾时所走的路程；（2）如果传令兵回到排尾时，全队正好前进了Lm，求传令兵行走的路程。【分析】正确给出（1）的解答，可以认为达到数学抽象素养水平一的要求；正确给出（2）的解答，可以认为达到数学抽象素养水平二的要求。这个问题也可以考查逻辑推理、数学运算等素养。本题可以作如下解答。（1）传令兵往返速度为2v，从排尾到排头所需时间为，从排头到排尾所需时间为.故传令兵往返共用时间为，往返路程为。（2）设传令兵的行进速度为，则传令兵从排尾刻排头所需时间为，从排头到排尾所需时间为，往返共用时间为，往返所走路程为。由传令兵回到排尾时全队正好前进了L，则，故传令兵往返路程为。【拓展】如果传令兵从排尾到排头的行进速度为整个队伍行进速度的，从排头再回到排尾的行进速度为整个队伍行进速度的，求传令兵行走的路程。案例22跑道问题【目的】说明数学直观想象素养的表现和水平，体会评价“能够在熟悉的情境中，建立实物的几何图形，能够建立简单图形与实物之间的联系，体会图形与图形、图形与数量的关系”的满意原则、加分原则。【情境】400 m标准跑道的内圈如图16所示，其中左右两边均是半径为36m的半圆弧。（注：400 m标准跑道最内圈约为400 m）（1）求每条直道的长度（圆周率取3.14，结果精确到1 m）；（2）建立平面直角坐标系xOy，写出跑道上半部分对应的函数解析式。图16 标准跑道内圈示意图【分析】回顾课程标准的要求：“在平面直角坐标系中，探索并掌握圆的标准方程与一般方程。”“能根据给定直线，圆的方程，判断直线与圆、圆与圆的位置关系。” 如果能够完成（1）的计算，可以认为达到直观想象素养水平一的要求，能够基本得到（2）所要求的表达式，可以认为达到直观想象素养水平二的要求。这个问题也可以考查数学运算等素养。本题解答如下。（1）因为跑道两端的弧形合起来是一个完整的圆周，所以弧形部分跑道的长度为2×3.14×36=226.08（m)，两条直道长度为400-226.08=173.92(m).所以每条直道长约为173.92÷2≈87（m）。（2）建立如图17所示的平面直角坐标系。当0≤x<36时，圆的方程为，函数解析式为；当36≤x<123时，函数解析或为y=36；当123≤x≤159时，函数解析式为。所以函数解析式为图17建立坐标系示意图【拓展】可以考虚以图形的中心为原点建立平面直角坐标系。案例23距离问题【目的】说明如何考查学生数学抽象、直观想象和数学运算等素养达成的综合情况，体会“要关注数学学科核心素养各要素的不同特征及要求，更要关注数学学科核心素养的综合性与整体性。” 【情境1】在数轴上，对坐标分别为x1和x2的两点A和B，用绝对值定义两点间的距离，表示为d（A，B）=\| x1- x2\|。回答下面的问题：（1）在数轴上任意取三点A，B，C，证明 d(A,B)≤d(A,C)+d(B,C)。（2）设A和B两点的坐标分别为和2，找出满足d（A，B)=d(A,C)+d(B,C)的点C的范围，再找出满足d（A,B)<d(A,C)+d(B,C)的点C的范围。【情境2】城市的许多街道是相互垂直或平行的，因此，往往不能沿直线行走到达目的地，只能按直角拐弯的方式行走。如果按照街道的垂直和平行方向建立平面直角坐标系，对两点A（x1，y1）和B（x2,y2),类比“情境1”中的方式定义两点间距离为 d（A，B)= \| x1- x2\|+\| y1- y2\|, 回答类似的问题：（1）在平面直角坐标系中任意取三点A，B，C，证明 d（A，B)≤d（A,C)+d(B,C)。（2）设A和B两点坐标分别为（x1，y1）和（x2,y2),找出满足d（A，B)=d（A,C)+d(B,C)的点C的范围，再找出满足d（A，B)< d（A,C)+ d(B,C)的点C的范围。【分析】考虑下面数学学科核心素养达成的等级划分标准。对于“情境1”中的问题，基本上给出（1)或（2)的证明，可以认为达到数学抽象、逻辑推理、直观想象和数学运算素养水平一的要求。对于“情境2”中的问题，关键点是通过理解特殊的“两点间距离”定义，考查学生的直观想象和数学抽象素养。对于问题（1），如果学生能够对平面上固定的三点A，B，C，说明d（A，B)≤d（A,C)+d(B,C)，可以认为达到数学抽象、逻辑推理、直观想象和数学运算等素养水平一的要求，进一步地，如果学生对任意的三点A，B，C，得到该结果，可以认为达到相应素养水平二的要求。对于问题（2），只要学生画出基本符合要求的图形，就可以认为达到相应素养水平二的要求；进一步地，如果学生还能给出清晰的证明，可以适当加分。【拓展】在“情境2”中的距离意义下，画出到定点O（0，0）的距离等于1的点P（x，y）所形成的图形。从上述距离的定义出发，给出“点到直线的距离”的定义，并计算已知点到已知直线的距离。案例24 四棱锥中的平行问题【目的】以空间中的平行关系为知识载体，以探索作图的可能性为数学任务，依托判断、说理等数学思维活动，说明逻辑推理素养水平一、水平二的表现，体会满意原则和加分原则。【情境】如图18，在四棱锥P-ABCD的底面ABCD中，AB∥DC。回答下面的问题：（1）在侧面PAB内能否作一条直线段使其与DC平行？如果能，请写出作图过程并给出证明；如果不能，请说明理由。（2）在侧面PBC中能否作出一条直线段使其与AD平行?如果能，请写出作图的过程并给出证明；如果不能，请说明理由。图18四棱锥示意图【分析】直线与直线、直线与平面、平面与平面的平行和垂直等位置关系是高中立体几何内容的重点，也是教学的难点。设计开放性问题，让学生在运用与平行和垂直的相关定理进行判断、说理的活动过程中，提升直观想象和逻辑推理素养；通过这样的活动也可以对学生达到的相应素养水平进行评价。（1)能作出平行线。具体作法是，在侧面PAB内作AB的平行线；因为AB与DC平行，依据平行公理，这条平行线也必然平行于DC。完成这个过程，说明学生知道在平面内作与平面外直线平行的直线，需要寻求平面外直线与这个平面之间的关联，依据满意原则，可以认为达到逻辑推理素养水平一的要求。（2）需要分别判断。如果AD与BC平行，可以参照（1）的方法作出平行线。如果AD与BC不平行，不能作出平行线。用反证法进行说理如下：假设侧面PBC内存直线与AD平行，可推证AD与侧面PBC平行，依据性质定理，可推证AD与BC平行，这与条件矛盾。完成这个过程，说明学生能够理解直线与平面平行的相关定理以及定理之间的逻辑关系，依据满意原则，可以认为达到逻辑推理素养水平二的要求。案例25覆盖问题【目的】以平面几何为知识载体，以证明“周长一定的四边形中正方形所围面积最大”为数学任务，说明逻辑推理素养水平一、水平二、水平三和数学抽象素养水平一、水平二的表现，体会满意原则和加分原则。【情境】设桌面上有一个由铁丝围成的封闭曲线，周长是2L。回答下面的问题：（1）当封闭曲线为平行四边形时，用直径为L的圆形纸片是否能完全覆盖这个平行四边形?请说明理由。（2)求证：当封闭曲线是四边形时，正方形的面积最大。【分析】虽然问题涉及的知识不难，但由于问题中的封闭曲线是动态的、问题是开放的，因此需要一定的数学抽象和逻辑推理素养才可能抓住问题的本质。如果学生能够构建过渡性命题、完成概念的抽象过程，并且论证途径清晰、推理过程表述严谨，可以认为达到逻辑推理素养水平三的要求。（1）首先，需要从生活语言到数学语言，表达清楚什么是完全覆盖。最初的生活语言可以是，周长为2L的平行四边形包含的点都在直径为L的圆面内，显然这个层面的表达是无法进行论证的；用数学语言可以表述为，周长为2L的平行四边形内的任意一点到圆心的距离不大于，可是，这样的表述又脱离了完全覆盖的背景；因此需要在表述中加上条件，例如让平行四边形的对称中心与圆的圆心重合。鼓励学生回顾并表述上面的思维过程。如果学生能够完成前两个过程，根据满意原则，可以认为达到数学抽象素养水平一的要求，如果学生能够完成三个过程，根据加分原则，可以认为达到数学抽象素养水平二的要求。如果学生能够得到可以完全覆盖的结论，但只是证明了平行四边形对角线的长度不大于L，说明学生已经有了论证的思路，但还没有理解完全覆盖的几何本质，依据满意原则，可以认为达到逻辑推理素养水平一的要求。如果学生进一步证明平行四边形四个顶点到对称中心距离不大于圆的半径，但没有说明平行四边形内其他点的情况，说明学生理解了完全覆盖的几何本质，但证明过程还不够严谨，依据满意原则，可以认为达到逻辑推理素养水平二的要求。如果学生能够完整证明平行四边形上的点到对称中心距离部不大于圆的半径，说明学生基本掌握了数学证明，依据加分原则，可以认为达到逻辑推理素养水平三的要求。（2）可以启发学生，采用列举、筛选的方法考察各种形式的四边形，逐一排除面积较小的四边形，构建一个递进式的证明路径，如图19所示。图19探索证明路径如果学生能够独立完成上面的过程，说明对较复杂的新问题，能够直观想象、创造性地构建证明路径，依据满意原则，可以认为达到逻辑推理素养水平二的要求，如果学生能够进一步用数学语言严谨地论证所得到的结论，根据加分原则，可以认为达到逻辑推理素养水平三的要求。案例26 鞋号问题【目的】在寻求变量简单变化规律的过程中，说明数学建模素养的表现和水平，体会评价过程中的满意原则和加分原则。【情境】网上购鞋常常看到下面的表格（表3）。表3 脚长与鞋号对应表脚长 220 225 230 235 240 245 250 255 260 265 鞋号 34 35 36 37 38 39 40 41 42 43 请解决下面的问题：（1）找出满足表3中对应规律的计算公式，通过实际脚长a计算出鞋号b；（2）根据计算公式，计算30号童鞋所对应的脚长是多少? （3）如果一个篮球运动员的脚长为282mm，根据计算公式，他该穿多大号的鞋? 【分析】数学建模素养的一个基本表现，就是能够针对具体的数据，选择合适的函数表达数量之间的关系，解决实际问题。在这样的活动中，可以体现数学建模素养不同水平的表现。（1）可以把表中的两行数据看成两个数列，分别为和。仔细观察可以知道，这两个数列分别满足下面的递推关系： ,； ,b1=34。由此得到=215+5n和=33+n，于是有=0.2-10。如果学生能够找到并且准确表达脚长与鞋号之间的线性关系，根据满意原则，可以认为达到数学建模素养水平一的要求。进一步，将脚长和对应的鞋号记作（a，b)，在平面直角坐标系中描点，观察到线性关系，然后建立关系式。这说明学生能够借助图形直观发现变化规律，并且能够用函数清晰表达变化规律，根据加分原则，可以加分。如果学生构建数据表，利用计算工具的电子表格作出散点图，选择几种函数模型进行拟合；对比拟合结果，发现线性函数的拟合效果最好，相关系数为1，进而确定计算公式是一个线性模型，最后确定模型中的参数，如图20所示。根据加分原则，可以针对“善于使用计算工具”加分。图20计算机模拟示意图（2）令b=30，代入公式，得a=200，脚的长度为200mm。虽然计算过程是套用已知结果，但由b求a涉及到简单的反函数，可以认为达到数学建模素养水平二的要求。（3）当a=282时，代入公式，得b=46.4。分两种情况：如果简单地进行“4舍5入”，选46号鞋或者直接选46.4号鞋，依然可以认为达到数学建模素养水平二的要求。如果知道作出的结论要符合实际，提出穿鞋要“不挤脚”，因此选47号鞋，或者提出要考虑脚型、鞋型，根据解答情况，可以加分。案例27　包装彩绳【目的】在把实际问题转化为数学问题的过程中，说明数学建模素养不同水平的表现，体会评价的满意原则和加分原则。【情境】春节期间，佳怡去探望奶奶，她到商店买了一盒点心，为了美观起见，售货员对点心盒做了一个捆扎（如图21（1）），并在角上配了一个花结.售货员说，这样的捆扎不仅漂亮，而且比一般的十字捆扎方式（如图21（2））包装更节省彩绳。你同意这种说法吗？请给出你的理由。（注：长方体点心盒的高小于长、宽）图21 点心盒的两种包装【分析】在数学建模的过程中，常常要把实际问题数学化。特别是，需要借助几何直观才能论证的问题，这通常是学生数学建模的难点。因此，对于这样一类问题，难点处理的差异能够反映数学建模素养的不同水平。如果学生能够结合几个具体的长方体盒子，通过捆扎操作、测量比较的方法，得到针对这几个盒子的结论，并且能够通过归纳提出一般长方体盒子下的猜想，即使不能给出证明，根据满意原则，也可以认为达到数学建模水平一的要求。如果学生能够用字母表示各段绳长，将长方体盒子平面展开，把问题转化为平面上的折线长度的比较，把“扎紧”的表述转化为两点间直线段，最后得出一般性的结论，可以认为达到数学建模水平二的要求。如果不考虑花结用绳，或者认为两种捆扎方法中花结的用绳长度相同，一个推理过程的返利可以表述如下，设长方体点心盒子的长、宽、高分别为x，y，z，依据图21（2）的捆扎方式，把彩绳的长度记作l，因为长方体的每个面上的那一段绳都与相交的棱垂直，所以。依据图21（1）的捆扎方式，可以想象将长方体盒子展开在一个平面上，则彩绳的平面展开图是一条由A到A的折线；在“扎紧”的情况下，彩绳的平面展开图是一条由A到A的线段，记为A′A″（如图22），这时用绳最短，绳长记作m，则在△A′BA″中，由三角形中两边之和大于第三边，得，即，因此，图21（1）所示的捆扎方式节省材料。图22 长方体盒子的平面展开示意图如果学生能够完成以上工作，可以认为达到数学建模水平二的要求。如果思路清晰、表达准确，还可以适当加分。案例28　体重与脉搏【目的】在构建“比例模型”解决实际问题的过程中，给出数学建模素养水平二、水平三的表现，体会评价的满意原则和加分原则。【情境】生物学家认为，睡眠中的恒温动物依然会消耗体内能量，主要是为了保持体温。研究表明，消耗的能量E与通过心脏的血流量Q成正比，并且根据生物学常识知道，动物的体重与体积成正比。表4给出一些动物体重与脉搏率对应的数据。表4 一些动物的体重和脉搏率回答下面的问题：（1）请你根据生物学常识，给出血流量与体重之间关系的数学模型。（2）从表4可以看到，体重越轻的动物脉搏率越高。请根据上面所提供的数据寻求数量之间的比例关系，建立脉搏率与体重关系的数学模型。（3）根据表4，作出动物的体重和脉搏率的散点图，验证建立的数学模型。【分析】为了建立数学模型，需要进一步理解一些生物学概念，例如，血流量Q是单位时间流过的血量，脉搏率是单位时间心跳的次数；还需要进一步知道一些生物学假设，例如，心脏每次收缩挤压出来的血量与心脏大小成正比，动物心脏的大小与这个动物体积的大小成正比。因为数学建模只用到“比例分析”，因此在知识层面上学生困难不大，但学生通常对比例模型的分析思路比较陌生；同时，这个数学活动体现了跨学科的应用，因此如果能很好地解决问题，可以认为达到数学建模素养水平二、甚至水平三的要求。例如，下面的建模过程，（1）因为动物体温通过身体表面散发热量，表面积越大，散发的热量越多，保持体温需要的能量也就越大，所以动物体内消耗的能量E与审题表面积S成正比，可以表示为。又因为动物体内消耗的能量E与通过心脏的血流量Q成正比，可以表示为,因此得到，其中，和均为正的比例系数。另一方面，因为体积V与体重W成正比，可以表示为，又因为表面积大约与体积的次方成正比，可以表示为，因此得到，其中，和均为正的比例系数。所以可以构建血流量与体重关系的数学模型，其中为正的比例系数。根据脉搏的定义，再根据生物学假设（为正的比例系数），最后得到，也就是，其中为正的待定系数。（2）脉搏率与体重关系的数学模型说明，恒温动物体重越大，脉搏率越低，脉搏率与体重的次方成反比。表4中的数据基本上反映了这个反比例的关系。（3）图23是原始数据的散点图，图24是以和为坐标的散点图。可以看出，数据取对数之后基本满足线性关系，因此得到体重和脉搏率的对数线性模型，可以把这个模型表达为。图23 脉搏率与体重W的散点图图24 与的散点图如果学生在上述分析过程中思路清晰、表达准确，可以认为达到数学建模素养水平三的于鏊求，如果在分析或者论证过程中还有一些创意，例如，对脉搏率与体重关系的模型两边取对数，形成对数线性模型，能够用相关系数进行线性相关性判断；能够用方差分析方法建议模型的是适合程度等，则根据加分原则，可以进行相应加分。因为这个问题的分析线索比较长，学生在建模求解的过程中，可能会得到一些有价值的中间结论，或者有些学生最终也不能把整个过程进行到底，甚至有些学生不经过任何分析就给出拟合函数（如图25所示）。这些情况都是数学建模和数据分析素养水平达成程度的表现，可以适度加分或者扣分。图25 案例29　估算地球周长【目的】说明直观想象素养水平的表现和水平，体会评价“在现实情境中，建立实物的几何图形，能够根据图形想象实物”的满意原则和加分原则。【情境】古希腊地理学家埃拉托色尼（Eratosthenes，前275—前193）用下面的方法估算地球的周长（即赤道周长）。他从书中得知，位于尼罗河第一瀑布的塞伊尼（现在的阿斯旺，在北回归线上），夏至那天正午立杆无影；同样在夏至那天，他所在的城市——埃及北部的亚历山大城，立杆可测得日影角大约为（如图26），埃拉托色尼猜想造成这个差异的原因是地球是圆的，并且因为太阳距离地球很远（现代科学观察得知，太阳光到达地球表面需要8.3 s，光速300000 km/s），太阳光平行照射在地球上。根据平面几何知识，平行线内错角相等，因此日影角与两地对应的地心角相等，他又派人测得两地距离大约5000希腊里，约合800 km；因为大约为的50倍，于是他估算地球周长约为（km），这与地球实际周长40076 km相差无几. （1）试画出平面示意图；（2）试由埃拉托色尼的估算结果，给出你的推理过程。图26 估算地球周长示意图【分析】如果学生能够画出基本合理的草图，可以认为达到直观想象素养水平一的要求；能够画出清晰合理的示意图，可以认为达到直观想象素养水平二的要求，本题也考查逻辑推理等素养。例如，下面的分析过程。（1）如图26，记塞伊尼为点A，亚历山大城为点B。在两个点处太阳光平行，因为内错角相等得到对应的地心角为，的长度为800km。（2）用的长乘可以近似得到地球周长。案例30　影子问题【目的】说明直观想象素养的表现和水平，体会满意原则和加分原则。【情境】如图27，广场上有一盏路灯挂在高10 m的电线杆上，记电线杆的底部为A。把路灯看作一个点光源，身高1.5 m的女孩站在离点A5 m的点B处。回答下面的问题：（1）若女孩以5 m为半径绕着电线杆走一个圆圈，人影扫过的是什么图形，求这个图形的面积；（2）若女孩向点A前行4 m到达点D，然后从点D出发，沿着以BD为对角线的正方形走一圈，画出女孩走一圈时头顶影子的轨迹，说明轨迹的形状。【分析】回顾课程标准中相关内容的要求：从空间几何体的整体观察入手，认识空间图形。如果学生能够在问题（1）中回答出人影扫过的图形是环形，或者在问题（2）的解答中提到棱锥，可以认为达到直观想象素养水平二的要求。如果学生能够清晰准确地回答两个问题，可以认为达到直观想象素养水平三的要求。例如，下面的回答。（1）如图28所示，S为路灯位置，C为女孩头顶部，女孩的影子为线段BP。女孩绕着电线杆走一个圆圈，人影扫过的是一个环形。已知SA=10 m，AB=5 m，BC=1.5 m。设BP=x，则由BC∥SA，得，即，解得x=。因此环形面积为π（AP2-AB2）=[(x+5)2-52]=≈30.166（m2）。（2）如图29，女孩头顶运动的轨迹是以CE为对角线的正方形（CE与BD平行且相等），且该正方形平行于地面，则在点光源S的投射下，投影应与原图形相似，因此女孩头顶影子的轨迹也是一个正方形。【拓展】如果这个女孩绕一个边长为2 m的正六边形走一圈，那么人影扫过的图形是什么？人影扫过的面积是多少？案例31　圆柱体截面问题【目的】说明直观想象素养的表现和水平，体会满意原则和加分原则。【情境】在一个密闭透明的圆柱桶内装一定体积的水。（1）将圆柱桶分别竖直、水平、倾斜放置时，指出圆柱桶内的水平面可能呈现出的所有几何形状，画出直观示意图。（2）参考图30，对上述结论给出证明。【分析】回顾课程标准中相关内容要求，“利用实物、计算机软件等观察空间图形，认识柱、锥、台、球及简单组合体的结构特征，能运用这些特征描述现实生活中简单物体的结构。” 如果学生能够比较完整地回答（1）的第一个问题，可以认为达到直观想象素养水平一的要求；能比较完整地回答（1）的第二个问题，可以认为达到直观想象素养水平二的要求；能比较完整地回答（2）的问题，可以认为达到直观想象素养水平三的要求。此题也考查逻辑推理等素养。例如，下面的回答。（1）圆柱桶竖直放置时，水平面为圆面；水平放置时，水平面为矩形面；倾斜放置时，水平面为椭圆面或者部分椭圆面。可能呈现的所有类型的几何图形，如图31所示。（2）圆柱桶竖直放置时，水平面相当于平行于底面的截面，因此水平面是圆面。圆柱桶水平放置时，水平面与圆柱侧面的两条交线是圆柱的母线，它们平行且相等，且垂直于水平面与圆柱底面的两条交线，所以水平面是矩形面。圆柱桶倾斜放置时，水平面相当于用平面斜截圆柱时所得到的截面。如图32所示，上下两球与截面和圆柱侧面均相切，两球面与圆柱侧面分别相切于以BC，DE为直径且平行于圆柱底面的大圆O1和O2，两球面与斜截面分别相切于点F和F′，斜截面与BD，CE分别交于点A和A′，P为所得截面边缘上一点（即斜截面与圆柱侧面交线上一点）。设过点P的圆柱的母线与圆O1和O2分别交于点M和N，则PM和PN分别是两球面的一条切线。由于PM和PF是同一个球面的切线，故PM=PF，同理PN=PF'，于是有PF+PF'=PM+PN=MN为定值，即点P到F和F'距离之和为定值，所以这时的截面是椭圆面。案例32　过河问题【目的】以平面向量的运算为知识载体，以确定游船的航向、航程为数学任务，借助理解运算对象、运用运算法则、探索运算思路、设计运算程序、实施运算过程等一系列数学思维活动，说明数学运算素养水平一、水平二和水平三的表现，体会满意原则和加分原则。【情境】长江某地南北两岸平行。如图33所示，江面宽度d=1 km，一艘游船从南岸码头A出发航行到北岸。假设游船在静水中的航行速度v1的大小为\|v1\|=10 km/h，水流的速度v2的大小为\|v2\|=4 km/h．设v1和v2的夹角为θ（0<θ<180°），北岸的点A′在A的正北方向。回答下面的问题：（1）当θ=120°时，判断游船航行到达北岸的位置在A′的左侧还是右侧，并说明理由。（2）当cosθ多大时，游船能到达A′处？需要航行多长时间？【分析】回答这个问题需要几何直观下的代数运算。（1）首先要知道游船航行速度是静水速度与水流速度之和，然后会按比例画示意图判断航行方向。如果学生能够用向量加法的平行四边形法则画出示意图、并给出合理解释，根据满意原则，可以认为达到数学运算素养水平一的要求。如果学生把航行速度即速度之和表示为v，可以通过计算航行速度向量v与水流速度向量v2之间的夹角进行判断，由，判断游船到达的位置在A'的左侧。说明学生不仅能够理解向量的加法，还能够根据题意，运用向量数量积运算求解向量之间的夹角，根据加分原则，可以认为达到数学运算素养水平二的要求。（2）首先要将“游船能到达A'处”抽象为游船的实际航向与河岸垂直，即游船的静水速度和水流速度的合速度方向与相同，将合速度运算与平面向量的加法运算联系起来，画出速度合成示意图（如图34）。根据满意原则，学生能够面出向量加法示意图，可以认为达到数学运算素养水平一的要求。通过解三角形，求得cosθ值为-；通过\|v\|=\|v1\|sinθ=2，得到航行时间 h。说明学生能够将题目中提供的数据信息与几何图形有机联系，并且能够明晰运算途径、得到运算结果，根据加分原则，可以认为达到数学运算素养水平二的要求。进一步地，如果学生能够通过直角三角形计算出cosθ=-，由勾股定通，通过(10t)2-(4t)2=1解得t= h。说明学生能够运用勾股定理建立方程求解，根据加分原则，可以认为达到数学运算素养水平三的要求。【拓展】在本题背景下，可以设计数学运算素养拓展问题，例如当θ=120°时，游船航行到北岸的实际航程是多少? 为了回答这个问题，可以先依据题意画出向量加法的示意图，如图36所示，然后利用向量数量积运算求得 \|tv\|2=t2(v1+v2)2=t2(102+2×10×4×cos120°+42)=76t2。在Rt△AA'C中，因为t\|v1\|cos30°=1，从而t=，所以AB=km。如果学生能够完成这个过程，说明学生能够综合运用向量的加法、数乘、数量积运算和勾股定理，恰到好处地设计运算程序、完成问题求解，根据加分原测，可以在数学运算素养水平三的基础上加分。案例33 隧道长度【目的】以解三角形为知识载体，以隧道测量为数学任务，借助明确运算对象、探索运算思路、设计运算程序、实施运算等一系列数学思维活动，说明数学运算素养的水平一和水平二的表现，体会满意原则。【情境】如图37所示，A，B，C为山脚两侧共线的三点，在山顶P处测得三点的俯角分别为α，β，γ。计划沿直线AC开通穿山隧道，为求出隧道DE的长度，你认为还需要直接测量出AD，EB，BC中的哪些线段的长度？根据条件，并把你认为需要测量的线段长度作为已知量，写出计算隧道DE长度的运算步骤。【分析】由已经测料的三个角α，β，γ，依据平面几何知识可以知道，△PAB，△PBC的三个内角已经确定，进而形状已经确定，因此还需要通过确定三角形的边长来确定三角形的大小。进一步，为了能够计算隧道DE的长度，由解三角形的知识，可以推断出还需要确定所有线段AD，EB，BC的长度。首先在△PBC中进行运算，依据正弦定理写出BC与PB（或PC）之间的等量关系式，表达出PB（或PC）。如果学生能够完成这个步骤，说明学生已经熟悉常规的解三角形问题及其解法，根据满意原则，可以认为达到数学运算素养水平一的要求。如果学生能够继续在△PAB（或△PAC）中，由正弦定理写出PB与AB（或PC与AC）之间的等量关系式，用已知角度α，β，γ和测量得的线段AD，EB，BC长度正确写出线段DE长度的表达式，说明学生能够清晰表达图37中多个三角形之间的关系，并且能够探索出运算程序、正确实施，根据满意原则，可以认为达到数学运算素养水平二的要求。案例34 迭代计算问题 .【目的】迭代方法县现代计算教学的基本方法。借助用“牛顿切线法”和“二分法”求一元二次方程解的问题，考查理解运算对象、把握运算规律、表达运算过程、设计运算程序等一系列数学运算的思维活动，说明数学运算素养水平三的表现，体会满意原则和加分原则。【情境】研究一元二次方程x2+x-1=0的求解问题，这是经典的求黄金分割的方程式。令f（x)= x2+x-1，对抛物线y= f（x)，持续实施下面“牛顿切线法”的步骤：在点（1，1）处作抛物线的切线交x轴于（x1，0）；在点（x1，f（x1))处作抛物线的切线，交x轴于点（x2，0），在点（x2，f（x2)）处作抛物线的切线，交x轴于点（x3，0）； …… 得到一个数列{xn}。回答下列问题：（1）求x1的值；（2）设xn+1=g（xn），求g（xn）的解析式；（3）用“二分法”求方程的近似解，给出前四步结果。比较“牛顿切线法”和“二分法”的求解速度。【分析】在数值计算中，“牛顿切线法”和“二分法”是最为常用的两种方法。（1）求出抛物线在点（1，1）处切线方程y-1=f'（1）（x-1），得到y=3x-2。令y=0，得到x1=。如果完成这个步骤，说明学生能够正确运用求导运算得到抛物线的切线方程，理解切线与x轴交点横坐标是所要求的运算对象，根据满意原则，可以认为达到数学运算素养水平一的要求，（2）求出抛物线在点（xn，f（xn)）处的切线方程y=（2xn +1）（x-xn）+（+xn-1）。说明学生能够一般性地理解运算对象，并能够正确地予以数学表达，根据满意规则，可以认为达到数学运算素养水平二的要求。令y=0，得到xn+1=，进而g（xn）=。如果完成这个步骤，说明学生能够很好地理解选代运算，并且能够正确地运用代数式予以表达，根据满意原则，可以认为达到数学运算素养水平三的要求。（3）用求根公式可以得到一元二次方程的正根为，近似解为0.518，就是著名的黄金分制数。用“二分法”求方程近似解的前四步为：因为f（0）=-1；f（1）=1，所以f（x）在区间（0，1）内至少有一个零点；因为f（0.5）=-0.25，所以f（x）在区间（0.5，1）内至少有一个零点；因为f（0.75）=0.3125，所以f（x）在区间（0.5，0.75）内至少有一个零点；因为f（0.625）=0.015625，所以f（x）在区阀（0.5，0.625）内至少有一个零点。可以看到，用“二分法”计算前四步得到近似解为0.625，同样从x=1出发，用“牛级切线法”解析式可求得第二步和第三步的近似解分别为x2≈0.619，x3≈0.618，比较“牛顿切线法”与“二分法”前几步的结果，可以看到“牛顿切线法”要比“二分法”快得多。如果学生完成这个步骤，根据加分原则，可以在数学运算素养水平三的基础上加分。【拓展】对于函数f（x）=x2+x-1，由f（0）<0和f（1）>0，可知函数在（0，1）内至少有一个零点，设该零点为x0。若x0<x，求证x0<g（xn）<xn。案例35估计考生总数【目的】分别说明数学建模素养和数据分析素养水平一、水平二的表现，体会评价的满意原则和加分原则。【情境】某大学美术系平面设计专业的报考人数连创新高，今年报名刚结束，某考生想知道报考人数。考生的考号按0001，0002，…的顺序从小到大依次排列。这位考生随机地了解了50个考生的考号，具体如下：请给出一种方法，根据这50个随机抽取的考号，帮助这位考生估计考生总数。【分析】用样本空间的数字特征估计总体的数字特征或性质，是统计建模的基本思想和基本手法，既可以表现数学建模素养水平，也可以表现数据分析素养水平。如果学生给出的方法体现了用样本估计总体的思想，并且述说的理由合理，即使表述得不完整、不清楚、不到位，根据满意原则，都可以认为达到数据分析素养水平一的要求。例如，用给出数据的最大值986（与0986对应）估计考生总数；用数据的最大值与最小值的和（986+17=1003)估计考生总数；借助数据中的部分数据的信息（如平均值、中位数等）估计考生的总数；等等。如果学生能够理解数据分析的思想，过程述说比较清楚，数学表达比较到位，可以认为达到数据分析素养水平二的要求。例如：设考生总数为N，即N是最大考号。方法一随机抽取的50个数的平均值应该和所有考号的平均值接近，即用祥本的平均值估计总体的平均值。这50个数的算术平均值是24 671÷50=493.42，它应该与接近。因此，估计今年报考这所大学美术系平面设计专业的考生总数为 N≈493.42×2≈987(人）。类似地，可以通过样本中位数得到N的估计。方法二把这50个数据从小到大排列，这50个数把区间[0，N] 分成51个小区间。由于N未知，除了最右边的区间外，其他区间都是已知的。可以利用这些区间长度来估计N。由于这50个数是随机抽取的，一般情况下可以认为最右边区间的长度近似等于[0，N]长的，并且可以用前50个区间的平均长度近似代替这个区间的长度。因为这50个区间长度的和，恰好是这50个数中的最大值986，因此得到。因为这是一道开放题，允许有不同的答案。只要学生能够对自己提出的方法给出合理的解释，可以认为达到相应水平的要求。案例36 函数单调性主题教学设计【目的】说明如何进行跨章节的主题教学没计。【情境】函数单调性是函数的重要性质之一,不仅与函数概念、函数的其他性质有关,也与基本初等函数、不等式、数列、导数等内容有关,在表述过程中还与常用逻辑用语中的量词有关。所以，函数单调性可以作为跨章节的主题进行整体教学设计。【分析】一来说,主题教学的整体设计可分为前期准备、开发设计、评价修改三个阶段,具体可以包括以下几个步骤,如图38所示。 (1)确定主题内容； (2)分析教学要素； (3)编制主题教学目标； (4)设计主题教学流程； (5)评价、反思及修改。作为案例,下面给出教学设计的具体方法。为了确定主题内容,下面的两种策略可供选择。无论采取哪种策略组织内容，在教学设计中，都要关注数学抽象、逻辑推理、直观想象、数学运算等素养水平的提升。分析教学要素是确定主题教学目标的前提，是主题教学设计的重点环节。教学要素分析主要包括以下方面：数学内容分析、课程标准分析、学情分析、教材分析、重难点分析以及教学方式分析。其体分析内容如表5所示。表5 教学要素分析的内容要素内容教学内容分析 1.本主题内容的数学本质、数学文化以及所渗透的数学思想等； 2.本主题内容在本学段数学课程中的地位； 3.本主题内容在整个中小学数学课程中的地位和作用； 4.本主题内容在数学整体中的地位； 5.本主题内容与本学段、前后学段以及大学其他知识之间的联系。课程标准分析 1.课程标准中对本主题内容的要求； 2.课程标准中对本主题内不同内容要求的关联。学情分析 1,学生学习新知识的预备状态； 2.学生对即将要学习的内容是否有所涉猎； 3.学生学习新知识的情感态度； 4.学生的学习方法、习惯以及风格。教材分析 1.比较不同版本教材的对本主题内容在概念引入、情境创设、例题习题的编排方式等方面的异同,分析各自的特点； 2.根据学情选择适当的内容及其处理方式。重难点分析 1.主题整体教学重难点; 2.具体课时重难点。教学方式分析从主题整体角度出发,选择合适的教学方式（体现学生的主体性)。编制主题教学目标和设计教学流程。因为是主题教学设计，教学内容将涉及若干节、甚至涉及若干章，因此在教学实施的过程中，可以划分为几个不同的阶段。例如，如果内容选取采用以函数单调性知识的前后逻辑为线索，其教学实施过程可分为以下几个阶段：第一阶段，从图形语言到符号语言的过渡，让学生感悟从直观想象到数学表达的抽象的过程，感悟常用逻辑用语中的量词与数学严谨性的关系；第二阶段，结合对几种初等函数单调性的研究，理解用代数方法证明函数单调性的基本思路与论证方式，增强逻辑推理和数学运算能力；第三阶段，利用导函数一般性地研究函数的单调性，感悟导数是研究函数性质强有力的工具，理解函数单调性的本质；第四阶段，通过利用函数单调性刻画现实问题的若干实例分析，理解为什么函数可以成为构建数学模型的有效的数学语言，从而理解研究函数的单调性不仅仅是为了数学本身的需要，也是为了更好地表达现实世界的需要。案例37 “互联网+”促进高中生数学学科核心素养发展的路径【目的】阐述“互联网+数学教育”背景下，教学、学习、评价一体化的实践，体会“互联网十数学教育”对教学带来的变革，感受在“互联网+”、大数据的支持下，数学教学由知识教学向核心素养教学转化的路径，实现课堂教学的精准化、高效化、个性化，提高学生数学学习的实际获得感。【分析】在有条件的地区或学校，可以借助“互联网+”、大数据等现代信息技术优化学科教育生态。采用“互联网+数学教育”的思路破解数学课堂模式化培养与个性化学习之间的矛盾，在教学中落实数学学科核心素养。“互联网+教育”的关键因素之一是用好在线教育平台，改善教师教学方式，促进学生线上线下（O2O）融合学习，使得学习更加丰富多彩、生动有趣，实线教学的精准化和个性化。学生、教师角色的在线平台界面如图39，40所示。图40 教师界面依据课程标准中数学学科核心素养内涵与水平划分，实线数学教学、学习、评价一体化（如图41），这是支撑测评和教学改进的关键。【情境1】数学学习的个性化评价反馈系统【分析】根据课程标准“除了考查全班学生在数学学科核心素养上的整体发展水平外，更需要根据学生个体的发展水平和特征进行个性化的反馈”的要求，在线测评可以实现学生学习的精准化、个性化诊断（如图42，43）。通过采集学生学习过程的大数据，可视化呈现学生数学学科核心素养，基于数据的学情分析，让学生更加精准地了解自己的学习问题和学科优势。图43 学生个性化诊断报告示例课程标准要求“重视评价的整体性与阶段性”。在线测评不仅实现了学生的个性化诊断反馈，而且同时兼顾了评价的整体性与阶段性。整体性评价以学段、学年、学期为时间单位，测量学生的数学学科核心素养发展水平。阶段性评价以知识单元为单位，测量每个知识单元的个体学习状况。测评数据的积累反映了学生数学学科核心素养发展的成长过程（如图44）。客观题的自动批阅为教师教学带来了便利，提高了效率。未来，人工智能技术的发展将会实现主观题的自动识别和批阅（如图45）。宏、微结合的整体性和阶段性测量诊断实现了学生数学学科核心素养水平诊断的精准化（如图46）。通过期中、期末考试诊断出学生某个知识单元学习存在问题之后。进一步进行知识单元诊断，精准发现学习问题。【情境2】基于精准诊断的个性化学习支持系统【分析】由于学生数学学习问题的诊断落实到了核心素养水平,所以,可以按照核心素养水平智能推荐对应的微课程；也可以智能推荐在线教师(如图47),学生在学习过程中连线,教师在线答疑(如图48)。推荐的微课程可以分“问题改进型”“优势增强型”等不同类型,以服务不同核心素养水平学生学习的需要。【情境3】基于精准诊断的教学支持系统【分析】在“互联网+数学教育”中,在线教育平台上丰富的教学资源,为教学提供大量不同核心素养水平的教学素材,教师可参考创设教学问题情境。信息技术全程融合应用教学,有多种教学模式可供选择。可以采用翻转课堂教学模式,也可以采用其他的教学模式。教师在课前利用网络平台,发布学习任务。学生通过微课程学习,网络社区互动,自我测验等环节,解决部分学习任务。通过在线教育平台中学生学习的数据报告,汇聚共性问题,线下课堂中采用合作探究等方式解决问题,课后利用在线平台以作业形式检测学习效果,确保学习的针对性和高效率,优化教学。在具备信息技术条件的学校,教师还可以充分利用在线平台开展其他类型的、丰富多彩的教学活动,不断探索创新，改进教学方式。本作品是中华人民共和国的法律、法规，国家机关的决议、决定、命令和其他具有立法、行政、司法性质的文件，及其官方正式译文。根据《中华人民共和国著作权法》第五条第一项，本作品不适用于该法，在中国大陆，也可能在其他管辖区，属于公有领域。注：中文维基文库社群认为，中华人民共和国公务演讲，不总是具有立法、行政、司法性质的文件。 Public domainPublic domainfalsefalse 检索自“ 分类：中华人民共和国公有领域中华人民共和国教育部的通知 2017年
2557	https://mathhelpforum.com/t/a-point-inside-an-equilateral-triangle.292936/	A point inside an equilateral triangle... \| Math Help Forum Math Help Forum Search [x] Search titles only By: SearchAdvanced search… [x] Search titles only By: SearchAdvanced… Math Help Forum HomeAlgebraPre-CalculusGeometryTrigonometryCalculusAdvanced AlgebraDiscrete MathDifferential GeometryDifferential EquationsNumber TheoryStatistics & ProbabilityBusiness MathChallenge ProblemsMath Software ForumsHigh School Math HelpHigh School STEMUniversity Math Homework HelpUniversity STEMGeneral MathematicsSearch forums Trending Physics Help Chemistry Help LoginRegister Search [x] Search titles only By: SearchAdvanced search… [x] Search titles only By: SearchAdvanced… High School Math Help High School STEM University Math Homework Help University STEM General Mathematics Search forums Menu Install the app Install How to install the app on iOS Follow along with the video below to see how to install our site as a web app on your home screen. Note: this_feature_currently_requires_accessing_site_using_safari Home Forums High School Math Geometry A point inside an equilateral triangle... Thread starterbeanspog Start dateSep 17, 2021 Tagsequilateral triangle 1 2 3 Next 1 of 3 Go to page Go NextLast beanspog Joined Sep 2021 16 Posts \| 0+ isb Discussion Starter Sep 17, 2021 #1 This question doesn't seem to get anywhere for me and my maths teacher. Let ABC be an equilateral triangle and P be a point inside this triangle such that PA=x, PB=y, PC=z. If z² = x² + y², Find the length of the sides of the triangle ABC in terms of x and y. MacstersUndead Joined Jan 2009 757 Posts \| 440+ Sep 17, 2021 #2 I considered using Viviani's theorem for this, but doing some digging, I needed to instead approach it like the proof of the theorem. First, drop perpendiculars from P then add the areas of the triangles inside ABC (the perpendiculars will be the heights) and compare it against the known formula for the area of an equilateral triangle based on side length. All credit for the solution goes to cosmonavt, post #5 Pythagorean theorem and an equilateral triangle Let ABC be an equilateral triangle and P be a point inside this triangle such that PA=x, PB=y and PC=z. If z^2 = x^2 + y^2, find the length of the sides of triangle ABC in terms of x and y. How do we go about this? AFAIK, the Pythagorean theorem can be applied to Right Angle Triangles. I am... mathhelpforum.com next stay CC Settings Off Arabic Chinese English French German Hindi Portuguese Spanish Font Color white Font Opacity 100%Font Size 100%Font Family Arial Text Shadow none Background Color black Background Opacity 50%Window Color black Window Opacity 0% White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25% 200%175%150%125%100%75%50% Arial Georgia Garamond Courier New Tahoma Times New Roman Trebuchet MS Verdana None Raised Depressed Uniform Drop Shadow White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25%0% White Black Red Green Blue Yellow Magenta Cyan 100%75%50%25%0% beanspog Joined Sep 2021 16 Posts \| 0+ isb Discussion Starter Sep 17, 2021 #3 MacstersUndead said: I considered using Viviani's theorem for this, but doing some digging, I needed to instead approach it like the proof of the theorem. First, drop perpendiculars from P then add the areas of the triangles inside ABC (the perpendiculars will be the heights) and compare it against the known formula for the area of an equilateral triangle based on side length. All credit for the solution goes to cosmonavt, post #5 Pythagorean theorem and an equilateral triangle Let ABC be an equilateral triangle and P be a point inside this triangle such that PA=x, PB=y and PC=z. If z^2 = x^2 + y^2, find the length of the sides of triangle ABC in terms of x and y. How do we go about this? AFAIK, the Pythagorean theorem can be applied to Right Angle Triangles. I am... mathhelpforum.com Click to expand... Hi! I actually didnt understand how h1+h2+h3 = x+y+z, can you explain me that please MacstersUndead Joined Jan 2009 757 Posts \| 440+ Sep 17, 2021 #4 Do'h. I missed that unjustified line. Let me try to figure out if that's true or not. I still think using Viviani's or something like it is the way to go for this problem though. beanspog Joined Sep 2021 16 Posts \| 0+ isb Discussion Starter Sep 17, 2021 #5 MacstersUndead said: Do'h. I missed that unjustified line. Let me try to figure out if that's true or not. I still think using Viviani's or something like it is the way to go for this problem though. Click to expand... Oh Alright! I have never heard about something like Viviani's theorem. I will be very thankful for your help. MacstersUndead Joined Jan 2009 757 Posts \| 440+ Sep 17, 2021 #6 Viviani's theorem - Wikipedia en.wikipedia.org Turns out you shouldn't use Viviani's Theorem. The proof is much more involved. First you find an interior angle (in the post with 0 upvotes finding angle PBC) then use the cosine rule to find the side length Given an equilateral triangle $ABC$ and $P$ is any point inside the triangle such that $PA^2 = PB^2 + PC^2$. Then what is angle $BPC$? I could only come up with a rough diagram but I couldn't move any further. math.stackexchange.com cross referenced with Help me solve this geometry question Seems like a question that I bit off more than I can chew but I can try to help you more after work if it's still unclear. Reactions:1 users beanspog Joined Sep 2021 16 Posts \| 0+ isb Discussion Starter Sep 17, 2021 #7 MacstersUndead said: Viviani's theorem - Wikipedia en.wikipedia.org Turns out you shouldn't use Viviani's Theorem. The proof is much more involved. First you find an interior angle (in the post with 0 upvotes finding angle PBC) then use the cosine rule to find the side length Given an equilateral triangle $ABC$ and $P$ is any point inside the triangle such that $PA^2 = PB^2 + PC^2$. Then what is angle $BPC$? I could only come up with a rough diagram but I couldn't move any further. math.stackexchange.com cross referenced with Help me solve this geometry question Seems like a question that I bit off more than I can chew but I can try to help you more after work if it's still unclear. Click to expand... I see, i am trying to understand how PH=x and then PH= L3^1/2? Also thank you so much for your help.. Attachments Capture.PNG 53.6 KB · Views: 3 L Lev888 Joined Nov 2019 466 Posts \| 386+ Boston Sep 17, 2021 #8 MacstersUndead said: Do'h. I missed that unjustified line. Let me try to figure out if that's true or not. I still think using Viviani's or something like it is the way to go for this problem though. Click to expand... That can't be true. x, y, z are hypotenuses in the small right triangles, h1, h2, h3 are legs in the corresponding triangles. MacstersUndead Joined Jan 2009 757 Posts \| 440+ Sep 17, 2021 #9 beanspog said: I see, i am trying to understand how PH=x and then PH= L3^1/2? Also thank you so much for your help.. Click to expand... Squeezing this in before going back to work from my break. The post actually presupposes that P is on the altitude of the equilateral triangle, which needs to be justified. There's also a typo. If PA (not PH) is the altitude, then A H=h=3 2 a where a is the side length and L=a 2 in the image. So A H=h=P A+x⟹h−x=P A⟹L 3−x=P A Again supposes P is on the altitude which I you can't assume without proof. I think you have to use an inequality proof to show if P isn't on the altitude then you can't have your initial conditions @Lev888 yeah thinking about it more it certainly can't be possible because of triangle inequality. Reactions:1 users beanspog Joined Sep 2021 16 Posts \| 0+ isb Discussion Starter Sep 17, 2021 #10 MacstersUndead said: Squeezing this in before going back to work from my break. The post actually presupposes that P is on the altitude of the equilateral triangle, which needs to be justified. There's also a typo. If PA (not PH) is the altitude, then A H=h=3 2 a where a is the side length and L=a 2 in the image. So A H=h=P A+x⟹h−x=P A⟹L 3−x=P A Again supposes P is on the altitude which I you can't assume without proof. I think you have to use an inequality proof to show if P isn't on the altitude then you can't have your initial conditions @Lev888 yeah thinking about it more it certainly can't be possible because of triangle inequality. Click to expand... Thanks! I understood it now but does moving the point P anywhere else change the angle PBC?? I mean when PH or PA or HA is not perpendicular to the base. 1 2 3 Next 1 of 3 Go to page Go NextLast Login or Register / Reply Similar Discussions V Let P0 be an equilateral triangle of area 10. 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2558	https://www.mdpi.com/2075-4418/14/19/2198	Next Article in Journal The Usefulness of the Regression-Based Normed SKT Short Cognitive Performance Test in Detecting Cognitive Impairment in a Community Sample Next Article in Special Issue Clinico–Pathological Features of Diffuse Midline Glioma, H3 K27-Altered in Adults: A Comprehensive Review of the Literature with an Additional Single-Institution Case Series Previous Article in Journal Polycystic Ovary Syndrome Accompanied by Hyperandrogenemia or Metabolic Syndrome Triggers Glomerular Podocyte Injury Active Journals Find a Journal Journal Proposal Proceedings Series ## Topics For Authors For Reviewers For Editors For Librarians For Publishers For Societies For Conference Organizers Open Access Policy Institutional Open Access Program Special Issues Guidelines Editorial Process Research and Publication Ethics Article Processing Charges Awards Testimonials ## Author Services Sciforum MDPI Books Preprints.org Scilit SciProfiles Encyclopedia JAMS Proceedings Series Overview Contact Careers News Press Blog Sign In / Sign Up Notice clear Notice You are accessing a machine-readable page. 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Nicoleta Sîrbu, C. Adella Axelerad, A. Pleșa, F. Cristina on Google Scholar Pleșa, A. Antochi, F. Anca Macovei, M. Laura Vîrlan, A. Georgescu, R. Beuran, D. Bucurica, S. Nicoleta Sîrbu, C. Adella Axelerad, A. Pleșa, F. Cristina on PubMed Pleșa, A. Antochi, F. Anca Macovei, M. Laura Vîrlan, A. Georgescu, R. Beuran, D. Bucurica, S. Nicoleta Sîrbu, C. Adella Axelerad, A. Pleșa, F. Cristina /ajax/scifeed/subscribe Article Views Citations - Table of Contents Altmetric share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles Need Help? Support Find support for a specific problem in the support section of our website. Get Support Feedback Please let us know what you think of our products and services. Give Feedback Information Visit our dedicated information section to learn more about MDPI. Get Information clear JSmol Viewer clear first_page Download PDF settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessArticle Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon by Andreea Pleșa Andreea Pleșa SciProfiles Scilit Preprints.org Google Scholar 1,2, Florina Anca Antochi Florina Anca Antochi SciProfiles Scilit Preprints.org Google Scholar 3,, Mioara Laura Macovei Mioara Laura Macovei SciProfiles Scilit Preprints.org Google Scholar 4,5, Alexandra-Georgiana Vîrlan Alexandra-Georgiana Vîrlan SciProfiles Scilit Preprints.org Google Scholar 6, Ruxandra Georgescu Ruxandra Georgescu SciProfiles Scilit Preprints.org Google Scholar 2, David-Ionuț Beuran David-Ionuț Beuran SciProfiles Scilit Preprints.org Google Scholar 5, Săndica Nicoleta Bucurica Săndica Nicoleta Bucurica SciProfiles Scilit Preprints.org Google Scholar 7, Carmen Adella Sîrbu Carmen Adella Sîrbu SciProfiles Scilit Preprints.org Google Scholar She graduated from the University of Medicine and Pharmacy, Carol Davila, in Bucharest in 1989 and a [...] She graduated from the University of Medicine and Pharmacy, Carol Davila, in Bucharest in 1989 and obtained the title of Doctor of Medicine in 2004.The activity record as a neurologist in 2024 is as follows: 15 books signed as author, co-author, or coordinator (including four awards), over 20 chapters written as sole author or co-author, of which two are in international publishing houses, and 86 publications on the Web of Science, Hirsh = 12 and 648 citations. Between 2014 and 2021, she was a professor at the Titu Maiorescu University of Bucharest. In 2023, she won the title of professor at Carol Davila, University of Medicine and Pharmacy and became the head of neurology within the Central Military Emergency University Hospital, Dr Carol Davila. Since 2024, she has been an Academy of Romanian Scientists corresponding member. Between 2006 and 2015, she was part of the subcommittees of Neuro-otology and Neuro-ophthalmology of the European Neurological Society. She was also a member of the subcommittee of Neuro-oncology of EFNS between 2006 and 2015 and of the working group of the European Brain Council in the project "Creating a Policy Narrative on Neurodegenerative Disorders across the Board Europe" (2018). In 2011, she joined the MSBase study group, publishing with other investigators in Brain, Neurology and Multiple Sclerosis Journal journals. She has participated in international clinical studies and projects such as COST, EUROMENE, and InternetAndMe. Read moreRead less 1,2,8,, Any Axelerad Any Axelerad SciProfiles Scilit Preprints.org Google Scholar 9,10 and Florentina Cristina Pleșa Florentina Cristina Pleșa SciProfiles Scilit Preprints.org Google Scholar 1,2 1 Clinical Neurosciences Department, “Carol Davila” University of Medicine and Pharmacy, 050474 Bucharest, Romania 2 Neurology Department, “Dr. Carol Davila” Central Military Emergency University Hospital, 010825 Bucharest, Romania 3 Neurology Department, University Emergency Hospital, 050098 Bucharest, Romania 4 Ophthalmology Department, “Carol Davila” University of Medicine and Pharmacy, 050474 Bucharest, Romania 5 Ophthalmology Department, “Dr. Carol Davila” Central Military Emergency University Hospital, 010825 Bucharest, Romania 6 Pediatric Neurology Department, “Prof. Dr. Alexandru Obregia” Clinical Psychiatric Hospital, 041914 Bucharest, Romania 7 Departament 5, “Carol Davila” University of Medicine and Pharmacy, 050474 Bucharest, Romania 8 Academy of Romanian Scientists, 050045 Bucharest, Romania 9 Department of Neurology, General Medicine Faculty, Ovidius University, 900470 Constanta, Romania 10 Department of Neurology, St. Andrew County Clinical Emergency Hospital of Constanta, 900591 Constanta, Romania Authors to whom correspondence should be addressed. Diagnostics 2024, 14(19), 2198; Submission received: 7 August 2024 / Revised: 28 September 2024 / Accepted: 30 September 2024 / Published: 2 October 2024 (This article belongs to the Special Issue Pathology and Diagnosis of Neurological Disorders) Download keyboard_arrow_down Download PDF Download PDF with Cover Download XML Download Epub Browse Figures Versions Notes Abstract Background/Objectives: This study investigated the frequency and timing of optic neuritis (ON) episodes in relation to the onset of multiple sclerosis (MS) and examined the occurrence of Uhthoff’s phenomenon and Lhermitte’s sign to understand their roles in early diagnosis and disease progression. Methods: A longitudinal study was conducted with 127 MS patients. Clinical data, including ophthalmological examinations for ON, were collected and questionnaires assessed the presence of Uhthoff’s phenomenon and Lhermitte’s sign. Results: Results showed that 37% of patients experienced demyelinating retrobulbar ON, with 25.53% having ON as the initial symptom of MS. Residual visual acuity impairment (below 20/40) and dyschromatopsia were reported by 25.53% and 17.02% of patients, respectively. Uhthoff’s phenomenon and Lhermitte’s sign were present in 26.77% and 36.22% of patients, respectively. The findings underscore the importance of early ophthalmological assessments in diagnosing MS, as ON can be an initial indicator of the disease. Conclusions: The study highlights the need for precise diagnostic tools and personalized therapeutic strategies focused on specific biomarkers and pathways involved in neuroinflammation and demyelination. Early diagnosis through vigilant ophthalmologic evaluation can lead to interventions that significantly alter disease progression, improving patient outcomes and quality of life. Keywords: multiple sclerosis; optic neuritis; Uhthoff’s phenomenon; Lhermitte’s sign; neuroinflammation; neurodegeneration; demyelination; visual acuity impairment; dyschromatopsia; ophthalmological examination; early diagnosis 1. Introduction Multiple sclerosis (MS) is a chronic inflammatory and degenerative disease of the central nervous system, with a progressive evolution that can lead to significant disability. This disease mainly affects young adults, being the main cause of non-traumatic disability among them . Worldwide, 2.8 million people are diagnosed with multiple sclerosis and 30,000 of them are under the age of 18 . Patients with multiple sclerosis have an extensive spectrum of neurological signs and symptoms as a result of variable localization of the central nervous system lesions. The most common of these symptoms are: vision loss, double vision, visual field disorders, nystagmus, speech and swallowing disorders, sensory impairments, gait disorders, ataxia, spasticity, and bladder and sphincter disorders . Ophthalmic symptoms appear frequently, sometimes even before the neurological signs. About 50% of all patients diagnosed with multiple sclerosis will develop one or more episodes of acute demyelinating optic neuritis . Even the first case of multiple sclerosis ever described in the literature was one of a young Dutch woman who suffered from visual impairment as part of the natural course of the disease . The most common manifestations are due to optic neuritis, nystagmus, and internuclear ophthalmoplegia [6,7]. The causes of optic neuritis (ON) are varied and can be classified into different categories based on their underlying mechanisms. These include demyelinating, infectious, inflammatory, and other less common etiologies (detailed in Table 1 ). Understanding the full spectrum of potential causes is essential for accurate diagnosis and management. In patients with demyelinating ON, it is crucial to consider the differential diagnosis of multiple sclerosis (MS), neuromyelitis optica spectrum disorder (NMOSD), and the newly discovered myelin oligodendrocyte glycoprotein antibody-associated disease (MOGAD). All of these conditions can present with similar initial symptoms, such as acute visual impairment due to optic nerve inflammation. However, they differ significantly in their underlying pathophysiology, clinical manifestations, prognosis, and treatment strategies . The characteristics and distinctions of optic neuritis in MS, NMOSD, and MOGAD are detailed in Table 2 . Although profound vision loss is unusual in multiple sclerosis, even a subtle visual deterioration may be hazardous in patients who already have sensory or motor dysfunctions. Identifying the most common ophthalmological manifestations that may occur, together with a careful clinical and paraclinical examination of a young patient who complains of visual disturbances can lead to an early diagnosis of multiple sclerosis. The Uhthoff phenomenon and Lhermitte’s sign are clinical features observed in multiple sclerosis. Recognizing these signs can serve as a clinical argument for early neurological assessment, facilitating timely intervention. The Uhthoff phenomenon, first described by ophthalmologist Wilhelm Uhthoff in 1890, is most commonly noticed in MS, with 60% to 80% of MS patients experiencing this phenomenon when exposed to heat. However, it can also occur in other optic disorders affecting the afferent pathways . Uhthoff’s phenomenon is characterized by a stereotyped worsening of neurological symptoms, especially visual disturbances, that is temporary and lasts less than 24 h . It is triggered by elevated body temperature due to exercise, fever, or hot weather. It is important to differentiate this transient worsening from an exacerbation of MS or a true relapse. Uhthoff’s phenomenon is most commonly associated with multiple sclerosis (MS); however, other rare conditions may present with similar symptoms. Although these conditions are uncommon, they should be included in the differential diagnosis to ensure a comprehensive evaluation and appropriate management. A detailed list of conditions to consider in the differential diagnosis of Uhthoff’s phenomenon is provided in Table 3 . Lhermitte’s sign, characterized by an electric shock-like sensation that radiates down the spine and into the limbs when the neck is flexed, is a common manifestation in patients with MS, appearing in up to 41% of cases . However, it is not specific to MS and can be found in other pathologies involving the cervical spinal cord. Table 4 categorizes the differential diagnosis of Lhermitte’s sign, emphasizing the importance of considering these diverse conditions in clinical practice. Research indicates that its prevalence among MS patients is notably lower compared to those with neuromyelitis optica (NMO) . This sensation results from miscommunication between demyelinated nerves in the dorsal column of the spinal cord, particularly at the cervical level . The aim of this study was to determine the frequency of ON episodes and their timing relative to the onset of MS, as well as to assess the frequency of Uhthoff phenomenon and Lhermitte’s sign in these patients. 2. Materials and Methods This was a descriptive, observational, retrospective, and longitudinal study involving 127 patients diagnosed with multiple sclerosis (MS) who were undergoing treatment with disease-modifying therapy. The patients were admitted to the Neurology Department at “Dr. Carol Davila” Central Military Emergency University Hospital between July 2022 and June 2024. Patient selection followed specific inclusion and exclusion criteria. Patients were excluded if they had experienced an acute relapse of optic neuritis (ON) within the last six months, as this could affect the assessment of residual impairment. Additionally, patients with a disease duration shorter than one year were excluded to ensure that the study encompassed chronic cases, including primary progressive MS. Finally, patients with other eye conditions, such as glaucoma, which could independently cause visual impairment, were also excluded from the study. Data collection was conducted retrospectively using patient records from the hospital’s archives. The clinical data had been previously gathered through anamnesis, clinical examination, and paraclinical investigations. An eye examination, which included visual acuity testing with a 4 m Snellen chart and color vision testing using Ishihara Color Blindness Test Plates, was performed by an ophthalmologist. Additionally, the presence of Uhthoff phenomenon and Lhermitte’s sign was assessed using a self-administered questionnaire completed by the patients during their medical visits. The collected data were stored and subsequently processed and statistically analyzed using Microsoft Office Excel 2021. Quantitative variables were analyzed using descriptive statistics and measures of central tendency (mean), with standard errors and deviations calculated. The results of the categorical variable analysis were graphically represented using pie charts and bar charts. Quantitative variables were converted into categorical variables and also graphically represented in the same way. To determine the statistical significance of the processed data, the predictive value (p) was calculated using the Student’s t-test for quantitative variables and the Chi-square test for qualitative variables. A p-value ≤ 0.05 was considered statistically significant. 3. Results 3.1. Clinical and Demographic Characteristics Our analysis of the clinical data revealed a significant prevalence of MS-related ON, with 37% (47 out of 127) of patients having a history of this condition. We divided the patients into two categories: those with and those without a history of ON, and evaluated their age, gender, residency, age at onset of the disease, type of MS, and treatment. Patients with ON were younger at the onset of MS (31.60 ± 9.38 years) compared to those without (35.76 ± 10.51 years), the difference being statistically significant (p = 0.039). This finding aligns with previous studies suggesting that ON often presents earlier in the disease course of MS. There were no significant differences between the two groups concerning other characteristics such as gender, residency, type of MS, and treatment (Table 5). When analyzing the demographic data for patients with a history of ON, we found out that the mean age was 41.87 ± 11.958 years (range 22 to 68 years). Seventy-four percent of patients were women and the mean age at the onset of MS was 31.60 ± 9.380 years. These data were consistent with the literature, considering that MS predominantly affects young women . Most of the patients lived in urban areas, with an urban-to-rural ratio of 4:1. The higher proportion of urban residents in our sample (4:1 urban-to-rural ratio) can be attributed to the greater ease with which urban residents can access hospitals and doctors compared to rural residents. This is noteworthy, especially considering that the urban and rural populations in Romania are almost equal. The majority of patients (96%) presented with the relapsing–remitting form of MS, while 4% had the secondary progressive form. This distribution aligns with expectations, given that relapsing–remitting MS is the most common form of the disease . Regarding treatment, beta interferon was the most prescribed drug, followed by ocrelizumab and teriflunomide. To assess disease severity, we used the Expanded Disability Status Scale (EDSS), which ranges from 0 to 10 in 0.5 increments based on neurological examination and patient history [17,18]. The mean EDSS score in the ON group was 2.20 ± 1.51, slightly lower than the score in the non-ON group (2.35 ± 1.80). Despite the higher score in the latter group, patients with ON showed a considerable impact on overall disability, indicating the significant role of visual symptoms in the disease burden. 3.2. Optic Neuritis Characteristics Visual impairment was reported as the initial symptom of MS by 63% of patients with ON (30 out of 47). This represented 25.53% of all study participants, underscoring the importance of comprehensive ophthalmological examinations for early diagnosis. In an additional six cases, ON led to the diagnosis of MS, even though it was not the first relapse (see Figure 1). The majority of patients (72.34%) did not experience recurrences of ON. Among those who did, eight had two episodes, two had three episodes, and three had more than three episodes. Four patients experienced bilateral optic nerve involvement at disease onset (see Figure 2). Following ON, 74.47% of patients experienced recovery with normal or near-normal visual acuity. Furthermore, 25.53% had lasting visual impairment, with their best-corrected visual acuity in the worst-affected eye being less than 20/40. Visual dysfunction was classified according to the International Classification of Diseases 11 (2018), with five patients (10.63%) having mild impairment and seven (14.89%) having moderate impairment (see Figure 3). Color vision dysfunction, assessed via Ishihara plates, was found in eight patients (17.02%) and is represented in Figure 4. Bilateral dyschromatopsia was present in two patients, while the remaining cases involved unilateral impairment. The most common type was red–green color vision deficiency, present in seven patients. 3.3. Uhthoff Phenomenon and Lhermitte’s Sign The presence of the Uhthoff phenomenon and Lhermitte’s sign was evaluated via a self-administered questionnaire. The Uhthoff phenomenon was observed in 34 out of 127 patients (26.77%), with a slightly higher frequency in the group without ON (27.5% vs. 25.53%, RR = 0.93). Lhermitte’s sign was present in 46 out of 127 patients (36.22%), with no significant difference between the two groups (36.17% in the group with ON and 36.25% in the group without, RR = 0.99). The prevalence of these phenomena is represented in Figure 5. These symptoms are common in MS and further underscore the complex and varied symptomatology of the disease. 4. Discussion 4.1. Optic Neuritis in Multiple Sclerosis Optic neuritis (ON) in multiple sclerosis arises from complex molecular mechanisms involving immune-mediated damage to the optic nerve. The optic nerve is part of the central nervous system (CNS). It consists of approximately 1.2 million axons originating from the retinal ganglion cells . These ganglion cell axons are unmyelinated fibers along the optic nerve head, becoming myelinated immediately posterior to the lamina cribrosa . Oligodendrocytes are responsible for myelination and the maintenance of saltatory conduction, which facilitates efficient nerve impulse transmission along the axons within the CNS. The localization of oligodendrocytes in the posterior compartment explains the pattern of inflammatory damage to the optic nerve during ON, as retinal inflammation is not typical. In MS, retrobulbar ON is characteristic . In MS, damage to the optic nerve occurs through several mechanisms, including demyelination mediated by the death of oligodendroglial cells, activation of glial cells (such as microglia and astrocytes), and axonal degeneration . The body’s natural response to demyelination involves the activation of immunomodulatory networks that limit inflammation and initiate repair processes, resulting in partial remyelination and clinical remission . However, endogenous remyelination has limitations. Remyelinated axons often have thinner myelin sheaths and shorter internodal lengths, leading to slower axonal conduction velocities. Moreover, recurrent demyelination can further impair remyelination processes, resulting in permanently demyelinated axons and chronic damage in MS . ON is a common clinical manifestation in patients with MS. Various studies have reported differing prevalence rates of ON in MS, reflecting variability due to study design, follow-up period, and population characteristics. Studies indicate that optic neuritis (ON) occurs in approximately half of multiple sclerosis (MS) patients over the course of the disease [24,25]. In our study, 37% of patients experienced ON. This lower prevalence compared to the literature may be influenced by the variability in follow-up duration, as our cohort included patients diagnosed between one and over twenty years ago. While some patients were followed for a long period, others, particularly those more recently diagnosed, had shorter follow-up times. This design limits the ability to capture the lifetime prevalence of ON for all patients. Given the known frequency of ON in MS, it is possible that additional cases may emerge with further follow-up. Additionally, ON may be underrecognized, particularly in milder or transient cases where medical attention may not have been sought. Given the high prevalence, the presence of ophthalmological manifestations in a young patient requires a complete ophthalmological examination and referral to a neurologist for early treatment and diagnosis of a possible MS onset. ON is often the first symptom of MS. Studies have reported that ON is the initial presenting symptom in up to 25% of cases . Our results are consistent with this, as we found that 25.53% of MS patients had ON as their initial symptom. However, older studies reported lower rates, such as 15–20% [27,28]. This discrepancy could be due to the under-recognition of isolated ON, leading to delayed referrals to neurologists. Additionally, symptoms of ON are often transient, and patients might not seek immediate medical attention, delaying diagnosis until subsequent relapses occur. Moreover, variability might arise from differences in healthcare access, diagnostic criteria, and awareness of ON among primary care providers. The high rate of visual impairment as an initial symptom of MS underscores the need for early and thorough ophthalmological evaluation and referral to a neurologist in suspected cases. Recurrent ON is not uncommon. In the ten-year follow-up of the Optic Neuritis Treatment Trial, recurrences of ON were observed in 35% of the patients . In our study, 27.66% of patients with a history of ON experienced recurrences. The lower rate in our study can also be explained by the shorter follow-up period for some of the patients. There is a risk of residual visual impairment following an episode of ON. In the ten-year follow-up of the Optic Neuritis Treatment Trial, 9% of patients had a visual acuity worse than 20/40 in the affected eye . It is important to note that in this study, not all patients with ON developed MS. Visual function was generally poorer in patients with multiple sclerosis. Since the reported percentage refers to the entire group of patients, it may underestimate the rate of residual visual dysfunction specifically in MS patients. This could explain the much higher rate of residual visual dysfunction observed in our group (25.53%). Additionally, some patients were followed for a longer period and experienced recurrences of ON, which likely contributed to the higher rate of residual visual impairment. A limitation of the study in assessing residual visual dysfunction is that we only used the Snellen chart for assessment. We did not evaluate optic nerve atrophy using OCT or MRI measurements, which can reveal structural damage that correlates with visual dysfunction. In a small study, color vision was evaluated in a group of patients with ON, and color blindness, as diagnosed by the Ishihara test, was found in four out of forty-four patients (20.45%) . In our study, color vision dysfunction was identified in a similar percentage, 17.02%. This suggests that color vision deficits are a common, though often overlooked, consequence of ON in MS. The frequency of ON recurrences and the residual impairment observed in our patients highlight the disease burden in this population. Continuous ophthalmological monitoring is crucial for patients with MS, given that ON can occur at any stage of the disease. The management of multiple sclerosis (MS) involves a comprehensive approach tailored to the individual’s disease form, severity, and other factors such as age, overall health, and response to previous treatments and family planning considerations. Disease-modifying therapies (DMTs) are at the forefront of MS management, with proven efficacy in reducing the frequency and severity of relapses, slowing disease progression, and limiting the development of new lesions as seen via MRI. The early initiation of disease-modifying therapies can slow the progression of the disease and reduce the degree of disability in these patients . The treatment of acute relapse in multiple sclerosis typically involves corticosteroid therapy to reduce inflammation, significantly shorten the duration of the relapse, and improve symptoms. Prompt administration is crucial. For severe relapses that do not respond to steroid treatment, plasmapheresis can be an effective alternative. This is of utmost importance when addressing ON, as the resulting visual dysfunction can be a highly debilitating condition. Prompt and effective treatment is essential to reduce the impact on vision and overall quality of life. Symptomatic therapy plays a crucial role in improving the quality of life for individuals with MS, addressing issues like spasticity, pain, fatigue, and bladder dysfunction . Although these treatments do not modify the disease course, they are essential for maintaining daily comfort and functioning. Interest is also growing in complementary and alternative therapies, such as cannabis, vitamin D, dietary changes, physical activity, exercise, and yoga [33,34,35]. While preliminary evidence suggests potential benefits, more research is needed to confirm their effectiveness and safety. Neurorehabilitation is a cornerstone of MS management, aiming to optimize physical and cognitive function through a comprehensive, multidisciplinary approach [36,37]. 4.2. Uhthoff’s Phenomenon and Lhermitte Sign Uhthoff’s Phenomenon occurs because demyelination causes the nodes of Ranvier, which are rich in sodium channels, to widen . This widening disrupts the organization of sodium channels and impairs depolarization. As a compensatory mechanism, new sodium channels form along the axonal membrane, but they have altered properties . Even a minimal increase in temperature (as little as 0.5 °C) is enough to close these sodium channels and halt depolarization. Furthermore, the exposed potassium channels lead to hyperpolarization. These combined effects result in delayed or blocked conduction that clinically manifest as worsening MS symptoms . The underlying mechanism behind Lhermitte’s sign involves a disruption in normal nerve conduction due to myelin loss, which is crucial for the rapid transmission of nerve impulses. In demyelinated fibers, conduction becomes erratic and can lead to the misfiring of sensory signals. The pathophysiology also includes activation of neuropathic pain pathways, influenced by the impaired function of inhibitory GABAergic interneurons, which normally regulate sensory input and prevent excessive excitability . Additionally, microglia, the central nervous system’s immune cells, play a key role. Their activation increases the production of inflammatory cytokines, which exacerbate the inflammatory environment in MS, exacerbating the abnormal neuronal signaling and leading to the clinical manifestation of Lhermitte’s sign . Uhthoff’s phenomenon is reported to affect approximately 60% to 80% of MS patients in various studies . Lhermitte’s sign is observed in 9 to 41% of MS patients . In our study, Uhthoff’s phenomenon was observed in 26.77% of patients, while Lhermitte’s sign was present in 36.22%. These findings are consistent with the prevalence range for Lhermitte’s sign but notably lower for Uhthoff’s phenomenon compared to the literature. This discrepancy might be attributed to our use of a standardized self-assessment questionnaire, which could result in underreporting due to patients’ difficulty recognizing or recalling their symptoms. 5. Conclusions MS, a neurological disorder which mainly affects women and young people, frequently causes ocular manifestations, sometimes as the first clinical sign of the disease. Uhthoff phenomenon and Lhermitte’s sign are clinical phenomena that may occur in MS and can lead to timely diagnosis. The intricate molecular mechanisms underlying MS significantly contribute to neuroinflammation and the development of conditions such as ON, Uhthoff’s phenomenon, and Lhermitte’s sign. Understanding these molecular mechanisms is crucial for advancing diagnostic and monitoring techniques in MS. By identifying specific biomarkers and pathways involved in neuroinflammation and demyelination, clinicians can develop more accurate diagnostic tools and personalized therapeutic strategies. Tailored treatments that target these molecular processes could potentially enhance remyelination, reduce neuroinflammation, and prevent the progression of chronic damage, ultimately improving the quality of life for individuals with multiple sclerosis. Our study adds to existing knowledge by providing detailed clinical and demographic insights into Romanian MS patients with ON, emphasizing the importance of early diagnosis and comprehensive management of visual symptoms. The role of ophthalmologists in the early detection and referral to neurologists is critical, as it ensures timely and effective treatment for MS patients. Early diagnosis facilitated by vigilant ophthalmological evaluation can lead to interventions that may significantly alter the disease’s progression and improve patient outcomes and quality of life. Author Contributions Conceptualization, A.P., R.G., C.A.S. and F.C.P.; data curation, F.A.A. and A.-G.V.; formal analysis, A.P. and F.C.P.; funding acquisition, F.C.P.; investigation, A.P., M.L.M., A.-G.V., R.G., D.-I.B. and F.C.P.; methodology, F.A.A., M.L.M., R.G. and S.N.B.; project administration, S.N.B., C.A.S. and A.A.; resources, A.P., M.L.M., A.A. and F.C.P.; software, D.-I.B. and S.N.B.; supervision, F.A.A. and C.A.S.; validation, C.A.S. and A.A.; visualization, M.L.M. and A.A.; writing—original draft preparation, A.P. and D.-I.B.; writing—review and editing, F.A.A., A.-G.V., R.G., C.A.S. and F.C.P. All authors have read and agreed to the published version of the manuscript. Funding Publication of this paper was supported by the University of Medicine and Pharmacy Carol Davila, through the institutional program Publish not Perish. Institutional Review Board Statement The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Ethics Committee of “Dr. Carol Davila” Central Military Emergency University Hospital, Bucharest, Romania (PV nr. 534/08.06.2022). Informed Consent Statement Given the retrospective nature of the study, informed consent was waived as the data used were anonymized to ensure the privacy and confidentiality of the patients. All procedures adhered to institutional and national ethical standards, ensuring that patient rights and dignity were protected throughout the research process. Data Availability Statement The data presented in this study are available on request from the corresponding author due to privacy reasons. Conflicts of Interest The authors declare no conflicts of interest. References Sormani, M.P.; Naismith, R.T. Measuring disability in multiple sclerosis: Walking plus much more. Neurology 2019, 93, 919–920. 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Distribution of visual acuity outcomes after optic neuritis. Figure 3. Distribution of visual acuity outcomes after optic neuritis. Figure 4. Frequency of dyschromatopsia among patients. Figure 4. Frequency of dyschromatopsia among patients. Figure 5. Prevalence of Uhthoff phenomenon (left) and Lhermitte sign (right) in multiple sclerosis (MS) patients with and without optic neuritis (ON). Figure 5. Prevalence of Uhthoff phenomenon (left) and Lhermitte sign (right) in multiple sclerosis (MS) patients with and without optic neuritis (ON). Table 1. Causes of optic neuritis. Table 1. Causes of optic neuritis. \| Demyelinating \| Infections \| Inflammatory \| Others \| --- --- \| \| Multiple sclerosis NMOSD MOGAD \| Viral Bacterial \| Parainfectious Sarcoidosis CRION Systemic autoimmune diseases \| Compressive Genetic causes Toxic and metabolic Trauma \| Abbreviations: NMOSD: Neuromyelitis optica spectrum disorder; MOGAD: myelin oligodendrocyte glycoprotein antibody-associated disease; CRION: chronic relapsing inflammatory optic neuropathy. Table 2. Comparison of ON Features in MS, NMOSD, and MOGAD. Table 2. Comparison of ON Features in MS, NMOSD, and MOGAD. \| ON Features \| MS \| NMOSD \| MOGAD \| --- --- \| \| Bilateral presentation \| rare \| common \| common \| \| Visual acuity impairment \| mild to moderate \| moderate to severe \| moderate to severe \| \| Ocular pain \| common \| rare \| common \| \| Optic nerve involvement \| segmental (<50% of optic nerve) \| extensive (>50% of optic nerve) \| extensive (>50% of optic nerve) \| \| Optic chiasm involvement \| infrequent \| common \| uncommon \| \| \| \| Perineuritis \| absent or mild \| less common \| common \| \| RNFL measured via OCT \| typically normal in acute phase \| typically normal in acute phase \| acutely thickening \| \| Relapse recovery \| intermediate \| generally poor \| usually good \| Abbreviations: ON: optic nerve; MS: multiple sclerosis; NMOSD: neuromyelitis optica spectrum disorder; MOGAD: myelin oligodendrocyte glycoprotein antibody disease; RNFL: retinal nerve fiber layer; OCT: optical coherence tomography. Table 3. Differential diagnosis of Uhthoff’s phenomenon. Table 3. Differential diagnosis of Uhthoff’s phenomenon. \| Category \| Specific Causes \| --- \| \| Demyelinating Diseases \| MS, NMOSD \| \| Autoimmune Disorders \| Antiphospholipid antibody syndrome, Behçet’s disease, CNS lupus, CNS vasculitis, Sjogren’s syndrome \| \| Infectious Diseases \| HIV, HTLV, Lyme disease \| \| Hematologic Conditions \| CNS lymphoma \| \| Nutritional Deficiencies \| Copper deficiency \| \| Genetic Disorders \| Leukodystrophies \| \| Vascular Disorders \| Small vessel disease \| \| Other Conditions \| Sarcoidosis, osmotic demyelination syndrome \| Abbreviations: MS: multiple sclerosis; NMOSD: neuromyelitis optica spectrum disorder; CNS: central nervous system; HIV: human immunodeficiency virus; HTLV: human T-lymphotropic virus. Table 4. Differential diagnosis of Lhermitte’s sign. Table 4. Differential diagnosis of Lhermitte’s sign. \| Category \| Specific Causes \| --- \| \| Demyelinating Diseases \| MS \| \| Autoimmune Disorders \| Transverse myelitis, CNS lupus, Behçet’s disease \| \| Infectious Diseases \| Herpes zoster toxicity, parasitic invasion of the cord \| \| Nutritional Deficiencies \| Vitamin B12 deficiency \| \| Toxic/Drug-related \| Radiation myelopathy, high dose chemoradiation (cisplatin), nitric oxide toxicity \| \| Mechanical/Structural \| Tumor, radiculopathy, cervical spondylitis, Arnold–Chiari malformation, syringomyelia, trauma, arachnoiditis, post-dural puncture headache \| Abbreviations: MS: multiple sclerosis; CNS: central nervous system. Table 5. General characteristics of the cohort. Table 5. General characteristics of the cohort. \| Parameter \| Group 1—Patients with ONN = 47 \| Group 2—Patients without ONN = 80 \| p Value \| --- --- \| \| Age \| Mean age ± SD \| 41.87 ± 11.958 years \| 44.44 ± 10.918 years \| 0.223 (NS) \| \| GenderN, % \| Male \| 12 (26%) \| 25 (31%) \| 0.557 (NS) \| \| Female \| 35 (74%) \| 55 (69%) \| \| Residency \| Urban \| 37 (79%) \| 69 (86%) \| 0.273 (NS) \| \| Rural \| 10 (21%) \| 11 (14%) \| \| MS age of onset \| Mean age of onset ± SD \| 31.60 ± 9.380 years \| 35.76 ± 10.511 years \| 0.039 (S) \| \| Type of MS \| Recurrent remissive \| 45 (96%) \| 65 (81%) \| 0.132 (NS) \| \| Primary progressive \| 0 (0%) \| 8 (10%) \| \| Secondary progressive \| 2 (4%) \| 6 (9%) \| \| EDSS score \| Mean EDSS ± SD \| 2.20 ± 1.51 \| 2.35 ± 1.801 \| 0.639 (NS) \| \| Treatment \| Beta-interferon \| 18 (38%) \| 18 (23%) \| Ocrelizumab \| 7 (15%) \| 18 (23%) \| \| Teriflunomide \| 5 (11%) \| 13 (16%) \| \| Natalizumab \| 5 (11%) \| 6 (8%) \| \| Glatiramer acetate \| 4 (9%) \| 3 (4%) \| \| Cladribine \| 4 (9%) \| 6 (8%) \| \| Dimethyl fumarate \| 2 (4%) \| 7 (9%) \| \| Fingolimod \| 1 (2%) \| 5 (6%) \| \| Siponimod \| 1 (2%) \| 4 (5%) \| Abbreviations: ON: optic nerve; MS: multiple sclerosis; EDSS: Expanded Disability Status Scale; SD: standard deviation. NS: not significant; S: significant. \| \| \| Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. \| © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( Share and Cite MDPI and ACS Style Pleșa, A.; Antochi, F.A.; Macovei, M.L.; Vîrlan, A.-G.; Georgescu, R.; Beuran, D.-I.; Bucurica, S.N.; Sîrbu, C.A.; Axelerad, A.; Pleșa, F.C. Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics 2024, 14, 2198. AMA Style Pleșa A, Antochi FA, Macovei ML, Vîrlan A-G, Georgescu R, Beuran D-I, Bucurica SN, Sîrbu CA, Axelerad A, Pleșa FC. Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics. 2024; 14(19):2198. Chicago/Turabian Style Pleșa, Andreea, Florina Anca Antochi, Mioara Laura Macovei, Alexandra-Georgiana Vîrlan, Ruxandra Georgescu, David-Ionuț Beuran, Săndica Nicoleta Bucurica, Carmen Adella Sîrbu, Any Axelerad, and Florentina Cristina Pleșa. 2024. "Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon" Diagnostics 14, no. 19: 2198. APA Style Pleșa, A., Antochi, F. A., Macovei, M. L., Vîrlan, A.-G., Georgescu, R., Beuran, D.-I., Bucurica, S. N., Sîrbu, C. A., Axelerad, A., & Pleșa, F. C. (2024). Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics, 14(19), 2198. Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. Article Metrics No Article Access Statistics For more information on the journal statistics, click here. Multiple requests from the same IP address are counted as one view. Zoom \| Orient \| As Lines \| As Sticks \| As Cartoon \| As Surface \| Previous Scene \| Next Scene Export citation file: BibTeX) MDPI and ACS Style Pleșa, A.; Antochi, F.A.; Macovei, M.L.; Vîrlan, A.-G.; Georgescu, R.; Beuran, D.-I.; Bucurica, S.N.; Sîrbu, C.A.; Axelerad, A.; Pleșa, F.C. Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics 2024, 14, 2198. AMA Style Pleșa A, Antochi FA, Macovei ML, Vîrlan A-G, Georgescu R, Beuran D-I, Bucurica SN, Sîrbu CA, Axelerad A, Pleșa FC. Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics. 2024; 14(19):2198. Chicago/Turabian Style Pleșa, Andreea, Florina Anca Antochi, Mioara Laura Macovei, Alexandra-Georgiana Vîrlan, Ruxandra Georgescu, David-Ionuț Beuran, Săndica Nicoleta Bucurica, Carmen Adella Sîrbu, Any Axelerad, and Florentina Cristina Pleșa. 2024. "Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon" Diagnostics 14, no. 19: 2198. APA Style Pleșa, A., Antochi, F. A., Macovei, M. L., Vîrlan, A.-G., Georgescu, R., Beuran, D.-I., Bucurica, S. N., Sîrbu, C. A., Axelerad, A., & Pleșa, F. C. (2024). Eyes as Windows: Unveiling Neuroinflammation in Multiple Sclerosis via Optic Neuritis and Uhthoff’s Phenomenon. Diagnostics, 14(19), 2198. Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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2559	https://byjus.com/maths/existence-and-uniqueness-theorems-for-initial-value-problems/	The existence and uniqueness theorem for initial value problems of ordinary differential equations implies the condition for the existence of a solution of linear or non-linear initial value problems and ensures the uniqueness of the obtained solution. Learn Ordinary Differential Equations Open Rectangle:An open rectangle R is a set of points (x, y) on a plane, such that for any fixed points a, b, c and d a < x < b and c < y < d It is called “open” because points a, b, c and d are not included in the region R. Theorem Statement Existence Theorem: If f is a continuous function in an open rectangle R = {(x, y) \| a < x < b and c < y < d } that contains a point (x o, y o), then the initial value problem y’ = f(x, y), y(x o) = y o has atleast a solution in some open sub-interval of (a, b) which contains the point x o. Uniqueness Theorem:If f and f y are continuous function in an open rectangle R = {(x, y) \| a < x < b and c < y < d } that contains a point (x o, y o), then the initial value problem y’ = f(x, y), y(x o) = y o has a unique solution on some open sub-interval of (a, b) which contains the point x o. Some important points that the existence and uniqueness theorem directly implies: It provides information about the existence of the solution to the initial value problem but does not state how to find the solution or find which open interval. The existence theorem does not provide information about how many solutions that the initial value problem may have. The uniqueness theorem ensures the uniqueness of the solution if the interval (a, b) ≠is not so large. Also Read: Differential Equations Solution of Linear Differential Equation Second Order Linear Differential Equation with Variable Coefficients Balanced and Unbalanced Transportation Problems Wave Equation Partial Differential Equations Solved Examples on Existence and Uniqueness Theorem Example 1: Check the existence and uniqueness of the solution for the initial value problem y’ = x – y + 1, y(1) = 2. Solution: Given the initial value problem y’ = x – y + 1, y(1) = 2. where f(x, y) = x – y + 1 and its partial derivative with respect to y, f y = – 1, which is continuous in every real interval. Hence the existence and uniqueness theorem ensures that in some open interval centred at 1, the solution of the given ODE exists. Now, y’ + y = x + 1 is a linear differential equation of the form y’ + P(x)y = Q(x), where P = 1 and Q = x + 1. Hence, I.F. = e∫P dx = e∫ 1 . dx = e x The solution is y e x = ∫ (x + 1). e x dx ⇒ y e x = ∫ x.e x dx + ∫ e x dx + C ⇒ y = x + C e–x At x = 1 ⇒ y = 2, we get 2 = 1 + C e–1 ⇒ C = e Thus, the solution of given ODE is y = x + e 1 – x, which exists for all x ∈ R. Example 2: Check the existence and uniqueness of the solution for the initial value problem y’ = y 2 , y(0) = 1. Solution: Given the initial value problem y’ = y 2 , y(0) = 1. where f(x, y) = y 2 and its partial derivative with respect to y, f y = 2y, which is continuous in every real interval. Hence the existence and uniqueness theorem ensures that in some open interval centred at 0, the solution of the given ODE exists. Now, separating the variables y – 2 dy = dx Integrating both sides, we get ∫ y – 2 dy = ∫ dx + C 1 ⇒ – 2/ y = x + C 1 ⇒ y = – 2/ (x + C 1) At x = 0, y = 1 1 = –2/C 1 or C = 1 {Let – 2/C 1 = C} Thus, the solution of given ODE is y = 1/ (1 – x), which exists for all x ∈ ( – ∞, 1). Practice Problems Check the existence and uniqueness of the solution for the initial value problem y’ = 1 + y 2 , y(0) = 0. Check the existence and uniqueness of the solution for the initial value problem y’ = y 1/3 , y(0) = 0. Frequently Asked Questions – FAQs Q1 How do you know if an initial value problem has a unique solution? By the uniqueness theorem, if f and f y are continuous functions in an open rectangle R that contains a point (x o, y o), then for the initial value problem y’ = f(x, y), y(x o) = y o has a unique solution on some open sub-interval of (a, b) which contains the point x o. Q2 What is the condition for the existence of a solution for an initial value problem? By the existence theorem, if f is a continuous function in an open rectangle R that contains a point (x o, y o), then the initial value problem y’ = f(x, y), y(x o) = y o has atleast a solution in some open sub-interval of (a, b) which contains the point x o. Q3 What is meant by an open rectangle? An open rectangle R is a set of points (x, y) on a plane, such that for any fixed points a, b, c and d, and a Comments Leave a Comment Cancel reply Your Mobile number and Email id will not be published.Required fields are marked Send OTP Did not receive OTP? Request OTP on Voice Call Website Post My Comment Register with BYJU'S & Download Free PDFs Send OTP Download Now Register with BYJU'S & Watch Live Videos Send OTP Watch Now
2560	https://brilliant.org/wiki/parallelogram/	Parallelogram Sign up with Facebook or Sign up manually Already have an account? Log in here. Niranjan Khanderia, Lawrence Chiou, Karthik ., and Hamza Waseem Andrew Ellinor ZhiJie Goh Jimin Khim contributed A parallelogram is a quadrilateral whose opposite sides are parallel. A parallelogram, in its most general form, looks something like this: Note that the arrowheads are used to indicate which pair of sides are parallel. Contents Special Cases Basic Properties Area Some Cooler Properties Properties of Parallelograms Properties Special Parallelograms Special Cases Basic Properties Area Some Cooler Properties Properties of Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. Properties The fundamental definition of a parallelogram is as follows: A parallelogram is a quadrilateral whose opposite sides are parallel. In the diagram of a general parallelogram above, AB∣∣DC and AD∣∣BC. Several important properties then follow. First, it is clear that the opposite angles must be equal (∠A=∠C and ∠B=∠D) since they make corresponding angles with the two sets of parallel lines. Meanwhile, consecutive angles are supplementary. Property 1. The opposite angles of a parallelogram are equal. Property 2. Consecutive angles of a parallelogram are supplementary. One can also show that the opposite sides are equal (AB=DC and AD=BC): the two triangles formed by drawing in a diagonal of the parallelogram (i.e. either the segment AC or BD above) must be congruent by angle-side-angle, so the corresponding sides of the two triangles must be congruent as well. Property 3. The lengths of the opposite sides of a parallelogram are equal. Now that the lengths of all of the sides are known, it is easy to compute the perimeter of a parallelogram. Property 4. A parallelogram whose side lengths are a and b has perimeter 2a+2b. Meanwhile, the area of the parallelogram can be found by computing the sum of the area of the two triangles formed. Property 5. The area of a parallelogram with side lengths a and b, with the acute angle formed between them θ, is given by absinθ. Drawing in both diagonals simultaneously produces four congruent triangles. Therefore, the intersection of the two diagonals must be the midpoint of each diagonal. Property 6. The diagonals of a parallelogram bisect each other. One last result is left as an exercise for the reader. Property 7. Lines that connect midpoints of opposite sides with opposite vertices trisect the diagonal. In a parallelogram ABCD, the length of AB is twice the length of AD. E is the midpoint of AB. Find the measure of ∠DEC in degrees. The correct answer is: 90 31 83 21 53 In the given parallelogram, the midpoints of two adjacent sides of the parallelogram are joined and then connected to the opposite vertex to form a triangle. What fraction of the total parallelogram is the shaded area? Click here for more from this set. The correct answer is: 83 A parallelogram has sides measuring 7 and 9. Its shorter diagonal has a length of 8. Find the measure of the longer diagonal. The correct answer is: 14 A point X is drawn on the diagonal of a parallelogram ABCD. The lines parallel to the sides through X are constructed, and quadrilaterals P and Q are formed, as shown in the diagram. Given that Q has base 12 and perpendicular height 4 and that the area of ABCD is 200, find the sum of all possible values of the base of P. Special Parallelograms Commonly encountered special cases of parallelograms include rectangles, all of whose angles are equal; rhombuses, all of whose sides are equal; squares, all of whose sides and angles are equal. Because opposite angles of a cyclic quadrilateral are supplementary, all cyclic parallelograms are rectangles. Furthermore, the only rectangles with an incircle are squares, so the only bicyclic parallelogram is a square. Cite as: Parallelogram. Brilliant.org. Retrieved 03:04, September 28, 2025, from
2561	https://www.msdmanuals.com/professional/hematology-and-oncology/anemias-caused-by-deficient-erythropoiesis/anemia-of-chronic-disease	Anemia of Chronic Disease - Hematology and Oncology - MSD Manual Professional Edition honeypot link skip to main content Professional Consumer Professional edition active ENGLISH MSD Manual Professional Version MEDICAL TOPICSRESOURCESCOMMENTARYPROCEDURESQUIZZESABOUT US MEDICAL TOPICSRESOURCESCOMMENTARYPROCEDURESQUIZZES Professional/ Hematology and Oncology/ Anemias Caused by Deficient Erythropoiesis/ Anemia of Chronic Disease/ IN THIS TOPIC Etiology Diagnosis Treatment Key Points OTHER TOPICS IN THIS CHAPTER Overview of Decreased Erythropoiesis Anemia of Chronic Disease Anemia of Renal Disease Aplastic Anemia Iron Deficiency Anemia Megaloblastic Macrocytic Anemias Myelodysplasia and Iron-Transport Deficiency Anemia Myelophthisic Anemia Pure Red Blood Cell Aplasia Sideroblastic Anemias Anemia of Chronic Disease (Anemia of Chronic Inflammation) ByGloria F. Gerber, MD, Johns Hopkins School of Medicine, Division of Hematology Reviewed ByJerry L. Spivak, MD; MACP, , Johns Hopkins University School of Medicine Reviewed/Revised Modified Mar 2025 v969267 View Patient Education Anemia of chronic disease is a multifactorial anemia. Diagnosis generally requires the presence of a chronic inflammatory condition, such as infection, autoimmune disease, kidney disease, or cancer. It is characterized by a microcytic or normocytic anemia and low reticulocyte count. Values for serum iron and transferrin are typically low, while the serum ferritin value can be normal or elevated. Treatment is to reverse the underlying disorder and in some cases, to give erythropoietin. Etiology\| Diagnosis\| Treatment\| Key Points\| (See also Overview of Decreased Erythropoiesis.) Worldwide, the anemia of chronic disease is the second most common anemia. Early on, the red blood cells (RBCs) are normocytic; with time they may become microcytic. The major issue is that erythropoiesis is restricted due to inappropriate iron sequestration. Etiology of Anemia of Chronic Disease Anemia of chronic disease occurs as part of a chronic inflammatory disorder, most often chronic infection, an autoimmune disease (especially rheumatoid arthritis), kidney disease, heart failure, or cancer; however, the same process appears to begin acutely during virtually any infection or inflammation, including during trauma and critical illness or post-surgically (1). (See also Anemia of Renal Disease.) Three pathophysiologic mechanisms have been identified: Slightly shortened RBC survival, thought to be due to increased hemophagocytosis by macrophages, occurs in patients with inflammatory diseases in the acute setting. Erythropoiesis is impaired because of decreases in both erythropoietin (EPO) production and marrow responsiveness to EPO. Further, inflammatory cytokines can impair erythroid proliferation and differentiation via radical formation and/or induction of apoptosis. Iron metabolism is altered due to an increase in hepcidin, which inhibits iron absorption and recycling, leading to iron sequestration. Reticuloendothelial cells retain iron from senescent RBCs, making iron unavailable for hemoglobin (Hb) synthesis. There is thus a failure to compensate for the anemia with increased RBC production. Macrophage-derived cytokines (eg, interleukin-1-beta, interleukin-6, tumor necrosis factor-alpha, interferon-gamma) in patients with infections, inflammatory states, and cancer contribute to the decrease in EPO production and impaired iron availability by increased hepatic hepcidin synthesis. Etiology reference Weiss G, Ganz T, Goodnough LT. Anemia of inflammation.Blood. 2019;133(1):40-50. doi:10.1182/blood-2018-06-856500 Diagnosis of Anemia of Chronic Disease Symptoms and signs of the underlying disorder Complete blood count (CBC) and serum iron, ferritin, transferrin (or total iron binding capacity), and reticulocyte count Clinical findings in the anemia of chronic disease are usually those of the underlying disorder (infection, inflammation, cancer). The anemia of chronic disease should be suspected in patients with microcytic or normocytic anemia who also have chronic illness, infection, inflammation, or cancer. If anemia of chronic disease is suspected, serum iron, transferrin, reticulocyte count and serum ferritin are measured. Hb usually is > 8 g/dL (> 80 g/L) unless an additional mechanism contributes to anemia, such as concomitant iron deficiency (see table Differential Diagnosis of Microcytic Anemia Due to Decreased RBC Production) or iatrogenic phlebotomy. A serum ferritin level of < 100 ng/mL (< 224.7 pmol/L) in a patient with inflammation (< 200 ng/mL [< 449.4 pmol/L] in patients with chronic kidney disease) suggests that iron deficiency may be superimposed on anemia of chronic disease, because serum ferritin is usually elevated as an acute-phase reactant. If the diagnosis is not clear following standard iron studies, soluble transferrin receptor (sTFR) and sTFR-ferritin index (elevated in iron deficiency) and/or reticulocyte hemoglobin content (ret-He), which is low in iron deficiency, may help identify concomitant iron deficiency and anemia of chronic disease, although these test results potentially may also be subject to confounding effects of inflammation or analytical variables, such as differences in measurement methods and reference ranges that may not be standardized between labs (1). In patients with possible inflammation and in whom other causes of anemia have been excluded, erythrocyte sedimentation rate (ESR) and/or C-reactive protein (CRP) may be obtained because these test results are nonspecific markers of inflammation. Diagnosis reference Günther F, Straub RH, Hartung W, et al. Association of Serum Soluble Transferrin Receptor Concentration With Markers of Inflammation: Analysis of 1001 Patients From a Tertiary Rheumatology Center.J Rheumatol. 2024;51(3):291-296. Published 2024 Mar 1. doi:10.3899/jrheum.2023-0654 Treatment of Anemia of Chronic Disease Treatment of underlying disorder Sometimes iron supplements in patients with concomitant iron deficiency Treatment of the anemia of chronic disease requires treating the underlying disorder. Because the anemia is generally mild, transfusions usually are not required. Iron supplementation may be helpful because iron deficiency can occur in patients with anemia of chronic disease, and iron studies are often difficult to interpret when these conditions coexist. However, in patients without suspected concomitant iron deficiency and in patients with acute, uncontrolled infection, iron supplementation is generally avoided. Recombinant human erythropoietin or erythropoiesis-stimulating agents (ESAs) may be considered in patients with end-stage or chronic kidney disease, selected patients with chemotherapy-induced anemia, and some patients before elective surgery. Key Points Almost any chronic infection, inflammation, or cancer can cause anemia; hemoglobin usually is > 8 g/dL (> 80 g/L) unless an additional mechanism contributes. Multiple factors are involved, including shortened red blood cell survival, impaired erythropoiesis, and impaired iron availability. Anemia is initially normocytic and then can become microcytic. Serum iron and transferrin are typically decreased, while ferritin is normal to increased. Treat the underlying disorder. Test your Knowledge Take a quiz on this topic Copyright © 2025 Merck & Co., Inc., Rahway, NJ, USA and its affiliates. All rights reserved. About us Disclaimer Cookie Preferences Copyright© 2025 Merck & Co., Inc., Rahway, NJ, USA and its affiliates. All rights reserved. Find In Topic This Site Uses Cookies and Your Privacy Choice Is Important to Us We suggest you choose Customize my Settings to make your individualized choices. Accept Cookies means that you are choosing to accept third-party Cookies and that you understand this choice. See our Privacy Policy Customize my Settings Reject Cookies Accept Cookies
2562	https://brainly.com/question/63281455	[FREE] Simplify. $\sqrt{12}$ - brainly.com 6 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +34,6k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +28,8k Ace exams faster, with practice that adapts to you Practice Worksheets +5,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified Simplify. 12 1 See answer Explain with Learning Companion NEW Asked by nunya346 • 08/09/2025 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 1183120948 people 1183M 0.0 0 Upload your school material for a more relevant answer Find the prime factorization of 12: 12=2 2⋅3. Rewrite the square root using the prime factorization: 12=2 2⋅3. Separate the square root: 2 2⋅3=2 2⋅3. Simplify: 12=2 3. The simplified expression is 2 3. Explanation Understanding the Problem We are asked to simplify the expression 12. This means we want to find the simplest form of the square root of 12. To do this, we will find the prime factorization of 12 and simplify the square root. Prime Factorization First, we find the prime factorization of 12. We can write 12 as a product of its prime factors: 12=2⋅2⋅3=2 2⋅3. Rewriting the Square Root Now we rewrite the square root using the prime factorization: 12=2 2⋅3. Separating the Square Root We use the property a⋅b=a⋅b to separate the square root: 2 2⋅3=2 2⋅3. Simplifying the Perfect Square We simplify the square root of the perfect square: 2 2=2. Final Result Finally, we write the simplified expression: 2 3. Examples Square roots are used in many areas of math and science. For example, when calculating the distance between two points in a plane, we use the distance formula, which involves square roots. If we have two points (1, 2) and (4, 6), the distance between them is (4−1)2+(6−2)2=3 2+4 2=9+16=25=5. Simplifying square roots helps us to express these distances in their simplest form. Answered by GinnyAnswer •8M answers•1.2B people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) 0.0 0 Upload your school material for a more relevant answer The simplified form of 12 is 2 3. This was achieved by finding the prime factorization of 12 and using properties of square roots to separate and simplify the expression. Following the steps led us to express it in its simplest form. Explanation To simplify 12, we will follow these steps: Find the Prime Factorization The first step is to break down 12 into its prime factors. We have: 12=2×2×3=2 2×3. Rewrite the Square Root Using the prime factorization, we can express the square root as: 12=2 2×3. Separate the Square Root We can use the property of square roots that states a×b=a×b. Therefore, we can write: 2 2×3=2 2×3. Simplify the Perfect Square Now we can simplify 2 2. Since the square root of a square is the number itself, we have: 2 2=2. Final Result Putting it all together, we get: 12=2 3. Thus, the simplified form of 12 is 2 3. Examples & Evidence For further understanding, consider another example: to simplify 20, we find its prime factorization is 20=2 2×5, thus 20=2 2×5=2 5. The methods used to simplify the square root demonstrated are based on established mathematical principles regarding square roots and prime factorization, which are regularly taught in school curricula. Thanks 0 0.0 (0 votes) Advertisement nunya346 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer Simplify: [tex]\sqrt{12}[/tex] Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. New questions in Mathematics Factor completely n 2−5 n+4 A. (n+4)(n+1) B. (n−3)(n+4) C. (n−4)(n−1) D. $(n-4)(n+1) The manager of a juice bar polled 56 random customers to get feedback on a new smoothie. The results are shown in the frequency table. \| { 2 - 4 } \| Like \| Dislike \| Total \| :---: :---: \| \| Adult \| 23 \| 6 \| 29 \| \| Child \| 18 \| 9 \| 27 \| \| Total \| 41 \| 15 \| 56 \| Complete the statements using the table. The number of children who disliked the smoothie is □ The total number of children is □ The conditional relative frequency that a customer disliked the new smoothie, given that the person is a child is approximately □ Solve the equation x 3−13 x 2+47 x−35=0 given that 1 is a zero of f(x)=x 3−13 x 2+47 x−35. Graph the function f by starting with the graph of y=x 2 and using transformations (shifting, compressing, stretching, and/or reflecting). [Hint: If necessary, write f in the form f(x)=a(x−h)2+k.]f(x)=x 2+10 x+22 Which transformations are needed to graph the function f(x)=x 2+10 x+22 ? A. The graph of y=x 2 should be horizontally shifted to the left by 3 units and shifted vertically up by 5 units. B. The graph of y=x 2 should be horizontally shifted to the right by 3 units and shifted vertically down by 5 units. C. The graph of y=x 2 should be horizontally shifted to the left by 5 units and shifted vertically down by 3 units. D. The graph of y=x 2 should be horizontally shifted to the right by 5 units and shifted vertically up by 3 units. Compute the derivative of the function y=(x 2+x+5)6 50 using the Chain Rule. d x d y= Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
2563	https://math.stackexchange.com/questions/1548329/finding-mass-of-a-sphere-given-density-1-rho2-and-radius-1	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Finding mass of a sphere given density = $1-\rho^2$ and radius = 1 Ask Question Asked Modified 9 years, 8 months ago Viewed 18k times 0 $\begingroup$ I'm currently working on what should be a relatively simple problem but I'm not sure if I'm right in my answer. This is for Calc3 homework that I'm just trying to get a grasp on the concepts before the final. So the question is a sphere of radius 1 has a density of $1-\rho^2$ at a point distance $\rho$ from the center. How can I get the total mass of the sphere given this information? I know $Mass = Volume \cdot Density$ and $Volume = r^3(4\pi/3)$ since $r = 1 $ $Volume = \frac{4\pi}{3}$ So that leaves me with $m_{total} = \frac{4\pi}{3(1-\rho^2)}$ What should I be doing from here? Or am I even where I should be at this point in the problem? calculus integration Share edited Nov 27, 2015 at 7:27 Ben Longo 1,08666 silver badges1616 bronze badges asked Nov 27, 2015 at 7:21 J0hnJ0hn 12711 gold badge11 silver badge66 bronze badges $\endgroup$ 1 $\begingroup$ Consider the mass of a small spherical shell. $\endgroup$ Element118 – Element118 2015-11-27 07:24:34 +00:00 Commented Nov 27, 2015 at 7:24 Add a comment \| 2 Answers 2 Reset to default 2 $\begingroup$ Your formula of mass = volume $\times$ density needs to be a bit modified here since the density is non-uniform. Every bit of volume of the sphere has a different density so you have to integrate it appropriately as follows: $$M=\int_0^1 \text{density}\cdot dV = \int_0^1 (1-r^2) \cdot dV$$ and we know that $V=\frac{4}{3}\pi r^3$ where $r$ is the radius of the sphere So $dV=4\pi r^2dr$ Hence $$M=\int_0^1 (1-r^2) \cdot 4\pi r^2dr$$ $$=4\pi \int_0^1(r^2-r^4)dr$$ $$=4\pi(\frac{1}{3}-\frac{1}{5})=\frac{8\pi}{15}$$ Share edited Jan 13, 2016 at 23:05 ws04 1766 bronze badges answered Nov 27, 2015 at 7:35 SchrodingersCatSchrodingersCat 24.9k77 gold badges4747 silver badges9090 bronze badges $\endgroup$ 4 $\begingroup$ Very helpful, thank you. $\endgroup$ J0hn – J0hn 2015-11-27 07:37:13 +00:00 Commented Nov 27, 2015 at 7:37 $\begingroup$ You're welcome, $\endgroup$ SchrodingersCat – SchrodingersCat 2015-11-27 07:37:31 +00:00 Commented Nov 27, 2015 at 7:37 $\begingroup$ schrodinger's cat shouldn't we do $ 2\int_0^{2\pi}\int_0^1\int_0^{\sqrt{1-x^2-y^2}}(1-r^2)rdrd\theta$. Using the cylindrical cooedinates $\endgroup$ Upstart – Upstart 2017-07-07 22:59:19 +00:00 Commented Jul 7, 2017 at 22:59 $\begingroup$ @Upstart Why use cylindrical coordinates? All of a sudden while working with a sphere? $\endgroup$ SchrodingersCat – SchrodingersCat 2017-07-08 04:31:26 +00:00 Commented Jul 8, 2017 at 4:31 Add a comment \| 0 $\begingroup$ You are wrong. The density is not uniform so you need to integrate the following: $$\int_0^r(4\pi\rho^2)(1-\rho^2)d\rho$$ Share answered Nov 27, 2015 at 7:25 cr001cr001 12.8k1919 silver badges2929 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus integration See similar questions with these tags. 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2564	https://centlib.mui.ac.ir/sites/centlib/files/amin-lib/ebooks/Atlas%20of%20Anatomy.pdf	To access the additional media content available with this e-book via Thieme MedOne, please use the code and follow the instructions provided at the back of the e-book. Based on the work of Michael Schuenke, MD, PhD Institute of Anatomy Christian Albrechts University Kiel Kiel, Germany Erik Schulte, MD Department of Functional and Clinical Anatomy University Medicine Johannes Gutenberg University Mainz, Germany Udo Schumacher, MD, FRCPath, CBiol, FSB, DSc Institute of Anatomy and Experimental Morphology Center for Experimental Medicine University Cancer Center University Medical Center Hamburg-Eppendorf Hamburg, Germany Thieme New York · Stuttgart · Delhi · Rio de Janeiro Atlas of Anatomy Fourth Edition Edited by Anne M. Gilroy, MA Professor Emeritus Department of Radiology University of Massachusetts Medical School Worcester, Massachusetts Brian R. MacPherson, PhD Professor and Vice Chair Department of Neuroscience University of Kentucky College of Medicine Lexington, Kentucky Jamie C. Wikenheiser, PhD Associate Professor Department of Anatomy and Neurobiology UC Irvine School of Medicine Irvine, California Illustrations by Markus Voll Karl Wesker 2113 illustrations Illustrators: Markus Voll and Karl Wesker Development Editor: Judith Tomat Production Editor: Barbara Chernow Compositor: Carol Pierson, Chernow Editorial Services, Inc. Library of Congress Cataloging-in-Publication Data Names: Gilroy, Anne M., editor. \| MacPherson, Brian R., editor. \| Wikenheiser, Jamie C., editor. \| Voll, Markus M., illustrator. \| Wesker, Karl, illustrator. \| Schünke, Michael. Thieme atlas of anatomy. Title: Atlas of anatomy / edited by Anne M. Gilroy, Brian R. MacPherson, Jamie C. Wikenheiser ; based on the work of Michael Schuenke, Erik Schulte, Udo Schumacher ; illustrations by Markus Voll, Karl Wesker. Other titles: Atlas of anatomy (Gilroy) Description: Fourth edition. \| New York : Thieme, \| Includes index. \| Summary: “An updated atlas that provides a clear, accurate, and fully illustrated guide to human anatomy”— Provided by publisher. Identifiers: LCCN 2019058797 (print) \| LCCN 2019058798 (ebook) \| ISBN 9781684202034 (paperback) \| ISBN 9781684202041 (ebook) Subjects: MESH: Anatomy \| Atlas Classification: LCC QM25 (print) \| LCC QM25 (ebook) \| NLM QS 17 \| DDC 611.0022/3—dc23 LC record available at LC ebook record available at Important note: Medicine is an ever-changing science undergoing continual development. Research and clinical experience are continually expanding our knowledge, in particular our knowledge of proper treat-ment and drug therapy. 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Ltd. 5 4 3 2 1 ISBN 978-1-68420-203-4 Also available as an e-book: eISBN 978-1-68420-204-1 To the thousands of former students who have populated every specialty and migrated to every corner of this country, while dedicating their careers to making life better for thousands more. I am inspired by their empathy and kindness, and grateful to have been a small part of their journey. And as always, to Colin and Bryan. Anne M. Gilroy To my friend and mentor Dr. Ken McFadden, who was responsible for my early training in gross anatomy and was a role model for success in teaching. I deeply appreciate the feedback I have received over the past 40 plus years from the thousands of students I have taught and who have made me an even better teacher. However, none of the success I have enjoyed in my life would have been possible without the constant support, assistance, and encouragement of my late wife, Cynthia Long. Brian R. MacPherson To my wife Jen and my son Quinn. Jamie C. Wikenheiser Dedications Table of Contents 1 Surface Anatomy Surface Anatomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Bones, Ligaments & Joints Vertebral Column: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Vertebral Column: Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cervical Vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Thoracic & Lumbar Vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Sacrum & Coccyx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Intervertebral Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Joints of the Vertebral Column: Overview . . . . . . . . . . . . . . . . 16 Joints of the Vertebral Column: Craniovertebral Region . . . . . 18 Vertebral Ligaments: Overview & Cervical Spine . . . . . . . . . . 20 Vertebral Ligaments: Thoracolumbar Spine . . . . . . . . . . . . . . 22 3 Muscles Muscles of the Back: Overview . . . . . . . . . . . . . . . . . . . . . . . . . 24 Intrinsic Muscles of the Cervical Spine . . . . . . . . . . . . . . . . . . . 26 Intrinsic Muscles of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Neurovasculature Arteries & Veins of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Nerves of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Spinal Cord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Spinal Cord Segments & Spinal Nerves . . . . . . . . . . . . . . . . . . 42 Arteries & Veins of the Spinal Cord . . . . . . . . . . . . . . . . . . . . . 44 Neurovascular Topography of the Back . . . . . . . . . . . . . . . . . . 46 5 Sectional & Radiographic Anatomy Radiographic Anatomy of the Back (I) . . . . . . . . . . . . . . . . . . . 48 Radiographic Anatomy of the Back (II). . . . . . . . . . . . . . . . . . . 50 Back Thorax Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Preface to the First Edition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 6 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7 Thoracic Wall Thoracic Skeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Sternum & Ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Joints of the Thoracic Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Thoracic Wall Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Neurovasculature of the Diaphragm . . . . . . . . . . . . . . . . . . . . 66 Arteries & Veins of the Thoracic Wall . . . . . . . . . . . . . . . . . . . . 68 Nerves of the Thoracic Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Neurovascular Topography of the Thoracic Wall . . . . . . . . . . . 72 Female Breast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Lymphatics of the Female Breast . . . . . . . . . . . . . . . . . . . . . . . 76 8 Thoracic Cavity Divisions of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . 78 Arteries of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 80 Veins of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Lymphatics of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . 84 Nerves of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 86 vii Table of Contents 12 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13 Abdominal Wall Bony Framework for the Abdominal Wall . . . . . . . . . . . . . . . 142 Muscles of the Anterolateral Abdominal Wall . . . . . . . . . . . . 144 9 Mediastinum Mediastinum: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Mediastinum: Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Heart: Functions & Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pericardium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Heart: Surfaces & Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Heart: Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Arteries & Veins of the Heart . . . . . . . . . . . . . . . . . . . . . . . . . 100 Conduction & Innervation of the Heart . . . . . . . . . . . . . . . . . 102 Pre- & Postnatal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Esophagus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Neurovasculature of the Esophagus . . . . . . . . . . . . . . . . . . . 108 Lymphatics of the Mediastinum . . . . . . . . . . . . . . . . . . . . . . . 110 10 Pulmonary Cavities Pulmonary Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Pleura: Subdivisions, Recesses & Innervation . . . . . . . . . . . . 114 Lungs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Bronchopulmonary Segments of the Lungs . . . . . . . . . . . . . 118 Trachea & Bronchial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Respiratory Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Pulmonary Arteries & Veins . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Neurovasculature of the Tracheobronchial Tree . . . . . . . . . . 126 Lymphatics of the Pleural Cavity . . . . . . . . . . . . . . . . . . . . . . 128 11 Sectional & Radiographic Anatomy Sectional Anatomy of the Thorax . . . . . . . . . . . . . . . . . . . . . . 130 Radiographic Anatomy of the Thorax (I). . . . . . . . . . . . . . . . . 132 Radiographic Anatomy of the Thorax (II). . . . . . . . . . . . . . . . 134 Radiographic Anatomy of the Thorax (III). . . . . . . . . . . . . . . . 136 Rectus Sheath & Posterior Abdominal Wall . . . . . . . . . . . . . . 146 Abdominal Wall Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . 148 Inguinal Region & Canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Inguinal Region & Inguinal Hernias. . . . . . . . . . . . . . . . . . . . . 152 Scrotum & Spermatic Cord . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 14 Abdominal Cavity & Spaces Divisions of the Abdominopelvic Cavity . . . . . . . . . . . . . . . . 156 Peritoneum, Mesenteries & Omenta . . . . . . . . . . . . . . . . . . . 158 Mesenteries & Peritoneal Recesses . . . . . . . . . . . . . . . . . . . . 160 Lesser Omentum & Omental Bursa . . . . . . . . . . . . . . . . . . . . 162 Mesenteries & Posterior Abdominal Wall. . . . . . . . . . . . . . . . 164 15 Internal Organs Stomach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Duodenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Jejunum & Ileum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Cecum, Appendix & Colon . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Liver: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Liver: Lobes & Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Gallbladder & Bile Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Pancreas & Spleen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Kidneys & Suprarenal Glands (I) . . . . . . . . . . . . . . . . . . . . . . . 182 Kidneys & Suprarenal Glands (II) . . . . . . . . . . . . . . . . . . . . . . 184 16 Neurovasculature Arteries of the Abdominal Wall & Organs . . . . . . . . . . . . . . . 186 Abdominal Aorta & Renal Arteries . . . . . . . . . . . . . . . . . . . . . 188 Celiac Trunk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Superior & Inferior Mesenteric Arteries . . . . . . . . . . . . . . . . . 192 Veins of the Abdominal Wall & Organs . . . . . . . . . . . . . . . . . 194 Inferior Vena Cava & Renal Veins . . . . . . . . . . . . . . . . . . . . . . 196 Portal Vein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Superior & Inferior Mesenteric Veins . . . . . . . . . . . . . . . . . . . 200 Lymphatics of the Abdominal Wall & Organs . . . . . . . . . . . . 202 Lymph Nodes of the Posterior Abdominal Wall . . . . . . . . . . . 204 Lymph Nodes of the Supracolic Organs . . . . . . . . . . . . . . . . . 206 Lymph Nodes of the Infracolic Organs . . . . . . . . . . . . . . . . . . 208 Nerves of the Abdominal Wall . . . . . . . . . . . . . . . . . . . . . . . . 210 Autonomic Innervation: Overview . . . . . . . . . . . . . . . . . . . . . 212 Autonomic Innervation & Referred Pain. . . . . . . . . . . . . . . . . 214 Innervation of the Foregut & Urinary Organs . . . . . . . . . . . . 216 Innervation of the Intestines . . . . . . . . . . . . . . . . . . . . . . . . . 218 17 Sectional & Radiographic Anatomy Sectional Anatomy of the Abdomen . . . . . . . . . . . . . . . . . . . 220 Radiographic Anatomy of the Abdomen (I). . . . . . . . . . . . . . 222 Radiographic Anatomy of the Abdomen (II). . . . . . . . . . . . . . 224 Abdomen viii Table of Contents 18 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 19 Bones, Ligaments & Muscles Pelvic Girdle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Female & Male Pelvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Female & Male Pelvic Measurements. . . . . . . . . . . . . . . . . . . 234 Pelvic Ligaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Muscles of the Pelvic Floor & Perineum . . . . . . . . . . . . . . . . . 238 Pelvic Floor & Perineal Muscle Facts. . . . . . . . . . . . . . . . . . . . 240 20 Spaces Contents of the Pelvis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Peritoneal Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Pelvis & Perineum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 21 Internal Organs Rectum & Anal Canal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Ureters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Urinary Bladder & Urethra . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Overview of the Genital Organs. . . . . . . . . . . . . . . . . . . . . . . 254 Uterus & Ovaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Ligaments & Fascia of the Deep Pelvis . . . . . . . . . . . . . . . . . . 258 Vagina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Female External Genitalia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Penis, Testis & Epididymis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Male Accessory Sex Glands. . . . . . . . . . . . . . . . . . . . . . . . . . . 266 22 Neurovasculature Overview of the Blood Supply to Pelvic Organs & Wall. . . . . 268 Arteries & Veins of the Male Pelvis. . . . . . . . . . . . . . . . . . . . . 270 Arteries & Veins of the Female Pelvis. . . . . . . . . . . . . . . . . . . 272 Arteries & Veins of the Rectum & External Genitalia. . . . . . . 274 Lymphatics of the Pelvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Lymph Nodes of the Genitalia. . . . . . . . . . . . . . . . . . . . . . . . . 278 Autonomic Innervation of the Genital Organs. . . . . . . . . . . . 280 Autonomic Innervation of the Urinary Organs & Rectum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Neurovasculature of the Male & Female Perineum . . . . . . . . 284 23 Sectional & Radiographic Anatomy Sectional Anatomy of the Pelvis & Perineum. . . . . . . . . . . . . 286 Radiographic Anatomy of the Female Pelvis. . . . . . . . . . . . . 288 Radiographic Anatomy of the Male Pelvis . . . . . . . . . . . . . . . 290 Pelvis & Perineum Upper Limb 24 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 25 Shoulder & Arm Bones of the Upper Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Clavicle & Scapula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Humerus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Joints of the Shoulder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Joints of the Shoulder: Glenohumeral Joint . . . . . . . . . . . . . . 304 Subacromial Space & Bursae . . . . . . . . . . . . . . . . . . . . . . . . . 306 Anterior Muscles of the Shoulder & Arm (I) . . . . . . . . . . . . . . 308 Anterior Muscles of the Shoulder & Arm (II) . . . . . . . . . . . . . 310 Posterior Muscles of the Shoulder & Arm (I) . . . . . . . . . . . . . 312 Posterior Muscles of the Shoulder & Arm (II) . . . . . . . . . . . . . 314 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Muscle Facts (IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 26 Elbow & Forearm Radius & Ulna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Elbow Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Ligaments of the Elbow Joint . . . . . . . . . . . . . . . . . . . . . . . . . 328 Radioulnar Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Muscles of the Forearm: Anterior Compartment . . . . . . . . . 332 Muscles of the Forearm: Posterior Compartment . . . . . . . . . 334 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 27 Wrist & Hand Bones of the Wrist & Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Carpal Bones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Joints of the Wrist & Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 ix Table of Contents Ligaments of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Ligaments & Compartments of the Wrist . . . . . . . . . . . . . . . 350 Ligaments of the Fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Muscles of the Hand: Superficial & Middle Layers . . . . . . . . . 354 Muscles of the Hand: Middle & Deep Layers . . . . . . . . . . . . . 356 Dorsum of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 28 Neurovasculature Arteries of the Upper Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Veins & Lymphatics of the Upper Limb . . . . . . . . . . . . . . . . . 366 Nerves of the Upper Limb: Brachial Plexus . . . . . . . . . . . . . . 368 Supraclavicular Branches & Posterior Cord . . . . . . . . . . . . . . 370 Posterior Cord: Axillary & Radial Nerves . . . . . . . . . . . . . . . . 372 Medial & Lateral Cords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Median & Ulnar Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Superficial Veins & Nerves of the Upper Limb . . . . . . . . . . . . 378 Posterior Shoulder & Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Anterior Shoulder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Axilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Anterior Arm & Cubital Region . . . . . . . . . . . . . . . . . . . . . . . . 386 Anterior & Posterior Forearm . . . . . . . . . . . . . . . . . . . . . . . . . 388 Carpal Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Palm of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Dorsum of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 29 Sectional & Radiographic Anatomy Sectional Anatomy of the Upper Limb. . . . . . . . . . . . . . . . . . 396 Radiographic Anatomy of the Upper Limb (I). . . . . . . . . . . . . 398 Radiographic Anatomy of the Upper Limb (II). . . . . . . . . . . . 400 Radiographic Anatomy of the Upper Limb (III). . . . . . . . . . . . 402 Radiographic Anatomy of the Upper Limb (IV) . . . . . . . . . . . 404 Hip Joint: Ligaments & Capsule . . . . . . . . . . . . . . . . . . . . . . . 416 Anterior Muscles of the Hip, Thigh & Gluteal Region (I) . . . . 418 Anterior Muscles of the Hip, Thigh & Gluteal Region (II) . . . 420 Posterior Muscles of the Hip, Thigh & Gluteal Region (I) . . . 422 Posterior Muscles of the Hip, Thigh & Gluteal Region (II) . . . 424 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 32 Knee & Leg Tibia & Fibula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Knee Joint: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Knee Joint: Capsule, Ligaments & Bursae . . . . . . . . . . . . . . . 436 Knee Joint: Ligaments & Menisci . . . . . . . . . . . . . . . . . . . . . . 438 Cruciate Ligaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Knee Joint Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Muscles of the Leg: Anterior & Lateral Compartments . . . . . 444 Muscles of the Leg: Posterior Compartment . . . . . . . . . . . . . 446 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 33 Ankle & Foot Bones of the Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Joints of the Foot (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Joints of the Foot (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Joints of the Foot (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Ligaments of the Ankle & Foot . . . . . . . . . . . . . . . . . . . . . . . . 460 Plantar Vault & Arches of the Foot . . . . . . . . . . . . . . . . . . . . . 462 Muscles of the Sole of the Foot . . . . . . . . . . . . . . . . . . . . . . . 464 Muscles & Tendon Sheaths of the Foot . . . . . . . . . . . . . . . . . 466 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 34 Neurovasculature Arteries of the Lower Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Veins & Lymphatics of the Lower Limb . . . . . . . . . . . . . . . . . 474 Lumbosacral Plexus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Nerves of the Lumbar Plexus . . . . . . . . . . . . . . . . . . . . . . . . . 478 Nerves of the Lumbar Plexus: Obturator & Femoral Nerves . 480 Nerves of the Sacral Plexus . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Nerves of the Sacral Plexus: Sciatic Nerve . . . . . . . . . . . . . . . 484 Superficial Nerves & Veins of the Lower Limb . . . . . . . . . . . . 486 Topography of the Inguinal Region . . . . . . . . . . . . . . . . . . . . 488 Topography of the Gluteal Region . . . . . . . . . . . . . . . . . . . . . 490 Topography of the Anterior, Medial & Posterior Thigh . . . . . 492 Topography of the Posterior Compartment of the Leg & Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Topography of the Lateral & Anterior Compartments of the Leg & Dorsum of the Foot . . . . . . . . . . . . . . . . . . . . 496 Topography of the Sole of the Foot . . . . . . . . . . . . . . . . . . . . 498 35 Sectional & Radiographic Anatomy Sectional Anatomy of the Lower Limb . . . . . . . . . . . . . . . . . . 500 Radiographic Anatomy of the Lower Limb (I) . . . . . . . . . . . . 502 Lower Limb 30 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 31 Hip & Thigh Bones of the Lower Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Femur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Hip Joint: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 x Table of Contents 36 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 37 Neck Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 Arteries & Veins of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . 520 Lymphatics of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Innervation of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Larynx: Cartilage & Structure . . . . . . . . . . . . . . . . . . . . . . . . . 526 Larynx: Muscles & Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Neurovasculature of the Larynx, Thyroid & Parathyroids . . . 530 Topography of the Neck: Regions & Fascia . . . . . . . . . . . . . . 532 Topography of the Anterior Cervical Region . . . . . . . . . . . . . 534 Topography of the Anterior & Lateral Cervical Regions . . . . 536 Topography of the Lateral Cervical Region . . . . . . . . . . . . . . 538 Topography of the Posterior Cervical Region . . . . . . . . . . . . 540 38 Bones of the Head Anterior & Lateral Skull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Posterior Skull & Calvaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 Base of the Skull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Neurovascular Pathways Exiting or Entering the Cranial Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Ethmoid & Sphenoid Bones . . . . . . . . . . . . . . . . . . . . . . . . . . 550 39 Muscles of the Skull & Face Muscles of Facial Expression & of Mastication . . . . . . . . . . . . 552 Muscle Origins & Insertions on the Skull . . . . . . . . . . . . . . . . 554 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 40 Cranial Nerves Cranial Nerves: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 CN I & II: Olfactory & Optic Nerves . . . . . . . . . . . . . . . . . . . . 562 CN III, IV & VI: Oculomotor, Trochlear & Abducent Nerves . . . 564 CN V: Trigeminal Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 CN VII: Facial Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 CN VIII: Vestibulocochlear Nerve . . . . . . . . . . . . . . . . . . . . . . 570 CN IX: Glossopharyngeal Nerve . . . . . . . . . . . . . . . . . . . . . . . 572 CN X: Vagus Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 CN XI & XII: Accessory & Hypoglossal Nerves . . . . . . . . . . . . 576 Autonomic Innervation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 41 Neurovasculature of the Skull & Face Innervation of the Face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 Arteries of the Head & Neck . . . . . . . . . . . . . . . . . . . . . . . . . . 582 External Carotid Artery: Anterior, Medial & Posterior Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 External Carotid Artery: Terminal Branches . . . . . . . . . . . . . . 586 Veins of the Head & Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 Meninges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 Dural Sinuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Topography of the Superficial Face . . . . . . . . . . . . . . . . . . . . 594 Topography of the Parotid Region & Temporal Fossa . . . . . . 596 Topography of the Infratemporal Fossa . . . . . . . . . . . . . . . . . 598 Neurovasculature of the Infratemporal Fossa . . . . . . . . . . . . 600 42 Orbit & Eye Bones of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Muscles of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 Neurovasculature of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . 606 Topography of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Orbit & Eyelid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Eyeball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Cornea, Iris & Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 43 Nasal Cavity & Nose Bones of the Nasal Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Paranasal Air Sinuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Neurovasculature of the Nasal Cavity . . . . . . . . . . . . . . . . . . 620 Pterygopalatine Fossa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 44 Temporal Bone & Ear Temporal Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 External Ear & Auditory Canal . . . . . . . . . . . . . . . . . . . . . . . . . 626 Middle Ear: Tympanic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 628 Middle Ear: Ossicular Chain & Tympanic Membrane . . . . . . . 630 Arteries of the Middle Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 Inner Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 45 Oral Cavity & Pharynx Bones of the Oral Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Temporomandibular Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 Oral Cavity Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Innervation of the Oral Cavity . . . . . . . . . . . . . . . . . . . . . . . . 644 Tongue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Topography of the Oral Cavity & Salivary Glands . . . . . . . . . 648 Head & Neck Radiographic Anatomy of the Lower Limb (II) . . . . . . . . . . . . 504 Radiographic Anatomy of the Lower Limb (III). . . . . . . . . . . . 506 Radiographic Anatomy of the Lower Limb (IV). . . . . . . . . . . . 508 xi Table of Contents Brain & Nervous System 47 Brain Nervous System: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 Nervous System: Development . . . . . . . . . . . . . . . . . . . . . . . 676 Brain, Macroscopic Organization . . . . . . . . . . . . . . . . . . . . . . 678 Diencephalon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 Brainstem & Cerebellum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 Ventricles & CSF Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 48 Blood Vessels of the Brain Veins & Venous Sinuses of the Brain . . . . . . . . . . . . . . . . . . . 686 Arteries of the Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 49 Functional Systems Anatomy & Organization of the Spinal Cord. . . . . . . . . . . . . . 690 Sensory & Motor Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 50 Autonomic Nervous System Autonomic Nervous System (I): Overview . . . . . . . . . . . . . . . 694 Autonomic Nervous System (II) . . . . . . . . . . . . . . . . . . . . . . . 696 51 Sectional & Radiographic Anatomy Sectional Anatomy of the Nervous System . . . . . . . . . . . . . . 698 Radiographic Anatomy of the Nervous System. . . . . . . . . . . 700 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 Tonsils & Pharynx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 Pharyngeal Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Neurovasculature of the Pharynx . . . . . . . . . . . . . . . . . . . . . . 654 46 Sectional & Radiographic Anatomy Sectional Anatomy of the Head & Neck (I). . . . . . . . . . . . . . . 656 Sectional Anatomy of the Head & Neck (II) . . . . . . . . . . . . . . 658 Sectional Anatomy of the Head & Neck (III). . . . . . . . . . . . . . 660 Sectional Anatomy of the Head & Neck (IV). . . . . . . . . . . . . . 662 Sectional Anatomy of the Head & Neck (V). . . . . . . . . . . . . . 664 Radiographic Anatomy of the Head & Neck (I). . . . . . . . . . . . 666 Radiographic Anatomy of the Head & Neck (II). . . . . . . . . . . 668 Radiographic Anatomy of the Head & Neck (III). . . . . . . . . . . 670 xii Table of Contents Acknowledgments We would like to thank the authors of the original award-winning Thieme Atlas of Anatomy three-volume series, Michael Schuenke, Erik Schulte, and Udo Schumacher, and the illustrators, Karl Wesker and Marcus Voll, for their work over the course of many years. We thank the many instructors, students, and translators for our non-English versions, who have taken the time to point out to us what we have done well and brought to our attention errors, ambiguities, and new information, or have suggested how we could present a topic more effectively. This input, combined with our experience teaching with the Atlas, have guided our work on this edition. We cordially thank the reviewers of the third edition, especially those who provided in-depth feedback: • Jennifer Brueckner-Collins, PhD University of Louisville School of Medicine Louisville, Kentucky • Jennifer Carr, PhD Salem State University Salem, Massachussetts • C. Cem Denk, MD, PhD Hacettepe University Faculty of Medicine Ankara, Turkey • Gary J. Farkas, PhD University of California, San Francisco School of Medicine San Francisco, California • Derek Harmon, PhD University of California, San Francisco School of Medicine San Francisco, California • Lindsey Kent (Class of 2020) West Virginia School of Osteopathic Medicine Lewisburg, West Virginia • Barbie Klein, PhD University of California, San Francisco School of Medicine San Francisco, California • Nancy Lin (Class of 2021) CUNY School of Medicine New York, New York • Luís Otávio Carvalho de Moraes, PhD Federal University of São Paulo São Paulo, Brazil • F. Baker Mills IV, MS (Class of 2021) University of South Carolina School of Medicine Columbia, South Carolina • Stephen M. Novak, MD, JD Harvard University Cambridge, Massachusetts • Joy R. Patel (Class of 2021) NYIT College of Osteopathic Medicine Old Westbury, New York • Paisley Lynae Pauli, MHA (Class of 2021) University of the Incarnate Word School of Osteopathic Medicine San Antonio, Texas • Guenevere Rae, MS, PhD Tulane University School of Medicine New Orleans, Louisiana • Sherese Richards, MD The College of St. Scholastica Duluth, Minnesota • William J. Swartz, PhD LSU Health Sciences Center New Orleans, Louisiana Foreword This Atlas of Anatomy, in my opinion, is the finest single-volume atlas of human anatomy that has ever been created. Two factors make it so: the images and the way they have been organized. The artists, Markus Voll and Karl Wesker, have created a new standard of excellence in anatomical art. Their graceful use of transparency and their sensitive representation of light and shadow give the reader an accurate three-dimensional understanding of every structure. The authors have organized the images so that they give just the flow of information a student needs to build up a clear mental image of the human body. Each two-page spread is a self-contained lesson that un-obtrusively shows the hand of an experienced and thoughtful teacher. I wish I could have held this book in my hands when I was a student; I envy any student who does so now. Robert D. Acland, 1941–2016 Louisville, Kentucky December 2015 Preface In this new fourth edition of the Atlas of Anatomy, we are proud to offer what we believe is our best effort at presenting a clear and accurate story of human anatomy. A significant part of this effort is the addition of our newest co-author, Dr. Jamie C. Wikenheiser from the University of Cali-fornia, Irvine. Jamie’s love of anatomy, attention to detail, and proud background in teaching excellence in anatomy at all student levels makes him a highly qualified addition to the editorship of the Atlas that will ensure its continued development. As with previous editions, we have made every attempt to respond to the requests, comments, and critiques of our world-wide users. As al-ways, we recognize that anatomy is a changing science. As concepts and terminology evolve, we feel a responsibility to pass this on and keep these aspects of the Atlas updated. Thus, our initial task for this edition was to update and further clarify the material already present in the Atlas. Among these modifications was a major revision of the many au-tonomic innervation wiring schematics. These are now uniformly de-signed to clearly differentiate between sympathetic and parasympathetic components and pre-and post-ganglionic fibers. We improved many tables by reorganizing and rewording the content and enlarging labels. Sectional and radiographic chapters in each unit, established in the third edition, have been expanded with more than forty additional MR and CT images, now accompanied, as are all sectional images throughout the Atlas, by new simplified navigators. Another focus of this edition was to provide more written and schematic- based information that addresses complex anatomic concepts. This in-cludes new schematics that complement other images, expanded legends that accompany images, and most notably, the addition of almost thirty new clinical boxes (most with illustrations) in every unit. These focus on function, pathology, anatomic variations, clinical procedures, diagnostic techniques, embryological development, and aging. We continue to try to make difficult areas of anatomy more easily under-stood through better organization of chapter content and new diagram-matic approaches. The two-page spread that has been so popular in previous editions has been maintained in this edition, but an effort was made to improve their layouts by tabulating some content and adding more than 120 new illustrations and images. In this edition, the reader will notice major changes in two regions. In the abdomen and pelvic units, a greater focus is placed on the peritoneum, mesenteries, and peritoneal spaces. The inguinal region, a difficult area for students, is also expanded with new images and tables, as well as new and revised images of perineal structures. The head and neck unit is the second area of major revisions. In an effort to bring this material into alignment with the way it is usually encountered in the dissection lab, the chapter on the neck now precedes those on the head and includes new artwork that promotes the dissection views. Students will appreciate the reorgani zation and additional clarifying images of areas such as the cavernous sinus, pterygopalatine and infratemporal fossae, and oral and nasal cavi-ties. Finally, a new expanded overview introduces the brain and nervous system chapter. As always, we are extremely grateful for the contributions of the many colleagues and reviewers who provide important feedback on earlier editions, alert us to inaccuracies and ambiguities, and share suggestions for new material. We recognize that our efforts, though important, are just one part of the process that brings this textbook to its final production. The entire Thieme Publishers team has encouraged and supported our efforts throughout this process. Our deep appreciation is extended to the most important contributors: Judith Tomat, Developmental Editor; Delia DeTurris, Acqui-sitions Editor, and Barbara Chernow, PhD, Production Manager, for their dedication and expertise in their respective fields and their confidence in our ability to produce a quality manuscript. Anne M. Gilroy Worcester, Massachusetts Brian R. MacPherson Lexington, Kentucky Jamie C. Wikenheiser Irvine, California December 2019 Preface to the First Edition Each of the authors was amazed and impressed with the extraordinary detail, accuracy, and beauty of the illustrations that were created for the Thieme Atlas of Anatomy. We feel these images are one of the most significant additions to anatomical education in the past 50 years. It was our intent to use these exceptional illustrations as the cornerstone of our effort in creating a concise single volume Atlas of Anatomy for the curious and eager health science student. Our challenge was first to select from this extensive collection those images that are most instructive and illustrative of current dissec-tion approaches. Along the way, however, we realized that creating a single-volume atlas was much more than choosing images: each im-age has to convey a significant amount of detail while the appeal and labeling need to be clean and soothing to the eye. Therefore, hundreds of illustrations were drawn new or modified to fit the approach of this new atlas. In addition, key schematic diagrams and simplified sum-mary-form tables were added wherever needed. Dozens of applicable radiographic images and important clinical correlates have been added where appropriate. Additionally, surface anatomy illustrations are accompanied by questions designed to direct the student’s atten-tion to anatomic detail that is most relevant in conducting the phys-ical exam. Elements from each of these features are arranged in a regional format to facilitate common dissection approaches. Within each region, the various components are examined systemically, followed by topographical images to tie the systems together within the region. In all of this, a clinical perspective on the anatomical struc-tures is taken. The unique two facing pages “spread” format focuses the user to the area/topic being explored. We hope these efforts — the results of close to 100 combined years experience teaching the discipline of anatomy to bright, enthusiastic students — has resulted in a comprehensive, easy-to-use resource and reference. We would like to thank our colleagues at Thieme Publishers who so professionally facilitated this effort. We cannot thank enough Cathrin E. Schulz, MD, Editorial Director, Educational Products, who so graciously reminded us of deadlines, while always being available to “trouble shoot” problems. More importantly, she encouraged, helped, and complimented our efforts. We also wish to extend very special thanks and appreciation to Bridget Queenan, Developmental Editor, who edited and developed the manuscript with an outstanding talent for visualization and intuitive flow of information. We are very grateful to her for catching many de-tails along the way while always patiently responding to requests for artwork and labeling changes. Cordial thanks to Elsie Starbecker, Senior Production Editor, who with great care and speed produced this atlas with its over 2,200 illustra-tions. Finally, thanks to Rebecca McTavish, Developmental Editor, for joining the team in the correction phase. So very much of their hard work has made the Atlas of Anatomy a reality. Anne M. Gilroy Worcester, Massachusetts Brian R. MacPherson Lexington, Kentucky Lawrence M. Ross Houston, Texas March 2008 Back 1 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Bones, Ligaments & Joints Vertebral Column: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Vertebral Column: Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cervical Vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Thoracic & Lumbar Vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Sacrum & Coccyx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Intervertebral Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Joints of the Vertebral Column: Overview . . . . . . . . . . . . . . . . 16 Joints of the Vertebral Column: Craniovertebral Region . . . . . 18 Vertebral Ligaments: Overview & Cervical Spine . . . . . . . . . . 20 Vertebral Ligaments: Thoracolumbar Spine . . . . . . . . . . . . . . 22 3 Muscles Muscles of the Back: Overview . . . . . . . . . . . . . . . . . . . . . . . . . 24 Intrinsic Muscles of the Cervical Spine . . . . . . . . . . . . . . . . . . . 26 Intrinsic Muscles of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Neurovasculature Arteries & Veins of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Nerves of the Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Spinal Cord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Spinal Cord Segments & Spinal Nerves . . . . . . . . . . . . . . . . . . 42 Arteries & Veins of the Spinal Cord . . . . . . . . . . . . . . . . . . . . . 44 Neurovascular Topography of the Back . . . . . . . . . . . . . . . . . . 46 5 Sectional & Radiographic Anatomy Radiographic Anatomy of the Back (I) . . . . . . . . . . . . . . . . . . . 48 Radiographic Anatomy of the Back (II). . . . . . . . . . . . . . . . . . . 50 Vertebra prominens (C7) Scapular spine Medial border, scapula Inferior angle, scapula Iliac crest Posterior superior iliac spine Ischial tuberosity Greater trochanter, femur Sacrum Anterior superior iliac spine 6th through 12th ribs Greater tubercle, humerus Acromion Teres minor Thoracolumbar fascia Gluteus maximus Gluteus medius External oblique Latissimus dorsi Triceps brachii Teres major Deltoid Trapezius Back 2 1 Surface Anatomy Surface Anatomy Fig. 1.1 Palpable structures of the back Posterior view. B Musculature. A Bony prominences. Posterior midline Scapular line Paravertebral line Cervicothoracic junction S2 spinous process Posterior superior iliac spine Iliac crest L4 spinous process T12 spinous process Inferior angle of scapula T7 spinous process Scapular spine T3 spinous process C7 spinous process (vertebra prominens) 12th rib Vertebral region Gluteal region Sacral region Lumbar triangle Infrascapular region Lateral pectoral region Deltoid region Scapular region Interscapular region Suprascapular region Anal region 1 Surface Anatomy 3 Fig. 1.2 Regions of the back and buttocks Posterior view. Fig. 1.3 Spinous processes and landmarks of the back Posterior view. Table 1.1 Reference lines of the back Posterior midline Posterior trunk midline at the level of the spinous processes Paravertebral line Line at the level of the transverse processes Scapular line Line through the inferior angle of the scapula Table 1.2 Spinous processes that provide useful posterior landmarks Vertebral spinous process Posterior landmark C7 Vertebra prominens (the projecting spinous process of C7 is clearly visible and palpable) T3 The scapular spine T7 The inferior angle of the scapula T12 Just below the 12th rib L4 The summit of the iliac crest S2 The posterior superior iliac spine (recognized by small skin depressions directly over the iliac spines) C1–C7 vertebrae Coccyx L1–L5 vertebrae Sacrum (S1–S5 vertebrae) T1–T12 vertebrae Spinous process Costal facets Articular processes Interverte-bral foramina Sacral promontory Interverte-bral disk Lumbosacral junction Thoracolumbar junction Cervicothoracic junction Craniocervical junction Sacrum (sacral spine) Lumbar spine Thoracic spine Cervical spine Kyphotic spine of the newborn Transitional phase Adult spinal column Sacral kyphosis Lumbar lordosis Thoracic kyphosis Cervical lordosis 4 2 Bones, Ligaments & Joints Vertebral Column: Overview Back Fig. 2.1 Vertebral column Left lateral view. The vertebral column (spine) is divided into four regions: the cervical, thoracic, lumbar, and sacral spines. Both the cervical and lumbar spines demonstrate lordosis (inward curvature); the thoracic and sacral spines demonstrate kyphosis (outward curvature). A Regions of the spine. B Bony vertebral column. Clinical box 2.1 Spinal development The characteristic curvatures of the adult spine appear over the course of postnatal development, being only partially present in a newborn. The newborn has a “kyphotic” spinal curvature (A); lumbar lordosis develops later and becomes stable at puberty (C). A B C Whole-body center of gravity Inflection points Line of gravity External auditory canal Dens of axis (C2) Tongue Larynx Trachea Ascending aorta Heart Diaphragm Liver Abdominal aorta Stomach Bladder Coccyx Sacral promontory Cauda equina Conus medullaris Body of L1 Intervertebral disk Spinal cord Spinous process of vertebra prominens (C7) Dens of axis (C2) Rectum Vertebral canal Spinous process Esophagus 5 2 Bones, Ligaments & Joints B Midsagittal section through an adult male. Fig. 2.2 Normal anatomical position of the spine Left lateral view. A Line of gravity. The line of gravity passes through certain anatomical landmarks, including the inflection points at the cer-vicothoracic and thoracolumbar junctions. It continues through the center of gravity (anterior to the sacral promontory) before passing through the hip joint, knee, and ankle. Clinical box 2.2 Abnormal Vertebral Column Curvatures A Normal B Excessive kyphosis C Excessive lordosis D Scoliosis Scoliotic curve Asymmetrical waistline E Right convex thoracic scoliosis Vertebral body Transverse processes Costal processes Coccyx (Co1—Co4 vertebrae) Sacrum (fused S1—S5 vertebrae) L1—L5 vertebrae T1—T12 vertebrae C1—C7 vertebrae Anterior sacral foramina Intervertebral disk Atlas (C1) Axis (C2) Posterior sacral foramina Coccyx Transverse processes Spinous processes Atlas (C1) Dens of axis (C2) Sacrum Vertebra prominens (C7) L1 6 Vertebral Column: Elements Back Fig. 2.3 Bones of the vertebral column The transverse processes of the lumbar vertebrae are originally rib rudiments and so are named costal processes. A Anterior view. B Posterior view. Vertebral arch Pedicle Lamina Inferior articular process Spinous process Superior articular process Transverse process Vertebral body Vertebral foramen Lamina Pedicle Transverse process with groove for spinal n. Body Anterior tubercle Transverse foramen Posterior tubercle Superior articular facet Vertebral arch Spinous process Lamina Pedicle Inferior costal facet Superior costal facet Body Superior articular facet Transverse process Spinous process Costal facet Vertebral foramen Accessory process Vertebral arch Body Superior vertebral notch Transverse process Superior articular process Superior articular facet Spinous process Wing of sacrum Superior articular process Promontory Base of sacrum Lateral part of sacrum Sacral canal Median sacral crest 7 2 Bones, Ligaments & Joints Fig. 2.4 Structural elements of a vertebra Left posterosuperior view. With the exception of the atlas (C1) and axis (C2), all vertebrae consist of the same structural elements. Fig. 2.5 Typical vertebrae Superior view. A Cervical vertebra (C4). B Thoracic vertebra (T6). C Lumbar vertebra (L4). D Sacrum. Table 2.1 Structural elements of vertebrae Vertebrae Body Vertebral foramen Transverse processes Articular processes Spinous process Cervical vertebrae C3–C7 Small (kidney-shaped) Large (triangular) Small (may be absent on C7); anterior and posterior tubercles enclose transverse foramen Superoposteriorly and inferoanteriorly; oblique facets: most nearly horizontal Short (C3–C5); bifid (C3–C6); long (C7) Thoracic vertebrae T1–T12 Medium (heart-shaped); includes costal facets Small (circular) Large and strong; length decreases T1–T12; costal facets (T1–T10) Posteriorly (slightly laterally) and anteriorly (slightly medially); facets in coronal plane Long, sloping postero-inferiorly; tip extends to level of vertebral body below Lumbar vertebrae L1–L5 Large (kidney-shaped) Medium (triangular) Called costal processes, long and slender; accessory process on posterior surface Posteromedially (or medially) and anterolaterally (or laterally); facets nearly in sagittal plane; mammillary process on posterior surface of each superior articular process Short and broad Sacral vertebrae (sacrum) S1–S5 (fused) Decreases from base to apex Sacral canal Fused to rudimentary rib (ribs, see pp. 56–59) Superoposteriorly (SI) superior surface of lateral sacrum-auricular surface Median sacral crest C1 (atlas) and C2 (axis) are considered atypical (see pp. 8–9). Uncinate process Anterior tubercle C1 (atlas) C2 (axis) Groove for spinal n. Vertebral body Anterior tubercle Posterior tubercle Transverse process C7 (vertebra prominens) Transverse foramen Superior articular process Inferior articular process Zygapo-physeal joint Spinous process Posterior arch of atlas Posterior tubercle Spinous process Groove for spinal n. Uncovertebral joint Superior articular facet Anterior tubercle Transverse foramen Inferior articular facet Transverse process Posterior arch of atlas Posterior tubercle Groove for vertebral a. Vertebral arch Anterior articular facet Superior articular facet Transverse foramen Body Transverse process Inferior articular facet Spinous process Posterior articular facet Dens Superior articular process Transverse process Body Groove for spinal n. Inferior articular facet Inferior articular process Spinous process Superior articular facet Transverse foramen C1 (atlas) C2 (axis) C7 spinous process 8 Cervical Vertebrae Back Fig. 2.6 Cervical spine Left lateral view. A Bones of the cervical spine, left lateral view. B Radiograph of the cervical spine, left lateral view. Fig. 2.7 Atlas (C1) Fig. 2.8 Axis (C2) Fig. 2.9 Typical cervical vertebra (C4) A Left lateral view. A Left lateral view. A Left lateral view. The seven vertebrae of the cervical spine differ most conspicuously from the common vertebral morphology. They are specialized to bear the weight of the head and allow the neck to move in all directions. C1 and C2 are known as the atlas and axis, respectively. C7 is called the vertebra prominens for its long, palpable spinous process. Anterior arch Superior articular facet Transverse foramen Inferior articular facet Anterior tubercle Transverse process Posterior arch Superior articular facet Anterior arch Anterior tubercle Facet for dens Lateral masses Transverse process Transverse foramen Groove for vertebral a. Posterior tubercle Anterior articular facet Superior articular facet Body Inferior articular facet Transverse process Dens Transverse process Superior articular facet Anterior articular facet Dens Transverse foramen Vertebral foramen Vertebral arch Spinous process Inferior articular process Uncinate process Trans-verse process Spinous process Inferior articular facet Anterior tubercle Groove for spinal n. Posterior tubercle Superior articular process Body Vertebral foramen Lamina Pedicle Transverse process with groove for spinal n. Body Anterior tubercle Transverse foramen Posterior tubercle Superior articular facet Vertebral arch Spinous process Anterior displace-ment of body of C2 vertebra Vertebral body of C3 Spinous process of C1 Spinous process of C2 Fractured vertebral arch of C2 9 2 Bones, Ligaments & Joints B Anterior view. C Superior view. B Anterior view. C Superior view. B Anterior view. C Superior view. Clinical box 2.3 Injuries in the cervical spine The cervical spine is prone to hyperextension injuries, such as “whiplash,” which can occur when the head extends back much farther than it normally would. The most common injuries of the cervical spine are fractures of the dens of the axis, traumatic spondylolisthesis (anterior slippage of a vertebral body), and atlas fractures. Patient prognosis is largely dependent on the spinal level of the injuries (see p. 42). This patient hit the dashboard of his car while not wearing a seat belt. The resulting hyperextension caused the traumatic spondylolisthesis of C2 (axis) with fracture of the vertebral arch of C2, as well as tearing of the ligaments between C2 and C3. This injury is often referred to as “hangman’s fracture.” Superior costal facet Vertebral body Inter-vertebral foramen Inferior vertebral notch Superior vertebral notch Inferior articular facet Zygapo-physeal joint Costal facet on transverse process Transverse process Superior articular process Inferior articular process Spinous process 1st thoracic vertebra (T1) 12th thoracic vertebra (T12) Inferior costal facet Superior vertebral notch Superior costal facet Body Inferior costal facet Inferior vertebral notch Inferior articular facet Spinous process Costal facet on transverse process Transverse process Superior articular facet Superior articular process Superior costal facet Inferior costal facet Spinous process Body Transverse process Inferior articular facet Costal facet on transverse process Lamina Pedicle Inferior costal facet Superior costal facet Body Superior vertebral notch Superior articular facet Transverse process Spinous process Costal facet on transverse process 10 Fig. 2.10 Thoracic spine Left lateral view. Fig. 2.11 Typical thoracic vertebra (T6) A Left lateral view. B Anterior view. C Superior view. Thoracic & Lumbar Vertebrae Back Inter-vertebral foramen Inferior vertebral notch Superior vertebral notch Vertebral body 5th lumbar vertebra (L5) Inferior articular process Inferior articular facet Zygapophyseal joint Spinous process Transverse process Superior articular process 1st lumbar vertebra (L1) Body Inferior articular process Inferior articular facet Spinous process Transverse process Superior articular process Mammillary process Inferior vertebral notch Superior articular process Inferior articular process Spinous process Transverse process Body Inferior articular facet Vertebral foramen Accessory process Vertebral arch Body Superior vertebral notch Transverse process Mammillary process Superior articu-lar process Superior articular facet Spinous process 11 Fig. 2.12 Lumbar spine Left lateral view. Fig. 2.13 Typical lumbar vertebra (L4) A Left lateral view. B Anterior view. C Superior view. Clinical box 2.4 A Radiograph of a normal lumbar spine, left lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) B Radiograph of an osteoporotic lumbar spine with a compression fracture at L1 (arrow). Note that the vertebral bodies are decreased in density, and the internal trabecular structure is coarse. (Reproduced from Jallo J, Vaccaro AR. Neurotrauma and Critical Care of the Spine, 1st ed. New York, NY: Thieme; 2009.) Osteoporosis The spine is the structure most affected by degenerative diseases of the skeleton, such as arthrosis and osteoporosis. In osteoporosis, more bone material gets reabsorbed than built up, resulting in a loss of bone mass. Symptoms include compression fractures and resulting back pain. 2 Bones, Ligaments & Joints Wing of sacrum Promontory Anterior sacral foramina Coccyx Sacrococcygeal joint Transverse lines Lateral part Superior articular process Apex of sacrum Medial sacral crest Coccygeal cornu Sacrococcygeal joint Sacral cornua Sacral hiatus Median sacral crest Lateral sacral crest Auricular surface Sacral tuberosity Superior articular facet Sacral canal Coccyx Posterior sacral foramina Lateral part 12 Back Sacrum & Coccyx Fig. 2.14 Sacrum and coccyx A Anterior view. B Posterior view. The sacrum is formed from five postnatally fused sacral vertebrae. The base of the sacrum articulates with the 5th lumbar vertebra, and the apex articulates with the coccyx, a series of three or four rudimen-tary vertebrae. See Fig. 19.1, p. 230. Anterior sacral foramen Coccyx Pelvic surface Lateral part Posterior sacral foramen Median sacral crest Sacral canal Lateral sacral crest Base of sacrum Sacral promon-tory Anterior (pelvic) surface Posterior surface Sacral tuberosity Superior articular process Auricular surface Coccyx Sacro-iliac joint Sacral promon-tory Wing of sacrum Superior articular process Promontory Lateral part of sacrum Sacral canal Median sacral crest 13 2 Bones, Ligaments & Joints Fig. 2.15 Sacrum Superior view. B Transverse section through second sacral vertebra demonstrating anterior and posterior sacral foramina, superior view. A Base of sacrum, superior view. C Left lateral view. D Radiograph of sacrum, anteroposterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) 14 Intervertebral Disks Back Inter-vertebral disk Anulus fibrosus Nucleus pulposus Spinous process Ligamentum flavum Vertebral arch Superior articular facet Vertebral canal Vertebral body Interspinous lig. Fig. 2.16 Intervertebral disk in the vertebral column Midsagittal section of T11–T12, left lateral view. The intervertebral disks occupy the spaces between vertebrae (intervertebral joints, see p. 16). Intervertebral surface Anulus fibrosus Nucleus pulposus Marginal ridge (epiphyseal ring) Body Hyaline cartilage end plate Transverse process Superior articular process Fig. 2.17 Structure of intervertebral disk Anterosuperior view with the anterior half of the disk and the right half of the end plate removed. The intervertebral disk consists of an external fibrous ring (anulus fibrosus) and a gelatinous core (nucleus pulposus). Superior vertebral notch Nucleus pulposus Anulus fibrosus Transverse process Superior articular process Spinous process Vertebral foramen Intervertebral foramen Inner zone Outer zone Fig. 2.18 Relation of intervertebral disk to vertebral canal Fourth lumbar vertebra, superior view. Superior articular process Marginal ridge (epiphyseal ring) Vertebral bodies Inferior articular process Spinous process Crossing fiber systems of the anulus fibrosus Transverse process Fig. 2.19 Outer zone of the annulus fibrosus Anterior view of L3–L4 with intervertebral disk. Fat in the epidural space Herniated disk Cauda equina in CSF-filled dural sac Sacrum L3 L4 Pedicle (cut surface) Intervertebral disk Dural sac Compressed nerve roots Posterolateral herniation Dural sleeve with spinal n. Central herniation Intervertebral foramen Nucleus pulposus Cauda equina Epidural fat Dural sleeve with spinal n. Spinal dura mater Dural sleeve with spinal n. Posterolateral herniation Spondylophyte Nucleus pulposus 15 2 Bones, Ligaments & Joints A Superior view. B Midsagittal T2-weighted MRI (magnetic resonance image). C Superior view. D Posterior view, vertebral arches removed. Clinical box 2.5 As the stress resistance of the anulus fibrosus declines with age, the tissue of the nucleus pulposus may protrude through weak spots under loading. If the fibrous ring of the anulus ruptures completely, the herniated material may compress the contents of the intervertebral foramen (nerve roots and blood vessels—see posterolateral herniation below). These patients often suffer from severe local back pain. Pain is also felt in the associated dermatome (see p. 42). When the motor part of the spinal nerve is affected, the muscles served by that spinal nerve will show weakening. It is an important diagnostic step to test the muscles innervated by a nerve from a certain spinal segment, as well as the sensitivity in the specific dermatome. Example: The first sacral nerve root innervates the gastrocnemius and soleus muscles; thus, standing or walking on toes can be affected (see p. 446). Disk herniation in the lumbar spine Posterior herniation (A, B) In the MRI, a conspicuously herniated disk at the level of L3–L4 protrudes posteriorly (transligamentous herniation). The dural sac is deeply indented at that level. CSF (cerebrospinal fluid). Posterolateral herniation (C, D) A posterolateral herniation may compress the spinal nerve as it passes through the intervertebral foramen. If more medially positioned, the herniation may spare the nerve at that level but impact nerves at inferior levels. L3 L4 L5 S1 Bone drill Microsurgical instrument Compressed nerve Herniated disc Microdiscectomy surgery (E, F) is performed in order to remove a portion of a herniated disc that is irritating the nerve root. Through a small incision, the erector spinae muscles are reflected laterally to expose the ligamen-tum flavum, which is then removed in order to access the nerve roots in the spinal canal. A small portion of the facet joint may be removed to both facilitate access and relieve pressure on the nerve roots. Only the herniated portion of the disk is removed with the remaining tissue left intact. E F Groove for spinal n. Anterior tubercle Posterior tubercle Transverse process Superior articular process Inferior articular process Zygapophyseal joint Spinous process Transverse foramen Transverse process Superior articular facet Zygapophyseal joint Inferior articular facet Costal facet Zygapophyseal joint Transverse process Superior articular process Spinous process Inferior articular process Vertebral foramen ① ② ③ ④ ⑤ 16 Joints of the Vertebral Column: Overview Back Table 2.2 Joints of the vertebral column Craniovertebral joints ① Atlanto-occipital joints Occiput–C1 ② Atlantoaxial joints C1–C2 Joints of the vertebral bodies ③ Uncovertebral joints C3–C7 ④ Intervertebral joints C2–S1 Joints of the vertebral arch ⑤ Zygapophyseal joints C2–S1 Fig. 2.20 Zygapophyseal (intervertebral facet) joints The orientation of the zygapophyseal joints differs between the spinal regions, influencing the degree and direction of movement. A Cervical region, left lateral view. The zygapophyseal joints lie 45 degrees from the horizontal. B Thoracic region, left lateral view. The joints lie in the coronal plane. C Lumbar region, posterior view. The joints lie in the sagittal plane. Atlas (C1) Dens Inferior articular facet Groove for spinal n. Intervertebral disk Vertebral body Uncinate processes Axis (C2) Lateral atlantoaxial joint Trans-verse process Posterior tubercle Anterior tubercle C1 spinal n. Vertebral a. in transverse foramen C7 spinal n. Vertebral body (C7) Spinal n. in groove Transverse process Uncinate processes Vertebral a. Axis (C2) Atlas (C1) Spinal n. Vertebral foramen Lamina Spinal cord Superior articular facet Posterior root (spinal) ganglion Vertebral a. Transverse foramen Vertebral body Uncinate process Transverse process Spinous process Anulus fibrosus Nucleus pulposus Uncovertebral joint 17 2 Bones, Ligaments & Joints Fig. 2.21 Uncovertebral joints Anterior view. Uncovertebral joints form during childhood between the uncinate processes of C3–C7 and the vertebral bodies immediately superior. The joints may result from fissures in the cartilage of the disks that assume an articular character. If the fissures become complete tears, the risk of nucleus pulposus herniation is increased (see p. 15). A Uncovertebral joints in the cervical spine of an 18-year-old man, anterior view. B Uncovertebral joint (enlarged), anterior view of coronal section. C Uncovertebral joints, split intervertebral disks, anterior view of coronal section. Proximity of the spinal nerve and vertebral artery to the uncinate process The spinal nerve and vertebral artery pass through the intervertebral and transverse foramina, respectively (A and B). Bony outgrowths (osteophytes) on the uncinate process (C) resulting from uncovertebral arthrosis (degeneration) may compress both the nerve and the artery and can lead to chronic pain in the cervical region. Clinical box 2.6 A Cervical spine, anterior view. B Fourth cervical vertebra, superior view. C Advanced uncovertebral arthrosis of the fourth cervical vertebra, superior view. Vertebral body Uncinate process Transverse foramen Superior articular facet Spinous process Spondylo-phytes Inferior articular process Transverse process Lateral mass of the atlas Posterior tubercle of the atlas Spinous process of axis Vertebral foramen Superior articular facet Dens Transverse lig. of atlas Apical lig. of the dens Alar ligs. Anterior tubercle Longitudinal fascicles Median atlantoaxial joint Longitudinal fascicles Capsule of lateral atlanto-occipital joint Groove for vertebral a. Spinous process Nuchal lig. Posterior arch of atlas Intertransverse lig. Transverse process Posterior atlanto-occipital membrane Transverse lig. of atlas Tectorial membrane Alar ligs. Apical lig. of the dens Superior articular facet Superior nuchal line Mastoid process (temporal bone) Styloid process (temporal bone) Dens of axis (C2) Atlas (C1) Occipital condyle External occipital protuberance Axis (C2) Groove for vertebral a. Spinous process Transverse process Median atlantoaxial joint Superior articular facet (lateral mass of atlas) Dens of axis (C2) 18 Joints of the Vertebral Column: Craniovertebral Region Back Fig. 2.22 Craniovertebral joints A Posterior view. B Atlas and axis, posterosuperior view. Fig. 2.23 Ligaments of the craniovertebral joints A Ligaments of the median atlantoaxial joint, superior view. The fovea of the atlas is hid-den by the joint capsule. B Ligaments of the craniovertebral joints, posterosuperior view. The dens of the axis is hidden by the tectorial membrane. Lateral atlantoaxial joint (capsule) Styloid process Ligamentum flavum Nuchal lig. Atlas (C1) Posterior atlanto-occipital membrane External occipital protuberance Axis (C2) Posterior atlanto-occipital membrane Vertebral arch Tectorial membrane (posterior longitudinal lig.) Posterior arch of atlas Atlanto-occipital joint Nuchal lig. Spinous process Atlanto-occipital capsule Posterior longitudinal lig. Transverse lig. of atlas Longitudinal fascicles Alar ligs. Lateral mass of C1 Dens, posterior articular surface Alar lig. Apical lig. of dens 19 2 Bones, Ligaments & Joints Fig. 2.24 Dissection of the craniovertebral joint ligaments A Nuchal ligament and posterior atlanto-occipital membrane. B Posterior longitudinal ligament. Removed: Spinal cord; vertebral canal windowed. C Cruciform ligament of atlas (). Removed: Tectorial membrane, posterior atlanto- occipital membrane, and vertebral arches. D Alar and apical ligaments. Removed: Transverse ligament of atlas. The atlanto-occipital joints are the two articulations between the convex occipital condyles of the occipital bone and the slightly concave superior articular facets of the atlas (C1). The atlantoaxial joints are the two lateral and one medial articulations between the atlas (C1) and axis (C2). Anterior longitudinal lig. Posterior longitudinal lig. Vertebral arch Pedicle Lamina Inferior articular process Superior articular process Spinous process Supra-spinous lig. Inter-transverse lig. Transverse process Ligamentum flavum Interspinous lig. P A ① ② ③ ④ Atlanto-occipital capsule Posterior atlanto-occipital membrane Posterior longitudinal lig. Vertebral arch Tectorial membrane Atlanto-occipital joint External occipital protuberance 20 Vertebral Ligaments: Overview & Cervical Spine Back The ligaments of the spinal column bind the vertebrae and enable the spine to withstand high mechanical loads and shearing stresses and limit the range of motion. The ligaments are subdivided into vertebral body ligaments and vertebral arch ligaments. Fig. 2.25 Vertebral ligaments Viewed obliquely from the left posterior view. Intervertebral disk Atlanto-occipital joint (atlanto-occipital capsule) Atlas (C1) Transverse foramina Axis (C2) Anterior longitudinal lig. Vertebra prominens (C7) Zygapophyseal joint (capsule) Lateral atlantoaxial joint (capsule) Transverse process Anterior atlanto-occipital membrane Occipital bone, basilar part Internal occipital protuberance Anterior tubercle Posterior tubercle Groove for spinal nerve Fig. 2.26 Anterior longitudinal ligament Anterior view with base of skull removed. Fig. 2.27 Posterior longitudinal ligament Posterior view with vertebral canal opened via laminectomy and spinal cord removed. The tectorial membrane is a broadened expansion of the posterior longitudinal ligament. Table 2.3 Vertebral ligaments Ligament Location Vertebral body ligaments Anterior longitudinal lig. Along anterior surface of vertebral body Posterior longitudinal lig. Along posterior surface of vertebral body Vertebral arch ligaments ① Ligamentum flavum Between laminae ② Interspinous lig. Between spinous process ③ Supraspinous lig. Along posterior ridge of spinous processes ④ Intertransverse lig. Between transverse processes Nuchal lig. Between external occipital protuberance and spinous process of C7 Corresponds to a supraspinous ligament that is broadened superiorly. P A Posterior atlanto-occipital membrane Sphenoid sinus Maxilla Occipital bone, basilar part Apical lig. of the dens Anterior arch of atlas (C1) Dens of axis (C2) Transverse lig. of atlas Intervertebral disk Anterior longitudinal lig. Posterior longitudinal lig. C7 vertebral body (vertebra prominens) Supraspinous lig. Interspinous lig. Spinous process Ligamenta flava Vertebral arch Zygapophyseal joint capsule Nuchal lig. Posterior arch of atlas, posterior tubercle External occipital protuberance Tectorial membrane Longitu-dinal fascicles Hypoglossal canal Sella turcica Anterior atlanto-occipital membrane Intervertebral foramen Apex of dens Posterior longitu-dinal lig. Vertebral body Intervertebral disk Vertebra prominens (C7) Subarachnoid space Supraspinous lig. Posterior tubercle of atlas Cerebellomedullary cistern Nuchal lig. Spinal cord Body of axis Anterior longitudinal lig. 21 2 Bones, Ligaments & Joints Fig. 2.28 Ligaments of the cervical spine Mid-sagittal view. A Midsagittal section, left lateral view. The nuchal ligament is the broadened, sagittally oriented part of the supraspinous ligament that extends from the vertebra prominens (C7) to the external occipital protuberance. B Midsagittal T2-weighted MRI, left lateral view. Zygapophyseal joint capsule Posterior longitudinal lig. Intervertebral disk Anulus fibrosus Nucleus pulposus Anterior longitudinal lig. Vertebral body Inferior articular facet Supraspinous lig. Intertransverse ligs. Transverse process Interspinous ligs. Spinous processes Superior articular process Ligamenta flava Vertebral arch Superior articular facet Vertebral canal Clinical box 2.7 Spinal fusion is a surgical procedure used to restore stability to the vertebral column or to eliminate painful motion. The basic idea involves fusing two or more vertebrae so they will heal into a single, solid bone. Fusions can take place at any part of the vertebral column. Spinal fusion procedure 22 Back Vertebral Ligaments: Thoracolumbar Spine Fig. 2.29 Ligaments of the vertebral column: Thoracolumbar junction Left lateral view of T11–L3, with T11–T12 sectioned in the midsagittal plane. A Midline cutaway B Posterior view Anterior longitudinal lig. Vertebral body Transverse process Intervertebral disk Transverse process Posterior longitudinal lig. Anterior longitudinal lig. Spinous process Inferior articular facet Superior articular process Ligamenta flava Lamina Inter-transverse ligs. Superior articular process Intervertebral foramen Posterior longitudinal lig. Intervertebral disk Gap in ligamentous reinforcement of the disk Spinous process Inferior articular process Transverse process Superior articular facet Vertebral body Pedicles (cut) of vertebral arches Nutrient foramina Vertebral canal 23 2 Bones, Ligaments & Joints Fig. 2.31 Ligamenta flava and intertransverse ligaments Anterior view of opened vertebral canal at level of L2–L5. Removed: L2–L4 vertebral bodies. Fig. 2.32 Posterior longitudinal ligament Posterior view of opened vertebral canal at level of L2–L5. Removed: L2–L4 vertebral arches at pedicular level. Fig. 2.30 Anterior longitudinal ligament Anterior view of L3–L5. Lumbar triangle, internal oblique Trapezius (descending part) Trapezius (transverse part) Trapezius (ascending part) Scapular spine Deltoid Teres major Latissimus dorsi Triceps brachii Olecranon Iliac crest Gluteal aponeurosis Gluteus maximus Thoracolumbar fascia, posterior layer Serratus posterior inferior Serratus anterior Teres major Infraspinatus Scapula, medial border Supraspinatus Clavicle Rhomboid major Levator scapulae Rhomboid minor Sternocleido-mastoid Thoracolumbar fascia (= deep layer of nuchal fascia) External oblique Internal oblique Latissimus dorsi (cut) Acromion Aponeurotic origin of latissimus dorsi 24 3 Muscles Muscles of the Back: Overview Back The muscles of the back are divided into two groups, the ex trinsic and the intrinsic muscles, which are separated by the posterior layer of the thoracolumbar fascia. The superficial extrinsic muscles are considered muscles of the upper limb that have migrated to the back; these muscles are discussed in the Upper Limb, pp. 312–317. Fig. 3.1 Superficial extrinsic muscles of the back Posterior view. Removed: Trapezius and latissimus dorsi (right). Revealed: Thoracolumbar fascia. Note: The posterior layer of the thoracolumbar fascia is reinforced by the aponeurotic origin of the latissimus dorsi. Carotid sheath Sternocleidomastoid Internal jugular v. Common carotid a. Levator scapulae Superficial layer Deep layer Nuchal fascia Intrinsic back muscles Scalene muscles Prevertebral layer Pretracheal muscular layer Investing (superficial) layer Deep cervical fascia Thyroid gland Pretracheal visceral layer Infrahyoid muscles Trapezius Vagus n. Esophagus Trachea Brachial plexus C6 vertebra Spinal cord Longus colli Intrinsic back muscles Quadratus lumborum Serratus posterior inferior Latissimus dorsi Renal fascia, posterior layer Psoas fascia Fibrous capsule Lateral abdominal wall muscles Renal fascia, anterior layer Parietal peritoneum Kidney Inferior vena cava Abdominal aorta Psoas major L2 vertebra Transverse process of L2 Vertebral arch Spinous process of L1 Posterior layer Middle layer Thoracolumbar fascia Transversalis fascia Anterior layer (quadratus lumborum fascia) A B 25 3 Muscles A Transverse section at level of C6 vertebra, superior view. B Transverse section at level of L2, superior view. Removed: Cauda equina and anterior trunk wall. Fig. 3.2 Thoracolumbar fascia Transverse section, superior view. The intrinsic back muscles are sequestered in an osseofibrous canal, formed by the thoracolumbar fascia, the vertebral arches, and the spinous and transverse processes of associated vertebrae. The thoracolumbar fascia consists of a posterior and middle layer that unite at the lateral margin of the intrinsic back muscles. In the neck, the posterior layer blends with the nuchal fascia (deep layer), becoming continuous with the deep cervical fascia (pre-vertebral layer). Splenius capitis Splenius cervicis Semispinalis cervicis Semispinalis capitis Longissimus capitis Obliquus capitis inferior Rectus capitis posterior major Rectus capitis posterior minor Obliquus capitis superior Mastoid process Splenius capitis Semispinalis capitis Atlas (C1), transverse process Axis (C2), spinous process Parietal bone External occipital protuberance Superior nuchal line Semispinalis capitis Sternocleido-mastoid Splenius capitis Trapezius Occipital bone Sternocleido-mastoid 26 Intrinsic Muscles of the Cervical Spine Back Fig. 3.3 Muscles in the nuchal region Posterior view. Removed: Trapezius, sternocleidomastoid, splenius, and semispinalis muscles (right). Revealed: Nuchal muscles (right). Transverse process of atlas (C1) Trapezius Sternocleido-mastoid External occipital protuberance Rectus capitis posterior minor Semispinalis capitis Obliquus capitis superior Splenius capitis Longissimus capitis Rectus capitis posterior major Obliquus capitis inferior Spinous process of axis (C2) Intertransversarii cervicis Spinous process of C7 Interspinales cervicis Rectus capitis posterior major Posterior arch of atlas (C1) Posterior atlanto-occipital membrane (pierced by vertebral a.) Mastoid process Obliquus capitis superior Inferior nuchal line Superior nuchal line Transverse process of C7 Longissimus capitis Splenius capitis Semispinalis capitis Interspinales cervicis Obliquus capitis inferior Rectus capitis posterior major Obliquus capitis superior Sternocleido-mastoid Trapezius Rectus capitis posterior minor Intertransversarii cervicis 27 3 Muscles A Course of the short nuchal muscles. B Suboccipital region. Muscle origins are shown in red, insertions in blue. Fig. 3.4 Short nuchal muscles Posterior view. See Fig. 3.6. Three of the short nuchal muscles (obliquus capitis inferior, obliquus capitis superior and the rectus capitis posterior major) form the boundaries of the suboccipital triangle (region). Thoracolumbar fascia (= deep layer of nuchal fascia) Internal oblique External oblique (cut) Gluteus maximus Thoraco-lumbar fascia, posterior layer Latissimus dorsi (cut) aponeurosis External oblique Serratus posterior inferior Trapezius (cut) Serratus posterior superior Rhomboids major and minor (cut) Iliac crest External intercostal muscles Semispinalis capitis Splenius capitis Splenius cervicis Spinalis Ilio-costalis Longissi-mus Internal oblique External oblique Thoracolumbar fascia (= deep layer of nuchal fascia) Thoracolumbar fascia, posterior layer Iliac crest External intercostal muscles 28 Intrinsic Muscles of the Back Back The extrinsic muscles of the back (trapezius, latissimus dorsi, leva-tor scapulae, and rhomboids) are discussed in the Upper Limb, pp. 312–313. The serratus posterior, considered an intermediate extrinsic back muscle, has been included with the superficial intrinsic muscles in this unit. Fig. 3.5 Intrinsic muscles of the back Posterior view. Sequential dissection of the thoracolumbar fascia, superficial intrinsic muscles, intermediate intrinsic muscles, and deep intrinsic muscles of the back. A Thoracolumbar fascia. Removed: Shoulder girdles and extrinsic back muscles (except serratus posterior and aponeurotic origin of latis-simus dorsi). Revealed: Posterior layer of thoracolumbar fascia. B Superficial and intermediate intrinsic back muscles. Removed: Tho-racolumbar fascia, posterior layer (left). Revealed: Erector spinae and splenius muscles. Splenius capitis (cut) Longissimus capitis Iliocostalis cervicis External intercostal muscles Iliocostalis thoracis Levatores costarum Spinalis Iliocostalis lumborum Transversus abdominis Gluteus maximus Multifidus Longissimus thoracis Splenius cervicis Splenius capitis Semispinalis capitis Iliac crest Thoracolumbar fascia, posterior layer Internal oblique Quadratus lumborum Splenius capitis Obliquus capitis superior Rectus capitis posterior major Longissimus capitis Semispinalis capitis Spinalis cervicis External intercostal muscles Levatores costarum longi Levatores costarum breves Spinalis thoracis Thoracolumbar fascia, posterior layer Iliac crest Multifidus Interspinales lumborum Medial inter-transversarii lumbora Transversus abdominis Lateral inter-transversarii lumbora Rotatores thoracis breves Rotatores thoracis longi Superior nuchal line 12th rib Costal processes Interspinales cervicis Obliquus capitis inferior Rectus capitis posterior minor 29 3 Muscles C Intermediate and deep intrinsic back muscles. Removed: Longissimus thoracis and cervicis, splenius muscles (left); iliocostalis (right). Note: The posterior layer of the thoracolumbar fascia gives origin to the internal oblique and transversus abdominis. Revealed: Deep muscles of the back. D Deep intrinsic back muscles. Removed: Superficial and intermediate intrinsic back muscles (all); deep fascial layer and multifidus (right). Revealed: Intertransversarii and quadratus lumborum (right). ① ② ③ ④ Posterior tubercle of atlas (C1) Superior nuchal line Inferior nuchal line Obliquus capitis superior Transverse process of atlas (C1) Obliquus capitis inferior Spinous process of axis (C2) Rectus capitis posterior major Rectus capitis posterior minor Mastoid process Transverse process of atlas (C1) Spinous process of axis (C2) Obliquus capitis inferior Rectus capitis posterior major Rectus capitis posterior minor Obliquus capitis superior External occipital protuberance Mastoid process Mandible Atlas (C1) Axis (C2) 30 Back Muscle Facts (I) Fig. 3.6 Short nuchal and craniovertebral joint muscles A Posterior view, schematic. B Suboccipital muscles, posterior view. C Suboccipital muscles, left lateral view. Table 3.1 Short nuchal and craniovertebral joint muscles Muscle Origin Insertion Innervation Action Rectus capitis posterior ① Rectus capitis posterior major C2 (spinous process) Occipital bone (inferior nuchal line, middle third) C1 (posterior ramus = suboccipital n.) Bilateral: Extends head Unilateral: Rotates head to same side ② Rectus capitis posterior minor C1 (posterior tubercle) Occipital bone (inferior nuchal line, inner third) Obliquus capitis ③ Obliquus capitis superior C1 (transverse process) Occipital bone (inferior nuchal line, middle third; above rectus capitis posterior major) Bilateral: Extends head Unilateral: Flexes head to same side; rotates to opposite side ④ Obliquus capitis inferior C2 (spinous process) C1 (transverse process) Bilateral: Extends head Unilateral: Rotates head to same side Atlas (C1) Axis (C2) C7 vertebra T3 vertebra Occipital bone, basilar portion ① ② ③ ④ Anterior tubercle T3 vertebra 1st rib C7 vertebra Transverse process of atlas (C1) Rectus capitis lateralis Rectus capitis anterior Occipital bone, basilar portion Axis (C2) Longus capitis Vertical part Mastoid process Superior oblique part Inferior oblique part Longus colli 31 3 Muscles Fig. 3.7 Prevertebral muscles A Anterior view, schematic. B Prevertebral muscles, anterior view. Removed: Longus capitis (left); cervical viscera. Table 3.2 Prevertebral muscles Muscle Origin Insertion Innervation Action ① Longus capitis C3–C6 (transverse processes, anterior tubercles) Occipital bone (basilar part) Direct branches from cervical plexus (C1–C3) Bilateral: Flexes head Unilateral: Flexes and slightly rotates head to same side ② Longus colli (cervicis) Vertical (medial) part C5–T3 (anterior sides of vertebral bodies) C2–C4 (anterior sides of vertebral bodies) Direct branches from cervical plexus (C2–C6) Bilateral: Flexes cervical spine Unilateral: Flexes and rotates cervical spine to same side Superior oblique part C3–C5 (transverse processes, anterior tubercles) C1 (transverse process, anterior tubercle) Inferior oblique part T1–T3 (anterior sides of vertebral bodies) C5–C6 (transverse processes, anterior tubercles) ③ Rectus capitis anterior C1 (lateral mass) Occipital bone (basilar part) C1 (anterior ramus) Bilateral: Flexion at atlanto-occipital joint Unilateral: Lateral flexion at atlanto-occipital joint ④ Rectus capitis lateralis C1 (transverse process) Occipital bone (basilar part, lateral to occipital condyles) ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ① ② 32 Back Muscle Facts (II) The intrinsic back muscles are divided into superficial, intermediate, and deep layers. The serratus posterior muscles are extrinsic back muscles, innervated by the anterior rami of intercostal nerves, not the posterior rami, which innervate the intrinsic back muscles. They are included here as they are encountered in dissection of the back musculature. Table 3.3 Superficial intrinsic back muscles Muscle Origin Insertion Innervation Action Serratus posterior ① Serratus posterior superior Nuchal lig.; C7–T3 (spinous processes) 2nd–4th ribs (superior borders) Spinal nn. T2–T5 (anterior rami) Elevates ribs ② Serratus posterior inferior T11–L2 (spinous processes) 8th–12th ribs (inferior borders, near angles) Spinal nn. T9–T12 (anterior rami) Depresses ribs Splenius ③ Splenius capitis Nuchal lig.; C7–T3 or T4 (spinous processes) Lateral 1/3 nuchal line (occipital bone); mastoid process (temporal bone) Spinal nn. C1–C6 (posterior rami, lateral branches) Bilateral: Extends cervical spine and head Unilateral: Laterally flexes and rotates head to the same side ④ Splenius cervicis T3–T6 or T7 (spinous processes) C1–C3/4 (transverse processes) Fig. 3.8 Superficial intrinsic back muscles, schematic Right side, posterior view. Fig. 3.9 Intermediate intrinsic back muscles, schematic Right side, posterior view. These muscles are collectively known as the erector spinae. A Iliocostalis muscles. B Longissimus muscles. C Spinalis muscles. Table 3.4 Intermediate intrinsic back muscles (erector spinae) Muscle Origin Insertion Innervation Action Iliocostalis ⑤ Iliocostalis cervicis 3rd–7th ribs C4–C6 (transverse processes) Spinal nn. C8–L1 (posterior rami, lateral branches) Bilateral: Extends spine Unilateral: Flexes spine laterally to same side ⑥ Iliocostalis thoracis 7th–12th ribs 1st–6th ribs ⑦ Iliocostalis lumborum Sacrum; iliac crest; thoracolumbar fascia (posterior layer) 6th–12th ribs; thoracolumbar fascia (posterior layer); upper lumbar vertebrae (transverse processes) Longissimus ⑧ Longissimus capitis T1–T3 (transverse processes); C4–C7 (transverse and articular processes) Temporal bone (mastoid process) Spinal nn. C1–L5 (posterior rami, lateral branches) Bilateral: Extends head Unilateral: Flexes and rotates head to same side ⑨ Longissimus cervicis T1–T6 (transverse processes) C2–C5 (transverse processes) Bilateral: Extends spine Unilateral: Flexes spine laterally to same side ⑩ Longissimus thoracis Sacrum; iliac crest; lumbar vertebrae (spinous processes); lower thoracic vertebrae (transverse processes) 2nd–12th ribs; thoracic and lumbar vertebrae (transverse processes) Spinalis ⑪ Spinalis cervicis C5–T2 (spinous processes) C2–C5 (spinous processes) Spinal nn. (posterior rami) Bilateral: Extends cervical and thoracic spine Unilateral: Flexes cervical and thoracic spine to same side ⑫ Spinalis thoracis T10–L3 (spinous processes, lateral surfaces) T2–T8 (spinous processes, lateral surfaces) B Splenius muscles. A Serratus posterior. Spinalis thoracis Spinalis cervicis Longissimus capitis Iliocostalis cervicis Longissimus cervicis Iliocostalis thoracis Longissimus thoracis Iliocostalis lumborum 33 3 Muscles Fig. 3.10 Superficial and intermediate intrinsic back muscles Posterior view. Superior nuchal line Spinous process of C7 4th rib Splenius cervicis Splenius capitis Mastoid process 12th rib Serratus posterior inferior Serratus posterior superior 8th rib L2 A Superficial back muscles: Splenius and serratus posterior muscles. B Intermediate intrinsic back muscles (erector spinae): Iliocostalis, longissimus, and spinalis muscles. ② A ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑬ 34 Muscle Facts (III) Back The deep intrinsic back muscles are divided into two groups: trans versospinalis and deep segmental muscles. The transversospinalis muscles pass between the transverse and spinous processes of the vertebrae. Table 3.5 Transversospinalis muscles Muscle Origin Insertion Innervation Action Rotatores ① Rotatores breves T1–T12 (between transverse and spinous processes of adjacent vertebrae) Spinal nn. (posterior rami) Bilateral: Extends thoracic spine Unilateral: Rotates thoracic spine to opposite side ② Rotatores longi T1–T12 (between transverse and spinous processes, skipping one vertebra) Multifidus ③ Sacrum, ilium, mamillary processes of L1–L5, transverse and articular processes of T1–T4, C4–C7 Superomedially to spinous processes, skipping two to four vertebrae Bilateral: Extends spine Unilateral: Flexes spine to same side, rotates it to opposite side Semispinalis ④ Semispinalis capitis C4–T7 (transverse and articular processes) Occipital bone (between superior and inferior nuchal lines) Bilateral: Extends thoracic and cervical spines and head (stabilizes craniovertebral joints) Unilateral: Flexes head, cervical and thoracic spines to same side, rotates to opposite side ⑤ Semispinalis cervicis T1–T6 (transverse processes) C2–C5 (spinous processes) ⑥ Semispinalis thoracis T6–T12 (transverse processes) C6–T4 (spinous processes) Fig. 3.11 Transversospinalis muscles Posterior view, schematic. A Rotatores muscles. B Multifidus. C Semispinalis. Fig. 3.12 Deep segmental muscles Posterior view, schematic. Table 3.6 Deep segmental back muscles Muscle Origin Insertion Innervation Action Interspinales ⑦ Interspinales cervicis C1–C7 (between spinous processes of adjacent vertebrae) Spinal nn. (posterior rami) Extends cervical and lumbar spines ⑧ Interspinales lumbora L1–L5 (between spinous processes of adjacent vertebrae) Bilateral: Stabilizes and extends the cervical and lumbar spines Unilateral: Flexes the cervical and lumbar spines laterally to same side Inter- transversarii Anterior intertransversarii cervices C2–C7 (between anterior tubercles of adjacent vertebrae) Spinal nn. (anterior rami) ⑨ Posterior intertransversarii cervices C2–C7 (between posterior tubercles of adjacent vertebrae) Spinal nn. (posterior rami) ⑩ Medial intertransversarii lumbora L1–L5 (between mammillary processes of adjacent vertebrae) ⑪ Lateral intertransversarii lumbora L1–L5 (between transverse processes of adjacent vertebrae) Spinal nn. (anterior rami) Levatores costarum ⑫ Levatores costarum breves C7–T11 (transverse processes) Costal angle of next lower rib Spinal nn. (posterior rami) Bilateral: Extends thoracic spine Unilateral: Flexes thoracic spine to same side, rotates to opposite side ⑬ Levatores costarum longi Costal angle of rib two vertebrae below Both the interspinales and intertransversarii muscles traverse the entire spine; only their clinically relevant components have been included. Interspinales lumbora Interspinales cervicis Posterior intertrans-versarii cervices Levatores costarum longi Levatores costarum breves Medial intertransversarii lumbora Lateral intertransversarii lumbora 35 3 Muscles Fig. 3.13 Deep intrinsic back muscles Posterior view. Transverse processes Sacrum Superior nuchal line Inferior nuchal line Spinous process of C7 Rotatores breves Spinous process Rotatores longi Transverse process Semispinalis thoracis Multifidus Semispinalis capitis Semispinalis cervicis A Transversospinalis muscles: Rotatores, multifidus, and semispinalis. B Deep segmental muscles: Interspinales, intertransversarii, and levatores costarum. Vertebral a. Right subclavian a. External carotid a. Internal carotid a. Right common carotid a. Thyrocervical trunk Internal thoracic a. Costocervical trunk 1st posterior intercostal a. 2nd posterior intercostal a. Posterior intercostal a. Sternal brs. Thoracic aorta Internal thoracic a. Medial cutaneous br. Dorsal branch of posterior intercostal a. Anterior intercostal a. Lateral cutaneous br. Posterior intercostal a. Lateral cutaneous br. Spinal br. Anterior cutaneous br. External iliac a. Common iliac a. Abdominal aorta Median sacral a. Lateral sacral a. Internal iliac a. Coccyx Right subclavian a. Posterior intercostal aa. External iliac a. Subcostal a. Anterior intercostal aa. Aortic arch Brachiocephalic trunk Right common carotid a. Thoracic aorta Abdominal aorta 36 Back 4 Neurovasculature Arteries & Veins of the Back Fig. 4.1 Arteries of the back The structures of the back are supplied by branches of the posterior intercostal arteries, which arise from the thoracic aorta or from the subclavian artery. A Arteries of the trunk, right lateral view. B Vascular supply to the nuchal region, pos-terolateral view. Note: The first and second posterior intercostal arteries arise from the costocervical trunk, a branch of the subclavian artery. C Posterior intercostal arteries, oblique pos-terosuperior view. The posterior intercostal arteries give rise to cutaneous and muscu-lar branches, as well as spinal branches that supply the spinal cord. D Vascular supply to the sacrum, anterior view. External iliac v. Subcostal v. Inferior vena cava Anterior intercostal vv. Superior vena cava Right brachiocephalic v. Right subclavian v. Right internal jugular v. Azygos v. Posterior intercostal vv. Lateral cutaneous br. Azygos v. Anterior internal vertebral venous plexus Posterior intercostal v. Anterior external vertebral venous plexus Posterior internal vertebral venous plexus Intervertebral v. Hemiazygos v. Medial cutaneous br. External iliac v. Ascending lumbar v. Anterior internal vertebral venous plexus External vertebral venous plexus External vertebral venous plexus Hemiazygos v. Accessory hemiazygos v. Superior vena cava Left brachiocephalic v. Internal jugular v. Superior sagittal sinus Azygos v. Posterior intercostal vv. Internal iliac v. Posterior internal vertebral venous plexus Transverse sinus Lumbar v. Sigmoid sinus Emissary v. Right brachiocephalic v. 37 4 Neurovasculature Fig. 4.2 Veins of the back The veins of the back drain into the azygos vein via the posterior inter-costal veins, hemiazygos vein, and ascending lumbar veins. The interior of the spinal column is drained by the vertebral venous plexus that runs the length of the spine. B Vertebral venous plexus, posterior view with vertebral canal win-dowed in the lumbar and sacral spine. The external vertebral venous plexus communicates with the sigmoid sinus through emissary veins in the skull. The external vertebral venous plexus is divided into an anterior and a posterior portion that run along the exterior of the vertebral column. The anterior and posterior internal vertebral venous plexus run in the vertebral foramen and drain the spinal cord. C Intercostal veins and anterior vertebral venous plexus, anterosupe-rior view. The intercostal veins follow a similar course to the intercos-tal nerves and arteries (see pp. 36, 38). Note: The anterior external vertebral venous plexus can be seen communicating with the azygos vein. A Veins of the trunk, right lateral view. Sympathetic (paravertebral) ganglion Meningeal br. Posterior (dorsal) ramus Anterior (ventral) ramus Spinal n. Posterior root Spinal ganglion Anterior root Inner layer, arachnoid mater Outer layer, dura mater Medial cutaneous br. White and gray rami communicans Aorta Esophagus Lateral cutaneous br. 38 Back Nerves of the Back Fig. 4.3 Nerves of the back Cross section of the vertebral column and spinal cord with surrounding musculature, superior view. The back receives its innervation from branches of the spinal nerves. The posterior (dorsal) rami of the spinal nerves supply most of the intrinsic muscles of the back. The extrinsic muscles of the back are supplied by the anterior (ventral) rami of the spinal nerves. C5 spinal n., posterior ramus Supraclavicular nn. 3rd occipital n. (C3) Greater occipital n. (C2) Great auricular n. Lesser occipital n. Suboccipital n. (C1) Fig. 4.4 Nerves of the nuchal region Right side, posterior view. Fig. 4.5 Cutaneous innervation of the back Color denotes the skin areas innervated by (A) particular peripheral nerves or (B) particular pairs of segmental spinal nerves. Patterns of loss of cutaneous sensation can be helpful in diagnosis of nerve lesions. Greater occipital n. Medial cutaneous brs. Lateral cutaneous brs. Superior clunial nn. Middle clunial nn. Iliohypogastric n. Intercostal nn., (anterior rami, lateral cutaneous brs.) Supraclavicular nn. Great auricular n. Lesser occipital n. Spinal nn., (posterior rami) Axillary n. L1 T1 C5 T1 C4 C3 C2 C6 C8 C5 A Cutaneous innervation patterns of specific peripheral nerves. B Dermatomes: Dermatomes are bilateral band-like areas of skin re-ceiving innervation from a single pair of spinal nerves (from a single segment of the spinal cord). Note: Spinal nerve C1 is purely motor; consequently there is no C1 dermatome. Table 4.1 Nerves of the nuchal region Branches Function Posterior (dorsal) ramus Suboccipital n. (C1) Innervates the rectus capitis posterior major and minor; and obliquus capitis superior and inferior Greater occipital n. (C2) Assits in the innervation of the semispinalis capitis muscle and supplies skin behind the auricle and the scalp to the coronal suture Third occipital n. (C3) Assists in the innervation of the semispinalis capitis muscle, the C2-C3 facet joint, and supplies a small area of skin just below the superior nuchal line Anterior (ventral) ramus Lesser occipital n. (C2) Cutaneous only, supplies an area of scalp posterolateral to the auricle, and the skin on the upper third of the medial aspect of the auricle Greater auricular n. (C2, C3) Cutaneous only, supplies an area of skin over the parotid gland, the majority of the pinna, lateral neck, and posterior to the auricle The anterior rami of C1-C3 also give rise to the ansa cervicalis, which innervates the infrahyoid muscles (see p. 524). 39 4 Neurovasculature Spinal ganglion Anterior internal vertebral venous plexus Posterior root Vertebral vv. Intervertebral foramen Spinal n. Vertebral a. Anterior root Posterior internal vertebral venous plexus Denticulate lig. Epidural space Subarachnoid space Dura mater Arachnoid (mater) Root sleeve Spinal cord Anterior spinal a. Pia mater Arachnoid (mater) Subdural space Dura mater Anterior spinal vv. Subarachnoid space Posterior root Spinal ganglion Posterior ramus Denticulate lig. Anterior rootlets Anterior ramus Anterior root White and gray rami communicans Spinal n. C1 spinal n. Atlas (C1) Vertebra prominens (C7) T1 spinal n. T12 vertebra L1 spinal n. L5 vertebra S1 spinal n. Sacral hiatus Cauda equina Conus medullaris Lumbosacral enlargement Cervical enlargement Medulla oblongata Arachnoid (mater) Dura mater 40 Clinical box 4.1 Spina bifida is a neural tube defect that occurs when the spine and spinal cord do not form properly. In the United States, it affects about one out of every 1,500 newborns. There are three main types. • Spina bifida occulta (A) is the most common congenital anomaly of the vertebral column in which the laminae of L5 and/or S1 fail to develop. The defect is often hidden and most individuals are unaware they have the condition because there is only a small defect in the vertebrae. There is generally no disturbance of spinal function. • Spina bifida (meningocele) (B) occurs when one or more vertebral arches fail to develop and presents with a herniation or sac of only the meninges. The spinal cord and nerves are normal and not severely affected. • Spina bifida (myelomeningocele) (C) occurs when multiple vertebral arches fail to develop resulting in a herniation of both the meninges and spinal nerves. This is the most severe form exposing the newborn to life threatening infections, bowel and bladder dysfunction, and total paralysis of the lower extremities. Spina Bifida Back Spinal Cord Fig. 4.6 Spinal cord in situ Posterior view with vertebral canal windowed. Fig. 4.7 Spinal cord and its meningeal layers Posterior view. The dura mater is opened and the arachnoid is sectioned. The detailed anatomy of the spinal cord can be found on pp. 690–691. Fig. 4.8 Cervical spinal cord in situ: Transverse section Superior view. Spinal cord at level of C4 vertebra. The dura mater of the cranial cavity is composed of two layers, the periosteal and meningeal. Only the meningeal layer extends into the vertebral canal with the spinal cord. The periosteal layer of dura terminates at the foramen magnum and is replaced in the vertebral canal with the periosteum of the vertebral bone. Due to this structural difference in the two regions, the dural sac is not adherent to the bone of the vertebral canal as it is in the cranial cavity. C B A Anterior internal vertebral venous plexus Spinal ganglion Spinal dura mater Fatty tissue Cauda equina Epidural space Posterior internal vertebral venous plexus Dural sac Conus medullaris Cauda equina (posterior and anterior spinal roots) Filum terminale Sacral hiatus Arachnoid (mater) Dura mater Spinal ganglion L1 vertebra T12 L1 Conus medullaris (newborn) Dural sac (lumbar cistern) Conus medullaris (adult) 1 2 3 Sacral hiatus Cauda equina Conus medullaris 41 4 Neurovasculature Fig. 4.9 Cauda equina in the vertebral canal Posterior view. The lamina and posterior surface of the sacrum have been partially removed. Fig. 4.10 Cauda equina in situ: Transverse section Superior view. Cauda equina at level of L2 vertebra. Fig. 4.11 Spinal cord, dural sac, and vertebral column at different ages. Anterior view. Longitudinal growth of the spinal cord lags be-hind that of the vertebral column. At birth, the distal end of the spinal cord, the conus medullaris, is at the level of the L3 vertebral body, but in the average adult it extends to the level of L1/L2. The dural sac always extends into the upper sacrum. Clinical box 4.2 Lumbar puncture A needle introduced into the dural sac (lumbar cistern) generally slips past the spinal nerve roots without injuring the spinal cord or spinal nerves. Cerebrospinal fluid (CSF) samples are therefore taken between the L3 and L4 vertebrae (2), once the patient has leaned forward to separate the spinous processes of the lumbar spine. Anesthesia Lumbar anesthesia may be administered in a similar fashion (2). Epidural anesthesia is administered by placing a catheter in the epidural space without penetrating the dural sac (1). This may also be done by passing a needle through the sacral hiatus (3). Atlas (C1) Axis (C2) C7 T1 T12 L1 L5 S1 Coccyx S1 L1 T12 T1 C8 C1 Spinal cord segment Vertebra T2 T1 T1 L1 L5 S1 S5 C2 C5 C7 C8 C6 C3 C4 Cervical cord lesion Thoracic cord lesion Lumbar cord lesion Conus/cauda equina lesion Anterior rootlets Meningeal br. Anterior ramus Splanchnic nn. Sympathetic (paravertebral) ganglion White ramus communicans Gray ramus communicans Posterior ramus Posterior root (with spinal ganglion) Gray matter, posterior horn Gray matter, anterior horn Posterior rootlets Anterior root Spinal n. White matter Sympathetic trunk 42 Back Spinal Cord Segments & Spinal Nerves Fig. 4.12 Spinal cord segment The spinal cord consists of 31 segments, each innervating a specific area of the skin (a dermatome) of the head, trunk, or limbs. Afferent (sensory) posterior rootlets and efferent (motor) anterior rootlets form the posterior and anterior roots of the spinal nerve for that segment. The two roots fuse to form a mixed (motor and sensory) spinal nerve that exits the intervertebral foramen and immediately thereafter divides into an anterior and posterior ramus. Fig. 4.13 Spinal cord segments, dermatomes, and effects of spinal cord lesions The spinal cord is divided into four major regions: cervical, thoracic, lumbar, and sacral. The regions of the spinal cord are designated by colors: red, cervical; brown, thoracic; green, lumbar; blue, sacral. A Spinal cord segments. Initially spinal nerves pass out above the vertebrae for which they are numbered. However, since there is an 8th cervical spinal nerve but no 8th cervical vertebrae, C8 passes out above vertebral level T1, and the spinal nerve for T1, and those following, pass out below the verte-bral level for which they are numbered. C Effects of lesions in each region of the spinal cord. B Dermatomes, band-like areas of skin receiv-ing sensory innervation from a single pair of spinal nerves (from a single segment of the spinal cord). Note: Spinal nerve C1 is purely motor; consequently there is no C1 dermatome. Anterior root Cauda equina Posterior root Posterior sacral foramen Lateral br. (to the clunial nerves) Posterior ramus Anterior ramus (to sacral plexus) Anterior sacral foramen Spinal ganglion Spinal cord Posterior ramus Anterior ramus Anterior cutaneous br. Lateral cutaneous br. Sympathetic ganglion Meningeal br. Spinal ganglion Articular br. White and gray rami communicans Lateral br. Medial br. Sympathetic trunk 43 4 Neurovasculature Fig. 4.14 Spinal nerve branches B Spinal nerve branches in the sacral foramina. Superior view of transverse section through right half of sacrum. A Superolateral view of a thoracic spinal nerve. The posterior (dor-sal) rami of the spinal nerves give rise to muscular and cutaneous branches, as well as articular branches to the zygapophyseal joints. The anterior (ventral) rami of the spinal nerves form the cervical plexus (C1–C4), the brachial plexus (C5–T1), the lumbar plexus (T12–L4), and the sacral plexus (L4–S3). The anterior rami of spinal nerves T1–T11 produce the intercostal nerves (T12 produces the subcostal nerve). Table 4.2 Branches of a spinal nerve Branches Territory Meningeal br. Spinal meninges; ligaments of spinal column Posterior (dorsal) ramus Medial brs. Articular br. Zygapophyseal joints Muscular br. Intrinsic back muscles Cutaneous br. Skin of posterior head, neck, back, and buttocks Lateral brs. Cutaneous br. Muscular br. Intrinsic back muscles Anterior (ventral) ramus Lateral cutaneous brs. Skin of lateral chest wall Anterior cutaneous brs. Skin of anterior chest wall The white and gray rami communicans carry pre- and postganglionic fibers between the sympathetic trunk and spinal n. Spinal br. Anterior spinal a. Sulcal a. Vasocorona Posterior spinal aa. Anterior horn Posterior horn Posterior segmental medullary a. Anterior segmental medullary a. Basilar a. Vertebral a. Anterior spinal a. Lumbar aa. Great anterior segmental medullary a. Ascending cervical a. Posterior spinal a. Anterior segmental medullary a. Subclavian a. Posterior segmental medullary a. Segmental a. Posterior intercostal aa. Posterior segmental medullary a. Vertebral a. Posterior segmental medullary a. Spinal br. Posterior (dorsal) br. Anterior segmental medullary a. Thoracic aorta Lateral cutaneous br. Medial cutaneous br. Posterior intercostal a. Posterior inter-costal a. 44 Arteries & Veins of the Spinal Cord Like the spinal cord itself, the arteries and veins of the spinal cord consist of multiple horizontal systems (blood vessels of the spinal cord seg-ments) that are integrated into a vertical system. Fig. 4.15 Arteries of the spinal cord The unpaired anterior and paired posterior spinal arteries typically arise from the vertebral arteries. As they descend within the vertebral canal, the spinal arteries are reinforced by anterior and posterior segmental medullary arteries. Depending on the spinal level, these reinforcing branches may arise from the vertebral, ascending or deep cervical, posterior intercostal, lumbar, or lateral sacral arteries. A Spinal and segmental medullary arteries. B Origins of the segmental medullary arter-ies. In the thorax, the segmental medullary arteries arise from the spinal branch of the posterior intercostal arteries (see p. 36). C Arterial supply system. Back Accessory hemiazygos v. Right deep cervical v. Right vertebral v. Right subclavian v. Right internal jugular v. Superior vena cava Hemiazygos v. Azygos v. Left brachio-cephalic v. Anterior spinal v. Intercostal vv. Posterior radicular v. Anterior radicular v. Inferior vena cava Common iliac v. Anterior spinal v. Ascending lumbar v. Sulcal v. Anterior spinal v. Anterior radicular v. Posterior spinal v. Spinal v. Venous ring Posterior radicular v. Ascending lumbar v. Subcostal v. Anterior internal vertebral venous plexus Inter-vertebral v. Basivertebral v. Posterior internal vertebral venous plexus Anterior external vertebral venous plexus External iliac v. Internal iliac v. Medial and lateral epidural vv. Basivertebral v. Ascending lumbar v. Posterior internal vertebral venous plexus (in epidural space) Intervertebral v. Anterior internal vertebral venous plexus 45 Fig. 4.16 Veins of the spinal cord The interior of the spinal cord drains via venous plexuses into an anterior and a posterior spinal vein. The radicular and spinal veins connect the veins of the spinal cord with the internal vertebral venous plexus. The intervertebral and basivertebral veins connect the internal and external venous plexuses, which drain into the azygos system. A Venous drainage system. B Spinal and radicular veins. C Vertebral venous plexuses. D Veins in the sacral and lumbar canals. 4 Neurovasculature Obliquus capitis superior Rectus capitis posterior minor Greater occipital n. Rectus capitis posterior major 3rd occipital n. Obliquus capitis inferior Axis (C2), spinous process Deep cervical a. Semispinalis cervicis Semispinalis capitis Suboccipital n. Longissimus capitis Sternocleidomastoid Great auricular n. Atlas (C1), transverse process Lesser occipital n. Vertebral a. Occipital a. 46 Neurovascular Topography of the Back Back Fig. 4.17 Neurovasculature of the nuchal region Posterior view. Removed: Trapezius, sterno-cleidomastoid, and semispinalis capitis. Revealed: Suboccipital region. Intercostal nn. and posterior intercostal aa. and vv. (lateral cutaneous brs.) Iliolumbar triangle (of Petit) Inferior clunial nn. Middle clunial nn. Superior clunial nn. Latissimus dorsi Iliac crest Internal oblique External oblique Fibrous lumbar triangle (of Grynfeltt) Serratus posterior inferior Thoracolumbar fascia, posterior layer Rhomboid major Accessory n. Transverse cervical a. Splenius capitis 3rd occipital n. Deltoid Trapezius Spinal nn., posterior rami (lateral cutaneous brs.) Dorsal scapular n. 47 4 Neurovasculature Fig. 4.18 Neurovasculature of the back Posterior view. Removed: Muscle fascia (except posterior layer of thoracolumbar fascia); latis-simus dorsi (right). Reflected: Trapezius (right). Revealed: Transverse cervical artery in the deep scapular region. See p. 72 for the course of the intercostal vessels. Back 48 5 Sectional & Radiographic Anatomy Radiographic Anatomy of the Back (I) Fig. 5.1 MRI of the spine Sagittal view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Cervical vertebrae C1-C7 Thoracic vertebrae T1-T12 Lumbar vertebrae L1-L5 Sacrum (sacral vertebrae) S1-S5 Coccyx (coccygeal vertebrae) Co1-Co3 or Co4 Dens of axis (C2) Vertebra prominens (C7) Body of thoracic vertebra T1 Vertebral canal Spinal cord (thoracic part) Body of lumbar vertebra L1 Intervertebral disk Supraspinous lig. Cauda equina Interspinous ligs. Conus medullaris Spinous process Dural sac Nuchal lig. Sacrum (S1) Sacral promontory Coccyx Fig. 5.2 MRI of the lumbar spine Parasagittal view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Thoracic vertebral body T12 Mammillary process Erector spinae Inferior vena cava Spinal ganglion L2 Intervertebral disk L3/L4 (nucleus pulposus) Lamina Intervertebral foramen Superior articular process Inferior articular process Common iliac a. Zygapophyseal joint Promontory of sacrum Multifidus Gluteus maximus Lumbar vertebral body L2 Sacrum (S1) 5 Sectional & Radiographic Anatomy 49 Fig. 5.3 Radiograph of the cervical spine Lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 5.4 Radiograph of the thoracic spine Anteroposterior view. Lower thoracic region. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Transverse process Intervertebral disk Body of vertebra Spinous process Pedicle Superior articular facet Superior vertebral end plate Base of the skull Odontoid process Posterior arch of the atlas Inferior vertebral end plate Mandible Spinous process Anterior arch of the atlas Body of the axis Inferior articular facet Anterior inferior margin of the vertebra Transverse process Anterior superior margin of the vertebra Trachea Spinous process Articular pillar Lamina Intervertebral disk space Intervertebral facet joint Back 50 Radiographic Anatomy of the Back (II) Fig. 5.5 Radiograph of the lumbar spine Lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Inferior vertebral end plate Superior vertebral end plate Intervertebral foramen Facet joint Invertebral disk space Superior articular process Inferior articular process Promontory of sacrum Pedicle Fig. 5.6 Radiograph of the lumbar spine Oblique view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Inferior articular process Ribs Intervertebral disk space Ipsilateral transverse process Spinous process Pedicle Body of vertebra Intervertebral foramen Interarticular part Lamina Contralateral transverse process Superior articular process Body of vertebra Ribs Intervertebral disk space Ipsilateral transverse process Pedicle Interarticular part Lamina Contralateral transverse process Superior articular process Intravertebral foramen Inferior articular process Spinous process A B 5 Sectional & Radiographic Anatomy 51 Fig. 5.7 MRI of the sacrum I Oblique view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) External oblique Ileum Internal oblique Iliac aa. Transversus abdominis Common iliac a. and v. Psoas major Descending colon Iliacus Ilium (wing) 5th lumbar nerve root L5 vertebra, body Anterior sacroiliac ligs. Sacroiliac joint Gluteus medius Sacrum (lateral mass) Gluteus maximus Interosseous sacroiliac ligs. Anterior sacral foramina Posterior sacroiliac ligs. Sacral canal Fig. 5.8 MRI of the sacrum II Oblique view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Ascending colon L5 vertebra, body Sacrum (S1, body) Posterior sacroiliac ligs. Descending colon Psoas major Iliacus Dural sac Ilium, wing Gluteus maximus Spinal n. roots in sacral spinal canal 6 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7 Thoracic Wall Thoracic Skeleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Sternum & Ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Joints of the Thoracic Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Thoracic Wall Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Neurovasculature of the Diaphragm . . . . . . . . . . . . . . . . . . . . 66 Arteries & Veins of the Thoracic Wall . . . . . . . . . . . . . . . . . . . . 68 Nerves of the Thoracic Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Neurovascular Topography of the Thoracic Wall . . . . . . . . . . . 72 Female Breast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Lymphatics of the Female Breast . . . . . . . . . . . . . . . . . . . . . . . 76 8 Thoracic Cavity Divisions of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . 78 Arteries of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 80 Veins of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Lymphatics of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . 84 Nerves of the Thoracic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 86 9 Mediastinum Mediastinum: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Mediastinum: Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Thorax Heart: Functions & Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pericardium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Heart: Surfaces & Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Heart: Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Arteries & Veins of the Heart . . . . . . . . . . . . . . . . . . . . . . . . . 100 Conduction & Innervation of the Heart . . . . . . . . . . . . . . . . . 102 Pre- & Postnatal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Esophagus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Neurovasculature of the Esophagus . . . . . . . . . . . . . . . . . . . 108 Lymphatics of the Mediastinum . . . . . . . . . . . . . . . . . . . . . . . 110 10 Pulmonary Cavities Pulmonary Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Pleura: Subdivisions, Recesses & Innervation . . . . . . . . . . . . 114 Lungs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Bronchopulmonary Segments of the Lungs . . . . . . . . . . . . . 118 Trachea & Bronchial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Respiratory Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Pulmonary Arteries & Veins . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Neurovasculature of the Tracheobronchial Tree . . . . . . . . . . 126 Lymphatics of the Pleural Cavity . . . . . . . . . . . . . . . . . . . . . . 128 11 Sectional & Radiographic Anatomy Sectional Anatomy of the Thorax . . . . . . . . . . . . . . . . . . . . . . 130 Radiographic Anatomy of the Thorax (I). . . . . . . . . . . . . . . . . 132 Radiographic Anatomy of the Thorax (II). . . . . . . . . . . . . . . . 134 Radiographic Anatomy of the Thorax (III). . . . . . . . . . . . . . . . 136 Thyroid cartilage Jugular notch Serratus anterior Pectoralis major Deltopectoral groove Deltoid Supraclavicular fossa Sternocleidomastoid Presternal region Epigastric region (epi-gastrium) Hypochondriac region Lateral pectoral region Inframammary region Deltoid region Axillary region Pectoral region Clavipectoral triangle Infraclavicular fossa Midclavicular line (MCL) Greater and lesser tubercles Subcostal plane Xiphoid process Sternal angle Clavicle, medial head Coracoid process Supraclavicular fossa Thorax 54 6 Surface Anatomy Surface Anatomy Fig. 6.1 Regions of the thorax Anterior view. Fig. 6.2 Palpable structures of the thorax Anterior view. A Bony prominences. B Musculature. Anterior axillary line Midclavi-cular line Parasternal line Sternal line Anterior midline Anterior axillary line Posterior axillary line Midaxillary line Left lung Costomediastinal recesses of pleural cavities Cervical pleura (cupola) Parietal pleura Right lung Inferior border of lung Costo-diaphragmatic recesses of pleural cavities Inferior border of lung Parietal pleura Right lung Left lung Costo-diaphragmatic recesses of pleural cavities 6 Surface Anatomy 55 Fig. 6.3 Vertical reference lines of the thorax Fig. 6.4 Pleural cavities and lungs projected onto the thoracic skeleton A Anterior view. A Anterior view. B Right lateral view. B Posterior view. Clavicular notch Inferior thoracic aperture Costal cartilage Jugular notch Superior thoracic aperture Xiphoid process Sternum Body Sternal angle Manubrium Costal margin (arch) 1st rib Clavicular notch Sternum Vertebral body, L1 12th rib Vertebral body, T12 Intervertebral disk Spinous process, T1 Vertebral body, T1 Costal cartilage Costal margin (arch) Costal angle Costal tubercle 12th rib Spinous process, L1 Costo-transverse joint Transverse process T1 spinous process Spinous process, T12 56 7 Thoracic Wall Thoracic Skeleton Thorax The thoracic skeleton consists of 12 thoracic vertebrae (p. 10), 12 pairs of ribs with costal cartilages, and the sternum. In addition to participat-ing in respiratory movements, it provides a measure of protection to vital organs. The female thorax is generally narrower and shorter than the male equivalent. Fig. 7.1 Thoracic skeleton A Anterior view. B Left lateral view. C Posterior view. Vertebral body Vertebral foramen Costal angle Body (shaft) of rib Sternum Costal cartilage Costal tubercle Neck of rib Head of rib Transverse process Spinous process 57 7 Thoracic Wall Fig. 7.2 Structure of a thoracic segment Superior view of 6th rib pair. Fig. 7.3 Types of ribs Left lateral view. Rib type Ribs Anterior articulation True ribs 1–7 Sternum (costal notches) False ribs 8–10 Rib above Floating ribs 11, 12 None Elements of a thoracic segment Vertebra Rib Bony part (costal bone) Head Neck Costal tubercle Body (including costal angle) Costal part (costal cartilage) Sternum (articulates with costal cartilage of true ribs only; see Fig. 7.3) Table 7.1 Manubrium Sternal angle Body Xiphoid process Clavicular notch Jugular notch Manubrium Sternal angle Body 2nd through 7th costal notches 1st costal notch Clavicular notch Xiphoid process 58 Sternum & Ribs Thorax Fig. 7.4 Sternum The sternum is a dagger-like bone consisting of the manubrium (the handle), body (blade), and xiphoid process (tip of blade). The junction of the manubrium and body (the sternal angle) is typically elevated and marks the articulation of the second rib. The sternal angle (projected posteriorly to intersect with T4/5) is an important landmark for internal structures. A Anterior view. B Left lateral view. The costal notches are sites of articulation with the costal cartilage of the true ribs (see Fig. 7.3). Atlas (C1) Dens of axis (C2) Sternoclavicular joint Scapular notch Supraspinous fossa Acromion Acromio-clavicular joint Coracoid process Clavicle 1st rib 2nd rib Manubrium 12th rib 5th rib Head Neck Tubercle for anterior scalene Costal tubercle Body (shaft) Groove for subclavian a. Groove for subclavian v. Tuberosity for serratus anterior Crest of neck Head Neck Head Neck Crest of neck Costal tubercle Costal angle Head Costal angle Costal tubercle Body (shaft) 2nd rib 11th rib 5th rib 59 7 Thoracic Wall Fig. 7.5 Ribs Right ribs, superior view. See pp. 298–299 for joints of the shoulder. A Variations in rib size and shape. B 1st rib. Most ribs have a costal groove along the inferior border (See Fig. 7.24), which protects the intercostal vessels and nerves. C Right ribs, superior view. Infrasternal angle Transverse thoracic dimension Anteroposterior (AP) dimension Increase in Anteroposterior (AP) dimension Increase in transverse dimension Neck of rib Axis of movement Axis of movement Upper rib Lower rib Infrasternal angle Transverse thoracic dimension Anteroposterior (AP) dimension 60 Thorax Joints of the Thoracic Cage A Anterior view. B Left lateral view. C Position of diaphragm during respiration. Blue line = expiration, red line = inspiration. D Axes of rib movement, superior view. Expiration Inspiration The diaphragm is the chief muscle for quiet respiration (see p. 64). The muscles of the thoracic wall (see p. 62) contribute to deep (forced) inspiration. Fig. 7.6 Rib cage movement Full inspiration (red); full expiration (blue). In deep inspiration, there is an increase in transverse and anteroposterior (AP) dimensions, as well as the infrasternal angle. Note the red lines (inspiration dimensions) on the diagrams are longer than the blue ones (expiration dimensions) below. The downward movement of the diaphragm further increases the volume of the thoracic cavity. Clavicular notch 1st rib Sternum Costal cartilage Xiphoid process Costoxiphoid lig. Radiate sternocostal ligs. Joint space Superior articular facet Lateral costotrans-verse lig. Superior costotrans-verse lig. (cut) Costotransverse lig. Radiate lig. Intervertebral disk T8 Joint of head of rib Costal tubercle, articular surface Facet for rib tubercle Neck of 8th rib Costal tubercle Costotransverse joint Crest of rib head T5 Costal facets Intra-articular lig. Intervertebral disk Radiate lig. T8 8th rib (neck) Costal tubercle Superior costotransverse lig. Articular facets (on head of 7th rib) Costotransverse lig. (cut) Spinous process Lateral costotrans-verse lig. (cut) Facet for rib tubercle Transverse process 61 7 Thoracic Wall A Costotransverse joint. Superior view with joints of the left rib transversely sectioned. B Costovertebral joints. Left lateral view with the joint head of the 7th rib viewed in sagittal section.. Fig. 7.7 Sternocostal joints Anterior view with right half of sternum sectioned frontally. True joints are generally found only at ribs 2 to 5; ribs 1, 6, and 7 attach to the sternum by synchondroses. Fig. 7.8 Costovertebral joints Two synovial joints make up the costovertebral articulation of each rib. The costal tubercle of each rib articulates with the costal facet of its accompanying vertebra (A). The head of most ribs articulates with the vertebra of its own number and the vertebra immediately superior. Ribs 1, 11, and 12 typically articulate only with their own vertebrae. ②③ ① ④ ④ ⑥ ⑥ ⑤ ⑤ ⑦ 62 Thoracic Wall Muscle Facts Thorax The muscles of the thoracic wall are primarily responsible for ribcage movement during respiration, although other muscles aid in deep in-spiration: the pectoralis major and serratus anterior are discussed with the shoulder (see pp. 318–319), and the serratus posterior is discussed with the back (see p. 32). Fig. 7.9 Muscles of the thoracic wall A Scalene muscles, anterior view. B Intercostal muscles, anterior view. C Transversus thoracis, posterior view. Table 7.2 Muscles of the thoracic wall Muscle Origin Insertion Innervation Action Scalene mm. ① Anterior scalene m. C3–C6 (transverse processes, anterior tubercles) 1st rib (anterior scalene tubercle) Anterior rami of C4–C6 spinal nn. With ribs mobile: Elevates upper ribs (inspiration) With ribs fixed: Flexes cervical spine to same side (unilateral); flexes neck (bilateral) ② Middle scalene m. C1–C2 (transverse processes) C3–C7 (transverse processes, posterior tubercles) 1st rib (posterior to groove for subclavian a.) Anterior rami of C3–C8 spinal nn. ③ Posterior scalene m. C5–C7 (transverse processes, posterior tubercles) 2nd rib (outer surface) Anterior rami of C6–C8 spinal nn. Intercostal mm. ④ External intercostal mm. Lower margin of rib to upper margin of next lower rib (courses obliquely forward and downward from costal tubercle to chondro-osseous junction) 1st to 11th intercostal nn. Elevates ribs (inspiration); supports intercostal spaces; stabilizes chest wall ⑤ Internal intercostal mm. Lower margin of rib to upper margin of next lower rib (courses obliquely forward and upward from costal angle to sternum) Depresses ribs (expiration); supports intercostal spaces, stabilizes chest wall ⑥ Innermost intercostal mm. Subcostal mm. Lower margin of lower ribs to inner surface of ribs two to three ribs below Adjacent intercostal nn. Depresses ribs (expiration) ⑦ Transversus thoracis m. Sternum and xiphoid process (inner surface) 2nd to 6th ribs (costal cartilage, inner surface) 2nd to 6th intercostal nn. Weakly depresses ribs (expiration) Internal intercostal muscles External intercostal muscles Subcostal muscles Posterior scalene Middle scalene Anterior scalene External intercostal membrane Radiate sternocostal ligs. Atlas (C1) Axis (C2) 1st rib Sternum Internal intercostal muscles Anterior longi-tudinal lig. Costal cartilage External intercostal muscles Manubrium Anterior longi-tudinal lig. Anterior scalene Middle scalene Posterior scalene External intercostal muscles Manubrium Xiphoid process Costal cartilage Transversus thoracis Innermost intercostal muscles Chondro-osseous junction Internal intercostal muscles 7 Thoracic Wall 63 Fig. 7.10 Muscles of the thoracic wall Anterior view. The external intercostal muscles are replaced anteriorly by the external inter-costal membrane. The internal intercostal muscles are replaced posteriorly by the internal intercostal membrane. Fig. 7.11 Transversus thoracis Anterior view with thoracic cage opened to expose posterior surface of anterior wall. The external and internal intercostal membranes have been removed. Central tendon Diaphragm, right dome Aortic hiatus Transverse process, L1 10th rib Diaphragm, left dome Xiphoid process Sternum Right crus Left crus Diaphragm, sternal part (attaching posterior to the sternum) Diaphragm, costal part Diaphragm, lumbar part Caval opening Lumbocostal triangle Transverse process, L1 Scapula Clavicle 12th rib Diaphragm, costal part Diaphragm, lumbar part Esophageal hiatus Median arcuate lig. Lateral arcuate lig. Medial arcuate lig. Quadratus lumborum Psoas minor Psoas major Transversus abdominis Aortic hiatus Left dome Central tendon Caval opening Right crus Right dome Diaphragm, costal part Diaphragm, lumbar part Left crus 64 Thorax Diaphragm Fig. 7.12 Diaphragm The diaphragm, which separates the thorax from the abdomen, has two asymmetric domes and three apertures (for the aorta, vena cava, and esophagus; see Fig. 7.13C). A Anterior view. B Posterior view. C Coronal section with diaphragm in intermediate position. Table 7.3 Diaphragm Muscle Origin Insertion Innervation Action Diaphragm Costal part 7th to 12th ribs (inner surface; lower margin of costal arch) Central tendon Phrenic n. (C3–C5, cervical plexus) Principal muscle of respiration (diaphragmatic and thoracic breathing); aids in compressing abdominal viscera (abdominal press) Lumbar part Medial part: L1–L3 vertebral bodies, intervertebral disks, and anterior longitudinal lig. as right and left crura Lateral parts: lateral and medial arcuate ligs. Sternal part Xiphoid process (posterior surface) Intercostal muscles Aortic hiatus Sternum Diaphragm, costal part T8 Intrinsic back muscles Rib Parietal pleura, costal part Caval opening Esophageal hiatus Diaphragm, sternal part Central tendon Latissimus dorsi Rectus abdominis Sternocostal triangle Sternum Central tendon Caval opening Median arcuate lig. Aortic hiatus External oblique Internal oblique Transversus abdominis Quadratus lumborum Psoas major Vertebral body Intrinsic back muscles Medial arcuate lig. Lateral arcuate lig. Lumbocostal (Bochdalek’s) triangle Left crus Esophageal hiatus Diaphragm, costal part Diaphragm, sternal part Right crus Aorta T12 Esophagus T10 Inferior vena cava T8 65 7 Thoracic Wall Fig. 7.13 Diaphragm in situ B Superior view. A Inferior view. C Diaphragmatic apertures, left lateral view. Internal thoracic a. Azygos v. Musculophrenic a. Superior phrenic aa. Inferior phrenic aa. Left phrenic n. Hemiazygos v. Superior vena cava Left brachiocephalic v. Left subclavian a. and v. Left external jugular v. Left internal jugular v. Inferior thyroid v. Inferior vena cava Celiac trunk Left phrenic n. Left common carotid a. Posterior intercostal vv. Pericardiacophrenic a. Right phrenic n. Accessory hemiazygous v. Afferent (somatic sensory) fibers Efferent (somatic motor) fibers From parietal pleura, mediastinal part Pericardial brs. From parietal pleura, diaphragmatic part Intercostal nn. Left phrenic n. Rib Intercostal mm. Diaphragm Anterior scalene C3 C4 C5 Phrenico-abdominal br. of phrenic n. Right phrenic n. 66 Thorax Neurovasculature of the Diaphragm Fig. 7.14 Neurovasculature of the diaphragm Anterior view of opened thoracic cage. Fig. 7.15 Innervation of the diaphragm Anterior view. The phrenic nerves lie on the lateral surfaces of the fibrous pericardium together with the pericardiacophrenic arter-ies and veins. Note: The phrenic nerves also innervate the pericardium. Esophagus Parietal pleura, mediastinal part Left superior phrenic a. (from thoracic aorta) Parietal pleura, diaphragmatic part Hemiazygos v. Thoracic aorta Azygos v. Right superior phrenic a. Inferior vena cava Parietal pleura, costal part Musculophrenic a. (from internal thoracic a.) Pericardium Internal thoracic a. and vv. Phrenic n., pericardiacophrenic a. and v. Sympathetic trunk Spinal cord Intercostal n. Parietal pleura, costal part Left superior suprarenal a. Quadratus lumborum Psoas major Diaphragm, central tendon Celiac trunk Splenic a. Intrinsic back muscles Lumbar vertebra Left ascending lumbar v. Greater splanchnic n. Left inferior phrenic a. Left phrenic n. Esophageal hiatus Phrenic n., phrenico- abdominal br. Sternum Abdominal aorta Right superior suprarenal a. Caval opening Right phrenic n. Diaphragm, costal part Right inferior phrenic a. Diaphragm, lumbar part External and internal oblique muscles, tranversus abdominis Rectus abdominis Common hepatic a. Spinal cord 67 7 Thoracic Wall Fig. 7.16 Arteries and nerves of the diaphragm Note: The margins of the diaphragm receive sensory innervation from the lowest intercostal nerves. A Superior view (~T8). B Inferior view (~T12). Removed: Parietal peritoneum. Table 7.4 Blood vessels of the diaphragm Artery Origin Vein Drainage Inferior phrenic aa. (chief blood supply) Abdominal aorta; occasionally from celiac trunk Inferior phrenic vv. Inferior vena cava Superior phrenic aa. Thoracic aorta Superior phrenic vv. Azygos v. (right side), hemiazygos v. (left side) Pericardiacophrenic aa. Internal thoracic aa. Pericardiacophrenic vv. Internal thoracic vv. or brachiocephalic vv. Musculophrenic aa. Musculophrenic vv. Internal thoracic vv. Superior thoracic a. Thoracoacromial a. Axillary a. Lateral thoracic a. Aortic bifurcation 2nd and 3rd lumbar aa. Abdominal aorta Anterior intercostal a. Medial mammary br. Dorsal br. Thoracic aorta 2nd intercostal a. Internal thoracic a. Left sub-clavian a. Vertebral a. Left common carotid a. Musculophrenic a. Superior epigastric a. Anterior intercostal aa. Thyrocervical trunk Posterior intercostal a. Collateral br. Lateral cutaneous br. Thoraco-dorsal a. Internal thoracic a. Posterior intercostal a. Sternal brs. Thoracic aorta Internal thoracic a. Medial cutaneous br. Dorsal branch of posterior intercostal a. Anterior intercostal a. Lateral cutaneous br. Posterior intercostal a. Lateral cutaneous br. Spinal br. Anterior cutaneous br. 68 Arteries & Veins of the Thoracic Wall The posterior intercostal arteries anastomose with the anterior intercostal arteries to supply the structures of the thoracic wall. The posterior intercostal arteries branch from the thoracic aorta, with the exception of the 1st and 2nd, which arise from the superior intercostal artery (a branch of the costocervical trunk). Fig. 7.17 Arteries of the thoracic wall Anterior view. Table 7.5 Arteries of the thoracic wall Origin Branch Axillary a. Lateral thoracic a. Thoracoacromial a. Subclavian a. Posterior intercostal aa. (1st and 2nd; see Fig. 4.1, p. 36) Superior thoracic a. Thoracic aorta Posterior intercostal aa. (3rd through 12th) Internal thoracic a. Anterior intercostal aa. Musculophrenic a. Superior epigastric a. Fig. 7.18 Branches of the intercostal arteries Superior view. Table 7.6 Branches of the intercostal arteries Artery Branches Supplies Posterior intercostal aa. Dorsal br. Spinal br. Spinal cord Medial cutaneous br. Posterior thoracic wall Lateral cutaneous br. Collateral br. Lateral thoracic wall Lateral cutaneous br. Anterior thoracic wall Anterior intercostal aa. The lateral mammary br. from the lateral cutaneous br. supplies the breast along with the medial mammary br. from the internal thoracic a. Thorax Right internal jugular v. Right subclavian v. Right brachiocephalic v. Superior vena cava Accessory hemiazygos v. Inferior vena cava 1st lumbar v. Anterior intercostal vv. Subcostal v. (12th intercostal v.) Hemiazygos v. Azygos v. Internal thoracic v. Internal thoracic vv. Posterior intercostal vv. Left brachiocephalic v. Left subclavian v. Azygos v. Posterior intercostal vv. Anterior external venous plexus Anterior and posterior internal vertebral venous plexus Great saphenous v. External pudendal v. Femoral v. Superficial circumflex iliac v. Superficial epigastric v. Periumbilical vv. Thoracoepigastric v. Areolar venous plexus Cephalic v. Axillary v. Subclavian v. Internal jugular v. External jugular v. External iliac v. Common iliac v. Inferior vena cava Azygos v. Superior vena cava 69 7 Thoracic Wall The intercostal veins drain primarily into the azygos system, but also into the internal thoracic vein. This blood ultimately returns to the heart via the superior vena cava. The intercostal veins follow a similar course to their arterial counterparts. However, the veins of the vertebral column form an external vertebral venous plexus that traverses the entire length of the spine (see p. 37). Fig. 7.19 Veins of the thoracic wall A Anterior view with rib cage opened. B Vertebral venous plexus, anterior view. Fig. 7.20 Superficial veins Anterior view. The thoracoepigastric veins are a potential superficial collateral venous drain-age route in the event of superior or inferior vena cava obstruction. Anterior cutaneous brs. Iliohypogastric n., lateral cutaneous br. Supraclavicular nn. Intercostal nn. Lateral cutaneous brs. Medial cutaneous brs. Lateral cutaneous brs. Intercostal nn., lateral cutaneous brs. Supraclavicular nn. Spinal nn., dorsal rami Superior clunial nn. Posterior ramus T1 Intercosto-brachial nn. Anastomosis with medial brachial cutaneous n. Lateral cuta-neous br. Anterior cuta-neous br. 1st and 2nd intercostal nn. 3rd and 4th intercostal nn. Sternal brs. Subcostal n. (12th intercostal n.) 70 Thorax Nerves of the Thoracic Wall Fig. 7.21 Intercostal nerves Anterior view. The 1st rib has been removed to reveal the 1st and 2nd intercostal nerves. Fig. 7.22 Cutaneous innervation of the thoracic wall A Anterior view. B Posterior view. L1 T10 T2 C3 C4 T6 T4 L1 T2 C4 C5 C3 C5 Parietal pleura, costal part Intercostal v., a., and n. Visceral pleura Liver Right lung 8th rib Costal groove External intercostal Internal intercostal Innermost intercostal Endothoracic fascia Diaphragm Posterior root Posterior ramus Anterior ramus (intercostal n.) Anterior root Anterior cutaneous br. Lateral cutaneous br. Sympathetic ganglion Meningeal br. Sensory (spinal) ganglion White ramus communicans Gray ramus communicans 71 7 Thoracic Wall Fig. 7.23 Spinal nerve branches Superior view. The spinal nerve is formed by the union of posterior (dorsal) and anterior (ventral) roots. The posterior root contains sen-sory fibers and the anterior root contains motor fibers. The spinal nerve and all its subsequent branches are mixed nerves, containing both motor and sensory fibers. The spinal nerve exits the vertebral canal via the intervertebral foramen. Its posterior ramus innervates the skin and intrinsic muscles of the back; its anterior ramus forms the cervical, brachial, lumbar, and sacral plexuses, and the intercostal nerves. See p. 38 for more details. Fig. 7.24 Arrangement of intercostal neurovascular bundle Coronal section, anterior view. Fig. 7.25 Dermatomes of the thoracic wall Landmarks: T4 generally includes the nipple; T6 innervates the skin over the xiphoid. A Anterior view. B Posterior view. Visceral pleura Rib Pleural space Innermost intercostal Internal and external intercostal muscles Puncture site Endothoracic fascia Intercostal v., a., and n. Costal groove Parietal pleura Pleural effusion Chest tube Pectoralis major Lateral thoracic a. and v. Intercostal a., v., and n. Internal oblique Rectus abdominis Superior epigastric a. and v. Thoracoepigastric v. Cephalic v. Internal thoracic a. and v. Axillary a. and v. External jugular v. Deltoid Median n. Ulnar n. External oblique Anterior cutaneous brs. Lateral cutaneous brs. Intercostal aa., vv., and nn. 72 Thorax Neurovascular Topography of the Thoracic Wall Fig. 7.26 Anterior structures Anterior view (see Chapter 4 for neurovasculature of the back). Clinical box 7.1 Abnormal fluid collection in the pleural space (e.g., pleural effusion due to bronchial carcinoma) may necessitate the insertion of a chest tube. Generally, the optimal puncture site in a sitting patient is at the level of the 4th or 5th intercostal space in the mid to anterior axillary line, immediately behind the lateral edge of the pectoralis major. The drain should always be introduced at the upper margin of a rib to avoid injuring the intercostal vein, artery, and nerve. See Clinical box 10.5 on p. 123 for details on collapsed lungs. Insertion of a chest tube A Coronal section, anterior view. B Drainage tube is inserted perpendicular to chest wall. D At the superior margin of the rib, the tube is passed through the intercostal muscles and advanced into the pleural cavity. C At ribs, the tube is angled and advanced parallel to the chest wall in the subcutaneous plane. Central tendon of diaphragm Innermost intercostal Internal intercostal Intercostal n., collateral br. Parietal pleura, costal part Serratus anterior Intercostal nn., anterior rami Diaphragm Intercostal n., lateral cutaneous br. External oblique Intercostal n., anterior cutaneous br. Azygos v. Internal thoracic a. and vv. Inferior vena cava Esophagus Thoracic aorta Posterior intercostal aa. and vv. Intercostal v., posterior br. Spinal cord (with spinal ganglion) Intrinsic back muscles External intercostal Latissimus dorsi Pericardial sac Parietal pleura, diaphragmatic part Sternum Anterior perforating br. Costal groove Phrenic n., pericardiaco-phrenic a. and v. Musculophrenic a. (from internal thoracic a.) Right superior phrenic a. 73 7 Thoracic Wall Fig. 7.27 Intercostal structures in cross section Transverse section, anterosuperior view. The relationship of the inter-costal vessels in the costal groove, from superior to inferior, is vein, artery, and nerve (see clinical box, p. 72). Nipple Areolar glands Areola Intercostal nn., lateral mammary brs. Intercostal nn., medial mammary brs. Supraclavicular nn. Axillary a. and v. Lateral thoracic a. and v. Lateral mammary brs. Medial mammary brs. Perforating brs. Internal thoracic a. and v. Subclavian a. and v. Mammary brs. 74 Female Breast Thorax The female breast, a modified sweat gland in the subcutaneous tissue layer, consists of glandular tissue, fibrous stroma, and fat. The breast extends from the 2nd to the 6th rib and is loosely attached to the pec- toral, axillary, and superficial abdominal fascia by connective tissue. The breast is additionally supported by suspensory ligaments. An extension of the breast tissue into the axilla, the axillary tail, is generally present. Fig. 7.28 Female breast Right breast, anterior view. Fig. 7.29 Mammary ridges Rudimentary mammary glands form in both sexes along the mammary ridges. Occasionally, these may persist in humans to form accessory nipples (polythelia), although only thoracic nipples normally remain. Fig. 7.30 Blood supply to the breast Fig. 7.31 Sensory innervation of the breast Pectoralis major Pectoralis minor Intercostal muscles Intercostal v., a., and n. Superficial thoracic fascia Interlobular connective tissue Lactiferous duct Lactiferous sinus Nipple Mammary lobes Suspensory (Cooper’s) ligs. Pectoral fascia Mammary lobes Terminal duct lobular unit (TDLU) Acini Terminal duct Lactiferous sinus Lactiferous duct Lobules 75 7 Thoracic Wall The glandular tissue is composed of 10 to 20 individual lobes, each with its own lactiferous duct. The gland ducts open on the elevated nipple at the center of the pigmented areola. Just proximal to the duct opening is a dilated portion called the lactiferous sinus. Areolar elevations are the openings of the areolar glands (sebaceous). The glands and lactiferous ducts are surrounded by firm, fibrofatty tissue with a rich blood supply. Fig. 7.32 Structures of the breast A Sagittal section along midclavicular line. B Duct system and portions of a lobe, sagittal section. In the nonlac-tating breast (shown here), the lobules contain clusters of rudimen-tary acini. C Terminal duct lobular unit (TDLU). The clustered acini composing the lobule empty into a terminal ductule; these structures are collec-tively known as the TDLU. Axillary lymphatic plexus Subclavian a. Omohyoid, inferior belly (cut) Apical axillary l.n. Central axillary l.n. Axillary a. Brachial v. Pectoralis major Cubital l.n. Supratrochlear l.n. Brachial a. Brachial l.n. Basilic v. Subscapular axillary l.n. Humeral axillary l.n. Pectoral axillary l.n. Lateral thoracic v. Interpectoral axillary l.n. Pectoralis major Pectoralis minor Clavicle Internal jugular v. Cervical l.n. Supraclavicular l.n. Latissimus dorsi Biceps brachii Level III Level II Level I Interpectoral axillary l.n. Parasternal l.n. 76 Thorax Lymphatics of the Female Breast The lymphatic vessels of the breast (not shown) are divided into three systems: superficial, subcutaneous, and deep. These drain primar-ily into the axillary lymph nodes, which are classified based on their relationship to the pectoralis minor (Table 7.7). The medial portion of the breast is drained by the parasternal lymph nodes, which are associ-ated with the internal thoracic vessels. Fig. 7.33 Axillary lymph nodes A Lymphatic drainage of the breast. See Table 7.7 for explanation of level I, II, and III. B Anterior view. Table 7.7 Levels of axillary lymph nodes Level Position Lymph nodes (l.n.) I Lower axillary group Lateral to pectoralis minor Pectoral axillary l.n. Subscapular axillary l.n. Humeral axillary l.n. II Middle axillary group Along pectoralis minor Central axillary l.n. Interpectoral axillary l.n. III Upper infraclavicular group Medial to pectoralis minor Apical axillary l.n. Terminal duct lobular unit (TDLU) Acini Terminal duct Lactiferous sinus Lactiferous duct Lobules ≈15% ≈ 5% ≈10% ≈10% ≈ 60% Nipple 77 7 Thoracic Wall Clinical box 7.2 Stem cells in the intralobular connective tissue give rise to tremendous cell growth, necessary for duct system proliferation and acini differentiation. This makes the terminal duct lobular unit (TDLU) the most common site of origin of malignant breast tumors. Breast cancer A Terminal duct lobular unit. B Origin of malignant tumors by quadrant. Tumors originating in the breast spread via the lymphatic vessels. The deep system of lymphatic drainage (level III) is of particular importance, although the parasternal lymph nodes provide a route by which tumor cells may spread across the midline. The survival rate in breast cancer correlates most strongly with the number of lymph nodes involved at the axillary nodal level. Metastatic involvement is gauged through scintigraphic mapping with radiolabeled colloids (technetium [Tc] 99m sulfur microcolloid). The downstream sentinel node is the first to receive lymphatic drainage from the tumor and is therefore the first to be visualized with radiolabeling. Once identified, it can then be removed (via sentinel lymphadenectomy) and histologically examined for tumor cells. This method is 98% accurate in predicting the level of axillary nodal involvement. C Normal mammogram. D Mammogram of invasive ductal carcinoma (irregular white areas, arrows). The large lesion has changed the architecture of the neighboring breast tissue. Metastatic involvement 5-year survival rate Level I 65% Level II 31% Level III ~0% Diaphragm Thoracic inlet Superior mediastinum Left lung in left pulmonary cavity Inferior mediastinum Right lung in right pulmonary cavity Thoracic outlet Pericardiacophrenic a. and v., pericardial branches Left vagus n. Left phrenic n. Thyroid cartilage Thyroid gland, right lobe Right vagus n. Right common carotid a. Internal jugular v. Anterior scalene Right phrenic n. Brachial plexus First rib Right brachio-cephalic v. Brachiocephalic trunk Superior vena cava Right lung Thymus Pericardiaco-phrenic a. and v., phrenic n. Phrenic n., pericardial branches Fibrous pericardium Left pulmonary a. Left vagus n. Aortic arch Left brachio-cephalic v. Internal thoracic a. and v. Trachea Recurrent laryngeal n. Left subclavian a. and. v. Parietal pleura (diaphragmatic part) Parietal pleura, mediastinal part Left lung Thorax 78 The thoracic cavity is divided into three large spaces: the mediastinum (p. 90) and the two pleural (pulmonary) cavities (p. 112). 8 Thoracic Cavity Divisions of the Thoracic Cavity Fig. 8.1 Thoracic cavity Coronal section, anterior view. A Divisions of the thoracic cavity. B Opened thoracic cavity. Removed: Thoracic wall; connective tissue of anterior mediastinum. Table 8.1 Major structures of the thoracic cavity Mediastinum Superior mediastinum Thymus, great vessels, trachea, esophagus, and thoracic duct Inferior mediastinum Anterior Thymus (especially in children) Middle Heart, pericardium, and roots of great vessels Posterior Thoracic aorta, thoracic duct, esophagus, and azygos venous system Pulmonary cavities Right pulmonary cavity Right lung Left pulmonary cavity Left lung Left lung Esophagus Middle mediastinum Sternum Anterior mediastinum Posterior mediastinum Right lung Descending aorta Thoracic vertebra Thoracic part Esophagus (cervical part) Esophagus (thoracic part) Diaphragm Middle mediastinum Anterior mediastinum Sternum Superior mediastinum Cervical part Posterior mediastinum Trachea Thoracic inlet Superior vena cava Ascending aorta Right and left main bronchi Esophagus Descending aorta Inferior vena cava Azygos v. Esophagus Descending aorta 8 Thoracic Cavity 79 Fig. 8.2 Divisions of the mediastinum Fig. 8.3 Transverse sections of the thorax Computed tomography (CT) scan of thorax, inferior view. B Transverse section, inferior view. A Superior mediastinum. B Inferior mediastinum. A Midsagittal section, lateral view. Thyroid cartilage Right common carotid a. Anterior scalene Middle scalene Internal thoracic a. 1st rib Brachiocephalic trunk Ascending aorta Right main bronchus Esophageal branch Posterior intercostal aa. Diaphragm Inferior phrenic a. Celiac trunk Lumbar a. Aortic hiatus Left main bronchus Bronchial a. Aortic arch Esophagus Left subclavian a. Thyrocervical trunk Left common carotid a. Trachea Right vertebral a. Right subclavian a. Thoracic aorta Abdominal aorta Thorax 80 The arch of the aorta has three major branches: the brachiocephalic trunk, left common carotid artery, and left subclavian artery. After the aortic arch, the aorta begins its descent, becoming the thoracic aorta at the level of the sternal angle and the abdominal aorta once it passes through the aortic hiatus in the diaphragm. Arteries of the Thoracic Cavity Fig. 8.4 Thoracic aorta A Thoracic aorta in situ, anterior view. Removed: Heart, lungs, portions of diaphragm. Left pulmonary a. Descending aorta Aortic arch Left subclavian a. Left common carotid a. Esophagus Trachea Brachio-cephalic trunk Pulmonary trunk Left main bronchus Ascending aorta “False lumen” Descending aorta “False lumen” Ascending aorta Intima 8 Thoracic Cavity 81 B Parts of the aorta, left lateral view. Note: The aortic arch begins and ends at the level of the sternal angle (see p. 58). Clinical box 8.1 A tear in the inner wall (intima) of the aorta allows blood to separate the layers of the aortic wall, creating a “false lumen” and potentially resulting in life-threatening aortic rupture. Symptoms are dyspnea (shortness of breath) and sudden onset of excruciating pain. Acute aortic dissections occur most often in the ascending aorta and generally require surgery. More distal aortic dissections may be treated conservatively, provided there are no complications (e.g., obstruction of blood supply to the organs, in which case a stent may be inserted to restore perfusion). Aortic dissections occurring at the base of a coronary artery may cause myocardial infarction. Aortic dissection A Aortic dissection. Parts of the intima are still attached to the connective tissue in the wall of the aorta (arrow). B The flow in the coronary arteries is intact (arrow). Table 8.2 Branches of the thoracic aorta The thoracic organs are supplied by direct branches from the thoracic aorta, as well as indirect branches from the subclavian arteries. Part of aorta Branches Region supplied Ascending aorta Right and left coronary aa. Heart Bronchi, trachea, esophagus Arch of aorta Brachiocephalic trunk Right subclavian a. See left subclavian a. Right common carotid a. Head and neck Left common carotid a. Left subclavian a. Vertebral a. Internal thoracic a. Anterior intercostal aa. Anterior chest wall Thymic brs. Thymus Mediastinal brs. Posterior mediastinum Pericardiacophrenic a. Pericardium, diaphragm Thyrocervical trunk Inferior thyroid a. Esophagus, trachea, thyroid gland Costocervical trunk Superior intercostal a. Chest wall Descending aorta Visceral brs. Bronchi, trachea, esophagus Parietal brs. Posterior intercostal aa. Posterior chest wall Superior phrenic aa. Diaphragm Right internal thoracic v. Right supreme intercostal v. Right subclavian v. 1st rib Right brachiocephalic v. Azygos v. Posterior intercostal vv. Diaphragm, central tendon Diaphragm, costal part Right ascending lumbar v. Lumbar vv. Esophageal hiatus Caval opening Hemiazygos v. Accessory hemiazygos v. Superior vena cava Left brachiocephalic v. Left subclavian v. Left external jugular v. Middle scalene Anterior scalene Left internal jugular v. Inferior thyroid v. Inferior vena cava Aortic hiatus Left ascending lumbar v. Thorax 82 The superior vena cava is formed by the union of the two brachio cephalic veins at the level of the T2–T3 junction. It receives blood drained by the azygos system (the inferior vena cava has no tributaries in the thorax). Veins of the Thoracic Cavity Fig. 8.5 Superior vena cava and azygos system A Veins of the thoracic cavity (viscera removed), anterior view of opened thorax (posterior thoracic wall). Left pulmonary vv. Left brachio-cephalic v. Right brachio-cephalic v. Right internal jugular v. Right subclavian v. Superior vena cava Right pulmonary vv. Inferior vena cava Posterior inter-costal vv. Hepatic vv. Azygos v. Inferior vena cava Right ascending lumbar v. Left common iliac v. Lumbar vv. Left ascending lumbar v. Hemiazygos v. Accessory hemiazygos v. Superior vena cava Left brachio-cephalic v. Inferior thyroid v. Right internal jugular v. Right subclavian v. Right testicular/ ovarian v. Left renal v. Diaphragm 8 Thoracic Cavity 83 B Projection of venae cavae onto chest, anterior view. Table 8.3 Thoracic tributaries of the superior vena cava Major vein Tributaries Region drained Brachiocephalic vv. Inferior thyroid v. Esophagus, trachea, thyroid gland Internal jugular vv. Head, neck, upper limb External jugular vv. Subclavian vv. Supreme intercostal vv. Pericardial vv. Left superior intercostal v. Azygos system (left side: accessory hemiazygos v.; right side: azygos v.) Visceral brs. Trachea, bronchi, esophagus Parietal brs. Posterior intercostal vv. Inner chest wall and diaphragm Superior phrenic vv. Right superior intercostal v. Internal thoracic v. Thymic vv. Thymus Mediastinal tributaries Posterior mediastinum Anterior intercostal vv. Anterior chest wall Pericardiacophrenic v. Pericardium Musculophrenic v. Diaphragm Note: Structures of the superior mediastinum may also drain directly to the brachiocephalic veins via the tracheal, esophageal, and mediastinal veins. Fig. 8.6 Azygos system Anterior view. The left testicular/ovarian vein drains to the left renal vein. Right lymphatic duct Jugular trunk Subclavian a. Subclavian trunk Right subclavian v. Brachiocephalic trunk Right brachio-cephalic v. Bronchomediastinal trunk Superior vena cava Ascending aorta Thoracic duct Azygos v. Right lumbar trunk Left lumbar trunk Abdominal aorta Celiac trunk Aortic hiatus Hemiazygos v. Intercostal lymphatics Thoracic aorta Bronchomediastinal trunk Left brachio-cephalic v. Left subclavian v. Subclavian trunk Thoracic duct Jugular trunk Internal jugular v. Cisterna chyli Diaphragm Common carotid a. Accessory hemiazygos v. Thorax 84 The body’s chief lymph vessel is the thoracic duct. Beginning in the abdomen at the level of L1 at the cisterna chyli, the thoracic duct empties into the junction of the left internal jugular and subclavian veins. The right lymphatic duct drains to the right junction of the internal jugular and subclavian veins. Lymphatics of the Thoracic Cavity Fig. 8.7 Lymphatic trunks in the thorax Anterior view of opened thorax. Paraesophageal l.n. Paratracheal l.n. Intercostal l.n. Lymphatics in trunk wall Parasternal l.n. Tracheobronchial l.n. Bronchopulmonary l.n. Intrapulmonary l.n. Thoracic duct Right lymphatic duct Anterior mediastinum Anterior intercostal spaces Anterior thoracic wall Mammary gland Paravertebral I.n. Parasternal I.n. Paramammary I.n. Brachiocephalic I.n. Pre-pericardial I.n. Lateral pericardial I.n. Paraesophageal I.n. Paratracheal I.n. Tracheobronchial I.n. Bronchopulmonary I.n. Intrapulmonary I.n. Superior phrenic I.n. Abdomen, pelvis, and lower limb Head and neck Posterior thoracic wall Posterior intercostal spaces Superior mediastinum Right jugular trunk Right lymphatic duct Right subclavian trunk Right broncho-mediastinal trunk Left jugular trunk Left internal jugular v. Left subclavian trunk Left broncho-mediastinal trunk Left subclavian v. Thoracic duct Cisterna chyli Diaphragm 8 Thoracic Cavity 85 Fig. 8.9 Lymphatic pathways in the thorax Fig. 8.8 Lymphatic drainage pattern Fig. 8.10 Thoracic lymph nodes Transverse section at the level of the tracheal bifurcation (T4–T5), superior view. The tho-racic lymph nodes can be divided into three broad groups: nodes of the thoracic wall (pink), pulmonary nodes (blue), and medias-tinal nodes (green). For details of lymphatics of the mediastinum, see pp. 110–111. Right vagus n. Right phrenic n. Sympathetic trunk, thoracic ganglion Left phrenic n. Left vagus n. Posterior intercostal nn. Greater splanchnic n. Right subclavian a. Esophagus, thoracic part Innermost intercostals Sympathetic trunk, middle cervical ganglion Right vagus n. Right recurrent laryngeal n. Brachiocephalic trunk Sympathetic trunk Posterior intercostal a. Intercostal n. Anterior vagus br. Stomach Anterior esophageal plexus Thoracic aorta Aortic arch 1st rib Left subclavian a. Brachial plexus Scalene mm. Left vagus n. Common carotid a. Left recurrent laryngeal n. Esophagus, cervical part Trachea Sympathetic trunk Diaphragm Thorax 86 Thoracic innervation is mostly autonomic, arising from the para vertebral sympathetic trunks and parasympathetic vagus nerves. There are two exceptions: the phrenic nerves innervate the pericardium and diaphragm (p. 66) and the intercostal nerves innervate the thoracic wall (p. 70). Nerves of the Thoracic Cavity Fig. 8.11 Nerves in the thorax Anterior view of opened thorax. A Thoracic innervation. B Nerves of the thorax in situ. Lungs, pericardial sac, heart and costal pleura removed. Note: The recurrent laryngeal nerves have been slightly anteriorly retracted; normally, they occupy the groove between the trachea and the esopha gus, making them vulnerable during thyroid gland surgery. 8 Thoracic Cavity 87 Fig. 8.12 Sympathetic and parasympathetic nervous systems in the thorax Table 8.4 Peripheral sympathetic nervous system Origin of pre- ganglionic fibers Ganglion cells Course of post- ganglionic fibers Target Spinal cord Sympathetic trunk Follow intercostal nn. Blood vessels and glands in chest wall Accompany intrathoracic aa. Visceral targets Gather in greater and lesser splanchnic nn. Abdomen The axons of preganglionic neurons exit the spinal cord via the anterior roots and synapse with postganglionic neurons in the sympathetic ganglia. Table 8.5 Peripheral parasympathetic nervous system Origin of pre- ganglionic fibers Course of preganglionic motor axons Target Brainstem Vagus n. (CN X) Cardiac brs. Cardiac plexus Esophageal brs. Esophageal plexus Tracheal brs. Trachea Bronchial brs. Pulmonary plexus (bronchi, pulmonary vessels) The ganglion cells of the parasympathetic nervous system are scattered in microscopic groups in their target organs. The vagus n. thus carries the preganglionic motor axons to these targets. CN = cranial n. The autonomic nervous system innervates smooth muscle, cardiac muscle, and glands. It is subdivided into the sympathetic (red) and parasympathetic (blue) nervous systems, which together regulate blood flow, secretions, and organ function. Sympathetic nervous system T10 T9 T8 T7 T6 T5 T4 T3 T2 T1 L1 T12 T11 L2 Superior cervical ganglion Middle cervical ganglion Stellate ganglion Cervical cardiac nn. External carotid plexus Internal carotid plexus Common carotid plexus Vertebral plexus Subclavian plexus Thoracic aortic plexus Pulmonary plexus Cardiac plexus Greater and lesser splanchnic n. Sympathetic trunk To abdomen Cardiac brs. Vagal trunks Pulmonary plexus Esophageal plexus Pharyngeal plexus Larynx Recurrent laryngeal n. Superior laryngeal n. Vagus n. (CN X) Parasympathetic nervous system Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Esophageal inlet Azygos v. Left main bronchus, origin Tracheobronchial l. n. Right pulmonary a. Esophagus, thoracic part Left atrium Superior phrenic l. n. Diaphragm Site of attachment between liver and diaphragm (bare area) Liver Xiphoid process Pericardial cavity Sternum Aortic valve Thymus (retrosternal fat pad) Ascending aorta Manubrium Left brachiocephalic v. Investing layer Brachiocephalic l. n. Pretracheal layer Thyroid cartilage Trachea Esophagus, cervical part Deep cervical fascia Thoracic inlet Esophagus (cervical part) Esophagus, thoracic part Diaphragm Middle mediastinum Anterior mediastinum Sternum Superior mediastinum Posterior mediastinum Thoracic part Cervical part Trachea 88 9 Mediastinum Mediastinum: Overview Thorax Fig. 9.1 Divisions of the mediastinum A Schematic. B Midsagittal section, right lateral view. The mediastinum is the space in the thorax between the pleural sacs of the lungs. It is divided into two parts: superior and inferior. The inferior mediastinum is further divided into anterior, middle, and posterior portions. Table 9.1 Contents of the mediastinum ● Superior mediastinum Inferior mediastinum ● Anterior ● Middle ● Posterior Organs • Thymus • Trachea • Esophagus • Thymus, inferior aspects (especially in children) • Heart • Pericardium • Esophagus Arteries • Aortic arch • Brachiocephalic trunk • Left common carotid a. • Left subclavian a. • Smaller vessels • Ascending aorta • Pulmonary trunk and brs. • Pericardiacophrenic aa. • Thoracic aorta and brs. Veins and lymph vessels • Superior vena cava • Brachiocephalic vv. • Thoracic duct and right lymphatic duct • Smaller vessels, lymphatics, and l.n. • Superior vena cava • Azygos v. • Pulmonary vv. • Pericardiacophrenic vv. • Azygos v. • Accessory hemiazygos and hemiazygos vv. • Thoracic duct Nerves • Vagus nn. • Left recurrent laryngeal n. • Cardiac nn. • Phrenic nn. • None • Phrenic nn. • Vagus nn. Brachial plexus Pulmonary trunk Superior vena cava Right pulmonary vv. Right pleural cavity Parietal pleura, diaphragmatic part Fibrous pericardium Esophagus, thoracic part Parietal pleura, mediastinal part Thoracic aorta Left pleural cavity Superior and inferior lobar bronchi Left pulmonary a. Ligamentum arteriosum Aortic arch Parietal pleura, cervical part Left subclavian a. and v. Left internal jugular v. Left brachio-cephalic v. Caval opening Pericardiacophrenic a. and v., phrenic n. Thyroid cartilage Thyroid gland, right lobe Anterior scalene Phrenic n. Parietal pleura, diaphragmatic part Diaphragm Attachment between fibrous pericardium and central tendon of the diaphragm Fibrous pericardium Parietal pleura, mediastinal part Left pulmonary a. Aorta Internal thoracic a. and v. Vagus n. (CN X) Left common carotid a. Left recurrent laryngeal n. Trachea Inferior thyroid v. Superior vena cava Pericardiaco-phrenic a. and v., phrenic n. Left recurrent laryngeal n. Left vagus n. Thymus Right pulmonary a. Left common carotid a. Posterior intercostal aa. Right pulmonary vv. Left subclavian a. and v. Fibrous pericardium, left atrium Fibrous pericardium, right atrium Inferior pharyngeal constrictor Left internal jugular v. Left pulmonary a. Left pulmonary vv. Diaphragm Inferior vena cava (in caval opening) Esophagus, thoracic part Right main bronchus Azygos v. Superior vena cava Trachea Esophagus, cervical part Thyroid gland, right lobe Thoracic aorta Esophageal hiatus Aortic arch Fibrous pericardium, left ventricle 89 9 Mediastinum Fig. 9.2 Contents of the mediastinum A Anterior view. The thymus extends into the anterior division of the inferior mediastinum and grows throughout childhood. At puberty, high levels of cir-culating sex hormones cause the thymus to atrophy leaving indistinguishable pieces embedded in the fat that now occupies the anterior mediastinum. B Anterior view with lungs, heart, pericardium, and thymus removed. C Posterior view. Brachiocephalic l.n. Left brachiocephalic v. 1st rib Right vagus n. Intercostal v., a., and n. White and gray rami communicantes Azygos v. Superior lobar bronchus Right pulmonary a. Parietal pleura, costal part Common trunk of middle and inferior lobar bronchi Intercostal mm. Esophagus Greater splanchnic n. Diaphragm (covered by parietal pleura, diaphragmatic part) Right pulmonary vv. Phrenic n., pericardiacophrenic a. and v. Fibrous pericardium Thymus (retrosternal fat pad) Right phrenic n. Superior vena cava Trachea Right brachiocephalic v. Right subclavian a. and v. Brachial plexus Clavicle Sympathetic trunk, thoracic ganglion Posterior intercostal v. and a., intercostal n. Brachiocephalic trunk Right recurrent laryngeal n. Mediastinal pleura 90 Mediastinum: Structures Thorax Fig. 9.3 Mediastinum A Right lateral view, parasagittal section. Note the many structures passing between the superior and inferior (middle and posterior) mediastinum. Left vagus n. Brachial plexus Clavicle Left subclavian a. and v. Ligamentum arteriosum Left superior intercostal v. Left vagus n. Left pulmonary a. Left phrenic n. Left pulmonary vv. Phrenic n., pericardiaco-phrenic a. and v. Lateral pericardial l.n. Parietal pleura, mediastinal part Superior phrenic l.n. Diaphragm (covered by parietal pleura, diaphragmatic part) Thoracic aorta (descending aorta) Intercostal mm. Left main bronchus Rami communicantes 1st rib Intercostal v., a., and n. Aortic arch Thoracic duct Hemiazygos v. Parietal pleura, costal part Sympathetic trunk Accessory hemiazygos v. Left recurrent laryngeal n. Esophagus Posterior intercostal v. and a., intercostal n. Greater splanchnic n. Splanchnic nn. 91 9 Mediastinum B Left lateral view, parasagittal section. Removed: Left lung and parietal pleura. Revealed: Posterior mediastinal structures. Cardiac apex Pulmonary trunk Ascending aorta Left subclavian a. and v. Left internal jugular v. Right common carotid a. Right brachio-cephalic v. Superior vena cava Diaphragm Superior vena cava Upper body circulation Hepatic vv. Right ventricle Pulmonary v. Left ventricle Portal v. Inferior vena cava Aorta Ascending aorta Right atrium Pulmonary circulation Portal circulation Lower body circulation Left atrium Pulmonary a. Brachiocephalic trunk Superior vena cava 2nd rib Sternum, body Pulmonary trunk Left pulmonary vv. Diaphragm Thoracic aorta Esophagus Left main bronchus Aortic arch Trachea Pericardial sac Abdominal aorta Left common carotid a. Left subclavian a. Sternum, manubrium Stomach 92 Thorax Heart: Functions & Relations B Left lateral view. Removed: Left thoracic wall and left lung. A Projection of the heart and great vessels onto chest, anterior view. Fig. 9.4 Circulation Oxygenated blood is shown in red; deoxygenated blood in blue. See p. 104 for prenatal circulation. The heart pumps the blood: unoxygenated blood to the lungs and oxygenated blood throughout the body. It is located posterior to the sternum in the middle portion of the mediastinum in the pericardial cavity, located between the right and left pleural cavities containing the lungs. The apex of the cone-shaped heart points anteriorly and to the left in the thoracic cavity. Fig. 9.5 Topographical relations of the heart Serous pericardium, visceral layer (epicardium) Aortic arch Superior vena cava Diaphragm Cardiac surface Parietal pleura, mediastinal part Left lung Serous pericardium, parietal layer Fibrous pericardium (= external layer) Brachio-cephalic trunk Right brachio-cephalic v. Right phrenic n. Superior vena cava Right lung Parietal pleura, mediastinal part Diaphragm Right auricle Fibrous pericardium Right ventricle Stomach Left ventricle Left auricle Pulmonary trunk Left pulmonary a. Ascending aorta Ligamentum arteriosum Left vagus n. Left brachio-cephalic v. Cardiac apex Anterior interventric-ular a. (LAD) Aortic arch Left phrenic n. 93 9 Mediastinum Fig. 9.6 Heart in situ A Anterior view of the opened thorax with the thymus removed and flaps of the anterior layer of the pericardial sac reflected to reveal the heart. B Anterior view of the opened thorax with thymus and anterior pericardium removed to reveal the heart. Superior vena cava Left auricle Heart, diaphragmatic surface Coronary sinus Pulmonary trunk Left vagus n. Inferior vena cava Left pulmonary vv. Right pulmonary v. Oblique pericardial sinus Pericardiacophrenic a. and v., left phrenic n. Ascending aorta Superior vena cava Right pulmonary vv. Inferior vena cava Sternum Serous pericardium, parietal layer Left pulmonary vv. Fibrous pericardium Parietal pleura, mediastinal part Left phrenic n. Attachment of fibrous pericardium to central tendon of diaphragm Pulmonary trunk Ligamentum arteriosum Left vagus n. Left recurrent laryngeal n. Ascending aorta Oblique pericardial sinus Transverse pericardial sinus 94 Pericardium Thorax Fig. 9.7 Posterior pericardial cavity Anterior view of opened thorax with the anterior pericardium removed. The heart has been partially elevated to reveal the posterior pericardial cavity and the oblique pericardial sinus. Fig. 9.8 Posterior pericardium Anterior view of the opened thorax with the anterior pericardium and heart removed to reveal the posterior pericardium and the oblique pericardial sinus. The transverse pericardial sinus is the passage between the reflections of the serous layer of the pericardium around the arterial and venous great vessels of the heart. Cut edge of fibrous pericardium surrounding origin of a. Superior vena cava Inferior vena cava Ascending aorta Parietal pleura, mediastinal part Left pulmonary a. Left phrenic n. Left vagus n. Sternum Attachment of fibrous pericardium to central tendon of diaphragm Anterior vagal trunk Cut edge of fibrous pericardium surrounding termination of vv. Esophagus Posterior vagal trunk Left pulmonary vv. Anterior esophageal plexus Right pulmonary a. Transverse pericardial sinus Left atrium Superior phrenic l.n. Attachment of fibrous pericardium to central tendon of diaphragm Pericardial cavity Parietal layer Aortic valve Ascending aorta Left brachiocephalic v. Attachment of liver (bare area) to diaphragm Trachea Esophagus Visceral layer Serous pericardium 95 9 Mediastinum Clinical box 9.1 Rapid increases of fluid or blood within the pericardial sac inhibits full expansion of the heart, reducing cardiac blood return, thus decreasing cardiac output. This condition, cardiac tamponade (compression), is potentially fatal, unless relieved. The fluid or blood must first be removed to restore cardiac function and then the cause of the fluid or blood accumulation corrected. Cardiac Tamponade Fig. 9.9 Posterior rela-tions of the heart Anterior view of the opened thorax with the anterior peri-cardium and heart removed and a window cut in the poste-rior pericardium to reveal the structures immediately pos-terior to the heart. This shows the close relationship of the esophagus to the heart, which is used in the transesophageal sonogram to assess the left atrium of the heart. Fig. 9.10 Pericardium, pericardial cavity, and transverse pericardial sinus Sagittal section through the mediastinum. The fibrous pericardium is attached to the central tendon of the dia-phragm and is continuous superiorly with the outer layer of the great vessels. The parietal layer of serous pericar-dium lines the inner surface of the fibrous pericardium and the visceral layer adheres to the heart. The pericardial cavity, the space between the parietal and visceral layers of serous pericardium around the heart, is filled with a thin layer of serous fluid that allows for frictionless move-ment. Where the parietal and visceral layers of serous pericardium reach and reflect around the great vessels, they are continuous with one another. The passage between the arterial and venous reflections of the serous pericardium is the transverse pericardial sinus. Left subclavian a. Left common carotid a. Brachio-cephalic trunk Right pulmonary a. Superior vena cava Ascending aorta Right atrium Coronary (right atrioventricular) sulcus Inferior vena cava Right ventricle Anterior inter-ventricular sulcus Cardiac apex Left ventricle Fibrous pericardium (cut edge) Left auricle Pulmonary trunk Left pulmonary vv. Left pulmonary a. Ligamentum arteriosum Aortic arch Right auricle Left pulmonary a. Left pulmonary vv. Left atrium Coronary sinus Left ventricle Cardiac apex Right atrium Right pulmonary vv. Superior vena cava Right pulmonary a. Aortic arch Inferior vena cava Right ventricle Posterior interventricular sulcus Crux of heart Left auricle Left ventricle Right atrium Left pulmonary a. Left pulmonary vv. Coronary sinus Left atrium Inferior vena cava Visceral layer of serous pericardium (reflected edge) Right pulmonary vv. Superior vena cava Right pulmonary a. Aortic arch Brachiocephalic trunk Left common carotid a. Left subclavian a. 96 Thorax Heart: Surfaces & Chambers Note the reflection of visceral serous pericardium to become parietal serous pericardium. Fig. 9.11 Surfaces of the heart The heart has three surfaces: anterior (sternocostal), posterior (base), and inferior (diaphragmatic). A Anterior (sternocostal) surface. B Posterior surface (base). C Inferior (diaphragmatic) surface. Right atrioventric-ular orifice with atrioventricular valve Terminal crest Right ventricle Right auricle Superior vena cava Right pulmonary a. Right pulmonary vv. Left atrium Inferior vena cava Oval fossa Limbus of oval fossa Pectinate mm. Interatrial septum Pulmonary trunk Ascending aorta Valve of inferior vena cava Valve of coronary sinus Aortic arch Pulmonary trunk Left auricle Pectinate mm. Cardiac apex Posterior papillary m. Left atrioventric-ular valve, cusp Inferior vena cava Interatrial septum Left superior pulmonary v. Left pulmonary a. Right pulmonary a. Anterior papillary m. Trabeculae carneae of interventricular septum Tendinous cords Left atrium Valve of oval fossa Conus arteriosus (infundibulum) Supraventricular crest Posterior papillary m. Interventricular septum Trabeculae carneae Right pulmonary a. Superior vena cava Right atrium Coronary sulcus Inferior vena cava Right atrioventric-ular valve, anterior cusp Anterior papillary m. Septomarginal trabecula (moderator band) Cardiac apex Left ventricle Septal papillary m. Valve of pulmonary trunk, cusps Left pulmonary vv. Pulmonary trunk Ligamentum arteriosum Aortic arch Tendinous cords 97 9 Mediastinum Fig. 9.12 Chambers of the heart C Left atrium and ventricle, left lateral view. Note the irregular trabeculae carneae characteristic of the ventricular wall. B Right atrium, right lateral view. A Right ventricle, anterior view. Note the supraventricular crest, which marks the adult boundary between the embryonic ventricle and the bulbus cordis (now conus arteriosus). Circumflex a. Anterior cusp Posterior cusp Left atrio-ventricular (bicuspid or mitral) valve Coronary sinus Posterior cusp Septal cusp Anterior cusp Right atrio-ventricular (tricuspid) valve Right coronary a. Posterior cusp Right cusp Left cusp Aortic valve Right cusp Anterior cusp Left cusp Pulmonary valve Anterior interventricular a. (LAD) Left coronary a. Left fibrous trigone Left fibrous anulus Right fibrous anulus Right fibrous trigone Fibrous ring of aortic valve Tendon of conus Fibrous ring of pulmonary valve Opening for the bundle of His Circumflex a. Anterior cusp Posterior cusp Left atrio-ventricular valve Coronary sinus Posterior cusp Septal cusp Anterior cusp Right atrio-ventricular valve Right coronary a. Posterior cusp Right cusp Left cusp Aortic valve Right cusp Anterior cusp Left cusp Pulmonary valve Left coronary a. Anterior interventricular a. (LAD) 98 Heart: Valves Thorax The cardiac valves are divided into two types: semilunar and atrio ventricular. The two semilunar valves (aortic and pulmonary) located at the base of the two great arteries of the heart regulate passage of blood from the ventricles to the aorta and pulmonary trunk. The two atrioventricular valves (left and right) lie at the interface between the atria and ventricles. Fig. 9.13 Cardiac valves Plane of cardiac valves, superior view. Removed: Atria and great arteries. A Ventricular diastole (relaxation of the ventricles). Closed: Semilunar valves. Open: Atrioventricular valves. B Ventricular systole (contraction of the ventricles). Closed: Atrioventricular valves. Open: Semilunar valves. C Cardiac skeleton, superior view. The cardiac skeleton is formed by dense fibrous connective tissue. The fibrous anuli (rings) and inter-vening trigones separate the atria from the ventricles. This provides mechanical stability, electrical insulation (see p. 102 for cardiac conduction system), and an attachment point for the cardiac muscles and valve cusps. Table 9.2 Position and auscultation sites of cardiac valves Valve Anatomical projection Auscultation site Aortic valve Left sternal border (at level of 3rd rib) Right 2nd intercostal space (at sternal margin) Pulmonary valve Left sternal border (at level of 3rd costal cartilage) Left 2nd intercostal space (at sternal margin) Left atrioventricular valve Left 4th/5th costal cartilage Left 5th intercostal space (at midclavicular line) or cardiac apex Right atrioventricular valve Sternum (at level of 5th costal cartilage) Left 5th intercostal space (at sternal margin) Posterior papillary m. Opening of left coronary a. Left cusp Lunule Ascending aorta Nodule Posterior cusp Opening of right coronary a. Aortic sinus Right cusp Anterior cusp Lunule Nodule Pulmonary trunk Left cusp Opening of right pulmonary a. Right cusp Trabeculae carnae Posterior cusp Left atrium Commissural cusp Anterior cusp Interatrial septum Membranous part Muscular part Anterior papillary m. Cardiac apex Tendinous cords Posterior papillary m. Inter-ventricular septum Septal cusp Posterior cusp Anterior cusp Tendinous cords Anterior papillary m. Septomarginal trabecula Interventricular septum Septal papillary m. Posterior papillary m. 99 9 Mediastinum Fig. 9.14 Semilunar valves Valves have been longitudinally sectioned and opened. A Aortic valve. B Pulmonary valve. Fig. 9.15 Atrioventricular valves Anterior view during ventricular systole. A Left atrioventricular valve. B Right atrioventricular valve. Aortic valve Right atrio-ventricular valve Left atrio-ventricular valve Pulmonary valve Clinical box 9.2 Auscultation of the cardiac valves Heart sounds, produced by closure of the semilunar and atrioventricular valves, are carried by the blood flowing through the valve. The resulting sounds are therefore best heard “downstream,” at defined auscultation sites (dark circles on diagram). Valvular heart disease causes turbulent blood flow through the valve; this produces a murmur that may be detected in the region of ascultation. Anterior vv. of right ventricle Great cardiac v. Left marginal a. and v. Right marginal a. and v. Atrial br. Conus br. Br. to sinoatrial node Lateral br. Anterior inter-ventricular br. (left anterior descending) Atrial brs. Circumflex br. Right ventricle Small cardiac v. Right coronary a. Cardiac apex Left auricle (atrial appendage) Superior left pulmonary v. Left coronary a. Pulmonary valve Ascending aorta with aortic sinus Superior vena cava Left ventricle Right auricle (atrial appendage) Left posterior ventricular v. Right posterolateral a. Posterior interventricular a. (posterior descending a.) Middle cardiac v. Br. to sinoatrial node Atrial brs. Circumflex br. Great cardiac v. Right pulmonary vv. Left pulmonary vv. Small cardiac v. Inferior vena cava Superior vena cava Left atrium Right coronary a. Right atrium Coronary sinus Right ventricle Left ventricle Oblique v. of left atrium Left marginal v. 100 Thorax Arteries & Veins of the Heart Fig. 9.16 Coronary arteries and cardiac veins A Anterior view. B Posteroinferior view. Note: The right and left coronary arteries typically anastomose posteriorly at the left atrium and ventricle. Table 9.3 Branches of the coronary arteries Left coronary artery Right coronary artery Circumflex br. • Atrial br. • Left marginal a. • Posterior left ventricular br. Br. to SA node Conus br. Atrial br. Right marginal a. Anterior interventricular br. (left anterior descending) • Conus br. • Lateral br. • Interventricular septal brs. Posterior interventricular br. (posterior descending) • Interventricular septal brs. Br. to AV node Right posterolateral a. AV, atrioventricular; SA, sinoatrial. Divisions of the cardiac veins Table 9.4 Vein Tributaries Drainage to Anterior cardiac vv. (not shown) Right atrium Great cardiac v. Anterior interventricular v. Coronary sinus Left marginal v. Oblique v. of left atrium Left posterior ventricular v. Middle cardiac v. (posterior interventricular v.) Small cardiac v. Anterior vv. of right ventricle Right marginal v. Posterior inter-ventricular br. Right coronary a. Left coronary a. Posterior left ventricular br. Posterior inter-ventricular br. Right coronary a. Circumflex br. Left ventricle Interventricular septum Right ventricle Posterior inter-ventricular br. Posterior left ventricular br. Posterior inter-ventricular br. Circumflex br. L A R P Posterior inter-ventricular br. Posterior left ventricular br. Right coronary a. Area of deficient blood flow Left coronary a. Ascending aorta Right coronary a. 101 9 Mediastinum Fig. 9.17 Distribution of the coronary arteries Anterior and posterior views of the heart, with superior views of transverse sections through the ventricles. The “distribution” of the coronary arteries refers to the area of the myocardium supplied by each artery, as seen in the transverse views, but the term “dominance” refers to the artery that gives rise to the posterior interventricular artery, as seen in the anterior and posterior views. Right coronary artery and branches (green); left coronary artery and branches (red). A Left coronary dominance (15–17%). B Balanced distribution, right coronary artery dominance (67–70%). C Right coronary dominance (~15%). Clinical box 9.3 Disturbed coronary blood flow Although the coronary arteries are connected by structural anastomoses, they are end arteries from a functional standpoint. The most frequent cause of deficient blood flow is athero sclerosis, a narrowing of the coronary lumen due to plaque-like deposits on the vessel wall. When the decrease in luminal size (stenosis) reaches a critical point, coronary blood flow is restricted, causing chest pain (angina pectoris). Initially, this pain is induced by physical effort, but eventually it persists at rest, often radiating to characteristic sites (e.g., medial side of left upper limb, left side of head and neck). A myocardial infarction occurs when deficient blood supply causes myocardial tissue to die (necrosis). The loc-ation and extent of the infarction depends on the stenosed vessel (see A–E, after Heinecker). A Supra-apical anterior infarction. B Apical anterior infarction. C Anterior lateral infarction. D Posterior lateral infarction. E Posterior infarction. Pulmonary trunk Interventricular septum Cardiac apex Left ventricle Left bundle br. Left atrium Middle fascicle Posterior fascicle Anterior fascicle Aortic arch Subendocardial brs. Pulmonary trunk Aortic arch Superior vena cava Sinoatrial (SA) node Right atrium Atrioven-tricular (AV) node Atrioventricular (AV) bundle (of His) Purkinje fibers Anterior papillary m. Septomarginal trabecula, (moderator band) Right ventricle Right bundle br. Interventricular septum Left bundle br. Interatrial bundle Atrioventricular (AV) bundle (of His) Superior vena cava Sinoatrial (SA) node Anterior, middle, and posterior internodal bundles Atrioventricular (AV) node Right bundle br. Left bundle br. Interventricular septum 102 Thorax Conduction & Innervation of the Heart Contraction of cardiac muscle is modulated by the cardiac conduction system. This system of specialized myocardial cells (Purkinje fibers) generates and conducts excitatory impulses in the heart. The conduc-tion system contains two nodes, both located in the right atrium: the sinoatrial (SA) node, known as the pacemaker, and the atrioventricular (AV) node. Fig. 9.18 Cardiac conduction system A Anterior view. Opened: All four chambers. B Right lateral view. Opened: Right atrium and ventricle. C Left lateral view. Opened: Left atrium and ventricle. Q wave S wave ST segment T wave R wave P wave Lead vectors in Einthoven’s triangle Recording electrodes I II III Clinical box 9.4 Electrocardiogram (ECG) The cardiac impulse (a physical dipole) travels across the heart and may be detected with electrodes. The use of three electrodes that separately record elec- trical activity of the heart along three axes or vectors (Einthoven limb leads) generates an electrocardiogram (ECG). The ECG graphs the cardiac cycle (“heartbeat”), reducing it to a series of waves, segments, and intervals. These ECG components can be used to determine whether cardiac impulses are normal or abnormal (e.g., myocardial infarction, chamber enlargement). Note: Although only three leads are required, a standard ECG examination includes at least two others (Goldberger, Wilson leads). A ECG recording electrodes. B ECG. Cervical cardiac nn. Sympathetic trunk, inferior cervical ganglion Cardiac brs. to cardiac plexus Cardiac plexus Cardiac plexus (along the coronary aa.) Pulmonary a. and pulmonary vv. with pulmonary plexus Aortic arch with thoracic aortic plexus Subclavian a. Thoracic aortic plexus Superior laryngeal n. Right vagus n. Anterior scalene Brachial plexus Right recurrent laryngeal n. Trachea Right vagus n. Right phrenic n. Superior vena cava Ascending aorta Phrenic n. (on diaphragm) Fibrous pericardium (opened) Gastric plexus Pulmonary trunk Pulmonary plexus Sympathetic trunk, thoracic ganglion Left vagus n. Left phrenic n. Brachiocephalic trunk Left recurrent laryngeal n. Left common carotid a. Sympathetic trunk, middle cervical ganglion Left vagus n. Thyroid gland Thyroid cartilage Hyoid bone Cardiac plexus Ligamentum arteriosum 103 9 Mediastinum Sympathetic innervation: Preganglionic neurons from T1 to T6 spinal cord segments send fibers to synapse on postganglionic neurons in the cervical and upper thoracic sympathetic ganglia. The three cervical cardiac nerves and thoracic cardiac branches contribute to the cardiac plexus. Parasympathetic innervation: Preganglionic neurons and fibers reach the heart via cardiac branches, some of which also arise in the cervical region. They synapse on postganglionic neurons near the SA node and along the coronary arteries. Fig. 9.19 Autonomic innervation of the heart A Schematic. B Autonomic plexuses of the heart, right lateral view. Note the conti-nuity between the cardiac, aortic, and pulmonary plexuses. C Autonomic nerves of the heart. Anterior view of opened thorax. Dorsal motor (vagal) nucleus Sympathetic trunk Stellate ganglion Superior, middle, and inferior cervical cardiac nn. Middle cervical ganglion Superior cervical ganglion T1 spinal cord segment Thoracic cardiac brs. Sinoatrial (SA) node Atrioventricular (AV) node Myocardium Cardiac plexus Superior and inferior cervical cardiac brs. Vagus n. (CN X) Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Portal v. Ductus arteriosus (patent) Aortic arch Oval foramen (patent) Superior vena cava Right atrium Hepatic vv. Ductus venosus Anastomosis between umbilical v. and portal v. Liver Umbilical v. Umbilical aa. Placenta Umbilicus Inferior vena cava Abdominal aorta Right ventricle Pulmonary trunk Left ventricle Left atrium Left pulmonary vv. (very little blood flow) Pulmonary aa. (very little blood flow) ④ ③ ② ① ⑤ Common iliac a. Internal iliac a. Umbilical aa. 104 Thorax Pre- & Postnatal Circulation Fig. 9.20 Prenatal circulation After Fritsch and Kühnel. ① Oxygenated and nutrient-rich fetal blood from the placenta passes to the fetus via the umbilical vein. ② Approximately half of this blood bypasses the liver (via the ductus venosus) and enters the inferior vena cava. The remainder enters the portal vein to supply the liver with nutrients and oxygen. ③ Blood entering the right atrium from the inferior vena cava bypasses the right ventricle (as the lungs are not yet functioning) to enter the left atrium via the oval foramen, a right-to-left shunt. ④ Blood from the superior vena cava enters the right atrium, passes to the right ventricle, and moves into the pulmonary trunk. Most of this blood enters the aorta via the ductus arteriosus, a right-to-left shunt. ⑤ The partially oxygenated blood in the aorta returns to the placenta via the paired umbilical arteries that arise from the internal iliac arteries. Obliterated umbilical aa. (medial umbilical ligs.) Aortic arch Superior vena cava Oval foramen (closed) Right atrium Hepatic vv. Ligamentum venosum (obliterated ductus venosus) Liver Umbilical cord Umbilicus Right ventricle Pulmonary trunk Left ventricle Left atrium Left pulmonary vv. (perfused) Pulmonary aa. (perfused) Portal v. Inferior vena cava Abdominal aorta Ligamentum arteriosum (obliterated ductus arteriosus) Round lig. of liver (obliterated umbilical v.) ② ② ④ ③ ① 105 9 Mediastinum Fig. 9.21 Postnatal circulation After Fritsch and Kühnel. ① As pulmonary respiration begins at birth, pulmonary blood pressure falls, causing blood from the right pulmonary trunk to enter the pulmonary arteries. ② The foramen ovale and ductus arteriosus close, eliminating the fetal right-to-left shunts. The pulmonary and systemic circulations in the heart are now separate. ③ As the infant is separated from the placenta, the umbilical arteries occlude (except for the proximal portions), along with the umbilical vein and ductus venosus. ④ Blood to be metabolized now passes through the liver. Table 9.5 Derivatives of fetal circulatory structures Fetal structure Adult remnant Ductus arteriosus Ligamentum arteriosum Foramen ovale Oval fossa (fossa ovalis) Ductus venosus Ligamentum venosum Umbilical v. Round lig. of the liver (ligamentum teres) Umbilical a. Medial umbilical lig. RV LV Clinical box 9.5 Septal defects Septal defects, the most common type of congenital heart defect, allow blood from the left chambers of the heart to improperly pass into the right chambers during systole. Ventricular septal defect (VSD, shown below) is a defect in either the membranous or muscular portion of the ventricular septum—most commonly the membranous portion. Patent foramen ovale, the most prevalent form of atrial septal defect (ASD), results from improper closure of the fetal shunt. LV, left ventricle; RV, right ventricle. Brachial plexus Pulmonary trunk Right vagus n. Brachiocephalic trunk Anterior scalene Right brachiocephalic v. Right pulmonary a. Azygos v., arch Azygos v. Thoracic duct Right pulmonary vv. Parietal pleura, diaphragmatic part Central tendon of diaphragm Stomach Esophagus, thoracic part Parietal pleura, mediastinal part Anterior esophageal plexus Thoracic aorta Superior and inferior lobar bronchi Left pulmonary a. Left vagus n. Ligamentum arteriosum Aortic arch Parietal pleura, cervical part Left subclavian a. and v. Left internal jugular v. Left brachiocephalic v. Esophagus, cervical part Trachea, cervical part Cricoid cartilage Esophageal inlet Trachea, thoracic part Lower esophageal (phrenic) constriction Diaphragm Middle esophageal (thoracic) constriction Sternum Upper esophageal (pharyngo-esophageal) constriction C6 T4 T10 Aorta Diaphragm Cervical part Thoracic part Abdominal part 106 Thorax Esophagus The esophagus is divided into three parts: cervical (C6–T1), thoracic (T1 to the esophageal hiatus of the diaphragm), and abdominal (the diaphragm to the cardiac orifice of the stomach). It descends slightly to the right of the thoracic aorta and pierces the diaphragm slightly to the left, just below the xiphoid process of the sternum. Fig. 9.22 Esophagus: Location and constrictions A Projection of esophagus onto chest wall. Esophageal constrictions are indicated with arrows. B Esophageal constrictions, right lateral view. Fig. 9.23 Esophagus in situ Anterior view. Esophageal hiatus Peritoneal cavity Parietal peritoneum Mucosa, longitudinal folds Circular layer Longitudinal layer Gastroesophageal junction (Z line) Visceral peritoneum Gastric folds (rugae) Mediastinal part Gastric fundus Diaphragmatic part Gastric cardia Parietal pleura Muscularis Esophagus Mucosa Submucosa Muscular coat, circular layer Muscular coat, circular layer Inferior pharyngeal constrictor, crico-pharyngeal part Inferior pharyngeal constrictor, thyro-pharyngeal part Pharyngeal raphe Muscular coat, longitudinal layer Thyroid cartilage Cricoid cartilage Trachea Killian’s triangle Right main bronchus Left main bronchus Parabronchial diverticulum Diaphragm Esophagus, abdominal part Epiphrenic diverticulum Esophagus (thoracic part) Inferior pharyngeal constrictor Trachea Zenker’s diverticulum 107 9 Mediastinum Fig. 9.24 Structure of the esophagus A Esophageal wall, oblique left posterior view. Pharynx (p. 650); trachea (p. 120). B Esophagogastric junction, anterior view. A true sphincter is not identifiable at this junction; instead, the diaphragmatic muscle of the esophageal hiatus functions as a sphincter. It is often referred to as the “Z line” because of its zigzag form. C Functional architecture of esophageal muscle. Clinical box 9.6 Esophageal diverticula Diverticula (abnormal outpouchings or sacs) generally develop at weak spots in the esophageal wall. There are three main types of esophageal diverticula: • Hypopharyngeal (pharyngo-esophageal) diverticula: Outpouchings occurring at the junction of the pharynx and the esophagus. These include Zenker’s diverticula (70% of cases). • “True” traction diverticula: Protrusion of all wall layers, not typically occurring at characteristic weak spots. However, they generally result from an inflammatory process (e.g., lymphangitis) and are thus com-mon at sites where the esophagus closely approaches the bronchi and bronchial lymph nodes (thoracic or parabronchial diverticula). • “False” pulsion diverticula: Herniations of the mucosa and submucosa through weak spots in the muscular coat due to a rise in esophageal pressure (e.g., during normal swallowing). These include parahiatal and epiphrenic diverticula occurring above the esophageal aperture of the diaphragm (10% of cases). Left vagus n. Left recurrent laryngeal n. Right recurrent laryngeal n. Right vagus n. Right sympathetic trunk 3rd through 6th thoracic ganglia Esophageal plexus Anterior vagal trunk Anterior gastric plexus Stomach Esophageal brs. Left sympathetic trunk Greater splanchnic n. Right subclavian a. Caval opening Esophagus, thoracic part Right main bronchus Left main bronchus Sympathetic trunk, middle cervical ganglion Right vagus n. Right recurrent laryngeal n. Brachiocephalic trunk Sympathetic trunk Posterior intercostal a. Intercostal n. Anterior gastric plexus Stomach Anterior vagal trunk with esophageal plexus Thoracic aorta Aortic arch First rib Left subclavian a. Brachial plexus Left vagus n. Left common carotid a. Left recurrent laryngeal n. Esophagus, cervical part Trachea Sympathetic trunk Diaphragm 108 Thorax Neurovasculature of the Esophagus Sympathetic innervation: Preganglionic fibers arise from the T2–T6 spinal cord segments. Postganglionic fibers arise from the sympathetic trunk to join the esophageal plexus. Parasympathetic innervation: Preganglionic fibers arise from the dorsal vagal nucleus and travel in the vagus nerves to form the extensive esophageal plexus. Note: The postganglionic neurons are in the wall of the esophagus. Fibers to the cervical portion of the esophagus travel in the recurrent laryngeal nerves. A Anterior view. Note the postgangli-onic sympathetic contribution to the esophageal plexus. B Esophageal plexus in situ. Anterior view. Fig. 9.25 Autonomic innervation of the esophagus Fig. 9.26 Esophageal plexus The left and right vagus nerves initially descend on the left and right sides of the esophagus. As they begin to contribute to the esophageal plexus, they shift to anterior and posterior positions, respectively. As the vagus nerves continue into the abdomen, they are named the anterior and posterior vagal trunks. Sympathetic trunk T2 spinal cord segment T6 spinal cord segment Esophageal brs. Vagal trunk Esophagus, abdominal part Esophagus, thoracic part Esophageal plexus Esophagus, cervical part Recurrent laryngeal n. Vagus n. (CN X) Dorsal motor (vagal) nucleus Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Inferior thyroid a. Thyrocervical trunk Trachea Internal thoracic a. Esophageal brs. (from thoracic aorta) Celiac trunk Common hepatic a. Esophageal br. Gastric fundus Diaphragm Thoracic aorta Posterior intercostal aa. Left main bronchus Aortic arch Left common carotid a. Left subclavian a. Esophageal brs. Vertebral a. Anterior scalene Abdominal aorta Left gastric a. Left inferior phrenic a. Right common carotid a. Brachiocephalic trunk Splenic a. Inferior thyroid v. Anterior scalene Right brachio-cephalic v. Superior vena cava Esophageal vv. Azygos v. Left gastric v. Esophageal vv. Diaphragm Hemiazygos v. Posterior intercostal vv. Accessory hemiazygos v. Left sub-clavian v. Left external jugular v. Left internal jugular v. Esophageal vv. Left brachio-cephalic v. Stomach Esophageal plexus Posterior gastric plexus Posterior vagal trunk Right vagus n. Left vagus n. 109 9 Mediastinum C Posterior view. Fig. 9.27 Esophageal arteries Anterior view. Fig. 9.28 Esophageal veins Anterior view. Table 9.6 Blood vessels of the esophagus Part Origin of esophageal arteries Drainage of esophageal veins Cervical Inferior thyroid a. Inferior thyroid v. Rarely direct brs. from thyrocervical trunk or common carotid a. Left brachiocephalic v. Thoracic Aorta (four or five esophageal aa.) Upper left: Accessory hemiazygos v. or left brachiocephalic v. Lower left: Hemiazygos v. Right side: Azygos v. Abdominal Left gastric a. Left gastric v. Celiac trunk Esophagus Trachea Left brachiocephalic vein Right brachio-cephalic vein Brachio-cephalic l.n. Superior vena cava Fibrous pericardium Diaphragm Abdominal aorta Stomach Celiac l.n. Superior phrenic l.n. Tracheo-bronchial l.n. Intercostal lymphatics Paratracheal l.n. Thoracic duct at junction of left subclavian and internal jugular veins Left jugular trunk Thoracic duct Prevertebral l.n. Right coronary trunk Right ventricle Left ventricle Aortic arch Junction of left subclavian and internal jugular vv. Trachea Superior vena cava Bronchopulmonary l.n. Left coronary trunk Right ventricle Left ventricle Aortic arch Trachea Junction of right subclavian and internal jugular vv. Superior vena cava Inferior tracheobronchial l.n. Right coronary trunk Left coronary trunk Trachea Inferior tracheo-bronchial l.n. Bronchopulmonary l.n. Left atrium Right atrium Bronchopulmonary l.n. Superior vena cava 110 Thorax Lymphatics of the Mediastinum The superior phrenic lymph nodes drain lymph from the diaphragm, pericardium, lower esophagus, lung, and liver into the broncho mediastinal trunk. The inferior phrenic lymph nodes, found in the abdomen, collect lymph from the diaphragm and lower lobes of the lung and convey it to the lumbar trunk. Note: The pericardium may also drain superiorly to the brachiocephalic lymph nodes. Fig. 9.29 Lymph nodes of the mediastinum and thoracic cavity Left anterior oblique view. Fig. 9.30 Lymphatic drainage of the heart A unique “crossed” drainage pattern exists in the heart: lymph from the left atrium and ventricle drains to the right venous junction, whereas lymph from the right atrium and ventricle drains to the left venous junction. B Lymphatic drainage of the right chambers, anterior view. A Lymphatic drainage of the left chambers, anterior view. C Posterior view. Esophagus Left common carotid a. Left subclavian a. and v. Left internal jugular v. Aortic arch Paraesophageal l.n. Fibrous pericardium Diaphragm Inferior vena cava Bronchopulmonary l.n. Azygos v. Superior vena cava Paraesophageal l.n. Trachea Bronchopulmonary l.n. Left main bronchus Left pulmonary a. Tracheobronchial l.n. Paraesophageal l.n. Right main bronchus Inferior tracheo-bronchial l.n. Paraesophageal l.n. Diaphragm Inferior phrenic l.n. Cardiac lymphatic ring (inconstant) Stomach Left main bronchus Paratracheal l.n. Trachea Esophagus 111 9 Mediastinum The paraesophageal nodes drain the esophagus. Lymphatic drainage of the cervical part of the esophagus is primarily cranial, to the deep cervical lymph nodes and then to the jugular trunk. The thoracic part of the esophagus drains to the bronchomediastinal trunks in two parts: the upper half drains cranially, and the lower half drains inferiorly via the superior phrenic lymph nodes. The bronchopulmonary and paratracheal nodes drain lymph from the lungs, bronchi, and trachea into the bron-chomediastinal trunk (see p. 128). Fig. 9.31 Mediastinal lymph nodes A Anterior view of opened thorax. B Posterior view of mediastinal lymph nodes. Sternal line Midclavicular line Paravertebral line Scapular line Midaxillary line Costodia-phragmatic recess of pulmonary cavity Parietal pleura Midaxillary line Costodia-phragmatic recess of pulmonary cavity Parietal pleura Thorax 112 10 Pulmonary Cavities Pulmonary Cavities The paired pulmonary cavities contain the left and right lungs. They are completely separated from each other by the media stinum and are under negative atmospheric pressure (see respiratory mechanics, pp. 122–123). The left pulmonary cavity is slightly smaller than the right, especially anteriorly, due to the asymmetrical position of the heart in the mediastinum, with the greater mass on the left. This causes a shift of some of the boun daries of the parietal pleura and lung on the left side at the level of the heart, as reflected in the difference in thoracic landmarks found at the intersection of the anterior border of the pulmonary cavities with certain reference lines on the left and right. A Anterior view. C Right lateral view. B Posterior view. D Left lateral view. Table 10.1 Pulmonary cavity boundaries and reference points Fig. 10.1 Boundaries of the lungs and pulmonary cavities The upper red dot on each reference line is the inferior boundary of the lung and the lower blue dot is the inferior boundary of the pulmonary cavity. Reference line Right lung Right parietal pleura Left lung Left parietal pleura Sternal line (STL) 6th rib 7th rib 4th rib 4th rib Midclavicular line (MCL) 6th rib 8th costal cartilage 6th rib 8th rib Midaxillary line (MAL) 8th rib 10th rib 8th rib 10th rib Scapular line (SL) 10th rib 11th rib 10th rib 11th rib Paravertebral line (PV) 10th rib T12 vertebra 10th rib T12 vertebra Costal part Mediastinal part Diaphragmatic part Cervical part Pericardial sac Sympathetic trunk Visceral pleura Costomediastinal recess Left lung, superior lobe Oblique fissure Left lung, inferior lobe Thoracic aorta Hemiazygos v. Azygos v., thoracic duct, and left vagus n. Right lung, inferior lobe Oblique fissure Right lung, middle lobe Horizontal fissure Right lung, superior lobe Parietal pleura, mediastinal part Parietal pleura, costal part Spinal cord Fibrous pericardium Serous pericardium Left main broncus Right main bronchus Phrenic n. Parietal pleura, costal part Parietal peritoneum Intercostal v., a., and n. Visceral pleura Liver Diaphragm Parietal pleura, diaphragmatic part Right lung 8th rib Costal groove External intercostal Visceral peritoneum Costodiaphragmatic recess Endothoracic fascia Internal intercostal Innermost intercostal Parietal pleura, costal part 10 Pulmonary Cavities 113 B Costodiaphragmatic recess, coronal section, anterior view. Reflection of the diaphragmatic pleura onto the inner thoracic wall (becoming the costal pleura) forms the costodiaphragmatic recess. A Parts of the parietal pleura. Opened: Right pleural cavity, anterior view. Fig. 10.2 Parietal pleura The pulmonary cavity is bounded by two serous layers. The visceral pleura covering the lungs, and parietal pleura lining the inner surfaces of the thoracic cavity. The four divisions of the parietal pleura (costal, diaphragmatic, mediastinal, and cervical) are continuous. C Transverse section at T7, inferior view. Reflection of the costal pleura onto the pericardium forms the costomediastinal recess. Superior lobe Middle lobe Inferior lobe Fibrous pericardium Parietal pleura, mediastinal part Lung with visceral pleura Parietal pleura, costal part Parietal pleura, mediastinal part Internal thoracic a. and v. Parietal pleura, diaphragmatic part Thorax 114 Pleura: Subdivisions, Recesses & Innervation Fig. 10.4 Innervation of the pleura The costal and cervical portions and the periphery of the diaphragmatic portion of the parietal pleura are innervated by the intercostal nerves. The mediastinal and central portions of the diaphragmatic pleura are innervated by the phrenic nerves. The visceral pleura covering the lung itself receives its innervation from the autonomic nervous system. Fig. 10.3 Pleura and its divisions The anterior thoracic wall and costal portion of the parietal pleura have been removed to show the lungs in situ. Parietal pleura innervated by intercostal nn. Parietal pleura innervated by phrenic n. Visceral pleura innervated by autonomic nn. 10 Pulmonary Cavities 115 Parietal pleura, mediastinal part Pericardiacophrenic a. and v., phrenic n. Fibrous pericardium Costodiaphragmatic recess Internal thoracic a. and v. Costomediastinal recess Parietal pleura, costal pleura Anterior mediastinum Serous pericardium, parietal layer Phrenic n., pericardiacophrenic a. and v. Parietal pleura, diaphragmatic part Parietal pleura, costal part Esophagus Parietal pleura, mediastinal part Left sympathetic trunk Hemiazygos v. Thoracic aorta Thoracic duct Azygos v. Right sympathetic trunk Diaphragm, central tendon Inferior vena cava Diaphragm, costal part Costo-diaphragmatic recess Phrenic n., pericardiaco-phrenic a.and v. Internal thoracic a. and v. Sternum, body Costomediastinal recess Fibrous pericardium Costomediastinal recess Costodiaphragmatic recess Fig. 10.6 Pleural recesses Transverse section at T8, superior view. Fig. 10.5 Costomediastinal and costodiaphragmatic recesses On the left side of the thorax, an examiner’s fingertips are placed in the costomediastinal and costodiaphragmatic recesses. These recesses are formed by the acute reflection of the costal part of the parietal pleura onto the fibrous pericardium as mediastinal pleura (costomediastinal) or on to the diaphragm as diaphragmatic pleura (costodiaphragmatic). Right lung Esophagus Mediastinum Left lung Descending aorta Superior lobe Horizontal fissure Middle lobe Oblique fissure Inferior lobe Superior lobe Oblique fissure Inferior lobe Pulmonary trunk Brachiocephalic trunk Superior vena cava Right pulmonary a. Right pulmonary vv. Right lung middle lobe Parietal pleura, diaphragmatic part Right lung, inferior lobe Central tendon of diaphragm Esophagus, thoracic part Parietal pleura, mediastinal part Thoracic aorta Left lung, superior lobe Superior and inferior lobar bronchi Left pulmonary a. Aortic arch Pulmonary apex Parietal pleura, cervical part Left subclavian a. and v. Left brachio-cephalic v. Costodia-phragmatic recess Parietal pleura, costal part Right lung, superior lobe Right lung, horizontal fissure , Right lung, oblique fissure Diaphragm Left lung, oblique fissure Left lung, inferior lobe Parietal layer of serous pericardium Fibrous pericardium Thorax 116 Lungs Fig. 10.7 Lungs in situ The left and right lungs occupy the full vol-ume of the pleural cavity. Note that the left lung is slightly smaller than the right due to the asymmetrical position of the heart. A Topographical relations of the lungs, transverse section, inferior view. B Anterior view with lungs retracted. Costal surface Inferior border (inserts into costodiaphragmatic recess) Inferior lobe Oblique fissure Middle lobe Horizontal fissure Anterior border (inserts into costo-mediastinal recess) Superior lobe Apex Lingula Costal surface Inferior border (inserts into costodiaphragmatic recess) Anterior border (inserts into costo-mediastinal recess) Inferior lobe Superior lobe Apex Oblique fissure Inferior and middle lobar bronchi (common origin) Inferior border (inserts into costodiaphragmatic recess) Mediastinal surface Anterior border Horizontal fissure Branches of right pulmonary vv. Pulmonary lig. Diaphragmatic surface (base of lung) Costal surface, vertebral part Inferior lobe Hilum Oblique fissure Superior lobar bronchus Branches of right pulmonary a. Superior lobe Apex Cardiac impression Middle lobe Cardiac impression Cardiac notch Lingula Diaphragmatic surface (base of lung) Inferior border (inserts into costo-diaphragmatic recess) Superior lobe Oblique fissure Branches of left pulmonary a. Superior and inferior lobar bronchi Hilum Aortic impression Costal surface (vertebral part) Inferior lobe Pulmonary lig. Branches of left pulmonary vv. Anterior border Mediastinal surface Apex 10 Pulmonary Cavities 117 Fig. 10.8 Gross anatomy of the lungs The oblique and horizontal fissures divide the right lung into three lobes: superior, middle, and inferior. The oblique fissure divides the left lung into two lobes: superior and inferior. The apex of each lung extends into the root of the neck. The hilum is the location at which the bronchi and neurovascular structures connect to the lung. A Right lung, lateral view. B Left lung, lateral view. C Right lung, medial view. D Left lung, medial view. I II III IV V VIII I II III IV V VII, VIII Left lung Right lung Horizontal fissure Oblique fissure Oblique fissure I II III IV V VIII X VII VI I II III IV V IX X VII, VIII VI IX Trachea and bronchial tree I II III IV V VIII I II III IV V VII, VIII Left lung Right lung Horizontal fissure Oblique fissure Oblique fissure Thorax 118 Bronchopulmonary Segments of the Lungs The lung lobes are subdivided into bronchopulmonary segments, the smallest resectable portion of a lung, each supplied by a tertiary Table 10.2 Segmental architecture of the lungs Each segment is supplied by a segmental bronchus of the same name (e.g., the apical segmental bronchus supplies the apical segment). See pp. 120–121 for details of the trachea and bronchial tree. Right lung Left lung Superior lobe I Apical segment Apicoposterior segment I II Posterior segment II III Anterior segment III Middle lobe Lingula IV Lateral segment Superior lingular segment IV V Medial segment Inferior lingular segment V Inferior lobe VI Superior segment VI VII Medial basal segment VII VIII Anterior basal segment VIII IX Lateral basal segment IX X Posterior basal segment X Fig. 10.9 Segmentation of the lung Anterior view. See pp. 120–121 for details of the trachea and bronchial tree. (segmental) bronchus. Note: These subdivisions are not defined by surface boundaries but by origin. Fig. 10.10 Anteroposterior bronchogram Anterior view of right lung. I II VI X IX VII VIII V III Horizontal fissure Oblique fissure I II VI VIII IX X I II III IV V VIII IX X VI Segment I of right lung Right lung Left lung Trachea Superior lobe of right lung I III IV V VII, VIII IX X VI II Oblique fissure I II III IV VII, VIII IX X VI I II VI X IX VII, VIII V IV III 10 Pulmonary Cavities 119 Fig. 10.11 Right lung: Bronchopulmonary segments A Medial view. B Posterior view. C Lateral view. Fig. 10.12 Left lung: Bronchopulmonary segments A Medial view. B Posterior view. C Lateral view. Clinical box 10.1 Lung cancer, emphysema, or tuberculosis may necessitate the surgical removal of damaged portions of the lung. Surgeons exploit the anatomical subdivision of the lungs into lobes and segments when excising damaged tissue. Lung resections A Segmentectomy (wedge resection): Removal of one or more segments. B Lobectomy: Removal of lobe. C Pneumonectomy: Removal of entire lung. I II III IV V VI VII VIII IX X I II III IV V VI VII VIII X IX Right main bronchus Left main bronchus Membranous posterior wall (with tracheal glands) Tracheal cartilages Arytenoid cartilage Thyroid cartilage Cricoid cartilage Position of carina (at tracheal bifurcation) Mucosa I II III IV V VI VII VIII X IX I II III IV V VI VII VIII IX X Tracheal bifurcation Right main bronchus Left main bronchus Anular ligs. Cricoid cartilage Thyroid cartilage Right superior lobar bronchus Right middle lobar bronchus Right/left inferior lobar bronchi Left superior lobar bronchus Tracheal cartilages Median cricothyroid lig. Cervical part Thoracic part Right main bronchus Left main bronchus Tracheal bifurcation Trachea Thorax 120 Trachea & Bronchial Tree Fig. 10.13 Trachea See p. 530 for the structures of the thyroid. A Projection of trachea onto chest. Clinical box 10.2 Toddlers are at particularly high risk of potentially fatal aspiration of foreign bodies. In general, foreign bodies are more likely to become lodged in the right main bronchus than the left: the left bronchus diverges more sharply at the tracheal bifurcation to pass more horizontally over the heart, whereas the right bronchus is relatively straight and more in line with the trachea. Foreign body aspiration B Anterior view. At or near the level of the sternal angle (T4/T5), the lowest tracheal cartilage extends anteroposteriorly, forming the carina. The trachea C Posterior view with opened posterior wall. bifurcates at the carina into the right and left main bronchi. Each bronchus gives off lobar branches to the corresponding lung. Alveolar sacs Respiratory bronchiole Terminal bronchiole Bronchiole (cartilage-free wall) Small subsegmental bronchus Large subsegmental bronchus Cartilaginous plate Segmental bronchus see B Conduction portion of airway Respiratory portion of airway Pulmonary alveoli Alveolar sac Alveolar duct Respiratory bronchioles Pulmonary alveolus Elastic fibers Smooth muscle (lattice arrangement) Interalveolar septum Acinus Alveolus Elastic fibers in the interalveolar septum Erythrocyte Capillary endothelial cell Fusion of the basement membranes Type I pneumocyte Capillary lumen Alveolar lumen Surfactant Type II pneumocyte Alveolar macro-phage 10 Pulmonary Cavities 121 The conducting portion of the bronchial tree extends from the tracheal bifurcation to the terminal bronchiole, inclusive. The respiratory portion consists of the respiratory bronchiole, alveolar ducts, alveolar sacs, and alveoli. Fig. 10.14 Bronchial tree A Divisions of the bronchial tree. B Respiratory portion of the bronchial tree. C Epithelial lining of the alveoli. Clinical box 10.3 The most common cause of respiratory compromise at the bronchial level is asthma. Compromise at the alveolar level may result from increased diffusion distance, decreased aeration (emphysema), or fluid infiltration (e.g., pneumonia). Diffusion distance: Gaseous exchange takes place between the alveolar and capillary lumens in the alveoli (see Fig. 10.14C). At these sites, the basement membranes of capillary endothelial cells are fused with those of type I alveolar epithelial cells, lowering the exchange distance to 0.5 µm. Diseases that increase this diffusion distance (e.g., edematous fluid collection or inflammation) result in compromised respiration. Condition of alveoli: In diseases like emphysema, which occurs in chronic obstructive pulmonary disease (COPD), alveoli are destroyed or damaged. This reduces the surface area available for gaseous exchange. Production of surfactant: Surfactant is a protein-phospholipid film that lowers the surface tension of the alveoli, making it easier for the lung to expand. The immature lungs of a preterm infant often fail to produce sufficient surfactant, leading to respiratory problems. Surfactant is produced and absorbed by alveolar epithelial cells (pneumocytes). Type I alveolar epithelial cells absorb surfactant; type II produce and release it. Respiratory compromise Sternum Inspiratory position Expiratory position T12 vertebra 1st rib Diaphragm Expansion (transverse axis) Diaphragm Expansion (vertical axis) Expansion (sagittal axis) Thoracic inlet Epigastric angle Inspiration 1st rib Inferior border of lung (full inspiration) Inferior border of lung (full expiration) Contraction (transverse axis) Contraction (sagittal axis) Contraction (vertical axis) Expiration Thorax 122 Respiratory Mechanics The mechanics of respiration are based on a rhythmic increase and decrease in thoracic volume, with an associated expansion and contrac-tion of the lungs. Inspiration (red): Contraction of the diaphragm leaflets lowers the diaphragm into the inspiratory position, increasing the volume of the pleural cavity along the vertical axis. Contraction of the thoracic muscles (external intercostals with the scalene, intercartilaginous, and posterior serratus muscles) elevates the ribs, expanding the pleural cavity along the sagittal and transverse axes (Fig. 10.16A,B). Surface tension in the pleural space causes the visceral and parietal pleura to adhere; thus, changes in thoracic volume alter the volume of the lungs. This is particularly evident in the pleural recesses: at functional residual capacity (resting position between inspiration and expiration), the lung does not fully occupy the pleural cavity. As the pleural cavity expands, a negative intrapleural pressure is generated. The air pressure differential results in an influx of air (inspiration). Expiration (blue): During passive expiration, the muscles of the thoracic cage relax and the diaphragm returns to its expiratory position. Contraction of the lungs increases the pulmo-nary pressure and expels air from the lungs. For forcible expiration, the internal intercostal muscles (with the transverse thoracic and subcostal mucosa) can actively lower the rib cage more rapidly and to a greater extent than through passive elastic recoil. Fig. 10.15 Respiratory changes in thoracic volume Inspiratory position (red); expiratory position (blue). Fig. 10.16 Inspiration: Pleural cavity expansion A Anterior view. B Left lateral view. C Anterolateral view. Fig. 10.17 Expiration: Pleural cavity contraction A Anterior view. B Left lateral view. C Anterolateral view. Fig. 10.18 Respiratory changes in lung volume Left lung Right lung Normal airflow during inspiration Normal airflow during expiration “Empty” pleural cavity at atmospheric pressure Airflow into pleural defect Cardiac shift Airflow out of pleural defect Cardiac shift Collapsed lung Positive pressure in pleural cavity Pleural defect during inspiration Cardiac shift Collapsed lung One-way “valve” Right lung (full inspiration) Diaphragm Costodiaphragmatic recess Diaphragm Costodiaphragmatic recess Pleural space Right lung (full expiration) Lung (full inspiration) Lung (full expiration) Trachea 10 Pulmonary Cavities 123 Fig. 10.19 Inspiration: Lung expansion Fig. 10.20 Expiration: Lung contraction Fig. 10.21 Movements of the lung and bronchial tree As the volume of the lung changes with the volume of the thoracic cavity, the entire bronchial tree moves within the lung. These structural movements are more pronounced in portions of the bronchial tree distant from the pulmonary hilum. Inspiration Expiration A Normal respiration. B Pneumothorax. C Tension pneumothorax. Clinical box 10.4 The pleural space is normally sealed from the outside environment. Injury to the parietal pleura, visceral pleura, or lung allows air to enter the pleural cavity (pneumothorax). The lung collapses due to its inherent elasticity, and the patient’s ability to breathe is compromised. The uninjured lung continues to function under normal pressure variations, resulting in “mediastinal flutter”: the mediastinum shifts toward the normal side during inspiration and returns to the midline during expiration. Tension (valve) pneumothorax occurs when traumatically detached and displaced tissue covers the defect in the thoracic wall from the inside. This mobile flap allows air to enter, but not escape, the pleural cavity, causing a pressure buildup. The mediastinum shifts to the normal side, which may cause kinking of the great vessels and prevent the return of venous blood to the heart. Without treatment, tension pneumothorax is invariably fatal. Pneumothorax Pulmonary trunk Left pulmonary a. Right pulmonary a. Left subclavian v. Left internal jugular v. Left brachio-cephalic v. Right internal jugular v. Right subclavian v. Right brachio-cephalic v. Superior vena cava Right pulmonary vv. Inferior vena cava Left pulmonary vv. Right main bronchus Right lung Superior lobe Superior right pulmonary v. Inferior right pulmonary v. Superior vena cava Middle lobe Inferior lobe Right atrium Inferior vena cava Right pulmonary a. Ascending aorta Right ventricle Pulmonary trunk Left pulmonary a. Cardiac apex Inferior lobe Inferior left pulmonary v. Superior left pulmonary v. Superior lobe Left lung Aortic arch Left main bronchus Trachea Left ventricle Thorax 124 Pulmonary Arteries & Veins The pulmonary trunk arises from the right ventricle and divides into a left and right pulmonary artery for each lung. The paired pulmonary veins open into the left atrium on each side. The pulmonary arteries accompany and follow the branching of the bronchial tree, whereas the pulmonary veins do not, being located at the margins of the pulmonary lobules. Fig. 10.22 Pulmonary arteries and veins Anterior view. A Projection of pulmonary arteries on chest wall. B Projection of pulmonary veins on chest wall. C Distribution of the pulmonary arteries and veins, anterior view. ① ② ③ ④ ⑤ ⑥ ⑦ ⑧⑨ ⑩ ⑪ ⑫ ⑬ ⑭ ⑮ ⑯ ⑰ ⑱ ⑲ Right pulmonary a. Middle lobe a. Pulmonary trunk Left pulmonary a. Ligamentum arteriosum Aortic arch Left subclavian a. Left common carotid a. Brachio-cephalic trunk S K J H V N A D F G L Ö Ä Y X CB Right/left inferior pulmonary v. Right/left superior pulmonary v. 10 Pulmonary Cavities 125 Fig. 10.23 Pulmonary arteries A Schematic. B Pulmonary arte-riogram, arterial phase, anterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Ra-diographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Table 10.3 Pulmonary arteries and their branches Right pulmonary artery Left pulmonary artery Superior lobe arteries ① Apical segmental a. ⑪ ② Posterior segmental a. ⑫ ③ Anterior segmental a. ⑬ Middle lobe arteries ④ Lateral segmental a. Lingular a. ⑭ ⑤ Medial segmental a. Inferior lobe arteries ⑥ Superior segmental a. ⑮ ⑦ Anterior basal segmental a. ⑯ ⑧ Lateral basal segmental a. ⑰ ⑨ Posterior basal segmental a. ⑱ ⑩ Medial basal segmental a. ⑲ Fig. 10.24 Pulmonary veins A Schematic. B Pulmonary arte-riogram, venous phase, anterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Ra-diographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Table 10.4 Pulmonary veins and their tributaries Right pulmonary vein Left pulmonary vein Superior pulmonary veins ① Apical v. Apicoposterior v. ⑩ ② Posterior v. ③ Anterior v. Anterior v. ⑪ ④ Middle lobe v. Lingular v. ⑫ Inferior pulmonary veins ⑤ Superior v. ⑬ ⑥ Common basal v. ⑭ ⑦ Inferior basal v. ⑮ ⑧ Superior basal v. ⑯ ⑨ Anterior basal v. ⑰ Clinical box 10.5 Potentially life-threatening pulmonary embolism occurs when blood clots migrate through the venous system and become lodged in one of the arteries supplying the lungs. Symptoms include dyspnea (difficulty breathing) and tachycardia (increased heart rate). Most pulmonary emboli originate from stagnant blood in the veins of the lower limb and pelvis (venous thromboemboli). Causes include immobilization, disordered blood coagulation, and trauma. Note: A thromboembolus is a thrombus (blood clot) that has migrated (embolized). Pulmonary embolism Fibrous septum between pulmonary lobules Pulmonary alveolus Subpleural connective tissue Pulmonary alveolus Capillary bed on an alveolus Tributary of pulmonary v. (oxygenated blood) Smooth muscle Respiratory bronchiole Br. of pulmonary a. (deoxygenated blood) Bronchial a. Trachea Brachio-cephalic trunk Ascending aorta Posterior intercostal a. Right main bronchus Superior lobe bronchus Bronchial brs. (from a posterior inter-costal a.) Middle lobe bronchus Inferior lobe bronchus Thoracic aorta Posterior intercostal aa. Inferior lobe bronchus Left main bronchus Superior lobe bronchus Bronchial brs. (from the thoracic aorta) Aortic arch Left subclavian a. Left common carotid a. Thorax 126 Neurovasculature of the Tracheobronchial Tree Fig. 10.25 Pulmonary vasculature The pulmonary system is responsible for gaseous exchange within the lung. Pulmo-nary arteries (shown in blue) carry deoxygen-ated blood and follow the bronchial tree. The pulmonary vein and its tributaries (red) is the only vein in the body carrying oxygenated blood, which it receives from the alveolar capillaries at the periphery of the lobule. Fig. 10.26 Arteries of the tracheobron-chial tree The bronchial tree receives its nutrients via the bronchial arteries, found in the adventitia of the airways. Typically, there are one to three bronchial arteries arising directly from the aorta. Origin from a posterior intercostal artery may also occur. T1 spinal cord segment Postganglionic fibers (to cardiac plexus ) Middle cervical ganglion Cervicothoracic (stellate) ganglion 2nd - 5th thoracic sympathetic ganglia Greater splanchnic (to abdomen) Pulmonary plexus Bronchial brs. in pulmonary plexus Left main bronchus Right main bronchus Autonomic brs. to trachea Trachea Laryngopharygeal brs. Larynx, thyroid cartilage Recurrent laryngeal n. Superior laryngeal n. Vagus n. (CN X) Dorsal motor (vagal) nucleus Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Trachea Inferior thyroid v. Right brachio-cephalic v. Superior vena cava Superior lobe bronchus Bronchial vv. (opening into the azygos v.) Middle lobe bronchus Inferior lobe bronchus Azygos v. Inferior lobe bronchus Accessory hemiazygos v. Left brachio-cephalic v. Hemiazygos v. Bronchial vv. (opening into the accessory hemiazygos v.) Left main bronchus Superior lobe bronchus 10 Pulmonary Cavities 127 Fig. 10.28 Autonomic innervation of the tracheobronchial tree Sympathetic innervation (red); parasympathetic innervation (blue). Fig. 10.27 Veins of the tracheobronchial tree Trachea To right broncho-mediastinal trunk Paratracheal l.n. Right lung Inferior tracheo-bronchial l.n. Diaphragm Inferior phrenic l.n. Drainage through diaphragm To inferior tracheobronchial l.n. Left lung To left broncho-mediastinal trunk Superior tracheo-bronchial l.n. Paratracheal l.n. Intrapulmonary l.n. Bronchopulmonary l.n. Parasternal l.n. Sternum Tracheobronchial l.n. Peribronchial network Subpleural network Trachea Intercostal l.n. Lymphatics in the trunk wall Thorax 128 Lymphatics of the Pleural Cavity The lungs and bronchi are drained by two lymphatic drainage systems. The peribronchial network follows the bronchial tree, draining lymph from the bronchi and most of the lungs. The subpleural network col-lects lymph from the peripheral lung and visceral pleura. Fig. 10.29 Lymphatic drainage of the pleural cavity and thoracic wall A Peribronchial network, coronal section, an-terior view. (Intra)pulmonary nodes along the bronchial tree drain lymph from the lungs into the bronchopulmonary (hilar) nodes. Lymph then passes sequentially through the inferior and superior tracheo-bronchial nodes, paratracheal nodes, bronchomediastinal trunk, and finally to the right lymphatic or thoracic duct. Note: Significant amounts of lymph from the left lower lobe drain to the right supe-rior tracheobronchial nodes. B Subpleural and thoracic wall networks, transverse section, superior view. Paratracheal l.n. Right internal jugular v. Right jugular trunk Right subclavian v. Right subclavian trunk Right broncho-mediastinal trunk Superior tracheo-bronchial l.n. Right main bronchus Right lung Inferior tracheo-bronchial l.n. Left main bronchus Thoracic aorta Broncho-pulmonary l.n. Left lung Intra-pulmonary l.n. Left broncho-mediastinal trunk Left subclavian trunk Thoracic duct Deep cervical l.n. Left jugular trunk Trachea 10 Pulmonary Cavities 129 Fig. 10.30 Lymph nodes of the pleural cavity Anterior view of pulmonary nodes. Clinical box 10.6 Carcinoma of the lung accounts for ≈ 20% of all cancers and is mainly caused by cigarette smoking. It arises first in the lining of the bronchi and metastasizes quickly to bronchopulmonary lymph nodes and subsequently to other node groups, including supraclavicular nodes. It can also spread via the blood to the lungs, brain, bone, and suprarenal glands. Lung cancer can invade adjacent structures such as the phrenic nerve, resulting in paralysis of a hemidiaphragm, or the recurrent laryngeal nerve, resulting in hoarseness due to paralysis of the vocal cord. Carcinoma of the Lung Thorax 130 11 Sectional & Radiographic Anatomy Sectional Anatomy of the Thorax Fig. 11.1 Transverse section through the thoracic inlet region of the thorax Inferior view. Right brachio-cephalic v. Trachea Esophagus T3 vertebra Spinal cord Sympathetic trunk Left subclavian a. Left vagus n. Left common carotid a. Phrenic n. Clavicle Left brachio-cephalic v. Manubrium Brachiocephalic trunk First rib Second rib Third rib Right lung, superior lobe Left lung, superior lobe Right ventricle Sympathetic trunks Interventricular septum Left atrium Right atrium Costomediastinal recess Left vagus n. (anterior vagal trunk) Left lung, superior lobe Oblique fissure of left lung Phrenic n. (between fibrous pericardium and parietal pleura, mediastinal part) Left lung, inferior lobe Thoracic (descending) aorta Hemiazygos v. Thoracic duct Azygos v. Esophagus Right lung, inferior lobe Oblique fissure of right lung Right lung, middle lobe Horizontal fissure of right lung Right lung, superior lobe Internal thoracic a. and v. Sternum Left ventricle Fig. 11.2 Transverse section through the mid region of the thorax Inferior view. 11 Sectional & Radiographic Anatomy 131 Fig. 11.3 Coronal section through heart and similar MRI A Image displays left ventricular outflow tract (LVOT) during diastole. B Corresponding coronal (frontal) anatomical cross section of the heart, anterior view. Fig. 11.4 Transverse section through heart and similar MRI A Image displays the atrioventricular connections of both the right and left sides of the heart during diastole (four-chamber view). B Corresponding transverse anatomical cross section of the heart, inferior view. Fig. 11.5 Sagittal section of the heart and similar MRI A Image displays the right ventricular outflow tract (RVOT) during diastole. B Corresponding sagittal anatomical cross section of the heart, viewed from the left side. A Superior vena cava Pulmonary trunk Ascending aorta LV RA B Superior vena cava Ascending aorta Lung Pulmonary trunk Aortic valve Left ventricle (LV) Diaphragm Right atrium (RA) Liver A Interventricular septum Cardiac apex Opening of pulmonary veins RV LV LA RA Interatrial septum B Right ventricle (RV) Interventric-ular septum Left ventricle (LV) Mitral valve Ascending aorta Left atrium (LA) Esophagus Pulmonary veins Right atrium (RA) Tricuspid valve A LA RV Aortic arch Pulmonary trunk Infundibulum of the RVOT Aortic valve B Left atrium (LA) Right ventricle (RV) Aortic valve Pulmonary trunk Aortic arch Left main bronchus Pulmonary veins Thorax 132 Radiographic Anatomy of the Thorax (I) Fig. 11.6 (Reproduced from Lange S. Radiologische Diagnostik der Thoraxerkrankungen, 4th ed. Stuttgart: Thieme; 2010.) A Posterior-anterior (PA) chest radiograph. Anterior view. B Left lateral chest radiograph. Lower lobe aa. Superior vena cava Right atrium Clavicle Apex of heart Left ventricle Auricle of left atrium Pulmonary a. Aortic arch Trachea Sternum, manubrium Dome of diaphragm Sternum Left upper lobe bronchus Dome of diaphragm Left ventricle Aortic arch Trachea Right ventricle Retrocardiac space Right pulmonary a. Retrosternal space Scapula Left atrium 11 Sectional & Radiographic Anatomy 133 Fig. 11.7 Left bronchogram Anteroposterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Apical-posterior segment of upper lobe Main bronchus Lobar bronchus Superior segment of lower lobe Posterior basal segment of lower lobe Anterior segment of upper lobe Lingula of the superior segment of upper lobe Lingula of the inferior segment of upper lobe Segmental bronchus Anterior-medial basal segment of lower lobe Lateral basal segment of lower lobe Fig. 11.8 MRI of the thorax Coronal view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 3rd ed. New York, NY: Thieme; 2007.) Right pulmonary a. Right main bronchus Right pulmonary v. Right lung Liver Spinal cord Left lung Aortic arch Left pulmonary a. Left main bronchus Left pulmonary v. Esophagus Descending aorta Spleen Thoracic vertebrae, T11 Intervertebral disc T11–T12 Thorax 134 Radiographic Anatomy of the Thorax (II) Fig. 11.9 Selective coronary angiography of the left coro-nary artery in a right anterior oblique position Fig. 11.10 Selective coronary angiography of the right coronary artery in a left anterior oblique projection (Reproduced from Thelen M. et al. Bildgebene Kardiodiagnostik. Stuttgart: Thieme; 2007.) Left coronary a. Left marginal a. Left posterolateral a. Left circumflex a. Left diagonal aa. Left interventricular a. Right coronary a. Right marginal a. Right posterolateral a. Posterior interventricular a. Fig. 11.11 CT of the heart CT angiography. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Right auricle Ascending aorta Right coronary a. (RCA) Pulmonary trunk Anterior interventricular br. of LCA Left atrium Left coronary a. (LCA) Left pulmonary v. Left auricle Circumflex br. of LCA Marginal br. 11 Sectional & Radiographic Anatomy 135 Fig. 11.12 MRI of the heart (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A Left ventricular outflow tract. Fig. 11.13 Aortic arch angiogram Left lateral view. Right thyrocervical trunk Right common carotid a. Right vertebral a. Right subclavian a. Brachiocephalic trunk Aortic arch Ascending aorta Left thyrocervical trunk Left common carotid a. Left vertebral a. Left subclavian a. Descending aorta B Two chamber view of right ventricle. Branchiocephalic trunk Ascending aorta Right atrium Interventricular septum Right ventricle Pulmonary a. Aortic valve Left ventricle Right pulmonary vv. Superior vena cava Right atrium Right atrio-ventricular (tricuspid) valve Brachiocephalic trunk Ascending aorta Pulmonary trunk Right ventricle Thorax 136 Radiographic Anatomy of the Thorax (III) Fig. 11.14 CT of the thorax (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A B C Rib Anterior edge of lung in costomediastinal recess Superior vena cava Right main bronchus Pectoralis major Sternum Ascending aorta Left lung Left main bronchus Descending aorta Scapula Subscapularis Infraspinatus Pulmonary trunk Left pulmonary v. Left atrium Left inferior lobar bronchus Esophagus Azygos v. Descending aorta Ascending aorta Superior vena cava Right pulmonary a. Right inferior lobar bronchus Conus arteriosus Ascending aorta Left ventricle Left pulmonary v. Descending aorta Right auricle Superior vena cava Left atrium Esophagus C A B 11 Sectional & Radiographic Anatomy 137 D E Conus arteriosus Left ventricle Left atrium Left pulmonary v. Descending aorta Right atrium Aortic valve Right pulmonary v. Esophagus Right ventricle Right atrioventricular (tricuspid) valve Right atrium Left atrium Esophagus Interventricular septum Left ventricle Descending aorta Sympathetic trunk Inferior vena cava Azygos v. Esophagus Descending aorta F F D E 12 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13 Abdominal Wall Bony Framework for the Abdominal Wall . . . . . . . . . . . . . . . 142 Muscles of the Anterolateral Abdominal Wall . . . . . . . . . . . . 144 Rectus Sheath & Posterior Abdominal Wall . . . . . . . . . . . . . . 146 Abdominal Wall Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . 148 Inguinal Region & Canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Inguinal Region & Inguinal Hernias. . . . . . . . . . . . . . . . . . . . . 152 Scrotum & Spermatic Cord . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 14 Abdominal Cavity & Spaces Divisions of the Abdominopelvic Cavity . . . . . . . . . . . . . . . . 156 Peritoneum, Mesenteries & Omenta . . . . . . . . . . . . . . . . . . . 158 Mesenteries & Peritoneal Recesses . . . . . . . . . . . . . . . . . . . . 160 Lesser Omentum & Omental Bursa . . . . . . . . . . . . . . . . . . . . 162 Mesenteries & Posterior Abdominal Wall. . . . . . . . . . . . . . . . 164 15 Internal Organs Stomach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Duodenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Jejunum & Ileum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Cecum, Appendix & Colon . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Liver: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Liver: Lobes & Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Gallbladder & Bile Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Pancreas & Spleen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Kidneys & Suprarenal Glands (I) . . . . . . . . . . . . . . . . . . . . . . . 182 Kidneys & Suprarenal Glands (II) . . . . . . . . . . . . . . . . . . . . . . 184 16 Neurovasculature Arteries of the Abdominal Wall & Organs . . . . . . . . . . . . . . . 186 Abdominal Aorta & Renal Arteries . . . . . . . . . . . . . . . . . . . . . 188 Celiac Trunk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Superior & Inferior Mesenteric Arteries . . . . . . . . . . . . . . . . . 192 Veins of the Abdominal Wall & Organs . . . . . . . . . . . . . . . . . 194 Inferior Vena Cava & Renal Veins . . . . . . . . . . . . . . . . . . . . . . 196 Portal Vein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Superior & Inferior Mesenteric Veins . . . . . . . . . . . . . . . . . . . 200 Lymphatics of the Abdominal Wall & Organs . . . . . . . . . . . . 202 Lymph Nodes of the Posterior Abdominal Wall . . . . . . . . . . . 204 Lymph Nodes of the Supracolic Organs . . . . . . . . . . . . . . . . . 206 Lymph Nodes of the Infracolic Organs . . . . . . . . . . . . . . . . . . 208 Nerves of the Abdominal Wall . . . . . . . . . . . . . . . . . . . . . . . . 210 Autonomic Innervation: Overview . . . . . . . . . . . . . . . . . . . . . 212 Autonomic Innervation & Referred Pain. . . . . . . . . . . . . . . . . 214 Innervation of the Foregut & Urinary Organs . . . . . . . . . . . . 216 Innervation of the Intestines . . . . . . . . . . . . . . . . . . . . . . . . . 218 17 Sectional & Radiographic Anatomy Sectional Anatomy of the Abdomen . . . . . . . . . . . . . . . . . . . 220 Radiographic Anatomy of the Abdomen (I). . . . . . . . . . . . . . 222 Radiographic Anatomy of the Abdomen (II). . . . . . . . . . . . . . 224 Abdomen Transumbilical plane (L3–4 disk) Pubic tubercle Anterior superior iliac spine (ASIS) Inguinal lig. Pubic symphysis Linea alba Semilunar line Superficial inguinal ring Quadriceps femoris Sartorius Anterior superior iliac spine (ASIS) External oblique Tendinous intersections Rectus abdominis Small intestine (jejunum and ileum) Liver Transverse colon Stomach Right upper quadrant (RUQ) Right lower quadrant (RLQ) Periumbilical region Left lower quadrant (LLQ) Left upper quadrant (LUQ) Costal margin (arch) Gallbladder Duodenum Descending colon Pancreas Spleen Ascending colon with cecum and vermiform appendix Suprarenal glands Kidneys Ureters Abdominal aorta Urinary bladder A Quadrants, defined by the in-tersection of the median plane and the transumbilical plane through the L3–L4 disk. B Organs of the anterior layer. D Organs of the posterior layer. C Organs of the middle layer. Abdomen 140 12 Surface Anatomy Surface Anatomy Fig. 12.1 Palpable structures of the abdomen and pelvis Anterior view. See pp. 2–3 for structures of the back. Fig. 12.2 Quadrants and layers of the abdominopelvic cavity Anterior view. The location of the organs of the abdomen and pelvis can be described by quadrant and layer. B Musculature. A Bony prominences. ① ⑤ ④ ③ ② Superior border of pubic symphysis Superior border of manubrium Upper abdomen Mid-abdomen Lower abdomen Inguinal lig. Iliac crest Costal margin (arch) Subcostal plane Supracrestal plane Midclavicular line ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ 12 Surface Anatomy 141 Table 12.1 Transverse planes through the abdomen ① Transpyloric plane Transverse plane midway between the superior borders of the pubic symphysis and the manubrium ② Subcostal plane Plane at the lowest level of the costal margin (the inferior margin of the tenth costal cartilage) ③ Supracrestal plane Plane passing through the summits of the iliac crests ④ Transtubercular plane Plane at the level of the iliac tubercles (the iliac tubercle lies ~5 cm posterolateral to the anterior superior iliac spine) ⑤ Interspinous plane Plane at the level of the anterior superior iliac spines Table 12.2 Regions of the abdomen ① Epigastric region ② Umbilical region ③ Pubic region ④ Left hypochondriac region ⑤ Left lateral (lumbar) region ⑥ Left inguinal region ⑦ Right hypochondriac region ⑧ Right lateral (lumbar) region ⑨ Right inguinal region 4th rib 6th rib 8th rib 10th rib 12th rib Wing (ala) of ilium Iliac crest Anterior superior iliac spine Iliopubic eminence Superior pubic ramus Pubic tubercle Pubic symphysis L5 L4 L3 L2 L1 T12 Sacrum Ischial spine Arcuate line Sacral promontory Iliac tuberosity Transverse processes of lumbar vertebrae Costal margin Xiphoid process Body of sternum Pubic tubercle Inguinal lig. Anterior superior iliac spine Pubic symphysis Fig. 13.1 Bony framework of the abdomen Anterior view. These bones are the site of attachment for the muscles and ligaments of the anterolateral abdominal wall and form a bony cage that protects certain abdominal organs. Fig 13.2 The inguinal ligament Male pelvis, anterosuperior view. The inguinal ligament is a palpable landmark that forms the demarcation between the abdominal wall and thigh. It is formed by the inferior edge of the external oblique apo-neurosis, the most superficial of the anterior abdominal wall muscles. The inguinal ligament attaches laterally to the anterior superior iliac spine and medially to the pubic tubercle. It is important clinically as it forms the floor of the inguinal canal (see Table 13.2) and the roof of the retro-inguinal space (see Fig. 34.31). Abdomen 142 13 Abdominal Wall Bony Framework for the Abdominal Wall Quadratus lumborum Conjoint tendon Rectus abdominis, lateral head Pyramidalis Rectus sheath, anterior layer Rectus abdominis, medial head Transversus abdominis Internal oblique External oblique Psoas minor Iliac crest Anterior superior iliac spine Anterior inferior iliac spine Arcuate line Superior pubic ramus Auricular surface of ilium Quadratus lumborum Transversus abdominis Psoas minor Posterior superior iliac spine Superior pubic ramus Ischial tuberosity Ischial spine Greater sciatic notch Fig. 13.3 Abdominal wall muscle attachment sites Left hip bone. Muscle origins are in red, insertions in blue. 13 Abdominal Wall 143 A Superior view. B Medial view. Linea alba Pectoralis major (sternocostal part) Pectoralis major (abdominal part) Serratus anterior External oblique Inguinal lig. Superficial inguinal ring Fundiform lig. of the penis Rectus sheath, anterior layer Umbilicus Spermatic cord, cremaster muscle Sternum External oblique aponeurosis Anterior superior iliac spine Internal oblique aponeurosis External intercostals Rectus abdominis External oblique Internal oblique Inguinal lig. Spermatic cord, cremaster muscle Linea alba Umbilicus Xiphoid process Sternum Costal cartilage Internal intercostals Rectus sheath, anterior layer Abdomen 144 Muscles of the Anterolateral Abdominal Wall The muscles of the anterolateral abdominal wall consist of the external and internal obliques and the transversus abdominis. The posterior or deep abdominal wall muscles (notably the psoas major) are functionally hip muscles (see p. 148). Fig. 13.4 Muscles of the abdominal wall Right side, anterior view. A Superficial abdominal wall muscles. B Removed: External oblique, pectoralis major, and serratus anterior. Transversus abdominis aponeurosis External oblique Internal oblique Transversus abdominis Rectus abdominis Inguinal lig. Pyramidalis Linea alba Umbilicus Tendinous intersections Sternum Spermatic cord, cremaster muscle External oblique Semilunar line Transversalis fascia Arcuate line Spermatic cord Deep inguinal ring Umbilicus Rectus sheath, posterior layer Sternum Linea alba Rectus sheath, anterior layer Inguinal lig. Rectus abdominis Transversus abdominis Internal oblique 13 Abdominal Wall 145 C Removed: External and internal obliques. D Removed: Rectus abdominis. Costal part of diaphragm Transversus abdominis Iliacus Transversus abdominis External oblique Rectus abdominis Iliopubic tract Internal oblique Arcuate line Transversalis fascia Umbilicus Parietal peritoneum Linea alba Diaphragm Parietal pleura, diaphragmatic part Central tendon Plane of section in figure B Plane of section in figure C Rectus sheath, posterior layer Abdomen 146 Rectus Sheath & Posterior Abdominal Wall Fig. 13.5 The rectus sheath The rectus sheath encloses the rectus abdom-inis and pyramidalis muscles on either side of the midline. Its anterior and posterior layers are formed by the aponeuroses of the antero-lateral muscles as they split to pass around the rectus muscles. An arcuate line marks the inferior extent of the posterior layer, the point at which all of the aponeuroses pass anterior to the rectus muscles. A Posterior (interior) view of the anterior abdominal wall. Peritoneum and trans-versalis fascia have been removed on the left side to reveal the rectus sheath. B Section through the abdominal wall superior to the arcuate line. C Section through the abdominal wall inferior to the arcuate line. Membranous layer, subcutaneous tissue Parietal peritoneum Preperitoneal fat Rectus sheath, anterior layer External oblique aponeurosis Internal oblique aponeurosis Transversus abdominis aponeurosis Rectus sheath, posterior layer Linea alba Transversalis fascia Transversus abdominis Internal oblique External oblique Rectus abdominis Membranous layer Subcutaneous tissue Parietal peritoneum External oblique aponeurosis Internal oblique aponeurosis Transversus abdominis aponeurosis Linea alba Skin Fatty layer Transversalis fascia Rectus sheath, anterior layer Manubrium Transversus thoracis Central tendon Body of sternum Diaphragm, costal part Costal arch Transversus abdominis Rectus sheath, posterior layer Iliacus Pubic symphysis Obturator internus Gluteus maximus Piriformis Psoas major L5 vertebral body Iliac crest T12 vertebral body Aortic hiatus T10 vertebral body Esophageal hiatus T8 vertebral body Caval opening Internal intercostals Sacrospinous lig. Esophageal hiatus Median arcuate lig. Lateral arcuate lig. Medial arcuate lig. Quadratus lumborum Psoas minor Iliopsoas Iliacus Psoas major Transversus abdominis Aortic hiatus Diaphragm, lumbar part, and left crus Diaphragm, costal part Central tendon Caval opening Right crus 13 Abdominal Wall 147 Fig. 13.6 Muscles of the posterior abdominal wall A Midsagittal section with diagraphm in intermediate position. B Coronal section with diaphragm in intermediate position. Median arcuate lig. Right crus Left crus Aorta Crural sling Esophagus Central tendon Inferior vena cava C Apertures of the diaphragm with vessels transected. Anterior view. The caval opening is located to the right of the midline, those for the esophagus and aorta are to the left. Note that the crura of the diaphragm typically extend inferiorly as far as the L3 vertebra on the right and L2 vertebra on the left. Clinical box 13.1 In diaphragmatic hernias, abdominal viscera prolapse into the thorax through a congenital or acquired opening in the diaphragm. By far the most common herniation site is the esophageal hiatus, accounting for 90% of cases. “Sliding” hernias, which account for 85% of these hiatal hernias, occur when the distal end of the esophagus and the cardia of the stomach slide upward into the thorax through the esophageal hiatus. Diaphragmatic hernias A S Linea alba D F G ⑦ ⑥ ⑧ Abdomen 148 Abdominal Wall Muscle Facts Fig. 13.7 Anterior abdominal wall muscles Anterior view. Fig. 13.8 Anterolateral abdominal wall muscles Anterior view. A External oblique. B Internal oblique. C Transversus abdominis. Fig. 13.9 Posterior abdominal wall muscles Anterior view. The psoas major and iliacus are together known as the iliopsoas inferiorly. Table 13.1 Abdominal wall muscles Muscle Origin Insertion Innervation Action Anterior abdominal wall muscles ① Rectus abdominis Lateral head: Crest of pubis to pubic tubercle Medial head: Anterior region of pubic symphysis Cartilages of 5th to 7th ribs, xiphoid process of sternum Intercostal nn. (T5– T11) , subcostal n. (T12) Flexes trunk, compresses abdomen, stabilizes pelvis ② Pyramidalis Pubis (anterior to rectus abdominis) Linea alba (runs within the rectus sheath) Subcostal n. (T12) Tenses linea alba Anterolateral abdominal wall muscles ③ External oblique 5th to 12th ribs (outer surface) Linea alba, pubic tubercle, anterior iliac crest Intercostal nn. (T7–T11) , subcostal n. (T12) Unilateral: Flexes trunk to same side, rotates trunk to opposite side (external oblique) or same side (internal oblique) Bilateral: Flexes trunk, compresses abdomen, stabilizes pelvis ④ Internal oblique Thoracolumbar fascia (deep layer), iliac crest (intermediate line), anterior superior iliac spine, iliopsoas fascia 10th to 12th ribs (lower borders), linea alba (anterior and posterior layers) Intercostal nn. (T7–T11) , subcostal n. (T12) iliohypogastric n., ilioinguinal n. ⑤ Transversus abdominis 7th to 12th costal cartilages (inner surfaces), thoracolumbar fascia (deep layer), iliac crest, anterior superior iliac spine (inner lip), iliopsoas fascia Linea alba, pubic crest Unilateral: Rotates trunk to same side Bilateral: Compresses abdomen Posterior abdominal wall muscles Psoas minor (see Fig. 31.19) T12, L1 vertebrae and intervertebral disk (lateral surfaces) Pectineal line, iliopubic ramus, iliac fascia; lowermost fibers may reach inguinal lig. L1–L2 (L3) spinal nn. Weak flexor of the trunk ⑥ Psoas major Superficial layer T12–L4 vertebral bodies and associated intervertebral disks (lateral surfaces) Femur (lesser trochanter), joint insertion as iliopsoas muscle Hip joint: Flexion and external rotation Lumbar spine (with femur fixed): Unilateral: Contraction flexes trunk laterally Bilateral: Contraction raises trunk from supine position Deep layer L1–L5 (costal processes) ⑦ Iliacus Iliac fossa Femoral n. (L2–L4) ⑧ Quadratus lumborum Iliac crest and iliolumbar lig. (not shown) 12th rib, L1–L4 vertebrae (costal processes) Subcostal n. (T12), L1–L4 spinal nn. Unilateral: Flexes trunk to same side Bilateral: Bearing down and expiration, stabilizes 12th rib Approximately 50% of the population has this muscle. For the diaphragm see pp. 64–65. Quadratus lumborum Iliac crest Psoas major Iliopsoas Lesser trochanter Pubic tubercle Pubic symphysis Pyramidalis Inguinal lig. Rectus abdominis Tendinous intersections Linea alba Xiphoid process 5th rib Iliac fossa Iliacus Xiphoid process Linea alba Umbilical ring Anterior superior iliac spine Inguinal lig. Superficial inguinal ring External oblique aponeurosis 5th rib Outer lip of iliac crest External oblique Xiphoid process Anterior superior iliac spine Pubic symphysis Inguinal lig. Iliac crest, intermediate line Internal oblique Internal oblique aponeurosis 10th rib Linea alba Rectus sheath, anterior layer Rectus sheath, posterior layer Sternum Xiphoid process Linea alba Pubic symphysis Inguinal lig. Anterior superior iliac spine Iliac crest Transversus abdominis aponeurosis Transversus abdominis Arcuate line 13 Abdominal Wall 149 Fig. 13.10 Anterior, anterolateral, and posterior abdominal wall muscles Anterior view. B External oblique. C Internal oblique. D Transversus abdominis. A Anterior and posterior muscles. Rectus abdominis Internal oblique External oblique Transversus abdominis Linea alba Body of sternum Fig. 13.11 Anterior and lateral abdominal wall muscles as a functional unit Transversus abdominis Medial crus Intercrural fibers Internal oblique Superficial inguinal ring Lateral crus Iliopectineal arch Iliopsoas Femoral n. Femoral a. and v. Inguinal lig. External oblique aponeurosis Membranous layer, subcutaneous tissue Pectineus Lacunar lig. Reflected inguinal lig. Rectus sheath Spermatic cord Genitofemoral n., genital br. Linea alba Ilioinguinal n. Rectus abdominis Plane of section in Table 13.2 External oblique Cremaster m. and cremasteric fascia External spermatic fascia Pubic tubercle External oblique aponeurosis Intercrural fibers Medial crus Fat pad Ilioinguinal n. Round lig. of uterus V. of round lig. A. of round lig. Genitofemoral n., genital br. Site of inguinal lig. Lateral crus Superficial inguinal ring Abdomen 150 Inguinal Region & Canal Fig. 13.12 Inguinal region Right side, anterior view. A Male. B Female. The inguinal region is the junction of the anterior abdominal wall and the anterior thigh. The inguinal canal in the male is an important site for the passage of structures into and out of the abdominal cavity (e.g., components of the spermatic cord). Clinical box 13.2 The inguinal canal is an oblique pathway through the inferior part of the anterior abdominal wall. In the male, it represents the path through which the testis migrated from the posterior abdominal wall into the scrotum during the perinatal period. It’s traversed by the spermatic cord (see Table 13.4), which connects the testis to the rest of the genitourinary system. The female inguinal canal is smaller and contains only the round ligament with its neurovasculature. This ligament is a remnant of the distal portion of the embryonic gubernaculum, a structure originally present in both males and females but which regresses in the male with descent of the testes. Gender differences of the inguinal canal ② ① ③ ④ ⑤ ⑥ Membranous layer, subcutaneous tissue Ilioinguinal n. Spermatic cord Fascia lata Pectineus Superior pubic ramus Lacunar lig. Pectineal lig. Iliopubic tract Spermatic cord with cremaster m. Internal oblique Inguinal lig. Spermatic cord with internal spermatic fascia Transversalis fascia Transversus abdominis Rectus abdominis Inguinal lig. Spermatic cord Lateral crus Superficial inguinal ring Intercrural fibers Medial crus Location of deep inguinal ring External oblique aponeurosis 13 Abdominal Wall 151 Fig. 13.13 Dissection of the male inguinal region Right side, anterior view. A Superficial layer. B Removed: External oblique aponeurosis. C Removed: Internal oblique m. Superficial inguinal ring (cut open) External oblique aponeurosis Spermatic cord with cremaster m. and cremasteric fascia Iliohypogastric n., anterior cutaneous br. Internal oblique Ilioinguinal n. Genito-femoral n., genital br. Transversus abdominis Internal oblique Ilioinguinal n. Femoral a. and v. Spermatic cord, internal spermatic fascia Reflected inguinal lig. Hesselbach’s triangle Inferior epigastric a. and v., interfoveolar lig. Transversalis fascia Deep inguinal ring Fig. 13.14 Opening of the inguinal canal Right side, anterior view. A Divided: External oblique aponeurosis. B Divided: Internal oblique and cremaster mm. Table 13.2 Structures of the inguinal canal Structures Formed by Wall Anterior wall ① External oblique aponeurosis Roof ② Internal oblique m. ③ Transversus abdominis m. Posterior wall ④ Transversalis fascia ⑤ Parietal peritoneum Floor ⑥ Inguinal lig. (densely interwoven fibers of the lower external oblique aponeurosis and adjacent fascia lata of the thigh) Openings Superficial inguinal ring Opening in external oblique aponeurosis; bounded by medial and lateral crus, intercrural fibers, and reflected inguinal lig. Deep inguinal ring Outpouching of the transversalis fascia lateral to the lateral umbilical fold (inferior epigastric vessels) Sagittal section through plane in Fig. 13.12A. Parietal peritoneum Transversalis fascia Rectus sheath, posterior layer Deep circumflex iliac a. and v. Inferior epigastric a. and v. External iliac a. and v. Ductus deferens Supravesical fossa Medial inguinal fossa (Hesselbach‘s triangle) Lateral inguinal fossa (deep inguinal ring) Psoas major Iliacus Lateral umbilical fold Medial umbilical fold Median umbilical fold Umbilicus Rectus abdominis Round lig. of liver, paraumbilical veins Diaphragm Falciform lig. Arcuate line Bladder Prostate Abdomen 152 Inguinal Region & Inguinal Hernias Fig. 13.15 Sites of herniation through the anterior abdominal wall Coronal section, male, posterior (internal) view. A The three fossae of the anterior abdominal wall (circled) are sites of potential herniation through the wall. B Internal hernial openings in the male inguinal region. Detail from A. Peritoneum and transversalis fascia have been removed to reveal the hernia openings. Color shading indicates openings for supravesical (green), indirect (teal) and direct (purple) hernias (see Table 13.3). Iliopsoas Transversalis fascia Iliopubic tract Femoral n. Femoral ring Medial umbilical fold Inferior epigastric a. and v. Peritoneum Interfoveolar lig. Lateral inguinal fossa (deep inguinal ring) Supravesical fossa Medial inguinal fossa (Hesselbachʼs triangle) External iliac a. and v. Iliopectineal arch Arcuate line Rectus abdominis Pectineal lig. Ductus deferens Testicular a. and v. Iliacus Psoas major Transversus abdominis 13 Abdominal Wall 153 Fig. 13.16 Schematic of the male inguinal canal and its relation to structures of the abdominal wall Right side, anterior view. Scrotal skin with dartos m. and fascia Superficial inguinal ring Inferior epigastric a. and v. Subcutaneous tissue (fatty and membranous layers) External oblique aponeurosis Superficial investing fascia Internal oblique Transversus abdominis Transversalis fascia Deep inguinal ring (lateral inguinal fossa) Lateral umbilical fold Medial inguinal fossa (Hesselbach's triangle) Medial umbilical fold (obliterated umbilical a.) Rectus abdominis Rectus sheath, anterior layer Cremaster m. and cremasteric fascia Pampiniform plexus, testicular a. and ductus deferens Internal spermatic fascia Epididymis Testis Scrotal cavity External spermatic fascia B Direct inguinal hernia Table 13.3 Hernias of the inguinal region Most inguinal hernias occur in males. All are located above the inguinal ligament and, if large enough, protrude externally through the superficial ring. However, the internal site of origin, and therefore structure of the hernia sac (covering), differ among types. Femoral hernias, more common in women, originate at the femoral ring below the inguinal ligament and emerge at the saphenous opening in the thigh. Hernia type Site of origin Hernia sac Indirect inguinal (congenital or acquired) Lateral inguinal fossa (deep inguinal ring) lateral to inferior epigastric vessels Peritoneum, transversalis fascia, cremaster m. Direct inguinal (acquired) Medial inguinal fossa (Hesselbach’s triangle), medial to inferior epigastric vessels Peritoneum, transversalis fascia Femoral Femoral ring, inferior to inguinal lig. Cribiform fascia of saphenous opening A Indirect inguinal hernia Superficial inguinal ring Femoral a. and v. Testicular a. Pampiniform plexus (testicular vv.) Parietal layer Visceral layer Epididymis Processus vaginalis (obliterated) Scrotum Dartos m. External spermatic fascia Ductus deferens Testicular plexus Internal spermatic fascia Cremasteric fascia and cremaster m. Superficial inguinal ring External spermatic fascia Tunica vaginalis B Fascial and muscular layers of the spermatic cord have been opened to reveal its contents. A Structure and contents of the scrotum. Abdomen 154 Fig. 13.17 Scrotum and spermatic cord Anterior view. Scrotum & Spermatic Cord The coverings of the scrotum, testis, and spermatic cord are continuations of muscular and fascial lay-ers of the anterior abdominal wall, as are those of the inguinal canal. Superficial inguinal ring Scrotal skin Dartos m. and fascia Scrotal septum Root of the penis Spermatic cord Femoral a. and v. Fundiform lig. of the penis Gubernaculum External spermatic fascia covering spermatic cord and testis Superficial penile (Colles’) fascia Testis with tunica vaginalis, visceral layer Scrotum Pampiniform plexus Cremasteric fascia and cremaster m. Dartos fascia Membranous layer, subcutaneous tissue Testicular a. Testicular plexus Tunica vaginalis, parietal layer Glans of penis Epididymis, head Internal spermatic fascia Epididymis, body External spermatic fascia Skin ① ④ Fibrous stroma ③ ② External spermatic fascia Cremasteric fascia Cremaster m. Internal spermatic fascia ⑨ ⑧ ⑤ ⑥ ⑦ 13 Abdominal Wall 155 Fig. 13.18 Testis and epididymis Left lateral view. Ductus deferens Testicular a. Mediastinum testis with rete testis Epididymis, head Lobule Scrotal septum Tunica albuginea Septum Cavity of tunica vaginalis Pampiniform plexus 6b 6a ① ② ③ ④ ⑤ Table 13.5 Coverings of the testis Covering layer Derived from ① Scrotal skin Abdominal skin ② Dartos m. and fascia Membranous layer, subcutaneous tissue ③ External spermatic fascia External oblique aponeurosis and superficial investing fascia ④ Cremaster m. and cremasteric fascia Internal oblique m. ⑤ Internal spermatic fascia Transversalis fascia Tunica vaginalis, parietal layer Peritoneum Tunica vaginalis, visceral layer The transversus abdominis has no contribution to the spermatic cord or covering of the testis. Transverse section through right testis, superior view. Table 13.4 Contents of the spermatic cord Surrounding layer Contents External spermatic fascia ① Ilioinguinal n. Cremasteric muscle ② Cremasteric a. and v. ③ Genitofemoral n., genital br. Internal spermatic fascia ④ A. and v. of ductus deferens ⑤ Ductus deferens ⑥ Testicular a. ⑦ Processus vaginalis (obliterated) ⑧ Testicular (nerve) plexus ⑨ Pampiniform (venous) plexus 6b 6a Rectus abdominis Abdominal aorta Liver, bare area Liver Sternum Hepatogastric lig. (lesser omentum) Omental bursa (lesser sac) Pancreas, neck Stomach Middle colic a. Transverse mesocolon Transverse colon Greater omentum Jejunum and ileum Urinary bladder Scrotum, septum Bulbospongiosus Deep transverse perineal Prostate Ductus deferens, ampulla Rectum Rectovesical pouch L5 vertebra Left common iliac a. and v. Mesentery Duodenum, horizontal part Pancreas, uncinate process Left renal v. Superior mesenteric a. Left renal a. Splenic a. and v. Celiac trunk Omental foramen Esophagus Abdomen 156 14 Abdominal Cavity & Spaces Divisions of the Abdominopelvic Cavity Fig. 14.1 Organs of the abdominopelvic cavity Midsagittal section, male, viewed from the left. Clinical box 14.1 Acute abdominal pain (“acute abdomen”) may be so severe that the abdominal wall becomes extremely sensitive to touch (“guarding”) and the intestines stop functioning. Causes include organ inflammation such as appendicitis, perforation due to a gastric ulcer (see p. 167), or organ blockage by a stone, tumor, etc. In women, gynecological processes or ectopic pregnancies may produce severe abdominal pain. Acute abdominal pain 14 Abdominal Cavity & Spaces 157 Fig. 14.2 Divisions of the pelvic and abdominal cavities Each column of diagrams shows a midsagittal section viewed from the left side, as well as two axial sections, one at the L1 level and the other at the lower part of the sacrum, both viewed from below. Linea terminalis A B C Omental bursa Rectovesical pouch D Omental bursa E Rectovesical pouch F G Retroperitoneal space H Subperitoneal space I A–C Topography of body cavities: abdominal cavity and pelvic cavity (imaginary line separating the two cavities is the linea terminalis). D–F Serous cavities (peritoneal spaces): abdominal peritoneal cavity and pelvic peritoneal cavity. G–I Connective tissue spaces (extraperito-neal spaces): retroperitoneal space and subperitoneal space; serous cavities and extraperitoneal spaces are sepa-rated by peritoneum. Abdomen 158 Peritoneum, Mesenteries & Omenta Parietal peritoneum Greater omentum Lesser omentum Parietal peritoneum Visceral peritoneum Transverse mesocolon Mesentery (of the small intestine) Bare area of the liver Rectovesical space Omental bursa Fig. 14.3 Peritoneal cavity A Midsagittal section through the male abdominopelvic cavity, viewed from the left. The peritoneum is shown in red. Organs in the abdominopelvic cavity are classified by the presence of surrounding peritoneum (the serous membrane lining the cavity) and a mesentery (a double layer of peritoneum that connects the organ to the abdominal wall) (see Table 14.1). Clinical box 14.2 Bacterial contamination of the perito neum following surgery or rupture of an inflamed organ (duode num, gallbladder, appendix) results in peritonitis, inflammation of the peritoneum. It is accompanied by severe abdominal pain, tenderness, nausea, and fever and can be fatal when generalized throughout the peritoneal cavity. It often results in ascites, the accumulation of excess peritoneal fluid due to a change in concen tration gradients that results in loss of capillary fluid. Ascites can also accompany other pathologic conditions, such as metastatic liver cancer and portal hypertension. In these cases, many liters of ascitic fluid can accumulate in the peritoneal cavity. The fluid is aspirated by paracentesis. The needle is carefully inserted through the abdominal wall so as to avoid the urinary bladder and inferior epigastric vessels. Peritonitis and ascites 14 Abdominal Cavity & Spaces 159 Table 14.2 Organs of the abdominopelvic cavity classified by their relationship to the peritoneum Location Organs Intraperitoneal organs: These organs have a mesentery and are completely covered by the peritoneum. Abdominal peritoneal • Stomach • Gallbladder • Small intestine (jejunum, ileum, some of • Cecum with vermiform appendix (portions of variable the superior part of the duodenum) size may be retroperitoneal) • Spleen • Large intestine (transverse and sigmoid colons) • Liver Pelvic peritoneal • Uterus (fundus and body) • Ovaries • Uterine tubes Extraperitoneal organs: These organs either have no mesentery or lost it during development. Retroperitoneal Primarily • Kidneys and ureters • Suprarenal glands • Uterine cervix Secondarily • Duodenum (descending, horizontal, and ascending) • Ascending and descending colon and cecum • Pancreas • Rectum (upper 2/3) Infraperitoneal/subperitoneal • Urinary bladder • Distal ureters • Prostate • Seminal glands • Uterine cervix • Vagina • Rectum (lower 1/3) Fig. 14.4 Schematic showing peritoneal relations of intraperitoneal and extraperitoneal (primary and secondary retroperitoneal) organs of the abdomen Transverse section, superior view (See Table 14.2). Intraperitoneal organ Visceral layer Parietal layer Mesentery Peritoneal cavity Peritoneum Fig. 14.5 Relationship of an intraperitoneal organ to the mesentery and peritoneum Arrows indicate blood vessels in the mesentery. Fig. 14.6 Structure of the greater and lesser omenta and their relation to the omental bursa Sagittal section, left lateral view. Gastrocolic lig. Transverse mesocolon Transverse colon Mesentery (of the small intestine) Pancreas Duodenum Omental bursa Lesser omentum Liver Greater omentum Stomach Table 14.1 Mesenteries and omenta Mesenteries Mesentery (of the small intestine) Transverse mesocolon Sigmoid mesocolon Mesoappendix Omenta Lesser omentum Greater omentum Reflections of the peritoneum that connect organs to the body wall or to another organ allow normal mobility of the gastrointestinal tract while preventing excessive movement. A mesentery is a double layer of peritoneum that connects intraperitoneal organs to the posterior abdominal wall and transmits nerves and vessels. An omentum is a double layer of peritoneum that connects the stomach and duodenum to another organ or to the posterior abdominal wall. Visceral peritoneum Mesentery Abdominal aorta Lumbar spine Retroperitoneal space Secondarily retroperitoneal organ (e.g., ascending colon) Parietal peritoneum Peritoneal cavity Intraperitoneal organ (e.g., jejunum) Extraperitoneal or primarily retroperitoneal organ (e.g., kidney) Transverse mesocolon with middle colic a. and v. Ileum Jejunum (covered by visceral peritoneum) Greater omentum (reflected superiorly) Ascending colon Tenia coli Parietal peritoneum Transverse colon Left colic flexure Gallbladder Round lig. of liver Liver, right lobe Ascending colon Tenia coli Ileum Medial umbilical fold (with obliterated umbilical a.) Lateral umbilical fold (with inferior epigastric a. and v.) Greater omentum Transverse colon Stomach Liver, left lobe Falciform lig. of liver Arcuate line Median umbilical fold (with obliterated urachus) Rectus abdominis Abdomen 160 Mesenteries & Peritoneal Recesses The peritoneal cavity is divided into the large greater sac and small omental bursa (lesser sac). The greater omentum is an apron-like fold of peri toneum suspended from the greater curvature of the stomach and covering the anterior surface of the transverse colon. The attach-ment of the transverse mesocolon on the anterior surface of the de-scending part of the duodenum and the pancreas divides the peritoneal cavity into a supracolic compartment (liver, gallbladder, and stomach) and an infracolic compartment (intestines). Fig. 14.7 Dissection of the peritoneal cavity Anterior view. A Greater sac. Retracted: Abdominal wall. B Infracolic compartment, the portion of the peritoneal cavity below the attachment of the transverse mesocolon. Reflected: Greater omentum and transverse colon. Mesentery, root Retrocecal recess Inferior iliocecal recess Mesoappendix Intersigmoidal recess Inferior duodenal recess Descending colon Sigmoid colon Appendix Sigmoid mesocolon Superior duodenal recess Left colic flexure Greater omentum (reflected superiorly) 14 Abdominal Cavity & Spaces 161 D Mesenteries and mesenteric recesses in the infracolic compartment. Reflected: Greater omentum, transverse colon, small intestines, and sigmoid colon. C Mesentery (of the small intestine). Reflected: Greater omentum, transverse colon, small intestine. Ascending colon Cecum Superior iliocecal recess Convoluted small intestine Root of mesentery Transverse colon Greater omentum The omental bursa, or lesser sac, is the portion of the peritoneal cav-ity behind the stomach and the lesser omentum (a double-layered peritoneal structure connecting the lesser curvature of the stomach and the proximal part of the duodenum to the liver). The omental bursa communicates with the greater sac via the omental (epiploic) foramen, located posterior to the free edge of the lesser omentum. Omental foramen Gallbladder Liver, right lobe Duodenum, descending part Right kidney Right colic flexure Ascending colon Gastrocolic lig. Greater omentum Middle colic a. and v. Descending colon Transverse colon Transverse mesocolon Pancreas Phrenicocolic lig. Gastrosplenic lig. Spleen Left kidney, superior pole Splenic a. Left suprarenal gland Left gastric a. Stomach, posterior surface Gastrocolic lig. Stomach, greater curvature Vestibule of omental bursa Celiac trunk Common hepatic a. Fig. 14.8 The lesser omentum Anterior view with liver retracted superiorly. The arrow points to the omental foramen, the opening into the omental bursa, posterior to the lesser omentum. Abdomen 162 Gallbladder Duodenum Hepatogastric lig. Diaphragm Stomach Greater omentum Hepatoduodenal lig. Lesser omentum Lesser Omentum & Omental Bursa Fig. 14.9 Omental bursa in situ Anterior view. Divided: Gastrocolic ligament. Retracted: Liver. Reflected: Stomach. Liver, caudate lobe Hepatoduodenal lig. (lesser omentum) Gallbladder Liver, right lobe Duodenum Greater omentum Transverse mesocolon Gastrocolic lig. (cut) Pancreas Spleen Cardiac orifice Liver, left lobe Inferior recess of omental bursa Vestibule of omental bursa Splenic recess of omental bursa Superior recess of omental bursa Diaphragm Inferior vena cava Diaphragm, hepatic surface Hepatoduodenal lig. (lesser omentum, cut) Transverse colon Duodenum Pancreas Spleen Abdominal aorta Inferior vena cava Liver Pancreas Omental bursa Spleen Splenic recess of omental bursa Stomach Left kidney Fig. 14.10 Location of the omental bursa Transverse section, inferior view. 14 Abdominal Cavity & Spaces 163 Fig. 14.11 Boundaries and walls of the omental bursa (lesser sac) Anterior view. A Boundaries of the omental bursa (lesser sac). B Posterior wall of the omental bursa (lesser sac). Table 14.3 Boundaries of the omental bursa Direction Boundary Recess Anterior Lesser omentum, gastrocolic lig. — Inferior Transverse mesocolon Inferior recess Superior Liver (with caudate lobe) Superior recess Posterior Pancreas, aorta (abdominal part), celiac trunk, splenic a. and v., gastrosplenic fold, left suprarenal gland, left kidney (superior pole) — Right Liver, duodenal bulb — Left Spleen, gastrosplenic lig. Splenic recess Table 14.4 Boundaries of the omental foramen The communication between the greater sac and lesser sac (omental bursa) is the omental (epiploic) foramen (see arrow in Fig. 14.9). Direction Boundary Anterior Hepatoduodenal lig. with the portal v., proper hepatic a., and bile duct Inferior Duodenum (superior part) Posterior Inferior vena cava, diaphragm (right crus) Superior Liver (caudate lobe) Liver, right lobe Gallbladder Omental foramen Hepatoduodenal lig. (lesser omentum) Duodenum, superior part Stomach, pyloric part Right colic flexure Greater omentum (cut) Ascending colon Terminal ileum Cecum Tenia coli Duodenum, horizontal part Rectum Mesentery (cut) Descending colon Transverse colon Transverse colon Left colic flexure Duodenojejunal flexure Transverse mesocolon, root Pancreas Gastrosplenic lig. Gastric surface Superior border Cardiac orifice Liver, left lobe Hepatogastric lig. (lesser omentum, cut) Round lig. of liver Median umbilical fold (with obliterated urachus) Rectus abdominis Lateral umbilical fold (with inferior epigastric a. and v.) Medial umbilical fold (with obliterated umbilical a.) Transversus abdominis, internal and external oblique Sigmoid mesocolon (cut) Spleen Abdomen 164 Mesenteries & Posterior Abdominal Wall Fig. 14.12 Mesenteric attachments of intraperitoneal organs Anterior view. Removed: Stomach, jejunum and ileum, and transverse and sigmoid colons. Retracted: Liver. Fig. 14.13 Location of mesenteric sites of connection to the abdominal wall L4 Sigmoid mesocolon Transverse mesocolon Mesentery Superior mesenteric a. and v. Right suprarenal gland Right kidney Superior part Pancreas, head Descending part Ascending part Horizontal part Mesenteric root Ascending colon (site of attachment) Right common iliac a. and v. Abdominal aorta Splenic a. and v. Mesoappendix Right ureter Parietal peritoneum External iliac a. Left ureter Diaphragm, hepatic surface Sigmoid mesocolon Descending colon (site of attachment) Paracolic gutter Left colic a. and v. Left kidney Pancreas, body and tail Gastrosplenic lig. Left suprarenal gland Cardiac orifice of stomach Inferior vena cava Hepatic vv. Rectum Hepatoduodenal lig. (with portal v., hepatic a., and bile duct) Transversus abdominis, internal and external oblique Inferior mesenteric a. Parietal peritoneum Duodenum Duodenum 14 Abdominal Cavity & Spaces 165 Fig. 14.14 Posterior wall of the peritoneal cavity Anterior view. Removed: All intraperitoneal organs. Revealed: Structures of the retro-peritoneum (see Table 14.2 and p. 250). Fig. 14.15 Drainage spaces and recesses within the peritoneal cavity Anterior view. Subphrenic recess Subhepatic recess Hepato-renal recess Right paracolic gutter Right infracolic space Left infracolic space Left paracolic gutter Hepatoduodenal ligament Right kidney Duodenum Site of attachment of ascending colon Mesentery (root) Superior ileocecal recess Retrocecal recess Rectovesical pouch Intersigmoid recess Sigmoid mesocolon (root) Left paracolic gutter Site of attachment of descending colon Inferior duodenal recess Superior duodenal recess Transverse mesocolon (root) Left kidney Spleen Inferior vena cava Inferior ileocecal recess Hepatic surface of diaphragm A Anterior view with the greater omentum and small intes-tine removed; preferred metastatic sites (see blue stars). B Posterior wall of the peritoneal cavity, anterior view. The mesenteric roots and sites of organ attachment create partially bounded spaces (recesses or sulci) where peritoneal fluid can flow freely. Esophagus, muscular coat, longitudinal layer Rugal folds Inner oblique layer Outer longi-tudinal layer Muscular coat Fundus Duodenum, superior part Pyloric sphincter Esophagus, adventitia Endoscopic light source Middle circular layer Trans-pyloric plane RUQ LUQ Liver Stomach Omental bursa Spleen Left kidney Pancreas Abdominal aorta Inferior vena cava Lesser omentum (hepatogastric lig.) Epigastric surface Esophagus Phrenic surface Hepatic surface Colomesocolic surface Pancreatic surface Renal surface Suprarenal surface Splenic surface Hepatic surface Phrenic surface Pyloric canal Fundus Duodenum Angular notch Lesser curvature Cardia Esophagus Body Greater curvature Pyloric antrum Pyloric orifice Duodenum Pyloric sphincter Angular notch Cardia Esophagus Body with longitudinal rugal folds Abdomen 166 15 Internal Organs Stomach Fig. 15.1 Stomach: Location A Anterior view. B Transverse section, inferior view. Fig. 15.2 Relations of the stomach A Anterior view. B Posterior view. Fig. 15.3 Stomach Anterior view. A Anterior wall. B Muscular layers. Removed: Serosa and subserosa. Windowed: Muscular coat. C Interior. Removed: Anterior wall. Hepato-duodenal lig. Right kidney Transversus abdominis, internal and external oblique Falciform lig. of liver Round lig. of liver Liver, right lobe Gallbladder Lesser omentum Lesser curvature Right colic flexure Duodenum Pyloric canal Pyloric antrum Greater omentum Descending colon Greater curvature Stomach, body Spleen Stomach, cardia Stomach, fundus Diaphragm Parietal peritoneum Liver, left lobe Hepato-gastric lig. Esophagus Ascending colon Hepato-esophageal lig. 15 Internal Organs 167 Fig. 15.4 Stomach in situ Anterior view of the opened upper abdomen. Arrow indi-cates the omental foramen. Rugal folds Gastric antrum Gastric ulcer Clinical box 15.1 Gastritis and gastric ulcers, the two most common diseases of the stomach, are associated with increased acid production and are caused by alcohol, drugs such as aspirin, and the bacterium Helicobacter pylori. Symptoms include lessened appetite, pain, and even bleeding, which manifests as black stool or dark brown material, often described as resembling “coffee grounds,” in vomit. Gastritis is limited to the inner surface of the stomach, whereas gastric ulcers extend into the stomach wall. In these endoscopic images, the gastric ulcer in C is covered with fibrin and shows hematin spots. Gastritis and gastric ulcers A Body of normal stomach. B Normal pyloric antrum. C Gastric ulcer. The stomach, an intraperitoneal organ, resides primarily in the left upper quadrant. Double layers of peritoneum extend superiorly from its lesser curvature as the lesser omentum and inferiorly from its greater curvature as the greater omentum. Jejunum and ileum Duodenum Duodeno-jejunal flexure RUQ LUQ Duodenal bulb Inferior duodenal flexure Superior duodenal flexure Superior (1st) part Descending (2nd) part Horizontal (3rd) part Ascending (4th) part Jejunum Superior mesenteric a. Suspensory lig. of duodenum Celiac trunk Diaphragm, left crus Diaphragm, right crus Esophagus Inferior vena cava Pyloric orifice Jejunum Duodenum, horizontal part Circular layer Longitudinal layer Submucosa Pyloric sphincter Circular folds (valves of Kerckring) Main pancreatic duct Major duodenal papilla Duodenum, descending part Minor duodenal papilla Accessory pancreatic duct Duodenum, superior part Bile duct Muscular coat Pancreas Duodenojejunal flexure Superior mesenteric a. and v. Abdomen 168 Duodenum The small intestine consists of the duodenum, jejunum, and ileum. The duodenum is primarily retroperitoneal and is divided into four parts: superior, descending, horizontal, and ascending. Fig. 15.5 Duodenum: Location Anterior view. Fig. 15.6 Parts of the duodenum Anterior view. Fig. 15.7 Duodenum Anterior view with the anterior wall opened. Duodenal papilla Circular folds Inferior duodenal recess Superior duodenal recess Abdominal aorta Left colic a. and v. Right colic a. Inferior vena cava Hepatic vv. Diaphragm Parietal peritoneum Hepatic surface of diaphragm Hepatoduodenal lig. (with portal triad) Right suprarenal gland Right kidney Duodenum, superior part Right colic flexure Transverse colon Ascending colon Duodenum, descending part Root of mesentery Duodenum, horizontal part Superior mesenteric a. and v. Duodenum, ascending part Jejunum Pancreas Descending colon Left kidney Left colic flexure Splenic a. Left suprarenal gland Spleen Phrenicosplenic lig. Esophagus Left gastric a. Common hepatic a. Stomach Duodenal diverticula 15 Internal Organs 169 Fig. 15.8 Duodenum in situ Anterior view. Removed: Stomach, liver, small intestine, and large portions of the transverse colon. Thinned: Retroperitoneal fat and connective tissue. Clinical box 15.2 Two important ducts end at a common exit site in the descending portion of the duodenum: the common bile duct and the pancreatic duct (see Fig. 15.27). These ducts may be examined by X-ray through endoscopic retrograde cholangiopancreatography (ERCP), in which dye is injected endoscopically into the duodenal papilla. Duodenal diverticula (generally harmless outpouchings) may complicate the procedure. Endoscopy of the papillary region A Endoscopic appearance. B Radiograph. Rectum Duodeno-jejunal flexure RUQ LUQ RLQ LLQ Jejunum and ileum Circular folds Lymphatic follicles (Peyer’s patches) Transverse mesocolon (with middle colic a. and v.) Round lig. of liver Epiploic appendices Tenia coli Greater omentum (reflected superiorly) Transverse colon Ascending colon Tenia coli Cecum Ileum Rectus abdominis Arcuate line Median umbilical fold (with obliterated urachus) Lateral umbilical fold (with inferior epigastric a. and v.) Medial umbilical fold (with obliterated umbilical a.) Transversus abdominis, internal and external oblique Jejunum Abdomen 170 Jejunum & Ileum Fig. 15.9 Jejunum and ileum: Location Anterior view. The intraperitoneal jejunum and ileum are enclosed by the mesentery proper. Fig. 15.10 Mucosal appearance of the jejunum and ileum Macroscopic views of the longitudinally opened small intestine. A Jejunum. B Ileum. Fig. 15.11 Jejunum and ileum in situ Anterior view. Reflected: Transverse colon. Liver, right lobe Gallbladder Omental foramen Lesser omentum, hepatoduodenal lig. Duodenum, superior part Stomach, pyloric part Right colic flexure Greater omentum Ascending colon Terminal ileum Cecum Tenia coli Duodenum, horizontal part Rectum Mesentery (cut edge) Descending colon Transverse colon Transverse colon Left colic flexure Duodenojejunal flexure Transverse mesocolon, root Pancreas Gastrosplenic lig. Spleen Esophagus Liver, left lobe Hepatogastric lig. Round lig. of liver Sigmoid mesocolon (cut edge) 15 Internal Organs 171 Fig. 15.12 Mesentery of the small intestine Anterior view. Removed: Stomach, jejunum, and ileum. Reflected: Liver. Clinical box 15.3 Crohn’s disease, a chronic inflammation of the digestive tract, occurs most often in the terminal ileum (30% of cases). Patients are generally young and suffer from abdominal pain, nausea, elevated body temperature, and diarrhea. Initially, these symptoms can be confused with appendicitis. Complications of the chronic inflammation in Crohn’s disease often lead to fistula formation (seen here in figure B as an abnormal passage between two gastrointestinal regions). Crohn’s disease A MRI showing thickened wall of terminal ileum. (arrow). B Double-contrast radiograph, show-ing ileorectal fistula (arrow). Tenia coli Greater omentum (cut) Tenia coli Sigmoid colon Sigmoid mesocolon Haustra Ileum, terminal part Mesentery (with anterior cecal a.) Tenia coli Ascending colon Cecum Mesoappendix (with appendicular a.) Rectum (with peritoneal reflection) Epiploic appendices Descending colon Left colic (splenic) flexure Transverse colon Right colic (hepatic) flexure Transverse mesocolon Iliocecal labrum, superior and inferior lips Vermiform appendix (with orifice) Tenia coli Semilunar folds Haustra Ileocecal orifice Right colic flexure Ascending colon Rectum Sigmoid colon Descending colon Left colic flexure Transverse colon Cecum RUQ LUQ RLQ LLQ Superior lip Ileocecal orifice Inferior lip Muscular coat Ascending colon Ileal papilla, ileocolic labrum Inner circular layer Outer longitudinal layer Abdomen 172 Cecum, Appendix & Colon Fig. 15.13 Large intestine: Location Anterior view. Fig. 15.14 Ileocecal orifice Anterior view of longitudinal coronal section. Fig. 15.15 Large intestine Anterior view. The ascending and descending colon are normally secondarily retro-peritoneal, but are sometimes suspended by a short mesentery from the posterior abdominal wall. Note: In the clinical setting, the left colic flexure is often referred to as the splenic flexure and the right colic flexure, as the hepatic flexure. Transverse colon Right colic (hepatic) flexure Sigmoid colon Sigmoid mesocolon Descending colon Left colic (splenic) flexure Duodenojejunal flexure Greater omentum Terminal ileum Cecum Rectum Mesentery (cut) Transverse mesocolon Ascending colon Rectus abdominis Clinical box 15.4 Colitis Ulcerative colitis is a chronic inflammation of the large intestine, often starting in the rectum. Typical symptoms include diarrhea (sometimes with blood), pain, weight loss, and inflammation of other organs. Patients are also at higher risk for colorectal carcinomas. B Early-phase colitis. Double-contrast radiograph, anterior view. A Colonoscopy of ulcerative colitis. Clinical box 15.5 Colon carcinoma Malignant tumors of the colon and rectum are among the most frequent solid tumors. More than 90% occur in patients over the age of 50. In early stages, the tumor may be asymptomatic; later symptoms include loss of appetite, changes in bowel movements, and weight loss. Blood in the stools is particularly incriminating, necessitating a thorough examination. Hemorrhoids are not a sufficient explanation for blood in stools unless all other tests (including a colonoscopy) are negative. Colonoscopy of colon carcinoma. The tumor (black arrows) partially blocks the lumen of the colon. 15 Internal Organs 173 Fig. 15.16 Large intestine in situ Anterior view. Reflected: Transverse colon and greater omentum. Removed: Intraperitoneal small intestine. Liver Duodenum Ascending colon Small intestine Descending colon Transverse colon Stomach Spleen RUQ LUQ Pancreas Spleen Left kidney and supra-renal gland Ascending colon Right kidney and suprarenal gland Liver RUQ LUQ Omental bursa, vestibule Liver Stomach Omental bursa, splenic recess Spleen Left kidney Pancreas Abdominal aorta Inferior vena cava Lesser omentum Mediastinal pleura Liver, left lobe Stomach Transverse colon Greater omentum Ascending colon Gallbladder Round lig. of liver Liver, right lobe Falciform lig. Diaphragm Transversus abdominis, internal and external oblique Esophagus Aorta Diaphragmatic pleura Fibrous pericardium Duodenal impression Colic impression Renal impression Suprarenal impression Gastric impression Fig. 15.18 Relations of the liver Visceral (inferior) surface, inferior view. Abdomen 174 A Anterior view. B Posterior view. C Transverse section, inferior view. Fig. 15.19 Liver in situ Anterior view. The liver is intraperitoneal except for its “bare area” (see Fig. 15.21); its mesenteries include the falciform, coronary, and triangular ligaments (see Fig. 15.22). Liver: Overview Fig. 15.17 Liver: Location Fig. 15.20 Liver in situ: Inferior surface The liver is retracted to show the gallbladder on its inferior surface. Bare area Coronary lig. Left triangular lig. Right triangular lig. Right suprarenal gland Duodenum Stomach Abdominal aorta Inferior vena cava Parietal peritoneum Hepato-duodenal lig. Pancreas Spleen Hepatic surface of diaphragm (no parietal peritoneum) Right kidney Fig. 15.21 Attachment of liver to diaphragm A Diaphragmatic surface of the liver, posterior view. B Hepatic surface of the diaphragm, anterior view. Hepato-duodenal lig. Right kidney Falciform lig. of liver Liver, right lobe Gallbladder Lesser omentum Right colic flexure Duodenum Stomach, body Liver, left lobe Hepato-gastric lig. Clinical box 15.6 A T2W sequence. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) B T1W sequence. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) C Fat-suppressed T1W sequence. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) Hepatic cirrhosis is a condition leading to irreversible fibrosis of the liver parenchyma. Alcohol abuse is the leading cause (70% of cases) followed by hepatitis B. Portal hypertension with the development of collateral vessels is a common result arising in approximately 30% of cases. Hepatic cirrhosis Changes associated with advanced hepatic cirrhosis. All three sequences show multiple regenerating nodules in the liver, creating a nodular surface contour. Only the caudate lobe (B, arrow) is less affected by the changes and still shows a relatively normal signal. 15 Internal Organs 175 Inferior border Gallbladder, fundus Round lig. of liver (ligamentum teres hepatis) Falciform lig. Left lobe, diaphragmatic surface Fibrous appendix of liver Left triangular lig. Bare area (diaphragmatic surface of liver) Coronary lig. Right lobe, diaphragmatic surface Right triangular lig. Round lig. of liver (ligamentum teres hepatis) Quadrate lobe Right hepatic a. Bile duct Cystic duct Cystic a. Gallbladder Portal v. Coronary lig. Bare area Caudate process Inferior vena cava Caudate lobe Fibrous appendix of liver Lig. of vena cava Right hepatic duct Proper hepatic a. Left hepatic a. Left hepatic duct Left lobe, visceral surface Right lobe, visceral surface Abdomen 176 Liver: Lobes & Segments Fig. 15.22 Surfaces of the liver The liver is divided into four lobes by its ligaments: right, left, caudate, and quadrate. The falciform ligament, a double layer of parietal peritoneum that reflects off the anterior abdominal wall and extends to the liver, spreading out over its surface as visceral peritoneum, A Anterior view. B Inferior view. divides the liver into right and left anatomical lobes. The round ligament of the liver is found in the free edge of the falciform ligament and is the obliterated umbilical vein, which once extended from the umbilicus to the liver. Branch of cystic a. Quadrate lobe Caudate process Caudate lobe Right lobe, visceral surface Left lobe, visceral surface Ligamentum venosum Portal v. Left hepatic a. Round lig. of liver Proper hepatic a. Right hepatic duct Gallbladder Right triangular lig. Bare area Right hepatic v. Left and intermediate hepatic vv. Left triangular lig. Fibrous appendix of liver Bile duct Right hepatic a. Cystic duct Coronary lig. Groove for inferior vena cava 15 Internal Organs 177 C Posterior view. Branches of hepatic vv. Branches of portal v. Branches of proper hepatic a. Branches of hepatic duct IV II III V VI VII VIII Round lig. of liver VII VI V IV III II Fibrous appendix Gallbladder Round lig. of liver Inferior vena cava I Fig. 15.23 Segmentation of the liver The liver is divided into functional divisions, which are further divided into segments (see Table 15.1). Each segment is served by tertiary branches of the hepatic artery, the portal vein, and the common hepatic duct, which together make up the portal triad. A Diaphragmatic surface, ante-rior view. B Visceral sur-face, inferior view. Table 15.1 Hepatic segments Part Division Segment Left part Posterior part I Caudate lobe Left lateral division II Left posterolateral III Left anterolateral Left medial division IV Left medial Right part Right medial division V Right anteromedial VI Right anterolateral Right lateral division VII Right posterolateral VIII Right posteromedial Common hepatic duct Gallbladder Bile duct Left hepatic duct Right hepatic duct Cystic duct Liver, right lobe Liver, left lobe Right duct of caudate lobe Left duct of caudate lobe Infundibulum Neck Right hepatic duct Minor duodenal papilla Duodenum, descending part Major duodenal papilla Pancreatic duct Accessory pancreatic duct Bile duct Duodenum, superior part Left hepatic duct Cystic duct Common hepatic duct Duodenum, horizontal part Fundus Body Gall-bladder Cystic duct Left hepatic duct Right hepatic duct Common hepatic duct Bile duct Gallbladder RUQ Bare area Portal v. Common hepatic duct Liver, caudate lobe Liver, left lobe Left hepatic duct Liver, quadrate lobe Bile duct Cystic duct Gallbladder Right hepatic duct Inferior vena cava Sphincter of hepatopancreatic ampulla Sphincter of pancreatic duct Sphincter of bile duct Duodenum wall Hepato-pancreatic ampulla Sphincter of hepato-pancreatic ampulla Pancreatic duct Longitudinal slips of duodenal muscle on bile duct Bile duct Abdomen 178 Gallbladder & Bile Ducts A Anterior view. B Inferior view. A Sphincters of the pancreatic and bile ducts. B Sphincter system in the duodenal wall. Fig. 15.24 Gallbladder: Location Fig. 15.25 Hepatic bile ducts: Location Projection onto surface of the liver, anterior view. Fig. 15.26 Biliary sphincter system Fig. 15.27 Extrahepatic bile ducts Anterior view. Opened: Gallbladder and duodenum. Gallstones Pancreatic duct Cystic duct Gallbladder Hepato-pancreatic duct (opening on major duodenal papilla) Proper hepatic a. Bile duct Celiac trunk Common hepatic duct Esophagus Inferior vena cava Hepatic vv. Common hepatic a. Spleen Abdominal aorta Left suprarenal gland Splenic a. Left colic flexure Left kidney Jejunum Superior mesenteric a. and v. Duodenum, ascending part Pancreas Duodenum, descending part Right colic flexure Liver, right lobe Left hepatic duct Right hepatic duct 15 Internal Organs 179 Fig. 15.28 Biliary tract in situ Anterior view. Removed: Stomach, small intestine, transverse colon, and large portions of the liver. The gallbladder is intraperitoneal, covered by visceral peritoneum where it is not attached to the liver. Clinical box 15.7 As bile is stored and concentrated in the gallbladder, certain substances, such as cholesterol, may crystallize, resulting in the formation of gallstones. Migration of gallstones into the bile duct causes severe pain (colic). Gallstones may also block the pancreatic duct in the papillary regions, causing highly acute or even life-threatening pancreatitis. Obstruction of the bile duct Ultrasound appearance of two gallstones. Black arrows mark the echo-free area behind the stones. Right hepatic duct Cystic duct (spiral fold) Body of gallbladder Neck of gallbladder Duodenum Left hepatic duct Common hepatic duct Pancreatic duct Common bile duct Duodenal papilla Fig. 15.29 MR Cholangiopancreatography (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 3rd ed. New York, NY: Thieme; 2007.) Pancreas Spleen RUQ LUQ 10th rib Splenorenal lig. Left kidney Pancreas Abdominal aorta Inferior vena cava Liver Lesser omentum (hepatogastric lig.) Stomach Gastrosplenic lig. Omental bursa, splenic recess Spleen Diaphragmatic surface Anterior extremity Superior border Posterior extremity Inferior border Renal surface Colic surface Hilum Splenic a. Splenic v. Inferior border Anterior extremity Gastric surface Superior border Posterior extremity Pancreas, uncinate process Superior mesenteric a. and v. Duodenum, ascending part Jejunum Pancreas, tail Pancreas, body Pancreatic duct Pancreas, head Accessory pancreatic duct Duodenum, superior part Pancreatic duct Duodenum, horizontal part Duodenum, descending part Pancreas, neck Abdomen 180 Pancreas & Spleen Fig. 15.30 Pancreas and spleen: Location A Anterior view. B Left lateral view. C Transverse section through L1 vertebra, inferior view. Fig. 15.31 Pancreas Anterior view with dissection of the pancreatic duct. Fig. 15.32 Spleen A Costal surface. B Visceral surface. Left colic a. and v. Inferior vena cava Hepatic vv. Diaphragm Parietal peritoneum Hepatic surface of diaphragm Hepatoduodenal lig. Right suprarenal gland Right kidney Mesentery (root) Duodenum, horizontal part Superior mesenteric a. and v. Duodenum, ascending part Jejunum Pancreas, body Descending colon Left kidney Left colic flexure Pancreas, tail Splenic a. and v. Left supra-renal gland Superior border Common hepatic a. Celiac trunk Transverse mesocolon, root Pancreas, uncinate process Duodenum, superior part Pancreas, head Duodenum, descending part Gastric surface Spleen Left gastric a. L1 vertebra Spinal cord (in vertebral canal) Abdominal aorta Splenic a. and v. Left kidney and peri-renal fat Pancreas, tail Spleen Gastrosplenic lig. Stomach Visceral peritoneum (cut edge) Greater omentum Omental bursa (lesser sac) Lesser omentum (hepatoduodenal and hepatogastric ligs.) Proper hepatic a. Bile duct Portal vein Omental (epiploic) foramen Duodenum Transverse colon Right (hepatic) colic flexure Portal triad Parietal peritoneum cut edge Right kidney Inferior vena cava Pancreas, head 15 Internal Organs 181 Fig. 15.33 Pancreas and spleen in situ Anterior view. Removed: Liver, stomach, small intestine, and large intestine. The pancreas is retroperitoneal, whereas the spleen is intraperitoneal. Fig. 15.34 Pancreas and spleen: Transverse section Superior view. Section through L1 vertebra. Ilioinguinal n. Iliohypogastric n. Right kidney Subcostal n. 12th rib Iliac crest Right suprarenal gland Right colic flexure surface Right ureter Duodenal surface Left ureter Descending colic surface Pancreatic surface Left renal hilum Gastric surface Splenic surface Left suprarenal gland Hepatic surface Right renal hilum Diaphragm Right lung Perirenal fat capsule Right suprarenal gland Renal fibrous capsule Right kidney Renal hilum Renal fascia, posterior layer Transverse colon Duodenum Renal fascia, anterior layer Liver Iliac crest Hepatorenal recess Parietal peritoneum Right suprarenal gland Right kidney Urinary bladder Left ureter RUQ LUQ A Anterior view. Abdomen 182 Kidneys & Suprarenal Glands (I) Fig. 15.35 Kidneys and suprarenal glands: Location Fig. 15.36 Relations of the kidneys: areas of organ contact. Anterior view. Fig. 15.37 Right kidney in the renal bed B Posterior view. Right side windowed. A Sagittal section at approximately the level of the renal hilum, viewed from the right side. B Transverse section through the abdomen at approxi-mately the L1/L2 level, viewed from above. Renal fascia, posterior layer Fat capsule Right kidney Liver Renal fascia, anterior layer Parietal peritoneum Inferior vena cava Abdominal aorta L 1 vertebra A Removed: Intraperitoneal organs, along with portions of the ascending and descending colon. B Removed: Peritoneum, spleen and gastro - intestinal organs, along with fat capsule (left side) Retracted: Esophagus Right suprarenal v. Left superior suprarenal a. Left suprarenal v. Left middle and inferior suprarenal aa. Inferior vena cava Right superior suprarenal a. Right suprarenal gland Right kidney Right inferior suprarenal a. Right renal a. and v. Superior mesenteric a. Right ureter Right ovarian/ testicular a. and v. Inferior mesenteric a. Ilioinguinal n. Iliohypogastric n. Left ureter Left ovarian/testicular a. and v. Left renal a. and v. Left suprarenal gland Celiac trunk Abdominal aorta Esophagus Diaphragm Perirenal fat capsule Costodia-phragmatic recess Left suprarenal gland Splenic a. Esophagus Left gastric a. Proper hepatic a. Inferior vena cava Hepatic vv. Portal v. Hepatoduodenal lig. Diaphragm, hepatic surface Right suprarenal gland Hepatic ducts Attachment of ascending colon Duodenum, horizontal part Mesentery, root Abdominal aorta Superior mesenteric a. and v. Duodenum, ascending part Left colic a. and v. Left renal a. and v. Left kidney Pancreas Splenic fossa Attachment of descending colon Transverse mesocolon, root Duodenum, superior part Parietal peritoneum Right kidney 15 Internal Organs 183 Fig. 15.38 Kidneys and suprarenal glands in the retroperitoneum Anterior view. Both the kidneys and suprarenal glands are retro-peritoneal. Inferior pole Renal pelvis Right renal a. and v. Right suprarenal gland Pararenal fat pad Superior pole Right suprarenal v. Anterior surface Inferior suprarenal a. Middle suprarenal a. Renal hilum Lateral border Medial border Superior suprarenal aa. Right ureter Inferior pole Renal pelvis Right renal a. and v. Right suprarenal v. Inferior suprarenal a. Middle suprarenal a. Medial border Superior suprarenal aa. Right ureter Right suprarenal gland Fibrous capsule Posterior surface Renal cortex Renal hilum Renal papilla Arcuate a. and v. Minor calyx Major calyx Renal a. and v. Renal pelvis Fibrous capsule Renal column Interlobar a. and v. Medullary rays Ureter Renal pyramid Renal medulla Renal cortex Renal sinus Medullary rays Segmental aa. and vv. Renal papilla Renal a. and v. Renal pelvis Ureter Major calyx Renal pyramid Minor calyx Renal column Renal cortex Fibrous capsule Abdomen 184 Kidneys & Suprarenal Glands (II) Fig. 15.39 Kidney: Structure Right kidney with suprarenal gland. C Posterior view with upper half partially removed. D Posterior view, midsagittal section. A Anterior view. B Posterior view. Anastomosis between inferior phrenic v. and suprarenal v. Inferior phrenic v. Left suprarenal gland Left suprarenal v. Superior suprarenal aa. Middle suprarenal a. Inferior phrenic a. Esophagus Left gastric a. Abdominal aorta Inferior vena cava Portal v. Proper hepatic a. Common hepatic a. Bile duct Pancreas, neck Duodenum Superior mesenteric a. and v. Splenic a. and v. Left testicular/ ovarian a. and v. Genitofemoral n. Left ureter Left renal a. and v. Ilioinguinal n. Iliohypogastric n. Subcostal n. Left kidney Pancreas, tail Inferior suprarenal a. Diaphragm Transversus abdominis, internal and external oblique Fig. 15.41 Left kidney and suprarenal gland Anterior view. Removed: Perirenal fat capsule. Retracted: Pancreas. Fig. 15.40 Right kidney and suprarenal gland Anterior view. Removed: Perirenal fat capsule. Retracted: Inferior vena cava. Suprarenal v. Superior suprarenal aa. Right suprarenal gland Diaphragm Iliohypogastric n. Right renal a. and v. Ilioinguinal n. Right ureter Right testicular/ ovarian a. and v. Left renal v. Superior mesenteric a. Inferior suprarenal a. Abdominal aorta Celiac trunk Middle suprarenal a. Inferior vena cava Inferior phrenic a. and v. Subcostal n. (12th intercostal n.) Right kidney 15 Internal Organs 185 Left gastric a. Splenic a. Left superior suprarenal a. Gastroduodenal a. Proper hepatic a. Right gastric a. Common hepatic a. Right superior suprarenal a. Left inferior suprarenal a. Celiac trunk Right inferior phrenic a. Left inferior phrenic a. Left middle suprarenal a. Superior mesenteric a. Left renal a. Left testicular/ ovarian a. Left common iliac a. Median sacral a. Right lumbar a. Inferior mesenteric a. Right common illiac a. Celiac trunk (T12) Superior mesenteric a. (L1) Renal aa. (L1/L2) Inferior mesenteric a. (L3) Aortic bifurcation (L4) Left common iliac a. Abdomen 186 Subclavian artery Posterior intercostal aa. Femoral a. External iliac a. Inferior epigastric a. 1st–4th lumbar aa. Superior epigastric a. Musculophrenic a. Anterior intercostal aa. Subcostal a. Internal thoracic a. Aortic arch Thoracic aorta Abdominal aorta 16 Neurovasculature Arteries of the Abdominal Wall & Organs Fig. 16.2 Abdominal aorta and major branches Anterior view. The abdominal aorta extends from T12 to its bifurcation at L4. It gives off visceral branches to the kidneys, suprarenal glands, gonads, and organs of the gas trointestinal system, and parietal branches to the body wall. Table 16.1 Branches of the abdominal aorta The abdominal aorta gives rise to three major unpaired trunks (bold) and the unpaired median sacral artery, as well as six paired branches. Branch from abdominal aorta Branches Inferior phrenic aa. (paired) Superior suprarenal aa. Celiac trunk Left gastric a. Splenic a. Common hepatic a. Proper hepatic a. Right gastric a. Gastroduodenal a. Middle suprarenal aa. (paired) Superior mesenteric a. Renal aa. (paired) Inferior suprarenal aa. Lumbar aa. (1st through 4th, paired) Testicular/ovarian aa. (paired) Inferior mesenteric a. Common iliac aa. (paired) External iliac a. Internal iliac a. Median sacral a. Deep circumflex iliac a. Superficial epigastric a. Superficial circumflex iliac a. Femoral a. External iliac a. Inferior epigastric a. Superior epigastric a. Subclavian a. Internal thoracic a. Superior thoracic a. Axillary a. Lateral thoracic a. Thoraco dorsal a. Fig 16.1 Arteries of the abdominal wall In addition to thoracic and abdominal aortic branches, the abdominal wall is supplied by branches of the subclavian, external iliac and femoral arteries. Numerous potential anastomoses exist between these vessels, which allows the potential for blood to bypass the abdominal aorta. A Anterior view. B Lateral view. Abdominal aorta Celiac trunk Splenic a. Left gastric a. Right gastro omental a. Pancreatic brs. Inferior pancreatico duodenal a. Superior mesenteric a. Proper hepatic a. Common hepatic a. Right gastric a. Gastroduodenal a. Anterior/posterior superior pancreatico duodenal a. Duodenal br. Left gastro omental a. Middle colic a. Left colic a. Superior rectal a. Middle/inferior rectal a. Internal iliac a. supplies: reproductive organs Celiac trunk supplies: Esophagus Stomach Duodenum Liver Spleen Gallbladder Pancreas Superior mesenteric a. supplies: Pancreas Duodenum Jejunum Ileum Cecum Ascending and transverse colon Inferior mesenteric a. supplies: Transverse, descending and sigmoid colon Rectum Anal canal 1 2 3 Pancreatico duodenal aa. Posterior superior pancreatico duodenal a. Anterior superior pancreatico duodenal a. Great pancreatic a. Splenic a. with pancreatic brs. Dorsal pancreatic a. Left gastric a. Celiac trunk Common hepatic a. Gastro duodenal a. Inferior pancreatico duodenal a., anterior br. Superior mesenteric a. Inferior pancreatic a. Abdominal aorta A. of pan creatic tail Superior mesenteric a. Middle colic a. Jejunal and ileal aa. Appendicular a. Ileocolic a. Right colic a. Inferior pancreatico duodenal a. Left colic flexure Inferior mesenteric a. Superior rectal a. Left colic aa. Sigmoid aa. Left colic flexure 16 Neurovasculature 187 Fig. 16.3 Celiac trunk A Celiac trunk distribution. B Arterial supply to the pancreas Fig. 16.4 Superior mesenteric artery Fig. 16.5 Inferior mesenteric artery Fig. 16.6 Abdominal arterial anastomoses Three major anastomoses provide overlap in the arterial supply to abdominal areas to ensure adequate blood flow. Between the: 1–celiac trunk and the superior mesenteric artery via the pan creaticoduodenal arteries. 2–superior and inferior mesenteric arteries via the middle and left colic arteries. 3–inferior mesenteric and the internal iliac arteries via the superior and middle or inferior rectal arteries. Left inferior phrenic a. Left superior suprarenal aa. Left middle suprarenal a. Left renal a. Left 1st lumbar a. Left ovarian a. (testicular a. in males) Inferior mesenteric a. Median sacral a. Left superior gluteal a. Sacral plexus Obturator branch of right inferior epigastric a. Right inferior gluteal a. Right internal pudendal a. Right middle rectal a. Right inferior vesical a. Right obturator a. Right umbilical a. Right external iliac a. Right internal iliac a. Right common iliac a. Superior mesenteric a. Abdominal aorta Uterine a. Femoral a. and v. Left inferior suprarenal a. Celiac trunk Esophagus Inferior vena cava Aortic hiatus (median arcuate lig.) Left ureter Left iliolumbar a. Left lateral sacral a. Left deep circumflex iliac a. Left inferior epigastric a. Right superior vesical a. Abdomen 188 Abdominal Aorta & Renal Arteries Fig. 16.7 Abdominal aorta Anterior view of the female abdomen. Removed: All organs except the left kidney and suprarenal gland. The abdominal aorta is the distal continuation of the thoracic aorta (see p. 80). It enters the abdomen at the T12 level through the aortic hiatus and courses anterior to the vertebral bodies to the left of the midline before bifurcating into the common iliac arteries at L4. Interlobar a. (between the medullary pyramids) Medullary (renal) pyramid Major calyx Arcuate a. (at base of medullary pyramids) Anterior superior segmental a. Interlobular a. Fibrous capsule Left ureter Ureteral brs. Inferior segmental a. Left renal a., main trunk Left renal a., posterior br. Anterior inferior segmental a. Superior segmental a. Inferior suprarenal a. Capsular brs. Left renal a., anterior br. Branch of posterior segmental a. Renal pelvis Minor calyx Clinical box 16.1 Anterior view of the right kidney. As the kidneys ascend from their site of origin in the pelvis to the lumbar region, new renal arteries are formed as older ones regress. Commonly some fail to regress, resulting in multiple arteries to one or both kidneys. Variants of the renal artery B Aberrant renal arteries do not enter the kidney at the hilum. Aberrant right renal a. Abdominal aorta Inferior vena cava Accessory renal a. in front of inferior vena cava Accessory renal a. behind inferior vena cava Inferior vena cava Abdominal aorta A Accessory renal arteries pass from the aorta to the renal hilum. Note: one of the accessory arteries is passing anterior to the inferior vena cava. 16 Neurovasculature 189 Fig. 16.8 Renal arteries Left kidney, anterior view. The renal arteries arise at approximately L1/L2. Each artery divides into an anterior and a posterior branch. The anterior branch further divides into four seg mental arteries (circled). Clinical box 16.2 The kidney is an important blood pressure sensor and regulator. Stenosis (narrowing) of the renal artery reduces blood flow through the kidney and stimulates increased production of renin, an enzyme that cleaves angiotensinogen to form angiotensin I. Subsequent cleavage yields angiotensin II, which induces vasoconstriction and an increase in blood pressure. Renal hypertension should be excluded (or confirmed) when diagnosing high blood pressure. Renal hypertension Stenosis of the right renal artery (arrow), visible via arteriography. Celiac trunk Common hepatic a. Right hepatic a. Cystic a. Gallbladder Proper hepatic a. Right gastric a. Posterior superior pancreatico duodenal a. Gastro duodenal a. Duodenum Anterior superior pancreaticoduodenal a. Right gastro omental a. Pancreas Left gastro omental a. Splenic a. Spleen Left gastric a. Abdominal aorta Left hepatic a. Portal v. Inferior vena cava Bile duct Liver Stomach Greater omentum Lesser omentum Gastro duodenal a. Abdomen 190 Celiac Trunk Fig. 16.9 Celiac trunk: Stomach, liver, and gallbladder Anterior view. Opened: Lesser omentum. Incised: Greater omentum. The celiac trunk arises from the abdominal aorta at about the level of T12. It supplies the structures of the foregut, the proximal part of the alimentary canal and the spleen. The foregut consists of the esophagus (distal 1.25 cm), stomach, duodenum (proximal half), liver, gallbladder, and pancreas (superior portion). Inferior vena cava Common hepatic a. Left gastric a. Celiac trunk Splenic a. Short gastric aa. Left gastro omental a. A. of pancre atic tail Great pancreatic a. Inferior pancreatic a. Dorsal pancreatic a. Anastomosis between superior mesenteric a. and inferior pancreatic a. Superior mesenteric a. and v. Inferior pancreatico duodenal a. Anterior br. Posterior br. Anterior superior pancreatico duodenal a. Posterior superior pancreatico duodenal a. Supra duodenal a. (variant) Gastro duodenal a. Right gastric a. Proper hepatic a. Cystic a. Portal v. Transverse mesocolon, root Posterior gastric a. Splenic a., pancreatic brs. Duodenal br. Inferior pancreatico duodenal a. Splenic v. Right gastro omental a. 16 Neurovasculature 191 Fig. 16.10 Celiac trunk: Pancreas, duodenum, and spleen Anterior view. Removed: Stomach (body) and lesser omentum. Middle colic a. (cut) Proper hepatic a. Right gastric a. Gastroduodenal a. Right gastro omental a. Anterior superior pancreatico duodenal a. Inferior pancreatico duodenal a., anterior and posterior brs. Right colic a. Ileocolic a. Ileocolic a., colic br. Ileocolic a., ileal br. Anterior cecal a. Posterior cecal a. Ileal aa. Jejunal aa. Left renal a. Superior mesenteric a. Splenic a. Left gastric a. Common hepatic a. Inferior vena cava Portal v. Left renal v. Vasa recta Marginal a. Clinical box 16.3 A decrease in blood flow to the intestine (ischemia) can result from occlusion of the superior mesenteric artery (SMA) by a thrombus or embolus (acute) or may be secondary to severe atherosclerosis (chronic). In the acute condition, the embolus can obstruct the SMA at its origin or, if small enough, may travel further to obstruct a more peripheral branch. Acute ischemia results in necrosis of the affected part of the intestine. Chronic ischemia is less threatening since obstruction of the vessels occurs gradually, allowing the formation of collateral vessels that will supply the affected intestine. Because of the extensive anastomoses between intestinal arteries, chronic vascular ischemia is rare. Symptoms occur only if two of the three major vessels (celiac trunk or superior or inferior mesenteric arteries) are compromised. Mesenteric ischemia Abdomen 192 Superior & Inferior Mesenteric Arteries Fig. 16.11 Superior mesenteric artery Anterior view. Partially removed: Stomach, duodenum, and peritoneum. Reflected: Liver and gallbladder. Note: The middle colic artery has been truncated (see Fig. 16.12). The superior mesenteric artery arises from the aorta opposite L1. It supplies the structures of the midgut: the duodenum (distal half), jejunum and ileum, cecum and appendix, ascending colon, right colic flexure, and the proximal two thirds of the transverse colon. Superior mesenteric a. (cut) Inferior mesenteric a. Aortic bifurcation Left colic a. Sigmoid aa. Superior rectal a. Posterior cecal a. Anterior cecal a. Right common iliac a. Ileocolic a., ileal br. Ileocolic a., colic br. Ileocolic a. (cut) Right colic a. Middle colic a. Abdominal aorta Inferior vena cava Ascending colon Duodenum Sigmoid colon Transverse colon Greater omentum Descending colon Left colic (splenic) flexure Marginal a. Marginal a. Clinical box 16.4 Anastomoses between branches of the superior mesenteric and inferior mesenteric arteries can compensate for abnormally low blood flow in either of the arteries. Two of these anastomoses, although variable, are of significant value: Riolan’s arcade (arc of Riolan) – a connection between the middle colic artery and the left colic artery that arises close to their origins from the superior and inferior mesenteric arteries, respectively. Marginal artery (of Drummond) – a connection between all arteries of the colon that runs along the periphery of the mesentery close to the intestinal tube. Anatomoses between arteries of the large intestine Middle colic a. Superior mesenteric a. Inferior mesenteric a. Left colic a. Arc of Riolan Marginal a. (of Drummond) Left colic flexure 16 Neurovasculature 193 Fig. 16.12 Inferior mesenteric artery Anterior view. Removed: Jejunum and ileum. Reflected: Transverse colon. The inferior mesenteric artery arises from the aorta opposite L3. It supplies structures of the hindgut: the transverse colon (distal third), left colic flexure, descending and sigmoid colons, rectum, and anal canal (upper part). Abdominal aorta Inferior vena cava L4 vertebra Common iliac vv. A Anterior view. Great saphe nous v. External pu dendal vv. Femoral v. Superficial circumflex iliac v. Superficial epigastric v. Periumbilical vv. Thoracoepigastric v. Areolar venous plexus Cephalic v. Axillary v. Femoral v. External iliac v. Inferior epigastric v. Subcostal v. Lumbar vv. Inferior vena cava Musculophrenic v. Superior epigastric v. Anterior intercostal vv. Internal thoracic v. Superior vena cava Subclavian v. Azygos v. Posterior intercostal vv. B Lateral view. Fig 16.13 Veins of the abdominal wall The abdominal wall is drained by veins that accompany the arteries and are tributaries of the azygos system and inferior vena cava. Additionally, a large thoracoepigastric vein connects the femoral and axillary veins. Abdomen 194 Veins of the Abdominal Wall & Organs Fig. 16.14 Inferior vena cava Anterior view. The inferior vena cava arises at L5 with the convergence of the common iliac veins. It ascends along the right side of the vertebral column, passes through the caval opening in the diaphragm at T8 and terminates in the thorax in the right atrium of the heart. Azygos v. Hemiazygos v. Inferior vena cava ② ⑨ 1R 3R 4R 5R 7R 6R 8R 1L 3L 4L 5L 6L 7L 8L Table 16.2 Tributaries of the inferior vena cava Inferior phrenic vv. (paired) ② Hepatic vv. (3) Suprarenal vv. (the right vein is a direct tributary) Renal vv. (paired) Testicular/ovarian vv. (the right vein is a direct tributary) Ascending lumbar vv. (paired), not direct tributaries Lumbar vv. Common iliac vv. (paired) ⑨ Median sacral v. 3R 4R 5R 6R 7R 8R 3L 4L 5L 6L 7L 8L 1R 1L Portal v. Splenic v. Superior mesen teric v. L4 Inferior mesen teric v. Superior mesenteric v. Posterior superior pancreatico duodenal v. Inferior pancreatico duodenal v. Right gastro omental v. Ileal vv. Ileocolic v. Right colic v. Middle colic v. Inferior mesenteric v. Left colic v. Sigmoid vv. Superior rectal v. Splenic v. Left gastro omental v. Pancreatic vv. Short gastric vv. Cystic v. Left gastric v. (with esophageal vv.) Right gastric v. Jejunal vv. Appendicular v. Portal v. From hepatic vv. within the liver to IVC Portal v. Left gastric v. Right gastric v. Esophageal vv. Azygos/ hemi azygos v. Superior vena cava Subclavian v. Inferior vena cava Superior epigastric v. Periumbilical vv. Inferior epigastric v. Common iliac v. Ascending lumbar v. Colic vv. Superior rectal v. Internal thoracic v. Colic vv. Middle/inferior rectal v. Paraumbilical vv. Superior mesenteric v. Inferior mesenteric v. D F S S D A 16 Neurovasculature 195 Fig. 16.15 Portal vein The portal vein (see p. 198) drains venous blood from the abdominopelvic organs sup plied by the celiac trunk and superior and inferior mesenteric arteries. A Location, anterior view. B Portal vein distribution. C Portocaval anastomotic collateral pathways between the portal and systemic systems. When the portal system is compromised, the portal vein can divert blood away from the liver back to its supplying veins, which return this nutrient-rich blood to the heart via the venae cavae. The red arrows indicate the flow reversal in the (1) esophageal veins, (2) paraumbilical veins, (3) the colic veins, and (4) the middle and infe rior rectal veins. Clinical box 16.5 Cancer metastases Tumors in the region drained by the superior rectal vein may spread through the portal venous system to the capillary bed of the liver (hepatic metastasis). Tumors drained by the middle or inferior rectal veins may metastasize to the capillary bed of the lung (pulmonary metastasis) via the inferior vena cava and right heart. Femoral a. and v. Right suprarenal v. Right renal v. Right ovarian a. and v. Right common iliac v. Right internal iliac v. Right obturator v. Right inferior epigastric a. and v. Right inferior vesical v. Right uterine v. Right middle rectal v. Right internal pudendal v. Right inferior gluteal v. Vesical venous plexus Median sacral a. and v. Uterine venous plexus Left superior gluteal v. Left lateral sacral v. Deep circumflex iliac a. and v. Left common iliac a. and v. Inferior mesenteric a. Left 3rd lumbar v. Abdominal aorta Ureter Left ovarian a. and v. Left ascending lumbar v. Left renal a. and v. Superior mesenteric a. Left suprarenal v. Celiac trunk Left inferior phrenic v. Inferior vena cava Hepatic vv. Rectum (and rectal venous plexus) Urethra Vagina Esophagus Right external iliac v. Abdomen 196 Inferior Vena Cava & Renal Veins Fig. 16.16 Inferior vena cava Anterior view of the female abdomen. Removed: All organs except the left kidney and suprarenal gland. The inferior vena cava courses along the right side of the vertebral bodies from its origin at L5 to the caval opening in the diaphragm at T8. Unlike the branches of the aorta, vis ceral and parietal drainages to the inferior vena cava are asymmetrical (note drainages of the suprarenal glands, gonads and azgygos veins). It communicates with the azygos system through lumbar veins and receives blood from the portal venous system via the hepatic veins. Right suprarenal v. (typically opens directly into inferior vena cava) Right middle suprarenal a. Right superior suprarenal a. Right renal a. and v. Right testicular/ ovarian a. and v. Abdominal aorta Right ureter Ureteral branches (from testicular/ ovarian a. or common iliac a.) Inferior mesenteric a. Left testicular/ ovarian a. and v. Left renal a. and v. Left inferior suprarenal a. Left middle suprarenal a. Left suprarenal v. (typically opens into left renal v.) Left inferior phrenic a. Left superior suprarenal aa. Left inferior phrenic v. (anastomosis with left suprarenal v.) Superior mesenteric a. Celiac trunk Inferior vena cava Right inferior phrenic a. and v. Right inferior suprarenal a. Clinical box 16.6 On the right side, the suprarenal and testicular/ovarian veins drain directly into the inferior vena cava. The corresponding veins on the left side, however, drain into the left renal vein. (This is a remnant from early development when there were both right and left sided venae cave.) It is believed that this asymmetrical drainage pattern is the cause of the varicose dilations of the veins in the spermatic cord (varicoceles) that occur more commonly on the left side. Tributaries of the left renal vein Anastomosis Right inferior phrenic v. Right suprarenal v. Right renal v. Right testicular/ ovarian v. Left testicular/ ovarian v. Left renal v. Left suprarenal v. Left inferior phrenic v. Inferior vena cava 16 Neurovasculature 197 Fig. 16.17 Renal veins Anterior view. See p. 189 for the renal arteries in isolation. Removed: All organs except kidneys and suprarenal glands. Inferior vena cava Esophageal vv. Celiac trunk Portal v. Posterior superior pancreatico duodenal a. Gastro duodenal a. Anterior superior pancreatico duodenal a. Middle colic v. Superior mesenteric a. and v. Right gastro omental a. and v. Left gastro omental a. and v. Splenic a. and v. Short gastric vv. Left gastric a. and v. Hepatic vv. Pancreatico duodenal v. Greater omentum Spleen Splenic a. Inferior pancreatico duodenal a. Right kidney and supra renal gland Common hepatic a. Left and right hepatic aa. Right gastric a. and v. Proper hepatic a. Abdomen 198 Portal Vein Fig. 16.18 Portal vein: Stomach and duodenum Anterior view. Removed: Liver, lesser omentum, and peritoneum. Opened: Greater omentum. The portal vein is typically formed by the union of the superior mesenteric and the splenic veins posterior to the neck of the pancreas. Hepatic vv. Gastro duodenal a. Left and right hepatic aa. Celiac trunk Inferior vena cava Left ureter Portal v. Right gastric a. Pancreatico duodenal v. Middle colic v. Inferior pancreatic a. Left gastric a. and v. Inferior mesenteric v. Left ovarian/ testicular a. and v. Left renal a. and v. Left supra renal v. Left gastro omental a. and v. Splenic a. and v. Splenic v. Short gastric vv. Superior mesenteric a. and v. Right gastric v. Posterior and anterior superior pancreatico duodenal aa. Inferior pancreatico duodenal a., anterior and posterior brs. Right gastro omental a. and v. Left ascending lumbar v. Proper hepatic a. Clinical box 16.7 Upper esophageal veins drain into the azygos system, but the lower esophagus drains into the portal system via the left gastric veins. As a result of this portocaval anastomosis, venous varices (dilations, arrows) of the esophageal wall may develop in patients with portal hypertension. Severe acute hemorrhage is the greatest risk associated with this condition. Esophageal varices 16 Neurovasculature 199 Fig. 16.19 Portal vein: Pancreas and spleen Anterior view. Partially removed: Liver, stomach, pancreas, and peritoneum. Portal v. Right gastric a. and v. Pancreatico duodenal a. and vv. Right colic a. and v. Ileocolic a. and v. Cecal vv. Jejunal aa. and vv. Right gastro omental a. and v. Inferior vena cava Middle colic a. and v. Inferior mesenteric v. Splenic a. and v. Left gastric a. and v. Left renal a. Proper hepatic a. Gastro duodenal a. Inferior vena cava Ileocolic a., colic br. Posterior cecal a., appendicular v. Anterior cecal a. Ileocolic a., ileal br. Ileal aa. and vv. Superior mesenteric a. and v. Abdomen 200 Superior & Inferior Mesenteric Veins Fig. 16.20 Superior mesenteric vein Anterior view. Partially removed: Stomach, duodenum, and peritoneum. Removed: Pancreas, greater omentum, and transverse colon. Reflected: Liver and gallbladder. Displaced: Small intestine. The superior mesen teric vein receives tributaries from the entire small intestine as well as the cecum, appendix, ascending colon, and two thirds of the transverse colon. It normally lies to the right of the superior mesenteric artery then joins with the splenic vein posterior to the neck of the pancreas to form the portal vein. Inferior vena cava Portal v. Right colic a. and v. Ileocolic a. and v. Cecal vv. Sigmoid aa. and vv. Left colic a. and v. Middle colic a. and v. Superior mesenteric a. and v. Inferior mesenteric v. Splenic a. and v. Right gastric a. and v. Left gastric a. and v. Inferior mesenteric a. and v. Left renal a. Proper hepatic a. Gastro duodenal a. Right gastro omental a. and v. Posterior cecal a. Anterior cecal a. Superior rectal a. and v. Left common iliac a. and v. Jejunal/ileal aa. and vv. (cut) 16 Neurovasculature 201 Fig. 16.21 Inferior mesenteric vein Anterior view. Partially removed: Stomach, duodenum, and peritoneum. Removed: Pancreas, greater omentum, transverse colon, and small intestine. Reflected: Liver and gallbladder. The inferior mesenteric vein drains a smaller territory than the superior mesenteric vein. It receives tributaries from the distal transverse colon, descending and sigmoid colons and upper rectum. It ascends in the retroperitoneum, separate from the artery, and generally joins with the splenic vein posterior to the stomach and pancreas. Note that the ascending and descending colons may also be drained by lumbar veins in the retroperitoneum, which empty into the inferior vena cava, constituting a portocaval collateral pathway. ⑧ ⑧ ② ④ ③ ⑤ ⑥ Left lumbar trunk Cisterna chyli Thoracic duct Intestinal trunk Right lumbar trunk Right common iliac l.n. Left common iliac l.n. Abdomen 202 Lymphatics of the Abdominal Wall & Organs ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ Cisterna chyli Lumbar l.n. Table 16.3 Parietal lymph nodes of the abdomen ① Inferior phrenic l.n. Lumbar l.n. Preaortic l.n. ② Celiac l.n. ③ Superior mesenteric l.n. ④ Inferior mesenteric l.n. ⑤ Left lateral aortic l.n. ⑥ Right lateral aortic (caval) l.n. ⑦ Retroaortic l.n. ⑧ Common iliac l.n. Fig. 16.23 Lymphatic drainage of the internal organs See Table 16.3 for numbering. Lymph drainage from the abdomen, pelvis, and lower limb ultimately passes through the lumbar lymph nodes (clinically, the aortic nodes), which consist of the right lateral aortic (caval) and left lateral aortic nodes, the preaortic nodes, and the retroaortic nodes. Efferent lymph vessels from the lateral aortic, retroaortic, and inferior mesenteric nodes form the lumbar trunks. Those from the remaining preaortic nodes form the intestinal trunks. The lumbar and intestinal trunks terminate in the cisterna chyli. Fig. 16.22 Lymphatic drainage of the anterior trunk wall Lymph from the skin of the trunk wall is collected mainly by the axil lary and superficial inguinal lymph nodes (arrows indicate direction of lymph flow). A curved line that lies between the umbilicus and costal arch defines the “watershed” zone between the two drainages. Lymph from the right upper quadrant (green) is drained by the right lymphatic duct. Lymph from the remainder of the body (blue) is drained by the thoracic duct. Axillary l.n. “Watershed” Superficial inguinal l.n. Parasternal l.n. Cervical l.n. 16 Neurovasculature 203 Fig. 16.24 Principal lymphatic pathways draining the digestive organs and spleen Lymph from the spleen and most digestive organs drains directly from regional lymph nodes or through intervening collecting nodes to the intestinal trunks, except for the descending and sigmoid colon and the upper part of the rectum, which are drained by the left lumbar trunk. The three large collecting nodes are: • Celiac lymph nodes collect lymph from the stomach, duodenum, pancreas, spleen, and liver. Topographically and at dissection they are often indistinguishable from the regional lymph nodes of the nearby upper abdominal organs. • Superior mesenteric lymph nodes collect lymph from the jejunum, ileum, ascending and transverse colon. • Inferior mesenteric lymph nodes collect lymph from the descending and sigmoid colon and rectum. These nodes drain principally through the intestinal trunks to the cisterna chyli, but there is an accessory drainage route by way of the left lumbar lymph nodes. Lymph from the pelvis also drains up into the inferior mesenteric and lateral aortic lymph nodes. A complete drainage pathway for lymph from the pelvis can be found on p. 276. Lateral aortic l.n. Preaortic l.n. Juxtaintestinal l.n. Prececal l.n. Retrocecal l.n. Ileocolic l.n. Appendicular l.n. Right colic l.n. Middle colic l.n. Sigmoid l.n. Superior rectal l.n. Left colic l.n. Supra, sub, and retropyloric l.n. Cystic l.n. Foraminal l.n. Splenic l.n. Hepatic l.n. Right/left gastric l.n. Pyloric l.n. Right left gastro-omental l.n. Superior/inferior pancreatic l.n. Superior/inferior pancreaticoduodenal l.n. Inferior mesenteric l.n. Mesocolic l.n. Mesocolic l.n. Celiac l.n. Superior mesenteric l.n. Left lumbar l.n. Intermediate lacunar l.n. Inferior mesenteric l.n. Right lateral caval l.n. Sacral l.n. Deep inguinal l.n. Superficial inguinal l.n. (horizontal and vertical groups) External iliac l.n. Internal iliac l.n. Common iliac l.n. Left lateral aortic l.n. Retroaortic l.n. Left lumbar trunk Intestinal trunk Celiac l.n. Inferior phrenic l.n. Right lumbar trunk Retrocaval l.n. Intermediate lumbar l.n. Superior mesenteric l.n. Cisterna chyli Diaphragm Inferior vena cava Esophagus Abdominal aorta Common iliac a. Inguinal lig. Abdomen 204 Lymph Nodes of the Posterior Abdominal Wall Lymph nodes in the abdomen and pelvis may be classified as either pa rietal or visceral. The majority of the parietal lymph nodes are located on the posterior abdominal wall. Fig. 16.25 Parietal lymph nodes in the abdomen and pelvis Anterior view. Removed: All visceral structures except vessels. Retrocaval l.n. Right lateral caval l.n. Intermediate lumbar l.n. Common iliac l.n. Promontory l.n. Preaortic l.n. Left lateral aortic l.n. Inferior phrenic l.n. Cisterna chyli Thoracic duct Intermediate lumbar l.n. Subaortic l.n. Promontory l.n. Lateral, medial, and intermediate common iliac l.n. Common iliac l.n. Left lumbar trunk Right lumbar trunk Lateral caval l.n. Precaval l.n. Retrocaval l.n. Right lumbar l.n. Lateral aortic l.n. Preaortic l.n. Retroaortic l.n. Left lumbar l.n. Obturator l.n. Lateral, medial, and intermediate external iliac l.n. Interiliac l.n. External iliac l.n. Sacral l.n. Superior and inferior gluteal l.n. Internal iliac l.n. Lacunar l.n. (lateral, medial, and intermediate) Deep inguinal l.n. Superficial inguinal l.n. Lower limb, uterus, and vagina 16 Neurovasculature 205 Fig. 16.26 Lymph nodes of the kidneys, ureters, and suprarenal glands Anterior view. Fig. 16.27 Lymphatic drainage of the kidneys and gonads (with pelvic organs) Inferior vena cava Portal v. Pancreatic l.n. Left gastro omental l.n. Hepatic l.n. Suprapyloric l.n. Subpyloric l.n. Right gastro omental l.n. Left gastric l.n. Splenic l.n. Cardiac lymphatic ring Celiac l.n. Abdomen 206 Lymph Nodes of the Supracolic Organs Fig. 16.28 Lymph nodes of the stomach and liver Anterior view. Removed: Lesser omentum. Opened: Greater omentum. Arrows show direction of lymphatic drainage. Fig. 16.29 Lymphatic drainage of the liver and biliary tract Anterior view. In the region of the liver, the major lymph-producing organ, the important pathways are: • Liver and intrahepatic bile ducts: Most lymph drains inferiorly through the hepatic nodes to the celiac nodes and then to the intestinal trunk and cisterna chyli, but it may take a more direct route bypassing the celiac nodes. A small amount drains cranially through the inferior phrenic nodes to the lumbar trunk. It also can drain through the diaphragm to the superior phrenic nodes and on to the bronchomediastinal trunk. • Gallbladder: Lymph drains initially to the cystic node, then follows one of the pathways described above. • Common bile duct: Lymph drains through the pyloric nodes (supra-, sub-, and retropyloric) and the foraminal node to the celiac nodes, then to the intestinal trunk. Bile duct m u n e d o u D Liver Pancreas Stomach Cystic l.n. Pyloric l.n. Celiac trunk with celiac l.n. Hepatic l.n. Inferior phrenic l.n. Inferior vena cava Diaphragm Superior phrenic l.n. Gallbladder Pancreaticoduodenal l.n. Suprapyloric l.n. Retropyloric l.n. Subpyloric l.n. Pancreatic l.n. (inferior) Superior mesenteric l.n. Pancreatic l.n. (superior) Splenic l.n. Left gastric l.n. Celiac l.n. Hepatic l.n. Cystic l.n. Cisterna chyli Celiac l.n. Cystic l.n. Foraminal l.n. Hepatic l.n. Intestinal trunks Splenic l.n. Gastric l.n. (right and left) Supra, sub, and retropyloric l.n. Pyloric l.n. Gastroomental l.n. (right and left) Pancreaticoduodenal l.n. (superior and inferior) Pancreatic l.n. (superior and inferior) Superior mesenteric l.n. Thoracic duct 16 Neurovasculature 207 Fig. 16.30 Lymph nodes of the spleen, pancreas, and duodenum Anterior view. Removed: Stomach and colon. Fig. 16.31 Lymphatic drainage of the stomach, liver, spleen, pancreas, and duodenum Ileocolic l.n. Juxtaintestinal l.n. Intermediate mesenteric l.n. Superior mesenteric l.n. Celiac l.n. Thoracic duct with cisterna chyli Abdominal aorta Ileum Duodenum Transverse colon Ascending colon Jejunum Left lumbar l.n. Lateral aortic l.n. Preaortic l.n. Cisterna chyli Thoracic duct Intestinal trunks Left lumbar trunk Sigmoid l.n. Superior rectal l.n. Inferior mesenteric l.n. Left colic l.n. Middle colic l.n. Right colic l.n. Mesocolic l.n. Juxtaintestinal l.n. Prececal l.n. Retrocecal l.n. Ileocolic l.n. Appendicular l.n. Superior mesenteric l.n. Abdomen 208 Lymph Nodes of the Infracolic Organs Fig. 16.32 Lymph nodes of the jejunum and ileum Anterior view. Removed: Stomach, liver, pancreas, and colon. Fig. 16.33 Lymphatic drainage of the intestines Clinical box 16.8 Regional lymphatic pathways in the large intestine have important clinical affects. • Ascending colon, cecum, and transverse colon: Lymph drains initially to the right and middle colic nodes, then to the superior mesenteric nodes, and finally to the intestinal trunk. • Descending colon: Lymph drains initially to the regional left colic nodes, then to the inferior mesenteric nodes, then via the left lumbar nodes into the left lumbar trunk. • Sigmoid colon: Lymph drains initially to sigmoid nodes then follows the pathway described above for the descending colon. • Upper rectum: Lymph drains initially to the superior rectal nodes then fol lows the pathway described above for the sigmoid colon. Thus, a malignant tumor undergoing lymphogenous spread must negotiate several lymph node groups (all of which should be removed in tumor resec- tions) before the malignant cells can reach the intestinal trunk and thoracic duct and finally enter the bloodstream. This long route of lymphogenous spread improves the prospects for a cure. Lymphatic drainage of the large intestine Middle colic l.n. Right colic l.n. Ileocolic l.n. Prececal l.n. Inferior mesenteric l.n. Superior rectal l.n. Sigmoid l.n. Left colic l.n. Superior mesenteric l.n. Intermediate colic l.n. Epicolic l.n. Paracolic l.n. 16 Neurovasculature 209 Fig. 16.34 Lymph nodes of the large intestine Anterior view. Reflected: Transverse colon and greater omentum. Sciatic n. Femoral n. Obturator n. Intercostal nn. Subcostal n. Genitofemoral n. Sacral plexus Lumbar plexus Intercostal nn. Iliohypogastric n. Ilioinguinal n. Abdomen 210 Nerves of the Abdominal Wall Fig. 16.35 Somatic nerves of the abdomen and pelvis Anterior view. The abdominal wall is innervated by somatic nerves that include the lower intercostal nerves and branches of the lumbar plexus. Fig. 16.36 Cutaneous innervation of the anterior trunk Anterior view. Fig. 16.37 Dermatomes of the anterior trunk Anterior view. Intercostal nn., lateral cutaneous brs. Iliohypogastric n., lateral cutaneous br. Lateral femoral cutaneous n. Femoral n., anterior cutaneous brs. Genitofemoral n., femoral br. Ilioinguinal n. Iliohypogastric n., anterior cutaneous br. Intercostal nn., anterior cutaneous brs. Supraclavicular nn. L3 L2 L1 T12 T10 T 4 T3 T2 C3 C4 S2 L4 C8 T1 C6 C5 C7 16 Neurovasculature 211 Fig. 16.38 Nerves of the lumbar plexus Anterior view. Genitofemoral n., genital br. Medial arcuate lig. Lateral arcuate lig. Subcostal n. Quadratus lumborum Transversus abdominis Ilioinguinal n. Iliacus Iliohypogastric n., lateral cutaneous br. Lateral femoral cutaneous n. Femoral n. Femoral n., anterior cutaneous br. Genitofemoral n., femoral br. Ilioinguinal n. Iliohypogastric n., anterior cutaneous br. Abdominal aorta Sympathetic trunk Diaphragm, lumbar part Inferior vena cava Femoral br. Genital br. Genitofe moral n. Iliohypogastric n. Psoas major and minor Median arcuate lig. Subcostal n. Iliohypogastric n. Ilioinguinal n. Lateral femoral cutaneous n. Obturator n. Femoral n. External iliac a. Sympathetic trunk Internal iliac a. Common iliac a. Abdominal aorta Inferior vena cava Femoral br. Genital br. Genitofemoral n. Lumbar plexus Genitofe moral n. A Lumbar plexus in situ. Removed: All visceral structures except vessels. B Lumbar plexus, dissection. Windowed: Psoas major and minor muscles. Sympathetic trunk Sacral splanchnic nn. Thoracic splanchnic nn. Lumbar splanchnic nn. {with intermesenteric plexus) Celiac ganglion Sympathetic C8 T1 T5 L1 Sympathetic (prevertebral) ganglia Superior cervical ganglion Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus Pelvic splanchnic nn. Superior and inferior mesenteric ganglia Inferior hypogastric plexus Parasympathetic S2 S4 Vagus n. Dorsal vagal nucleus Head and neck Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Sympathetic trunk Sacral splanchnic nn. Thoracic splanchnic nn. Lumbar splanchnic nn. {with intermesenteric plexus) Celiac ganglion Sympathetic C8 T1 T5 L1 Sympathetic (prevertebral) ganglia Superior cervical ganglion Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus Pelvic splanchnic nn. Superior and inferior mesenteric ganglia Inferior hypogastric plexus Parasympathetic S2 S4 Vagus n. Dorsal vagal nucleus Head and neck Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Sympathetic trunk Sacral splanchnic nn. Thoracic splanchnic nn. Lumbar splanchnic nn. {with intermesenteric plexus) Celiac ganglion Sympathetic C8 T1 T5 L1 Sympathetic (prevertebral) ganglia Superior cervical ganglion Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus Pelvic splanchnic nn. Superior and inferior mesenteric ganglia Inferior hypogastric plexus Parasympathetic S2 S4 Vagus n. Dorsal vagal nucleus Head and neck Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Abdomen 212 Autonomic Innervation: Overview Fig. 16.39 Sympathetic and parasympathetic nervous systems in the abdomen and pelvis Table 16.4 Effects of the autonomic nervous system in the abdomen and pelvis Organ (organ system) Sympathetic effect Parasympathetic effect Gastrointestinal tract Longitudinal and circular muscle fibers ↓ motility ↑ motility Sphincter muscles Contraction Relaxation Glands ↓ secretions ↑ secretions Splenic capsule Contraction No effect Liver ↑ glycogenolysis/gluconeogenesis Pancreas Endocrine pancreas ↓ insulin secretion Exocrine pancreas ↓ secretion ↑ secretion Urinary bladder Detrusor vesicae Relaxation Contraction Functional bladder sphincter Contraction Inhibits contraction Seminal glands and ductus deferens Contraction (ejaculation) No effect Uterus Contraction or relaxation, depending on hormonal status Arteries Vasoconstriction Vasodilation of the arteries of the penis and clitoris (erection) Suprarenal glands (medulla) Release of adrenalin No effect Urinary tract Kidney Vasoconstriction (↓ urine formation) Vasodilation impar Ganglion Sympathetic trunk with lumbar ganglia Intermesenteric plexus Lumbar splanchnics Sacral ganglia Hypogastric nn. Sacral splanchnic Iliac plexus 16 Neurovasculature 213 Table 16.5 Autonomic plexuses in the abdomen and pelvis Ganglia Subplexus Distribution Celiac plexus Celiac ganglia Hepatic plexus • Liver, gallbladder Gastric plexus • Stomach Splenic plexus • Spleen Pancreatic plexus • Pancreas Superior mesenteric plexus Superior mesenteric ganglion — • Pancreas (head) • Duodenum • Jejunum • Ileum • Cecum • Colon (to left colic flexure) • Ovary Suprarenal and renal plexus Aorticorenal ganglion Ureteral plexus • Suprarenal gland • Kidney • Proximal ureter Ovarian/testicular plexus — — • Ovary/testis Inferior mesenteric plexus Inferior mesenteric ganglion Left colic plexus • Left colic flexure Superior rectal plexus • Descending and sigmoid colon • Upper rectum Superior hypogastric plexus — Hypogastric nn. • Pelvic viscera Inferior hypogastric plexus Pelvic ganglia Middle and inferior rectal plexus • Middle and lower rectum Prostatic plexus • Prostate • Seminal gland • Bulbourethral gland • Ejaculatory duct • Penis • Urethra Deferential plexus • Ductus deferens • Epididymis Uterovaginal plexus • Uterus • Uterine tube • Vagina • Ovary Vesical plexus • Urinary bladder Ureteral plexus • Ureter (ascending from pelvis) Note: The two sacral sympathetic trunks converge and terminate in front of the coccyx in a small ganglion, the ganglion impar. Abdomen 214 Autonomic Innervation & Referred Pain Pain afferents from the viscera (visceral pain) and dermatomes (somatic pain) terminate at the same processing neurons in the posterior horn of the spinal cord. The convergence of these visceral and somatic affer ent fibers confuses the relationship between the pain’s origin and its perception. This phenomenon is called referred pain. The pain impulses from a particular internal organ are consistently projected to the same well-defined skin area. Thus, the area of skin that the pain is projected to provides crucial information regarding what organ is affected. Fig. 16.40 Autonomic innervation of the liver, gallbladder, and stomach A Schematic of celiac plexus distribution to the liver, gallbladder, and stomach. B Zones of referred pain from the liver, gallbladder, and stomach. Fig. 16.41 Autonomic innervation of the pancreas, duodenum, and spleen A Schematic of celiac plexus distribution to the pancreas, duodenum, and spleen. B Zones of referred pain from the pan creas. There are no zones associated with the duodenum and spleen. Gallbladder Liver and gallbladder Stomach Pyloric br. of posterior vagal trunk Right greater splanchnic n. Posterior vagal trunk Left greater splanchnic n. Celiac br. of anterior vagal trunk Posterior gastric plexus Anterior vagal trunk Sympathetic trunk Pyloric br. of anterior vagal trunk Hepatic br. of anterior vagal trunk Hepatic plexus Hepatic br. of posterior vagal trunk Celiac ganglia Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Posterior vagal trunk Left greater splanchnic n. Splenic plexus Pancreatic plexus Anterior vagal trunk Sympathetic trunk Superior mesenteric ganglion Brs. of superior mesenteric plexus to pancreas and duodenum Brs. of celiac plexus to duodenum Celiac ganglia Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Pyloric br. of posterior vagal trunk Right greater splanchnic n. Posterior vagal trunk Left greater splanchnic n Celiac br. of anterior vagal trunk Posterior gastric plexu Anterior vagal trunk Sympathetic trunk Pyloric br. of anterior vagal trunk Hepatic br. of anterior vagal trunk Hepatic plexus Hepatic br. of posterior vagal trunk Celiac ganglia Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Posterior vagal trunk Left g splan Splenic plexus Pancr plexu Anter vagal Symp trunk Superior mesente ganglion Brs. of superior mesenteric plexus to pancreas and duodenum Brs. of celiac plexus to duodenum Celiac ganglia Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers 16 Neurovasculature 215 A Schematic of superior mesenteric, inferior mesenteric, and inferior hypogastric plexuses distribution. B Zones of referred pain from the small and large intestine. Fig. 16.43 Autonomic innervation of the kidneys and upper ureters A Schematic of renal and ureteral plexuses distribution. B Zones of referred pain from the left kidney and bladder. Small intestine Large intestine Kidney Urinary bladder Greater splanchnic n. (T5-T9) Lesser splanchnic n. (T10-T11) Least splanchnic n. (T12) Lumbar splanchnic n. (L1-L2) Lumbar splanchnic n. (L3-L5) Sacral splanchnic nn. (S1-S3) Pelvic splanchnic nn. (S2-S4) Superior rectal plexus Inferior mesenteric plexus Inferior rectal plexus Inferior hypogastric plexus and pelvic ganglia Middle rectal plexus Intermesenteric plexus Superior mesenteric plexus Superior mesenteric ganglion Inferior mesenteric ganglion Superior hypogastric plexus Celiac ganglia Posterior vagal trunk Sympathetic trunk Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Sympathetic trunk Posterior vagal trunk Lesser splanchnic n. (T10-T11) Aorticorenal ganglion Upper ureter Ureteral plexus Renal plexus Renal ganglia First lumbar splanchnic n. Least splanchnic n. (T12) Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Sympathetic trunk Posterior vagal trunk Lesser splanchnic n. (T10-T11) Aorticoren ganglion Ureteral plexus Renal plexus Renal ganglia First lumbar splanchnic n. Least splanchnic n. (T12) Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Greater splanchnic n. (T5-T9) Lesser splanchnic n. (T10-T11) Least splanchnic n. (T12) Lumbar splanchnic n. (L1-L2) Lumbar splanchnic n. (L3-L5) Sacral splanchnic nn. (S1-S3) Pelvic splanchnic nn. (S2-S4) Inferior p Inferior hypogas plexus and pelvic ganglia Interm p Su me p Superior mesenteric ganglion Inferior mesenteric ganglion Superior hypogastric plexus Celiac ganglia Posterior vagal trunk Sympathetic trunk Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Greater splanchnic n. (T5-T9) Lesser splanchnic n. (T10-T11) Least splanchnic n. (T12) Lumbar splanchnic n. (L1-L2) Lumbar splanchnic n. (L3-L5) Sacral splanchnic nn. (S1-S3) Pelvic splanchnic nn. (S2-S4) Inferior p Inferior hypogas plexus and pelvic ganglia Interm p Su me p Superior mesenteric ganglion Inferior mesenteric ganglion Superior hypogastric plexus Celiac ganglia Posterior vagal trunk Sympathetic trunk Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Fig. 16.42 Autonomic innervation of the midgut and hindgut Brs. of gastric plexus (on gastro omental aa.) Pancreatic plexus (on pancreatico duodenal aa.) Margin of hepatoduodenal lig. Anterior vagal trunk, pyloric br. Superior mesenteric plexus (on superior mesenteric a.) Splenic plexus Celiac ganglia Posterior vagal trunk, hepatic br. Anterior vagal trunk, hepatic br. Hepatic plexus (on common hepatic a.) Left lesser splanchnic n. Left greater splanchnic n. Anterior vagal trunk Anterior gastric plexus Gastric plexus (on left gastric a.) Posterior vagal trunk, celiac br. Abdomen 216 Innervation of the Foregut & Urinary Organs Fig. 16.44 Innervation of the foregut and spleen Anterior view. Removed: Lesser omentum, ascending colon, and parts of the transverse colon. Opened: Omental bursa. The anterior and poste rior vagal trunks each produce a celiac, hepatic, and pyloric branch, and a gastric plexus. See p. 214 for schematic. Clinical box 16.9 The enteric plexus is the portion of the autonomic nervous system that specifically serves all the organs of the gastrointestinal tract. Located within the wall of the digestive tube (intramural nervous system), it is subject to both sympathetic and parasympathetic influences. Congenital absence of the enteric plexus leads to severe disturbances of gastrointestinal transit (e.g., Hirschsprung disease). The enteric plexus has basically the same organization throughout the gastrointestinal tract, although there is an area in the wall of the lower rectum that is devoid of ganglion cells. Three subsystems are distinguished in the enteric plexus: • Submucosal plexus (Meissner’s plexus) • Myenteric plexus (Auerbach’s plexus) • Subserosal plexus Organization of the enteric plexus Mucosa Submucosa Muscularis externa, circular layer Muscularis externa, longitudinal layer Serosa Subserosal plexus Myenteric plexus Submucosal plexus 1st sacral n., anterior ramus Pelvic splanchnic nn. Posterior vagal trunk Anterior vagal trunk Celiac ganglion Right greater splanchnic n. Right lesser splanchnic n. Suprarenal plexus Renal plexus Sympathetic trunk, lumbar ganglia Ureteral plexus Iliac plexus Right hypo gastric nn. Vesical plexus Prostatic plexus Sympathetic trunk, sacral ganglia Middle rectal plexus Inferior hypo gastric plexus Left hypo gastric n. Superior hypo gastric plexus Inferior mesen teric plexus Testicular plexus Inferior mesen teric ganglion Intermesenteric plexus Aorticorenal ganglia Superior mesen teric ganglion 16 Neurovasculature 217 Fig. 16.45 Innervation of the urinary organs Anterior view of the male abdomen and pelvis. Removed: Peritoneum, majority of stomach, and abdominal organs except kidneys, suprarenal glands, and bladder. See pp. 215 and 282 for schematic. Testicular (ovarian) plexus Right colic a. (with autonomic plexus) Ileocolic a. (with autonomic plexus) Jejunal and ileal aa. (with autonomic plexuses) Anterior vagal trunk, pyloric br. Posterior vagal trunk, celiac br. Aorticorenal ganglion Left lesser splanchnic n. Superior mesenteric plexus Renal plexus Superior mesen teric ganglion Splenic plexus Celiac ganglia Anterior vagal trunk Posterior vagal trunk Right greater splanchnic n. Anterior vagal trunk, hepatic br. Hepatic plexus Left greater splanchnic n. Abdomen 218 Innervation of the Intestines Fig. 16.46 Innervation of the small intestine Anterior view. Partially removed: Stomach, pancreas, and transverse colon (distal part). See p. 215 for schematic. Left colic a. (with autonomic plexus) Sigmoid aa. (with autonomic plexus) Right hypo gastric nn. Middle and right colic aa. (with autonomic plexuses) Ascending colon Superior hypo gastric plexus Superior rectal a. (with autonomic plexus) Inferior hypo gastric plexus, brs. to descending colon and sigmoid colon Inferior mesen teric plexus Inferior mesen teric ganglion Descending colon Intermesenteric plexus Transverse colon Ileocolic a. (with autonomic plexus) 16 Neurovasculature 219 Fig. 16.47 Innervation of the large intestine Anterior view. Removed: Small intestine. Reflected: Transverse and sigmoid colons. See p. 215 for schematic. Vertebral canal with spinal cord Spleen Stomach Left colic flexure Splenic a. Left kidney Lumbar l.n. (preaortic) Abdominal aorta T12 vertebra Common hepatic a. Right supra-renal gland Diaphragm, costal part Inferior vena cava Gallbladder Portal v. Liver, right lobe Falciform lig. of liver Liver, left lobe Diaphragm, costal part Left suprarenal gland Parietal peritoneum Visceral peritoneum Spinal cord (in vertebral canal) L1 vertebra Right suprarenal gland Greater omentum Pyloric part Anterior wall Posterior wall Omental bursa Transverse colon Descending colon Perirenal fat capsule Left kidney Lateral lumbar l.n. Splenic v. Abdominal aorta Kidney (with right renal a.) Intermediate lumbar l.n. Inferior vena cava Liver, right lobe Superior mesenteric a. and v. Gallbladder Duodenum Transverse colon Pancreas Spleen Common bile duct Stomach Left colic flexure Vertebral venous plexus Internal thoracic a. and v. Intercostal v., a., and n. Abdomen 220 17 Sectional & Radiographic Anatomy Sectional Anatomy of the Abdomen Fig. 17.1 Transverse sections of the abdomen Inferior view. A Section through T12 vertebra. B Section through L1 vertebra. Transverse colon Duodenum, descending part Gallbladder Liver Superior mesenteric a. and v. Right kidney Psoas major Inferior vena cava Spinal cord Celiac l.n. Ureter Abdominal aorta Duodenojejunal flexure Jejunum Transverse mesocolon Jejunal a. Stomach, body Pancreas, head Descending colon C Section through L2 vertebra. 17 Sectional & Radiographic Anatomy 221 C Transverse section through L2 vertebral level B Transverse section through L1 vertebral level Liver (right lobe) Portal vein (right br.) Gallbladder Common hepatic a. Splenic a. Jejunum Descending colon Portal v. Right suprarenal gland Inferior vena cava Abdominal aorta in aortic hiatus Stomach (pylorus) Pancreas (body) Diaphragm (lumbar part, left crus) Splenic a. and v. Spleen Pancreas (head) Duodenum Portal v. (confluence) Transverse colon Splenic v. Jejunum Pancreas (tail) Right hepatic v. Inferior vena cava Right suprarenal gland and (superior) suprarenal a. Abdominal aorta Celiac trunk Left kidney (superior pole) Left lung (costo-diaphragmatic recess) Liver (right lobe) Right renal a. and v. Duodenum (descending part) Inferior vena cava Pancreas (head) Superior mesenteric a. and v. Duodenum (ascending part) Inferior mesenteric v. Descending colon Right kidney (renal pyramid, medulla) Psoas major muscle Abdominal aorta Left renal v. Left kidney (hilum) A B C Abdomen 222 Radiographic Anatomy of the Abdomen (I) Fig. 17.2 CT of the abdomen: Transverse sections (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A Transverse section through T12 vertebral level D Transverse section through L3 vertebral level F Transverse section through L5 vertebral level E Transverse section through L4 vertebral level Liver (right lobe) Right kidney (pelvis) Small intestine Duodenum (horizontal part) Inferior vena cava Ileocolic a. and v. Root of mesentery Superior mesenteric a. and v. Abdominal aorta Jejunum Quadratus lumborum muscle Psoas major muscle Spinal canal with cauda equina Posterior pararenal space Posterior paracolic space External oblique muscle Internal oblique muscle Transverse abdominis muscle Right kidney (renal pyramid, medulla) Right testicular a. and v. Inferior vena cava Umbilicus Abdominal aorta Rectus abdominis muscle Right ureter Anterior superior iliac crest Ascending colon Inferior vena cava (confluence) Common iliac arteries Small intestine Descending colon Wing of ilium (superior border) Gluteus medius muscle Iliacus muscle Lumbar plexus Psoas major muscle D E F 17 Sectional & Radiographic Anatomy 223 Fig. 17.3 CT of the abdomen: Sagittal section through the aorta (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Fig. 17.5 Radiograph of intravenous pylegram Anterior view. Renal pelvis Inferior pole of left kidney Distal ureter Urinary bladder Right ureter 12th rib Major calyces Fig. 17.4 CT of the Abdomen: Coronal section through the kidneys (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Heart Liver (left lobe) Celiac trunk Stomach Superior mesenteric a. Right renal v. Lumbar vertebral body (L2) Pancreas Transverse colon Jejunum Duodenum (horizontal part) Abdominal aorta Promontory of sacrum Rectum Urinary bladder Stomach (fundus) Spleen with splenic a. and v. Liver (right lobe) Pancreas (tail) Inferior vena cava Left renal v. and a. Left kidney (renal cortex) Right kidney, superior pole, and right renal a. Renal pelvis Inferior mesenteric v. Psoas muscle Iliacus muscle Common iliac a. and v. (left) Gluteus medius muscle Abdomen 224 Radiographic Anatomy of the Abdomen (II) Fig. 17.6 Radiographs of double contrast barium enema. Anterior view. Ilium Cecum Sacrum Sigmoid colon Left colic flexure Transverse colon Right colic flexure Colonic haustra Ascending colon Descending colon Jejunum Ileum Circular folds A Small intestine. (Reproduced courtesy of Universitätsmedizin Mainz, Klinik und Poliklinik für Diagnostische und Interventio-nelle Radiologie.) B Large intestine. (Reproduced courtesy of Klinik für Diag-nostische Radiologie, Universitätsklinikum Schleswig Hol-stein, Campus Kiel: Prof. Dr. Med. S. Müller-Huelsbeck.) Fig. 17.7 MRI of the intestines Coronal view. Sectional imaging modalities like CT and MR have mostly replaced conventional radiographs in the evaluation of gastrointestinal disease. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) A Jejunum (arrow) B Ileum (arrow), transverse colon (TC), urinary bladder (B) C Ascending colon (AC), descending colon (DC), transverse colon (TC), small bowel and mesenteric structures. B TC TC AC DC B TC 17 Sectional & Radiographic Anatomy 225 18 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 19 Bones, Ligaments & Muscles Pelvic Girdle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Female & Male Pelvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Female & Male Pelvic Measurements. . . . . . . . . . . . . . . . . . . 234 Pelvic Ligaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Muscles of the Pelvic Floor & Perineum . . . . . . . . . . . . . . . . . 238 Pelvic Floor & Perineal Muscle Facts. . . . . . . . . . . . . . . . . . . . 240 20 Spaces Contents of the Pelvis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Peritoneal Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Pelvis & Perineum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 21 Internal Organs Rectum & Anal Canal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Ureters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Urinary Bladder & Urethra . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Overview of the Genital Organs. . . . . . . . . . . . . . . . . . . . . . . 254 Uterus & Ovaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Ligaments & Fascia of the Deep Pelvis . . . . . . . . . . . . . . . . . . 258 Vagina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Female External Genitalia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Penis, Testis & Epididymis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Male Accessory Sex Glands. . . . . . . . . . . . . . . . . . . . . . . . . . . 266 22 Neurovasculature Overview of the Blood Supply to Pelvic Organs & Wall. . . . . 268 Arteries & Veins of the Male Pelvis. . . . . . . . . . . . . . . . . . . . . 270 Arteries & Veins of the Female Pelvis. . . . . . . . . . . . . . . . . . . 272 Arteries & Veins of the Rectum & External Genitalia. . . . . . . 274 Lymphatics of the Pelvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Lymph Nodes of the Genitalia. . . . . . . . . . . . . . . . . . . . . . . . . 278 Autonomic Innervation of the Genital Organs. . . . . . . . . . . . 280 Autonomic Innervation of the Urinary Organs & Rectum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Neurovasculature of the Male & Female Perineum . . . . . . . . 284 23 Sectional & Radiographic Anatomy Sectional Anatomy of the Pelvis & Perineum. . . . . . . . . . . . . 286 Radiographic Anatomy of the Female Pelvis. . . . . . . . . . . . . 288 Radiographic Anatomy of the Male Pelvis . . . . . . . . . . . . . . . 290 Pelvis & Perineum Transumbilical plane (L3–4 disk) Pubic tubercle Anterior superior iliac spine (ASIS) Inguinal lig. Pubic symphysis Superficial inguinal ring Quadriceps femoris Sartorius Anterior superior iliac spine (ASIS) Pelvis & Perineum 228 18 Surface Anatomy Surface Anatomy Fig. 18.1 Palpable structures of the pelvis Anterior view. The structures are common to both male and female. See pp. 2–3 for structures of the back. B Musculature, male pelvis. A Bony prominences, female pelvis. Ischiopubic ramus Labium majus Ischial tuberosity Ischial spine Coccyx Anus Perineal raphe Posterior labial commissure Vagina (vaginal orifice) Labium minus External urethral orifice Glans of clitoris Prepuce of clitoris Mons pubis Sacrum Perineal region Anal triangle Urogenital triangle Posterior commissure of vagina Lateral crease of thigh Surgical gynecological perineum Anterior border of anus Pubic symphysis Coccyx Anus Perineal raphe Scrotum Glans of penis Penis Sacrum Ischiopubic ramus Ischial tuberosity Ischial spine Perineal region Anal triangle Urogenital triangle Posterior border of root of scrotum Lateral crease of thigh Surgical perineum Anterior border of anus 18 Surface Anatomy 229 Fig. 18.2 Regions of the female perineum Lithotomy position. Fig. 18.3 Regions of the male perineum Lithotomy position. A Perineal region. A Perineal region. B Surgical gynecological perineum. B Surgical perineum. The perineum is the inferiormost portion of the trunk, between the thighs and buttocks, extending from the pubis to the coccyx and superiorly to the inferior fascia of the pelvic diaphragm, including all of the structures of the anal and urogenital triangles (Fig. 18.2A). The bilateral boundaries of the perineum are the pubic symphysis, ischio pubic ramus, ischial tuberosity, sacrotuberous ligament, and coccyx. Sacroiliac joint Coxal bone Pubic symphysis Sacrum Iliac fossa Auricular surface of ilium Arcuate line Ischial spine Pectineal line Symphyseal surface Obturator foramen Acetabulum Acetabular rim Anterior inferior iliac spine Anterior superior iliac spine Iliac crest Ischial tuberosity Iliac tuberosity Iliac fossa Iliac tuberosity Posterior superior iliac spine Auricular surface of ilium Ilium, body Arcuate line Ischial spine Ischium, body Ischial tuberosity Inferior pubic ramus Symphyseal surface Pubis, body Pubic tubercle Pectineal line Superior pubic ramus Anterior inferior iliac spine Obturator foramen Anterior superior iliac spine Iliac crest Ischial ramus Posterior inferior iliac spine Pelvis & Perineum 230 19 Bones, Ligaments & Muscles Pelvic Girdle Fig. 19.1 Pelvic girdle Anterosuperior view. The pelvic girdle con-sists of the two coxal bones and the sacrum. Fig. 19.2 Coxal bone Right side (male). A Anterior view. B Medial view. The pelvis is the region of the body inferior to the abdomen and surrounded by the pelvic girdle, which is the two coxal (hip) bones and the sacrum that connect the vertebral column to the femur. The two coxal bones are connected to each other at the cartilaginous pubic symphysis and to the sacrum via the sacroiliac joints, creating the pelvic brim (red, Fig. 19.1). The stability of the pelvic girdle is necessary for the transfer of trunk loads to the lower limb, which occurs in normal gait. Iliac wing Acetabulum Pubis, body Superior pubic ramus Obturator foramen Inferior pubic ramus Ischial ramus Ischial tuberosity Ischium, body Ischial spine Ilium, body Iliac crest Triradiate cartilage Pubis Ischium Ilium Acetab-ulum Posterior gluteal line Iliac crest Anterior superior iliac spine Inferior gluteal line Anterior inferior iliac spine Acetabular rim Lunate surface Acetabular fossa Acetabular notch Pubic tubercle Obturator foramen Ischial tuberosity Lesser sciatic notch Ischial spine Greater sciatic notch Posterior superior iliac spine Gluteal surface Acetabulum Posterior inferior iliac spine Anterior gluteal line 19 Bones, Ligaments & Muscles 231 Fig. 19.3 Triradiate cartilage of the coxal bone Right coxal bone, lateral view. The coxal bone consists of the ilium, ischium, and pubis. A Junction of the triradiate cartilage. B Radiograph of a child’s acetabulum. Fig. 19.4 Coxal bone Right side (male), lateral view. Sacroiliac joint Iliac crest Anterior inferior iliac spine Ischial spine Obturator foramen Superior and inferior pubic rami Pubic symphysis Coccyx Pubic tubercle Ischial ramus Pubic arch Acetabular margin Anterior superior iliac spine Iliac fossa Sacrum Iliac crest Posterior superior iliac spine Posterior inferior iliac spine Ischial spine Superior pubic ramus Inferior pubic ramus Sacral hiatus Lesser sciatic notch Greater sciatic notch Iliac wing Median sacral crest Sacral canal Ischial tuberosity Iliac tuberosity Sacroiliac joint Iliac crest Inner lip Intermediate line Outer lip Coccyx Arcuate line Ischial spine Pubic tubercle Pectineal line (pecten pubis) Anterior superior and inferior iliac spines Iliac fossa Ala of sacrum Iliac tubercle Sacral canal Promontory Pubic crest Pelvis & Perineum 232 Female & Male Pelvis Fig. 19.5 Female pelvis A Anterior view. B Posterior view. C Superior view. Clinical box 19.1 Childbirth A non-optimal relation between the maternal pelvis and the fetal head may lead to complications during childbirth, potentially necessitating a caesarean section. Maternal causes include earlier pelvic trauma and innate malformations. Fetal causes include hydrocephalus (disturbed circulation of cerebrospinal fluid, leading to brain dilation and cranial expansion). Anterior sacral foramina Superior articular process Iliac crest Posterior inferior iliac spine Pubic tubercle Ischial spine Obturator foramen Pubic arch Pubic symphysis Pectineal line (pecten pubis) Ala Pelvic surface Promon-tory Acetabulum Anterior superior and inferior iliac spines Sacrum Acetabular margin Sacral hiatus Posterior sacral foramina Superior articular process Iliac crest Posterior superior and inferior iliac spines Ischial spine Coccyx Pubis Ischial tuberosity Median sacral crest Sacral canal Gluteal surface Illiac tuberosity Iliac tubercle Median sacral crest Arcuate line Ischial spine Pubic symphysis Pectineal line (pecten pubis) Iliopubic eminence Anterior superior and inferior iliac spines Iliac fossa Ala of sacrum Superior articular process Base of sacrum Iliac crest Inner lip Intermediate line Outer lip 19 Bones, Ligaments & Muscles 233 Fig. 19.6 Male pelvis A Anterior view. B Posterior view. C Superior view. Female Male Pubic symphysis Subpubic angle A Male vs. female pelvis. B Female. C Male. Table 19.1 Gender-specific features of the pelvis Structure ♀ ♂ False pelvis Wide and shallow Narrow and deep Pelvic inlet Transversely oval Heart-shaped Pelvic outlet Roomy and round Narrow and oblong Ischial tuberosities Everted Inverted Pelvic cavity Roomy and shallow Narrow and deep Sacrum Short, wide, and flat Long, narrow, and convex Subpubic angle 90–100 degrees 70 degrees Pelvis & Perineum 234 Female & Male Pelvic Measurements The pelvic inlet, the superior aperture of the pelvis, is the boundary between the abdominal and pelvic cavities. It is defined by the plane that passes through its edge, the pelvic brim, which is the prominence of the sacrum, the arcuate and pectineal lines, and the upper margin of the pubic symphysis. Occasionally, the terms pelvic inlet and pelvic brim are used interchangeably. The pelvic outlet is the plane of the inferior aperture, passing through the pubic arch, the ischial tuberosities, the inferior margin of the sacrotuberous ligament, and the tip of the coccyx. Fig. 19.7 True and false pelvis The pelvis is the region of the body inferior to the abdomen, surrounded by the pelvic girdle. The false pelvis is immediately inferior to the ab-dominal cavity, between the iliac alae, and superior to the pelvic inlet. The true pelvis is the bony-walled space between the pelvic inlet and the pelvic outlet. It is bounded inferiorly by the pelvic diaphragm, also called the pelvic floor. Coccyx Plane of pelvic inlet Symphyseal surface Plane of pelvic outlet Coccyx Plane of pelvic inlet Symphyseal surface Plane of pelvic outlet A Female. Midsagittal section, viewed from left side. B Male. Midsagittal section, viewed from left side. Linea terminalis Diagonal conjugate True conjugate Plane of pelvic inlet ~60° ~15° Plane of pelvic outlet Fig. 19.8 Narrowest diameter of female pelvic canal The true conjugate, the distance between the promontory and the most posterosuperior point of the pubic symphysis, is the narrowest AP (anteroposterior) diameter of the pelvic (birth) canal. This diameter is difficult to measure due to the viscera, so the diagonal conjugate, the distance between the promontory and the inferior border of the pubic symphysis, is used to estimate it. The linea terminalis is part of the border defining the pelvic inlet (pelvic brim). 19 Bones, Ligaments & Muscles 235 A Female pelvis, superior view. Pelvic inlet outlined in red. B Male pelvis, superior view. Pelvic inlet outlined in red. Left oblique diameter Right oblique diameter Interspinous diameter Transverse diameter of pelvic inlet plane Pelvic inlet plane Linea terminalis Pelvic inlet plane Interspinous distance Trans-tubercular distance Superior pubic ramus Inferior pubic ramus Ischial ramus Ischial tuberosity Coccyx Pubic symphysis Superior pubic ramus Coccyx Pubic symphysis Fig. 19.9 Pelvic inlet and outlet The measurements shown are applicable to both male and female. The transverse and oblique diameters of the female pelvic inlet are obstetrically important, as they are the measure of the diameter of the pelvic (birth) canal. The interspinous distance is the narrowest diameter of the pelvic outlet. C Female pelvis, inferior view. Pelvic outlet outlined in red. D Male pelvis, inferior view. Pelvic outlet outlined in red. Anterior longitudinal lig. Anterior inferior iliac spine Sacrotuberous lig. Ischial spine Sacrospinous lig. Coccyx Obturator membrane Pubic tubercle Inguinal lig. Anterior superior iliac spine Anterior sacroiliac ligs. Sacral promontory Iliolumbar lig. Pubic symphysis Pectineal lig. Iliolumbar lig. Long posterior sacroiliac lig. Posterior sacroiliac ligs. Greater sciatic foramen Lesser sciatic foramen Sacrotuberous lig. Obturator membrane Coccyx Sacrospinous lig. Ischial tuberosity Ischial spine Ilium, gluteal surface L4 spinous process Iliac crest Iliac tubercle Posterior superior iliac spine Posterior inferior iliac spine Short posterior sacroliac ligs. Pelvis & Perineum 236 Pelvic Ligaments Fig. 19.10 Ligaments of the pelvis Male pelvis. A Anterosuperior view. B Posterior view. On the right, the superficial part of the posterior sacroiliac ligament has been removed to reveal long and short posterior sacroiliac ligaments which blend with the deeper interosseous sacroiliac ligament. Acetabulum Posterior sacroiliac lig. Anterior sacroiliac lig. Anterior sacral foramina Sacro-spinous lig. Ischial spine Sacro-tuberous lig. Pubic symphysis Coccyx Anterior sacrococcygeal lig. Ilium Posterior superior iliac spine Sacral canal Sacrum Iliac tuberosity Sacral tuberosity Interosseous sacroiliac lig. Sacroiliac joint Greater sciatic foramen Lesser sciatic foramen Lesser sciatic foramen Greater sciatic foramen Sacral canal Sacral hiatus Promontory Anterior superior iliac spine Pectineal line Symphyseal surface Obturator membrane Ischial tuberosity Sacrotuberous lig. Coccyx Sacrospinous lig. Ischial spine Anterior sacroiliac lig. Sacrum L5 spinous process Intervertebral disk L4/5 Arcuate line Obturator canal 19 Bones, Ligaments & Muscles 237 Fig. 19.11 Ligaments of the sacroiliac joint Male pelvis, midsagittal section. A Right half of pelvis, medial view. B Oblique section, superior view. Fig. 19.12 Pelvic ligament attachment sites on the coxal bone Left coxal bone, medial view. Ligament attach-ments are shown in green. Interosseous sacroiliac lig. Sacrospinous lig. Sacrotuberous lig. Pubic symphysis Puborectalis Tendinous arch of levator ani Sacrum Anococcygeal raphe Coccygeus Piriformis Ischial spine Iliococcygeus Pubococcygeus Rectal hiatus Obturator fascia (obturator internus) Obturator canal Prerectal fibers Levator ani Urogenital hiatus Puborectalis Pubococcygeus Anococcygeal lig. Anterior sacroiliac lig. Arcuate line Tendinous arch of levator ani Obturator internus fascia Pubic symphysis Deep transverse perineal Iliococcygeus Ischial spine Coccygeus Piriformis Levator ani Obturator foramen Ischial spine Posterior superior iliac spine Coccygeus Sacrospinous lig. Coccyx Sacrotuberous lig. Pubic tubercle Piriformis Levator ani Pubo-rectalis Pubo-coccygeus Ilio-coccygeus Coccyx Coccygeus Piriformis Obturator internus Prerectal fibers Inferior pubic lig. Pubic symphysis Acetabulum Ischial tuberosity Urogenital hiatus Levator ani Rectal hiatus Pelvis & Perineum 238 Muscles of the Pelvic Floor & Perineum Fig. 19.13 Muscles of the pelvic floor A Superior view. B Inferior view. C Medial view of right hemipelvis. D Right lateral view. Perineal body Anococcygeal lig. Coccyx Anal cleft Superficial perineal (Colles’) fascia Ischial tuberosity Obturator fascia Inferior fascia of pelvic diaphragm External anal sphincter Levator ani Gluteus maximus Obturator internus Superficial transverse perineal Perineal membrane Ischiocavernosus Bulbospongiosus Superficial perineal (Colles’) fascia Ischial tuberosity Obturator fascia Inferior fascia of pelvic diaphragm Anococcygeal lig. External anal sphincter Levator ani Gluteus maximus Obturator internus Superficial transverse perineal Perineal membrane Ischiocavernosus Bulbospongiosus Anococcygeal lig. Connective tissue gaps External anal sphincter Levator ani 19 Bones, Ligaments & Muscles 239 A Female. B Male. Fig. 19.14 Muscles and fascia of the pelvic floor and perineum, in situ Lithotomy position. Removed on left side: Superficial perineal (Colles’) fascia, inferior fascia of the pelvic diaphragm, and obturator fascia. Note: The green arrows are pointing forward to the anterior recess of the ischioanal fossa. Fig. 19.15 Gender-related differences in structure of the levator ani Posterior view. Note the connective tissue gaps between muscular parts of the levator ani in the female. A Male. B Female. Pelvis & Perineum 240 Pelvic Floor & Perineal Muscle Facts ① ③ ② ④ Anococcygeal lig. Obturator internus Iliococcygeus Coccygeus Piriformis Fig. 19.16 Muscles of the pelvic floor Superior view. Fig. 19.17 Muscles of the perineum Inferior view. Table 19.2 Muscles of the pelvic floor Muscle Origin Insertion Innervation Action Muscles of the pelvic diaphragm Levator ani ① Puborectalis Superior pubic ramus (both sides of pubic symphysis) Anococcygeal lig. Nerve to levator ani (S4), inferior rectal n. Pelvic diaphragm: Supports pelvic viscera ② Pubococcygeus Pubis (lateral to origin of puborectalis) Anococcygeal lig., coccyx ③ Iliococcygeus Internal obturator fascia of levator ani (tendinous arch) ④ Coccygeus Lateral surface of coccyx and S5 segment Ischial spine Direct branches from sacral plexus (S4–S5) Supports pelvic viscera, flexes coccyx Muscles of the pelvic wall (parietal muscles) Piriformis Sacrum (pelvic surface) Femur (apex of greater trochanter) Direct branches from sacral plexus (S1–S2) Hip joint: External rotation, stabilization, and abduction of flexed hip Obturator internus Obturator membrane and bony boundaries (inner surface) Femur (greater trochanter, medial surface) Direct branches from sacral plexus (L5–S1) Hip joint: External rotation and abduction of flexed hip The piriformis and obturator internus are considered muscles of the hip (see p. 426). The female and male external genitalia are shown on pp. 262–265. A Muscles of the pelvic diaphragm. A Superficial and deep perineal muscles in the male. B Outermost layer of the pelvic floor. B Superficial and deep perineal muscles in the female. Bulbospongiosus Ischiocavernosus Compressor urethrae Superficial transverse perineal Urethrovaginal sphincter Deep transverse perineal Perineal membrane Bulbospongiosus Ischiocavernosus External urethral sphincter Deep transverse perineal Superficial transverse perineal External anal sphincter Perineal membrane ② ① ③ ⑥ ② ① ③ ⑥ 19 Bones, Ligaments & Muscles 241 Table 19.3 Muscles of the perineum Muscle Origin Insertion Innervation Action ① Ischiocavernosus Ischial ramus Crus of clitoris or penis Pudendal n. (S2–S4) Maintains erection by squeezing blood into corpus cavernosum of clitoris or penis ② Bulbospongiosus Runs anteriorly from perineal body to clitoris (females) or penile raphe (males) Females: Compresses greater vestibular gland Males: Assists in erection ③ Superficial transverse perineal Ischiopubic ramus Perineal body Helps hold perineal body in median plane, holds the pelvic organs in place, and supports visceral canals through the muscles of the perineum ④ Deep transverse perineal Ishiopubic ramus Perineal body and external anal sphincter ⑤ External urethral sphincter Encircles urethra (division of deep transverse perineal muscle), in males ascends anteriorly to neck of the bladder; in females, some fibers surround the vagina as the urethrovaginal sphincter, others extend laterally as the compressor urethrae (See Figs. 21.9 and 21.11) Closes urethra ⑥ External anal sphincter Encircles anus (runs posteriorly from perineal body to anococcygeal lig.) Closes anus Typically, this muscle is not developed in females and is replaced by smooth muscle tissue. When developed, it provides dynamic support to the pelvic organs. A Muscles of the superficial pouch in the male. A Muscles of the superficial pouch in the female. ⑤ ④ ⑤ ④ Compressor urethrae Urethrovaginal sphincter B Muscles of the deep pouch in the male. B Muscles of the deep pouch in the female. Fig. 19.18 Muscles of the male perineum Fig. 19.19 Muscles of the female perineum Visceral pelvic fascia on rectum Visceral pelvic fascia on bladder Inferior pubic ramus Superior pubic ramus Rectus abdominis Parietal peritoneum Right common iliac a. and v. L5 vertebra Right ductus deferens Rectovesical pouch Visceral peritoneum on rectum Rectum Levator ani Anus Perineal body Prostate Rectoprostatic fascia Right seminal gland Right ureter Urinary bladder Visceral peritoneum on bladder Sigmoid colon Sigmoid mesocolon Tenia coli External anal sphincter Ischiocavernosus Bulbospongiosus Pelvis & Perineum 242 20 Spaces Contents of the Pelvis Fig. 20.1 Male pelvis Parasagittal section, viewed from the right side. Round lig. of uterus Visceral pelvic fascia on rectum Inferior pubic ramus Superior pubic ramus Visceral pelvic fascia on bladder Lig. of ovary L5 vertebra Rectouterine pouch Visceral peritoneum on rectum Rectum Levator ani Anus Vagina Perineal body Right ureter Urinary bladder Vesico-uterine pouch Visceral peritoneum on bladder Uterine tube Uterus Sigmoid colon Sigmoid mesocolon Tenia coli Right common iliac a. and v. External anal sphincter Ischiocavernosus 20 Spaces 243 Fig. 20.2 Female pelvis Parasagittal section, viewed from the right side. Medial umbilical fold (with obliterated umbilical a.) Cecum Round lig. of uterus Parietal peritoneum Vesicouterine pouch Urinary bladder Median umbilical fold (with obliterated urachus) Supravesical fossa Rectus abdominis Transverse vesical fold Lateral umbilical fold (with inferior epigastric a. and v.) Fundus of uterus Deep inguinal ring Paravesical fossa Lig. of ovary Sigmoid colon Uterine tube Left ovary Suspensory lig. of ovary Rectouterine fold Rectouterine pouch Rectum Broad lig. of uterus Rectum Fundus of uterus Cardinal lig. Obturator fascia Superior and inferior fascia of pelvic diaphragm Vagina Superficial perineal fascia Perineal membrane Deep transverse perineal Ischioanal fossa, anterior recess Levator ani Obturator internus External iliac a. and v. Peritoneum, parietal layer Perineal body Vesicouterine pouch Urinary bladder Retropubic space Rectovaginal septum Retrorectal (presacral) space Rectum Rectouterine pouch Uterus Sigmoid colon Pelvis & Perineum 244 Peritoneal Relationships Fig. 20.3 Peritoneal relationships in the pelvis: Female Superior view. A Lesser pelvis, anterosuperior view. Retracted: Small intestine loops and colon (portions). B Muscles (red) of the pelvic floor. Coronal section, anterior view. C Peritoneal and subperitoneal spaces (green) in the pelvis. Midsagittal section, viewed from the left side. Lateral umbilical fold (with inferior epigastric a. and v.) Lateral inguinal fossa Cecum Parietal peritoneum Ductus deferens Vermiform appendix Transverse vesical fold Rectovesical pouch Rectum Ileum Rectus abdominis Urinary bladder Median umbilical fold (with obliterated urachus) Medial umbilical fold (with obliterated umbilical a.) Sigmoid colon Obturator internus Levator ani Ischioanal fossa, anterior recess External urethral sphincter Perineal membrane Bulb of penis Crus of penis Inferior pubic ramus Paravesical space Peritoneum, parietal layer Urinary bladder Prostate Superior and inferior fascia of pelvic diaphragm Retrorectal (presacral) space Rectum Rectovesical pouch Sigmoid colon Perineal body Urinary bladder Retropubic space Rectovesical septum 20 Spaces 245 Fig. 20.4 Peritoneal relationships in the pelvis: Male Superior view. A Lesser pelvis, anterosuperior view. Retracted: Small intestine and colon (portions). B Muscles (red) of the pelvic floor. Coronal section, anterior view. C Peritoneal and subperitoneal spaces (green) in the pelvis. Midsagittal section, viewed from the left side. Peritoneal cavity Subperitoneal space Ischioanal fossa Visceral pelvic fascia Parietal pelvic fascia Vestibular bulb and bulbospongiosus Obturator internus Crus of clitoris and ischio-cavernosus Superficial perineal (Colles’) fascia Uterus Peritoneum Vagina Perineal membrane Internal pudendal a. and v., pudendal n. Skin Vestibule of vagina Pelvic diaphragm (with fascia) Inferior pubic ramus Deep pouch Superficial pouch Superficial perineal (Colles’) fascia Bulb of penis Inferior fascia of pelvic diaphragm Urinary bladder Prostate Levator ani Obturator internus Crus of penis and ischio-cavernosus Peritoneum Bulbo-spongiosus Superior fascia of pelvic diaphragm Perineal membrane Urethra, spongy part Deep pouch Superficial pouch Pelvis & Perineum 246 Pelvis & Perineum Fig. 20.5 Pelvis and urogenital triangle A Female. Oblique section. B Male. Coronal section. Table 20.1 Divisions of the pelvis and perineum The levels of the pelvis are determined by bony landmarks (iliac alae and pelvic inlet/brim). The contents of the perineum are separated from the true pelvis by the pelvic diaphragm and two fascial layers. Iliac crest Pelvis False pelvis • Ileum (coils) • Cecum and appendix • Sigmoid colon • Common and external iliac aa. and vv. • Lumbar plexus (brs.) Pelvic inlet True pelvis • Distal ureters • Urinary bladder • Rectum ♀: Vagina, uterus, uterine tubes, and ovaries ♂: Ductus deferens, seminal gland, and prostate • Internal iliac a. and v. and brs. • Sacral plexus • Inferior hypogastric plexus Pelvic diaphragm (levator ani & coccygeus) Perineum Deep pouch • Sphincter urethrae and deep transverse perineal mm. • Urethra (membranous) • Vagina • Rectum • Bulbourethral gland • Ischioanal fossa • Internal pudendal a. and v., pudendal n. and brs. Perineal membrane Superficial pouch • Ischiocavernosus, bulbospongiosus, and superficial transverse perineal mm. • Urethra (penile) • Clitoris and penis • Internal pudendal a. and v., pudendal n. and branches Superficial perineal (Colles’) fascia Subcutaneous perineal space • Fat Skin The pelvis is the region of the body inferior to the abdomen, surrounded by the pelvic girdle. The false, or greater, pelvis is immediately inferior to the abdominal cavity, between the iliac alae, and superior to the pelvic inlet. The true, or lesser, pelvis is found between the pelvic inlet and the pelvic outlet and extends inferiorly to the pelvic diaphragm (levator ani and coccygeus ), a muscular sling attached to the boundaries of the pelvic outlet. The perineum is the inferior most portion of the trunk, be- tween the thighs and buttocks, extending from the pubis to the coccyx and superiorly to the pelvic diaphragm. The superficial perineal pouch lies between the membranous layer of the subcutaneous tissue (Colles’ fascia) and the perineal membrane. The deep perineal pouch lies between the perineal membrane and the inferior fascia of the pelvic diaphragm. Rectum Ovary Suspensory lig. of ovary Iliacus Uterine tube Cardinal (transverse cervical) lig. Obturator internus Ischioanal fossa, anterior recess Levator ani Deep transverse perineal Superficial perineal (Colles’) fascia Vestibule of vagina Vestibular bulb (with bulbospongiosus) Crus of clitoris (with ischiocavernosus) Inferior pubic ramus Vagina Paravaginal tissue (fascia) Cervix of uterus Round lig. of uterus External iliac a. and v. Fundus of uterus Right ureter Right ureter Subcutaneous perineal space Superficial perineal (Colles’) fascia Urinary bladder Quadratus femoris Crus of penis (with ischio-cavernosus) Seminal colliculus Urethra, membranous part Bulb of penis (with bulbospongiosus) Prostate Deep transverse perineal Adductor mm. Obturator externus Inferior pubic ramus Levator ani Obturator internus Femur, head Gluteus minimus Ureteral orifice Internal urethral orifice Venous plexus Paravesical fossa 20 Spaces 247 Fig. 20.6 Pelvis: Oblique section Anterior view. A Female. Oblique section. B Male. Coronal section. Rectum Sigmoid colon RLQ LLQ Ischium Perineal flexure Sacral flexure Sacrum Ilium Rectum Pubis Coccyx Pubococcygeus Perineal flexure Puborectalis Pubis Tenia coli Transverse rectal fold Ischioanal fossa Parietal peritoneum Obturator internus Levator ani (pelvic diaphragm) External anal sphincter Anal canal Internal anal sphincter Perineal n. Internal pudendal a. and v. Pudendal n. Rectouterine (uterosacral) fold Rectum Sigmoid colon Sigmoid mesocolon Ureter External iliac a. and v. Superior and inferior fascia of pelvic diaphragm Pelvis & Perineum 248 21 Internal Organs Rectum & Anal Canal Fig. 21.1 Rectum: Location A Anterior view. B Left anterolateral view. Fig. 21.2 Closure of the rectum Left lateral view. The puborectalis acts as a muscular sling that kinks the anorectal junc-tion. It functions in the maintenance of fecal continence. Fig. 21.3 Rectum in situ Coronal section, anterior view of the female pelvis. The upper third of the rectum is covered with visceral peritoneum on its anterior and lateral sides. The middle third is covered only anteriorly and the lower third is inferior to the parietal peritoneum. ① Anorectal junction Anocuta-neous line Pectinate line ② ③ ④ ⑤ Anal canal 21 Internal Organs 249 Table 21.1 Regions of the rectum and anal canal Region Epithelium ① Rectum Colon-like with crypts; simple columnar with goblet cells Anal canal ② Columnar zone Stratified squamous, nonkeratinized ③ Anal pecten ④ Cutaneous zone Stratified squamous, keratinized with sebaceous glands ⑤ Perianal skin (pigmented) Stratified squamous, keratinized with sebaceous glands, hairs, and sweat glands Anal pecten n i k s l a n a i r e P ) e n o z e t i h w ( Anus Anorectal junction Rectal ampulla Middle transverse rectal fold Parietal peritoneum Superior fascia of pelvic diaphragm Levator ani Inferior fascia of pelvic diaphragm Deep part Superficial part Subcutaneous part Anal sinuses Anocutaneous line Anal columns Subcutaneous venous plexus Corrugator cutis ani Internal anal sphincter Hemorrhoidal plexus Inferior transverse rectal fold Longitudinal layer Circular layer Superior transverse rectal fold Peritoneal covering of rectum Muscularis externa External anal sphincter Anal canal Anal valves Fig. 21.4 Rectum and anal canal Coronal section, anterior view with the anterior wall removed. Pubic symphysis Right suprarenal gland and v. Right kidney Perirenal fat capsule Superior mesenteric a. Inferior mesenteric a. Right testicular a. and v. Right common iliac a. Median sacral a. and v. Right internal iliac a. and v. Sacral plexus Right ductus deferens Urinary bladder Median umbilical lig. Rectum Anterior trunk of internal iliac a. and v. Inferior epigastric a. and v. Left external iliac a. and v. Left superior gluteal a. Left internal iliac a. and v. Iliacus Psoas major Ureter, abdominal part Left renal a. and v. Left kidney Left inferior suprarenal a. Left middle suprarenal a. Left suprarenal gland and v. Left superior suprarenal a. Left inferior phrenic a. and v. Celiac trunk Abdominal aorta Inferior vena cava Left testicular a. and v. Ureter, pelvic part Ureteropelvic junction Ureterovesical junction Pelvis & Perineum 250 Ureters Fig. 21.5 Ureters in situ Anterior view, male abdomen. Removed: Nonurinary organs and rectal stump. The ureters descend along the posterior abdominal wall in the retroperitoneal space. On each side, they enter the pelvis after crossing the common iliac artery at its bifurcation into the external and internal arteries. Pubovesical muscles Left ureter Left ductus deferens Rectum with peritoneal covering on anterior wall Right ductus deferens Tendinous arch of pelvic fascia Right ureter Pelvic diaphragm, superior fascia Tendinous arch of levator ani Pubic symphysis Bladder, apex Pubis Inferior (arcuate) pubic lig. Bladder, body Median umbilical lig. 21 Internal Organs 251 Fig. 21.7 Ureter in the female pelvis Superior view. The pelvic ureters pass under the uterine artery approximately 2 cm lateral to the cervix. Fig. 21.6 Ureter in the male pelvis Superior view with peritoneum removed. Passage of right ureter through broad lig. of uterus Passage of left ureter through broad lig. of uterus Median umbilical lig. Vesicouterine pouch Parietal peritoneum Left external iliac a. and v. Left uterine tube Left ovary Left ovarian a. and v. in ovarian suspensory lig. Left broad lig. of uterus Left ureter Uterosacral fold (with uterosacral lig.) Sacral promontory Rectum Right ureter Rectouterine pouch Round lig. of uterus Uterus, posterior surface Uterus, fundus Medial umbilical fold (occluded part of umbilical a.) Bladder, body Pubis Transverse vesical fold Pubis symphysis Clinical box 21.1 There are three normal anatomical constrictions where a pain-causing kidney stone from the renal pelvis is apt to become lodged: • Narrowing at the origin of the ureter from the renal pelvis (ureteropelvic junction) • Site where the ureter crosses over the external or common iliac vessels • Passage of the ureter through the bladder wall (ureterovesical junction). Occasionally a fourth constriction can be identified where the testicular or ovarian artery and vein pass anterior to the ureter. Anatomical constrictions of the ureter First constriction: narrowing of the ureter as it passes over inferior renal pole (abdominal part) Possible constriction where the testicular or ovarian vessels pass anterior to the ureter Second constriction: ureter crosses over external iliac vessels (pelvic part) Third constriction: ureter traverses the bladder wall (intramural part) Orifices of urethral glands Urinary bladder, neck Ureteral orifice Right ureter, intramural part Submucosa Muscularis Mucosa with longitudinal folds Internal urethral orifice with bladder uvula Adventitia with visceral pelvic fascia Detrusor m. Mucosa Urinary bladder, trigone Interureteral fold Urinary bladder, body Urethra Anterior vaginal fornix Suspensory lig. of ovary (with ovarian a. and v.) Levator ani External anal sphincter Right ovary and lig. of ovary Right uterine tube Right external iliac a. and v. Rectus abdominis Round lig. of uterus Fundus of uterus Body of uterus Urinary bladder Pubic symphysis Clitoris Urethra Vagina Rectum Posterior vaginal fornix Cervix of uterus Right ureter L5 vertebra Left common iliac a. and v. Perineal membrane External urethral orifice Median umbilical lig. Urogenital peritoneum Visceral pelvic fascia Left ureter Fundus of bladder Neck of bladder Apex of bladder Body of bladder Female urethra Urinary bladder External urethral sphincter Urethra Left ureter Vagina Compressor urethrae Urethrovaginal sphincter Pelvis & Perineum 252 Urinary Bladder & Urethra C Trigone and urethra, coronal section, anterior view. B Bladder and urethra, left lateral view. Fig. 21.8 Female urinary bladder and urethra A Midsagittal section of pelvis, viewed from the left side. Right hemipelvis. Fig. 21.9 Urethral sphincter mechanism in the female Anterolateral view. Penile fascia Rectovesical septum Urinary bladder Pubic symphysis Retropubic space Suspensory lig. of penis Penis, corpus cavernosum Penis, corpus spongiosum Scrotal septum Bulbospongiosus Bulbourethral gland External urethral sphincter Prostate Ductus deferens, ampulla Rectum Rectovesical pouch Prepuce Urethra, spongy part Ejaculatory duct 21 Internal Organs 253 B Bladder, urethra and prostate, left lateral view. Fig. 21.10 Male urinary bladder and urethra A Midsagittal section of pelvis, viewed from left side. Right hemipelvis. Median umbilical lig. Urogenital peritoneum Ampulla of ductus deferens Prostate Male urethra Visceral pelvic fascia Apex of bladder Body of bladder Left ureter Fundus of bladder Prostatic urethra Neck of bladder, internal urethral orifice Fundus of bladder, trigone Openings of ejaculatory ducts Prostatic utricle Seminal colliculus Prostate Internal urethral sphincter Detrusor muscle Ureteral orifice Interureteric crest Internal urethral sphincter Dilator urethrae Prostate External urethral sphincter Ejaculatory ducts Prostatic urethra Bulb of penis C Trigone, urethra and prostate, coronal section, anterior view. Fig. 21.11 Urethral sphincter mechanism in the male Lateral view. Labia minora Vestibule Greater vestibular (Bartholin’s) gland Vestibular bulb Glans and crus of clitoris Vagina Uterus Uterine tube Ovary Cervix of uterus Ureteral orifice Suspensory lig. of ovary Greater vestibular (Bartholin's) gland Vagina Uterus Right ureter Right kidney Median umbilical lig. Right uterine tube Right ovary Round lig. of uterus Urinary bladder Clitoris Urethra Labium majus Labium minus Pelvis & Perineum 254 Overview of the Genital Organs The genital organs can be classified topographically (external versus internal) and functionally (Tables 21.2 and 21.3). Fig. 21.12 Female genital organs A Internal and external genitalia. B Urogenital system. Note: The female urinary and genital tracts are functionally separate, though topographically close. Table 21.2 Female genital organs Organ Function Internal genitalia Ovary Germ cell and hormone production Uterine tube Site of conception and transport organ for zygote Uterus Organ of incubation and parturition Vagina (upper portion) Organ of copulation and parturition External genitalia Vulva Vagina (vestibule) Labia majora and minora Accessory copulatory organ Clitoris Greater and lesser vestibular glands Production of mucoid secretions Mons pubis Protection of the pubic bone Urinary bladder Ureter Ductus deferens Inguinal canal Ejaculatory duct Bulbourethral gland Penis Testis Epididymis Urethra Seminal gland Prostate Deep transverse perineal Excretory duct Ductus deferens, ampulla Ductus deferens Ureteral orifice Ejaculatory duct Bulbourethral gland Prostate Seminal gland Right ureter Right kidney Median umbilical lig. Glans of penis, corpus spongiosum Ductus deferens Urinary bladder Urethra, spongy part Scrotum Testis Epididymis Bulb of penis, corpus spongiosum Penis, corpus cavernosum Penis, corpus spongiosum 21 Internal Organs 255 Fig. 21.13 Male genital organs B Urogenital system. Note: The male urethra serves as a common urinary and genital passage. A Seminiferous structures. Table 21.3 Male genital organs Organ Function Internal genitalia Testis Germ cell and hormone production Epididymis Storage reservoir for sperm Ductus deferens Transport organ for sperm Accessory sex glands Prostate Production of secretions (semen) Seminal glands Bulbourethral gland External genitalia Penis Copulatory and urinary organ Urethra Conduit for urine and semen Scrotum Protection of testis Coverings of the testis Mesosalpinx Germinal epithelial covering Ovary Mesovarium Uterine tube Peritoneal covering Mesometrium Endometrium Myometrium Longitudinal body axis Longitudinal cervical axis (in cervical canal ) Longitudinal uterine axis (in uterine cavity) Vesicouterine pouch Anterior vaginal fornix Longitudinal vaginal axis Posterior vaginal fornix Rectouterine pouch Body of uterus Visceral peritoneum Fundus of uterus A S Uterus, posterior surface Proper ovarian lig. Uterine pole Follicular stigma (bulge from Graafian follicle) Mesometrium Mesovarium Mesovarial margin Uterine tube Vascular pole Ovarian suspensory lig. Ovarian a. and v. Medial surface Free margin Pelvis & Perineum 256 Uterus & Ovaries Fig. 21.14 The broad ligament Regions of the broad ligament, sagittal section. The uterus and ovaries are suspended by the broad ligament of the uterus, which is composed of a double layer of peritoneum, arranged as a combination of mesenteries: the mesosalpinx, meso varium, and mesometrium. Fig. 21.15 Ovary Right ovary, posterior view. Fig. 21.16 Normal curvature and posi-tion of the uterus Midsagittal section, left lateral view. The position of the uterus can be described in terms of: ① Flexion, the angle between the longitudinal cervical axis and the longitudinal uterine axis; the normal position is anteflexion. ② Version, the angle between the longitudinal cervical axis and longitudinal vaginal axis; the normal position is anteversion. Uterine isthmus Supravaginal part Uterine cervix Vaginal part Infundibulum, uterine tube Epoöphoron Lig. of ovary Mesosalpinx (with tubal brs. of uterine a. and v.) Left uterine tube Cervix of uterus Vagina Uterosacral lig. (in rectouterine fold) Right ureter Mesometrium Vesicular appendices Ovarian a. and v. (in suspensory lig. of ovary) Fimbriae at abdominal ostium Body of uterus Fundus of uterus Left ovary Isthmus Ampulla Embryonic remnants Uterine part Left uterine tube Cavity of uterus Ampulla Infundibulum Lig. of ovary Mesosalpinx Mesovarium Myometrium Cervical canal External os Vagina, anterior wall Vaginal part Vaginal fornix, lateral part Supravaginal part Internal os (at uterine isthmus) Fimbriae at abdominal ostium Uterine ostium Fundus of uterus Vascular pole Uterine pole Cervix of uterus Endometrium Right ovary Isthmus 21 Internal Organs 257 A Posterosuperior view. Fig. 21.17 Uterus and uterine tube B Coronal section, posterior view with uterus straightened. Removed: Mesome-trium. Clinical box 21.2 After fertilization in the ampulla of the uterine tube, the ovum usually implants in the wall of the uterine cavity. However, it may become implanted at other sites (e.g., the uterine tube or even the peritoneal cavity). Tubal pregnancies, the most common type of ectopic pregnancy, pose the risk of tubal wall rupture and potentially life-threatening bleeding into the peritoneal cavity. Tubal pregnancies are promoted by adhesion of the tubal mucosa, mostly due to inflammation. Ectopic pregnancy Piriformis Rectum Fascia over obturator internus Tendinous arch of levator ani Tendinous arch of pelvic fascia Levator ani Lateral lig. of the bladder Pubovesical lig. Uterosacral lig. Cardinal (transverse cervical) lig. Cervix Paracolpium Fascia over anterior vagina Obturator canal Urethra Pubic symphysis Pelvis & Perineum 258 Ligaments & Fascia of the Deep Pelvis Ilium Rectum Uterus Lig. of ovary Uterine tube Urinary bladder Pubic symphysis Pubis Pubovesical lig. Vesicouterine lig. Round lig. of uterus Cardinal (transverse cervical) lig. Rectouterine lig. Uterosacral lig. Sacrum Fig. 21.18 Ligaments of the female pelvis Superior view. Removed: Perito-neum, neurovasculature, and superior portion of the bladder to demonstrate only the fascial con-densations (ligaments). Deep pelvic ligaments support the uterus within the pelvic cavity and prevent uterine prolapse, the downward displace-ment of the uterus into the vagina. Fig. 21.19 Ligaments of the deep pelvis in the female Superior view. Removed: perito-neum, neurovasculature, uterus and bladder. Uterosacral ligaments and the paracolpium support and help maintain the positions of the cervix and vagina in the pelvis. 21 Internal Organs 259 Cardinal (transverse cervical) lig. Ureter Obturator a. Uterine a. Internal iliac a. and v. Mesorectal space Rectum Presacral space Retrorectal space Retroperitoneal space Psoas major muscle Ovarian a. and v. Parietal pelvic fascia Visceral pelvic fascia Neurovascular bundle of the rectum (with the middle rectal a.) Inferior vesical a. Rectovaginal space Superior vesical a. in lateral vesicular lig. Cervix Vesicovaginal space Urinary bladder Retropubic space Tendinous arch of levator ani Inferior epigastric a. and v. Tendinous arch of pelvic fascia Medial umbilical lig. Pubovesical lig. Uterosacral lig. Fig. 21.20 Fascia and ligaments of the female pelvis Transverse section, through cervix, superior view. Fascia of the pelvis plays an important role in the support of pelvic viscera. On either side of the pelvic floor, where the visceral fascia of the pelvic organs is continuous with the parietal fascia of the muscular walls, thickenings called tendinous arches of the pelvic fascia are formed. In females, the paracolpium—lateral connections between the visceral fascia and the tendinous arches—suspends and supports the vagina. Pubovesical ligaments (and puboprostatic ligaments in the male) are extensions of the tendinous arches that support the bladder and prostate. Endopelvic fascia, a loose areolar (fatty) tissue that fills the spaces between pelvic viscera, condenses to form “ligaments” (cardinal, lateral visceral, and lateral rectal ligaments; see Fig. 21.20) that provide passage for the ureters and neurovascular elements within the pelvis. Rectovaginal septum Vesicouterine pouch Urinary bladder Vagina, anterior wall Urethra Vesicovaginal septum (clinical term) Vaginal orifice External urethral orifice Vaginal vestibule with labium minus Deep transverse perineal Vagina, posterior wall Rectum Anterior part Cervix of uterus, vaginal part Posterior part Rectouterine pouch Visceral peritoneum on uterus Body of uterus Cervix of uterus, supravaginal part Vaginal fornix Urethrovaginal sphincter Anterior vaginal column Posterior lip of uterine os Anterior lip of uterine os Vagina, anterior wall Urethral carina External urethral orifice Clitoris Labium minor Vaginal rugae Uterine os Cervix of uterus, supravaginal part Vaginal vestibule Vaginal orifice Fig 21.22 Relationship of the vagina to the peritoneum and pelvic organs Midsagittal section, left lateral view. The vagina lies almost completely in the subperitoneal space. However, drainage of peritoneal fluid or pus from an abcess in the rectouterine space, a procedure known as culdo-centecis, can be achieved through an incision in the posterior fornix. Pelvic peritoneal cavity Uterine fundus Uterine body Rectum Peritoneum on anterior wall of rectum Uterine cervix Rectouterine pouch Vagina Parietal peritoneum Visceral peritoneum on the uterus Visceral peritoneum on the bladder Symphysis Bladder Vesicouterine pouch Pelvis & Perineum 260 Vagina Fig. 21.21 Location of vagina Midsagittal section, left lateral view. Fig. 21.23 Structure of vagina Posteriorly angled coronal section, posterior view. Transverse perineal lig. Inferior pubic ramus Dorsal clitoral a. and n. Perineal membrane Vagina Ischiocavernosus Female urethra Deep dorsal clitoral v. Pubic symphysis Round lig. of uterus Perineal a. A. of vestibular bulb Rectum Suspensory lig. of ovary Parietal peritoneum Cardinal (transverse cervical) lig. (with sections of the uterine a. and uterine venous plexus) Cervix with uterine os Obturator internus (with obturator fascia) Levator ani (with superior and inferior fascia of pelvic diaphragm) Ischiopubic ramus Crus of clitoris with ischiocavernosus Deep transverse perineal Superficial perineal (Colles’) fascia Vestibular bulb with bulbospongiosus Vestibule of vagina (with vaginal orifice) Labium minus Labium majus Vagina, posterior wall with vaginal rugae Vaginal arterial branches and venous plexus Pelvic retro-peritoneal space Round lig. of uterus Fundus of the uterus Left uterine tube Left ovary Iliacus Sigmoid colon Ilium Right external iliac a. and v. Internal iliac a. and v. Perineal membrane Urethovaginal sphincter 21 Internal Organs 261 Fig. 21.24 Female genital organs: Coronal section Anterior view. The vagina is both pelvic and perineal in location. It is also retroperitoneal. Fig. 21.25 Vagina: Location in the perineum Inferior view. Anterior labial commissure Labia minora Posterior labial commissure Perineal raphe Anus Opening of greater vestibular (Bartholin’s) glands Vaginal orifice Labia majora External urethral orifice Prepuce of clitoris Mons pubis Clitoris, glans Frenulum of clitoris Labia minora External urethral orifice Bulbospongiosus Bulb of vestibule Greater vestibular (Bartholin’s) gland Vestibule of vagina (space between labia minora) Vaginal orifice Symphysis Pubic tubercle Pars intermedia Crus of clitoris Urethral orifice Pubic ramus Vaginal orifice Body of clitoris (corpora cavernosa) Glans clitoris Corpus cavernosum Bulb of vestibule Greater vestibular gland Pelvis & Perineum 262 Female External Genitalia Fig. 21.26 Female external genitalia Lithotomy position with labia minora separated. Fig. 21.27 Vestibule and vestibular glands Lithotomy position with labia minora separated. Fig. 21.28 Erectile tissue in the female perineum Ischiocavernosus Prepuce of clitoris Bulbo-spongiosus Bulb of vestibule Ischiocavernosus Crus of clitoris Glans of clitoris Body of clitoris Superficial transverse perineal Perineal membrane Levator ani Greater ves-tibular gland Ischial tuberosity 21 Internal Organs 263 Fig. 21.29 Erectile tissue and muscles of the female Lithotomy position. Removed: Labia and skin. Removed from left side: Ischiocavernosus and bulbospongiosus muscles. Posterior commissure External anal sphincter Levator ani Superficial transverse perineal Ischio-cavernosus Bulbospongiosus Lateral episiotomy Mediolateral episiotomy Anus Perineum Midline episiotomy Clinical box 21.3 Episiotomy is a common obstetric procedure used to enlarge the birth canal during the expulsive stage of labor. The procedure is generally used to expedite the delivery of a baby at risk for hypoxia during the expulsive stage. Alternately, if the perineal skin turns white (indicating diminished blood flow), there is imminent danger of perineal laceration, and an episiotomy is often performed. More lateral incisions gain more room, but they are more difficult to repair. Episiotomy A Types of episiotomy. B Mediolateral episiotomy at height of contraction. C Pelvic floor with crowning of fetal head. Corona of glans Corpus spongiosum Corpus cavernosum Superior pubic ramus Ischiopubic ramus Ischiocavernosus Perineal membrane Bulbospongiosus Bulb of penis Deep transverse perineal Crus of penis Plane of section in C Plane of section in D Root of penis Body of penis Glans of penis Obturator foramen Prostatic ductules Brs. of deep penile a. Glans of penis Orifices of urethral glands External urethral orifice, urethral crest Urethra, prostatic part Seminal colliculus Navicular fossa Bulbourethral gland Urethra, membranous part Prostate Urethra, preprostatic part Urinary bladder Crus of penis Urethra, spongy part Urethral ampulla Corpus spongiosum Corpus cavernosum Pubic symphysis Corpus cavernosum Bulb of penis, corpus spongiosum Urethra, spongy part Bulbo-spongiosus Urethral a. Deep penile a. Dorsal penile a. and n. Deep dorsal penile v. Pelvis & Perineum 264 Penis, Testis & Epididymis A Inferior view. B Longitudinal section. C Cross section through the root of the penis. Fig. 21.30 Penis Superficial dorsal penile v. Dorsal penile a. and n. Deep dorsal penile v. Corpus cavernosum Corpus spongiosum Urethral a. Urethra, spongy part Penile septum Deep penile a. Tunica albuginea of corpus cavernosum Deep penile fascia Superficial penile fascia Penile skin Tunica albuginea of corpus spongiosum Tunica vaginalis, visceral layer (on testis) Scrotum Pampiniform plexus (testicular vv.) Cremaster m. and cremasteric fascia Tunica dartos Superficial fascia, deep layer Testicular a. Tunica vaginalis, parietal layer Glans of penis Epididymis, head Internal spermatic fascia Epididymis, body External spermatic fascia Rete testis in mediastinum testis Epididymis, head Efferent ductules Tunica albuginea Septum Lobule Epididymis, tail Ductus deferens Epididymis, body Pampiniform plexus (testicular vv.) Testicular a. Epididymis, appendix Testis, appendix Ductus deferens Epididymis, tail Epididymis, body Epididymis, head 21 Internal Organs 265 Fig. 21.31 Testis and epididymis Left lateral view. A Testis and epididymis in situ. B Surface anatomy of the testis and epididymis. C Sagittal section of the testis and epididymis. D Cross section through the body of the penis. The accessory male sex glands consist of the seminal, prostate, and bulbourethral glands, which contribute fluid to the ejaculate that provides nourishment for the spermatozoa as well as neutralizes the pH of the male urethra and the vaginal environment. Urinary bladder Ureter Ductus deferens, ampulla Urethra Seminal gland Prostate Bulbourethral glands Prostate Seminal colliculus Membranous part Deep transverse perineal Bulbourethral gland Prostatic part Prostatic capsule Neck of bladder Spongy part Urethra Base Apex Ejaculatory duct orifices Left lobe Right lobe Prostatic isthmus Urethra Prostatic capsule Urethra Ejaculatory duct Ductus deferens Seminal glands Neck of bladder A Coronal section, anterior view. B Sagittal section, left lateral view. C Transverse section, superior view. Pelvis & Perineum 266 Male Accessory Sex Glands Fig. 21.32 Accessory sex glands Posterior view. The ducts of the seminal gland and ductus deferens combine to form the ejaculatory duct. A Prostate and seminal glands. B Coronal section, anterior view. Seminal colliculus Bulbourethral gland Deep transverse perineal Prostatic urethra Bladder neck Plane of section in D Fig. 21.33 Anatomic divisions of the prostate Peripheral zone Central zone Anterior zone Transition zone Periurethral zone Urethra Ejaculatory ducts C Sagittal section, left lateral view. D Transverse section, superior view. Fig. 21.34 Clinical divisions of the prostate Urinary bladder, fundus Urinary bladder, neck Urinary bladder, body Urinary bladder, apex Superficial abdominal fascia, deep layer Pubic symphysis Retropubic space Superficial dorsal penile v. Urethra, spongy part Superficial and deep penile fascia Penis, corpus cavernosum Penis, corpus spongiosum Glans of penis Prepuce Urethra, navicular fossa Scrotal septum Scrotum Bulbospongiosus Bulbourethral gland Deep transverse perineal Rectoprostatic fascia Prostate Ejaculatory duct Rectum Seminal gland Rectovesical pouch Visceral peritoneum 21 Internal Organs 267 Fig. 21.35 Prostate in situ Sagittal section through the male pelvis, left lateral view. Rectum Rectovesical pouch Prostatic carcinoma, subcapsular Urinary bladder Clinical box 21.4 Prostatic carcinoma is one of the most common malignant tumors in older men, often growing at a subcapsular location (deep to the prostatic capsule) in the peripheral zone of the prostate. Unlike benign prostatic hyperplasia, which begins in the central part of the gland, prostatic carcinoma does not cause urinary outflow obstruction in its early stages. Being in the peripheral zone, the tumor is palpable as a firm mass through the anterior wall of the rectum during rectal examination. In certain prostate diseases, especially cancer, increased amounts of a protein, prostate-specific antigen or PSA, appear in the blood. This protein can be measured by a simple blood test. Prostatic carcinoma and hypertrophy A Most common site of prostatic carcinoma. B Prostatic carcinoma (arrows) with bladder infiltration. Pelvis & Perineum 268 22 Neurovasculature Overview of the Blood Supply to Pelvic Organs & Wall Fig 22.1 Branches of the right internal iliac artery Side wall of the male pelvis, left lateral view. The internal iliac artery arises from the common iliac artery. Its anterior trunk gives off visceral branches to pelvic organs and parietal branches to the pelvic wall. The posterior trunk gives off only parietal branches. Branches to the uterus and vagina in the female are the principal differences from the male vasculature. Right internal iliac a. L 5 vertebra Abdominal aorta Right common iliac a. Right external iliac a. Umbilical a., patent part Internal iliac a., anterior trunk Obturator n. Superior vesical a. Obturator br. of inferior epigastric a. Obturator a. Inferior vesical a. Middle rectal a. Obturator internus Pudendal n. Internal pudendal a. Coccygeus Sacral plexus Inferior gluteal a. Superior gluteal a. Lateral sacral a. Internal iliac a., posterior trunk Iliolumbar a. Median sacral a. Umbilical a., occluded part Inferior epigastric a. A. of ductus deferens ③ ② ① Sacrospinous lig. Sacrotuberous lig. ⑤ ④ ⑥ Inguinal lig. Obturator membrane Iliopectineal arch Piriformis Table 22.1 Neurovascular pathways in the pelvis There are six major neurovascular tracts on the pelvic walls, four of which () contain branches from the internal iliac artery. Tract Neurovascular structures transmitted Posterior ① Greater sciatic foramen, suprapiriform part (above the piriformis) Superior gluteal a. and v., superior gluteal n. ② Greater sciatic foramen, infrapiriform part (below the piriformis) Inferior gluteal a. and v., inferior gluteal n., sciatic n., internal pudendal a. and v., pudendal n., posterior femoral cutaneous n. On pelvic floor ③ Lesser sciatic foramen through pudendal canal Internal pudendal a. and v., pudendal n. Lateral ④ Obturator canal Obturator a. and v., obturator n. Anterior ⑤ Muscular lacuna (posterior to inguinal lig., lateral to iliopectineal arch) Femoral n., lateral femoral cutaneous n. ⑥ Vascular lacuna (posterior to inguinal lig., medial to iliopectineal arch) Femoral a. and v., lymphatic vessels (the femoral a. is a branch of the external iliac a.), femoral branch of genito- femoral n. 22 Neurovasculature 269 ④ ⑤ Superior vesical a. ⑥ Dorsal penile a. Posterior scrotal brs. Inferior rectal a. ⑧ ⑦ ⑨ A. of ductus deferens ③ ② ① Internal iliac a. External iliac a. ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ Inferior vena cava Right common iliac v. Right internal iliac v. Inferior rectal vv. Posterior scrotal vv. Vv. of penile bulb Deep penile vv. Deep dorsal penile v. ⑨ Right external iliac v. Inferior vena cava Uterine vv. ⑩ Right internal iliac v. Right external iliac v. ① ② ⑧ ⑦ ⑥ ⑤ ④ ③ A Male pelvis. B Female pelvis. A Male pelvis. B Female pelvis. Table 22.2 Branches of the internal iliac artery The internal iliac artery gives off five parietal (pelvic wall) and four visceral (pelvic organs) branches. Parietal branches are shown in italics. Branches ① liolumbar a. ② Superior gluteal a. ③ Lateral sacral a. ④ Umbilical a. A. of ductus deferens Superior vesical a. ⑤ Obturator a. ⑥ Inferior vesical a. ⑦ Middle rectal a. ⑧ Internal pudendal a. Inferior rectal a. Dorsal penile a. Posterior scrotal aa. ⑨ Inferior gluteal a. In the female pelvis, the origin of the uterine and vaginal arteries is highly variable. Table 22.3 Venous drainage of the pelvis Tributaries ① Superior gluteal v. ② Lateral sacral v. ③ Obturator vv. ④ Vesical vv. ⑤ Vesical venous plexus ⑥ Middle rectal vv. (rectal venous plexus) (also superior and inferior rectal vv., not shown) ⑦ Internal pudendal v. ⑧ Inferior gluteal vv. ⑨ Prostatic venous plexus ⑩ Uterine and vaginal venous plexus The male pelvis also contains veins draining the penis and scrotum. Abdominal aorta Left common iliac a. Right internal iliac a. Left ureter Uterine a. Vaginal a. Inferior rectal a. Coccygeus Piriformis Left internal iliac a. Left external iliac a. ⑤ ④ ② ⑨ ⑦ ⑧ ⑥ Median sacral a. Right inferior gluteal a. and v. Right lateral sacral v. Right external iliac a. and v. Seminal gland Prostate Perineal a. and v. Spermatic cord Abdominal aorta Inferior mesenteric a. Umbilical a. Right ureter Right obturator a. and v. Right ductus deferens and a. Dorsal penile a., deep dorsal penile v. Left superior and inferior vesical a. and v. Left internal pudendal a. and v. Left inferior rectal a. and v. Left middle rectal a. and v. Right middle rectal a. and v. Right inferior vesical a. and v. Right superior vesical a. and v. Superior rectal a. and v. (from/to inferior mesenteric a. and v.) Right iliolumbar a. Left common iliac a. and v. Right internal iliac a. and v. Left ureter Posterior scrotal a. and v. Internal pudendal a. and v. Fig 22.2 Blood vessels of the male pelvis Right hemipelvis, left lateral view. Pelvis & Perineum 270 Arteries & Veins of the Male Pelvis Left ductus deferens Right ductus deferens Psoas major Iliacus Sacral plexus Inguinal lig. Umbilical a. Inferior epigastric a. and v. Femoral a. and v. Pampiniform plexus (testicular vv.) Epididymis Testis Glans of penis Dorsal penile a., deep dorsal penile v. Dorsum of penis Suspensory lig. of penis Internal spermatic fascia Saphenous opening External pudendal a. and v. Right ureter External iliac a. and v. Internal iliac a. and v. Testicular a. and v. Urinary bladder Rectum Deep circumflex iliac a. and v. Fig 22.3 Blood vessels of the male genitalia Opened: Inguinal canal and coverings of the spermatic cord Pampiniform plexus (testicular vv.) Vv. of ductus deferens Cremasteric a. and v. A. of ductus deferens Testicular a. Fig 22.4 Blood vessels of the testis Left lateral view. Clinical box 22.1 The pampiniform plexus has an important cooling affect on the testis. Because drainage of the left testicular vein into the left renal vein is at a right angle, a physiological constriction may occur that can obstruct outflow from the testicular vein. This can result in enlargements, or “varicoceles,” of the left testicular vein and pampiniform plexus, which can disrupt the cooling function of the plexus and the fertility of the testis. Asymmetric venous drainage of the testes Inferior vena cava Testicular vv. (pampiniform plexus) Inguinal canal Left testicular v. Left renal v. 22 Neurovasculature 271 Pelvis & Perineum 272 Arteries & Veins of the Female Pelvis Median sacral a. Right vaginal a. Right inferior vesical a., vesical v. Left superior vesical a., vesical v. Left uterine a. and v. Left inferior vesical a., vesical v. Left ureter Right common iliac a. Right ovarian a. and v. (in ovarian suspensory lig.) Right external iliac a. and v. Right ureter Right umbilical a. Right superior vesical a. Right round lig. of uterus Right ovary and uterine tube Left internal pudendal a. and v. Vaginal venous plexus Left inferior rectal a. and v. Left middle rectal a. and v. Uterine venous plexus Right middle rectal a. and v. Right uterine a. and v. Right obturator a. and v. Superior rectal a. and v. Right iliolumbar a. Right internal iliac a. Internal iliac a. and v., anterior division Deep dorsal clitoral v. Perineal a. and v. Fig 22.5 Blood vessels of the female pelvis Right hemipelvis, left lateral view. Abdominal aorta Left external iliac a. and v. Inferior mesenteric a. Inferior vena cava Median sacral a. and v. Rectum Fundus of uterus Uterine tube Round lig. of uterus Urinary bladder Middle rectal a. Inferior vesical a. Uterine a. and v. Vaginal a. Superior vesical a., vesical v. Umbilical a., obliterated part Obturator a., v., and n. Umbilical a., patent part Ovary Uterine a., tubal br. Left internal iliac a. and v. Left common iliac a. and v. Left ureter Left ovarian a. and v. Mesometrium (of broad lig. of uterus) Fig 22.6 Blood vessels of the female genitalia Removed: peritoneum on left side; Retracted: uterus. Fig 22.7 Relationship of the uterine artery and ureter The uterine artery runs in the broad ligament to the uterus. The ureter passes inferior to the artery lateral to the cervix. Thus the ureter is at risk for injury during uterine surgery. 22 Neurovasculature 273 A Superior view of the pelvis. B Left lateral view of left ureter and left uterine artery. Visceral peritoneum on urinary bladder Visceral peritonium on posterior surface of uterus Round lig. of uterus Uterine tube Ovary Left ureter Broad lig. Rectouterine fold Rectum Inferior vescial a. Internal iliac a. and v. Right ureter Uterine a. External iliac a. and v. Obturator a. Superior vesical a. Urinary bladder Uterus, fundus Common iliac a. Rectum Vaginal br. Left uterine a. Left ureter Bladder Uterus Inferior mesenteric a. and v. Sigmoid aa. and vv. Superior rectal a. and v. Left obturator a. Left external iliac a. and v. Left inferior gluteal a. Left middle rectal a. Left internal pudendal a. Left inferior rectal a. Median sacral a. and v. Inferior vena cava Abdominal aorta Levator ani To portal v. Right inferior rectal v. Right internal pudendal v. Right inferior gluteal v. Right obturator v. Right internal iliac a. and v. Right superior gluteal a. and v. Right common iliac a. and v. Right middle rectal v. Rectal venous plexus Pelvis & Perineum 274 Arteries & Veins of the Rectum & External Genitalia Fig. 22.8 Blood vessels of the rectum Posterior view. The superior rectal arteries are the main blood supply to the rectum; the middle rectal arteries serve as an anastomosis between the superior and inferior rectal arteries. Similarly, the middle rectal veins provide an important portocaval collateral pathway between the superior and inferior rectal veins. Fig. 22.9 The hemorrhoidal plexus Longitudinal section of the anal canal with the hemorrhoidal plexus windowed. The hemorrhoidal plexus, supplied by branches of the superior rectal artery, is a permanently distended cavernous body that forms circular cushions in the area of the anal columns. When filled with blood, these cushions serve as an effective continence mechanism that ensures liquid and gas-tight closure. The sustained contraction of the muscular sphincter appa-ratus inhibits venous drainage, but when the sphincter relaxes during defecation, blood is allowed to drain via arteriovenous anatomoses to the inferior mesenteric vein and middle and inferior rectal veins. Hemorrhoidal plexus Muscularis mucosae of the anal canal Pectinate line Anal pecten Transsphincteric vv. Rectal vv. of external venous plexus Proctodeal gland External anal sphincter Puborectalis Superior rectal a. and v. 22 Neurovasculature 275 Superficial inguinal ring Femoral a. and v. Anterior scrotal a. and v. External spermatic fascia Dorsal penile a. and n. Superficial penile fascia Deep dorsal penile v. Deep penile fascia Superficial dorsal penile vv. Suspensory lig. of penis Ilioinguinal n. External pudendal a. and v. Fig. 22.10 Neurovasculature of the penis and scrotum A Anterior view. Partially removed: Skin and fascia. External pudendal a. and v. Superficial dorsal penile vv. Deep penile fascia Glans, penis Corona of glans Tunica albuginea Dorsal penile a. and n. Deep dorsal penile v. B Dorsal vasculature of the penis. Removed from left side: Deep penile fascia. Deep clitoral a. A. of vestibular bulb Posterior labial brs. Internal pudendal a. Inferior rectal a. Perineal a. Dorsal clitoral a. Vestibular bulb Superficial transverse perineal Deep dorsal clitoral v. Deep clitoral vv. V. of vestibular bulb Perineal vv. Inferior rectal vv. Internal pudendal v. Posterior labial vv. Venous plexus of vestibular bulb Crus of clitoris Fig. 22.11 Blood vessels of the female external genitalia Inferior view. A Arterial supply. B Venous drainage. Vertical group Horizontal group ③ ④ ② ⑤ ⑥ ⑧ ⑨ ① ⑦ ⑩ Common iliac a. Internal iliac a. and l.n. Superficial inguinal l.n. Superior rectal a. Inferior mesenteric a. and l.n. Abdominal aorta External iliac l.n. Internal iliac l.n. Common iliac l.n. Superficial and deep inguinal l.n. Pelvis & Perineum 276 Lymphatics of the Pelvis Table 22.4 Lymph nodes of the pelvis Lymph nodes of the pelvis are distributed along major blood vessels and anterior to the sacrum. Lymph from pelvic organs can drain to one or more of several groups of lymph nodes (inguinal, internal iliac, external iliac, sacral or common iliac) before passing to the preaortic or lateral aortic nodes. Lymph from the perineum may drain first to superficial or deep inguinal nodes before draining to the external iliac nodes. Note that the testes and ovaries drain directly to lateral aortic nodes. Preaortic l.n. ① Superior mesenteric l.n. ② Inferior mesenteric l.n. ③ Left lateral aortic l.n. ④ Right lateral aortic (caval) l.n. ⑤ Common iliac l.n. ⑥ Internal iliac l.n. ⑦ External iliac l.n. ⑧ Superficial inguinal l.n. Horizontal group Vertical group ⑨ Deep inguinal l.n. ⑩ Sacral l.n. Fig 22.12 Lymphatic drainage of the rectum Anterior view. Three zones of the rectum drain to different groups of lymph nodes. The upper zone drains to inferior mesenteric nodes. The middle zone and columnar part of the lower zone drains to internal iliac nodes. The cutaneous part of the lower zone drains to superficial inguinal nodes. Fig 22.13 Lymphatic drainage of the bladder and urethra Anterior view. Different parts of the bladder drain to inter-nal iliac or external iliac nodes or directly to the common iliac nodes. The urethra, as well as the penis in the male, is drained by superficial and deep inguinal nodes. Right lateral aortic (caval) l.n. Deep inguinal l.n. Common iliac l.n. Left lateral aortic l.n. Abdominal aorta Inferior vena cava Right common iliac a. Superficial inguinal l.n. (vertical group) External iliac l.n. Internal iliac l.n. Sacral l.n. Superficial inguinal l.n. (horizontal group) Lateral aortic l.n. Internal iliac l.n. External iliac l.n. Sacral l.n. 22 Neurovasculature 277 Fig 22.14 Lymphatic drainage of the male genitalia Male pelvis, anterior view. Male genitalia drain to the lumbar lymph nodes via several pathways: Testis and epididymis – drain via a direct pathway along the testicular vessels to the right and left lumbar lymph nodes. Some lymph from the epididymis may drain first to internal iliac nodes. Ductus deferens and seminal glands – drain to external iliac (primarily) and internal iliac nodes. Prostate – drain along multiple pathways including to external iliac, internal iliac, and sacral nodes Scrotum and coverings of the testes – drain to superficial inguinal nodes Fig 22.15 Lymphatic drainage of the female genitalia Female pelvis, anterior view. Female genitalia drain to the lumbar lymph nodes via several pathways: Ovary, uterine fundus and distal part of uterine tube – drain via a direct pathway along the ovarian vessels to right and left lumbar lymph nodes. Uterine fundus and body, and proximal part of uterine tube – drain to internal iliac, external iliac, and sacral nodes. Uterine cervix, middle and upper part of vagina – drain to deep inguinal nodes External genitalia (except anterior clitoris) – drain to superficial inguinal nodes Body and glans of the clitoris – drain to deep inguinal and internal iliac nodes A Lymphatic drainage of the prostate, epididymis, ductus deferens and testes. B Lymphatic drainage of the testes and scrotum. Lymphatic drainage from the testis and epididymis Lymphatic drainage from the scrotum and coverings of the testis Superficial inguinal l.n. Testicular a. Lumbar l.n. Abdominal aorta Femoral a. Lymphatic drainage of the anterior clitoris Deep inguinal l.n. Superficial inguinal l.n. Internal iliac lymph nodes A Lymphatic drainage of the ovary, uterus, uterine tube, vagina, and labia. B Lymphatic drainage of the clitoris. Left lumbar l.n. Right lumbar l.n. Promontory l.n. Sacral l.n. Superficial inguinal l.n., horizontal group Deep inguinal l.n. Superficial inguinal l.n., vertical group External iliac l.n. Common iliac l.n. Inferior mesenteric l.n. Intermediate lumbar l.n. Urinary bladder Penis Scrotum Testis Epididymis Rectum Abdominal aorta External iliac a. Pelvis & Perineum 278 Lymph Nodes of the Genitalia Fig. 22.16 Lymph nodes of the male genitalia Anterior view. Removed: Gastrointestinal tract (except rectal stump) and peritoneum. Obturator l.n. Promontory l.n. Intermediate lacunar l.n. Deep inguinal l.n. Superficial inguinal l.n., vertical group Superficial inguinal l.n., horizontal group External iliac l.n. Internal iliac l.n. Sacral l.n. Inferior mesenteric l.n. Intermediate lumbar l.n. Uterine tube Uterus Mesometrium Rectum Urinary bladder Ovary Common iliac l.n. Cisterna chyli Thoracic duct Intermediate lumbar l.n. Subaortic l.n. Promontory l.n. Lateral, medial, and intermediate common iliac l.n. Common iliac l.n. Left lumbar trunk Right lumbar trunk Lateral caval l.n. Precaval l.n. Retrocaval l.n. Right lumbar l.n. Lateral aortic l.n. Preaortic l.n. Retroaortic l.n. Left lumbar l.n. Obturator l.n. Lateral, medial, and intermediate external iliac l.n. Interiliac l.n. External iliac l.n. Sacral l.n. Superior and inferior gluteal l.n. Internal iliac l.n. Lacunar l.n. (lateral, medial, and intermediate) Deep inguinal l.n. Superficial inguinal l.n. Pararectal l.n. Parauterine l.n. Paravaginal l.n. Lateral vesical l.n. Pre- and retrovesical l.n. Visceral pelvic l.n. 22 Neurovasculature 279 Fig. 22.17 Lymph nodes of the female genitalia Anterior view. Removed: Gastrointestinal tract (except rectal stump) and peritoneum. Retracted: Uterus. Fig. 22.18 Lymphatic drainage of the pelvic organs Lumbosacral trunk Cavernous nn. of penis Lumbar splanchnic nn. Gray ramus communicans Pelvic splanchnic nn. Lumbar nn., anterior rami Seminal gland Prostatic plexus Intermesen-teric plexus Inferior mesen-teric plexus Ureteral plexus Superior hypo-gastric plexus Right hypo-gastric n. Iliac plexus Obturator n. Deferential plexus Vesical plexus Dorsal n. of the penis Posterior scrotal nn. Left hypogastric n. L5 vertebra Sympathetic trunk, lumbar ganglia Inferior rectal nn. Inferior rectal plexus Pudendal n. Middle rectal plexus Pelvis & Perineum 280 Autonomic Innervation of the Genital Organs Fig 22.19 Innervation of the male pelvis Intermesenteric plexus Inferior hypogastric plexus Superior hypogastric plexus Inferior mesenteric ganglion Superior mesenteric ganglion Lumbar splanchnic n. (L1-L2) Least splanchnic n. (T12) Prostate with prostatic plexus Bladder with vesical plexus Lesser splanchnic n. (T10-T11) Sympathetic trunk Renal ganglion Seminal vesicle Ductus deferens with deferential plexus Pelvic splanchnic nn. (S2-S4) Sacral splanchnic nn. (S1-S4) Testicular plexus Epididymis, testes Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers B Schematic of innervation of male genitalia. A Right pelvis, left lateral view. 1st sacral n., anterior ramus Lumbosacral trunk Ureteral plexus Ovarian plexus Right hypo-gastric n. Obturator n. Vesical plexus Right utero-vaginal plexus Right middle rectal plexus Pudendal n. Right inferior hypogastric plexus Pelvic splanchnic nn. Sacral plexus Left hypo-gastric n. Superior hypo-gastric plexus L5 vertebra Lumbar nn., anterior rami Sympathetic trunk, lumbar ganglia Lumbar splanchnic nn. Gray ramus communicans Intermesen-teric plexus Inferior mesen-teric plexus 22 Neurovasculature 281 Fig 22.20 Innervation of the female pelvis Intermesenteric plexus Uterine tube Inferior hypo-gastric plexus Hypogastric nn. Celiac ganglion Renal ganglion Superior mesenteric ganglion Inferior mesenteric ganglion Superior hypogastric plexus Ovarian plexus Ovary Vagina Uterus Uterovaginal plexus Sacral splanchnic nn. (S1-S4) Pelvic splanchnic nn. (S2-S4) Lumbar splanchnic n. (L1-L2) Least splanchnic n. (T12) Lesser splanchnic n. (T10-T11) Sympathetic trunk Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. A Right pelvis, left lateral view. B Schematic of innervation of female genitalia. Pelvis & Perineum 282 Autonomic Innervation of the Urinary Organs & Rectum 1st sacral n., anterior ramus Pelvic splanchnic nn. Sympathetic trunk, lumbar ganglia Ureteral plexus Iliac plexus Right hypo-gastric n. Vesical plexus Prostatic plexus Sympathetic trunk, sacral ganglia Middle rectal plexus Inferior hypo-gastric plexus Left hypo-gastric n. Superior hypo-gastric plexus Inferior mesen-teric plexus Testicular plexus Inferior mesen-teric ganglion Intermesen-teric plexus Fig 22.21 Innervation of the pelvic urinary organs See pp. 215 and 217 for innervation of the kidneys and upper ureters. Sympathetic trunk Lumbar splanchnic n. (L1-L2) Sacral splanchnic nn. (S1-S4) Inferior mesenteric ganglion Superior hypo-gastric plexus Ureter (abdominal and pelvic parts) Ureteral plexus Bladder Seminal vesicle Prostate Vesical plexus Inferior hypo-gastric plexus Pelvic splanchnic nn. (S2-S4) Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers A Anterior view of male pelvis and lower abdomen. B Schematic of the urinary bladder and ureter. Sympathetic trunk Lumbar splanchnic n. (L1-L2) Sacral splanchnic nn. (S1-S4) Inferior mesenteric ganglion Superior hypo-gastric plexus Ureter (abdominal and pelvic parts) Ureteral plexus Prostate Vesical plexus Inferior hypo-gastric plexus Pelvic splanchnic nn. (S2-S4) Minimal sympathetic preganglioni traveling through the sacral splanchn will synapse in the ganglia located in inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Sympathetic trunk Lumbar splanchnic n. (L1-L2) Sacral splanchnic nn. (S1-S4) Inferior mesenteric ganglion Superior hypo-gastric plexus Ureter (abdominal and pelvic parts) Ureteral plexus Bladder Seminal vesicle Prostate Vesical plexus Inferior hypo-gastric plexus Pelvic splanchnic nn. (S2-S4) Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. Sympathetic preganglionic fibers Sympathetic postganglionic fibers Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers Fig 22.22 Innervation of the anal sphincter mechanism A Somatomotor and somatosensory innervation: The pudendal nerves and inferior rectal branches provide active, partly voluntary innervation of the external anal sphincter and levator ani muscles, and sensation for the anus and perianal skin. B Visceromotor and viscerosensory innervation: Pelvic splanchnic nerves (S2-4) innervate the internal anal sphincter, which helps to maintain closure of the anal canal. They also supply sensation to the wall of the rectum, particularly the stretch receptors in the rectal ampulla, which when stretched trigger an awareness of the need to defecate. External anal sphincter Superficial and deep parts Subcutaneous part Lumbar splanchnic nn. Sacral splanchnic nn. L5 vertebra Sacral plexus Rectal plexus Pelvic splanchnic nn. Inferior hypo-gastric plexus Superior hypo-gastric plexus Sympathetic trunk Inferior rectal nn. Sacral plexus Pudendal n. Branches from S2–S4 Levator ani muscle Cutaneous zone Internal anal sphincter Rectum L5 vertebra Internal anal sphincter Rectum 22 Neurovasculature 283 Clinical box 22.2 Both defecation and continence are under central nervous system control involving such diverse structures as the cerebral cortex, muscles of the abdomen and pelvis and perianal skin. Filling of the rectal ampulla and stimulation of local stretch receptors in the ampullary wall. When the fecal bolus is propelled into the ampulla, mechanoreceptors detect distension and transmit the information to the sensory cortex, which perceives the urge to defecate. Rectoanal inhibitory reflex and relaxation of the voluntary innervated sphincters. When the ampulla fills, the intrarectal pressure increases and the internal anal sphincter relaxes. This is followed by voluntary relaxation of the puborectalis sling and the external anal sphincter, which results in the straightening of the anorectal angle and widening of the anal canal. Propulsion of the fecal column. Rectal evacuation is assisted by a direct involuntary increase in pressure in the rectal area and by simultaneous increase in pressure by the contraction of voluntarily innervated muscles in the abdomen wall, pelvic floor, and diaphragm. With propulsion of the fecal column, the hemorrhoidal cushions are drained and pushed out. Completion of defecation. After the sphincter apparatus allows the fecal column to pass through, it comes in contact with the highly sensitive anoderm, which perceives the volume, consistency and location of the stool. This perception initiates the voluntary process of completing defecation, which is marked by the contraction of the sphincter apparatus and filling of the hemorrhoidal plexus. Mechanism of defecation (after Wedel) Pelvis & Perineum 284 Neurovasculature of the Male & Female Perineum Anus Pudendal n. Perineal nn. (brs. of pudendal n.) Ischial tuberosity Posterior femoral cutaneous n. Adductor magnus Ischio-cavernosus Gracilis Dorsal n. of penis (br. of pudendal n.) Posterior scrotal nn. (brs. of pudendal n.) Bulbo-spongiosus Scrotum Superficial transverse perineal External anal sphincter Inferior rectal nn. (br. of pudendal n.) Levator ani Gluteus maximus Pudendal n. Anococcygeal nn. Middle clunial nn. Ilioinguinal n. and genitofemoral n., genital br. Posterior femoral cutaneous n. Superior clunial nn Inferior clunial nn. Perineal body Corpora cavernosa Corpus spongiosum Spermatic cord Posterior scrotal nn. Bulbo-spongiosus Perineal nn. Inferior rectal nn. Anus External anal sphincter Inferior rectal a. and v. Gluteus maximus Pudendal n. Internal pudendal a. and v. Ischial tuberosity Bulbourethral gland Muscular brs. Dorsal penile n. Dorsal penile a. Transverse perineal lig. Arcuate pubic lig. Deep dorsal penile v. Fig. 22.23 Nerves of the male perineum and genitalia Lithotomy position. Fig. 22.24 Neurovasculature of the male perineum Lithotomy position. Removed from left side: Perineal membrane, bulbospongiosus, and root of penis. Pudendal n. Perineal nn. (brs. of pudendal n.) Ischial tuberosity Posterior femoral cutaneous n. Adductor magnus o-ernosus of penis endal n.) Pudendal n. Anococcygeal nn. Middle clunial nn. Ilioinguinal n. and genitofemoral n., genital br. Posterior femoral cutaneous n. Superior clunial nn. Inferior clunial nn. Perineal nn. (brs. of pudendal n.) Labium minus Vaginal orifice Anus External anal sphincter Inferior rectal nn. (brs. of pudendal n.) Levator ani Gluteus maximus Inferior clunial nn. Pudendal n. Ischial tuberosity Posterior femoral cutaneous n. Superficial transverse perineal Posterior femoral cutaneous n., perineal brs. Perineal membrane Adductor magnus Ischiocavernosus Gracilis Posterior labial nn. (br. of pudendal n.) Glans of clitoris External urethral orifice Dorsal clitoral n. (br. of pudendal n.) Bulbo-spongiosus Anococcygeal nn. Middle clunial nn. Posterior femoral cutaneous n. Superior clunial nn. Inferior clunial nn. Perineal body Superficial transverse perineal Ischial tuberosity Perineal membrane Perineal nn. Inferior rectal nn. Levator ani Inferior rectal a. and v. Internal pudendal a. and v. Pudendal n. Posterior labial nn. A. of vestibular bulb Deep clitoral a. Dorsal clitoral a. and n. Anterior labial nn. Ischio-cavernosus Bulbo-spongiosus Vestibular bulb Crus of clitoris Perineal a. Greater vestibular gland Perineal nn. (brs. of pudendal n.) Labium minus Vaginal orifice Anus External anal sphincter Inferior rectal nn. (brs. of pudendal n.) Levator ani Gluteus maximus Inferior clunial nn. Pudendal n. Ischial tuberosity Posterior femoral cutaneous n. Superficial transverse perineal Posterior femoral cutaneous n., perineal brs. Perineal membrane Adductor magnus Ischiocavernosus Gracilis Posterior labial nn. (br. of pudendal n.) Glans of clitoris External urethral orifice Dorsal clitoral n. (br. of pudendal n.) Bulbo-spongiosus Pudendal n. Anococcygeal nn. Middle clunial nn. Ilioinguinal n. and genitofemoral n., genital br. and labial br. Posterior femoral cutaneous n. Superior clunial nn. Inferior clunial nn. Perineal body 22 Neurovasculature 285 Fig. 22.25 Nerves of the female perineum and genitalia Sensory innervation of the female perineum. Lithotomy position. Fig. 22.26 Neurovasculature of the female perineum Lithotomy position. Removed from left side: Bulbospongiosus and ischiocavernosus. Uterovaginal venous plexus Urinary bladder Pectineus Obturator canal (inlet) Lig. of head of femur Ischial spine Uterosacral lig. Cervix of uterus Rectouterine pouch Coccyx Rectum Sacrospinous lig. Gluteus maximus Sciatic n. Obturator internus Left ureter (cut obliquely) Head of femur Iliopsoas Femoral a., v., and n. Pubis Rectovesical septum Orifice of right ureter Vesicoprostatic venous plexus Iliopsoas Head of femur Obturator internus Inferior hypogastric plexus Gluteus maximus Sacrospinous lig. Coccyx Rectum Ischial spine Sciatic n. Seminal gland Obturator a., v., and n. Femoral a., v., and n. Ductus deferens Urinary bladder Rectus abdominis Inferior vesical a. Pelvis & Perineum 286 23 Sectional & Radiographic Anatomy Sectional Anatomy of the Pelvis & Perineum Fig. 23.1 Female pelvis Transverse section through the bladder and cervix of the uterus. Inferior view. Fig. 23.2 Male pelvis Transverse section through the bladder and seminal glands. Inferior view. Adductor mm. Femoral a., v. and n. Obturator membrane Rectum Sciatic n. Ischioanal fossa Pudendal n., internal pudendal a. and v. Rectoprostatic fascia Ischial tuberosity Obturator internus Seminal gland Obturator externus Prostatic urethra Inferior pubic ramus Pubic symphysis Spermatic cord Corpora cavernosa of penis Prostate Gluteus maximus Levator ani 23 Sectional & Radiographic Anatomy 287 Fig. 23.3 Male pelvis Transverse section through the prostate gland and anal canal. Inferior view. Sartorius Femoral a., v., and n. Urethra Pubic symphysis Pubis (body) Rectus femoris Iliopsoas Femur Sciatic n. Gluteus maximus Vagina Rectum Obturator internus Ischial tuberosity Obturator externus Pectineus Levator ani Bladder Cervix Rectouterine pouch Rectum Coccyx Gluteus maximus Obturator internus Head of femur Femoral v. Symphysis Bladder Sacrum Iliac vessels Acetabulum Ovary Follicle Femoral a. Round lig. Sigmoid colon Uterine body Proper ovarian lig. Ovary Follicle C Section through the lower vagina. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A Section through the body of the uterus. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) B Section through the cervical canal. The image shows the low-signal intensity cervical stroma (arrows), which surrounds the narrow high-signal intensity cervical canal. (Re-produced from Hamm B. et al. MRT von Abdomen und Becken, 2nd ed. Stuttgart: Thieme; 2006.) A B C Pelvis & Perineum 288 Radiographic Anatomy of the Female Pelvis Fig 23.4 MRI of the female pelvis Transverse section, inferior view. Psoas major L4 vertebra Iliac crest Sigmoid colon Uterus Head of femur Levator ani Iliacus Internal iliac a. and v. Gluteus medius Urinary bladder Obturator internus Ischium, ramus Labium minus Cervical canal Myometrium Rectum Coccyx Levator ani External anal sphincter Endometrium Bladder Pubic symphysis Urethra Vagina B Position of the uterus with a full bladder. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A Position of the uterus with a near empty bladder. The image shows the uterus in the first half of the menstrual cycle (proliferative phase) with narrow endometrium and relatively low-signal intensity of the mymoetrium. (Reproduced from Hamm B. et al. MRT von Abdomen und Becken, 2nd ed. Stuttgart: Thieme; 2006.) Body of uterus (myometrium) Uterus (junctional zone) Uterus (endometrium) Urinary bladder Urethra Pubis Uterus (cavity) Vagina (wall) Rectum Rectouterine pouch (of Douglas) 23 Sectional & Radiographic Anatomy 289 Fig. 23.5 MRI of the female pelvis Sagittal section, left lateral view Fig. 23.6 MRI of the female pelvis Coronal section, anterior view. (Repro-duced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Pubis Urethra Mediastinum of testis Tunica albuginea Pubic symphysis Testes Corpus cavernosum of penis Corpus spongiosum of penis Spermatic cord Rectus abdominis Prostate Seminal gland Sacrum Rectovesical (Denonvillier) fascia Ampulla of rectum Anal canal Bulb of penis Corpus spongiosum of penis Corpus cavernosum of penis Pubic symphysis Bladder Tunica albuginea Mediastinum testis with rete testis Tail of the epididymis Head of the epididymis Spermatic cord A Coronal section, anterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) B Parasagittal section, T2 W image. (Reproduced from Krombach GA, Mahnken AH. Body Imaging: Thorax and Abdomen. New York, NY: Thieme; 2018.) Pelvis & Perineum 290 Radiographic Anatomy of the Male Pelvis Fig. 23.7 MRI of the male pelvis Sagittal section, left lateral view. (Reproduced from Hamm B. et al. MRT von Abdomen und Becken, 2nd ed. Stuttgart: Thieme; 2006.) Fig. 23.8 MRI of the testes Fig. 23.10 MRI of the male pelvis Coronal section. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Fig. 23.9 MRI of the prostate (Reproduced from Krombach GA, Mahnken AH. Body Imag-ing: Thorax and Abdomen. New York, NY: Thieme; 2018.) Head of femur Obturator externus muscle Pubis Common iliac a. and v. Sigmoid colon Urinary bladder Prostate (central zone) Prostate (peripheral zone) Prostate (transition zone) Ureter Corpus spongiosum Corpus cavernosum Ischiocavernosus muscle Bulbospongiosus muscle Anterior fibromuscular connective tissue Coil inside rectum Obturator internus muscle Pubis Peripheral zone Transitional zone Prostatic urethra A Transverse section, T2W image. B Coronal section, T2W image. Peripheral zone Urethra Ductus deferens Seminal glands Transitional zone 23 Sectional & Radiographic Anatomy 291 24 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 25 Shoulder & Arm Bones of the Upper Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Clavicle & Scapula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Humerus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Joints of the Shoulder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 Joints of the Shoulder: Glenohumeral Joint . . . . . . . . . . . . . . 304 Subacromial Space & Bursae . . . . . . . . . . . . . . . . . . . . . . . . . 306 Anterior Muscles of the Shoulder & Arm (I) . . . . . . . . . . . . . . 308 Anterior Muscles of the Shoulder & Arm (II) . . . . . . . . . . . . . 310 Posterior Muscles of the Shoulder & Arm (I) . . . . . . . . . . . . . 312 Posterior Muscles of the Shoulder & Arm (II) . . . . . . . . . . . . . 314 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Muscle Facts (IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 26 Elbow & Forearm Radius & Ulna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Elbow Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Ligaments of the Elbow Joint . . . . . . . . . . . . . . . . . . . . . . . . . 328 Radioulnar Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Muscles of the Forearm: Anterior Compartment . . . . . . . . . 332 Muscles of the Forearm: Posterior Compartment . . . . . . . . . 334 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 27 Wrist & Hand Bones of the Wrist & Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Carpal Bones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Joints of the Wrist & Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Ligaments of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Ligaments & Compartments of the Wrist . . . . . . . . . . . . . . . 350 Ligaments of the Fingers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Muscles of the Hand: Superficial & Middle Layers . . . . . . . . . 354 Muscles of the Hand: Middle & Deep Layers . . . . . . . . . . . . . 356 Dorsum of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 28 Neurovasculature Arteries of the Upper Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Veins & Lymphatics of the Upper Limb . . . . . . . . . . . . . . . . . 366 Nerves of the Upper Limb: Brachial Plexus . . . . . . . . . . . . . . 368 Supraclavicular Branches & Posterior Cord . . . . . . . . . . . . . . 370 Posterior Cord: Axillary & Radial Nerves . . . . . . . . . . . . . . . . 372 Medial & Lateral Cords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Median & Ulnar Nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Superficial Veins & Nerves of the Upper Limb . . . . . . . . . . . . 378 Posterior Shoulder & Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Anterior Shoulder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Axilla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Anterior Arm & Cubital Region . . . . . . . . . . . . . . . . . . . . . . . . 386 Anterior & Posterior Forearm . . . . . . . . . . . . . . . . . . . . . . . . . 388 Carpal Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Palm of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Dorsum of the Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 29 Sectional & Radiographic Anatomy Sectional Anatomy of the Upper Limb. . . . . . . . . . . . . . . . . . 396 Radiographic Anatomy of the Upper Limb (I). . . . . . . . . . . . . 398 Radiographic Anatomy of the Upper Limb (II). . . . . . . . . . . . 400 Radiographic Anatomy of the Upper Limb (III). . . . . . . . . . . . 402 Radiographic Anatomy of the Upper Limb (IV) . . . . . . . . . . . 404 Upper Limb Clavicle Deltoid Cephalic v. (in deltopectoral groove) Cephalic v. Extensor carpi radialis longus Brachioradialis Flexor carpi radialis Palmaris longus tendon Thenar eminence Hypothenar eminence Flexor carpi ulnaris Median cubital v. Basilic v. Biceps brachii Pectoralis major Scapular spine Teres major Latissimus dorsi Olecranon Basilic v. Extensor carpi ulnaris Flexor carpi ulnaris Extensor digitorum tendons, dorsal venous network Extensor digitorum Cephalic v. Extensor carpi radialis longus Long head Lateral head Deltoid Triceps brachii Deltoid region Palm of the hand Anterior carpal region Anterior forearm region Anterior cubital region Anterior arm region Axillary region Infraclavicular fossa Clavipectoral triangle Clavipectoral triangle Axillary region (axillary fossa) Infraclavicular fossa Deltoid region Dorsum of the hand Posterior carpal region Posterior forearm region Posterior cubital region Posterior arm region Scapular region Upper Limb 294 24 Surface Anatomy Surface Anatomy Fig. 24.1 Regions of the upper limb Fig. 24.2 Palpable musculature of the upper limb A Left limb, anterior view. B Right limb, posterior view. A Right limb, anterior view. C Right limb, posterior view. B Right axilla, anterior view. Coracoid process Clavicle Medial epicondyle Greater and lesser tubercles Acromion Metacarpo-phalangeal joints Inter-phalangeal joints Hook of hamate Pisiform bone Lateral epicondyle Tubercle of scaphoid Tubercle of trapezium Inferior angle Scapular spine Greater tubercle Acromion Superior angle Triquetrum bone Styloid process of radius Head of radius Olecranon Metacarpals Capitate bone Styloid process of ulna Phalanges Shaft of ulna Pisiform Flexor retinaculum (roof of carpal tunnel) Median n. Palmar aponeurosis (cut) Ulnar tunnel (with ulnar a. and n.) Palmar carpal lig. DIP joint crease PIP joint crease MCP joint crease IP joint crease MCP joint crease Thenar eminence Thenar crease (“life line”) Proximal wrist crease Distal wrist crease Hypothenar eminence Middle crease Proximal trans-verse crease Distal transverse crease 24 Surface Anatomy 295 Fig. 24.3 Palpable bony prominences of the upper limb Except for the lunate and trapezoid bones, all of the bones in the upper limb are palpable to some degree through the skin and soft tissues. Fig. 24.4 Surface anatomy of the wrist and hand DIP, distal interphalangeal; IP, interphalangeal; MCP, metacarpophalangeal; PIP, proximal interphalangeal. A Right limb, anterior view. B Right limb, posterior view. A Left palm and wrist. B Palm and wrist showing the carpal and ulnar tunnels. Arm Elbow joint Forearm Hand Ulna Radius Humerus Scapula Clavicle Phalanges Carpals Shoulder girdle Metacarpals Scapular spine Humerus Ulna Radius 4th proximal phalanx 4th middle phalanx 4th distal phalanx 1st metacarpal Shoulder joint Acromio-clavicular joint Scapula Clavicle Olecranon Head of radius Carpal bones 296 Upper Limb 25 Shoulder & Arm Bones of the Upper Limb Fig. 25.1 Bones of the upper limb Right limb. The upper limb is subdivided into three regions: arm, forearm, and hand. The shoulder girdle (clavicle and scapula) joins the upper limb to the thorax at the sternoclavicular joint. A Anterior view. B Posterior view. A Anterior view. Acromion Inferior angle Humerus Phalanges 1st distal phalanx 1st proximal phalanx 1st metacarpal Coracoid process Clavicle 2nd metacarpal Radius Ulna Scapula 297 25 Shoulder & Arm Fig. 25.2 Bones of the shoulder girdle in normal relation to those of the trunk C Lateral view. C Lateral view. B Posterior view. Acromio-clavicular joint Acromion Coracoid process Glenoid cavity Scapula, posterior surface Lateral border Clavicle First rib Acromion Coracoid process Glenoid cavity Scapula, costal surface Body of sternum Manubrium Sterno-clavicular joint Clavicle First rib Inferior angle Medial border Scapula, posterior surface Acromion Superior angle Clavicle Acromial end Conoid tubercle Shaft of clavicle Sternal articular surface Sternal end Suprascapular fossa Acromion Acromioclavicular joint Coracoid process Clavicle Sternoclavicular joint Manubrium Acromial articular surface Acromial end Conoid tubercle Groove for subclavius muscle Impression for costoclavicular lig. Sternal end 298 Upper Limb Clavicle & Scapula The shoulder girdle (clavicle and scapula) connects the bones of the upper limb to the thoracic cage. Whereas the pelvic girdle (paired hip bones) is firmly integrated into the axial skeleton (see p. 230), the shoulder girdle is extremely mobile. Fig. 25.4 Clavicle Right clavicle. The S-shaped clavicle is visible and palpable along its entire length (generally 12 to 15 cm). Its medial end articulates with the sternum at the sternoclavicular joint. Its lateral end articu-lates with the scapula at the acromio clavicular joint (see Fig. 25.3). B Inferior view. A Superior view. Fig. 25.3 Shoulder girdle in situ Right shoulder, superior view. Coracoid process Acromion Supraglenoid tubercle Lateral angle Glenoid cavity Infraglenoid tubercle Neck Lateral border Inferior angle Medial border Subscapular fossa Superior angle Superior border Scapular notch Posterior surface Costal surface Inferior angle Lateral border Infraglenoid tubercle Glenoid cavity Supraglenoid tubercle Coracoid process Superior angle Acromion Superior border Scapular notch Scapular spine Superior angle Supra-spinous fossa Medial border Inferior angle Lateral border Infraspinous fossa Infraglenoid tubercle Glenoid cavity Acromial angle Acromion Coracoid process 299 25 Shoulder & Arm Fig. 25.5 Scapula Right scapula. In its normal anatomical position, the scapula extends from the 2nd to the 7th rib. C Posterior view. A Anterior view. Scapular foramen Scapular foramen Clinical box 25.1 The superior transverse ligament of the scapula (see Fig. 25.14) may become ossified, transforming the scapular notch into an anomalous bony canal, the scapular foramen. This can lead to compression of the suprascapular nerve as it passes through the canal (see p. 381). B Right lateral view. Lesser tubercle Intertubercular groove Deltoid tuberosity Lateral supracondylar ridge Radial fossa Lateral epicondyle Condyle of humerus Medial epicondyle Coronoid fossa Medial supracondylar ridge Antero-lateral surface Anteromedial surface Crest of lesser tubercle Surgical neck Anatom-ical neck Head of humerus Capitulum Trochlea Greater tubercle Crest of greater tubercle Greater tubercle Radial groove (for radial n.) Lateral border Lateral supra-condylar ridge Radial fossa Lateral epicondyle Capitulum Shaft of humerus, anterolateral surface Intertubercular groove Anatomical neck Lesser tubercle Surgical neck Medial epicondyle Medial supracondylar ridge Anatom-ical neck Head of humerus Medial border Ulnar groove (for ulnar n.) Olecranon fossa Trochlea Lateral epicondyle Lateral supracondylar ridge Lateral border Greater tubercle Shaft of humerus, posterior surface 300 Upper Limb Humerus C Posterior view. B Lateral view. A Anterior view. Fig. 25.6 Humerus Right humerus. The head of the humerus articulates with the scapula at the glenohumeral joint (see p. 302). The capitulum and trochlea of the humerus articulate with the radius and ulna, respectively, at the elbow (cubital) joint (see p. 326). Lesser tubercle Trochlea of humerus Medial epicondyle Olecranon fossa Medial supracondylar ridge Medial border Shaft of humerus, anteromedial surface Crest of lesser tubercle Anatomical neck Head of humerus Anatomical neck Greater tubercle Intertubercular groove Lesser tubercle Head of humerus Lateral epicondyle Capitulum Olecranon fossa Trochlea Ulnar groove Medial epicondyle Capitulotrochlear groove Anatomical neck Head of humerus Greater tubercle Surgical neck Lesser tubercle Inter-tubercular groove 301 25 Shoulder & Arm F Distal (inferior) view. E Proximal (superior) view. D Medial view. Clinical box 25.2 Fractures of the humerus Anterior view. Fractures of the proximal humerus are very common and occur predominantly in older patients who sustain a fall onto the outstretched arm or directly onto the shoulder. Three main types are distinguished. Extra-articular fractures and intra-articular fractures are often accompanied by injuries of the blood vessels that supply the humeral head (anterior and posterior circumflex humeral arteries), with an associated risk of post-traumatic avascular necrosis. Fractures of the surgical neck can damage the axillary nerve and fractures of the humeral shaft and distal humerus are frequently associated with damage to the radial nerve. A Extra-articular fracture. B Intra-articular fracture. C Comminuted fracture. Acromioclavicular joint Subacromial space Gleno-humeral joint Scapulothoracic joint Sternoclavicular joint Sub-scapularis Scapulothoracic joint Acromion Head of humerus Coracoid process Serratus anterior Clavicle Superior posterior serratus Scapulothoracic joint Acromioclavicular joint (with acromio-clavicular lig.) Coraco-acromial lig. Posterior sternoclavicular lig. Sternoclavicular joint (with anterior sternoclavicular lig.) Glenohumeral joint 302 Upper Limb Joints of the Shoulder Fig. 25.7 Joints of the shoulder: Overview Right shoulder, anterior view. Fig. 25.8 Joints of the shoulder girdle Right side, superior view. Fig. 25.9 Scapulothoracic joint Right side, superior view. In all movements of the shoulder girdle, the scapula glides on a curved surface of loose connective tissue be-tween the serratus anterior and the subscapu-laris muscles. This surface can be considered a scapulothoracic joint. Inter-clavicular lig. Artic-ular disk 1st rib Sternocostal joint Manubrium Costal cartilage Clavicle Anterior sterno-clavicular lig. Costo-clavic-ular lig. Superior angle Superior transverse lig. of scapula Scapular notch Scapula, costal (anterior) surface Humerus Head of humerus Lesser tubercle Greater tubercle Coracoid process Coracoacromial lig. Acromion Acromio-clavicular lig. Clavicle, acromial end Conoid lig. Trapezoid lig. Coracoclavicular lig. Medial border Intertuber-cular groove Glenoid cavity Coraco-acromial arch Clavicle, sternal end 303 25 Shoulder & Arm Fig. 25.11 Acromioclavicular joint Anterior view. The acromioclavicular joint is a plane joint. Because the articulating surfaces are flat, they must be held in place by strong ligaments, greatly limiting the mobility of the joint. Fig. 25.10 Sternoclavicular joint Anterior view with sternum coronally sectioned (left). Note: A fibrocartilaginous articular disk compensates for the mismatch of surfaces be-tween the two saddle-shaped articular facets of the clavicle and the manubrium. Clinical box 25.3 Injuries of the acromioclavicular joint A fall onto the outstretched arm or shoulder frequently causes dislocation of the acromioclavicular joint (often known as a “shoulder separation”) and damage to the coracoclavicular ligaments. A Stretching of acromio clavicular ligaments. B Rupture of acromioclavicular ligament. C Complete dislocation of acromioclavicular joint. Note rup-ture of acromioclavicular and coracoclavicular ligaments. Clavicle Supraglenoid tubercle Scapular notch Lateral border of scapula Infraglenoid tubercle Head of humerus Inter-tubercular groove Lesser tubercle Greater tubercle Acromion Coracoid process Glenoid cavity Clavicle Head of humerus Humerus Anatomical neck Infraspinous fossa Scapular spine Scapular notch Acromion Greater tubercle A Anterior view. Humerus Lesser tubercle Coracoid process Greater tubercle Scapular spine Acromion Clavicle Superior transverse lig. of scapula Axillary recess Intertuber-cular synovial sheath Transverse lig. of humerus Subcoracoid bursa Synovial membrane Coraco-clavicular lig. Coracoid process Coraco-acromial lig. Acromion Acromio-clavicular lig. Intertubercular groove Tendon of biceps brachii, long head Subtendinous bursa of subscapularis 304 Upper Limb Joints of the Shoulder: Glenohumeral Joint Fig. 25.12 Glenohumeral joint: Bony elements Right shoulder. B Posterior view. C Lateral view. Fig. 25.13 Glenohumeral joint cavity Superior transverse lig. of scapula Clavicle Coraco-clavicular lig. Acromio-clavicular lig. Acromion Greater tubercle Joint capsule Scapular notch Humerus Infraspinous fossa Scapular spine Axillary recess Coracoclavicular lig. Clavicle Scapular notch Scapula, costal surface Lateral border Neck of scapula Axillary recess Inter-tubercular groove Intertubercular synovial sheath Coracohumeral lig. Coracoid process Coraco-acromial lig. Acromion Acromioclavicular lig. Coraco-acromial arch Joint capsule, glenohumeral ligs. Coracohumeral lig. Anterior band Tendon of biceps brachii, long head Posterior band Axillary recess Inferior glenohumeral lig. Middle glenohumeral lig. Superior glenohumeral lig. Glenoid cavity Triceps brachii, long head Coracohumeral lig. Coracoacromial lig. Middle gleno-humeral lig. Inferior glenohumeral lig. Superior gleno-humeral lig. Subscapularis Supraspinatus Tendon of biceps brachii, long head Anatomic neck of humerus Acromion 305 25 Shoulder & Arm Fig. 25.14 Glenohumeral joint: Capsule and ligaments Right shoulder. B Posterior view. A Anterior view. Fig. 25.15 Ligaments reinforcing capsule Schematic representation of the ligaments reinforc-ing the capsule after removal of the humeral head. Right shoulder. B Posterior view. A Lateral view. Teres minor Tendon of biceps brachii, long head Acromion Subacromial bursa Supra-spinatus Infraspinatus Glenoid cavity Subtendinous bursa of infraspinatus Glenoid labrum Infraspinatus Lateral border of scapula Subscapularis Subscapularis Joint capsule Coracoid process Subtendinous bursa of subscapularis Coracoacromial lig. Axillary recess Coracoacromial arch Acromion Subacromial bursa Infraspinatus Greater tubercle Teres minor Humerus Intertubercular tendon sheath Coracoid process Subtendinous bursa of subscapularis Biceps brachii, short head Biceps brachii, long head Transverse lig. of humerus Coracoacromial lig. Subdeltoid bursa Coracoacromial arch Supraspinatus Scapula Superior transverse lig. of scapula Coracoacromial arch Acromion Coraco-acromial lig. Coracoid process Joint capsule Humerus Subacromial bursa Lesser tubercle Intertubercular groove Subdeltoid bursa Greater tubercle Acromial articular surface 306 Upper Limb Subacromial Space & Bursae Fig. 25.16 Subacromial space Right shoulder. Fig. 25.17 Subacromial bursa and glenoid cavity Right shoulder, lateral view of sagittal section with humerus removed. A Lateral view. B Superior view. Note the position of the sub-acromial bursa between the supraspinatus muscle and the coracoacromial arch. Axillary recess Acromion Supraspinatus tendon Head of humerus Subdeltoid bursa Deltoid Humerus Teres major Glenoid labrum Subscapularis Scapula Glenoid cavity Supraspinatus Subacromial bursa Trapezius Subcutaneous tissue Latissimus dorsi Skin Trapezius Coracoclavicular lig. Acromioclavicular lig. Subcuta-neous acromial bursa Subacromial bursa Subdeltoid bursa Glenohumeral joint capsule Tendon sheath in inter-tubercular groove Biceps brachii, long head Biceps brachii, short head Coraco-brachialis Teres major Subscapularis Subtendinous bursa of subscapularis 1st rib Superior transverse lig. of scapula Clavicle Deltoid Acromion Coracoacromial lig. Coracoid process Coracoacromial arch Humerus 307 25 Shoulder & Arm Fig. 25.18 Subacromial and subdeltoid bursae Right shoulder, anterior view. B Coronal section. The arrows are pointing at the supraspinatus tendon, which is frequently injured in a “rotator cuff tear” (for rotator cuff, see p. 317). A Location of bursae. Sternocleido-mastoid 1st rib Clavicle Trapezius Deltoid Coracobrachialis Teres major Long head Short head Medial epicondyle Serratus anterior External oblique Abdominal part Clavicular part Sternocostal part Body of sternum Manubrium Vertebra prominens (C7) Rectus sheath Brachialis Pectoralis major Biceps brachii Biceps brachii Latissimus dorsi 308 Upper Limb Anterior Muscles of the Shoulder & Arm (I) Fig. 25.19 Anterior muscles of the shoulder and arm Right side, anterior view. Muscle origins are shown in red, insertions in blue. A Superficial dissection. Clavicle Supra-spinatus Pectoralis minor Acromial part Subscapularis Coracobrachialis Teres major Biceps brachii Short head Lateral epicondyle Brachialis Medial epicondyle Serratus anterior T12 vertebral body Sternum, xiphoid process Sternum, body Costal cartilage Sternum, manubrium Pectoralis major, clavicular part Costal arch (margin) Latissimus dorsi Deltoid Pectoralis major Greater tubercle Coracoid process Clavicular part Trapezius Sternocleido-mastoid Subclavius Pectoralis major, sternocostal part Long head Deltoid 309 25 Shoulder & Arm B Deep dissection. Removed: Sternocleidomastoid, trapezius, pectoralis major, deltoid, and external oblique muscles. Biceps brachii, short head Bicipital aponeurosis Supra-spinatus Trapezius Deltoid Pectoralis major Biceps brachii tendon Common head of superficial flexors Pronator teres Brachialis Coraco-brachialis Sub-scapularis Serratus anterior Pectoralis minor Subclavius Teres major Latissimus dorsi Biceps brachii, long head Biceps brachii, short head Bicipital aponeurosis Supra-spinatus Pectoralis major Biceps brachii tendon Brachialis Coraco-brachialis Subscapularis Serratus anterior Teres major Biceps brachii, long head 310 Upper Limb Anterior Muscles of the Shoulder & Arm (II) Fig. 25.20 Anterior muscles of the shoulder and arm: Dissection Right arm, anterior view. Muscle origins are shown in red, insertions in blue. A Removed: Thoracic skeleton. Partially removed: Latissimus dorsi and serratus anterior. B Removed: Latissimus dorsi and serratus anterior. Biceps brachii, long head Supra-spinatus Sub-scapularis Pectoralis major Deltoid Biceps brachii, radial tuberosity Brachialis Coraco-brachialis Subscapularis Biceps brachii, short head Teres major Latissimus dorsi Biceps brachii, long head Teres major Trapezius Deltoid Supra-spinatus Sub-scapularis Latissimus dorsi Pectoralis major Deltoid Common head of extensors Biceps brachii Brachialis Common head of superficial flexors Pronator teres Brachialis Coraco-brachialis Subscapularis Serratus anterior Pectoralis minor Subclavius Biceps brachii, short head, and coracobrachialis Intertuber-cular groove Extensor carpi radialis longus Extensor carpi radialis brevis Brachioradialis Flexor digitorum profundus Supinator 311 25 Shoulder & Arm C Removed: Subscapularis and supraspinatus. Partially removed: Biceps brachii. D Removed: Biceps brachii, coracobrachialis, and teres major. Thoracolumbar fascia Iliac crest External oblique Extensor digitorum Extensor carpi ulnaris Flexor carpi ulnaris Anconeus Extensor carpi radialis longus Extensor carpi radialis brevis Olecranon Lateral head Latissimus dorsi Teres major Deltoid Scapular spine Descending part Splenius capitis Sternocleidomastoid Semispinalis capitis Ascending part Transverse part Internal oblique Long head Trapezius Triceps brachii 312 Upper Limb Posterior Muscles of the Shoulder & Arm (I) Fig. 25.21 Posterior muscles of the shoulder and arm Right side, posterior view. A Superficial dissection. Intrinsic back muscles, thoracolumbar fascia, posterior layer Trapezius (cut ) Latissimus dorsi (cut) Thoracolumbar fascia, posterior layer Internal oblique External oblique Serratus posterior inferior Serratus anterior Teres major Teres minor Infraspinatus Scapula, medial border Scapular spine Supraspinatus Acromion Clavicle Rhomboid major Levator scapulae Rhomboid minor Splenius cervicis Splenius capitis Semispinalis capitis Sternocleido-mastoid Superior nuchal line Latissimus dorsi (cut) 313 25 Shoulder & Arm B Deep dissection. Partially removed: Trape-zius and latissimus dorsi. Olecranon Flexor carpi ulnaris Extensor carpi ulnaris Extensor digitorum Extensor carpi radialis longus Brachioradialis Trapezius Latissimus dorsi (scapular part) Rhomboid major Rhomboid minor Levator scapulae Supra-spinatus Teres major Triceps brachii, long head Deltoid Triceps brachii, lateral head Anconeus Extensor carpi radialis brevis Teres minor Infra-spinatus Supra-spinatus Deltoid (clavicular part) Deltoid (acromial part) Teres major Deltoid (spinal part) Teres minor Infra-spinatus Triceps brachii, long head Triceps brachii, lateral head Triceps brachii, medial head Common head of superficial flexors Anconeus Extensor carpi radialis brevis Common head of extensors Flexor carpi ulnaris Flexor digitorum profundus Supi-nator 314 Posterior Muscles of the Shoulder & Arm (II) Upper Limb Fig. 25.22 Posterior muscles of the shoulder and arm: Dissection Right arm, posterior view. Muscle origins are shown in red, insertions in blue. A Removed: Rhomboids major and minor, serratus anterior, and levator scapulae. B Removed: Deltoid and forearm muscles. Triceps brachii, lateral head (cut edge) Supra-spinatus Supra-spinatus Teres major Triceps brachii, lateral head Deltoid Brachialis Triceps brachii, medial head Triceps brachii, long head Teres minor Infra-spinatus Teres minor Infra-spinatus Triceps brachii, lateral head Teres minor Levator scapulae Supra-spinatus Rhomboid minor Rhomboid major Infraspinatus Latissimus dorsi (scapular part) Teres major Teres minor Triceps brachii, long head Triceps brachii Common head of superficial flexors Anconeus Common head of extensors Extensor carpi radialis brevis Extensor carpi radialis longus Brachio-radialis Triceps brachii, medial head Brachialis Deltoid Infra-spinatus Supra-spinatus Deltoid (spinal part) Deltoid (acromial part) Deltoid (clavicular part) Trapezius Radial groove 315 25 Shoulder & Arm C Removed: Supraspinatus, infraspinatus, and teres minor. Partially removed: Triceps brachii. D Removed: Triceps brachii and teres major. A S D Acromion Deltoid, acromial part Deltoid, clavicular part Deltoid tuberosity Shaft of humerus Scapula, costal surface Clavicle Coracoid process Clavicle Scapular spine Scapula, posterior surface Deltoid, spinal part Shaft of humerus Deltoid tuberosity Deltoid, acromial part Acromion Deltoid, clavicular part Acromion Scapular spine Deltoid, acromial part Deltoid, spinal part Scapula Shaft of humerus Deltoid tuberosity Deltoid, clavicular part Clavicle 316 Upper Limb Muscle Facts (I) The actions of the three parts of the deltoid muscle depend on their relationship to the position of the humerus and its axis of motion. At less than 60 degrees, the muscles act as adductors, but at greater than 60 degrees, they act as abductors. As a result, the parts of the deltoid can act antagonistically as well as synergistically. Fig. 25.23 Deltoid Right shoulder. A Parts of the deltoid, right lateral view, schematic. B Right lateral view. C Anterior view. D Posterior view. Table 25.1 Parts of the deltoid Muscle Origin Insertion Innervation Action Deltoid ① Clavicular (anterior) part Lateral one third of clavicle Humerus (deltoid tuberosity) Axillary n. (C5, C6) Flexion, internal rotation, adduction ② Acromial (lateral) part Acromion Abduction ③ Spinal (posterior) part Scapular spine Extension, external rotation, adduction Between 60 and 90 degrees of abduction, the clavicular and spinal parts assist the acromial part with abduction. A S D F Acromion Infra-spinatus Teres minor Scapula, lateral border Scapula, inferior angle Shaft of humerus Subscap-ularis Greater tuberosity Coracoid process Supra-spinatus Scapular spine Supra-spinatus Medial border Inferior angle Infra-spinatus Teres minor Shaft of humerus Greater tubercle Acromion Coracoid process Superior angle Lateral border Coracoid process Acromion Greater tubercle Lesser tubercle Intertuber-cular groove Crest of greater tubercle Crest of lesser tubercle Shaft of humerus Subscapularis Inferior angle Medial border Superior angle Superior border Supra-spinatus Scapular notch 317 25 Shoulder & Arm Fig. 25.24 Rotator cuff Right shoulder. The rotator cuff consists of four muscles: supraspinatus, infraspinatus, teres minor, and subscapularis. A Posterior view, schematic. B Anterior view, schematic. C Anterior view. D Lateral view. E Posterior view. Table 25.2 Muscles of the rotator cuff Muscle Origin Insertion Innervation Action ① Supraspinatus Scapula Supraspinous fossa Humerus Humerus (greater tubercle) Suprascapular n. (C4–C6) Initiates abduction ② Infraspinatus Infraspinous fossa External rotation ③ Teres minor Lateral border Axillary n. (C5, C6) External rotation, weak adduction ④ Subscapularis Subscapular fossa Humerus (lesser tubercle) Upper and lower subscapular nn. (C5, C6) Internal rotation F A S D Clavicular part Abdominal part Sternocostal part Humerus Coracoid process Intertubercular groove Crest of greater tubercle Sternum Clavicle Pectoralis major, abdominal part Pectoralis major, sternocostal part Pectoralis major, clavicular part Coraco-brachialis Acromion Lesser tubercle 318 Upper Limb Muscle Facts (II) Fig. 25.25 Pectoralis major and coracobrachialis Anterior view. A Schematic. B Pectoralis major in neutral position (left) and elevation (right). C Pectoralis major and coracobrachialis. Table 25.3 Pectoralis major and coracobrachialis Muscle Origin Insertion Innervation Action Pectoralis major ① Clavicular part Clavicle (medial half) Humerus (crest of greater tubercle) Medial and lateral pectoral nn. (C5–T1) Entire muscle: adduction, internal rotation Clavicular and sternocostal parts: flexion; assist in respiration when shoulder is fixed ② Sternocostal part Sternum and costal cartilages 1–6 ③ Abdominal part Rectus sheath (anterior layer) ④ Coracobrachialis Scapula (coracoid process) Humerus (in line with crest of lesser tubercle) Musculocutaneous n. (C5–C7) Flexion, adduction, internal rotation A S Subclavius Clavicle Acromion Coracoid process Pectoralis minor 3rd through 5th ribs 1st rib D F G Acromion Glenoid cavity Medial border Scapula Serratus anterior 1st rib Coracoid process Inferior angle 9th rib 319 25 Shoulder & Arm Fig. 25.26 Subclavius and pectoralis minor Right side, anterior view. A Schematic. B Subclavius and pectoralis minor. Fig. 25.27 Serratus anterior Right lateral view. B Schematic. A Serratus anterior. Table 25.4 Subclavius, pectoralis minor, and serratus anterior Muscle Origin Insertion Innervation Action ① Subclavius 1st rib Clavicle (inferior surface) N. to subclavius (C5, C6) Steadies the clavicle in the sternoclavicular joint ② Pectoralis minor 3rd to 5th ribs Coracoid process Medial pectoral n. (C8, T1) Draws scapula downward, causing inferior angle to move posteromedially; rotates glenoid inferiorly; assists in respiration Serratus anterior ③ Superior part 1st to 9th ribs Scapula (costal and dorsal surfaces of superior angle) Long thoracic n. (C5–C7) Superior part: lowers the raised arm ④ Intermediate part Scapula (costal surface of medial border) Entire muscle: draws scapula laterally forward; elevates ribs when shoulder is fixed ⑤ Inferior part Scapula (costal surface of medial border and costal and dorsal surfaces of inferior angle) Inferior part: rotates inferior angle of scapula laterally forward (allows elevation of arm above 90°) A S D H F G C7 spinous process Acromion Scapular spine T12 spinous process Trapezius (ascending part) Trapezius (transverse part) Trapezius (descending part) Nuchal lig. Superior nuchal line External occipital protuberance C1 (atlas) C2 (axis) C7 spinous process Rhomboid minor T1–T4 spinous processes Rhomboid major Inferior angle Medial border Scapula, posterior surface Acromion Scapular spine Clavicle Superior angle Levator scapulae C1–C4 transverse processes 320 Upper Limb Muscle Facts (III) Fig. 25.28 Trapezius Posterior view. B Schematic. A Trapezius. Fig. 25.29 Levator scapulae with rhomboids major and minor Right side, posterior view. A Schematic. B Levator scapulae with rhomboids major and minor. Table 25.5 Trapezius, levator scapulae, and rhomboids major and minor Muscle Origin Insertion Innervation Action Trapezius ① Descending part Occipital bone; spinous processes of C1–C7 Clavicle (lateral one third) Accessory n. (CN XI); C3–C4 of cervical plexus Draws scapula obliquely upward; rotates glenoid cavity superiorly; tilts head to same side and rotates it to opposite ② Transverse part Aponeurosis at T1–T4 spinous processes Acromion Draws scapula medially ③ Ascending part Spinous processes of T5–T12 Scapular spine Draws scapula medially downward Entire muscle: steadies scapula on thorax ④ Levator scapulae Transverse processes of C1–C4 Scapula (superior angle) Dorsal scapular n. and cervical spinal nn. (C3–C4) Draws scapula medially upward while moving inferior angle medially; inclines neck to same side ⑤ Rhomboid minor Spinous processes of C6, C7 Medial border of scapula above (minor) and below (major) scapular spine Dorsal scapular n. (C4–C5) Steadies scapula; draws scapula medially upward ⑥ Rhomboid major Spinous processes of T1–T4 vertebrae CN, cranial nerve. D A S F G Latissimus dorsi (scapular part) Latissimus dorsi (vertebral part) Sacrum Ilium Iliac crest Latissimus dorsi (iliac part) T7 spinous processes Thoracolumbar fascia Scapula Teres major Humerus Coracoid process Crest of lesser tubercle Teres major Latissimus dorsi Inferior angle Scapula, costal surface Clavicle Acromion Intertuber-cular groove 321 25 Shoulder & Arm Fig. 25.30 Latissimus dorsi and teres major Posterior view. A Latissimus dorsi, schematic. C Teres major, schematic. B Latissimus dorsi and teres major. D Insertion of the latis-simus dorsi on the floor of the inter-tubercular groove and the teres major on the crest of the lesser tubercle of the humerus. Table 25.6 Latissimus dorsi and teres major Muscle Origin Insertion Innervation Action Latissimus dorsi ① Vertebral part Spinous processes of T7–T12 vertebrae; thoracolumbar fascia Floor of the intertubercular groove of the humerus Thoracodorsal n. (C6–C8) Internal rotation, adduction, extension, respiration (“cough muscle”) ② Scapular part Scapula (inferior angle) ③ Costal part 9th to 12th ribs ④ Iliac part Iliac crest (posterior one third) ⑤ Teres major Scapula (inferior angle) Crest of lesser tubercle of the humerus (anterior angle) Lower subscapular n. (C5, C6) Internal rotation, adduction, extension A S D Greater tubercle Lesser tubercle Intertuber-cular groove Biceps brachii, long head Radial tuberosity, biceps brachii tendon Ulnar tuberosity, brachialis tendon Bicipital aponeurosis Brachialis Biceps brachii Biceps brachii, short head Scapula, anterior surface Supraglenoid tubercle Coracoid process Lateral epicondyle Radial tuberosity Medial epicondyle Ulnar tuberosity Brachialis Shaft of humerus 322 Upper Limb Muscle Facts (IV) The anterior and posterior muscles of the arm may be classified respectively as flexors and extensors relative to the movement of the elbow joint. Although the coracobrachialis is topographically part of the anterior compartment, it is functionally grouped with the muscles of the shoulder (see p. 318). Fig. 25.31 Biceps brachii and brachialis Right arm, anterior view. A Schematic. B Biceps brachii and brachialis. C Brachialis. Table 25.7 Anterior muscles: Biceps brachii and brachialis Muscle Origin Insertion Innervation Action Biceps brachii ① Long head Supraglenoid tubercle of scapula Radial tuberosity and bicipital aponeurosis Musculocutaneous n. (C5–C6) Elbow joint: flexion; supination Shoulder joint: flexion; stabilization of humeral head during deltoid contraction; abduction and internal rotation of the humerus ② Short head Coracoid process of scapula ③ Brachialis Humerus (distal half of anterior surface) Ulnar tuberosity Musculocutaneous n. (C5–C6) and radial n. (C7, minor) Flexion at the elbow joint Note: When the elbow is flexed, the biceps brachii acts as a powerful supinator because the lever arm is almost perpendicular to the axis of pronation/supination. Long head tendon of origin Medial head Tendon of insertion Anconeus Lateral head Shaft of humerus Long head Tendon of insertion Anconeus Medial head Radial groove Lateral head tendon of origin Scapular spine Scapula, posterior surface Lateral border Infraglenoid tubercle Triceps brachii, long head Medial epicondyle Olecranon Ulna Radius Anconeus Lateral epicondyle Triceps brachii, lateral head Triceps brachii, medial head Shaft of humerus Greater tubercle Acromion A S D F 323 25 Shoulder & Arm Fig. 25.32 Triceps brachii and anconeus Right arm, posterior view. D Schematic. A Triceps brachii and anconeus. B Partially removed: Lateral head of triceps brachii. C Partially removed: Long head of triceps brachii. Table 25.8 Posterior muscles: Triceps brachii and anconeus Muscle Origin Insertion Innervation Action Triceps brachii ① Long head Scapula (infraglenoid tubercle) Olecranon of ulna Radial n. (C6–C8) Elbow joint: extension Shoulder joint, long head: extension and adduction ② Medial head Posterior humerus, distal to radial groove; medial intermuscular septum ③ Lateral head Posterior humerus, proximal to radial groove; lateral intermuscular septum ④ Anconeus Lateral epicondyle of humerus (variance: posterior joint capsule) Olecranon of ulna (radial surface) Extends the elbow and tightens its joint Articular fovea Anterior border Styloid process of radius Carpal articular surface Shaft of radius, anterior surface Interosseous border Radial tuberosity Neck of radius Head of radius, articular circumference Radial notch Head of ulna Styloid process of ulna Articular circum-ference Shaft of ulna, anterior surface Ulnar tuberosity Coronoid process Trochlear notch Radial notch Coronoid process Medial surface Head of ulna Posterior surface Posterior border Olecranon Radial tuberosity Dorsal tubercle Lateral surface Posterior border Neck of radius Interosseous border Head of radius, articular circumference Styloid process of ulna Styloid process of radius Upper Limb 324 26 Elbow & Forearm Radius & Ulna Fig. 26.1 Radius and ulna Right forearm. A Anterior view. B Posterior view. Articular fovea Head of radius Radial tuberosity Anterior border Shaft of radius, anterior surface Interosseous border Styloid process of radius Head of ulna Interosseous membrane Shaft of ulna, anterior surface Ulnar tuberosity Coronoid process Trochlear notch Olecranon Proximal radioulnar joint Cartilage-free strip Distal radioulnar joint Head of radius, articular circumference Articular fovea Radial notch Coronoid process Trochlear notch Olecranon Posterior Proximal radioulnar joint Posterior Posterior surface Medial surface Anterior surface Ulna Ulnar interosseous border Interosseous membrane Radial interosseous border Radius Anterior surface Anterior border Lateral surface Posterior surface Dorsal tubercle Styloid process of radius Carpal articular surface Ulnar notch of radius Head of ulna Styloid process of ulna Distal radioulnar joint Anterior 26 Elbow & Forearm 325 C Anterosuperior view. D Proximal (superior) view. E Transverse section, proximal view. F Distal (inferior) view. Radial fossa Lateral supracondylar ridge Lateral epicondyle Capitulum Head of radius Neck of radius Radial tuberosity Radius Ulna Ulnar tuberosity Coronoid process Trochlea Medial epicondyle Coronoid fossa Medial supracondylar ridge Humerus Capitulotrochlear groove Trochlea Capitulum Head of radius Radial tuberosity Radius Ulna Coronoid process Olecranon Medial epicondyle Medial supracondylar ridge Humerus Olecranon fossa Medial supracondylar ridge Medial epicondyle Ulnar groove Olecranon Ulna Radius Head of radius, articular circumference Lateral epicondyle Lateral supracondylar ridge Lateral border Humerus Olecranon Humeroulnar joint Humeroradial joint Proximal radioulnar joint Ulna Radius Head of radius Capitulum Lateral epicondyle Lateral supracondylar ridge Humerus Upper Limb 326 Elbow Joint Fig. 26.2 Elbow (cubital) joint Right limb. The elbow consists of three articulations between the humerus, ulna, and radius: the humeroulnar, humeroradial, and proximal radioulnar joints. A Anterior view. C Medial view. B Posterior view. D Lateral view. Sacciform recess Brachioradialis Lateral epicondyle Extensor carpi radialis longus Radial collateral lig. Humeroradial joint (capitulum of humerus and articular fovea) Anular lig. of radius Head of radius Biceps brachii tendon Supinator Ulna Forearm flexors Proximal radioulnar joint (articular circum-ference and radial notch of ulna) Ulnar collateral lig. Humeroulnar joint (humeral trochlea and trochlear notch) Medial epicondyle Triceps brachii Humerus Plane of section in b Plane of section in c Capitulotrochlear groove Brachialis Capitulum Brachioradialis Radius Head of radius Triceps brachii Humerus Articular circumference Supinator Anconeus Radial notch of ulna Fat pad Articular fovea Ulna Brachialis Coronoid fossa Coronoid process Trochlea Olecranon Olecranon bursa Olecranon fossa Triceps brachii Humerus Ulna Trochlear notch Fat pad 26 Elbow & Forearm 327 Fig. 26.3 Skeletal and soft-tissue elements of the right elbow joint C Sagittal section through the humeroulnar joint, medial view. A Posterior view of extended elbow: The epicondyles and olecranon lie in a straight line. B Medial view of flexed elbow: The epicondyles and olecranon lie in a straight line. C Posterior view of flexed elbow: The two epicondyles and the tip of the olecranon form an equilateral triangle. Fractures and dislocations alter the shape of the triangle. A Coronal section viewed from the front (note the planes of section shown in B and C). B Sagittal section through the humeroradial joint and proximal radioulnar joint, medial view. Clinical box 26.1 Assessing elbow injuries The fat pads between the fibrous capsule and synovial membrane are part of the normal anatomy of the elbow joint. The anterior pad is most readily seen on a sagittal MRI while the posterior pad is often hidden within the bony fossa (see Figs. 26.3 and 29.11). With an effusion of the joint space, the inferior edge of the anterior pad appears concave as it gets pushed superiorly by the intra-articular fluid. This causes the pad to resemble the shape of a ship’s sail, thus creating a characteristic “sail sign.” The alignment of the prominences in the elbow also aids in the identification of fractures and dislocations. Humerus Lesser tubercle, supracondylar ridge Lateral epicondyle Anular lig. of radius Neck of radius Radius Ulna Radial collateral lig. Olecranon Sacciform recess Humerus Medial epicondyle Olecranon Ulnar collateral lig. (posterior part) Ulnar collateral lig. (anterior part) Coronoid process Ulna Radius Radial tuberosity Anular lig. of radius Ulnar collateral lig. (transverse part) Humerus Lateral supracondylar ridge Lateral epicondyle Radial collateral lig. Olecranon Ulnar collateral lig. Ulnar groove Medial epicondyle Olecranon fossa Upper Limb 328 Ligaments of the Elbow Joint Fig. 26.4 Ligaments of the elbow joint Right elbow in flexion. A Posterior view. B Medial view. C Lateral view. Table 26.1 Joints and ligaments of the elbow Joint Articulating surfaces Ligament Humeroulnar joint Trochlea Ulna (trochlear notch) Ulnar collateral ligament Humeroradial joint Capitulum Radius (articular fovea) Radial collateral ligament Proximal radioulnar joint Radius (articular circumference) Ulna (radial notch) Anular ligament Clinical box 26.2 Epiphyseal plates Anular lig. Ulna Radial head Humerus Capitulum Subluxation of the radial head (nursemaid’s elbow) A common and painful injury of small children occurs when the arm is jerked upward with the forearm pronated, tearing the anular ligament from its loose attachment on the radial neck. As the immature radial head slips out of the socket, the ligament may become trapped between the radial head and the capitulum of the humerus. Supinating the forearm and flexing the elbow usually returns the radial head to the normal position. Humerus Medial epicondyle Ulnar collateral lig. Joint capsule Ulnar tuberosity Ulna Radius Radial tuberosity Anular lig. of radius Radial collateral lig. Lateral epicondyle Trochlea Coronoid process Anular lig. of radius Head of radius Lateral epicondyle Capitulotroch-lear groove Radial fossa Coronoid fossa Humerus Ulnar collateral lig. Ulna Radius Radial collateral lig. Sacciform recess Medial epicon-dyle Capitulum 26 Elbow & Forearm 329 Fig. 26.5 Joint capsule of the elbow Right elbow in extension, anterior view. A Intact joint capsule. B Windowed joint capsule. Axis of pronation/ supination Articular fovea Radial collateral lig. Anular lig. Radial tuberosity Anterior border Interosseous border of radius Styloid process of ulna Head of ulna Interosseous membrane Interosseous border of ulna Shaft of ulna Oblique cord Ulnar tuberosity Ulnar collateral lig. Coronoid process Styloid process of radius Palmar radioulnar lig. Neck of radius Interosseous membrane Styloid process of ulna Dorsal tubercle Trochlear notch Interosseous border Interosseous border of ulna Olecranon Radial tuberosity Ulnar tuberosity Proximal radio-ulnar joint Radial collateral lig. Anular lig. Head of ulna Posterior surface Posterior border Lateral surface Dorsal radioulnar lig. Axis of pronation/ supination Distal radioulnar joint Radius Upper Limb 330 Radioulnar Joints The proximal and distal radioulnar joints function together to enable pronation and supination movements of the hand. The joints are func-tionally linked by the interosseous membrane. The axis for pronation and supination runs obliquely from the center of the humeral capitulum through the center of the radial articular fovea down to the styloid process of the ulna. Fig. 26.6 Supination Right forearm, anterior view. Fig. 26.7 Pronation Right forearm, anterior view. Styloid process of radius Styloid process of ulna Ulnar notch Articular circumference Dorsal radioulnar lig. Palmar radioulnar lig. Styloid process of ulna Head of ulna Dorsal radioulnar lig. Palmar radioulnar lig. Dorsal tubercle Styloid process of ulna Head of ulna Radius, carpal articular surface Styloid process of radius Extensor carpi ulnaris tendon Distal radioulnar joint Dorsal Styloid process of radius 26 Elbow & Forearm 331 Fig. 26.9 Distal radioulnar joint rotation Right forearm, distal view of articular surfaces of radius and ulna. The dorsal and palmar radioulnar ligaments stabilize the distal radioulnar joint. A Supination. B Semipronation. C Pronation. Olecranon Trochlear notch Coronoid process Proximal radioulnar joint Articular fovea Head of radius, lunula Anular lig. Radial notch of ulna Anular lig. Olecranon Trochlear notch Coronoid process Fig. 26.8 Proximal radioulnar joint Right elbow, proximal (superior) view. A Proximal articular surfaces of radius and ulna. B Radius removed. Clinical box 26.3 Falls onto the outstretched arm often result in fractures of the distal radius. In a Colles’ fracture, the distal fragment is tilted dorsally. Radius fracture A B Biceps brachii tendon Bicipital aponeurosis Medial epicondyle, common head of flexors Pronator teres Brachialis Biceps brachii Brachioradialis Extensor carpi radialis longus Extensor carpi radialis brevis Abductor pollicis longus Palmaris longus Flexor carpi ulnaris Palmaris longus Flexor carpi radialis Triceps brachii Flexor digitorum superficialis Flexor digitorum profundus tendons Flexor digitorum superficialis tendons Flexor pollicis longus Flexor pollicis longus tendon Pronator teres Medial epicondyle, common head of flexors Biceps brachii Supinator Flexor pollicis longus Pronator quadratus Brachioradialis Abductor pollicis longus Flexor carpi ulnaris Flexor digitorum profundus tendons Brachialis Flexor digitorum superficialis Flexor digitorum superficialis tendons Flexor pollicis longus tendon Upper Limb 332 Muscles of the Forearm: Anterior Compartment Fig. 26.10 Anterior muscles of the forearm: Dissection Right forearm, anterior view. Muscle origins are shown in red, insertions in blue. A Superficial flexors and radialis muscles. B Removed: Radialis muscles (brachioradialis, extensor carpi radialis lon-gus, and extensor carpi radialis brevis), flexor carpi radialis, flexor carpi ulnaris, abductor pollicis longus, palmaris longus, and biceps brachii. Brachialis Pronator teres, humeral head Medial epicondyle, common head of flexors Flexor digitorum superficialis, ulnar head Biceps brachii Supinator Flexor digitorum superficialis, radial head Pronator teres Flexor digitorum profundus Pronator quadratus Flexor pollicis longus tendon Flexor digitorum profundus tendons Flexor pollicis longus Brachioradialis Extensor carpi radialis longus Extensor carpi radialis brevis Lateral epicondyle, common head of extensors, supinator Biceps brachii Flexor digitorum superficialis, radial head Pronator teres Flexor pollicis longus Pronator quadratus Brachioradialis Abductor pollicis longus Flexor carpi radialis Flexor pollicis longus Flexor digitorum profundus Flexor digitorum superficialis Flexor carpi ulnaris Supinator Brachialis Pronator teres, ulnar head Flexor digitorum superficialis, ulnar head Medial epicondyle, common head of flexors Pronator teres, humeral head Brachialis Flexor digitorum profundus 26 Elbow & Forearm 333 C Removed: Pronator teres and flexor digitorum superficialis. D Removed: Brachialis, supinator, pronator quadratus, and deep flexors. Olecranon Inter-tendinous connections Extensor digitorum tendons, dorsal digital expansion Extensor pollicis longus tendon Extensor digitorum Brachioradialis Extensor carpi radialis longus Extensor carpi radialis brevis Triceps brachii Anconeus Flexor carpi ulnaris Brachioradialis Extensor pollicis brevis Abductor pollicis longus Extensor carpi radialis brevis Extensor digiti minimi Extensor carpi ulnaris Dorsal (“Lister’s”) tubercle of radius Brachioradialis Extensor carpi radialis longus Extensor carpi radialis brevis Supinator Medial epicondyle, common head of flexors Triceps brachii Anconeus Flexor digitorum profundus Flexor carpi ulnaris Extensor carpi ulnaris Abductor pollicis longus Extensor pollicis longus Extensor indicis Extensor pollicis brevis Brachioradialis Extensor digitorum Extensor digiti minimi Extensor carpi radialis brevis tendon Extensor carpi radialis longus tendon Upper Limb 334 Muscles of the Forearm: Posterior Compartment Fig. 26.11 Posterior muscles of the forearm: Dissection Right forearm, posterior view. Muscle origins are shown in red, insertions in blue. A Superficial extensors and radialis group. B Removed: Triceps brachii, anconeus, flexor carpi ulnaris, extensor carpi ulnaris, and extensor digitorum. Brachioradialis Extensor carpi radialis longus Extensor carpi radialis brevis Lateral epicondyle, common head of extensors Supinator Flexor digitorum profundus Brachioradialis Pronator teres Abductor pollicis longus Extensor pollicis brevis Extensor pollicis longus Extensor indicis Extensor pollicis longus Abductor pollicis longus Extensor carpi radialis longus Extensor carpi radialis brevis Dorsal (“Lister’s”) tubercle Triceps brachii Medial epicondyle, common head of flexors Anconeus Flexor digitorum profundus Flexor carpi ulnaris Extensor carpi ulnaris Extensor digiti minimi Extensor digitorum Extensor indicis Extensor pollicis longus Extensor pollicis brevis Extensor carpi radialis longus Extensor carpi radialis brevis Abductor pollicis longus Brachioradialis Extensor indicis Extensor pollicis brevis Extensor pollicis longus Abductor pollicis longus Pronator teres Supinator Lateral epicondyle, common head of extensors Extensor carpi radialis brevis Extensor carpi radialis longus Brachioradialis Supinator, humeral head Interosseous membrane 26 Elbow & Forearm 335 C Removed: Abductor pollicis longus, extensor pollicis longus, and radialis muscles. D Removed: Flexor digitorum profundus, supinator, extensor pollicis brevis, and extensor indicis. ① ② ③ ④ Radial head Humeral-ulnar head ⑤ Upper Limb 336 Muscle Facts (I) Fig. 26.12 Anterior compartment of the forearm Right forearm, anterior view. Table 26.2 Anterior compartment of the forearm Muscle Origin Insertion Innervation Action Superficial muscles ① Pronator teres Humeral head: medial epicondyle of humerus Ulnar head: coronoid process Lateral radius (distal to supinator insertion) Median n. (C6, C7) Elbow: weak flexion Forearm: pronation ② Flexor carpi radialis Medial epicondyle of humerus Base of 2nd metacarpal (variance: base of 3rd metacarpal) Wrist: flexion and abduction (radial deviation) of hand ③ Palmaris longus Palmar aponeurosis Median n. (C7, C8) Elbow: weak flexion Wrist: flexion tightens palmar aponeurosis ④ Flexor carpi ulnaris Humeral head: medial epicondyle Ulnar head: olecranon Pisiform; hook of hamate; base of 5th metacarpal Ulnar n. (C7–T1) Wrist: flexion and adduction (ulnar deviation) of hand Intermediate muscles ⑤ Flexor digitorum superficialis Humeral-ulnar head: medial epicondyle of humerus and coronoid process of ulna Radial head: upper half of anterior border of radius Sides of middle phalanges of 2nd to 5th digits Median n. (C8, T1) Elbow: weak flexion Wrist, MCP, and PIP joints of 2nd to 5th digits: flexion Deep muscles ⑥ Flexor digitorum profundus Ulna (proximal two thirds of flexor surface) and interosseous membrane Distal phalanges of 2nd to 5th digits (palmar surface) Median n. (C8, T1, radial half of fingers 2 and 3) Ulnar n. (C8, T1, ulnar half of fingers 4 and 5) Wrist, MCP, PIP, and DIP joints of 2nd to 5th digits: flexion ⑦ Flexor pollicis longus Radius (midanterior surface) and adjacent interosseous membrane Distal phalanx of thumb (palmar surface) Median n. (C8, T1) Wrist: flexion and abduction (radial deviation) of hand Carpometacarpal joint of thumb: flexion MCP and IP joints of thumb: flexion ⑧ Pronator quadratus Distal quarter of ulna (anterior surface) Distal quarter of radius (anterior surface) Hand: pronation Distal radioulnar joint: stabilization DIP, distal interphalangeal; IP, interphalangeal; MCP, metacarpophalangeal; PIP, proximal interphalangeal. A Superficial. B Intermediate. ⑥ ⑦ ⑧ C Deep. Pisiform bone Radial tuberosity Pronator teres Flexor carpi radialis Flexor digitorum superficialis Palmar aponeurosis Base of 2nd metacarpal 2nd through 5th middle phalanges Base of 5th metacarpal Hook of hamate Flexor carpi ulnaris Palmaris longus Medial epicondyle, common head of flexors Interosseous membrane Flexor digitorum superficialis, radial head Flexor digitorum superficialis, humeral-ulnar head Radial tuberosity Interosseous membrane Radius Pronator quadratus Tubercle of trapezium Trapezium Base of 1st distal phalanx Hook of hamate Pisiform bone Medial epicondyle Ulnar tuberosity Flexor digitorum profundus Flexor pollicis longus Coronoid process 4th distal phalanx 26 Elbow & Forearm 337 Fig. 26.13 Anterior compartment of the forearm Right forearm, anterior view. B Intermediate muscles. A Superficial muscles. C Deep muscles. Clinical box 26.4 Lateral epicondylitis Lateral epicondylitis, or tennis elbow, involves the extensor muscles and tendons of the forearm that attach on the lateral epicondyle. The tendon most commonly involved is that of the extensor carpi radialis brevis, a muscle that helps stabilize the wrist when the elbow is extended. When the extensor carpi radialis brevis is weakened from overuse, microscopic tears form in the tendon where it attaches to the lateral epicondyle. This leads to inflammation and pain. There is some evidence that the inflammation can extend back along the tendon to the periosteum of the lateral epicondyle. Athletes are not the only people who get tennis elbow and are actually in the minority — leading some to suggest the condition be referred to as “lateral elbow syndrome”. Workers whose activities require repetitive and vigorous use of the forearm muscles, such as common to painters, plumbers, and carpenters, are particularly prone to developing this pathology. Studies show a high incidence also among auto workers, cooks, and butchers. Common signs and symptoms of tennis elbow include pain with wrist extension against resistance, point tenderness or burning on the lateral epicondyle, and weak grip strength. Symptoms are intensified with forearm activity. A D S Upper Limb 338 Muscle Facts (II) Fig. 26.14 Posterior compartment of the forearm: Radialis muscles Right forearm, posterior view, schematic. Table 26.3 Posterior compartment of the forearm: Radialis muscles Muscle Origin Insertion Innervation Action ① Brachioradialis Distal humerus (lateral surface), lateral intermuscular septum Styloid process of the radius Radial n. (C5, C6) Elbow: flexion Forearm: semipronation ② Extensor carpi radialis longus Lateral supracondylar ridge of distal humerus, lateral intermuscular septum 2nd metacarpal (base) Radial n. (C6, C7) Elbow: weak flexion Wrist: extension and abduction ③ Extensor carpi radialis brevis Lateral epicondyle of humerus 3rd metacarpal (base) Radial n. (C7, C8) Ulna Radius Lateral supra-condylar crest Lateral epicondyle Olecranon Base of 3rd metacarpal 3rd metacarpal 2nd metacarpal Base of 2nd metacarpal Styloid process of radius Extensor carpi radialis brevis Extensor carpi radialis longus Brachioradialis Humerus Medial epicondyle Shaft of 2nd metacarpal Humerus Olecranon Radius Interosseous membrane Ulna Base of 3rd metacarpal Base of 2nd metacarpal Styloid process of radius Brachioradialis tendon Extensor carpi radialis brevis Extensor carpi radialis longus Lateral epicondyle Brachioradialis 26 Elbow & Forearm 339 Fig. 26.15 Posterior compartment of the forearm: Radialis muscles Right forearm. A Lateral (radial) view. B Posterior view. ① ② ③ ④ ⑤ ⑦ ⑧ ⑥ Upper Limb 340 Muscle Facts (III) Fig. 26.16 Posterior compartment of the forearm Right forearm, posterior view. Table 26.4 Posterior compartment of the forearm Muscle Origin Insertion Innervation Action Superficial muscles ① Extensor digitorum Common head (lateral epicondyle of humerus) Dorsal digital expansion of 2nd to 5th digits Radial n. (C7, C8) Wrist: extension MCP, PIP, and DIP joints of 2nd to 5th digits: extension/abduction of fingers ② Extensor digiti minimi Dorsal digital expansion of 5th digit Wrist: extension, ulnar abduction of hand MCP, PIP, and DIP joints of 5th digit: extension and abduction of 5th digit ③ Extensor carpi ulnaris Common head (lateral epicondyle of humerus) Ulnar head (dorsal surface) Base of 5th metacarpal Wrist: extension, adduction (ulnar deviation) of hand Deep muscles ④ Supinator Olecranon, lateral epicondyle of humerus, radial collateral ligament, annular ligament of radius Radius (between radial tuberosity and insertion of pronator teres) Radial n. (C6, C7) Radioulnar joints: supination ⑤ Abductor pollicis longus Radius and ulna (dorsal surfaces, interosseous membrane) Base of 1st metacarpal Radial n. (C7, C8) Radiocarpal joint: abduction of the hand Carpometacarpal joint of thumb: abduction ⑥ Extensor pollicis brevis Radius (posterior surface) and interosseous membrane Base of proximal phalanx of thumb Radiocarpal joint: abduction (radial deviation) of hand Carpometacarpal and MCP joints of thumb: extension ⑦ Extensor pollicis longus Ulna (posterior surface) and interosseous membrane Base of distal phalanx of thumb Wrist: extension and abduction (radial deviation) of hand Carpometacarpal joint of thumb: adduction MCP and IP joints of thumb: extension ⑧ Extensor indicis Ulna (posterior surface) and interosseous membrane Posterior digital extension of 2nd digit Wrist: extension MCP, PIP, and DIP joints of 2nd digit: extension DIP, distal interphalangeal; IP, interphalangeal; MCP, metacarpophalangeal; PIP, proximal interphalangeal. A Superficial muscles. B Deep muscles. Olecranon Ulna Extensor carpi ulnaris Extensor digiti minimi Base of 5th metacarpal 5th proximal phalanx, base Dorsal digital expansion, intertendi-nous con-nections Radius Extensor digitorum Common head of extensor digitorum, extensor digiti minimi, and extensor carpi ulnaris Lateral epicondyle Dorsal tubercle Medial epicondyle Ulnar groove Olecranon Ulna Posterior border of ulna 1st distal phalanx, base 1st proximal phalanx, base 1st meta-carpal Base of 1st metacarpal Extensor indicis Extensor pollicis brevis Extensor pollicis longus Abductor pollicis longus Radius Supinator Lateral epicondyle 2nd meta-carpal 26 Elbow & Forearm 341 Fig. 26.17 Posterior compartment of the forearm: Superficial and deep muscles Right forearm, posterior view. A Superficial extensors. B Deep extensors with supinator. Meta-carpals Carpal bones Phalanges 1st metacarpal Ulna Radius Styloid process of radius Scaphoid Trapezium Trapezoid Hamate Capitate Triquetrum Lunate Styloid process of ulna 2nd distal phalanx 2nd middle phalanx 2nd proximal phalanx Upper Limb 342 27 Wrist & Hand Bones of the Wrist & Hand Fig. 27.1 Dorsal view Right hand. Table 27.1 Bones of the wrist and hand Phalanges 1st to 5th proximal phalanges 2nd to 5th middle phalanges 1st to 5th distal phalanges Metacarpal bones 1st to 5th metacarpals Carpal bones Trapezium Scaphoid Trapezoid Lunate Capitate Triquetrum Hamate Pisiform There are only four middle phalanges (the thumb has only a proximal and a distal phalanx). Proximal Distal Meta-carpal Hook of hamate Tuberosity of distal phalanx Head Shaft Base Pisiform Triquetrum Lunate Styloid process Head Trapezoid Sesamoid bones Head Shaft Ulna Radius Tubercle of scaphoid Styloid process of radius Capitate Middle phalanx Base Tubercle of trapezium Triquetrum Capitate Trapezium Pisiform Hook of hamate Scaphoid Lunate Trapezoid 27 Wrist & Hand 343 Clinical box 27.1 Scaphoid Fractures Scaphoid fractures are the most common carpal bone fractures, generally occurring at the narrowed waist between the proximal and distal poles (A, right scaphoid red line; B, white arrow). Because blood supply to the scaphoid is transmitted via the distal segment, fractures at the waist can compromise the supply to the proximal segment, often resulting in nonunion and avascular necrosis. A B Fig. 27.2 Palmar view Right hand. Fig. 27.3 Radiograph of the wrist Anteroposterior view of left limb. 1st to 5th metacarpals Trapezium Trapezoid Styloid process of radius Dorsal tubercle a n l U s u i d a R Styloid process of ulna Lunate Scaphoid Triquetrum Hamate Capitate Upper Limb 344 Carpal Bones Fig. 27.4 Carpal bones of the right wrist A Carpal bones of the right wrist with the wrist in flexion, proximal view. B Carpal and metacarpal bones of the right wrist with radius and ulna removed, proximal view. Tubercle of trapezium Pisiform Carpal tunnel Scaphoid Tubercle of scaphoid Styloid process of radius Dorsal tubercle Radius, carpal articular surface Articular capsule Ulnar collateral ligament of wrist joint Styloid process of ulna Articular disk (ulnocarpal disk) Pisiform Triquetrum Lunate Proximal row of carpal bones Capitate Trapezoid Trapezium Tubercle of trapezium Tubercle of scaphoid Scaphoid Lunate Triquetrum Pisiform Hook of hamate Hamate Distal row of carpal bones 1st to 5th metacarpals 27 Wrist & Hand 345 C Articular surfaces of the radiocarpal joint of the right wrist. The proximal row of carpal bones is shown from the proximal view. The articular surfaces of the radius and ulna, and the articular disk (ulnocarpal disk) are shown from the distal view. D Articular surfaces of the midcarpal joint of the right wrist. The distal row of carpal bones is shown from the proximal view. The proximal row is shown from the distal view. Carpometacarpal joint Midcarpal joint Interosseous membrane Ulnar collateral lig. of wrist joint Lunate Triquetrum Pisiform Hamate Abductor digiti minimi 5th metacarpal 4th dorsal interosseous Metacarpophalangeal joint Proximal interphalangeal joint Distal interphalangeal joint Metacarpophalangeal joint Carpometacarpal joint of the thumb Radiocarpal joint Distal phalanx Middle phalanx Proximal phalanx Collateral ligaments 1st dorsal interosseous 1st metacarpal Opponens pollicis Trapezium Trapezoid Radial collateral lig. of wrist joint Capitate Scaphoid Interphalangeal joint Distal radioulnar joint Articular disk (ulnocarpal) Midcarpal joint Radiocarpal joint Carpometacarpal joints Distal radioulnar joint Metacarpo-phalangeal joint Distal inter-phalangeal joint Proximal interphalangeal joint Inter-phalangeal joint of thumb Metacarpo-phalangeal joint of thumb Carpometacarpal joint of thumb Upper Limb 346 Joints of the Wrist & Hand Fig. 27.5 Joints of the wrist and hand A Right hand, posterior (dorsal) view. B Coronal section. Right hand, posterior (dorsal) view. Clinical box 27.2 Abduction and adduction movements are described in relation to the middle finger: all movements away from the middle finger are classified as abduction, all movements toward the middle finger as adduction. Abduction and adduction at the metacarpophalangeal joint Adduction Abduction Abduction/ adduction axis Flexion/ extension axis Trapezium 27 Wrist & Hand 347 Fig. 27.6 Carpometacarpal joint of the thumb Right hand, radial view. The 1st metacarpal bone has been moved slightly distally to expose the articular sur-face of the trapezium. Two cardinal axes of motion are shown here: (a) abduction/adduction and (b) flexion/ extension. Fig. 27.7 Movements of the carpometacarpal joint of the thumb Right hand, Palmar view. Tuberosity of distal phalanx Head Shaft Base 1st proximal phalanx Trapezium Styloid process of radius Radius Ulna Styloid process of ulna Scaphoid Lunate Capitate Trapezoid Base Shaft Head Proximal phalanx Middle phalanx Distal phalanx 1st distal phalanx 1st metacarpal a b Phalanx Meta-carpal A The neutral (0°) position. B Axes of motion in the carpo-metacarpal joint of the thumb. C Adduction. D Abduction. E Flexion. F Extension. G Opposition. Dorsal carpo-metacarpal ligs. Radial collateral lig. of wrist joint Dorsal radiocarpal lig. Ulnar collateral lig. of wrist joint Dorsal intercarpal ligs. Dorsal metacarpal ligs. Metacarpophalangeal joint (collateral ligs.) Proximal interphalangeal joint (collateral ligs.) Distal interphalangeal joint (collateral ligs.) Dorsal radioulnar lig. Upper Limb 348 Ligaments of the Hand Fig. 27.8 Ligaments of the hand Right hand. A Posterior (dorsal) view. Clinical box 27.3 Palmar flexion and dorsal extension occur around a transverse axis (A) that runs through the lunate bone (radiocarpal joint) and capitate bone (midcarpal joint). Radial and ulnar deviation (B) occur around a dorsopalmar axis that runs through the capitate bone. Movements at the radiocarpal and midcarpal joints 40–60° 0° 60–80° Dorsal extension Transverse axis Palmar flexion Radial deviation Dorsopalmar axis 30–40° Ulnar deviation 0° 20° A B 27 Wrist & Hand 349 Distal interphalangeal joint capsule Proximal interphalangeal joint capsule Deep transverse metacarpal ligs. Metacarpo-phalangeal joint capsule Palmar metacarpal ligs. Palmar intercarpal ligs. Flexor carpi ulnaris tendon Palmar ulnocarpal lig. Radial collateral lig. of wrist joint Palmar carpometa-carpal ligs. Palmar ligs. Palmar radiocarpal lig. Palmar radioulnar lig. B Anterior (palmar) view. Cut: Flexor retinaculum. Functional position of the hand Clinical box 27.4 The anatomic position of the hand, in which the palm is flat, the fingers are extended, and the forearm is supinated (palm facing forward), differs from the normal relaxed position of the hand. At rest, the forearm is in mid-supination/pronation (palm facing the body), the wrist is slightly extended, the fingers form an arcade of flexion, and the thumb is in the neutral position. Postoperative immobilization of the hand (by a cast or splint) fixes the fingers in flexion and the wrist in extension to prevent shortening of the ligaments and to maintain the ability of the hand to assume normal resting position. 10° 30° 50–60° 30° Flexor retinaculum (transverse carpal lig.) Radius Tubercle of trapezium Carpal tunnel entrance Ulna Pisiform bone Hook of hamate Flexor retinaculum (transverse carpal lig.) Tubercle of scaphoid Ulnar carpal eminence Pisiform Hook of hamate Triquetrum Lunate Capitate Trapezoid Carpometacarpal joint of the thumb Tubercle of trapezium Radial carpal eminence Carpal tunnel Tubercle of trapezium Trapezoid Capitate Hamate Hook of hamate Passage for flexor carpi radialis tendon Trapezium Flexor retinaculum A B Carpal tunnel Flexor retinac-ulum (trans-verse carpal lig.) Scaphoid Capitate Hamate Triquetrum Pisiform Ulnar tunnel Palmar carpal lig. Passage for flexor carpi radialis tendon Upper Limb 350 Ligaments & Compartments of the Wrist Fig. 27.9 Ligaments and bony boundaries of the carpal tunnel Right hand, anterior (palmar) view. A Carpal tunnel and flexor retinaculum. B Bony boundaries of the carpal tunnel. Fig. 27.10 Carpal tunnel Right hand, transverse section. The contents of the carpal and ulnar tunnels are discussed on p. 391. A Proximal part of the carpal tunnel. B Distal part of the carpal tunnel. Trapezoid Capitate Scaphoid Lunate Dorsal tubercle Radiotri-quetral lig. Radius Dorsal radioulnar lig. Ulna Ulnolunate lig. Styloid process of ulna Ulnotriquetral lig. Ulnocarpal meniscus homologue Ulnar collateral lig. of wrist joint Triquetrum Hamate Metacarpals Midcarpal joint Lunate Interosseous membrane Radius Ulnocarpal disk Radiocarpal joint Distal radioulnar joint Styloid process of ulna Extensor carpi ulnaris muscle tendon sheath Ulnocarpal meniscus homologue Ulnar collateral lig. of wrist joint Triquetrum Hamate Radius Dorsal tubercle Radius carpal articular surface Radiotriquetral lig. Dorsal radioulnar lig. Ulnocarpal meniscus homologue Styloid process of ulna Ulnar carpal collateral lig. Palmar radioulnar lig. Ulnocarpal disk (triangular disk) Ulnotriquetral lig. Ulnolunate lig. 27 Wrist & Hand 351 Fig. 27.11 Ulnocarpal complex Right hand. The ulnocarpal complex (triangular fibrocartilage com-plex) consists of ligaments and disks that connect the distal ulna, distal radioulnar joint, and the proximal carpal row. A Right wrist, posterior (dorsal) view. C Right wrist, distal view. B Schematic of a section through the triangular fibrocartilage (ulnocarpal) complex. Radial collateral lig. Interosseous ligs. Pisiform Ulnar collateral lig. Ulnocarpal meniscus homologue Ulnocarpal disk Distal radioulnar joint Radiocarpal joint Medial carpal compartment Thumb saddle joint Carpometacarpal compartment Intermetacarpal joint Fig. 27.12 Compartments of the wrist Right wrist, posterior view, schematic. Interosseous ligaments and the ulnocarpal disk divide the inter-articular space into compartments. A5 C3 A4 C2 A3 C1 A2 A1 Flexor digitorum superficialis tendon Flexor digitorum profundus tendon Metacarpo-phalangeal joint Proximal interphalangeal joint Distal inter-phalangeal joint Proximal phalanx A2 Phalangoglenoid lig. A1 Meta-carpal bone Accessory collateral lig. Collateral lig. Accessory collateral lig. Collateral lig. Phalangoglenoid lig. Anular ligs. (A1–A5) Deep transverse metacarpal lig. 3rd metacarpal Phalangoglenoid lig. Cruciform lig. (C1) Collateral ligs. Cruciform lig. (C3) Collateral ligs. Flexor digitorum superficialis tendon Flexor digitorum profundus tendon Upper Limb 352 Ligaments of the Fingers Fig. 27.13 Ligaments of the fingers: Lateral view Right middle finger. Joint capsules, ligaments, and digital tendon sheaths. The outer fibrous layer of the tendon sheaths (stratum fibro-sum) is strengthened by the anular and cruciform ligaments, which also bind the sheaths to the palmar surface of the phalanx and prevent palmar deviation of the sheaths during flexion. A Extension. Note: Whereas the 1st through 5th anular ligaments (A1–A5) have fixed positions, the cruciform ligaments (C1–C3) are highly variable in their course. B Flexion. C Extension of the metacarpophalangeal joint. Note: The collateral ligament is lax. D Flexion of the metacarpo-phalangeal joint. Note: The collateral ligament is taut. Fig. 27.14 Ligaments during extension and flexion of fingers: Lateral view Anular lig. (A1) Palmar lig. Collateral lig. Flexor digitorum superficialis tendon Flexor digitorum profundus tendon Deep transverse metacarpal lig. 3rd metacarpal bone Extensor digitorum tendon Dorsal Distal phalanx Nail Tuberosity of distal phalanx Flexor digitorum profundus tendon Palmar lig. Middle phalanx Extensor digitorum tendon (dorsal digital expansion) Distal interphalangeal joint Anular ligs. (A1–A5) Deep transverse metacarpal lig. Metacarpal bone Proximal phalanx Flexor digitorum superficialis tendon Proximal inter-phalangeal joints (collateral ligs.) Middle phalanx Cruciform lig. (C3) Distal inter-phalangeal joints (collateral ligs.) Flexor digitorum profundus tendon Flexor digitorum profundus tendon Flexor digitorum superficialis tendon Metacarpo-phalangeal joint (collateral ligs.) Cruciform lig. (C1) Plane of section in Fig. 27.16 27 Wrist & Hand 353 A Superficial ligaments. B Deep ligaments with digital tendon sheath removed. Fig. 27.16 Third metacarpal: Transverse section Proximal view. Fig. 27.17 Fingertip: Longitudinal section The palmar articular surfaces of the phalanges are enlarged proximally at the joints by the palmar ligament. This fibrocartilaginous plate, also known as the volar plate, forms the floor of the digital tendon sheaths. Fig. 27.15 Ligaments of the fingers: Anterior (palmar) view Right middle finger. Anular ligs. (A1–A5) Cruciform ligs. Superficial transverse metacarpal lig. Abductor digiti minimi Palmaris brevis Palmar aponeurosis Palmaris longus tendon Adductor pollicis Flexor pollicis brevis Abductor pollicis brevis Opponens pollicis Flexor digiti minimi brevis Transverse fascicles Antebrachial fascia Flexor retinaculum Flexor carpi ulnaris Deep transverse metacarpal lig. Longitudinal fascicles Flexor digitorum profundus tendons Flexor digitorum superficialis tendons Common flexor tendon sheath Flexor carpi radialis Flexor pollicis longus Flexor digitorum superficialis Pronator quadratus Flexor retinaculum Flexor pollicis longus tendon Upper Limb 354 Muscles of the Hand: Superficial & Middle Layers Fig. 27.18 Intrinsic muscles of the hand: Superficial and middle layers Right hand, palmar surface. A Palmar aponeurosis. B Carpal and digital tendon sheaths. Removed: Palmar aponeurosis, palmaris longus, antebrachial fascia, and palmaris brevis. Clinical box 27.5 Gradual atrophy of the palmar aponeurosis leads to progressive shortening of the palmar fascia, chiefly affecting the 4th and 5th digits. Over a period of years, the contracture may become so severe that the fingers assume a flexed position (with fingertips touching the palms), severely compromising the grasping ability of the hand. The causes of Dupuytren’s contracture are poorly understood, but it is a relatively common condition, most prevalent in men over 40 and associated with chronic liver disease (i.e., cirrhosis). Treatment generally consists of complete surgical removal of the palmar aponeurosis. Dupuytren’s contracture Also known as transverse carpal ligament. 1st dorsal interosseus Adductor pollicis (transverse head) Adductor pollicis (oblique head) Flexor pollicis brevis (super-ficial head) Abductor pollicis brevis Flexor retinaculum Opponens pollicis Abductor pollicis longus Opponens digiti minimi Flexor digiti minimi brevis Abductor digiti minimi Lumbricals Deep transverse metacarpal lig. Flexor digitorum profundus tendons Flexor retinaculum Flexor digitorum superficialis tendons Flexor carpi radialis tendon Abductor pollicis longus tendon Extensor pollicis brevis Abductor pollicis brevis Opponens pollicis Flexor pollicis brevis (superficial head) Flexor pollicis brevis (super-ficial head) Flexor pollicis longus tendon Flexor carpi ulnaris tendon Abductor digiti minimi Flexor digiti minimi brevis Lumbricals Flexor digitorum profundus Flexor pollicis longus Flexor digitorum profundus tendons 27 Wrist & Hand 355 C Superficial layer of muscles. Removed: Tendon sheaths. D Middle layer of muscles. Removed: Flexor digitorum superficialis, flexors carpi radialis and ulnaris, and pronator quadratus. Clinical box 27.6 The digital tendon sheath of the thumb is continuous with the carpal tendon sheath of the flexor pollicis longus. The remaining fingers show variable communication with the carpal tendon sheaths (A is the most common variation). Infections within the tendon sheaths from puncture wounds of the fingers can track proximally to communicating spaces of the hand. Tendon sheath communication A B C Flexor digitorum profundus tendons Flexor digitorum superficialis tendons Flexor pollicis longus tendon Flexor retinaculum Opponens pollicis Abductor pollicis brevis Flexor pollicis brevis (superficial head) Adductor pollicis (oblique head) Adductor pollicis (transverse head) Abductor digiti minimi Abductor digiti minimi Flexor digiti minimi brevis Opponens digiti minimi 2nd and 3rd palmar interossei Flexor digiti minimi brevis Lumbricals Palmar ligs. 1st through 4th dorsal interossei 1st through 3rd palmar interossei Opponens digiti minimi Flexor carpi ulnaris tendon Extensor pollicis brevis Abductor pollicis longus tendon Opponens pollicis Flexor pollicis brevis (deep head) Flexor pollicis brevis (superficial head) Adductor pollicis Flexor carpi radialis tendon Upper Limb 356 Muscles of the Hand: Middle & Deep Layers Fig. 27.19 Intrinsic muscles of the hand: Middle and deep layers Right hand, palmar surface. A Middle layer of muscles. Cut: Flexor digitorum profundus, lumbricals, flexor pollicis longus, and flexor digiti minimi brevis. B Deep layer of muscles. Cut: Opponens digiti minimi, opponens pollicis, flexor pollicis brevis, and adductor pollicis (transverse and oblique heads). Extensor digitorum Extensor digiti minimi Abductor digiti minimi Dorsal interossei Opponens digiti minimi Extensor carpi ulnaris Extensor carpi radialis brevis Extensor carpi radialis longus Abductor pollicis longus Adductor pollicis Extensor pollicis brevis Extensor pollicis longus Palmar and dorsal interossei Extensor indicis 27 Wrist & Hand 357 Fig. 27.20 Origins and insertions of muscles of the hand Right hand. Muscle origins shown in red, insertions in blue. B Palmar (anterior) view. A Dorsal (posterior) view. Flexor digitorum superficialis Interossei Flexor pollicis longus Adductor pollicis Flexor pollicis brevis and abductor pollicis brevis 1st dorsal interosseus Flexor carpi radialis Opponens pollicis Abductor pollicis longus Abductor pollicis brevis Flexor pollicis brevis Flexor carpi ulnaris Extensor carpi ulnaris Opponens digiti minimi Abductor digiti minimi Flexor digiti minimi brevis Abductor digiti minimi Flexor digitorum profundus Radius Ulna 1st palmar interosseus 2nd dorsal interosseus 3rd dorsal interosseus 2nd palmar interosseus 4th dorsal interosseus 3rd palmar interosseus 1 2 3 4 5 6 S A D F G H Extensor pollicis brevis Extensor carpi radialis longus tendon Extensor digitorum Extensor carpi ulnaris Extensor digiti minimi Extensor retinaculum Extensor indicis tendon Abductor digiti minimi 4th dorsal interosseus 3rd dorsal interosseus 2nd dorsal interosseus 1st dorsal interosseus Extensor carpi radialis longus tendon Extensor carpi radialis brevis tendon Extensor pollicis longus tendon Abductor pollicis longus tendon Brachioradialis tendon Dorsal tubercle Interten-dinous connections Dorsal carpal tendon sheaths Plane of section in Fig. 27.21B ④ ⑤ ⑥ ① ② ③ Dorsal tubercle Extensor carpi ulnaris tendon ⑥ ⑤ Extensor digiti minimi tendon ④ Extensor indicis tendon Extensor digitorum tendon ④ ③ Extensor pollicis longus tendon Extensor carpi radialis brevis tendon Extensor carpi radialis longus tendon Extensor pollicis brevis tendon Abductor pollicis longus tendon Radius Extensor retinaculum Ulna ② ① ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ Upper Limb 358 Dorsum of the Hand Fig. 27.21 Extensor retinaculum and dorsal carpal tendon sheaths B Posterior (dorsal) compartments, proximal view of section in Fig. 27.21A. Table 27.2 Dorsal compartments for extensor tendons ① Abductor pollicis longus Extensor pollicis brevis ② Extensor carpi radialis longus Extensor carpi radialis brevis ③ Extensor pollicis longus ④ Extensor digitorum Extensor indicis ⑤ Extensor digiti minimi ⑥ Extensor carpi ulnaris A Right hand, posterior (dorsal) view. Fig. 27.22 Muscles and tendons of the dorsum Right hand, posterior (dorsal) view. Central band Lateral bands Lumbrical slip Interosseous slip 2nd lumbrical Extensor digitorum tendon 3rd metacarpal 3rd dorsal interosseus 2nd dorsal interosseus Dorsal digital expansion Distal phalanx Deep transverse metacarpal lig. Plane of section in B Anular ligs. Flexor digitorum superficialis tendon Flexor digitorum profundus tendon 2nd lumbrical 3rd metacarpal Extensor digitorum tendon 2nd dorsal interosseus Interosseous slip Lumbrical slip Dorsal digital expansion Distal phalanx Anular lig. (A1) Palmar lig. Collateral ligs. 3rd dorsal interosseus (fibers attached to bone) 3rd dorsal interosseus (fibers attached to extensor tendon) 2nd dorsal interosseus Deep transverse metacarpal lig. 2nd lumbrical Flexor digitorum superficialis tendon Flexor digitorum profundus tendon Deep transverse metacarpal lig. 3rd metacarpal Extensor digitorum tendon Dorsal Distal interphalangeal joint Proximal interphalangeal joint Vincula brevia Metacarpo-phalangeal joint Vinculum longum Flexor digitorum profundus tendon Deep transverse metacarpal lig. Flexor digitorum superficialis tendon 27 Wrist & Hand 359 Fig. 27.23 Dorsal digital expansion Right hand, middle finger. The dorsal digital expansion permits the long digital flexors and the short muscles of the hand to act on all three finger joints. A Posterior view. C Radial view. B Cross section through 3rd metacarpal head, proximal view. D Radial view with common tendon sheath of flexor digitorum super ficialis and profundus opened. ② ① ③ ④ ⑦ ⑥ ⑤ Upper Limb 360 Muscle Facts (I) The intrinsic muscles of the hand are divided into three groups: the thenar, hypothenar, and metacarpal muscles (see p. 362). The thenar muscles are responsible for movement of the thumb, while the hypothenar muscles move the 5th digit. Fig. 27.24 Thenar and hypothenar muscles Right hand, palmar (anterior) view, schematic. Table 27.3 Thenar muscles Muscle Origin Insertion Innervation Action ① Adductor pollicis Transverse head: 3rd metacarpal (palmar surface) Thumb (base of proximal phalanx) via the ulnar sesamoid Via the ulnar sesamoid Ulnar n. (C8, T1) C8, T1 CMC joint of thumb: adduction MCP joint of thumb: flexion Oblique head: capitate bone, 2nd and 3rd metacarpals (bases) ② Abductor pollicis brevis Scaphoid bone and trapezium, flexor retinaculum Thumb (base of proximal phalanx) via the radial sesamoid Via the radial sesamoid Median n. (C8, T1) CMC joint of thumb: abduction ③ Flexor pollicis brevis Superficial head: flexor retinaculum Superficial head: median n. (C8, T1) CMC joint of thumb: flexion Deep head: capitate bone, trapezium Deep head: ulnar n. (C8, T1) ④ Opponens pollicis Trapezium 1st metacarpal (radial border) Median n. (C8, T1) CMC joint of thumb: opposition CMC, carpometacarpal; MCP, metacarpophalangeal. Table 27.4 Hypothenar muscles Muscle Origin Insertion Innervation Action ⑤ Opponens digiti minimi Hook of hamate, flexor retinaculum 5th metacarpal (ulnar border) Ulnar n. (C8, T1) Draws metacarpal in palmar direction (opposition) ⑥ Flexor digiti minimi brevis 5th proximal phalanx (base) MCP joint of little finger: flexion ⑦ Abductor digiti minimi Pisiform bone 5th proximal phalanx (ulnar base) and dorsal digital expansion of 5th digit MCP joint of little finger: flexion and abduction of little finger PIP and DIP joints of little finger: extension Palmaris brevis Palmar aponeurosis (ulnar border) Skin of hypothenar eminence Tightens the palmar aponeurosis (protective function) DIP, distal interphalangeal; MCP, metacarpophalangeal; PIP, proximal interphalangeal. 5th proximal phalanx 5th meta-carpal Opponens digiti minimi Abductor digiti minimi Hook of hamate (under tendon) Capitate (under tendon) Pisiform (under tendon) Scaphoid Trapezium (under tendon) Oblique head Abductor pollicis brevis Transverse head 1st proximal phalanx Adductor pollicis 5th proximal phalanx Flexor digiti minimi brevis Hook of hamate (under tendon) Capitate (under tendon) Trapezium (under tendon) Opponens pollicis 1st proximal phalanx Flexor pollicis brevis 27 Wrist & Hand 361 Fig. 27.25 Thenar and hypothenar muscles Right hand, palmar (anterior) view. B Removed: Adductor pollicis, abductor pollicis brevis, abductor digiti minimi, and opponens digiti minimi. A Removed: Flexor pollicis brevis, opponens pollicis, and flexor digiti minimi brevis. F D S A K J H G L Ö Ä Upper Limb 362 Muscle Facts (II) The metacarpal muscles of the hand consist of the lumbricals and interossei. They are responsible for the movement of the digits (with the hypothenars, which act on the 5th digit). Fig. 27.26 Metacarpal muscles of the hand Right hand, palmar view, schematic. Table 27.5 Metacarpal muscles Muscle group Muscle Origin Insertion Innervation Action Lumbricals ① 1st Tendons of flexor digitorum profundus (radial sides) 2nd digit (dde) Median n. (C8, T1) 2nd to 5th digits: • MCP joints: flexion • Proximal and distal IP joints: extension ② 2nd 3rd digit (dde) ③ 3rd Tendons of flexor digitorum profundus (bipennate from medial and lateral sides) 4th digit (dde) Ulnar n. (C8, T1) ④ 4th 5th digit (dde) Dorsal interossei ⑤ 1st 1st and 2nd metacarpals (adjacent sides, two heads) 2nd digit (dde) 2nd proximal phalanx (radial side) 2nd to 4th digits: • MCP joints: flexion • Proximal and distal IP joints: extension and abduction from 3rd digit ⑥ 2nd 2nd and 3rd metacarpals (adjacent sides, two heads) 3rd digit (dde) 3rd proximal phalanx (radial side) ⑦ 3rd 3rd and 4th metacarpals (adjacent sides, two heads) 3rd digit (dde) 3rd proximal phalanx (ulnar side) ⑧ 4th 4th and 5th metacarpals (adjacent sides, two heads) 4th digit (dde) 4th proximal phalanx (ulnar side) Palmar interossei ⑨ 1st 2nd metacarpal (ulnar side) 2nd digit (dde) 2nd proximal phalanx (base) 2nd, 4th, and 5th digits: • MCP joints: flexion • Proximal and distal IP joints: extension and adduction toward 3rd digit ⑩ 2nd 4th metacarpal (radial side) 4th digit (dde) 4th proximal phalanx (base) ⑪ 3rd 5th metacarpal (radial side) 5th digit (dde) 5th proximal phalanx (base) dde, dorsal digital expansion; IP, interphalangeal; MCP, metacarpophalangeal. A Lumbricals. B Dorsal interossei. C Palmar interossei. 1st palmar interosseus 3rd palmar interosseus 2nd palmar interosseus 2nd through 5th metacarpals 3rd lumbrical (often arises by two heads) 4th lumbrical (often arises by two heads) Hook of hamate Pisiform Ulna Flexor digitorum profundus tendons Radius Trapezoid 2nd lumbrical 1st lumbrical 2nd metacarpal 2nd proximal phalanx 2nd distal phalanx, base 1st metacarpal 1st dorsal interosseus 2nd dorsal interosseus 2nd through 5th proximal phalanges 2nd through 5th metacarpals 3rd dorsal interosseus 4th dorsal interosseus 27 Wrist & Hand 363 Fig. 27.27 Metacarpal muscles Right hand, palmar (anterior) view. A Lumbrical muscles. B Dorsal interosseus muscles. C Palmar interosseus muscles. Vertebral a. Left common carotid a. Brachiocephalic trunk Superior thoracic a. Thoracic aorta Internal thoracic a. Circumflex scapular a. Thoraco-dorsal a. Lateral thoracic a. Subscapular a. Superior and inferior ulnar collateral aa. Ulnar recurrent a. Common interosseous a. Ulnar a. Deep palmar arch Superficial palmar arch Common palmar digital aa. Palmar digital aa. Thyrocervical trunk Subclavian a. Suprascapular a. Acromial br. Deltoid br. Pectoral br. Thoraco-acromial a. Axillary a. Anterior and posterior circumflex humeral aa. Deep a. of arm Brachial a. Radial collateral a. Middle collateral a. Radial recurrent a. Posterior interosseous a. Radial a. Anterior interosseous a. Superficial palmar br. (radial a.) Left subclavian a. Axillary a. Brachial a. Radial a. Ulnar a. Subclavian a. Brachiocephalic trunk Upper Limb 364 28 Neurovasculature Arteries of the Upper Limb Fig. 28.1 Arteries of the upper limb Right limb with the forearm supinated, anterior view. A Main arterial segments. B Course of the arteries. Ascending cervical a. Anterior scalene Superficial cervical a. (superficial br.) Dorsal scapular a. (deep br.) Transverse cervical a. Middle scalene Posterior scalene Supra-scapular a. Subclavian a. Clavicle Deep cervical a. Supreme inter-costal a. Internal thoracic a. 1st rib Costocervical trunk Thyrocervical trunk Common carotid a. Inferior thyroid a. Vertebral a. Axillary a. Thoraco-dorsal a. Transverse cervical a. Dorsal scapular a. Subscapular a. Brachial a. Deep a. of arm Posterior circumflex humeral a. Anterior circumflex humeral a. Axillary a. Circumflex scapular a. Acromial brs. Suprascapular a. Subclavian a. Thyro-cervical trunk Vertebral a. Posterior interosseous a. Dorsal carpal network Dorsal carpal a. Perforating br. Dorsal metacarpal a. Dorsal and palmar digital aa. Radial a. Dorsal Palmar Palmar carpal network Deep palmar arch Metacarpal palmar a. Superficial palmar arch Recurrent interosseous a. Posterior interosseous a. Anterior interosseous a. Radial a. Deep palmar arch Proper palmar digital aa. Palmar digital aa. Common palmar digital aa. Perforating brs. Superficial palmar arch Palmar carpal brs. (to palmar carpal network) Interosseous membrane Ulnar a. Common interosseous a. Princeps pollicis a. Radialis indicis a. Interosseous membrane Common interosseous a. Posterior interosseous a. Anterior interosseous a. Radial a. Dorsal digital aa. Dorsal carpal a. Ulnar a. (dorsal carpal br.) Anterior interosseous a. (posterior br.) Dorsal metacarpal aa. Dorsal carpal network 28 Neurovasculature 365 Fig. 28.2 Branches of the subclavian artery Right side, anterior view. Fig. 28.3 Scapular arcade Right side, posterior view. Fig. 28.4 Arteries of the forearm and hand Right limb. The ulnar and radial arteries are interconnected by the super ficial and deep palmar arches, the perforating branches, and the dorsal carpal network. B Anterior (palmar) view. C Posterior (dorsal) view. A Right middle finger, lateral view. Cephalic v. Median ante-brachial v. Cephalic v. Perforator vv. Superficial palmar venous arch Intercapitular vv. Median cubital v. Basilic v. Basilic v. Deltopectoral groove Basilic hiatus Axillary v. Anterior inter-osseous vv. Radial vv. Palmar digital vv. Palmar metacarpal vv. Deep palmar venous arch Ulnar vv. Brachial vv. Basilic v. Subscapular v. Intercapitular vv. Dorsal digital vv. Dorsal venous network Cephalic v. Basilic v. Basilic v. Deep median cubital v. Median cubital v. Basilic v. Cephalic v. Median ante-brachial v. Median cephalic v. Perforator v. Median antebrachial v. Basilic v. Median basilic v. Cephalic v. Accessory cephalic v. Median cephalic v. Cephalic v. Basilic v. Median cubital v. Median antebrachial v. Median basilic v. Upper Limb 366 Veins & Lymphatics of the Upper Limb Fig. 28.5 Veins of the upper limb Right limb, anterior view. A Superficial veins. B Deep veins. Fig. 28.6 Veins of the dorsum Right hand, posterior view. Fig. 28.7 Cubital fossa Right limb, anterior view. The subcutaneous veins of the cubital fossa have a highly variable course. A M-shaped. B Accessory cephalic vein. C Absent median cubital vein. Clinical box 28.1 Venipuncture The veins of the cubital fossa are frequently used when drawing blood. In preparation, a tourniquet is applied above the cubital fossa. This allows arterial blood to flow, but blocks the return of venous blood. The resulting swelling makes the veins more visible and palpable. Pectoralis minor Level III Level II Level I Right lymphatic duct Radial group of lymphatics Dorsolateral arm territory Radial bundle territory Dorsal descending lymphatics Ulnar group of lymphatics Middle forearm territory Ulnar bundle territory Cubital l.n. Middle arm territory Dorsomedial arm territory Axillary lymph nodes Radial bundle territory Dorsolateral arm territory Radial bundle territory Radial group of lymphatics Lymph vessels ascending from the palmar to dorsal side 28 Neurovasculature 367 Fig. 28.9 Lymphatic drainage of the hand Right hand, radial view. Most of the hand drains to the axillary nodes via cubital nodes. However, the thumb, index finger, and dorsum of the hand drain directly. B Posterior view. A Anterior view. Fig. 28.8 Lymph vessels of the upper limb Right limb. Lymph from the upper limb and breast drains to the axillary lymph nodes. The superficial lymphatics of the upper limb lie in the subcutaneous tissue, while the deep lymphatics accompany the arteries and deep veins. Numerous anastomoses exist between the two systems. Fig. 28.10 Axillary lymph nodes Right side, anterior view. For surgical purposes, the axillary lymph nodes are divided into three levels with respect to their relationship with the pectoralis minor: lateral (level I), posterior (level II), or medial (level III). They have major clinical importance in breast cancer (see p. 76). C5 Suprascapular n. Dorsal scapular n. Medial brachial cutaneous n. Subscapular n. Thoraco-dorsal n. Medial cord Posterior cord Lateral cord Axillary n. Deep br. Superficial br. N. to the subclavius Anterior interosseous n. Ulnar n. Musculo-cutaneous n. Median n. Radial n. Axillary a. Medial and lateral pectoral nn. Long thoracic n. T1 Phrenic n. Radial n. Clinical box 28.2 Injuries of the brachial plexus can be complicated to diagnose but an understanding of the basic organization of the plexus is essential. The location of the injury can be determined by careful examination of the type and specificity of the deficit. Nerves of the upper plexus innervate muscles of the proximal limb such as those of the shoulder girdle and arm, while nerves of the lower plexus innervate muscles of the distal limb, such as the forearm and hand. Symptoms from injuries at the root and cord levels will demonstrate this anatomical arrangement. Additionally, a proximal injury to a nerve will elicit more broad-ranging symptoms than a distal injury to that nerve. Injuries to nerves of the brachial plexus Upper Limb 368 Nerves of the Upper Limb: Brachial Plexus Almost all muscles in the upper limb are innervated by the brachial plexus, which arises from spinal cord segments C5 to T1. The anterior rami of the spinal nerves give off direct branches (supraclavicular part of the brachial plexus) and merge to form three trunks, six divisions (three anterior and three posterior), and three cords. The infraclavicular part of the brachial plexus consists of short branches that arise directly from the cords and long (terminal) branches that traverse the limb. Table 28.1 Nerves of the brachial plexus Supraclavicular part Direct branches from the anterior rami or plexus trunks Dorsal scapular n. C4–C5 Suprascapular n. C5, C6 N. to the subclavius C5–C6 Long thoracic n. C5–C7 Infraclavicular part Short and long branches from the plexus cords Lateral cord Lateral pectoral n. C5–C7 Musculocutaneous n. Median n. Lateral root C6–C7 Medial cord Medial root C8–T1 Medial pectoral n. Medial antebrachial cutaneous n. Medial brachial cutaneous n. T1 Ulnar n. C7–T1 Posterior cord Upper subscapular n. C5–C6 Thoracodorsal n. C6–C8 Lower subscapular n. C5–C6 Axillary n. Radial n. C5–T1 Interscalene space Subscapular n. Thoracodorsal n. Dorsal scapular n. Upper trunk Middle trunk Lower trunk Suprascapular n. Lateral cord Posterior cord Medial cord Axillary a. Axillary n. Posterior circumflex humeral a. Musculo-cutaneous n. Radial n. Median n. Ulnar n. Medial antebrachial cutaneous n. Lateral pectoral n. Middle scalene Brachiocephalic trunk Intercostobrachial n. N. to the subclavius C5 spinal n. C8 spinal n. T1 spinal n. Medial brachial cutaneous n. Medial pectoral n. Subclavian a. Long thoracic n. 1st rib Common carotid a. Vertebra prominens (C7) Anterior scalene Phrenic n. Anterior root Posterior root Anterior rami Posterior divisions of C5–T1 Anterior divisions of C5–C7 Lateral cord Posterior cord Medial cord Axillary n. Musculo-cutaneous n. Radial n. Median n. Ulnar n. Axillary a. Upper trunk (C5–C6) T1 C8 C7 C6 C5 Posterior rami Lower trunk (C8–T1) Middle trunk (C7) Anterior divisions of C8–T1 Lateral root Medial root Median n. 28 Neurovasculature 369 Fig. 28.11 Brachial plexus Right side, anterior view. A Structure of the brachial plexus. B Course of the brachial plexus, stretched for clarity. Vertebra prominens (C7) Rhomboid minor Rhomboid major Scapula, medial border Scapula, superior angle Dorsal scap-ular n. Levator scapulae Transverse process of atlas (C1) C4 spinal n. Supra-spinatus Scapular spine Infra-spinatus Greater tubercle Acromion Suprascapular n. in the scapular notch Superior transverse lig. of scapular Suprascapular n. Atlas (C1) Vertebra prominens (C7) Serratus anterior 9th rib Long thoracic n. 1st rib Subclavius Clavicle N. to the subclavius C5 spinal n. Upper Limb 370 Supraclavicular Branches & Posterior Cord The supraclavicular branches of the brachial plexus arise directly from the plexus roots (anterior rami of the spinal nerves) or from the plexus trunks in the lateral cervical triangle. Fig. 28.12 Supraclavicular branches Right shoulder. A Dorsal scapular nerve. Posterior view. B Suprascapular nerve. Posterior view. C Long thoracic nerve and nerve to the sub-clavius. Right lateral view. Table 28.2 Supraclavicular branches Nerve Level Innervated muscle Dorsal scapular n. C4–C5 Levator scapulae Rhomboids major and minor Suprascapular n. C5, C6 Supraspinatus Infraspinatus N. to the subclavius C5–C6 Subclavius Long thoracic n. C5–C7 Serratus anterior Posterior cord Teres major Lower subscapular n. Upper subscapular n. 2nd rib (cut) C5 spinal n. Subscapularis T7 spinous process T12 spinous process Sacrum Iliac crest Thoraco-lumbar fascia Latissimus dorsi Thoracodorsal n. C6 spinal n. 28 Neurovasculature 371 The posterior cord gives off three short branches (arising at the level of the plexus cords) and two long branches (terminal nerves, see pp. 372–373). Fig. 28.13 Posterior cord: Short branches Right shoulder. A Subscapular nerves. Anterior view. B Thoracodorsal nerve. Posterior view. Table 28.3 Branches of the posterior cord Nerve Level Innervated muscle Short branches Upper subscapular n. C5–C6 Subscapularis Lower subscapular n. Subscapularis Teres major Thoracodorsal n. C6–C8 Latissimus dorsi Long (terminal) branches Axillary n. C5–C6 See p. 372 Radial n. C5–T1 See p. 373 Axillary a. Deltoid Superior lateral brachial cutaneous n. (terminal sensory br. of axillary n.) Axillary n. Teres minor Anterior scalene Phrenic n. Middle scalene C5 spinal n. Posterior cord Atlas (C1) Superior lateral brachial cutaneous n. (axillary n.) Supra-clavicular nn. Upper Limb 372 Posterior Cord: Axillary & Radial Nerves Fig. 28.14 Axillary nerve: Cutaneous distribution Right limb. A Anterior view. B Posterior view. Fig. 28.15 Axillary nerve Right side, anterior view, stretched for clarity. Table 28.4 Axillary nerve (C5–C6) Motor branches Innervated muscles Muscular brs. Deltoid Teres minor Sensory branch Superior lateral brachial cutaneous n. Clinical box 28.3 The axillary nerve may be damaged in a fracture of the surgical neck of the humerus. This results in limited ability to abduct the arm and may cause a loss of profile of the shoulder. Inferior lateral brachial cutaneous n. Radial n., superficial br. Posterior antebrachial cutaneous n. Posterior brachial cutaneous n. Radialis muscle group Extensor pollicis brevis Abductor pollicis longus Posterior interosseous n. Supinator Inferior lateral brachial cutaneous n. Posterior antebrachial cutaneous n. Dorsal digital nn. Radial n., superficial br. Radial n., deep br. (in supinator canal) Radial n. Axillary a. Anterior scalene Posterior cord Radial n. (in radial groove) Radial tunnel Brachialis Extensor pollicis longus Extensor digitorum Triceps brachii Brachioradialis Posterior brachial cutaneous n. 28 Neurovasculature 373 Fig. 28.16 Radial nerve: Cutaneous distribution A Anterior view. B Posterior view. Fig. 28.17 Radial nerve Right limb, anterior view with forearm pronated. Clinical box 28.4 Chronic radial nerve compression in the axilla (e.g., due to extended/ improper crutch use) may cause loss of sensation or motor function in the hand, forearm, and posterior arm. More distal injuries (e.g., during anesthesia) affect fewer muscles, potentially resulting in wrist drop with intact triceps brachii function. Table 28.5 Radial nerve (C5–T1) Motor branches Innervated muscles Muscular brs. Brachialis (partial) Triceps brachii Anconeus Brachioradialis Extensors carpi radialis longus and brevis Deep br. (terminal br.: posterior interosseous n.) Supinator Extensor digitorum Extensor digiti minimi Extensor carpi ulnaris Extensors pollicis brevis and longus Extensor indicis Abductor pollicis longus Sensory branches Articular brs. from radial n.: Capsule of the shoulder joint Articular brs. from posterior interosseous n.: Joint capsule of the wrist and four radial metacarpophalangeal joints Posterior brachial cutaneous n. Inferior lateral brachial cutaneous n. Posterior antebrachial cutaneous n. Superficial brs. Dorsal digital nn. Ulnar communicating br. Pectoralis minor Medial pectoral n. Brachial plexus (C5–T1) Lateral pectoral n. Pectoralis major Medial cord C7 spinal n. Posterior ramus T3 vertebral body Intercosto-brachial nn. Anastomosis with medial brachial cutaneous n. Lateral cutaneous br. of 4th intercostal n. Anterior cutaneous br. of 2nd intercostal n. 2nd and 3rd intercostal nn. Medial ante-brachial cutaneous n. Medial brachial cutaneous n., intercostobrachial n. Upper Limb 374 Medial & Lateral Cords The medial and lateral cords give off four short branches. The inter costobrachial nerves are included with the short branches of the brachial plexus, although they are actually the cutaneous branches of the 2nd and 3rd intercostal nerves. Fig. 28.18 Medial and lateral cords: Short branches Right side, anterior view. A Medial and lateral pectoral nerves. B Intercostobrachial nerves. Fig. 28.19 Short branches of medial and lateral cords: Cutaneous distribution A Anterior view. B Posterior view. Table 28.6 Branches of the medial and lateral cords Nerve Level Cord Innervated muscle Short branches Lateral pectoral n. C5–C7 Lateral cord Pectoralis major Medial pectoral n. C8–T1 Medial cord Pectoralis major and minor Medial brachial cutaneous n. T1 — (sensory brs., do not innervate any muscles) Medial antebrachial cutaneous n. C8–T1 Intercostobrachial nn. T2–T3 Long (terminal) branches Musculocutaneous n. C5–C7 Lateral cord Coracobrachialis Biceps brachii Brachialis Median n. C6–T1 See p. 376 Medial cord Ulnar n. C7–T1 See p. 377 Radius Coracobrachialis Musculo-cutaneous n. Coracoid process Intertubercular groove Biceps brachii, short head Biceps brachii, long head Brachialis Biceps brachii Brachialis Lateral antebrachial cutaneous n. Musculo-cutaneous n. Axillary a. Anterior scalene Lateral cord Ulna Lateral antebrachial cutaneous n. 28 Neurovasculature 375 Fig. 28.20 Musculocutaneous nerve Right limb, anterior view. Fig. 28.21 Musculocutaneous nerve: Cutaneous distribution A Anterior view. B Posterior view. Table 28.7 Musculocutaneous nerve (C5–C7) Motor branches Innervated muscles Muscular brs. Coracobrachialis Biceps brachii Brachialis Sensory branches Lateral antebrachial cutaneous n. Articular brs.: Joint capsule of the elbow (anterior part) Note: Musculocutaneous n. innervation of the arm is purely motor; innervation of the forearm is purely sensory. Common and proper palmar digital nn. Median n., palmar branch Articular br. Pronator teres, ulnar head Anterior interosseous n. Flexor pollicis longus Pronator quadratus Median n., palmar br. Common palmar digital nn. Flexor digitorum profundus Flexor digitorum superficialis Palmaris longus Flexor carpi radialis Pronator teres, humeral head Medial epicondyle Median n. Lateral root Axillary a. Medial cord Lateral cord Anterior scalene Medial root Recurrent br. Proper palmar digital nn. 1st and 2nd lumbricals Flexor retinaculum Median n. Proper palmar digital nn. Upper Limb 376 Median & Ulnar Nerves The median nerve is a terminal branch arising from both the medial and the lateral cords. The ulnar nerve arises exclusively from the medial cord. Fig. 28.22 Median nerve Right limb, anterior view. Fig. 28.23 Median nerve: Cutaneous distribution A Anterior view. B Posterior view. Table 28.8 Motor branches Innervated muscles Direct muscular brs. Pronator teres Flexor carpi radialis Palmaris longus Flexor digitorum superficialis Muscular brs. from anterior interosseous n. Pronator quadratus Flexor pollicis longus Flexor digitorum profundus (radial half) Recurrent br. Abductor pollicis brevis Flexor pollicis brevis (superficial head) Opponens pollicis Muscular brs. from common palmar digital nn. 1st and 2nd lumbricals Sensory branches Articular brs.: Capsules of the elbow and wrist joints Palmar br. of median n. (thenar eminence) Communicating br. to ulnar n. Common palmar digital nn. Proper palmar digital nn. Median nerve (C6–T1) Clinical box 28.5 Median nerve injury caused by fracture/ dislocation of the elbow joint may result in compromised grasping ability and sensory loss in the fingertips (see Fig. 28.23 for territories). See also carpal tunnel syndrome (p. 391). Common and proper palmar digital nn. Ulnar n., palmar br. Ulnar n., dorsal br. Dorsal digital nn. Dorsal br. Medial cord Palmar br. Deep br. Proper palmar digital nn. 4th common palmar digital n. Superficial br. Flexor carpi ulnaris Medial epicondyle Ulnar n. Axillary a. Ulnar groove Interossei Flexor retinaculum Flexor digitorum profundus 28 Neurovasculature 377 Fig. 28.24 Ulnar nerve: Cutaneous distribution A Anterior view. B Posterior view. Fig. 28.25 Ulnar nerve Right limb, anterior view. Clinical box 28.6 Ulnar nerve palsy is the most common peripheral nerve damage. The ulnar nerve is most vulnerable to trauma or chronic compression in the elbow joint and ulnar tunnel (see p. 391). Nerve damage causes “clawing” of the hand and atrophy of the interossei. Sensory losses are often limited to the 5th digit. Table 28.9 Ulnar nerve (C7–T1) Motor branches Innervated muscles Direct muscular brs. Flexor carpi ulnaris Flexor digitorum profundus (ulnar half) Muscular br. from superior ulnar n. Palmaris brevis Muscular brs. from deep ulnar n. Abductor digiti minimi Flexor digiti minimi brevis Opponens digiti minimi 3rd and 4th lumbricals Palmar and dorsal interosseous muscles Adductor pollicis Flexor pollicis brevis (deep head) Sensory branches Articular brs.: Capsules of the elbow, carpal, and metacarpophalangeal joints Dorsal br. (terminal brs.: dorsal digital nn.) Palmar br. Proper palmar digital n. (from superficial br.) Common palmar digital n. (from superficial br.; terminal brs.: proper palmar digital nn.) Cephalic v. Lateral antebrachial cutaneous n. (musculocutaneous n.) Cephalic v. Perforating brs. Radial n., superficial br. Median n., palmar br. Palmar aponeurosis Ulnar n., palmar br. Basilic v. Medial brachial cutaneous n. Intercosto-brachial n. Supra-clavicular nn. Median antebrachial v. Median cubital v. Medial antebrachial cutaneous n. Basilic hiatus Superior lateral brachial cutaneous n. (axillary n.) Inferior lateral brachial cutaneous n. (radial n.) Intercostal nn., anterior cutaneous brs. Medial antebrachial cutaneous n. Supra-clavicular nn. Intercosto-brachial n. Medial brachial cutaneous n. Medial antebrachial cutaneous n. Ulnar n., dorsal br. Dorsal digital vv. Intercapitular vv. Dorsal venous network Radial n., superficial br. Cephalic v. Accessory cephalic v. Posterior antebrachial cutaneous n. (radial n.) Inferior lateral brachial cutaneous n. (radial n.) Superior lateral brachial cutaneous n. (axillary n.) Basilic v. Posterior brachial cutaneous n. (radial n.) Lateral antebrachial cutaneous n. (musculo-cutaneous n.) Upper Limb 378 Superficial Veins & Nerves of the Upper Limb Fig. 28.26 Superficial cutaneous veins and nerves of the upper limb A Anterior view. See pp. 392–393 for nerves of the palm. B Posterior view. See pp. 394–395 for nerves of the dorsum. Common and proper palmar digital nn. Common and proper palmar digital nn. Axillary n. Radial n. Palmar br. Palmar br. Medial antebrachial cutaneous n. Medial brachial cutaneous n., intercosto-brachial n. Supra-clavicular nn. Anterior cutaneous brs. Lateral cutaneous brs. Median n. Ulnar n. Intercos-tal nn. Musculocutaneous n. C6 C7 C8 T1 T3 T2 C4 T4 T5 C5 Dorsal br. Radial n. Musculocutaneous n. Proper palmar digital nn. (median n.) Medial brachial cutaneous n., intercosto-brachial n. Medial antebrachial cutaneous n. Axillary n. Supraclavicular nn. Dorsal digital nn. Ulnar n. T3 C8 C7 C6 C5 C4 T1 T4 T5 T2 28 Neurovasculature 379 Fig. 28.27 Cutaneous innervation of the upper limb Fig. 28.28 Dermatomes of the upper limb A Anterior view. A Anterior view. B Posterior view. B Posterior view. Trapezius, descending part Trapezius, transverse part Trapezius (ascending part) Posterior brachial cutaneous n. (radial n.) Inferior lateral brachial cutaneous n. (radial n.) Superior lateral brachial cutaneous n. (axillary n.) Teres major Infraspinatus Deltoid Scapular spine Supraclavicular nn. Latissimus dorsi Posterior rami of spinal nn., lateral brs. Coracoclavicular lig. Suprascapular a. (with superior transverse lig. of scapula) Omohyoid Suprascapular n. (in scapular notch) Supraspinatus Accessory n. and brs. of cervical plexus Posterior rami of spinal nn., medial brs. Teres minor Upper Limb 380 Posterior Shoulder & Arm Fig. 28.29 Posterior shoulder Right shoulder, posterior view. Raised: Trapezius (transverse part). Windowed: Supraspinatus. Revealed: Suprascapular region. Inferior transverse lig. of scapula Superior transverse lig. of scapula ⑤ ④ ① ② ③ Shoulder joint capsule Deep a. of arm and radial n. (in radial groove) Supraspinatus Scapular spine Infraspinatus Medial border scapula Circumflex scapular a. in triangular space Teres major Radial n., muscular brs. Deltoid Axillary n. and posterior circumflex humeral a. in quadrangular space Teres minor Suprascapular a. and n. in scapular notch Triceps brachii Lateral head Long head Acromion Clavicle Lateral intermuscular septum Triceps brachii, lateral head In triceps hiatus 28 Neurovasculature 381 Fig. 28.30 Triangular and quadrangular spaces Table 28.10 Neurovascular tracts of the scapula Passageway Boundaries Transmitted structures ① Scapular notch Superior transverse lig. of scapula, scapula Suprascapular a., v. and n. ② Medial border Scapula Dorsal scapular a., v. and n. ③ Triangular space Teres major and minor, triceps brachii Circumflex scapular a. and v. ④ Triceps hiatus Triceps brachii, humerus, teres major Deep a. and v. of arm and radial n. ⑤ Quadrangular space Teres major and minor, triceps brachii, humerus Posterior circumflex humeral a. and v. and axillary n. A Right shoulder, posterior view. Windowed: Deltoid. B Right shoulder, posterior view. Windowed: Infraspinatus, triceps brachii (lateral head). Great auricular n. External jugular v. Trapezius Clavipectoral fascia Supraclavicular nn. Deltoid Pectoralis major (clavicular part) Subclavian v. Sternocleidomastoid Transverse cervical n. Cephalic v. (in deltopectoral groove) Infraclavicular fossa Thoracoacromial a. Biceps brachii Brachial fascia Latissimus dorsi Medial pectoral n. Lateral pectoral n. Pectoralis major (sternocostal part) Middle scalene m. Brachial plexus Omohyoid, inferior belly Posterior scalene m. Pectoralis minor Pectoralis major Axillary fascia Clavipectoral fascia Subclavian v. Subclavius Clavicle Superficial thoracic fascia Upper Limb 382 Anterior Shoulder Fig. 28.31 Anterior shoulder: Superficial dissection Right shoulder. A Sagittal section through anterior wall. B Anterior view. Removed: Platysma, muscle fasciae, superficial layer of cervical fascia, and pectoralis major (clavicular part). Revealed: Clavipectoral triangle. Subtendinous bursa of subscapularis Glenoid cavity Scapula Infra-spinatus Deltoid Subdeltoid bursa Head of humerus Glenoid labrum Tendon of biceps brachii, long head Coracobrachialis Deltoid Pectoralis major Pectoralis minor Axillary a. and v., cords of brachial plexus Ribs Serratus anterior Subscapularis Rhomboid major Anterior Posterior Interscalene space Brachial plexus Suprascapular a. Omohyoid, inferior belly (cut) Trapezius Axillary a. Thoracoacromial a. Musculo-cutaneous n. Deltoid Cephalic v. Pectoralis major (cut) Median n. Ulnar n. Axillary a. and v. Circumflex scapular a. Subscapular a. Thoraco-dorsal a. Lateral thoracic a. Medial and lateral pectoral nn. Pectoralis minor Long thoracic n. Pectoralis major Superior thoracic a. Subclavius Clavicle Subclavian v. Thyrocervical trunk External jugular v. Transverse cervical a. Ascending cervical a. Inferior thyroid a. Phrenic n. Scalene mm. Common carotid a. Internal jugular v. 28 Neurovasculature 383 Fig. 28.32 Shoulder: Transverse section Right shoulder, inferior view. Fig. 28.33 Anterior shoulder: Deep dissection Right limb, anterior view. Removed: Sternocleidomastoid, omohyoid, and pectoralis major. This dissection reveals the neurovascular contents of the lateral cervical triangle (see pp. 538–539) and axilla (see pp. 384–385). Axillary a. and v. Lateral cord Thoraco-acromial a. Cephalic v. Pectoralis major Biceps brachii Median n. Ulnar n. Brachial a. and v. Lateral thoracic a. Thoracodorsal a. and n. Long thoracic n. Medial pectoral n. Lateral pectoral n. Long thoracic n., superior thoracic a. Pectoralis major Subclavius Lower subscapular n. Deltoid Musculocutaneous n. Median n. roots Circumflex scapular a. Medial and lateral cords Coracobrachialis Biceps brachii, short head Biceps brachii, long head Axillary a. and v. Head of humerus Posterior cord Scapula Subscapularis Rib Pectoralis major Pectoralis minor Serratus anterior Upper Limb 384 Axilla Fig. 28.34 Axilla: Dissection Right shoulder, anterior view. A Removed: Pectoralis major and clavipectoral fascia. Walls of the axilla Table 28.11 Anterior wall Pectoralis major Pectoralis minor Clavipectoral fascia Lateral wall Intertubercular groove of humerus Posterior wall Subscapularis Teres major Latissimus dorsi Medial wall Lateral thoracic wall Serratus anterior Medial and lateral cord brs. Radial n. Median n. Ulnar n. Brachial v. Brachial a. Circumflex scapular a. Radial n., motor brs. Thoracodorsal a. and n. Lateral thoracic a. Upper subscapular n. Long thoracic n., superior thoracic a. Axillary a. Medial cord Axillary v. Lower subscapular n. Axillary n. Subscapular a. Thoracoacromial a. Lateral cord Posterior cord Lower subscapular n. Biceps brachii tendon of long head Pectoralis major Deltoid Coracobrachialis Biceps brachii Triceps brachii Long head Radial n., motor brs. Posterior brachial cutaneous n. Latissimus dorsi Thoracodorsal n. Serratus anterior Teres major Subscapularis Lateral and medial cords Upper subscapular n. Suprascapular n. (in scapular notch) Pectoralis minor Coracoid process Deltoid Radial n. Biceps fascia Medial head Axillary n. 28 Neurovasculature 385 B Removed: Anterior wall (pectoralis major and minor, and clavipectoral fascia). Retracted: Medial and lateral cords of the brachial plexus. C Removed: Medial and lateral cords, and axillary vessels. Revealed: Poste-rior cord. Median n. Medial ante-brachial cuta-neous n. Medial brach-ial cutaneous n. Medial intermuscular septum Musculocutaneous n. (piercing the coracobrachialis) Biceps brachii tendon of long head Pectoralis major Deltoid Brachial a. Biceps brachii Brachialis Bicipital aponeurosis Cubital fossa Ulnar n. (in ulnar groove) Medial head Inferior ulnar collateral a. Superior ulnar collateral a. Ulnar n. Posterior brachial cutaneous n. Latissimus dorsi Serratus anterior Teres major Sub-scapularis Axillary a. and v. Long head Pectoralis minor Coracoid process Triceps brachii Lateral cord Medial cord Upper Limb 386 Anterior Arm & Cubital Region Fig. 28.35 Brachial region Right arm, anterior view. Removed: Deltoid, pectoralis major and minor. Revealed: Medial bicipital groove. Deep median cubital v. (perforator v.) Lateral antebrachial cutaneous n. Median ante-brachial v. Cephalic v. Basilic v. Median basilic v. Median cubital v. Medial antebrachial cutaneous n. Basilic v. Biceps brachii Cephalic v. Medial epicondyle Skin Subcutaneous tissue Brachialis Lateral antebra-chial cutaneous n. (musculo-cutaneous n.) Biceps brachii tendon Perforator v. Extensor carpi radialis longus Radial a. Cephalic v. Median ante-brachial v. Bicipital aponeurosis Pronator teres Superior ulnar collateral a., ulnar n. Inferior ulnar collateral a. Brachial a. and v. Basilic v. Medial antebrachial cutaneous n. Superficial fascia Cephalic v. Biceps brachii (and fascia) Brachioradialis Median n. Biceps brachii Brachialis Radial n. Deep br. Superficial br. Brachioradialis Biceps brachii tendon Radial recurrent a. Ulnar a. Radial a. Supinator Pronator teres Flexor carpi ulnaris Palmaris longus Flexor carpi radialis Ulnar head Humeral head Musculocutaneous n. Superior ulnar collateral a., ulnar n. Brachial a., median n. Triceps brachii Median n. Muscular brs. Radial tunnel Pro-nator teres 28 Neurovasculature 387 Fig. 28.36 Cubital region Right elbow, anterior view. A Cutaneous neurovascular structures in the cubital fossa. B Superficial cubital fossa. Removed: Fasciae and epifascial neurovascular structures. C Deep cubital fossa. Removed: Biceps brachii (distal muscle belly). Retracted: Brachio radialis. Biceps brachii Brachialis Brachio-radialis Biceps brachii tendon Radial a. Abductor pollicis longus Flexor digitorum superficialis Radial a. Flexor carpi radialis Flexor pollicis longus Median n. Thenar muscles Palmar aponeurosis Hypothenar muscles Ulnar a. Palmaris longus tendon Flexor carpi ulnaris Bicipital aponeurosis Palmaris longus Flexor carpi radialis Pronator teres Medial epicondyle Brachial a. Superior ulnar collateral a., ulnar n. Inferior ulnar collateral a. Median n. Triceps brachii Extensor carpi radialis brevis Extensor carpi radialis longus Ulnar n. (in ulnar tunnel) Brachialis Radial n., superficial br. Brachio-radialis Biceps brachii tendon Common inter-osseous a. Posterior inter-osseous a. Anterior inter-osseous a. Pronator teres Radial a. Flexor pollicis longus Abductor pollicis longus Median n. Pronator quadratus Flexor carpi radialis tendon Flexor retinaculum Palmar br. of median n. Thenar muscles Hypothenar muscles Flexor digitorum superficialis tendons Flexor carpi ulnaris Flexor digitorum superficialis Palmaris longus Flexor carpi radialis Pronator teres, ulnar head Pronator teres, humeral head Medial epicondyle Inferior ulnar collateral a. Superior ulnar collateral a., ulnar n. Median n. Biceps brachii Ulnar a. and n. Recurrent interosseous a. Upper Limb 388 Anterior & Posterior Forearm Fig. 28.37 Anterior forearm Right forearm, anterior view. A Superficial layer. Removed: Fasciae and superficial neurovasculature. B Middle layer. Partially removed: Superficial flexors (pronator teres, flexor digitorum superficialis, palmaris longus, and flexor carpi radialis). Musculo-cutaneous n. Muscular brs. Brachio-radialis Radial a. Pronator teres Flexor digitorum superficialis, radial head Flexor pollicis longus Abductor pollicis longus Pronator quadratus Radial a. Flexor digitorum superficialis tendons Ulnar a. and n. Flexor digitorum profundus tendons Ulnar a. and n. Flexor digitorum superficialis, humeroulnar head Biceps brachii tendon Brachialis Brachial a. Median n. Biceps brachii Median n. Superficial br. Deep br. Radial n. Extensor carpi radialis longus Extensor indicis Triceps brachii, lateral head Olecranon Anconeus Interosseous recurrent a. Passage through interosseous membrane Posterior interosseous a. Extensor carpi ulnaris Interosseous membrane Ulnar a., dorsal carpal br. Extensor retinaculum Radial a., dorsal carpal br. Extensor carpi radialis brevis tendon Radial a. Extensor carpi radialis longus tendon Anterior interosseous a. (piercing the membrane) Extensor digitorum Extensor carpi radialis brevis and longus Supinator Arterial network of elbow and lateral epicondyle Radial collateral a. Extensor carpi ulnaris Extensor pollicis longus tendon Extensor pollicis longus Extensor pollicis brevis Abductor pollicis longus Posterior interosseous n. Brachio-radialis 28 Neurovasculature 389 C Deep layer. Removed: Deep flexors. Fig. 28.38 Posterior forearm Right forearm, anterior view during pronation. Reflected: Anconeus and triceps brachii. Removed: Extensor carpi ulnaris and extensor digitorum. Superficial palmar arch Flexor digiti minimi brevis Abductor digiti minimi Palmaris brevis Palmar aponeurosis (cut) Pisiform Palmar carpal lig. Ulnar a. and n. Flexor carpi ulnaris Palmaris longus tendon Flexor digitorum superficialis Radial a. Flexor pollicis longus Flexor carpi radialis Pronator quadratus Median n. Radial a., superficial palmar br. Flexor retinaculum (transverse carpal lig.) Median n., recurrent br. Opponens pollicis Abductor pollicis brevis Flexor pollicis brevis, superficial head Ulnar tunnel Superficial palmar arch Flexor digiti minimi brevis Abductor digiti minimi Superficial br. Deep br. Ulnar a. and n. Flexor digitorum superficialis Flexor carpi ulnaris Flexor carpi radialis Median n. Radial a., superficial palmar br. Flexor retinaculum (transverse carpal lig.) Opponens pollicis Median n., recurrent br. Abductor pollicis brevis Flexor pollicis brevis, superficial head Ulnar a., deep br. Extensor carpi radialis longus and brevis Radial a. Flexor pollicis longus Ulnar n. Upper Limb 390 Carpal Region Fig. 28.39 Anterior carpal region Right hand, anterior (palmar) view. A Ulnar tunnel and deep palm. B Carpal tunnel with flexor retinaculum transparent. Removed: palmaris brevis, palmaris longus, palmar aponeurosis, and palmar carpal ligament. Superficial br. Deep br. Pisiform Ulnar a. and n. Radial a. Hook of hamate Deep palmar arch Superficial palmar arch Ulnar n. Ulnar a. and n., superficial brs. Palmaris brevis Hypothenar muscles Ulnar a. and n., deep brs. Pisiform Flexor carpi ulnaris Ulnar a. and n. Flexor digitorum superficialis tendons Palmaris longus Palmar carpal lig. Ulnar tunnel (proximal hiatus) Ulnar tunnel (distal hiatus) Hook of hamate Palmar aponeurosis Flexor retinaculum (transverse carpal lig.) Ulnar a. and n. Hypothenar eminence Pisiform Triquetrum Extensor carpi ulnaris tendon Extensor digiti minimi tendon Extensor digitorum and extensor indicis tendons Radial n., superficial br. Extensor pollicis longus tendon Extensor pollicis brevis tendon Abductor pollicis longus tendon Trapezium Thenar eminence Median n. Close-up in B Extensor carpi radialis longus tendon Hamate Capitate Extensor carpi radialis brevis tendon Scaphoid Palmar carpal lig. Flexor retinaculum (transverse carpal lig.) Ulnar a. and n. Pisiform Triquetrum Synovial cavity Hamate Flexor digitorum profundus tendons Capitate Scaphoid Flexor carpi radialis tendon Superficial palmar a. and v. Flexor pollicis longus tendon Median n. Flexor digitorum superficialis tendons 28 Neurovasculature 391 Fig. 28.40 Ulnar tunnel Right hand, anterior (palmar) view. A Bony landmarks. B Apertures and walls of the ulnar tunnel. Fig. 28.41 Carpal tunnel: Cross section Right hand, proximal view. The tight fit of sensitive neurovascular structures with closely apposed, frequently moving tendons in the carpal tunnel often causes prob-lems (carpal tunnel syndrome) when any of the structures swell or degenerate. B Structures in the ulnar tunnel (green) and carpal tunnel (blue). A Cross section through the right wrist. Palmar digital n. (exclusive area of ulnar n.) Radial n., dorsal digital n. Median n., palmar br. Palmar digital nn. (exclusive area of median n.) Ulnar n., palmar br. Flexor retinaculum (transverse carpal lig.) Antebrachial fascia Palmar digital nn. Common palmar digital aa. Flexor digiti minimi brevis Abductor digiti minimi Palmar aponeurosis Palmaris brevis Ulnar a. and n. Palmaris longus tendon Radial a. Radial a., superficial palmar br. Abductor pollicis brevis Flexor pollicis brevis, super-ficial head Adductor pollicis Palmar digital nn. of thumb Palmar digital aa. Ulnar tunnel Vincula brevia Vincula longa Flexor digitorum superficialis tendon Flexor digitorum profundus tendon Metacarpal Digitopalmar branches Palmar digital a. Palmar digital n., dorsal branch Palmar digital n. Proper palmar digital a. and n. Common palmar digital a. Dorsal digital a. and n. Metacarpo-phalangeal joint Upper Limb 392 Palm of the Hand Fig. 28.42 Superficial neurovascular structures of the palm Right hand, anterior view. A Sensory territories. Extensive overlap exists between adjacent areas. Exclusive nerve territories indicated with darker shading. B Superficial arteries and nerves. Fig. 28.43 Neurovasculature of the finger Right middle finger, lateral view. A Nerves and arteries. B Blood supply to the flexor tendons in the tendon sheath. Palmar digital aa. and nn. Lumbricals Common palmar digital aa. Superficial palmar arch Flexor digiti minimi brevis Abductor digiti minimi Ulnar n., superficial br. Ulnar a. and n., deep brs. Ulnar a. and n. Palmar carpal lig. Flexor digitorum superficialis Flexor carpi ulnaris Brachioradialis Radial a. Flexor pollicis longus Flexor carpi radialis Pronator quadratus Median n. Flexor retinaculum Palmaris longus Opponens pollicis Radial a., super-ficial palmar br. Abductor pollicis brevis Flexor pollicis brevis, super-ficial head Adductor pollicis 1st dorsal interosseous Palmar digital nn. Radial a., superficial palmar br. Palmar digital nn. Palmar digital aa. Abductor digiti minimi Flexor digiti minimi brevis Palmar metacarpal aa. Superficial palmar arch Ulnar n., deep br. Deep palmar arch Ulnar n., superficial br. Ulnar a., deep br. Ulnar a. and n. Flexor carpi ulnaris Pronator quadratus Radial a., superficial palmar br. Sensitive terminal br. of the anterior interosseous n. Opponens pollicis Abductor pollicis brevis Flexor pollicis brevis Adductor pollicis, oblique head Adductor pollicis, transverse head Lumbricals Common palmar digital aa. Opponens digiti minimi Anterior interosseous a. Radial a. Ulnar communicating br. Median communicating br. 28 Neurovasculature 393 Fig. 28.44 Deep neuro-vascular structures of the palm Right hand, anterior view. A Superficial palmar arch. B Deep palmar arch. Fig. 28.45 Innervation patterns in the palm Right hand, anterior view. A Ulnar communi-cating br. (45% of cases). B Median and ulnar communicating brs. (20%). C No communicating brs. (20%). Dorsal digital nn. (radial n.) Radial n., superficial br. Posterior antebrachial cutaneous n. (radial n.) Ulnar n., dorsal br. Dorsal digital nn. (ulnar n.) Palmar digital nn., dorsal brs. (median n.) Exclusive area of median n. Radial n., superficial br. and dorsal digital nn. Ulnar n., dorsal br. Dorsal digital n. (exclusive area of ulnar n.) Median n., dorsal brs. of palmar digital nn. 1st metacarpal Trape-zium Scaphoid Radial a. Abductor pollicis longus tendon Extensor pollicis brevis tendon Extensor pollicis longus tendon Extensor carpi radialis brevis tendon Radial n., superficial br. Extensor digitorum and extensor indicis tendons 1st dorsal interosseous Extensor retinaculum Radial a. Extensor carpi radialis longus Upper Limb 394 Dorsum of the Hand Fig. 28.46 Cutaneous innervation of the dorsum of the hand Right hand, posterior view. A Nerves of the dorsum. B Sensory territories. Extensive overlap exists between adjacent areas. Exclusive nerve territories indicated with darker shading. Fig. 28.47 Anatomic snuffbox Right hand, radial view. The three-sided “anatomic snuffbox” (shaded green) is bounded by the tendons of insertion of the abductor pollicis longus and extensors pollicis brevis and longus. Extensor carpi radialis brevis tendon Extensor retinaculum Ulnar a., dorsal carpal br. Dorsal carpal a. (radial a.) Dorsal metacarpal aa. Dorsal digital aa. Extensor pollicis longus tendon Radial a. Extensor carpi radialis longus tendon Dorsal carpal network Extensor carpi ulnaris tendon Extensor retinaculum Ulnar a., dorsal carpal br. Extensor digitorum tendon Extensor digiti minimi tendon Radial a., dorsal carpal br. Dorsal metacarpal aa. Dorsal digital aa. Dorsal interossei Extensor pollicis brevis tendon Extensor pollicis longus tendon Radial a. Extensor carpi radialis brevis and longus tendons Dorsal carpal network 28 Neurovasculature 395 A Superficial structures. Fig. 28.48 Neurovascular structures of the dorsum B Deep structures. Upper Limb 396 29 Sectional & Radiographic Anatomy Sectional Anatomy of the Upper Limb Fig. 29.1 Windowed dissection of the arm and forearm Right limb, anterior view. B Right forearm A Dissection of the arm. Deltoid Coraco-brachialis Teres major Biceps brachii, long head Biceps brachii, short head Medial epicondyle Pectoralis major Humerus Brachio-radialis Brachialis Biceps brachii Palmaris longus Palmaris brevis Palmar aponeurosis Thenar muscles Flexor retinaculum (transverse carpal ligament) Brachialis Biceps brachii Flexor pollicis longus Abductor pollicis longus Flexor digitorum superficialis Flexor carpi ulnaris Palmaris longus Flexor carpi radialis Pronator teres Bicipital aponeurosis Medial epicondyle, common head of flexors Biceps brachii, tendon of insertion Brachio-radialis Extensor carpi radialis longus Extensor carpi radialis brevis Ulna Radius 29 Sectional & Radiographic Anatomy 397 Fig. 29.2 Cross-section through the arm and forearm Right limb, proximal view. B Forearm (plane of section in Fig. 29.1B) A Arm (plane of section in Fig. 29.1A) Triceps brachii, lateral head Triceps brachii, long head Triceps brachii, medial head Medial inter-muscular septum of the arm Ulnar nerve Brachial vein Median nerve Brachial artery Musculo-cutaneous nerve Biceps brachii, short head Brachialis Humerus Lateral inter-muscular septum of the arm Radial nerve Biceps brachii, long head Anterior Posterior Anterior (palmar) Posterior (dorsal) Extensor digiti minimi Extensor pollicis longus Extensor carpi ulnaris Ulna Interosseous membrane of forearm Flexor digitorum profundus Ulnar nerve Flexor carpi ulnaris Flexor digitorum superficialis Ulnar artery Palmaris longus Median nerve Flexor pollicis longus Radial artery Flexor carpi radialis Pronator teres Radial nerve (superficial branch) Brachioradialis Extensor carpi radialis brevis Extensor carpi radialis longus Radius Anterior interosseous nerve of forearm Extensor digitorum Extensor pollicis brevis Posterior interosseous nerve of forearm Abductor pollicis longus Upper Limb 398 Radiographic Anatomy of the Upper Limb (I) Fig. 29.3 MRI of the arm Transverse section, distal (inferior) view. Cephalic v. Biceps brachii, long head Brachialis Humerus (shaft) Radial n. Triceps brachii, lateral head Deep brachial a. and v. Triceps brachii, long head Biceps brachii, short head Musculocutaneous n. Median n. Brachial a. and v. Basilic v. Ulnar n. Triceps brachii, medial head C Distal arm. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) B Mid-arm. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anat-omy: The Musculoskeletal System. New York, NY: Thieme; 2009.) A Proximal arm. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A B C Pectoralis major Biceps brachii (short head, tendon) Coracobrachialis Humerus Axillary a. and v., brachial plexus Deltoid Anterior humeral circumflex a. and v. Axillary n. Subscapularis Posterior humeral circumflex a. and v. Triceps brachii (long head) Circumflex scapular a. and v. Teres minor Infraspinatus Scapula Cephaic v. Biceps brachii (long and short heads) Musculocutaneous n. Brachioradialis Brachial a. and v. Radial n., deep brachial a. and v. Median n. Brachialis Basilic v. Humerus (shaft) Triceps brachii Ulnar n., a. and v. 29 Sectional & Radiographic Anatomy 399 Fig. 29.4 MRI of the forearm Transverse section, distal view. Flexor carpi radialis Median n. Radial a. and vv. Brachioradialis (tendon) Anterior interosseous a., v., and n. Cephalic v. Flexor pollicis longus Extensor carpi radialis brevis Radius Abductor pollicis longus Extensor digitorum Flexor digitorum superficialis Ulnar a.,v., and n. Flexor carpi ulnaris Flexor digitorum profundus Ulna Basilic v. Extensor pollicis longus Interosseous membrane Extensor indicis Extensor carpi ulnaris C Distal forearm. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) B Mid-forearm. (Repro-duced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) A Proximal forearm. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) A B C Median cubital v. Brachioradialis Brachial a. and v. Median n. Pronator teres Brachialis Radial n. Flexor digitorum superficialis Extensor carpi radialis brevis Head of radius Ulnar n. Anular lig. Flexor carpi ulnaris Proximal radioulnar joint Flexor digitorum profundus Anconeus Ulna Extensor digitorum Extensor carpi radialis brevis (tendon) Extensor carpi radialis longus (tendon) Radius Extensor carpi ulnaris Ulna Pronator quadratus Flexor digitorum profundus Ulnar n. Ulnar a. and vv. Radial a. and vv. Flexor carpi ulnaris Flexor carpi radialis Median n. Flexor digitorum superficialis Upper Limb 400 Radiographic Anatomy of the Upper Limb (II) Fig. 29.5 Radiograph of the right shoulder Anteroposterior view. Fig. 29.6 Radiograph of the scapula Anteroposterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 29.7 Diagnostic ultrasound of the anterior region of the left shoulder Transverse section at the level of the intertubercular groove. (Reproduced from Konerman W, Gruber G. Ultraschalldiagnostik der Bewegungsorgane, 2nd ed. Stuttgart: Thieme; 2006.) Infraglenoid tubercle Glenoid cavity Head of humerus Greater tubercle Lesser tubercle Lesser tubercle Greater tubercle Tendon of biceps brachii, long head Deltoid Deltoid Greater tubercle Lesser tubercle Tendon of biceps brachii, long head Subscapularis tendon Acromioclavicular joint Acromion Coracoid process Humerus, head Articular surface Scapula, neck Scapula, lateral margin Clavicle Superior angle Scapular spine Scapula, medial margin Scapula, inferior angle A Sonogram. B Schematic of the transverse section. 29 Sectional & Radiographic Anatomy 401 Fig. 29.8 MRI of the right shoulder joint in three planes Supraspinatus Subscapularis Glenoid cavity Intercostal mm. Serratus anterior Tendon of supraspinatus Subacromial bursa Acromion Acromio-clavicular joint Suprascapular a., v., and n. Trapezius Latissimus dorsi Axillary n., posterior humeral circumflex a. and v. Biceps brachii, long head Deltoid Head of humerus Deltoid, clavicular part Subscapularis Head of humerus Pectoralis major Biceps brachii, short head Teres major Biceps brachii, long head Deltoid, scapular part Posterior humeral circumflex a. and v. Axillary n. Teres minor Infraspinatus Supraspinatus Subacromial bursa Acromion C Transverse section, inferior view. B Sagittal section, lateral view. A Coronal section, anterior view. Tendon of biceps brachii, long head Greater tubercle Deltoid, acromial part Head of humerus Glenoid cavity Subscapular a., v., and n. Deltoid, scapular part Infraspinatus Scapula Subsca-pularis Serratus anterior Brachial plexus Axillary a. and v. Subclavius Pectoralis major Pectoralis minor Deltoid, clavicular part Glenoid labrum Lesser tubercle Upper Limb 402 Radiographic Anatomy of the Upper Limb (III) Fig. 29.9 Radiograph of the elbow Anteroposterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 29.10 Radiograph of the elbow Lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Humerus Olecranon fossa Lateral epicondyle Olecranon Capitulum Humeroradial joint Radial head Medial epicondyle Trochlea Humeroulnar joint Coronoid process Proximal radioulnar joint Radial tuberosity Coronoid fossa Coronoid process Radial head Radial tuberosity Radius Ulna Olecranon fossa Lateral epicondyle Humeroulnar joint Olecranon Humeroradial joint Humerus 29 Sectional & Radiographic Anatomy 403 Fig. 29.11 MRI of the elbow (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) B Sagittal section through the humeroulnar and humeroradial joints. C Coronal section through the humeroulnar and humeroradial joints. Triceps brachii Humerus Posterior fat body of elbow Olecranon Brachialis Anterior fat body and coronoid fossa Humerus, trochlea Biceps brachii, tendon Radial a. and v. Coronoid process Brachioradialis Ulnar a. and v. Flexor digitorum profundus Median n. Pronator teres Radial n. Trochlear notch Biceps brachii Brachialis Humerus, capitulum Radial n. Radial head Humeroradial joint Proximal radioulnar joint Biceps brachii, tendon Flexor digitorum superficialis Supinator Radial tuberosity Flexor digitorum profundus Pronator teres, ulnar head Brachioradialis Brachialis Medial epicondyle Medial collateral lig. Ulna, coronoid process Brachialis Brachioradialis Extensor carpi radialis longus Lateral epicondyle Humeroradial joint Humeroulnar joint Radial head Supinator Pronator teres Flexor carpi radialis Radial tuberosity Extensor digitorum A Sagittal section through the humeroulnar joint. Upper Limb 404 Radiographic Anatomy of the Upper Limb (IV) Fig. 29.12 Radiograph of the hand (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) B Oblique view. Fig. 29.13 MRI of the right wrist Transverse section, distal view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Distal phalanx Distal interphalangeal joint Proximal interphalangeal joint Middle phalanx Proximal phalanx Metacarpophalangeal joint Metacarpal head Metacarpal base Capitate Trapezoid Hamate Trapezium Triquetrum Scaphoid Pisiform Radius, styloid process Ulna, styloid process Lunate Head of proximal phalanx Base of proximal phalanx Metacarpal Metacarpal base Capitate and hamate Trapezoid Triquetrum Trapezium Lunate Scaphoid Ulna, styloid process Radius, styloid process Capitate Trapezoid Radial a. and vv. Abductor pollicis longus, tendon Hamate Flexor digitorum profundus, tendon Flexor digitorum superficialis, tendon Ulnar n., deep branch Ulnar a. and vv. Ulnar n. Flexor carpi radialis, tendon Trapezium Flexor retinaculum Median n. Dorsal A Anteroposterior view. 29 Sectional & Radiographic Anatomy 405 Fig. 29.14 MRI of the hand (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) C Transverse section through the palm, distal view. Adductor pollicis Abductor pollicis Opponens pollicis Opponens digiti minimi Flexor digiti minimi Abductor digiti minimi Flexor pollicis longus, tendon Hamate, hook Flexor pollicis brevis, deep head Metacarpal I, base Trapezium Flexor digitorum profundus, tendons Pisiform Palmar radiocarpal lig. Scaphoid B Coronal section through the palm. Collateral lig. Metacarpal, head Dorsal digital aa. and nn. Metacarpophalangeal joint Interosseous muscles Proximal phalanx, base Hamate Carpometacarpal joint Metacarpal II, base Trapezoid Capitate Scaphoid Radius Ulna Dorsal interosseous mm. First proximal phalanx Metacarpal II-IV, shafts Dorsal (extensor) expansion Palmar interosseous mm. Collateral lig. Flexor digitorum superficialis, tendon Flexor digitorum profundus, tendon Metacarpal V, head Lumbricals Extensor digitorum, tendons Palmar digital aa. and nn. Flexor pollicis longus, tendon Dorsal A Coronal section through the carpal tunnel. 30 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 31 Hip & Thigh Bones of the Lower Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 Femur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 Hip Joint: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Hip Joint: Ligaments & Capsule . . . . . . . . . . . . . . . . . . . . . . . 416 Anterior Muscles of the Hip, Thigh & Gluteal Region (I) . . . . 418 Anterior Muscles of the Hip, Thigh & Gluteal Region (II) . . . 420 Posterior Muscles of the Hip, Thigh & Gluteal Region (I) . . . 422 Posterior Muscles of the Hip, Thigh & Gluteal Region (II) . . . 424 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 32 Knee & Leg Tibia & Fibula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Knee Joint: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Knee Joint: Capsule, Ligaments & Bursae . . . . . . . . . . . . . . . 436 Knee Joint: Ligaments & Menisci . . . . . . . . . . . . . . . . . . . . . . 438 Cruciate Ligaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Knee Joint Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Muscles of the Leg: Anterior & Lateral Compartments . . . . . 444 Muscles of the Leg: Posterior Compartment . . . . . . . . . . . . . 446 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 33 Ankle & Foot Bones of the Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Joints of the Foot (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Joints of the Foot (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Joints of the Foot (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Ligaments of the Ankle & Foot . . . . . . . . . . . . . . . . . . . . . . . . 460 Plantar Vault & Arches of the Foot . . . . . . . . . . . . . . . . . . . . . 462 Muscles of the Sole of the Foot . . . . . . . . . . . . . . . . . . . . . . . 464 Muscles & Tendon Sheaths of the Foot . . . . . . . . . . . . . . . . . 466 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 34 Neurovasculature Arteries of the Lower Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Veins & Lymphatics of the Lower Limb . . . . . . . . . . . . . . . . . 474 Lumbosacral Plexus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Nerves of the Lumbar Plexus . . . . . . . . . . . . . . . . . . . . . . . . . 478 Nerves of the Lumbar Plexus: Obturator & Femoral Nerves . . 480 Nerves of the Sacral Plexus . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Nerves of the Sacral Plexus: Sciatic Nerve . . . . . . . . . . . . . . . 484 Superficial Nerves & Veins of the Lower Limb . . . . . . . . . . . . 486 Topography of the Inguinal Region . . . . . . . . . . . . . . . . . . . . 488 Topography of the Gluteal Region . . . . . . . . . . . . . . . . . . . . . 490 Topography of the Anterior, Medial & Posterior Thigh . . . . . 492 Topography of the Posterior Compartment of the Leg & Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Topography of the Lateral & Anterior Compartments of the Leg & Dorsum of the Foot . . . . . . . . . . . . . . . . . . . . 496 Topography of the Sole of the Foot . . . . . . . . . . . . . . . . . . . . 498 35 Sectional & Radiographic Anatomy Sectional Anatomy of the Lower Limb . . . . . . . . . . . . . . . . . . 500 Radiographic Anatomy of the Lower Limb (I) . . . . . . . . . . . . 502 Radiographic Anatomy of the Lower Limb (II) . . . . . . . . . . . . 504 Radiographic Anatomy of the Lower Limb (III). . . . . . . . . . . . 506 Radiographic Anatomy of the Lower Limb (IV). . . . . . . . . . . . 508 Lower Limb Medial surface of tibia Tuberosity of 5th metatarsal Lateral malleolus Interphalangeal joints of the foot Medial malleolus Metatarso-phalangeal joints Navicular tuberosity Patella Lateral tibial condyle Medial tibial condyle Tibial tuberosity Iliac crest Anterior superior iliac spine Greater trochanter Pubic tubercle Pubic symphysis Ischial tuberosity Navicular tuberosity Medial epicondyle Head of fibula Lateral epicondyle Calcaneal tuberosity Posterior superior iliac spine Sacrum Iliac crest Tuberosity of 5th metatarsal Dorsum of the foot Anterior leg region Posterior leg region Anterior genual region Anterior thigh region Femoral trigone Lower Limb 408 30 Surface Anatomy Surface Anatomy Fig. 30.1 Palpable bony prominences of the lower limb Right limb. Fig. 30.2 Regions of the lower limb Right leg. A Anterior view. A Anterior view. B Posterior view. Tensor fascia lata Rectus femoris Sartorius Vastus lateralis Fibularis longus Tibialis anterior Extensor digitorum tendons Extensor hallucis longus Tibia Gastroc-nemius Vastus medialis Iliac crest Gluteus maximus Semimembranosus, semitendinosus Gastrocnemius Calcaneal (Achilles’) tendon Biceps femoris Iliotibial tract Gluteus medius Gluteal region Sole of the foot Calcaneal region Posterior leg region Posterior thigh region Lateral retro-malleolar region Popliteal region 30 Surface Anatomy 409 A Anterior view, left limb. B Posterior view, right limb. B Posterior view. Fig. 30.3 Palpable musculature of the lower limb Pelvic girdle (right side) Patella Tibial plateau Ankle mortise Foot Phalanges Metatarsals Tarsals Leg Thigh Coxal bone Femur Tibia Fibula Iliac crest Anterior superior iliac spine Hip joint Pubic tubercle Greater trochanter Femur Patella Knee joint Tibial tuberosity Tibia Fibula Lateral malleolus Calcaneus Head of fibula Coxal bone Ischial spine Lesser trochanter Ischial tuberosity Coxal bone Neck of femur Greater trochanter Lateral condyle Head of fibula Fibula Lateral malleolus Calcaneus Talocrural (ankle) joint Medial malleolus Knee joint Medial condyle Lesser trochanter Posterior superior iliac spine Iliac crest Lateral tibial condyle Medial epicondyle Medial tibial condyle Tuberosity of 5th metatarsal Lower Limb 410 31 Hip & Thigh Bones of the Lower Limb A Anterior view. Fig. 31.1 Bones of the lower limb B Right lateral view. C Posterior view. The skeleton of the lower limb consists of a coxal (hip) bone and a free limb. The paired coxal bones attach to the trunk at the sacroiliac joint to form the pelvic girdle (see p. 230), and the free limb, divided into a thigh, leg, and foot, attaches to the pelvic girdle at the hip joint. Stabil-ity of the pelvic girdle is important in the distribution of weight from the upper body to the lower limbs. Center of gravity Inflection points of vertebral column External auditory canal Dens of axis (C2) Knee joint Ankle joint Hip joint L4 Sacroiliac joint Coxal bone Sacrum Hip joint Coccyx Pubic symphysis L4 Greater trochanter Coxal bone Neck of femur Ischial tuberosity Sacrum 31 Hip & Thigh 411 A Anterior view. B Posterior view. Fig. 31.2 Line of gravity Right lateral view. The line of gravity runs verti-cally from the whole-body center of gravity to the ground with characteristic points of intersection. Fig. 31.3 The coxal bones and their relation to the vertebral column. The paired coxal bones and sacrum form the pelvic girdle (see p. 230). Head Neck Lesser trochanter Adductor tubercle Medial condyle Patellar surface Lateral condyle Lateral epicondyle Intertrochan-teric line Greater trochanter Lateral lip Linea aspera Gluteal tuberosity Intertrochan-teric crest Greater trochanter Trochanteric fossa Fovea Pectineal line Medial epicondyle Intercondylar notch Lateral condyle Lateral epicondyle Medial lip Shaft Intercondylar line Medial supracondylar line Popliteal surface Lateral supracondylar line Lower Limb 412 Femur Fig. 31.4 Femur Right femur. The femur articulates proximally with the acetabulum of the pelvis at the hip joint and distally with the tibia at the knee joint. A Anterior view. B Posterior view. Iliopectineal bursa Head of femur Fibrous membrane Neck of femur Greater trochanter Trochanteric bursa Acetabulum Ischium Lig. of head of femur Patella Patellar surface of femur Head of femur Neck of femur Greater trochanter Lateral condyle Medial condyle Acetabular labrum Fovea of femoral head Acetabulum Patellar surface of femur Medial condyle Intercondylar notch Lateral condyle Patellar surface of femur (femoral trochlea) 31 Hip & Thigh 413 C Proximal view. The acetabulum has been sectioned in the horizontal plane. D Distal view. Fig. 31.5 Hip joint: Transverse section Right hip joint, superior view. Clinical box 31.1 The acetabular rim is oriented anteroinferiorly relative to the sagittal plane. At birth, the aperture angle measures approximately 7 degrees but increases to 17 degrees by adulthood (A). This angle affects the stability and “seating” of the femoral head in the hip joint. When the femoral head is centered in the acetabulum, the distal femur and thus the knee joint, point slightly inward. Note how external (B) and internal (C) rotation of the femoral head affect the orientation of the knee joint. Rotation of the femoral head 17° Patella Acetabular inlet plane Sagittal (anterior) aperture angle Sagittal plane A B C Bony acetabular rim Neck of femur Lesser trochanter Intertrochan-teric line Greater trochanter Head of femur Anterior superior iliac spine Iliac crest Iliac crest Head of femur Neck of femur Greater trochanter Intertrochan-teric crest Linea aspera Lesser trochanter Ischial tuberosity Ischial spine Pectineal line Gluteal tuberosity Acetabular rim Posterior superior iliac spine Posterior inferior iliac spine Pubic tubercle Lower Limb 414 Hip Joint: Overview Fig. 31.6 Right hip joint The head of the femur articulates with the acetabulum of the pelvis at the hip joint, a special type of spheroidal (ball-and-socket) joint. The roughly spherical femoral head (with an average radius of curvature of approximately 2.5 cm) is largely contained within the acetabulum. A Anterior view. B Posterior view. Clinical box 31.2 Femoral fractures caused by falls in patients with osteoporosis are most frequently located in the neck of the femur. Femoral shaft fractures are less frequent and are usually caused by strong trauma (e.g., a car accident). Fractures of the femur Medial femoral neck fractures Subtrochanteric femoral fracture Lateral femoral neck fracture Peritrochanteric femoral fracture Acetabulum Head of femur Lig. of head of femur Acetabular fossa Acetabular labrum Epiphyseal line Neck of femur Shaft of femur Greater trochanter Ilium Trochanteric bursa Inferior margin of ilium Femur Ossification center Acetabular labrum Bony acetabular rim Ilium Inferior margin of ilium Femur Acetabular labrum Bony acetabular rim 31 Hip & Thigh 415 Fig. 31.7 Hip joint: Coronal section Right hip joint, anterior view. Clinical box 31.3 Ultrasonography, the most important imaging method for screening the infant hip, is used to identify morphological changes such as hip dysplasia and dislocation. Clinically, hip dislocation presents with instability and limited abduction of the hip joint, and leg shortening with asymmetry of the gluteal folds. Diagnosing hip dysplasia and dislocation A Normal hip joint in a 5-month-old. B Hip dislocation and dysplasia in a 3-month-old. L5 vertebra Iliac crest Anterior superior iliac spine Inguinal lig. Pubic tubercle Pubofemoral lig. Iliofemoral lig. Greater trochanter Femur Ischiofemoral lig. Sacrotuberous lig. Ischial spine Sacrospinous lig. Sacrum Posterior sacroiliac ligs. Posterior superior iliac spine Iliolumbar lig. L4 vertebra Anterior longi-tudinal lig. L5 vertebra Sacral promontory Sacrotuberous lig. Sacrospinous lig. Ischial spine Pubic symphysis Pubofemoral lig. Lesser trochanter Intertrochan-teric line Iliofemoral lig. Greater trochanter Inguinal lig. Anterior superior iliac spine Anterior sacroiliac ligs. Iliac crest Iliolumbar lig. Iliac crest Posterior superior iliac spine Ischial spine Iliofemoral lig. Greater trochanter Intertro-chanteric crest Lesser trochanter Ischiofemoral lig. Ischial tuberosity Sacrotuberous lig. Sacrospinous lig. Posterior sacroiliac ligs. L5 vertebra L4 vertebra Lower Limb 416 Hip Joint: Ligaments & Capsule Fig. 31.8 Ligaments of the hip joint Right hip joint. A Lateral view The hip joint has three major ligaments: iliofemoral, pubofemoral, and ischiofemoral. The iliofemoral, the strongest of these, provides an important constraint for the hip joint: it prevents the pelvis from tilting posteriorly in the upright stance, without the need for muscular effort. It also limits adduction of the extended limb and stabilizes the pelvis on the stance side during gait. A fourth ligament, the zona orbicularis (anular ligament) is not visible externally and encircles the femoral neck like a buttonhole. B Anterior view C Posterior view Acetabular roof Lig. of head of femur Transverse lig. of acetabulum Acetabular fossa Lunate surface Acetabular labrum Joint capsule Greater trochanter Fovea on femoral head Lesser trochanter Lig. of head of femur Obturator membrane Acetabular fossa Acetabular labrum Joint capsule Pubofemoral lig. Iliofemoral lig. Iliofemoral lig. Ischiofemoral lig. Reflection of synovial membrane Synovial membrane Lesser trochanter Neck of femur Intertrochan-teric line Greater trochanter Fibrous membrane Fibrous membrane Synovial membrane 31 Hip & Thigh 417 A The joint capsule has been divided and the femoral head dislocated to expose the cut ligament of the head of the femur. B Acetabulum of the hip joint. Note: The ligament of the head of the femur (cut) has no mechanical function, but transmits branches from the obturator artery that nourish the femoral head (see p. 473). A Anterior view B Posterior view Fig. 31.9 Weakness in the joint capsule Right hip joint. Weak spots in the joint capsule (color-shaded areas) are located between the joint ligaments. External trauma may cause the femoral head to dislocate from the acetabulum at these sites. Fig 31.10 Synovial membrane of the joint capsule A Anterior view B Posterior view Fig. 31.11 Ligament of the head of the femur in the acetabulum Right hip joint, lateral view. Iliac crest Iliacus Anterior superior iliac spine Tensor fasciae latae Iliopsoas Rectus femoris Vastus lateralis Vastus medialis Iliotibial tract Head of fibula Pes anserinus Patellar lig. Patella Adductor magnus Gracilis Sartorius Adductor longus Pectineus Pubic symphysis Inguinal lig. Piriformis Anterior longitudinal lig. Sacral promontory Psoas major Quadriceps femoris tendon Pes anserinus (common tendon of insertion) Semi-tendinosus Gracilis Sartorius Vastus intermedius Rectus femoris Sartorius Lower Limb 418 Anterior Muscles of the Hip, Thigh & Gluteal Region (I) Fig. 31.12 Anterior muscles of the hip and thigh (I) Right limb. Muscle origins are shown in red, insertions in blue. A Removed: Fascia lata of thigh (to the lateral iliotibial tract). B Removed: Inguinal ligament, sartorius and rectus femoris. Tensor fasciae latae Iliofemoral lig. Pes anserinus Patellar lig. Gluteus medius Gluteus minimus Piriformis Iliopsoas Vastus medialis Vastus intermedius Adductor hiatus Gracilis Adductor longus Adductor brevis Obturator externus Pectineus Iliotibial tract Articularis genus Vastus medialis Vastus lateralis Vastus lateralis Rectus femoris Piriformis Psoas major Iliacus Sartorius Adductor magnus Semi-tendinosus Sartorius Gracilis Semimembranosus Iliotibial tract Biceps femoris Quadriceps femoris Adductor hiatus Adductor magnus Gracilis Adductor longus Adductor brevis Obturator externus Rectus femoris Gluteus medius Gluteus minimus Piriformis Vastus lateralis Iliopsoas Vastus medialis Vastus intermedius Articularis genus Iliacus Sartorius Pectineus Piriformis Psoas major Pes anserinus (common tendon of insertion) Adductor minimus 31 Hip & Thigh 419 C Removed: Rectus femoris (completely), vastus lateralis, vastus medialis, iliopsoas, and tensor fasciae latae. D Removed: Quadriceps femoris (rectus femoris, vastus lateralis, vastus medialis, vastus intermedius), iliopsoas, tensor fasciae latae, pectineus, and midportion of adductor longus. Adductor tubercle Quadratus femoris Tendinous insertion of adductor magnus Adductor hiatus Adductor magnus Gracilis Adductor brevis Adductor longus Pectineus Piriformis Gracilis Semi-membranosus Rectus femoris Piriformis Gluteus minimus Vastus lateralis Obturator externus Iliopsoas Adductor minimus Iliacus Sartorius Rectus femoris Quadratus femoris Piriformis Gluteus minimus Vastus lateralis Iliopsoas Vastus medialis Vastus intermedius Articularis genus Iliotibial tract Biceps femoris Quadriceps femoris Semi-tendinosus Sartorius Gracilis Semi-membranosus Obturator externus Piriformis Psoas major Adductor magnus Adductor magnus Gracilis Adductor brevis Adductor longus Pectineus Lower Limb 420 Anterior Muscles of the Hip, Thigh & Gluteal Region (II) Fig. 31.13 Anterior muscles of the hip and thigh (II) Right limb. Muscle origins are shown in red, insertions in blue. A Removed: Gluteus medius and minimus, piriformis, obturator externus, adductor brevis and longus, and gracilis. B Removed: All muscles. Iliac crest Anterior superior iliac spine Iliacus Psoas minor Psoas major Pubic symphysis Sartorius Adductor longus Rectus femoris Gracilis Vastus medialis Patella Patellar lig. Pes anserinus (common tendon of insertion) Tibialis anterior Tibia Gastroc-nemius Semi-membranosus Semi-tendinosus Adductor magnus Gluteus maximus Obturator internus Piriformis Sacrum Sacral promontory L5 vertebral body Sacrospinous lig. 31 Hip & Thigh 421 Fig. 31.14 Medial muscles of the hip, thigh, and gluteal region Midsagittal section. Adductor magnus Popliteal fossa Iliotibial tract Gluteus medius Iliac crest Anterior superior iliac spine Tensor fasciae latae Biceps femoris, long head Plantaris Gracilis Semi-tendinosus Semi-membranosus Gastrocnemius, medial and lateral heads Gluteus maximus Greater trochanter L5 spinous process Sacro-tuberous lig. Piriformis Gemellus superior Ischial tuberosity Gracilis Semi-tendinosus Semi-membranosus Plantaris Gluteus minimus Tensor fasciae latae Anterior superior iliac spine Iliac crest Gastrocnemius, medial and lateral heads Gluteus medius Gluteus medius Iliotibial tract Gemellus inferior Obturator internus Biceps femoris, long head Quadratus femoris Gluteus maximus Adductor magnus Gluteus maximus Pes anserinus Lower Limb 422 Posterior Muscles of the Hip, Thigh & Gluteal Region (I) Fig. 31.15 Posterior muscles of the hip, thigh, and gluteal region (I) Right limb. Muscle origins are shown in red, insertions in blue. A Removed: Fascia lata (to iliotibial tract). B Partially removed: Gluteus maximus and medius. Biceps femoris, long head Gracilis Semi-tendinosus (cut) Plantaris Semi-membranosus Tensor fasciae latae Piriformis Adductor magnus Vastus intermedius Gluteus minimus Gluteus medius Gluteus maximus Gemellus superior Gemellus inferior Obturator internus Gluteus medius Vastus lateralis Gluteus maximus Quadratus femoris Biceps femoris, short head Adductor magnus Gastrocnemius, medial and lateral heads Sacro-tuberous lig. Vastus lateralis Adductor magnus Gluteus medius and minimus Piriformis Adductor hiatus Gluteus maximus Gemellus superior Gemellus inferior Obturator internus Semimem-branosus Biceps femoris (long head) and semitendinosus (cut) Gastrocnemius, medial and lateral heads Semi-membranosus Popliteus Flexor digitorum longus Soleus Tibialis posterior Biceps femoris Plantaris Biceps femoris, short head Gluteus maximus Quadratus femoris Rectus femoris Tensor fasciae latae Gluteus medius Gluteus minimus Vastus inter-medius 31 Hip & Thigh 423 C Removed: Semitendinosus and biceps femoris (partially); gluteus maximus and medius (completely). D Removed: Hamstrings (semitendinosus, semimembranosus, and biceps femoris), gluteus minimus, gastrocnemius, and muscles of the leg. Iliopsoas Vastus medialis Adductor magnus Obturator externus Gluteus medius and minimus, piriformis Quadratus femoris Adductor magnus Adductor longus Adductor brevis Adductor magnus Gemellus superior Gemellus inferior Obturator internus Obturator internus and externus, gemellus superior and inferior Soleus Tibialis posterior Flexor digitorum longus Popliteus Gemellus superior Gemellus inferior Obturator internus Adductor magnus Semi-membranosus Biceps femoris, long head and semitendinosus Pectineus Adductor magnus Semi-membranosus Biceps femoris Gastrocnemius, medial and lateral heads Plantaris Adductor longus Biceps femoris, short head Adductor magnus Vastus intermedius Vastus lateralis Adductor brevis Gluteus maximus Iliopsoas Quadratus femoris Gluteus medius and minimus, piriformis Rectus femoris Gluteus maximus Gluteus medius Tensor fasciae latae Gluteus minimus Vastus medialis Lower Limb 424 Posterior Muscles of the Hip, Thigh & Gluteal Region (II) Fig. 31.16 Posterior muscles of the hip, thigh, and gluteal region (II) Right limb. Muscle origins are shown in red, insertions in blue. A Removed: Piriformis, obturator internus, quadratus femoris, and adductor magnus. B Removed: All muscles. L4 spinous process Posterior superior iliac spine Gluteus maximus Long head Fibula, head Fibularis longus Tibial tuberosity Patellar lig. Patella Vastus lateralis Iliotibial tract Rectus femoris Sartorius Tensor fasciae latae Anterior superior iliac spine Gluteus medius Iliac crest Tibialis anterior Gastrocnemius Short head Biceps femoris 31 Hip & Thigh 425 Fig. 31.17 Lateral muscles of the hip, thigh, and gluteal region Note: The iliotibial tract (the thickened band of fascia lata) functions as a tension band to reduce the bending loads on the proximal femur. Iliotibial tract F G H J K A S D L Ö Ä Lower Limb 426 Muscle Facts (I) Table 31.1 Iliopsoas muscle Muscles Origin Insertion Innervation Action ③ Iliopsoas ① Psoas major Superficial: T12–L4 and associated intervertebral disks (lateral surfaces) Deep: L1–L5 vertebrae (costal processes) Femur (lesser trochanter) Lumbar plexus L1, L2(L3) • Hip joint: flexion and external rotation • Lumbar spine: unilateral contraction (with the femur fixed) flexes the trunk laterally to the same side; bilateral contraction raises the trunk from the supine position ② Iliacus Iliac fossa Femoral n. (L2–L3) The psoas minor, present in approximately 50% of the population, is often found on the superficial surface of the psoas major (see Fig. 31.19). It is not a muscle of the lower limb. It originates, inserts, and exerts its action on the abdomen (see Table 13.1, p. 148). C Horizontally oriented gluteal muscles, posterior view. Table 31.2 Gluteal muscles Muscle Origin Insertion Innervation Action ④ Gluteus maximus Sacrum (dorsal surface, lateral part), ilium (gluteal surface, posterior part), thoracolumbar fascia, sacrotuberous lig. • Upper fibers: iliotibial tract • Lower fibers: gluteal tuberosity Inferior gluteal n. (L5–S2) • Entire muscle: extends and externally rotates the hip in sagittal and coronal planes • Upper fibers: abduction • Lower fibers: adduction ⑤ Gluteus medius Ilium (gluteal surface below the iliac crest between the anterior and posterior gluteal line) Greater trochanter of the femur (lateral surface) Superior gluteal n. (L4–S1) • Entire muscle: abducts the hip, stabilizes the pelvis in the coronal plane • Anterior part: flexion and internal rotation • Posterior part: extension and external rotation ⑥ Gluteus minimus Ilium (gluteal surface below the origin of gluteus medius) Greater trochanter of the femur (anterolateral surface) ⑦ Tensor fasciae latae Anterior superior iliac spine Iliotibial tract • Tenses the fascia lata • Hip joint: abduction, flexion, and internal rotation ⑧ Piriformis Pelvic surface of the sacrum Apex of the greater trochanter of the femur Sacral plexus (S1, S2) • External rotation, abduction, and extension of the hip joint • Stabilizes the hip joint ⑨ Obturator internus Inner surface of the obturator membrane and its bony boundaries Medial surface of the greater trochanter Sacral plexus (L5, S1) External rotation and extension of the hip joint (also active in abduction, depending on the joint’s position) ⑩ Gemelli • Gemellus superior: ischial spine • Gemellus inferior: ischial tuberosity Jointly with obturator internus tendon (medial surface, greater trochanter) ⑪ Quadratus femoris Lateral border of the ischial tuberosity Intertrochanteric crest of the femur External rotation of the hip joint B Vertically oriented gluteal muscles, posterior view. Fig. 31.18 Muscles of the hip Right side, schematic. A Iliopsoas muscle, anterior view. Lesser trochanter Sacrospinous lig. Inter-trochanteric line Greater trochanter Inguinal lig. Anterior superior iliac spine Iliac crest Sacral promontory Psoas minor L5 vertebra Psoas major Iliacus Iliopsoas Thoraco-lumbar fascia Fibula Interosseous membrane Iliotibial tract Gluteus maximus Tensor fasciae latae Gluteus medius Iliac crest Axis of abduction/ adduction Tibia Sacrotuberous lig. Ischial tuberosity Quadratus femoris Obturator internus Gemellus superior and inferior Gluteal tuberosity Greater trochanter Piriformis Iliac crest Gluteus medius Anterior superior iliac spine Ilium, gluteal surface Lesser trochanter Gluteus minimus Iliac crest Ischial spine Quadratus femoris Gemellus superior and inferior Intertrochanteric crest Greater trochanter Piriformis Posterior gluteal line Obturator internus 31 Hip & Thigh 427 A Deep layer with gluteus maximus removed. B Deep layer with gluteus maximus and gluteus medius removed. Fig. 31.19 Psoas and iliacus muscles Right side, anterior view. Fig. 31.20 Superficial muscles of the gluteal region Right side, posterior view. Fig. 31.21 Deep muscles of the gluteal region Greater trochanter Lesser trochanter Pectineus Adductor longus Patella Fibula Tibia Femur Superior pubic ramus Iliac crest Promontory Gracilis tendon Adductor brevis Gracilis A S D F Lower Limb 428 Muscle Facts (II) Fig. 31.22 Medial thigh muscles: Superficial layer Right side, anterior view. A Schematic. B Superficial adductor group. Table 31.3 Medial thigh muscles: Superficial layer Muscle Origin Insertion Innervation Action ① Pectineus Pecten pubis Femur (pectineal line and the proximal linea aspera) Femoral n., obturator n. (L2, L3) • Hip joint: adduction, external rotation, and slight flexion • Stabilizes the pelvis in the coronal and sagittal planes ② Adductor longus Superior pubic ramus and anterior side of the pubic symphysis Femur (linea aspera, medial lip in the middle third of the femur) Obturator n. (L2–L4) • Hip joint: adduction and flexion (up to 70 degrees); extension (past 80 degrees of flexion) • Stabilizes the pelvis in the coronal and sagittal planes ③ Adductor brevis Inferior pubic ramus Obturator n. (L2, L3) ④ Gracilis Inferior pubic ramus below the pubic symphysis Tibia (medial border of the tuberosity, along with the tendons of sartorius and semitendinosus) • Hip joint: adduction and flexion • Knee joint: flexion and internal rotation Functionally, the medial thigh muscles are considered the adductors of the hip. A S S Greater trochanter Lesser trochanter Patella Fibula Tibia Tibial tuberosity Femur Superior pubic ramus Iliac crest Adductor tubercle Adductor magnus, tendinous part Obturator externus Adductor magnus Adductor hiatus 31 Hip & Thigh 429 Fig. 31.23 Medial thigh muscles: Deep layer Right side, anterior view. A Schematic. B Deep adductor group. Table 31.4 Medial thigh muscles: Deep layer Muscle Origin Insertion Innervation Action ① Obturator externus Outer surface of the obturator membrane and its bony boundaries Trochanteric fossa of the femur Obturator n. (L3, L4) • Hip joint: adduction and external rotation • Stabilizes the pelvis in the sagittal plane ② Adductor magnus Inferior pubic ramus, ischial ramus, and ischial tuberosity • Deep part (“fleshy insertion”): medial lip of the linea aspera • Deep part: obturator n. (L2–L4) • Hip joint: adduction, extension, and slight flexion (the tendinous insertion is also active in internal rotation) • Stabilizes the pelvis in the coronal and sagittal planes • Superficial part (“tendinous insertion”): adductor tubercle of the femur • Superficial part: tibial n. (L4) A S G D F Anterior superior iliac spine Sartorius Anterior inferior iliac spine Greater trochanter Rectus femoris Vastus lateralis Vastus medialis Quadriceps femoris tendon Patella Patellar lig. Fibula Tibial tuberosity Pes anserinus Acetabular roof Sartorius Vastus medialis Vastus intermedius Lesser trochanter Lateral patellar retinaculum Rectus femoris Sartorius Medial patellar retinaculum Greater trochanter Quadriceps femoris Intertrochanteric line Vastus lateralis Rectus femoris Lower Limb 430 Muscle Facts (III) Fig. 31.24 Anterior thigh muscles Right side, anterior view. A Schematic. B Superficial group. C Deep group. Removed: Sartorius and rectus femoris. Table 31.5 Anterior thigh muscles Muscle Origin Insertion Innervation Action ① Sartorius Anterior superior iliac spine Medial to the tibial tuberosity (together with gracilis and semitendinosus) Femoral n. (L2, L3) • Hip joint: flexion, abduction, and external rotation • Knee joint: flexion and internal rotation Quadriceps femoris ② Rectus femoris Anterior inferior iliac spine, acetabular roof of hip joint Tibial tuberosity (via patellar lig.) Femoral n. (L2–L4) • Hip joint: flexion • Knee joint: extension ③ Vastus medialis Linea aspera (medial lip), intertrochanteric line (distal part) Tibial tuberosity via patellar lig.; patella and tibial tuberosity via respective medial and lateral patellar retinacula Knee joint: extension ④ Vastus lateralis Linea aspera (lateral lip), greater trochanter (lateral surface) ⑤ Vastus intermedius Femoral shaft (anterior side) Tibial tuberosity (via patellar lig.) Articularis genus (distal fibers of vastus intermedius) Anterior side of femoral shaft at level of the suprapatellar recess Suprapatellar recess of knee joint capsule Knee joint: extension; retracts the suprapatellar bursa to prevent entrapment of capsule The entire muscle inserts on the tibial tuberosity via the patellar lig. The anterior and posterior muscles of the thigh can be classified as extensors and flexors, respectively, with regard to the knee joint. A S D Popliteus Sacro-tuberous lig. Ischial tuberosity Semi-membranosus Popliteus Biceps femoris, long head Biceps femoris, short head Gluteal tuberosity Ischial spine Semi-tendinosus Fibula, head Pes anserinus Semimembranosus tendon (part) Biceps femoris, long head Linea aspera Posterior inferior iliac spine Biceps femoris, short head Popliteus Interosseous membrane Semi-tendinosus Semi-membranosus Common head Posterior superior iliac spine Anterior superior iliac spine Medial tibial condyle 31 Hip & Thigh 431 Fig. 31.25 Posterior thigh muscles Right side, posterior view. A Schematic. B Superficial group. C Deep group. Removed: Biceps femoris (long head) and semitendinosus. Table 31.6 Posterior thigh muscles Muscle Origin Insertion Innervation Action ① Biceps femoris Long head: ischial tuberosity, sacrotuberous lig. (common head with semitendinosus) Head of fibula Tibial n. (L5–S2) • Hip joint (long head): extends the hip, stabilizes the pelvis in the sagittal plane • Knee joint: flexion and external rotation Short head: lateral lip of the linea aspera in the middle third of the femur Common fibular n. (L5–S2) Knee joint: flexion and external rotation ② Semimembranosus Ischial tuberosity Medial tibial condyle, oblique popliteal lig., popliteus fascia Tibial n. (L5–S2) • Hip joint: extends the hip, stabilizes the pelvis in the sagittal plane • Knee joint: flexion and internal rotation ③ Semitendinosus Ischial tuberosity and sacrotuberous lig. (common head with long head of biceps femoris) Medial to the tibial tuberosity in the pes anserinus (along with the tendons of gracilis and sartorius) See p. 451 for the popliteus. Tibial plateau Tibial tuberosity Lateral surface Medial surface Anterior border Tibia, shaft Ankle mortise Medial malleolus Lateral malleolus Tibiofibular syndesmosis Lateral surface Fibula, shaft Interosseous membrane Neck of fibula Head of fibula Tibiofibular joint Lateral condyle Medial condyle Medial surface Lateral condyle Tibiofibular joint Head of fibula Neck of fibula Soleal line Interosseous membrane Fibula, shaft Lateral malleolar fossa Lateral malleolus Medial malleolus Malleolar groove (for tibialis posterior tendon) Head of tibia Intercondylar eminence Tibial plateau Medial condyle Tibia, shaft Posterior surface 432 32 Knee & Leg Tibia & Fibula Lower Limb The tibia and fibula articulate at two joints, allowing limited motion (rotation). The crural interosseous membrane is a sheet of tough connective tissue that serves as an origin for several muscles in the leg. It also acts with the tibiofibular syndesmosis to stabilize the ankle joint. Fig. 32.1 Tibia and fibula Right leg. A Anterior view. B Posterior view. Posterior inter-condylar area Intercondylar eminence Anterior intercondylar area Tibial tuberosity Head of fibula Lateral condyle Medial condyle Medial surface Posterior surface Tibia Anterior border Interosseous membrane Fibula Lateral surface Medial surface Lateral surface Posterior surface Tibia Fibula Lateral malleolar fossa Lateral malleolus Articular surface of lateral malleolus Inferior articular surface Articular surface of medial malleolus Medial malleolus Fibula Lateral malleolus Calcaneus Talus Tibiofibular syndesmosis Medial malleolus Tibia 433 32 Knee & Leg D Transverse section, proximal view. C Proximal view. E Distal view. Clinical box 32.1 Fibular fracture When diagnosing a fibular fracture, it is important to determine whether the tibiofibular syndesmosis (see p. 432) is disrupted. Fibular fractures may occur distal to, level with, or proximal to the tibiofibular syndesmosis; the latter two frequently involve tearing of the syndesmosis. In this fracture located proximal to the syndesmosis (arrow), the syndesmosis is torn, as indicated by the widened medial joint space of the upper ankle joint (see pp. 456–457). Patella Medial epicondyle Medial femoral condyle Medial tibial condyle Tibia Fibula Tibial plateau Lateral tibial condyle Lateral femoral condyle Lateral epicondyle Tibial tuberosity Femur Head of fibula Lateral epicondyle Intercondylar notch Intercondylar eminence Tibia Soleal line Neck of fibula Head of fibula Tibiofibular joint Lateral femoral condyle Popliteal surface Fibula 434 Knee Joint: Overview Lower Limb In the knee joint, the femur articulates with the tibia and patella. Both joints are contained within a common capsule and have communicat-ing articular cavities. Note: The fibula is not included in the knee joint (contrast to the humerus in the elbow; see p. 326). Instead, it forms a separate rigid articulation with the tibia. Fig. 32.2 Right knee joint A Anterior view. B Posterior view. Anterior surface Base Apex Articular surface Apex Lateral tibial condyle Tibial tuberosity Tibia Head of fibula Lateral femoral condyle Patella Fibula Femoropatellar joint Prepatellar bursa Joint space Medial facet Synovial membrane Medial collateral lig. Medial femoral condyle Cruciate ligs. Gastrocnemius Popliteal a. and v. Lateral femoral condyle Common fibular n. Tibial n. Lateral collateral lig. Patellar surface of femur Fibrous membrane Lateral facet Patellar lig. (quadriceps tendon) Patella Radiographic view in Fig. 35.11B 435 32 Knee & Leg C Lateral view. Fig. 32.4 Patellofemoral joint: Transverse section Distal view with right knee in slight flexion. Fig. 32.3 Patella A Anterior view. B Posterior view. Femur Vastus intermedius tendon of insertion Vastus medialis Rectus femoris tendon of insertion Medial collateral lig. Medial transverse patellar retinaculum Medial longitudinal patellar retinaculum Patellar lig. Tibial tuberosity Tibia Interosseous membrane Fibula Head of fibula Lateral collateral lig. Lateral longitudinal patellar retinaculum Lateral transverse patellar retinaculum Vastus lateralis 436 Lower Limb Knee Joint: Capsule, Ligaments & Bursae Table 32.1 Ligaments of the knee joint Extrinsic ligaments Anterior side Patellar lig. Medial longitudinal patellar retinaculum Lateral longitudinal patellar retinaculum Medial transverse patellar retinaculum Lateral transverse patellar retinaculum Medial and lateral sides Medial (tibial) collateral lig. Lateral (fibular) collateral lig. Posterior side Oblique popliteal lig. Arcuate popliteal lig. Intrinsic ligaments Anterior cruciate lig. Posterior cruciate lig. Transverse lig. of knee Posterior meniscofemoral lig. Fig. 32.5 Ligaments of the knee joint Anterior view of right knee. Lateral subtendinous bursa of gastrocnemius Lateral collateral lig. Arcuate popliteal lig. Subpopliteal recess Popliteus Semi-membranosus bursa Medial collateral lig. Oblique popliteal lig. Medial subtendinous bursa of gastrocnemius Fibula Tibia Femur 437 32 Knee & Leg Fig. 32.6 Capsule, ligaments, and periarticular bursae Posterior view of right knee. The joint cavity communicates with peri- articular bursae at the subpopliteal recess, semimembranosus bursa, and medial subtendinous bursa of the gastrocnemius. Clinical box 32.2 Axial MRI of a Baker’s cyst in the right popliteal fossa, inferior view. Baker’s cysts often occur in the medial part of the popliteal fossa between the semimem branosus tendon and the medial head of the gas trocnemius at the level of the posteromedial femoral condyle. Gastrocnemio-semimembranosus bursa (Baker’s cyst) Painful swelling behind the knee may be caused by a cystic outpouching of the joint capsule (synovial popliteal cyst). This frequently results from an increase in intra-articular pressure (e.g., in rheumatoid arthritis). Fibula Femur Patellar surface of femur Patella Lateral femoral condyle Patellar lig. Lateral meniscus Tibial tuberosity Anterior lig. of fibular head Posterior lig. of fibular head Lateral collateral lig. Lateral epicondyle Quadriceps femoris tendon Medial epicondyle Medial meniscus Medial collateral lig. Tibia, medial surface Patellar lig. Femoropatellar joint Medial femoral condyle Fibula Quadriceps femoris tendon Femur 438 Lower Limb Knee Joint: Ligaments & Menisci A Medial view. B Lateral view. Fig. 32.7 Collateral and patellar ligaments of the knee joint Right knee joint. Each knee joint has medial and lateral collateral liga- ments. The medial collateral ligament is attached to both the capsule and the medial meniscus, whereas the lateral collateral ligament has no direct contact with either the capsule or the lateral meniscus. Both collateral ligaments are taut when the knee is in extension and stabilize the joint in the coronal plane. Lateral collateral lig. Transverse lig. of knee Tibio-fibular joint Head of fibula Lateral meniscus Posterior menisco-femoral lig. Posterior cruciate lig. Medial meniscus Medial collateral lig. Anterior cruciate lig. Patellar lig. Posterior cruciate lig. Synovial membrane Head of fibula Lateral meniscus Medial meniscus Anterior cruciate lig. Medial collateral ligament Lateral collateral ligament Patellar lig. Patella Lateral collateral lig. Medial collateral ligament Lateral collateral ligament Patellar ligament Patella Lateral collateral ligament Extension Flexion 439 32 Knee & Leg Fig. 32.8 Menisci in the knee joint Right tibial plateau, proximal view. A Right tibial plateau with cruciate, patellar, and collateral ligaments divided. B Attachment sites of menisci and cruciate ligaments. Red line indicates the tibial attachment of the synovial membrane that covers the cruciate ligaments. The cruciate ligaments lie in the subsynovial connective tissue. Fig. 32.9 Movements of the menisci Right knee joint. The medial meniscus, which is anchored more securely than the lateral meniscus, undergoes less displacement during knee flexion. B Flexion. A Extension. C Tibial plateau, proximal view. Clinical box 32.3 Injury to the menisci The less mobile medial meniscus (see Fig. 32.9) is more susceptible to injury than the lateral meniscus. Trauma generally results from sudden extension or rotation of the flexed knee while the leg is fixed. A Bucket-handle tear. B Radial tear of posterior horn. Patella Anterior lig. of fibular head Lateral meniscus Lateral collateral lig. Anterior cruciate lig. Patellar surface of femur Transverse lig. of knee Patellar lig. (reflected inferiorly) Fibula Posterior cruciate lig. Medial meniscus Medial collateral lig. Anterior cruciate lig. Lateral femoral condyle Medial femoral condyle Head of fibula Posterior lig. of fibular head Lateral collateral lig. Lateral meniscus Posterior menisco-femoral lig. Intercondylar notch Interosseous membrane Tibia 440 Lower Limb Cruciate Ligaments Fig. 32.10 Cruciate and collateral ligaments Right knee joint. The cruciate ligaments keep the articular surfaces of the femur and tibia in contact, while stabilizing the knee joint primarily in the sagittal plane. Portions of the cruciate ligaments are taut in every joint position. A Anterior view. B Posterior view. Patellar surface of femur Posterior cruciate lig. Medial femoral condyle Medial meniscus Medial collateral lig. Tibia Tibial tuberosity Head of fibula Anterior cruciate lig. Lateral meniscus Lateral collateral lig. Lateral femoral condyle Fibula 441 32 Knee & Leg Fig. 32.11 Right knee joint in flexion Anterior view with joint capsule and patella removed. Fig. 32.12 Cruciate and collateral ligaments in flexion and extension Right knee, anterior view. Taut ligament fibers shown in red. While most parts of the collateral ligaments are taut only in extension (A), the cruciate ligaments, or portions of them, are taut in flexion, extension and internal rotation (B,C). The cruciate ligaments thus help stabilize the knee in any joint position. A Extension. B Flexion. C Flexion and internal rotation. B Right knee in flexion, anterior “drawer sign,” medial view. During examination of the flexed knee, the tibia can be pulled forward. A Right knee in flexion, rupture of anterior cruciate ligament, anterior view. Clinical box 32.4 Rupture of cruciate ligaments Cruciate ligament rupture destabilizes the knee joint, allowing the tibia to move forward (anterior “drawer sign”) or backward (posterior “drawer sign”) relative to the femur. Anterior cruciate ligament ruptures are approximately 10 times more common than posterior ligament ruptures. The most common mechanism of injury is an internal rotation trauma with the leg fixed. A lateral blow to the fully extended knee with the foot planted tends to cause concomitant rupture of the anterior cruciate and medial collateral ligaments, as well as tearing of the attached medial meniscus. Femur Suprapatellar pouch Femur, patellar surface Anterior cruciate lig. Medial femoral condyle Medial meniscus Joint capsule (cut edge) Patella, articular surface Suprapatellar pouch Tibia Fibula Infra-patellar fat pad Lateral collateral lig. Alar folds Lateral meniscus Lateral femoral condyle 442 Lower Limb Knee Joint Cavity Fig. 32.13 Opened joint capsule Right knee, anterior view with patella reflected downward. Suprapatellar pouch Patella Patellar lig. Lateral meniscus Infrapatellar bursa Tibia Subpopliteal recess Femur Quadriceps tendon Lateral collateral lig. Fibula Fig. 32.14 Joint cavity Right knee, lateral view. The joint cavity was demonstrated by injecting liquid plastic into the knee joint and later removing the capsule. Fig 32.15 Relations of structures to the joint capsule and articular cavity Right knee joint, proximal view. Several joint structures provide strength or stability to the knee from outside of, or within, the joint space. • Extracapsular structures (lateral collateral ligament) remain outside of the joint capsule. • Intracapsular structures (medial collateral and cruciate ligaments) lie within the joint capsule but run in the subsynovial tissue outside the synovial membrane and are therefore also extra-articular. • Intra-articular structures (menici) lie within the articular cavity, en-closed by the synovial membrane, and are bathed in synovial fluid. Cruciate ligs. Intracapsular ligs. Fibrous membrane Subintima Intima Synovial membrane Joint capsule Lateral meniscus Extracapsular lig. (lateral [fibular] collateral lig.) Fibula Patellar lig. Medial meniscus Medial (tibial) collateral lig. Femur Infrapatellar bursa Tibia Infrapatellar fat pad Patellar lig. Prepatellar bursa Patella Quadriceps tendon Supra-patellar pouch Anterior cruciate lig. Anterior inter-condylar area Sites of attachment of the joint capsule Suprapatellar pouch Patella Patellar lig. Quadriceps femoris 443 32 Knee & Leg Fig. 32.17 Suprapatellar pouch during flexion Right knee joint, medial view. A Neutral (0-degree) position. B 80 degrees of flexion. C 130 degrees of flexion. Fig. 32.16 Right knee joint: Midsagittal section Lateral view. Clinical box 32.5 Intra-articular effusion due to inflammatory changes or injury can be differentiated from swelling of the joint capsule by pushing down on the patella of the extended knee. If there is excessive fluid in the joint, the patella will rebound when released, signifying a positive test. The ballottable patella sign of knee effusion Effusion Tibia Fibula Femur Patella Effusion Iliotibial tract Rectus femoris Vastus lateralis Vastus medialis Extensor digitorum longus Extensor hallucis longus Interossei Tibialis anterior Pes anserinus (common tendon of insertion of sartorius, gracilis, and semitendinosus) Patellar lig. Patella Sartorius Gracilis Extensor digitorum longus Tibial tuberosity Fibularis longus Extensor hallucis longus Gastrocnemius, medial head Soleus Tibia Fibularis tertius (variable) Extensor hallucis brevis Medial malleolus Fibularis longus Extensor digitorum brevis Fibularis brevis Fibularis tertius Extensor hallucis longus Tibialis anterior Extensor hallucis brevis Tibialis anterior Extensor digitorum longus Extensor digitorum longus Head of fibula Tibial tuberosity Femur 444 Lower Limb Muscles of the Leg: Anterior & Lateral Compartments Fig. 32.18 Muscles of the anterior compartment of the leg Right leg. Muscle origins shown in red, insertions in blue. A All muscles shown. B Removed: Tibialis anterior and fibularis longus; extensor digitorum longus tendons (distal portions). Note: The fibularis tertius is a divi-sion of the extensor digitorum longus. Fibularis brevis Fibularis tertius Extensor digitorum longus Extensor hallucis brevis Extensor hallucis brevis and extensor digitorum brevis Patella Extensor digitorum longus Fibularis brevis Fibularis tertius Tibialis anterior Tibialis anterior Extensor hallucis longus Fibularis longus Extensor hallucis longus Extensor digitorum brevis Interosseous membrane Biceps femoris, long head Biceps femoris, short head Head of fibula Gastrocnemius, lateral head Soleus Triceps surae Fibularis brevis Calcaneal (Achilles’) tendon Calcaneus Fibularis longus Fibularis brevis Fibularis tertius (variable) Extensor digitorum longus Extensor digitorum brevis Lateral malleolus, fibula Extensor hallucis longus Extensor digitorum longus Tibialis anterior Fibularis longus Patellar lig. Patella Iliotibial tract Vastus lateralis Rectus femoris Lateral tibial condyle Biceps femoris, common tendon of insertion 445 32 Knee & Leg C Removed: All muscles. Fig. 32.19 Muscles of the lateral compartment of the leg Right leg. The triceps surae is comprised of the soleus and two heads of the gastrocnemius. Gastroc-nemius, medial head Gastroc-nemius, lateral head Flexor digitorum longus Tibialis posterior Flexor hallucis longus Flexor digitorum longus Fibularis longus Fibularis brevis Calcaneus Flexor hallucis longus Fibularis brevis Gracilis Semi-tendinosus Semi-membranosus Calcaneal (Achilles’) tendon Medial malleolus Lateral malleolus Fibularis longus Soleus Plantaris Biceps femoris Iliotibial tract Popliteus Soleus Plantaris tendon Flexor digitorum longus Tibialis posterior Flexor hallucis longus Flexor digitorum longus Fibularis longus Fibularis brevis Calcaneus Flexor hallucis longus Fibularis longus Calcaneal (Achilles’) tendon Plantaris Fibularis longus Biceps femoris Gastrocnemius, lateral head Gastrocnemius, medial head Fibularis brevis 446 Lower Limb Muscles of the Leg: Posterior Compartment Fig. 32.20 Muscles of the posterior compartment of the leg Right leg. Muscle origins shown in red, insertions in blue. A Note: The bulge of the calf is produced mainly by the triceps surae (soleus and the two heads of the gastrocnemius). B Removed: Gastrocnemius (both heads). Soleus Tibialis posterior Tibialis anterior Flexor hallucis longus Fibularis brevis Triceps surae Plantaris Flexor hallucis longus Flexor digitorum longus Tibialis posterior Fibularis longus Biceps femoris Gastrocnemius, lateral head Gastrocnemius, medial head Plantaris Crural chiasm (intersection of two tendons) Plantar chiasm (intersection of two tendons) Popliteus Flexor digitorum longus Soleus Tibialis posterior Tibialis anterior Fibularis longus Flexor hallucis longus Flexor digitorum longus Fibularis brevis Triceps surae Fibularis brevis Interosseous membrane Flexor hallucis longus Flexor digitorum longus Tibialis posterior Fibularis longus Biceps femoris Popliteus Gastrocnemius, lateral head Gastrocnemius, medial head Plantaris Plantaris 447 32 Knee & Leg C Removed: Triceps surae, plantaris, popliteus, fibularis longus, and fibularis brevis muscles. D Removed: All muscles. ① ② Head of fibula Interosseous membrane Fibularis brevis Lateral malleolus Calcaneus Fibularis longus tendon Fibularis brevis tendon Cuboid Patella Lateral tibial condyle Femur Fibularis longus Lateral tibial surface Tuberosity of 5th metatarsal Cuboid Medial cuneiform Fibularis longus tendon 1st metatarsal 448 Lower Limb Muscle Facts (I) The muscles of the leg control the flexion/extension and inversion/ eversion of the foot, which provide stability to the lower limb during movements at the knee and hip joint. Fig. 32.21 Muscles of the lateral compartment of the leg Right leg and foot. A Fibularis muscles, anterior view, schematic. B Lateral compartment, right lateral view. C Course of the fibularis longus tendon, plantar view. Table 32.2 Lateral compartment Muscle Origin Insertion Innervation Action ① Fibularis longus Fibula (head and proximal two thirds of the lateral surface, arising partly from the intermuscular septa) Medial cuneiform (plantar side), 1st metatarsal (base) Superficial fibular n. (L5, S1) • Talocrural joint: plantar flexion • Subtalar joint: eversion (pronation) • Supports the transverse arch of the foot ② Fibularis brevis Fibula (distal half of the lateral surface), intermuscular septa 5th metatarsal (tuberosity at the base, with an occasional division to the dorsal aponeurosis of the 5th toe) • Talocrural joint: plantar flexion • Subtalar joint: eversion (pronation) 2 4 1 3 Lateral epicondyle Lateral tibial condyle Head of fibula Extensor digitorum longus Medial malleolus Extensor digitorum longus tendon 1st through 5th distal phalanges Extensor hallucis longus tendon Extensor hallucis longus Tibialis anterior Shaft of tibia Tibial tuberosity Femur Lateral malleolus Fibularis tertius tendon Fibularis tertius 449 32 Knee & Leg Fig. 32.22 Muscles of the anterior compartment of the leg Right leg, anterior view. A Schematic. B Anterior compartment. Table 32.3 Anterior compartment Muscle Origin Insertion Innervation Action ① Tibialis anterior Tibia (upper two thirds of the lateral surface), interosseous membrane, and superficial crural fascia (highest part) Medial cuneiform (medial and plantar surface), first metatarsal (medial base) Deep fibular n. (L4, L5) • Talocrural joint: dorsiflexion • Subtalar joint: inversion (supination) ② Extensor hallucis longus Fibula (middle third of the medial surface), interosseous membrane 1st toe (at the dorsal aponeurosis at the base of its distal phalanx) Deep fibular n. (L4, L5) • Talocrural joint: dorsiflexion • Subtalar joint: active in both eversion and inversion (pronation/supination), depending on the initial position of the foot • Extends the MTP and IP joints of the big toe ③ Extensor digitorum longus Fibula (head and medial surface), tibia (lateral condyle), and interosseous membrane 2nd to 5th toes (at the dorsal aponeuroses at the bases of the distal phalanges) Deep fibular n. (L4, L5) • Talocrural joint: dorsiflexion • Subtalar joint: eversion (pronation) • Extends the MTP and IP joints of the 2nd to 5th toes ④ Fibularis tertius Distal fibula (anterior border) 5th metatarsal (base) Deep fibular n. (L4, L5) • Talocrural joint: dorsiflexion • Subtalar joint: eversion (pronation) IP, interphalangeal; MTP, metatarsophalangeal. ③ ② ① Medial femoral condyle Soleus Plantaris tendon Medial malleolus Talus Navicular 1st metatarsal Calcaneal tuberosity Lateral malleolus Plantaris Calcaneal (Achilles’) tendon Triceps surae Gastrocnemius, medial head Gastrocnemius, lateral head Femur Medial epicondyle Lateral epicondyle Triceps surae Gastrocnemius, medial head (cut) Medial tibial condyle Tendinous arch of soleus Talocrural joint Gastrocnemius, lateral head Gastrocnemius, medial head Soleus Head of fibula Gastrocnemius, lateral head (cut) Plantaris Plantaris tendon Talus Subtalar joint Calcaneal (Achilles’) tendon Calcaneus 450 Lower Limb Muscle Facts (II) The muscles of the posterior compartment are divided into two groups: the superficial and deep flexors. These groups are separated by the transverse intermuscular septum. Fig. 32.23 Muscles of the posterior compartment of the leg: Superficial flexors Right leg, posterior view. A Foot in plantar flexion, schematic. B Superficial flexors. C Superficial flexors with gastrocnemius removed (portions of medial and lateral heads). Table 32.4 Superficial flexors of the posterior compartment Muscle Origin Insertion Innervation Action Triceps surae ① Gastrocnemius Femur (medial head: superior posterior part of the medial femoral condyle. lateral head: lateral surface of lateral femoral condyle) Calcaneal tuberosity via the calcaneal (Achilles’) tendon Tibial n. (S1, S2) • Talocrural joint: plantar flexion when knee is extended (gastrocnemius) • Knee joint: flexion (gastrocnemius) • Talocrural joint: plantar flexion (soleus) ② Soleus Fibula (head and neck, posterior surface), tibia (soleal line via a tendinous arch) ③ Plantaris Femur (lateral epicondyle, proximal to lateral head of gastrocnemius) Calcaneal tuberosity Negligible; may act with gastrocnemius in plantar flexion D S A Crural chiasm Plantar chiasm F Tibialis posterior tendon Medial cuneiform Tuberosity of 5th metatarsal Tuberosity of cuboid Calcaneal tuberosity Lateral malleolus Interosseous membrane Tibialis posterior Fibula Talus Medial tibial condyle Tibialis posterior Medial malleolus Flexor hallucis longus tendon Tibialis posterior tendon Flexor digitorum longus tendons Lateral malleolus Calcaneal tuberosity Head of fibula Flexor digitorum longus Flexor hallucis longus Popliteus Head of fibula Medial tibial condyle Soleal line Tibialis posterior Calcaneus Tibialis posterior tendons Tuberosity of 5th metatarsal 1st through 5th meta-tarsals Posterior surface of tibia Femur Medial malleolus Posterior surface of fibula 451 32 Knee & Leg Fig. 32.24 Posterior compartment of the leg: Deep flexors Right leg with foot in plantar flexion, posterior view. A Schematic. B Deep flexors. C Tibialis posterior. D Insertion of the tibialis posterior. Table 32.5 Deep flexors of the posterior compartment Muscle Origin Insertion Innervation Action ① Tibialis posterior Interosseous membrane, adjacent borders of tibia and fibula Navicular tuberosity; cuneiforms (medial, intermediate, and lateral); 2nd to 4th metatarsals (bases) Tibial n. (L4, L5) • Talocrural joint: plantar flexion • Subtalar joint: inversion (supination) • Supports the longitudinal and transverse arches ② Flexor digitorum longus Tibia (middle third of posterior surface) 2nd to 5th distal phalanges (bases) Tibial n. (L5–S2) • Talocrural joint: plantar flexion • Subtalar joint: inversion (supination) • MTP and IP joints of the 2nd to 5th toes: plantar flexion ③ Flexor hallucis longus Fibula (distal two thirds of posterior surface), adjacent interosseous membrane 1st distal phalanx (base) • Talocrural joint: plantar flexion • Subtalar joint: inversion (supination) • MTP and IP joints of the 1st toe: plantar flexion • Supports the medial longitudinal arch ④ Popliteus Lateral femoral condyle, posterior horn of the lateral meniscus Posterior tibial surface (above the origin at the soleus) Tibial n. (L4–S1) Knee joint: flexes and unlocks the knee by externally rotating the femur on the fixed tibia IP, interphalangeal; MTP, metatarsophalangeal. Hindfoot Midfoot Forefoot Tarsus (tarsal bones) Metatarsus (metatarsal bones) Antetarsus (phalanges) Body Neck Head Navicular Intermediate cuneiform Medial cuneiform Lateral cuneiform 1st metatarsal 5th distal phalanx 5th middle phalanx 5th proximal phalanx 5th metatarsal Tuberosity of 5th metatarsal Cuboid Medial process of calcaneal tuberosity Lateral process of calcaneal tuberosity Cal-caneal tuber-osity Calcaneus Posterior process Talus 5th distal phalanx 5th middle phalanx 5th proximal phalanx Lateral cuneiform Tuberosity of 5th metatarsal Cuboid Calcaneus Calcaneal tuberosity Body Neck Head Navicular Intermediate cuneiform Medial cuneiform Base Shaft Head Base Shaft Head 1st proximal phalanx 1st meta-tarsal Talus 1st distal phalanx 5th metatarsal 452 Lower Limb 33 Ankle & Foot Bones of the Foot Fig. 33.1 Subdivisions of the pedal skeleton Right foot, dorsal view. Descriptive anatomy divides the skeletal elements of the foot into the tarsus, metatarsus, and forefoot (ante tarsus). Functional and clinical criteria divide the pedal skeleton into hindfoot, midfoot, and forefoot. A Right foot, dorsal (superior) view. B Right foot, lateral view. Fig. 33.2 Bones of the foot 1st distal phalanx 1st proximal phalanx Sesamoids 1st metatarsal Medial cuneiform Intermediate cuneiform Lateral cuneiform Navicular Head Neck Body Posterior process Calcaneus Cuboid Tuberosity of cuboid Groove for fibularis longus tendon Tuberosity of 5th metatarsal 5th metatarsal 5th proximal phalanx 5th middle phalanx 5th distal phalanx Talus Sustentaculum tali Medial tubercle Head Neck Body Lateral tubercle Posterior process of talus Calcaneal tuberosity Medial process of calcaneal tuberosity Sustentac ulum tali Cuboid Navicular Medial cuneiform 1st distal phalanx Head Shaft Base 1st metatarsal 1st proximal phalanx Shaft Base Head Talus 453 33 Ankle & Foot C Right foot, plantar (inferior) view. D Right foot, medial view. Clinical box 33.1 Right foot, lateral view. In the neutral (0°) position, the skeleton of the foot is angled approximately 90° relative to the skeleton of the leg. This plantigrade foot position is termed the “functional position” and is an important basis for normal standing and walking. The functional position of the foot Transverse tarsal joint Distal inter phalangeal joints Talocrural (ankle) joint Talus Navicular Cuneonavicular joint Intermediate cuneiform Medial cuneiform Tarsometatarsal joints (Lisfranc’s joint line) Abductor hallucis 1st metatarsal 1st metatarso phalangeal joint 1st proximal phalanx Proximal inter phalangeal joints 1st distal phalanx 5th middle phalanx Abductor digiti minimi Interossei Intercuneiform joints Lateral cuneiform Cuboid Calcaneo cuboid joint Talonavicular joint Calcaneus Interosseous talocalcanean ligament Lateral malleolus Fibula Tibia Medial malleolus Talonavicular joint Talocrural (ankle) joint Tranverse tarsal joint Calcaneocuboid joint Intermetatarsal joints Cuneonavicular joint Distal interphalangeal joints Tarsometatarsal joints Metatarsophalangeal joints Cuneocuboid joint Intercuneiform joints Subtalar (talocalcaneal) joint Interphalangeal joint of the hallux Proximal interphalangeal joints 454 Lower Limb Joints of the Foot (I) Fig. 33.3 Joints of the foot Right foot with talocrural joint in plantar flexion. A Anterior view. B Superior view of coronal section. Base of 1st proximal phalanx Tuberosity of 5th metatarsal Base of 5th metatarsal Base of 1st metatarsal 1st through 5th metatarsals Lateral cuneiform Tuberosity of 5th metatarsal Cuboid Medial cuneiform Intermediate cuneiform Navicular Cuboid Superior trochlear surface of talus Medial malleolar surface Head of talus (with articular surface for navicular) Sustentaculum tali Calcaneus Calcaneus (with articular surface for cuboid) Lateral malleolar surface Talus Navicular tuberosity Calcaneus Calcaneus (with articular surface for cuboid) Navicular Talus Navicular Medial cuneiform Intermediate cuneiform Lateral cuneiform Calcaneus Cuboid Base Shaft Head Sesamoids 1st through 5th metatarsals 1st metatarsal 455 33 Ankle & Foot Fig. 33.4 Proximal articular surfaces Right foot, proximal view. Fig. 33.5 Distal articular surfaces Right foot, distal view. A Metatarsophalangeal joints. B Tarsometatarsal joints. C Cuneonavicular and calcaneocuboid joints. D Talonavicular and calcaneocuboid joints. A Talonavicular and calcaneocuboid joints. B Cuneonavicular and calcaneocuboid joints. C Tarsometatarsal joints. D Metatarsophalangeal joints. A B C D A B C D 1st metatarsal Sesamoids Sustentaculum tali Calcaneal tuberosity Tuberosity of 5th metatarsal Lateral malleolus Talus Medial malleolus Navicular Fibula Tibia Talocrural joint Subtalar (talo calcaneal) joint Ankle mortise Extensor hallucis Extensor digitorum Talocrural joint Tibiofibular syndesmosis Fibula Lateral malleolar articular surface Subtalar (talo calcaneal) joint Lateral malleolus Fibularis brevis Fibularis longus Calcaneus Flexor digitorum brevis Quadratus plantae Abductor hallucis Posterior tibial a. and vv. Flexor hallucis longus Flexor digitorum longus Tibialis posterior Talus, superior trochlear surface Medial malleolus Medial malleolar articular surface Ankle mortise Tibialis anterior Tibia 456 Lower Limb Joints of the Foot (II) A Posterior view with foot in neutral (0-degree) position. B Coronal section, proximal view. The talocrural joint is plantar flexed, and the subtalar joint has been sectioned through its posterior compartment. Fig. 33.6 Talocrural and subtalar joints Right foot. The talocrural (ankle) joint is formed by the distal ends of the tibia and fibula (ankle mortise) articulating with the trochlea of the talus. The subtalar joint consists of an anterior and a posterior compart ment (the talocalcaneal and talocalcaneonavicular joints, respectively) divided by the interosseous talocalcaneal ligament (see p. 458). Clinical box 33.2 Right foot, anterior view. Range of motion of the forefoot and hindfoot A Eversion and pronation of the forefoot. B Inversion and supina tion of the forefoot. 30° 60° Fibula Tibia Medial malleolus Superior trochlear surface of talus (anterior diameter) Lateral malleolus Navicular Medial malleolus Ankle mortise Tibia Fibula Talus Lateral malleolus Calcaneus Superior trochlear surface of talus (posterior diameter) Sustentac ulum tali Navicular Lateral tubercle Lateral malleolar surface Superior trochlear surface Neck Head Posterior diameter Anterior diameter Medial malleolar surface Inferior articular surface Tibia Medial malleolus Medial malleolar articular surface Lateral malleolar articular surface Lateral malleolus Fibula Interosseous talocalcaneal lig. Tibia Talocrural joint Talus Plantar calcaneo navicular lig. Bursa of calcaneal tendon Calcaneus Talocalcaneal joint (posterior compartment of subtalar joint) Plantar aponeurosis Short pedal muscles Cuneiforms Navicular Talocalcaneonavicular joint (anterior compartment of subtalar joint) 2nd metatarsal Calcaneal (Achilles’) tendon 457 A Anterior view. B Posterior view. C Proximal (superior) view of talus. D Distal (inferior) view of ankle mortise. 33 Ankle & Foot Fig. 33.7 Talocrural and subtalar joints: Sagittal section Right foot, medial view. Fig. 33.8 Talocrural joint Right foot. The talocrural (ankle) joint is tighter and more stable with the foot in dorsiflexion, when the wider, anterior part of the trochlea (of the talus) is wedged within the ankle mortise. Accordingly the joint is looser and less stable in plantar flexion. Tibia Medial malleolus Talus Interosseous talocalcaneal lig. Calcaneus Sustentaculum tali Plantar calcaneo navicular lig. Long plantar lig. Plantar aponeurosis 1st metatarsal Medial cuneiform Navicular Navicular surface Navicular Plantar calcaneonavicular lig. Talus Sustentaculum tali Calcaneus Tunnel for fibularis longus tendon Medial cuneiform 5th metatarsal Long plantar lig. Cuboid Cuboid Anterior compart ment Bifurcate lig. Dorsal calcaneo cuboid lig. Posterior compart ment Calcaneus Interosseous talocalcaneal lig. Talus Plantar calcaneo navicular lig. Navicular Subtalar joint Medial cuneiform 458 A Dorsal view. B Plantar view. The plantar calcaneonavicular (“spring”) ligament com pletes the bony socket of the talocalcaneal joint. The long plantar ligament converts the tuberosity of the cuboid bone into a tunnel for the fibularis longus tendon (arrow). C Medial view. The interosseous talocalcaneal ligament has been divided and the talus displaced upward. Note the course of the plantar calca neonavicular ligament, which functions with the long plantar ligament and plantar aponeurosis to support the longitudinal arch of the foot. Fig. 33.9 Subtalar joint and ligaments Right foot with opened subtalar joint. The subtalar joint consists of two distinct articulations separated by the interosseous talocalcaneal liga ment: the posterior compartment (talocalcaneal joint) and the anterior compartment (talocalcaneonavicular joint). Joints of the Foot (III) Lower Limb Lateral malleolar surface Anterior talar articular surface Middle talar articular surface Cuboid articular surface Sulcus calcanei Sustentac ulum tali Groove for flexor hallucis longus tendon Lateral tubercle Medial tubercle Superior trochlear surface Medial malleolar surface Navicular articular surface Posterior process of talus Sinus tarsi Posterior talar articular surface Calcaneal body Superior trochlear surface Navicular articular surface Lateral malleolar surface Middle talar articular surface Cuboid articular surface Posterior talar articular surface Posterior calcaneal articular surface Sinus tarsi Navicular articular surface Anterior calcaneal articular surface Middle calcaneal articular surface Sulcus tali Posterior calcaneal articular surface Groove for flexor hallucis longus tendon Lateral tubercle Groove for flexor hallucis longus tendon Medial process Calcaneal tuberosity Lateral process Cuboid articular surface Medial tubercle Medial malleolar surface Posterior talar articular surface Sustentac ulum tali Middle talar articular surface Cuboid articular surface Anterior talar articular surface Navicular articular surface Superior trochlear surface Calcaneus 459 33 Ankle & Foot Fig. 33.10 Talus and calcaneus The two tarsal bones have been separated at the subtalar joint to demonstrate their articular surfaces. A Dorsal (superior) view. B Lateral view. C Plantar view. D Medial view. Tibia Medial malleolus Deltoid lig. Talus Dorsal talonavicular lig. Navicular Dorsal tarsal ligs. 1st metatarsal Metatarso phalangeal joint capsules 1st proximal phalanx 1st distal phalanx Dorsal metatarsal ligs. Bifurcate lig. Cuboid Anterior talo fibular lig. Lateral malleolus Anterior tibiofibular lig. Fibula Tibia Medial malleolus Deltoid lig. Talus Calcaneus Calcaneofibular lig. Lateral malleolus Posterior talo fibular lig. Posterior tibio fibular lig. Fibula Interosseous membrane 460 Lower Limb Ligaments of the Ankle & Foot The ligaments of the foot are classified as belonging to the talocrural joint, subtalar joint, metatarsus, forefoot, or sole of the foot. The medial and lateral collateral ligaments, along with the syndesmotic ligaments, are of major importance in the stabilization of the subtalar joint. Fig. 33.11 Ligaments of the ankle and foot Right foot. See p. 458 for inferior view. A Anterior view with talocrural joint in plan tar flexion. B Posterior view in plantigrade foot position. Table 33.1 Ligaments of the talocrural joint Lateral ligs. Anterior talofibular lig. Posterior talofibular lig. Calcaneofibular lig. Medial ligs. Deltoid lig. Anterior tibiotalar part Posterior tibiotalar part Tibionavicular part Tibiocalcaneal part Syndesmotic ligs. of the ankle mortise Anterior tibiofibular lig. Posterior tibiofibular lig. The medial and lateral ligs. are also known as the medial and lateral collateral ligs. Tibia Anterior tibiofibular lig. Lateral malleolus Talus Dorsal talonavicular lig. Navicular Bifurcate lig. Dorsal tarsal ligs. Cuboid Metatarsophalangeal joint capsules 5th metatarsal Dorsal calcaneocuboid ligs. Anterior talo fibular lig. Interosseous talocalcaneal lig. Long plantar lig. Calcaneo fibular lig. Calcaneus Posterior talofibular lig. Fibula Posterior tibiofibular lig. Tibiofibular syndesmosis (syndesmotic ligs.) 461 33 Ankle & Foot Posterior tibio fibular lig. Medial malleolus Calcaneus Sustentac ulum tali Long plantar lig. Plantar calcaneo navicular lig. Medial cuneiform Dorsal tarsal ligs. Navicular Talus Dorsal talonavicular lig. Tibia Deltoid lig. Posterior tibiotalar part Tibiocalcaneal part Tibionavicular part Anterior tibio talar part 1st distal phalanx 1st proximal phalanx 1st metatarsal C Medial view. D Lateral view. Proximal phalanx of great toe Metatarso phalangeal joint of great toe 1st metatarsal Medial cuneiform Tibialis posterior Medial malleolus Sustentac ulum tali Cuboid Adductor hallucis, oblique head Adductor hallucis, transverse head Plantar ligs. Deep transverse metatarsal lig. Fibularis longus Calcaneus Talus Adductor hallucis, transverse head Base of 5th meta tarsal Adductor hallucis, oblique head Base of 1st metatarsal Lateral cuneiform Cuboid Fibularis longus Tuberosity of 5th metatarsal Tibialis posterior Medial cuneiform Intermediate cuneiform Plantar ligs. Deep transverse metatarsal lig. Base of 1st proximal phalanx 462 Lateral rays Cuboid Calcaneus Talus Navicular Cuneiforms Medial rays Lower Limb Plantar Vault & Arches of the Foot A Superior view. B Posteromedial view. A Plantar view. B Anterior arch (forefoot), proximal view. C Metatarsal arch, proximal view. D Tarsal region, proximal view. Fig. 33.12 The plantar vault Right foot. The forces of the foot are distributed among two lateral (fibular) and three medial (tibial) rays. The arrangement of these rays creates a longitudinal and a transverse arch in the sole of the foot, help ing the foot adapt to uneven terrain and absorb vertical loads. Fig. 33.13 Stabilizers of the transverse arch Right foot. The transverse pedal arch is supported by both active and passive stabilizing structures (muscles and ligaments, respectively). Note: The arch of the forefoot has only passive stabilizers, whereas the arches of the metatarsus and tarsus have only active stabilizers. C Superior view. The area outlined in red by interconnecting the bony points of support for the plantar vault forms a triangle. By contrast, the area of ground contact defined by the plantar soft tissues (the foot print or podogram) is considerably larger. Head of 5th metatarsal Calcaneal tuberosity Head of 1st metatarsal Talus Medial malleolus Flexor digitorum longus Flexor hallucis longus Medial tubercle Sustentac ulum tali Long plantar lig. Plantar aponeurosis Medial cuneiform Navicular Plantar calcaneonavicular lig. Plantar calcaneocuboid lig. Fibularis longus tendon Flexor digitorum brevis Quadratus plantae Abductor hallucis Lumbrical Dorsal interossei Plantar interossei Flexor hallucis brevis Adductor hallucis Calcaneal (Achilles’) tendon Plantar aponeurosis 463 33 Ankle & Foot Fig. 33.14 Stabilizers of the longitudinal arch Right foot, medial view. A Passive stabilizers of the longitudinal arch. The main passive stabili-zers of the longitudinal arch are the plantar aponeurosis (strongest component), the long plantar ligament, and the plantar calcaneona vicular ligament (weakest component). B Active stabilizers of the longitudinal arch. Sagittal section at the level of the second ray. The major active stabilizers of the foot are the abductor hallucis, flexor hallucis brevis, flexor digitorum brevis, quadratus plantae, and abductor digiti minimi. Flexor hallucis longus Flexor digitorum longus Tibialis posterior Flexor hallucis brevis Abductor hallucis Fibularis longus Flexor digiti minimi brevis 3rd plantar interosseus Abductor digiti minimi Tuberosity of 5th metatarsal Calcaneal tuberosity Transverse fascicles Superficial transverse metatarsal lig. Plantar aponeurosis Lateral plantar septum Medial plantar septum Cruciform ligs. Annular ligs. Flexor hallucis longus Flexor digitorum longus Tibialis posterior Fibularis longus Plantar aponeurosis Abductor digiti minimi Flexor digitorum brevis tendons 3rd plantar interosseus 4th dorsal interosseus Lumbricals Flexor hallucis longus tendon Flexor hallucis brevis Abductor hallucis Flexor digitorum brevis Flexor digiti minimi brevis 464 Muscles of the Sole of the Foot Lower Limb Fig. 33.15 Plantar aponeurosis Right foot, plantar view. The plantar aponeurosis is a tough aponeurotic sheet, thickest at the center, that blends with the dorsal fascia (not shown) at the borders of the foot. Fig. 33.16 Intrinsic muscles of the sole of the foot Right foot, plantar view. A Superficial (first) layer. Removed: Plantar aponeurosis, including the superficial transverse metacarpal ligament. Flexor digitorum brevis tendons Flexor digitorum longus tendons Lumbricals 3rd plantar interosseus 4th dorsal interosseus Flexor digiti minimi brevis Abductor digiti minimi Fibularis longus Flexor digitorum brevis Flexor hallucis longus Flexor digitorum longus Tibialis posterior Abductor hallucis Flexor hallucis brevis Adductor hallucis, transverse head Flexor hallucis longus tendon Flexor digitorum longus Quadratus plantae Fibularis longus tendon Long plantar lig. Flexor digitorum brevis tendons Lumbricals Opponens digiti minimi Fibularis longus Abductor digiti minimi Tibialis posterior tendon Fibularis longus tendon Transverse head Flexor hallucis longus Quadratus plantae Abductor hallucis Flexor digitorum longus tendons Flexor hallucis brevis, medial and lateral heads Adductor hallucis Abductor hallucis Flexor digiti minimi brevis Fibularis brevis Flexor digitorum longus Flexor hallucis longus Plantar and dorsal interossei Tuberosity of 5th metatarsal Oblique head 465 33 Ankle & Foot B Second layer. Removed: Flexor digitorum brevis. C Third layer. Removed: Abductor digiti minimi, abductor hallucis, quadratus plantae, lumbricals, and tendons of insertion of the flexors digitorum and hallucis longus. 1st through 4th lumbricals Opponens digiti minimi 3rd plantar interosseus Flexor digiti minimi brevis Quadratus plantae Fibularis longus Abductor digiti minimi Flexor digitorum brevis Plantar aponeurosis Tibialis posterior tendon Fibularis longus tendon Tibialis anterior tendon Abductor hallucis Flexor hallucis brevis Adductor hallucis, oblique head Transverse head Plantar ligs. Abductor hallucis 4th dorsal interosseus 1st plantar interosseus Flexor digiti minimi brevis Flexor hallucis brevis Oblique head 1st dorsal interosseus 2nd dorsal interosseus Long plantar lig. Fibularis brevis Adductor hallucis Plantar calcaneonavicular lig. Opponens digiti minimi Flexor digitorum brevis 1st through 4th dorsal interossei 1st through 3rd plantar interossei Abductor digiti minimi Flexor digiti minimi brevis 1st plantar interosseus 2nd plantar interosseus 3rd plantar interosseus Abductor digiti minimi Quadratus plantae Abductor hallucis Tibialis posterior Tibialis anterior Flexor hallucis brevis Adductor hallucis, oblique head Fibularis longus 4th dorsal interosseus 3rd dorsal interosseus 2nd dorsal interosseus 1st dorsal interosseus Adductor hallucis, transverse head Adductor hallucis Abductor hallucis Flexor hallucis brevis Abductor digiti minimi and fibularis brevis Flexor digiti minimi brevis Flexor hallucis longus Flexor digitorum longus Flexor digitorum brevis 466 Lower Limb Muscles & Tendon Sheaths of the Foot Fig. 33.17 Deep intrinsic muscles of the sole of the foot Right foot, plantar view. A Fourth layer. Removed: Adductor hallucis, flexor digiti minimi brevis, and flexor hal lucis brevis. B Muscle origins are shown in red, insertions in blue. Dorsal aponeurosis Calcaneal (Achilles’) tendon Superior fibular retinaculum Fibularis longus Inferior fibular retinaculum Fibularis brevis Tuberosity of 5th metatarsal Extensor digitorum brevis tendons Extensor hallucis longus tendon Extensor digitorum longus tendons Extensor digitorum brevis Fibularis tertius Inferior extensor retinaculum Superior extensor retinaculum Fibula Extensor digitorum longus Extensor hallucis longus Tibialis anterior Abductor digiti minimi Lateral malleolus Fibularis brevis Triceps surae Fibularis longus Tuberosity of 5th metatarsal Tibialis anterior Tibia Superior extensor retinaculum Inferior extensor retinaculum Extensor hallucis longus Flexor hallucis longus Tibialis anterior Flexor digitorum longus Tibialis posterior Flexor hallucis longus Calcaneal tuberosity Flexor retinaculum Calcaneal (Achilles’) tendon Tendon sheath Flexor hallucis longus Tibialis posterior Triceps surae Flexor digitorum longus Medial malleolus Fibularis longus Extensor digitorum longus Lateral malleolus Fibularis brevis Tuberosity of 5th metatarsal Extensor digitorum brevis Abductor digiti minimi Extensor digitorum longus tendons Extensor hallucis longus tendon Interossei Extensor hallucis brevis Tendon sheath Inferior extensor retinaculum Medial malleolus Superior extensor retinaculum Extensor hallucis longus Tibialis anterior Tibia Triceps surae Fibularis brevis Fibularis tertius 467 33 Ankle & Foot Fig. 33.18 Tendon sheaths and retinacula of the ankle Right foot. The superior and inferior extensor retinacula retain the long extensor tendons, the fibularis retinacula hold the fibular muscle ten dons in place, and the flexor retinaculum retains the long flexor tendons. A Anterior view with talocrural joint in plantar flexion. B Medial view. C Lateral view. Extensor hallucis brevis tendon Medial cuneiform Intermediate cuneiform Navicular Talus Calcaneus Superior trochlear surface Tuberosity of 5th metatarsal Extensor digitorum brevis Extensor hallucis brevis 5th metatarsal 5th proximal phalanx 5th middle phalanx 5th distal phalanx Extensor digitorum brevis tendons 1st proximal phalanx A S 468 Lower Limb Muscle Facts (I) The dorsal surface (dorsum) of the foot contains only two muscles, the extensor digitorum brevis and the extensor hallucis brevis. The sole of the foot, however, is composed of four complex layers that maintain the arches of the foot. Fig. 33.19 Intrinsic muscles of the dorsum of the foot Right foot, dorsal view. A Schematic. B Dorsal muscles of the foot. Table 33.2 Intrinsic muscles of the dorsum of the foot Muscle Origin Insertion Innervation Action ① Extensor digitorum brevis Calcaneus (dorsal surface) 2nd to 4th toes (at dorsal aponeuroses and bases of the middle phalanges) Deep fibular n. (L5, S1) Extension of the MTP and PIP joints of the 2nd to 4th toes ② Extensor hallucis brevis 1st toe (at dorsal aponeurosis and proximal phalanx) Extension of the MTP joints of the 1st toe MTP, metatarsophalangeal; PIP, proximal interphalangeal. A S D Calcaneal tuberosity Plantar aponeurosis Abductor digiti minimi Tuberosity of 5th metatarsal Sesamoids Abductor hallucis Flexor digitorum brevis Cruciform ligs. Tuberosity of cuboid 469 33 Ankle & Foot Fig. 33.20 Superficial intrinsic muscles of the sole of the foot Right foot, plantar view. A First layer, schematic. B Intrinsic muscles of the sole, first layer. Table 33.3 Superficial intrinsic muscles of the sole of the foot Muscle Origin Insertion Innervation Action ① Abductor hallucis Calcaneal tuberosity (medial process); flexor retinaculum, plantar aponeurosis 1st toe (base of proximal phalanx via the medial sesamoid) Medial plantar n. (S1, S2) • 1st MTP joint: flexion and abduction of the 1st toe • Supports the longitudinal arch ② Flexor digitorum brevis Calcaneal tuberosity (medial tubercle), plantar aponeurosis 2nd to 5th toes (sides of middle phalanges) • Flexes the MTP and PIP joints of the 2nd to 5th toes • Supports the longitudinal arch ③ Abductor digiti minimi 5th toe (base of proximal phalanx), 5th metatarsal (at tuberosity) Lateral plantar n. (S1–S3) • Flexes the MTP joint of the 5th toe • Abducts the 5th toe • Supports the longitudinal arch MTP, metatarsophalangeal; PIP, proximal interphalangeal. A S Flexor digitorum longus tendon ④ ③ ⑥ ⑤ J K 470 Lower Limb Muscle Facts (II) Fig. 33.21 Deep intrinsic muscles of the sole of the foot Right foot, plantar view, schematics. A Second layer. B Third layer. C Fourth layer. Table 33.4 Deep intrinsic muscles of the sole of the foot Muscle Origin Insertion Innervation Action ① Quadratus plantae Calcaneal tuberosity (medial and plantar borders on plantar side) Flexor digitorum longus tendon (lateral border) Lateral plantar n. (S1–S3) Redirects and augments the pull of flexor digitorum longus ② Lumbricals (four muscles) Flexor digitorum longus tendons (medial borders) 2nd to 5th toes (at dorsal aponeuroses) 1st lumbrical: medial plantar n. (S2, S3) • Flexes the MTP joints of 2nd to 5th toes • Extension of IP joints of 2nd to 5th toes • Adducts 2nd to 5th toes toward the big toe 2nd to 4th lumbrical: lateral plantar n. (S2, S3) ③ Flexor hallucis brevis Cuboid, lateral cuneiforms, and plantar calcaneocuboid lig. 1st toe (at base of proximal phalanx via medial and lateral sesamoids) Medial head: medial plantar n. (S1, S2) • Flexes the first MTP joint • Supports the longitudinal arch Lateral head: lateral plantar n. (S1, S2) ④ Adductor hallucis Oblique head: 2nd to 4th metatarsals (at bases) cuboid and lateral cuneiforms 1st proximal phalanx (at base, by a common tendon via the lateral sesamoid) Lateral plantar n., deep branch (S2, S3) • Flexes the first MTP joint • Adducts big toe • Transverse head: supports transverse arch • Oblique head: supports longitudinal arch Transverse head: MTP joints of 3rd to 5th toes, deep transverse metatarsal lig. ⑤ Flexor digiti minimi brevis 5th metatarsal (base), long plantar lig. 5th toe (base of proximal phalanx) Lateral plantar n., superficial branch (S2, S3) Flexes the MTP joint of the little toe ⑥ Opponens digiti minimi Long plantar lig.; fibularis longus (at plantar tendon sheath) 5th metatarsal Pulls 5th metatarsal in plantar and medial direction ⑦ Plantar interossei (three muscles) 3rd to 5th metatarsals (medial border) 3rd to 5th toes (medial base of proximal phalanx) Lateral plantar n. (S2, S3) • Flexes the MTP joints of 3rd to 5th toes • Extension of IP joints of 3rd to 5th toes • Adducts 3rd to 5th toes toward 2nd toe ⑧ Dorsal interossei (four muscles) 1st to 5th metatarsals (by two heads on opposing sides) 1st interosseus: 2nd proximal phalanx (medial base) • Flexes the MTP joints of 2nd to 4th toes • Extension of IP joints of 2nd to 4th toes • Abducts 3rd and 4th toes from 2nd toe 2nd to 4th interossei: 2nd to 4th proximal phalanges (lateral base), 2nd to 4th toes (at dorsal aponeuroses) IP, interphalangeal; MTP, metatarsophalangeal. May be absent. 1st through 4th lumbricals Quadratus plantae Flexor digitorum brevis Fibularis longus tendon 3rd plantar interosseus 1st dorsal interosseus Flexor digitorum longus tendons Flexor digitorum longus Sustentaculum tali Medial cuneiform Calcaneus Tuberosity of 5th metatarsal Long plantar lig. Medial head Lateral head Tibialis posterior tendon Fibularis longus tendon Oblique head Flexor digiti minimi brevis Adductor hallucis Metatarso phalangeal joint capsules Opponens digiti minimi Long plantar lig. Medial process Lateral sesamoid Medial sesamoid Lateral process Flexor hallucis brevis Transverse head Plantar calcaneonavicular lig. 471 33 Ankle & Foot Fig. 33.22 Deep intrinsic muscles of the sole of the foot Right foot, plantar view. A Intrinsic muscles of the sole, second and fourth layers. B Intrinsic muscles of the sole, third layer. Medial malleolar brs. Communicating br. Perforating br. Medial superior genicular a. Adductor hiatus Adductor magnus Anterior tibial recurrent a. Posterior tibial a. Middle genicular a. Medial inferior genicular a. Anterior tibial a. Medial plantar a. Calcaneal brs. Lateral malleolar brs. Muscular brs. Fibular a. Posterior tibial recurrent a. Lateral inferior genicular a. Sural aa. Lateral superior genicular a. Popliteal a. Lateral tarsal a. Anterior lateral malleolar a. Anterior medial malleolar a. Superficial circum-flex iliac a. Deep circum-flex iliac a. Superficial epigastric a. External pudendal aa. Inferior epigastric a. 1st through 4th perforating aa. Adductor hiatus Piriformis Lateral circum-flex femoral a. Deep a. of the thigh Lateral superior and inferior genicular aa. Arcuate a. Dorsal metatarsal aa. Dorsal pedal a. Medial superior and inferior genicular aa. Descending genicular a. Adductor canal (with adductor magnus) Femoral a. Medial circumflex femoral a. External iliac a. Superior and inferior gluteal aa. Internal iliac a. Common iliac a. Abdominal aorta Popliteal a. Anterior tibial a. Interosseous membrane Anterior tibial recurrent a. Common plantar digital aa. Plantar meta-tarsal aa. Deep plantar arch Lateral plantar a. Posterior tibial a. Abductor hallucis Medial plantar a. Deep br. Superficial br. Medial plantar a. Proper plantar digital aa. Lower Limb 472 34 Neurovasculature Arteries of the Lower Limb Fig. 34.1 Arteries of the lower limb and the sole of the foot A Right leg, anterior view. B Right leg, posterior view. C Sole of right foot, plantar view. Head of femur Medial circum flex femoral a. Iliopsoas tendon Deep a. of the thigh Lateral circumflex femoral a. Lesser trochanter Femoral neck vessels Lig. of head of femur Synovial membrane Fibrous membrane A. of lig. of head of femur Acetabu lar fossa Lig. of head of femur Zona orbicularis Medial circumflex femoral a. Synovial membrane Fibrous membrane Acetabular labrum Acetabular roof Obturator a. Popliteal a. passing through adductor hiatus Adductor magnus 1st perforating a. 2nd perforating a. 3rd perforating a. Adductor brevis Adductor longus Femoral a. Medial plantar a. Posterior tibial a. Fibular a. Adductor hiatus Internal iliac a. Common iliac a. External iliac a. Adductor canal Deep a. of the thigh Inguinal lig. Abdominal aorta Dorsal pedal a. Anterior tibial a. Femoral a. Popliteal a. Interosseous membrane 34 Neurovasculature 473 A Right femur. B Right femur, coronal section. Clinical box 34.1 Femoral head necrosis Dislocation or fracture of the femoral head (e.g., in patients with osteoporosis) may tear the femoral neck vessels, resulting in femoral head necrosis. Fig. 34.2 Segments of the femoral artery The blood supply to the lower limbs originates from the femoral artery. Color is used to identify the named distal segments of this vessel. Fig. 34.3 Deep artery of the thigh Right leg. The artery passes posteriorly through the adductor muscles of the medial thigh to supply the muscles of the posterior compart ment via three to five perforating branches. Ligation of the femoral artery proximal to the origin of the deep artery of the thigh (left) is well tolerated owing to the collateral blood supply (arrows) from branches of the internal iliac artery that anastomose with the perfor ating branches. Fig. 34.4 Arteries of the femoral head Anterior view. Superficial circumflex iliac v. Femoral v. (in saphenous opening) Anterior femoral cutaneous v. Dorsal venous network Dorsal venous arch Great saphenous v. Accessory saphenous v. External pudendal vv. Superficial epigastric v. Lateral circumflex femoral vv. Medial circumflex femoral vv. Adductor hiatus Inguinal lig. Piriformis Deep v. of thigh Femoral v. Small saphenous v. Great saphenous v. Genicular vv. Adductor magnus Popliteal v. Adductor canal Great saphenous v. Accessory saphenous v. External iliac v. Anterior tibial vv. Dorsal venous network of the foot Posterior tibial vv. Fibular vv. Anterior tibial v. Small saphenous v. Popliteal v. Small saphenous v. Lateral malleolus Femoro popliteal v. Great saphenous v. Posterior arch v. Small saphenous v. Popliteal v. Plantar digital vv. Plantar metatarsal vv. Plantar venous arch Lateral plantar v. Small saphenous v. Posterior tibial vv. Great saphenous v. Dorsal venous arch Medial plantar v. Lower Limb 474 Veins & Lymphatics of the Lower Limb A Right limb, anterior view. A Right limb, anterior view. B Right limb, posterior view. B Right limb, posterior view. Fig. 34.5 Superficial (epifascial) veins of the lower limb Fig. 34.6 Deep veins of the lower limb Fig. 34.7 Veins of the sole of the foot Right foot, plantar view. Posterior arch v. Posterior tibial vv. Dodd’s vv. Great saphenous v. Boyd’s vv. Cockett’s vv. Femoral v. External iliac v. Great saphenous v. Femoral v. Internal iliac v. External iliac v. Superolateral l.n. Superomedial l.n. Inferior l.n. Femoral v. Great saphenous v. Inguinal lig. Common iliac v. Inferior vena cava Small saphenous v. Popliteal v. Common iliac lymph nodes • Receive drainage from – Pelvic organs – Pelvic wall – Gluteal muscles – Erectile tissues – Deep perineal region • Receive drainage from – Skin of the limb (except the calf and the medial border of the foot) – Abdominal wall below the umbilicus – Lower back – Gluteal region, bowel, anal region – External genitalia (in women, also the uterine fundus along the round lig.) Internal iliac lymph nodes Superficial inguinal lymph nodes • Receive drainage from – Deep inguinal l.n. – Urinary bladder, shaft and glans of penis, uterus External iliac lymph nodes • Receive drainage from – Leg – Foot Deep popliteal lymph nodes • Receive drainage from – Lateral border of foot – Calf Superficial popliteal lymph nodes • Receive drainage from – Deep portions of the lower limb Deep inguinal lymph nodes Lumbar lymph nodes Superficial inguinal l.n. Antero medial bundle Great saphenous v. Small saphenous v. Anus Scrotum Postero lateral bundle Superficial popliteal l.n. 34 Neurovasculature 475 Fig. 34.8 Clinically important perforating veins Right leg, medial view. Fig. 34.9 Superficial lymph nodes Right limb. Arrows indicate the main directions of lymphatic drainage. A Anterior view. B Posterior view. Fig. 34.10 Lymph nodes and lymphatic drainage Right limb, anterior view. Arrows indicate direction of lymphatic drainage. Yellow shading: superficial nodes; green shading: deep nodes. Femoral n. Pudendal n. Inferior clunial nn. Posterior cutaneous n. of the thigh Sciatic n. Sural n. Medial and lateral plantar nn. Lateral sural cutaneous n. (with communicating br.) Superficial fibular n. Deep fibular n. Saphenous n. Lateral cutaneous n. of the thigh Obturator n. Genitofemoral n. Ilioinguinal n. Iliohypogastric n. Tibial n. Common fibular n. Tibial n. Subcostal n. Lower Limb 476 Lumbosacral Plexus The lumbosacral plexus supplies sensory and motor innervation to the lower limb. It is formed by the anterior (ventral) rami of the lumbar and sacral spinal nerves, with contributions from the subcostal nerve (T12) and coccygeal nerve (Co1). The lumbar plexus mainly supplies the anterior and medial parts of the thigh with a small contribution to the medial leg. The sacral plexus supplies the posterior thigh and most of the leg and foot. Table 34.1 Nerves of the lumbosacral plexus Lumbar plexus Iliohypogastric n. L1 p. 479 Ilioinguinal n. L1 Genitofemoral n. L1–L2 Lateral cutaneous n. of the thigh L2–L3 Obturator n. L2–L4 p. 480 Femoral n. p. 481 Sacral plexus Superior gluteal n. L4–S1 p. 483 Inferior gluteal n. L5–S2 Posterior cutaneous n. of the thigh S1–S3 p. 482 Sciatic n. Common fibular n. L4–S2 p. 484 Tibial n. L4–S3 p. 485 Pudendal n. S2–S4 pp. 284–285 Clinical box 34.2 Similar to nerve injuries of the upper limb, injuries involving nerves of the lumbosacral plexus are best understood though an appreciation of the plexus organization. The lumbar plexus arises from higher levels (L1–L4) of the spinal cord and supplies muscles of the abdominal wall and anterior and medial thigh. The sacral plexus arises from lower levels (L4–S4) of the spinal cord and supplies the perineum and, via the large sciatic nerve, the posterior thigh, entire leg and most of the foot. Nerves of the lumbar and sacral plexuses are less likely to be injured at the root level than those of the brachial plexus, although exceptions to this are the obturator and femoral nerves that may be compromised by herniation of intervertebral disks at L4 or L5 as they pass through the intervertebral foramina. Peripheral nerve injuries, such as that of the common fibular nerve, can occur in places where the nerve is superficial and passes close to a bony prominence. Injuries to nerves of the lumbar and sacral plexuses Co1 L1 L2 L3 L4 L5 S1 S2 Iliohypo gastric n. Ilioinguinal n. Genito femoral n. Lateral cutaneous n. of the thigh Obturator n. Femoral n. Superior gluteal n. Inferior gluteal n. Sciatic n. Common fibular n. Tibial n. Posterior cutaneous n. of the thigh Pudendal n. Coccygeal n. S5 S4 S3 Coccygeal plexus Lumbar plexus Sacral plexus Subcostal n. T12 vertebra 12th rib Iliohypogastric n. Ilioinguinal n. Genito femoral n. Obturator n. Lateral cutaneous n. of the thigh Sciatic n. Femoral n. Sciatic n. (common fibular n. and tibial n.) Muscular brs. Obturator n. Posterior br. Anterior br. Pudendal n. Muscular brs. Coccygeal plexus, anococcygeal nn. Superior and inferior gluteal nn. L5 vertebra L1 vertebra S1 vertebra Coccygeal n. Inguinal lig. Femoral n. Anterior femoral cutaneous brs. Muscular brs. Saphenous n. Lumbosacral trunk 34 Neurovasculature 477 Fig. 34.11 Lumbosacral plexus Right side, anterior view. Spinal nerve contributions to nerves of the lumbar and sacral plexuses. Separation of the anterior rami into anterior and posterior divisions are not as neatly demarcated in the lumbosacral plexus as they are in the brachial plexus of the upper limb. Where clearly separated into nerves they are indicated as: green = anterior division, blue = posterior divi sion. Note: Nerves of the sacral plexus not shown: n. to piriformis (S1, S2), n. to obturator internus (L5, S1), and n. to quadratus femoris (L5, S1). A Structure of the lumbosacral plexus. B Course of the lumbosacral plexus. Distribution of anterior rami of lumbar (yellow/orange) and sacral (blue/green) spinal nerves to the gluteal region and lower limb. Iliohypo gastric n. Lateral cutaneous br. Anterior cutaneous br. Lateral cutaneous n. of the thigh Femoral n., anterior cutaneous brs. Femoral br. Genital br. Ilioinguinal n. Genitofemoral n. Ilioinguinal n. Genito femoral n. Anterior scrotal brs. Superficial inguinal ring Lower Limb 478 Nerves of the Lumbar Plexus Table 34.2 Nerves of the lumbar plexus Nerve Level Innervated muscle Cutaneous branches Iliohypogastric n. L1 Transversus abdominis and internal oblique (inferior portions) Anterior and lateral cutaneous brs. Ilioinguinal n. L1 ♂: Anterior scrotal nn. ♀: Anterior labial nn. Genitofemoral n. L1–L2 ♂: Cremaster (genital br.) Genital br. Femoral br. Lateral cutaneous n. of the thigh L2–L3 — lateral cutaneous n. of the thigh Obturator n. L2–L4 See p. 480 Femoral n. L2–L4 See p. 481 Short, direct muscular brs. T12–L4 Psoas major Quadratus lumborum Iliacus Intertransversarii lumborum — Fig. 34.12 Cutaneous innervation of the inguinal region Right male inguinal region, anterior view. Clinical box 34.3 Ischemia (diminished blood flow) of the lateral cutaneous nerve of the thigh can result when the nerve is stretched or entrapped by the inguinal ligament (see Fig. 34.11B) during hyperextension of the hip or with increased lordosis (curvature) of the lumbar spine, as often occurs during pregnancy. This results in pain, numbness, or paresthesia (tingling or burning) on the outer aspect of the thigh. It is most commonly found in obese or diabetic individuals and in pregnant women. Entrapment of the lateral femoral cutaneous nerve (meralgia paresthetica) Transversus abdominis Internal oblique External oblique Lateral cutaneous br. Anterior cutane ous br. Superficial inguinal ring Iliacus Psoas major Iliohypogastric n. Quadratus lumborum Inguinal lig. Iliac crest Internal oblique Inguinal lig. Superficial inguinal ring Iliacus Psoas major Ilioinguinal n. Quadratus lumborum Transversus abdominis Ilioinguinal n. Spermatic cord Femoral br. Genital br. Iliacus Psoas major Genitofemoral n. Quadratus lumborum Inguinal lig. Spermatic cord Lateral cutaneous n. of the thigh Rectus abdominis Iliacus Psoas major Quadratus lumborum Spermatic cord Inguinal lig. Anterior superior iliac spine Fascia lata 34 Neurovasculature 479 Fig. 34.13 Nerves of the lumbar plexus Right side, anterior view with the anterior abdominal wall removed. A Iliohypogastric nerve. B Ilioinguinal nerve. C Genitofemoral nerve. D Lateral cutaneous nerve of the thigh. Adductor magnus Obturator externus Muscular brs. Gracilis Cutaneous br. Adductor longus Adductor brevis Posterior br. Anterior br. Pectineus L4 vertebra Linea terminalis Obturator n. Cutaneous br. Lower Limb 480 Nerves of the Lumbar Plexus: Obturator & Femoral Nerves Fig. 34.14 Obturator nerve: Cutaneous distribution Right leg, medial view. Fig. 34.15 Obturator nerve Right side, anterior view. Table 34.3 Obturator nerve (L2–L4) Motor branches Innervated muscles Direct br. Obturator externus Anterior br. Adductor longus Adductor brevis Gracilis Pectineus Posterior br. Adductor magnus Sensory branches Cutaneous br. Muscular br. Rectus femoris Anterior cutaneous brs. Muscular brs. Sartorius Anteromedial intermuscular septum Saphenous n. Pectineus Iliacus L4 vertebra Psoas major Saphenous n. Muscular brs. Inguinal lig. Infrapatellar br. Sartorius Femoral n. Iliopsoas Vastus intermedius Quadriceps femoris Vastus lateralis Rectus femoris Vastus medialis Saphenous n. Medial cutaneous brs. Infra patellar br. Anterior cutaneous brs. 34 Neurovasculature 481 Table 34.4 Femoral nerve (L2–L4) Motor branches Innervated muscles Muscular brs. Iliopsoas Pectineus Sartorius Quadriceps femoris Sensory branches Anterior cutaneous br. Saphenous n. Fig. 34.16 Femoral nerve Right side, anterior view. Fig. 34.17 Femoral nerve: Cutaneous distribution Right limb, anterior view. Superior clunial nn. (posterior rami of L1–L3) Middle clunial nn. (posterior rami of S1–S3) Inferior clunial nn. (Posterior cutaneous n. of the thigh) Lateral br. (iliohypogastric n.) Inferior clunial nn. Posterior cutaneous n. of the thigh Perineal brs. Anterior (ventral) root Cauda equina Posterior (dorsal) root Posterior sacral foramen Lateral br. (to the clunial nn.) Posterior (dorsal) ramus Anterior (ventral) ramus (to sacral plexus) Anterior sacral foramen Lower Limb 482 Nerves of the Sacral Plexus Fig. 34.18 Cutaneous innervation of the gluteal region Right limb, posterior view. Fig. 34.19 Posterior cutaneous nerve of the thigh: Cutaneous distribution Right limb, posterior view. Fig. 34.20 Emerging spinal nerve Horizontal section, superior view. Table 34.5 Nerves of the sacral plexus Nerve Level Innervated muscle Cutaneous branches Superior gluteal n. L4–S1 Gluteus medius Gluteus minimus Tensor fasciae latae — Inferior gluteal n. L5–S2 Gluteus maximus — Posterior cutaneous n. of the thigh S1–S3 — Posterior cutaneous n. of the thigh Inferior clunial nn. Perineal brs. Direct branches N. of piriformis S1–S2 Piriformis — N. of obturator internus L5–S1 Obturator internus Gemelli — N. of quadratus femoris Quadratus femoris — Sciatic n. Common fibular n. L4–S2 See p. 484 Tibial n. L4–S3 See p. 485 Pudenal n. S2–S4 See pp. 284–285 Gluteus medius Superior gluteal n. Iliotibial tract Tensor fasciae latae Anterior superior iliac spine Inferior gluteal n. Muscular brs. Sciatic n. Gluteus maximus Obturator internus (with n.) Sciatic n. Quadratus femoris (with n.) Gemellus inferior Gemellus superior Piriformis (with n.) Sacrotuberous lig. 34 Neurovasculature 483 Fig. 34.21 Nerves of the sacral plexus Right limb. A Superior gluteal nerve. Lateral view. B Inferior gluteal nerve. Posterior view. C Direct branches. Posterior view. Stance leg Gluteus medius and minimus Pelvis sags Insufficient small gluteals Shifted center of gravity Swing leg Clinical box 34.4 Small gluteal muscle weakness The small gluteal muscles on the stance side stabilize the pelvis in the coronal plane (A). Weakness or paralysis of the small gluteal muscles from damage to the superior gluteal nerve (e.g., due to a faulty intramuscular injection) is manifested by weak abduction of the affected hip joint. In a positive Trendelenburg’s test, the pelvis sags toward the normal, unsupported side (B). Tilting the upper body toward the affected side shifts the center of gravity onto the stance side, thereby elevating the pelvis on the swing side (Duchenne’s limp) (C). With bilateral loss of the small gluteals, the patient exhibits a typical waddling gait. A Normal gait. B Small gluteal muscle weakness. C Duchenne’s limp. Tibial n. Common fibular n. Deep fibular n. Superficial fibular n. Fibularis longus Fibularis brevis Lateral malleolus Intermediate dorsal cuta neous n. Extensor hallucis longus Extensor digitorum longus Tibialis anterior Head of fibula Biceps femoris, long head Biceps femoris, short head Sciatic n. Anterior superior iliac spine Neck of fibula Medial dorsal cutaneous n. Superficial fibular n. Intermediate dorsal cutaneous n. Lateral sural cutaneous n. Deep fibular n. Medial cutaneous n. of 2nd toe Lateral cutaneous n. of big toe Medial dorsal cutaneous n. Superficial fibular n. Intermediate dorsal cuta neous n. Medial dorsal cutaneous n. Fibular communicating br. Lower Limb 484 Nerves of the Sacral Plexus: Sciatic Nerve Table 34.6 Common fibular nerve (L4–S2) Nerve Innervated muscles Sensory branches Direct branches from sciatic n. Bicep femoris (short head) — Superficial fibular n. Fibularis brevis and longus Medial dorsal cutaneous n. Intermediate dorsal cutaneous n. Deep fibular n. Tibialis anterior Extensors digitorum brevis and longus Extensors hallucis brevis and longus Fibularis tertius Lateral cutaneous n. of big toe Medial cutaneous n. of 2nd toe The sciatic nerve gives off several direct muscular branches before dividing into the tibial and common fibular nerves proximal to the popliteal fossa. Fig. 34.22 Common fibular nerve: Cutaneous distribution A Right leg, anterior view. B Right leg, lateral view. Fig. 34.23 Common fibular nerve Right limb, lateral view. Sacro tuberous lig. Muscular brs. Semi tendinosus Semi membranosus Gastrocnemius Deep flexor tendons Tibial n. (in malleolar canal) Deep flexors Soleus Tibial n. Biceps femoris, long head Adductor magnus, medial part Sciatic n. Popliteal fossa Biceps femoris, short head Lateral malleolus Quadratus plantae Proper plantar digital nn. Lumbricals Common plantar digital nn. Lateral plantar n., super ficial br. Abductor digiti minimi Lateral plantar n. Flexor digitorum brevis and plantar aponeurosis Tibial n. Medial plantar n. Abductor hallucis Flexor digitorum longus tendon Muscular brs. Flexor hallucis longus tendon Adductor hallucis Lateral dorsal cutaneous n. Proper plantar digital nn. Fibular com municating br. Medial sural cutaneous n. Sural n. Medial calcaneal brs. Lateral calcaneal brs. 34 Neurovasculature 485 Fig. 34.24 Tibial nerve Right limb. A Posterior view. B Right foot, plantar view. Fig. 34.25 Tibial nerve: Cutaneous distribution Right lower limb, posterior view. Table 34.7 Tibial nerve (L4–S3) Nerve Innervated muscles Sensory branches Direct brs. from sciatic n. Semitendinosus Semimembranosus Biceps femoris (long head) Adductor magnus (medial part) — Tibial n. Triceps surae Plantaris Popliteus Tibialis posterior Flexor digitorum longus Flexor hallucis longus Medial sural cutaneous n. Medial and lateral calcaneal brs. Lateral dorsal cutaneous n. Medial plantar n. Adductor hallucis Flexor digitorum brevis Flexor hallucis brevis (medial head) 1st lumbricals Proper plantar digital nn. Lateral plantar n. Flexor hallucis brevis (lateral head) Quadratus plantae Abductor digiti minimi Flexor digiti minimi brevis Opponens digiti minimi 2nd to 4th lumbricals 1st to 3rd plantar interossei 1st to 4th dorsal interossei Adductor hallucis Proper plantar digital nn. Lateral sural cutaneous n. (common fibular n.) Deep fibular n. Accessory saphenous v. Fascia lata Inguinal lig. Superficial circumflex iliac v. Lateral cutaneous n. of the thigh Femoral n., anterior femoral cutaneous brs. Saphenous n. (femoral n.) Saphenous n., infrapatellar br. Superficial fibular n. Intermediate dorsal cutaneous n. Medial dorsal cutaneous n. Great saphenous v. External pudendal vv. Superficial inguinal ring Ilioinguinal n. Femoral a. and v. (in saphenous opening) Superficial epigastric v. Sural n. (tibial n.) Obturator n. Great saphenous v. Superior clunial nn. Middle clunial nn. Inferior clunial nn. (posterior cutaneous n. of the thigh) Posterior cutaneous n. of the thigh Obturator n., cutaneous br. Saphenous n. (femoral n.) Small saphenous v. Calcaneal brs. Medial and lateral plantar cutaneous brs. Lateral dorsal cutaneous n. (tibial n.) Sural n. (tibial n.) Lateral sural cutaneous n. (common fibular n.) Lateral cutaneous n. of the thigh Iliohypogastric n., lateral cutaneous br. Medial sural cutaneous n. (tibial n.) Lower Limb 486 Superficial Nerves & Veins of the Lower Limb Fig. 34.26 Superficial cutaneous veins and nerves of right lower limb A Anterior view. B Posterior view. Iliohypo gastric n. Lateral cutaneous n. of the thigh Common fibular n. Femoral n. Sciatic n. Obturator n. Genitofemoral n. Tibial n. Ilioinguinal n. L1 L2 L5 S1 T11 T12 S2 L3 L4 Sciatic n. Femoral n. Iliohypogastric n. Clunial nn. Posterior cutaneous n. of the thigh Lateral cutaneous n. of the thigh Tibial n. Common fibular n. Obturator n. L2 S4 S3 L4 L5 L3 S5 S2 S1 L4 L5 34 Neurovasculature 487 Fig. 34.27 Cutaneous innervation of the lower limb Right lower limb. A Anterior view. A Anterior view. B Posterior view. B Posterior view. Fig. 34.28 Dermatomes of the lower limb Right lower limb. Anterior femoral cutaneous v. Great saphenous v. Superficial and superolateral inguinal l.n. External iliac v. Deep inguinal l.n. Rosenmüller’s l.n. Superficial and superomedial inguinal l.n. External pudendal v. External iliac l.n. Saphenous opening Superficial epigastric v. Superficial circum flex iliac v. Superficial and inferior inguinal l.n. Femoral a. and v. Inguinal lig. Transversus abdominis Internal oblique Superficial inguinal ring Superficial circumflex iliac a. and v. Genitofemoral n., femoral br. Lateral cutaneous n. of the thigh Femoral a. and v. (deep to saphenous opening) Inguinal lig. External oblique aponeurosis Superficial abdominal fascia Great saphenous v. Lacunar lig. Reflected inguinal lig. Anterior rectus sheath Spermatic cord (cut) Genitofemoral n., genital br. Linea alba Arcuate line Ilioinguinal n. Rectus abdominis External oblique Pectineus, deep to fascia lata External pudendal a. and v. Anterior femoral cutaneous v. Saphenous opening in fascia lata Lower Limb 488 Topography of the Inguinal Region Fig. 34.29 Superficial veins and lymph nodes Right male inguinal region, anterior view. Removed: Cribriform fascia over the saphenous opening. Fig. 34.30 Inguinal region Right male inguinal region, anterior view. Reflected inguinal lig. Anterior superior iliac spine Ilioinguinal lig. Iliopectineal arch Iliopubic eminence Lacunar lig. Pubic tubercle ③ ② ① Reflected inguinal lig. Iliopectineal arch Lateral cutaneous n. of the thigh External oblique Inguinal lig. Muscular compartment Femoral n. Iliopsoas Iliacus Psoas major Iliopectineal bursa Acetabular fossa Vascular compartment Genitofemoral n., femoral br. Femoral a. and v. Rosenmüller’s l.n. Ischial tuberosity Ischial spine Pubic symphysis Lacunar lig. Superficial inguinal ring Intercrural fibers External oblique aponeurosis Lateral crus Medial crus Femoral ring 34 Neurovasculature 489 Fig. 34.31 Retro-inguinal space: Muscular and vascular compartments Right inguinal region, anterior view. Table 34.8 Structures in the inguinal region Region Boundaries Contents Retro-inguinal space ① Muscular compartment Anterior superior iliac spine Inguinal lig. Iliopectineal arch Femoral n. Lateral cutaneous n. of the thigh Iliacus Psoas major ② Vascular compartment Inguinal lig. Iliopectineal arch Lacunar lig. Femoral a. and v. Genitofemoral n., femoral br. Rosenmüller’s lymph node Inguinal canal ③ Superficial inguinal ring Medial crus Lateral crus Reflected inguinal lig. Ilioinguinal n. Genitofemoral n., genital br. Spermatic cord Middle clunial nn. Superior clunial nn. Inferior clunial nn. Adductor magnus Semitendinosus Semi membranosus Gluteus maximus Gluteal fascia (gluteus medius) Iliohypogastric n., lateral br. Posterior cutaneous n. of the thigh (with biceps femoris long head) Obturator internus Gemellus superior and inferior Piriformis Superior gluteal a., v., and n. Gluteus medius Pudendal n., perineal brs. Posterior cutaneous n. of the thigh, perineal brs. Sciatic n. (with a.) Inferior gluteal a., v., and n. Sacrotuberous lig. Ischial tuberosity Adductor magnus Gracilis Posterior cutaneous n. of the thigh Adductor magnus Gluteus maximus Quadratus femoris Gluteus maximus Obturator internus Posterior cutaneous n. of the thigh Fascia lata Gluteal sulcus Gluteal fascia (gluteus maximus) Gluteal fascia (gluteus medius) Lower Limb 490 Topography of the Gluteal Region Fig. 34.32 Gluteal region Right gluteal region, posterior view. A Fasciae and cutaneous neurovasculature. C Deep gluteal region. Partially removed: Gluteus maximus. B Gluteal region. Removed: Fascia lata. Posterior superior iliac spine Sacro spinous lig. Sacro tuberous lig. Greater sciatic notch Anterior superior iliac spine Piriformis Lesser sciatic notch D A S Posterior cutaneous n. of the thigh Biceps femoris, long head 1st perforating a. Adductor magnus Sciatic n. Quadratus femoris Br. of medial circumflex femoral a. Gemellus inferior Obturator internus Gemellus superior Piriformis Tensor fasciae latae Gluteus minimus Anterior superior iliac spine Semi membranosus Trochanteric bursa Posterior superior iliac spine Superior gluteal a. and n. Inferior gluteal n. Inferior gluteal aa. and vv. Pudendal n. Internal pudendal a. and v. Obturator internus Pudendal (Alcock’s) canal Sacrotuberous lig. Semitendinosus Gracilis Adductor magnus 34 Neurovasculature 491 Fig. 34.33 Gluteal region and ischioanal fossa Right gluteal region, posterior view. Removed: Gluteus maximus and medius. Table 34.9 Sciatic foramina Foramen Transmitted structures Boundaries Greater sciatic foramen ① Suprapiriform portion Superior gluteal a., v., and n. Greater sciatic notch Sacrospinous lig. Sacrum ② Infrapiriform portion Inferior gluteal a., v., and n. Internal pudendal a. and v. Pudendal n. Sciatic n. Posterior cutaneous n. of the thigh ③ Lesser sciatic foramen Internal pudendal a. and v. Pudendal n. Obturator internus Lesser sciatic notch Sacrospinous lig. Sacrotuberous lig. Anterior superior iliac spine Inguinal lig. Tensor fasciae latae Superficial circumflex iliac a. Iliopsoas Femoral n. Femoral a. and v. Deep a. of thigh Sartorius Patellar vascular network Femoral a. and v. in adductor canal Pectineus Spermatic cord External pudendal a. Superficial epigastric a. Descending genicular a. Gracilis Adductor longus Iliotibial tract Fascia lata Quadriceps femoris Rectus femoris External oblique aponeurosis Sacral plexus Obturator n., cutaneous br. Lateral cutaneous n. of the thigh Inguinal lig. Femoral n. Sartorius Rectus femoris Lateral circumflex femoral a., ascending br. Medial circumflex femoral a. Deep a. of thigh Perforating aa. Lateral circumflex femoral a., descending br. Saphenous n. Sartorius Femoral a. and v., saphenous n. (in adductor canal) Adductor magnus Adductor longus Adductor brevis Obturator n. Pectineus Femoral a. and v. Superior and inferior gluteal aa. External iliac a. and v. Rectus femoris Vastus medialis Vastus lateralis Vastus intermedius Lower Limb 492 Topography of the Anterior, Medial & Posterior Thigh Fig. 34.34 Anterior and medial thigh Right thigh, anterior view. A Femoral triangle. Removed: Skin, subcutaneous tissue, and fascia lata. Partially transparent: Sartorius. B Neurovasculature of the anterior thigh. Removed: Anterior abdominal wall. Partially removed: Sartorius, rectus femoris, adductor longus, and pectineus. Middle clunial nn. Superior clunial nn. Inferior clunial nn. Adductor magnus Tibial n. Sural n. Lateral sural cutaneous n. Common fibular n. Biceps femoris, long head Fascia lata, iliotibial tract Gluteus maximus Gluteal fascia (gluteus medius) Iliohypo gastric n., lateral br. Posterior cutaneous n. of the thigh Popliteal a. and v. Trochanteric bursa Inferior gluteal n. Sacrotuberous lig. Obturator internus Pudendal n. Adductor magnus Superior gluteal a., v., and n. Inferior gluteal a. Posterior cutaneous n. of the thigh Gracilis Semi tendinosus Popliteal a. and v. Semi membranosus Gastrocnemius Lateral sural cutaneous n. Medial sural cutaneous n. Plantaris Biceps femoris, long head Common fibular n. Iliotibial tract Biceps femoris, short head Adductor hiatus 3rd perforating a. 2nd perforating a. 1st perforating a. Sciatic n. (with a.) Quadratus femoris Gluteus maximus Medial circumflex femoral a. Piriformis Gluteus minimus Gluteus medius Gluteus maximus Biceps femoris, long head Adductor magnus Tibial n. 34 Neurovasculature 493 Fig. 34.35 Posterior thigh Right thigh, posterior view. A Gluteal region and thigh. Removed: Fascia lata. B Neurovasculature of the posterior thigh. Partially removed: Gluteus maxi mus, gluteus medius, and biceps femoris. Retracted: Semimembranosus. Great saphenous v. Small saphenous v. Saphenous n. Tibial n., medial calcaneal br. Gastrocnemius, medial head Gastrocnemius, lateral head Lateral sural cutaneous n. Medial sural cutaneous n. Common fibular n. Dorsal cutaneous n. of the foot Biceps femoris Plantaris Communi cating br. Sural n. Deep fascia of the leg Semi tendinosus Semi membranosus Tibial n. Fibularis longus Semi tendinosus Gracilis Semi membranosus Plantaris Gastroc nemius Tendinous arch of soleus Flexor digitorum longus Tibialis posterior Calcaneal (Achilles’) tendon Calcaneal rete Lateral malleolus Communi cating br. Perforating br. Fibularis brevis Fibular a. Soleus Popliteal a. and v. Popliteus Common fibular n. Tibial n. Biceps femoris Fibular a. Tibial n. Flexor retinaculum Flexor hallucis longus Posterior tibial a. Medial malleolus Lower Limb 494 Topography of the Posterior Compartment of the Leg & Foot Fig. 34.36 Posterior compartment of leg Right leg, posterior view. A Superficial neurovascular structures. B Deep neurovascular structures. Removed: Gastrocnemius. Windowed: Soleus. Tibia Medial plantar a. and n. Abductor hallucis Medial plantar a. and n. Flexor retinaculum Medial calcaneal br. Calcaneal (Achilles’) tendon Flexor hallucis longus Flexor digi torum longus Tibialis posterior Tibial n., posterior tibial a. Superficial flexors Deep flexors Fibula Fibularis group Extensor group Medial plantar a., superficial br. Tibialis anterior Medial tarsal aa. Extensor hallucis longus tendon 1st metatarsal Superior extensor retinaculum Medial malleolus (with subcutaneous bursa) Inferior extensor retinaculum Tarsal tunnel Lateral plantar a. and n. Medial malleolar brs. Semi membranosus Popliteal a. and v. Gastroc nemius Small saphenous v. Plantaris Deep popliteal l. n. Biceps femoris Medial superior genicular a. Oblique popliteal lig. Semi membranosus Semi tendinosus Gastrocnemius, medial head Medial subtendinous bursa of gastrocnemius Middle genicular a. Semimembranosus tendon Semimembranosus bursa Medial inferior genicular a. Tibial n. Plantaris tendon Popliteus Posterior tibial recurrent a. Lateral inferior genicular a. Gastroc nemius, lateral head Plantaris Lateral superior genicular a. Common fibular n. Biceps femoris, short head Sciatic n. Biceps femoris, long head Popliteal a. and v. Gracilis Gastrocnemius Soleus Triceps surae 34 Neurovasculature 495 B Deep lymph nodes. Fig. 34.37 Popliteal region Right leg, posterior view. A Deep neurovascular structures. Fig. 34.38 Ankle region Right ankle, medial view. Deep fascia of the leg Biceps femoris Short head Long head Common fibular n. Head of fibula Fibularis longus Commu nicating br. Soleus Sural n. Lateral calcaneal brs. Lateral dorsal cutaneous n. Medial dorsal cutaneous n. Superficial fibular n. Extensor digitorum longus Tibialis anterior Superficial fibular n. Deep fibular n. Anterior crural intermuscular septum Lateral tibial condyle Patella Iliotibial tract Lateral sural cutaneous n. Medial sural cutaneous n. (tibial n.) Gastroc nemius Lateral malleolus Deep fibular n., cutaneous br. Intermediate dorsal cutaneous n. A F S D Fibula Deep fibular n., anterior tibial a. and v. Saphenous n., great saphenous v. Tibia Deep fascia of the leg Sural n., small saphenous v. Transverse intermuscular septum Tibial n., posterior tibial a. and v. Fibular a. and v. Posterior intermuscular septum Superficial fibular n. Anterior intermuscular septum Interosseous membrane Head of fibula Lower Limb 496 Topography of the Lateral & Anterior Compartments of the Leg & Dorsum of the Foot Table 34.10 Compartments of the leg Compartment Muscular contents Neurovascular contents ① Anterior compartment Tibialis anterior Deep fibular n. Anterior tibial a. and v. Extensor digitorum longus Extensor hallucis longus Fibularis tertius ② Lateral compartment Fibularis longus Superficial fibular n. Fibularis brevis Posterior compartment ③ Superficial part Triceps surae (gastroc-nemius and soleus) — Plantaris ④ Deep part Tibialis posterior Tibial n. Posterior tibial a. and v. Fibular a. and v. Flexor digitorum longus Flexor hallucis longus Fig. 34.39 Neurovasculature of the lateral compartment of the leg Right limb. Removed: Origins of the fibularis longus and extensor digitorum longus. Patella Head of fibula Patellar lig. Fibularis longus Superficial fibular n. Fibularis brevis Superior extensor retinaculum Inferior extensor retinaculum Lateral dorsal cutaneous n. Intermediate dorsal cutaneous n. Medial dorsal cutaneous n. Dorsal metatarsal aa. Deep fibular n. Extensor hallucis brevis Extensor hallucis longus tendon Dorsalis pedis a. Medial malleolus Muscular brs. Tibialis anterior Gastrocnemius Pes anserinus (common insertion of sartorius, gracilis, and semitendinosus) Soleus Anterior tibial a. and v. Deep fibular n. Extensor hallucis longus Extensor digitorum longus Arcuate a. Fibula Fibular a., perforating br. Anterior lateral malleolar a. Deep fibular n. Extensor hallucis brevis Extensor digitorum brevis Lateral tarsal a. Dorsal interossei Dorsal meta tarsal aa. Extensors digitorum longus and brevis tendons Dorsal digital nn. Dorsal digital aa. Deep fibular n., cutaneous br. Extensors hallucis longus and brevis tendons Deep plantar a. Tibia Tibialis anterior tendon Anterior tibial a. Extensor hallucis longus Dorsalis pedis a. Lateral br. of deep fibular n. Medial br. of deep fibular n. 34 Neurovasculature 497 Fig. 34.40 Neurovasculature of the anterior compartment of the leg and foot Right limb with foot in plantar flexion. B Neurovasculature of the leg. Removed: Skin, subcutaneous tissue, and fasciae. Retracted: Tibialis anterior and extensor hallucis longus. A Neurovasculature of the dorsum of the foot. Clinical box 34.5 Compartment syndrome Muscle edema or hematoma can lead to a rise in tissue fluid pressure in the compartments of the leg. Subsequent compression of neurovascular structures due to this increased pressure may cause ischemia and irreversible muscle and nerve damage. Patients with anterior compartment syndrome, the most common form, suffer excruciating pain and cannot dorsiflex the toes. Emergency incision of the fascia of the leg may be performed to relieve compression. Common plantar digital nn. Lateral plantar n., superficial brs. Lateral plantar sulcus Medial plantar n., super ficial br. Medial plantar sulcus Plantar aponeurosis Medial plantar a., deep br. Medial plantar a., super ficial br. Medial plantar n. Proper plantar digital aa. Proper plantar digital nn. Lateral plantar a. Abductor hallucis Plantar metatarsal aa. Common plantar digital nn. Lateral plantar n., super ficial br. Flexor digi torum longus tendon Abductor digiti minimi Abductor hallucis Medial plantar n. Medial plantar a., deep br. Medial plantar a., super ficial br. Flexor hallucis longus tendon Flexor digitorum brevis Lateral plantar n., deep br. Quadratus plantae Lateral plantar a., v., and n. Flexor digi torum brevis tendons Proper plantar digital aa. and nn. Plantar aponeurosis Lower Limb 498 Topography of the Sole of the Foot Fig. 34.41 Neurovasculature of the sole of the foot Right foot, plantar view. A Superficial layer. Removed: Skin, subcutane ous tissue, and fascia. B Middle layer. Removed: Plantar aponeurosis and flexor digitorum brevis. Plantar aponeurosis Flexor digitorum brevis Abductor hallucis Medial plantar a., deep br. Flexor hallucis brevis Flexor hallucis longus tendon Lumbricals Flexor digitorum longus tendons Flexor digitorum brevis tendons Proper plantar digital aa. and nn. Transverse head Oblique head Plantar interossei Plantar metatarsal aa. Deep plantar arch Lateral plantar n., deep br. Quadratus plantae Lateral plantar a., v., and n. Medial plantar n. Medial plantar a. Adductor hallucis 3rd metatarsal Extensor digitorum longus Lateral dorsal cutaneous n. Dorsal meta tarsal a. Deep layer of plantar fascia Abductor digiti minimi Flexor digiti minimi brevis Opponens digiti minimi Lateral plantar a. and v. Lateral plantar n., superficial br. Lateral plantar septum Quadratus plantae Fibularis longus Flexor digitorum brevis Aponeurosis of flexor digitorum longus Deep plantar arch Plantar aponeurosis Medial plantar a. and n. Medial plantar septum Flexor hallucis longus Flexor hallucis brevis Abductor hallucis Saphenous n., cutaneous br. Lateral plantar n., deep br. Tibialis anterior Medial cuneiform Medial dorsal cutaneous n. Extensor hallucis longus 2nd metatarsal Extensor hallucis brevis Deep fibular n., dorsal pedal a. Interossei Intermediate dorsal cuta neous n. Extensor digitorum longus Extensor digitorum longus 34 Neurovasculature 499 C Deep layer. Removed: Flexor digitorum longus. Windowed: Adductor hallucis (oblique head). Fig. 34.42 Neurovasculature of the foot: Cross section Coronal section, distal view. Fig. 35.1 Windowed dissection of the thigh and leg Right limb, posterior view. Tibia Triceps surae Fibula Piriformis Gemellus superior Obturator internus Gemellus inferior Ischial tuberosity Adductor magnus Gracilis Semitendinosus Vastus medialis Sartorius Adductor brevis Adductor longus Gracilis Gracilis Gastrocnemius Plantaris Sciatic nerve Biceps femoris, short head Vastus intermedius Vastus lateralis Rectus femoris Biceps femoris, long head Iliotibial tract Tensor fasciae latae Gluteus medius Iliac crest Femur Adductor magnus Quadratus femoris Gluteus maximus Iliotibial tract Semimembranosus Semitendinosus Biceps femoris, long head Interosseous membrane Gluteus maximus Gluteus minimus Soleus Calcaneal (Achilles’) tendon Lower Limb 500 35 Sectional & Radiographic Anatomy Sectional Anatomy of the Lower Limb Fig. 35.2 Cross-section through the thigh and leg Right limb, proximal view. B Leg (plane of lower section in Fig. 35.1B) A Thigh (plane of upper section in Fig. 35.1A) Vastus medialis Medial intermuscular septum Femoral artery and vein Sartorius Adductor longus Adductor brevis Gracilis Adductor magnus Semi-membranosus Semi-tendinosus Sciatic nerve Biceps femoris, long head Biceps femoris, short head Lateral inter-muscular septum Iliotibial tract Vastus intermedius Vastus lateralis Rectus femoris Femur Quadriceps femoris Deep fibular nerve Anterior tibial artery and vein Tibia Tibial nerve Posterior tibial artery and vein Plantaris tendon Soleus Gastrocnemius, medial head Transverse intermuscular septum Gastrocnemius, lateral head Flexor hallucis longus Fibula Posterior in-termuscular septum Fibularis longus Fibularis brevis Interosseous membrane Extensor digitorum longus Extensor hallucis longus Tibialis anterior Anterior intermuscular septum Flexor digitorum longus Tibialis posterior 35 Sectional & Radiographic Anatomy 501 A B C Lower Limb 502 Radiographic Anatomy of the Lower Limb (I) Fig. 35.3 MRI of the thigh Transverse section, distal (inferior) view. C Distal thigh. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) B Mid-thigh. (Reproduced from Moeller TB, Reif E. Atlas of Sec-tional Anatomy: The Musculo-skeletal System. New York, NY: Thieme; 2009.) A Proximal thigh. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Rectus femoris Vastus medialis Vastus intermedius Sartorius Vastus lateralis Femur Great saphenous v. Femoral a. and v. Adductor longus Gracilis Sciatic n. Adductor magnus Biceps femoris, long head Semimembranous Semitendinous Sartorius Femoral a., v., and n. Rectus femoris Circumflex femoral a. and v. Deep femoral a. and v. Tensor fasciae latae Adductor longus Pectineus Vastus lateralis Iliotibial tract Femur Adductor magnus Lateral femoral intermuscular septum Sciatic n. Gluteus maximus Rectus femoris, tendon Vastus medialis Vastus intermedius Femur Sartorius Vastus lateralis Femoral a. and v. Biceps femoris, short head Gracilis Perforating a. and v. of deep a. and v. of thigh Semimembranosus Common fibular n. Semitendinosus Tibial n. Biceps femoris, long head A B C 35 Sectional & Radiographic Anatomy 503 Fig. 35.4 MRI of the leg Transverse section, distal (inferior) view. C Distal leg. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) B Mid-leg. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) A Proximal leg. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 2, 4th ed. New York, NY: Thieme; 2014.) Tibialis anterior Tibia Extensor halluces longus Tibialis posterior Extensor digitorum longus Flexor digitorum longus Deep fibular n. Posterior tibial a. and v. Anterior tibial a. and v. Tibial n. Interosseous membrane Fibular a. and v. Fibularis brevis Fibula Flexor hallucis longus Soleus Tibial tuberosity Tibia Anterior tibial muscle Sartorius, tendon Extensor digitorum longus Gracilis, tendon Semitendinosus, tendon Popliteus Common fibular n. Popliteal a. and v. Tibial n. Gastrocnemius, medial head Gastrocnemius, lateral head Extensor hallucis longus Anterior tibial a. and v. Extensor digitorum longus Great saphenous v. Tibia Tibialis posterior, tendon Fibula Fibularis longus, tendon Tibial n. Fibularis brevis Flexor hallucis longus Sural n. Soleus muscle Small saphenous v. Tendons of triceps surae and plantaris Fig. 35.5 Radiograph of the right hip joint Anteroposterior view. Fig. 35.6 Radiograph of right hip joint with limb abducted laterally (Lauenstein view) (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Anterior superior iliac spine Sacroiliac joint Superior acetabular rim Anterior acetabular rim Greater trochanter Pubic bone Ischium Superior pubic ramus Lesser trochanter Femoral neck Femoral head Inferior pubic ramus Ischial tuberosity Roof of the acetabulum Anterior acetabular rim Posterior acetabular rim Fovea of the femoral head Femoral head Köhler’s teardrop figure Greater trochanter Femoral neck Superior pubic ramus Intertrochanteric crest Obturator foramen Ischial tuberosity Lesser trochanter Lower Limb 504 Radiographic Anatomy of the Lower Limb (II) Fig. 35.7 MRI of the right hip joint Transverse section, distal (inferior) view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Fig. 35.8 MRI of the hip joints Coronal section, anterior view. (Repro-duced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Fig. 35.9 MRI of the right hip joint Sagittal section, medial view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Sartorius Femoral a., v., and n. Iliopsoas Tensor fascia lata Urinary bladder Pubis, superior ramus Obturator a., v., and n. Gluteus medius Head of femur Levator ani Obturator internus Ischium Sciatic n. Gluteus maximus L4 vertebra Psoas Iliacus Gluteus medius Urinary bladder Hip joint Head of femur Iliotibial tract Greater trochanter Obturator internus Obturator externus Gracilis Adductor longus Vastus lateralis Iliopsoas Ilium (roof of acetabulum) Hip joint Gluteus maximus Femur, head Lateral circumflex femoris a. (ascending branch) Pectineus Sartorius Adductor magnus Ischium Deep a. and v. of the thigh Biceps femoris Vastus medialis 35 Sectional & Radiographic Anatomy 505 Fig. 35.10 Radiograph of the right knee joint Anteroposterior view. (Reproduced courtesy of Klinik für Diagnostische Radiologie, Univer-sitätsklinikum Schleswig Holstein, Campus Kiel: Prof. Dr. Med. S. Müller-Huelsbeck.) Fig. 35.11 Radiograph of the knee in flexion (Reproduced courtesy of Klinik für Diagnostische Radiologie, Universitätsklinikum Schleswig Holstein, Campus Kiel: Prof. Dr. Med. S. Müller-Huelsbeck.) Femur Patella Lateral femoral epicondyle Medial femoral epicondyle Lateral femoral condyle Medial femoral condyle Lateral tibial condyle Medial tibial condyle Medial and lateral tubercles of intercondylar eminence Epiphyseal plate Fibular head Tibia Fibula Cortex Femur Patella Lateral femoral condyle Tibial plateau Tibial tuberosity Medial femoral condyle Fibular head A Lateral view. B Sunrise view. Lateral femoral condyle Intercondylar fossa Patella Patellofemoral joint Medial femoral condyle Lower Limb 506 Radiographic Anatomy of the Lower Limb (III) Fig. 35.12 MRI of the knee joint (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Fig. 35.13 MRI of the knee joint Sagittal section. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Patellar lig. Lateral femoral condyle Posterior cruciate lig. Medial collateral lig. Lateral collateral lig. Medial femoral condyle Anterior cruciate lig. Biceps femoris Gastrocnemius, medial head Popliteal a. and v. Gastrocnemius, lateral head Tibial n. Anterior cruciate lig. Lateral epicondyle Lateral femoral condyle Medial collateral lig. Posterior cruciate lig. Lateral meniscus, intermediate portion Medial femoral condyle Lateral tibial condyle Medial intercondylar tubercle Medial meniscus, intermediate portion Fibula, head Medial tibial condyle Vastus lateralis Biceps femoris Gastrocnemius, lateral head Femur, lateral condyle Lateral meniscus, posterior horn Lateral meniscus, anterior horn Tibiofibular joint Lateral tibial condyle Fibula, head A Transverse section, distal (inferior) view. B Coronal section. A B Quadriceps tendon Patella Anterior cruciate lig. Patellar lig. Infrapatellar fat pad Popliteal a. Popliteal v. Posterior cruciate lig. A B 35 Sectional & Radiographic Anatomy 507 Fig. 35.14 Radiograph of the ankle (Reproduced from Moeller TB, Reif E. Taschenatlas der Roentgenanatomie, 2nd ed. Stuttgart: Thieme; 1998.) Fig. 35.15 Anterior-posterior view of the forefoot Tibia Fibula Growth plate Fibular notch Talocrural joint Medial malleolus Lateral malleolus Trochlea of talus Subtalar joint Calcaneus Navicular Fibula Tibia Growth plate Talocrural joint Trochlea of talus Medial malleolus Talus Lateral malleolus Talonavicular joint Sinus tarsi Navicular Posterior tuberosity of calcaneus Cuboid Distal phalanx Distal phalanx Middle phalanx Proximal phalanx Proximal phalanx Distal interphalangeal joint Proximal interphalangeal joint Metatarsophalangeal joint Sesamoids Metatarsals Medial cuneiform Intermediate cuneiform Lateral cuneiform Intertarsal joint Metatarsophalangeal joint Base of 5th metatarsal Cuboid Navicular Talocalcaneonavicular joint Calcaneocuboid joint A Anteroposterior view. B Left lateral view. Lower Limb 508 Radiographic Anatomy of the Lower Limb (IV) Fig. 35.18 MRI of the right foot and ankle Sagittal section. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Fig. 35.16 MRI of the right ankle Coronal section, anterior view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculoskeletal System. New York, NY: Thieme; 2009.) Fig. 35.17 MRI of the right foot Coronal section, anterior (distal) view. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Muscu-loskeletal System. New York, NY: Thieme; 2009.) Fibula Tibia Talus Talocrural joint Talofibular joint Deltoid lig. (posterior tibiotalar part) Posterior talofibular lig. Subtalar joint Calcaneofibular joint Fibularis longus tendon Calcaneus Medial plantar a., v., and n. Flexor digitorum brevis Plantar aponeurosis Abductor hallucis Intermediate cuneiform Lateral cuneiform Metatarsal II (base) Medial cuneiform Metatarsal III (base) Abductor hallucis Metatarsal IV (base) Metatarsal V (base) Interosseous mm. Abductor digiti minimi Flexor digitorum brevis Dorsal Proximal, middle, and distal phalanx of second toe Metatarsal II, head Adductor hallucis Medial cuneiform Intermediate cuneiform Quadratus plantae Plantar calcaneonavicular lig. Plantar aponeurosis Subtalar joint Calcaneous Navicular Talonavicular joint Talocalcaneal interosseous lig. Talus Tibia Talocrural joint Calcaneal (Achilles) tendon) 35 Sectional & Radiographic Anatomy 509 36 Surface Anatomy Surface Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 37 Neck Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Muscle Facts (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 Arteries & Veins of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . 520 Lymphatics of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Innervation of the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Larynx: Cartilage & Structure . . . . . . . . . . . . . . . . . . . . . . . . . 526 Larynx: Muscles & Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Neurovasculature of the Larynx, Thyroid & Parathyroids . . . 530 Topography of the Neck: Regions & Fascia . . . . . . . . . . . . . . 532 Topography of the Anterior Cervical Region . . . . . . . . . . . . . 534 Topography of the Anterior & Lateral Cervical Regions . . . . 536 Topography of the Lateral Cervical Region . . . . . . . . . . . . . . 538 Topography of the Posterior Cervical Region . . . . . . . . . . . . 540 38 Bones of the Head Anterior & Lateral Skull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Posterior Skull & Calvaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 Base of the Skull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Neurovascular Pathways Exiting or Entering the Cranial Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Ethmoid & Sphenoid Bones . . . . . . . . . . . . . . . . . . . . . . . . . . 550 39 Muscles of the Skull & Face Muscles of Facial Expression & of Mastication . . . . . . . . . . . . 552 Muscle Origins & Insertions on the Skull . . . . . . . . . . . . . . . . 554 Muscle Facts (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Muscle Facts (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 40 Cranial Nerves Cranial Nerves: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 CN I & II: Olfactory & Optic Nerves . . . . . . . . . . . . . . . . . . . . 562 CN III, IV & VI: Oculomotor, Trochlear & Abducent Nerves . . . 564 CN V: Trigeminal Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 CN VII: Facial Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 CN VIII: Vestibulocochlear Nerve . . . . . . . . . . . . . . . . . . . . . . 570 CN IX: Glossopharyngeal Nerve . . . . . . . . . . . . . . . . . . . . . . . 572 CN X: Vagus Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 CN XI & XII: Accessory & Hypoglossal Nerves . . . . . . . . . . . . 576 Autonomic Innervation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 41 Neurovasculature of the Skull & Face Innervation of the Face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 Arteries of the Head & Neck . . . . . . . . . . . . . . . . . . . . . . . . . . 582 External Carotid Artery: Anterior, Medial & Posterior Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 External Carotid Artery: Terminal Branches . . . . . . . . . . . . . . 586 Veins of the Head & Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 Meninges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 Dural Sinuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Topography of the Superficial Face . . . . . . . . . . . . . . . . . . . . 594 Topography of the Parotid Region & Temporal Fossa . . . . . . 596 Topography of the Infratemporal Fossa . . . . . . . . . . . . . . . . . 598 Neurovasculature of the Infratemporal Fossa . . . . . . . . . . . . 600 42 Orbit & Eye Bones of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Muscles of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604 Neurovasculature of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . 606 Topography of the Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 Orbit & Eyelid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Eyeball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Cornea, Iris & Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 43 Nasal Cavity & Nose Bones of the Nasal Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Paranasal Air Sinuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 Neurovasculature of the Nasal Cavity . . . . . . . . . . . . . . . . . . 620 Pterygopalatine Fossa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 44 Temporal Bone & Ear Temporal Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 External Ear & Auditory Canal . . . . . . . . . . . . . . . . . . . . . . . . . 626 Middle Ear: Tympanic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 628 Middle Ear: Ossicular Chain & Tympanic Membrane . . . . . . . 630 Arteries of the Middle Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 Inner Ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 45 Oral Cavity & Pharynx Bones of the Oral Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 Temporomandibular Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 Oral Cavity Muscle Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Innervation of the Oral Cavity . . . . . . . . . . . . . . . . . . . . . . . . 644 Tongue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Topography of the Oral Cavity & Salivary Glands . . . . . . . . . 648 Tonsils & Pharynx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 Pharyngeal Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 Neurovasculature of the Pharynx . . . . . . . . . . . . . . . . . . . . . . 654 46 Sectional & Radiographic Anatomy Sectional Anatomy of the Head & Neck (I). . . . . . . . . . . . . . . 656 Sectional Anatomy of the Head & Neck (II) . . . . . . . . . . . . . . 658 Sectional Anatomy of the Head & Neck (III). . . . . . . . . . . . . . 660 Sectional Anatomy of the Head & Neck (IV). . . . . . . . . . . . . . 662 Sectional Anatomy of the Head & Neck (V). . . . . . . . . . . . . . 664 Radiographic Anatomy of the Head & Neck (I). . . . . . . . . . . . 666 Radiographic Anatomy of the Head & Neck (II). . . . . . . . . . . 668 Radiographic Anatomy of the Head & Neck (III). . . . . . . . . . . 670 Head & Neck Parietal region Vertebra prominens Posterior cervical region Temporal region Occipital region Supraorbital margin Philtrum Mental protuberance Submandibular gland Commissure of lips Sternal head Jugular notch Clavicle Suprasternal notch Infraorbital margin Thyroid cartilage Frontal bone Clavicular head Sternocleido-mastoid Omohyoid, inferior belly Mandible, inferior border Trapezius Mandibular angle Antitragus Tragus Antihelix Helix Zygomatic bone Sternocleido-mastoid External occipital protuberance Spinous process of C7 Trapezius Mandibular angle Mastoid process Occipital bone Nuchal lig. Parietal bone Frontal region Zygomatic region Temporal region Parietal region Infratemporal region Parotid-masseteric region Buccal region Retromandibular region Posterior cervical region Lateral cervical region Greater supra-clavicular fossa Sternocleidomastoid region Lesser supra-clavicular fossa Jugular fossa Mental region Oral region Nasal region Infraorbital region Orbital region Anterior cervical region Carotid triangle Muscular (omotracheal) triangle Submental triangle Submandibular triangle B Right posterolateral view. Head & Neck 512 36 Surface Anatomy Surface Anatomy Fig. 36.1 Regions of the head and neck Fig. 36.2 Surface anatomy of the head and neck A Right anterolateral view. A Right anterolateral view. B Right posterolateral view. Nasal bone Frontal notch Frontal bone Supraorbital notch Zygomatic arch Infraorbital foramen Maxilla Mandibular angle Mental foramen Mental protuberance Body of hyoid bone Larynx Clavicle Sterno-clavicular joint Manubrium sterni Laryngeal prominence Superior thyroid notch Cricoid cartilage Sagittal suture External occipital protuberance Spinous processes Vertebra prominens (C7) Scapula, superior angle Mandibular angle Mastoid process Temporal bone Occipital bone Lambdoid suture Parietal bone Transverse process of atlas (C1) 36 Surface Anatomy 513 Fig. 36.3 Palpable bony prominences of the head and neck A Anterior view. B Posterior view. A S D F G H J K L Ö Ä Y Head & Neck 514 37 Neck Muscle Facts (I) The bones, joints, and ligaments of the neck and the six topographic classes of neck muscles are covered here in this unit with the neck, or in Unit 1 with the back (see Table 37.1). However, some muscles in the same topographic class belong in different functional classes; for example, the platysma belongs to the muscles of facial expression; the trapezius, to the muscles of the shoulder girdle; and the nuchal mus-cles, to the intrinsic back muscles. Note that the suboccipital muscles (short nuchal and craniovertebral joint muscles) are covered with the lateral (deep) muscles of the neck. Fig. 37.1 Superficial neck muscles schematic See Table 37.2 for details. A Sternocleidomastoid. B Trapezius. Fig. 37.2 Nuchal muscles schematic A Semispinalis. B Splenius. C Longissimus. D Iliocostalis. Table 37.1 Bones, joints, ligaments, and muscles of the neck Bones, joints, and ligaments Bones of the cervical spine See pp. 8–9 Joints & ligaments of the craniovertebral junction See pp. 18–19 Joints & ligaments of the cervical spine See pp. 16– 17, 20-21 Hyoid bone & larynx Fig. 45.3, Fig. 37.18 Muscles I Superficial neck muscles III Suprahyoid muscles Platysma, ①, ② sternocleidomastoid, ③, ④, ⑤ trapezius Fig. 37.3 Digastric, geniohyoid, mylohyoid, stylohyoid Fig. 37.4A II Nuchal muscles (intrinsic back muscles) IV Infrahyoid muscles ⑥ Semispinalis capitis ⑦ Semispinalis cervicis See p. 34 Sternohyoid, sternothyroid, thyrohyoid, omohyoid Fig. 37.4B ⑧ Splenius capitis ⑨ Splenius cervicis See p. 32 V Prevertebral muscles ⑩ Longissimus capitis ⑪ Longissimus cervicis Longus capitis, longus coli, rectus capitis anterior and lateralis See p. 31 Fig. 37.6A ⑫ Iliocostalis cervicis VI Lateral (deep) neck muscles Suboccipital muscles (short nuchal and craniovertebral joint muscles) Fig. 37.6C Anterior, middle, and posterior scalenes Fig. 37.6B Clavicular head Sternal head Trapezius Sternocleido-mastoid Trapezius Sternocleido-mastoid Depressor anguli oris Platysma Descending part Transverse part Scapular spine Supraspinatus Clavicle Levator scapulae Rhomboid minor Sternocleido-mastoid Deep layer of nuchal fascia Acromion Trapezius 37 Neck 515 Fig. 37.3 Superficial neck muscles A Anterior view. B Left lateral view. C Posterior view. Removed: Trapezius (right side). Table 37.2 Superficial neck muscles Muscle Origin Insertion Innervation Action Platysma Skin over lower neck and upper lateral thorax Mandible (inferior border), skin over lower face and angle of mouth Cervical branch of facial n. (CN VII) Depresses and wrinkles skin of lower face and mouth, tenses skin of neck, aids forced depression of mandible Sternocleido-mastoid ① Sternal head Sternum (manubrium) Temporal bone (mastoid process), occipital bone (superior nuchal line) Motor: Accessory n. (CN XI) Proprioception: Cervical plexus (C2, C3, [C4]) Unilateral: Tilts head to same side, rotates head to opposite side Bilateral: Extends head, aids in respiration when head is fixed ② Clavicular head Clavicle (medial one third) Trapezius ③ D escending part Occipital bone, spinous processes of C1–C7 Clavicle (lateral one third) Draws scapula obliquely upward, rotates glenoid cavity superiorly The transverse ④ and ascending ⑤ parts are described on p. 320. D S F 1a 1b G H K J Head & Neck 516 Muscle Facts (II) Table 37.3 Suprahyoid muscles The suprahyoid muscles are also considered accessory muscles of mastication. Muscle Origin Insertion Innervation Action Digastric 1a Anterior belly Mandible (digastric fossa) Hyoid bone (body) Via an intermediate tendon with a fibrous loop Mylohyoid n. (from CN V3) Elevates hyoid bone (during swallowing), assists in opening mandible 1b Posterior belly Temporal bone (mastoid notch, medial to mastoid process) Facial n. (CN VII) ② Stylohyoid Temporal bone (styloid process) Via a split tendon ③ Mylohyoid Mandible (mylohyoid line) Via median tendon of insertion (mylohyoid raphe) Mylohyoid n. (from CN V3) Tightens and elevates oral floor, draws hyoid bone forward (during swallowing), assists in opening mandible and moving it side to side (during mastication) ④ Geniohyoid Mandible (inferior mental spine) Directly Anterior ramus of C1 via hypoglossal n. (CN XII) Draws hyoid bone forward (during swallowing), assists in opening mandible Fig. 37.4 Suprahyoid and infrahyoid muscles A Suprahyoid muscles, left lateral view. B Infrahyoid muscles, anterior view. Table 37.4 Infrahyoid muscles Muscle Origin Insertion Innervation Action ⑤ Omohyoid Scapula (superior border) – inferior belly Hyoid bone (body) – superior belly Ansa cervicalis (C1–C3) of cervical plexus Depresses (fixes) hyoid, draws larynx and hyoid down for phonation and terminal phases of swallowing ⑥ Sternohyoid Manubrium and sternoclavicular joint (posterior surface) ⑦ Sternothyroid Manubrium (posterior surface) Thyroid cartilage (oblique line) Ansa cervicalis (C1–C3) of cervical plexus ⑧ Thyrohyoid Thyroid cartilage (oblique line) Hyoid bone (body) Anterior ramus of C1 via hypoglossal n. (CN XII) Depresses and fixes hyoid, raises the larynx during swallowing The omohyoid also tenses the cervical fascia (via its intermediate tendon). Digastric, anterior belly Mylohyoid Intermediate tendon of omohyoid Omohyoid, superior and inferior bellies Sternothyroid Thyrohyoid Digastric, posterior belly Stylohyoid Sternohyoid Hyoid bone (body) Mandibular ramus Coronoid process Head of mandible Geniohyoid Mylohyoid line Mylohyoid Mylohyoid Omohyoid, superior and inferior bellies Sternohyoid Hyoid bone Stylohyoid Posterior belly Anterior belly Sternothyroid Mylohyoid raphe Thyrohyoid Thyroid cartilage Digastric 37 Neck 517 Fig. 37.5 Suprahyoid and infrahyoid muscles A Left lateral view. B Mylohyoid and geniohyoid (oral floor), posterosuperior view. C Anterior view. The sternohyoid has been cut (right). F S D A ⑤ ⑥ ⑦ KL Ö Ä Head & Neck 518 Muscle Facts (III) Fig. 37.6 Deep muscles of the neck A Prevertebral muscles, anterior view. B Scalene muscles, anterior view. C Suboccipital muscles, posterior view. Table 37.5 Deep muscles of the neck Muscle Origin Insertion Innervation Action Prevertebral muscles ① Longus capitis C3–C6 (anterior tubercles of transverse processes) Occipital bone (basilar part) Anterior rami of C1–C3 Flexion of head at atlanto-occipital joints ② Longus colli Vertical (intermediate) part C5–T3 (anterior surfaces of vertebral bodies) C2–C4 (anterior surfaces) Anterior rami of C2–C6 Unilateral: Tilts and rotates cervical spine to opposite side Bilateral: Forward flexion of cervical spine Superior oblique part C3–C5 (anterior tubercles of transverse processes) Atlas (anterior tubercle) Inferior oblique part T1–T3 (anterior surfaces of vertebral bodies) C5–C6 (anterior tubercles of transverse processes) ③ Rectus capitis anterior C1 (lateral mass) Occipital bone (basilar part) Anterior rami of C1 and C2 Unilateral: Lateral flexion of the head at the atlanto-occipital joint Bilateral: Flexion of the head at the atlanto-occipital joint ④ Rectus capitis lateralis C1 (transverse process) Occipital bone (basilar part, lateral to occipital condyles) Scalene muscles ⑤ Anterior scalene C3–C6 (anterior tubercles of transverse processes) 1st rib (scalene tubercle) Anterior rami of C4–C6 With ribs mobile: Elevates upper ribs (during forced inspiration) With ribs fixed: Flexes cervical spine to same side (unilateral), flexes neck (bilateral) ⑥ Middle scalene C1–C2 (transverse processes), C3–C7 (posterior tubercles of transverse processes) 1st rib (posterior to groove for subclavian a.) Anterior rami of C3–C8 ⑦ Posterior scalene C5–C7 (posterior tubercles of transverse processes) 2nd rib (outer surface) Anterior rami of C6–C8 Suboccipital muscles (short nuchal and craniovertebral joint muscles) ⑧ Rectus capitis posterior minor C1 (posterior tubercle) Occipital bone (inner third of inferior nuchal line) Posterior ramus of C1 (suboccipital n.) Unilateral: Rotates head to same side Bilateral: Extends head ⑨ Rectus capitis posterior major C2 (spinous process) Occipital bone (middle third of inferior nuchal line) ⑩ Obliquus capitis inferior C1 (transverse process) ⑪ Obliquus capitis superior C1 (transverse process) Occipital bone (above insertion of rectus capitis posterior major) Unilateral: Tilts head to same side, rotates it to opposite side Bilateral: Extends head Posterior tubercle of atlas (C1) Superior nuchal line Inferior nuchal line Obliquus capitis superior Transverse process of atlas (C1) Obliquus capitis inferior Spinous process of axis (C2) Rectus capitis posterior major Rectus capitis posterior minor Mastoid process Posterior scalene Middle scalene Interscalene space Anterior scalene Inferior oblique part Vertical part Superior oblique part Longus capitis Rectus capitis lateralis Rectus capitis anterior 1st rib 2nd rib Longus colli Atlas (C1) Anterior scalene (cut) Middle scalene Posterior scalene Groove for subclavian a. Scalene tubercle 37 Neck 519 Fig. 37.7 Deep muscles of the neck A Suboccipital muscles, posterior view. B Prevertebral and scalene muscles, anterior view. Removed: Longus capitis and anterior scalene (left). Head & Neck 520 Arteries & Veins of the Neck Fig. 37.8 Arteries of the neck Left lateral view. The structures of the neck are primarily supplied by the external carotid artery (anterior branches) and the subclavian artery (vertebral artery, thyrocervical trunk, and costocervical trunk). Ascending pharyngeal a. Superior laryngeal a. Infrahyoid br. Cricothyroid br. Common carotid a. External carotid a. Internal carotid a. Vertebral a. Inferior thyroid a. Ascending cervical a. Transverse cervical a. Axillary a. Costocervical trunk Internal thoracic a. Thyrocervical trunk Suprascapular a. Deep cervical a. Highest intercostal a. Superior thyroid a. Vertebral a. Left subclavian a. Vertebral v. External jugular v. Deep cervical v. Posterior auricular v. Superficial temporal v. Transverse sinus Superior sagittal sinus Maxillary v. Retromandibular v. Facial v. Superior thyroid v. Jugular venous arch Anterior jugular v. Pterygoid plexus Cavernous sinus Superior and inferior ophthalmic vv. Angular v. Lingual v. Left brachio-cephalic v. Occipital v. Subclavian v. Internal jugular v. 37 Neck 521 Fig. 37.9 Veins of the neck Left lateral view. The principal veins of the neck are the internal, exter-nal, and anterior jugular veins. External jugular v. Superior vena cava Internal jugular v. Jugular venous arch Left brachio-cephalic v. Subclavian v. External jugular v. Sternocleidomastoid Trapezius External jugular v. Superior vena cava Internal jugular v. Jugular venous arch Left brachio-cephalic v. Subclavian v. External jugular v. Sternocleidomastoid Trapezius Clinical box 37.1 When clinical factors (e.g., chronic lung disease, mediastinal tumors, or infections) impede the flow of blood to the right heart, blood dams up in the superior vena cava and, consequently, the jugular veins (A). This causes conspicuous swelling in the jugular (and sometimes more minor) veins (B). Impeded blood flow and veins of the neck A B Fig. 37.11 Lymphatic drainage of the tongue and oral floor Lymph flows into the submental and submandibular lymph nodes of the tongue and oral floor, which ultimately drain into the jugular lymph nodes along the internal jugular vein. Because the lymph nodes receive drainage from both the ipsilateral and contralateral sides (B), tumor cells may become widely disseminated in this region (e.g., metastatic squamous cell carcinoma, especially on the lateral border of the tongue, frequently metastasizes to the opposite side). Submental l.n. Submandibular l.n. Internal jugular v. Jugulofacial venous junction Superior deep cervical l.n. Inferior deep cervical l.n. Lingual v. Superior deep cervical l.n. Inferior deep cervical l.n. A Lateral view. B Coronal section showing that lymphatic drain-age from one side of the tongue can drain to either side of the neck. Jugulo-subclavian venous junction Jugulofacial venous junction Axillary Laryngo-tracheo-thyroidal Nuchal Along the accessory nerve Parallel to internal jugular vein Occipital Parotid-auricular Facial Buccal Submental-submandibular Head & Neck 522 Lymphatics of the Neck Fig. 37.10 Lymphatic drainage regions Right lateral view. Lateral superficial cervical l.n. External jugular v. Anterior superficial cervical l.n. Deep parotid l.n. Superficial parotid l.n. Retroauricular l.n. Occipital l.n. Mastoid l.n. Facial v. II V III IV I VI Submental l.n. Internal jugular v. Submandibular l.n. 37 Neck 523 Fig. 37.12 Superficial cervical lymph nodes Right lateral view. Fig. 37.13 Deep cervical lymph nodes Right lateral view. Table 37.6 Superficial cervical lymph nodes Lymph nodes (l.n.) Drainage region Retroauricular l.n. Occiput Occipital l.n. Mastoid l.n. Superficial parotid l.n. Parotid-auricular region Deep parotid l.n. Anterior superficial cervical l.n. Sternocleidomastoid region Lateral superficial cervical l.n. Deep cervical lymph nodes Table 37.7 Level Lymph nodes (l.n.) Drainage region I Submental l.n. Face Submandibular l.n. II Lateral jugular l.n. group Upper lateral group Nuchal region, laryngo-tracheo-thyroidal region III Middle lateral group IV Lower lateral group V L.n. in posterior cervical triangle Nuchal region VI Anterior cervical l.n. Laryngo-tracheo-thyroidal region C5 spinal n., posterior ramus Supraclavicular nn. 3rd occipital n. Greater occipital n. Great auricular n. Lesser occipital n. Suboccipital n. (in suboccipital triangle) Supraclavicular nn. Great auricular n. Lesser occipital n. Greater occipital n. Ophthalmic n. (CN V1) Posterior rami of spinal nn. C4 C3 C2 Ophthalmic n. (CN V1) Lesser occipital n. Hypoglossal n. (CN XII) Superior root Inferior root Phrenic n. Supraclavicular nn. Transverse cervical n. Great auricular n. To brachial plexus C1 C2 C3 C4 C5 Ansa cervicalis Head & Neck 524 Innervation of the Neck Branching of the cervical plexus. Fig. 37.14 Sensory innervation of the nuchal region Posterior view. A Dermatomes. B Cutaneous nerve territories. C Spinal nerve branches. Table 37.8 Branches of the spinal nerves in the neck Posterior (dorsal) ramus Nerve Sensory function Motor function C1 Suboccipital n. No C1 dermatome Innervate intrinsic nuchal muscles C2 Greater occipital n. Innervate C2 dermatome C3 3rd occipital n. Innervate C3 dermatome Anterior (ventral) ramus Sensory branches Sensory function Motor branches Motor function C1 — — Form ansa cervicalis (motor part of cervical plexus) Innervate infrahyoid muscles (except thyrohyoid) C2 Lesser occipital n. Form sensory part of cervical plexus, innervate anterior and lateral neck C2– C3 Great auricular n. Transverse cervical n. C3– C4 Supraclavicular nn. Contribute to phrenic n. Innervate diaphragm and pericardium The anterior roots of C3–C5 combine to form the phrenic nerve (see p. 66). Geniohyoid Sternohyoid Omohyoid Sternothyroid Middle scalene Anterior scalene Inferior root of ansa cervicalis Hypoglossal n. (CN XII) C1 C2 Superior root of ansa cervicalis C4 Infrahyoid muscles Thyrohyoid Phrenic n. C1, anterior ramus Ansa cervicalis Mandibular n. (CN V3) Transverse cervical n. Supraclavic-ular nn. Posterior rami of spinal nn.+ Great auricular n. Lesser occipital n. Greater occipital n.+ Maxillary n. (CN V2) Ophthalmic n. (CN V1) Lesser occipital n. Great auricular n. Transverse cervical n. Supraclavicular nn. 37 Neck 525 Fig. 37.15 Sensory innervation of the anterolateral neck Left lateral view. A Cutaneous nerve territories. Trigeminal nerve, CN V3 (orange), posterior rami (+), anterior rami (). B Sensory branches of the cervical plexus. Fig. 37.16 Motor innervation of the anterolateral neck Left lateral view. Innervated by the anterior ramus of C1 (distributed by the hypoglossal n.). Trachea Greater horn Epiglottis Lesser horn Thyroid cartilage Cricoid cartilage Hyoid bone (body) Epiglottic cartilage Stalk of epiglottis Right lamina Superior thyroid notch Laryngeal prominence Left lamina Inferior thyroid notch Inferior tubercle Inferior horn Oblique line Superior tubercle Superior horn Arch Articular facet for arytenoid cartilage Articular facet for thyroid cartilage Lamina Colliculus Corniculate cartilage Apex Vocal process Muscular process Anterolateral surface Arytenoid cartilage Medial surface Articular facet Corniculate cartilage Posterior surface Muscular process Vocal process Arch Articular facet for thyroid cartilage Articular facet for arytenoid cartilage Head & Neck 526 Larynx: Cartilage & Structure Fig. 37.17 Laryngeal cartilages Left lateral view. The larynx consists of five laryngeal cartilages: epiglot-tic, thyroid, cricoid, and the paired arytenoid and corniculate cartilages. They are connected to each other, the trachea, and the hyoid bone by elastic ligaments. Fig. 37.18 Epiglottic cartilage The elastic epiglottic cartilage comprises the internal skeleton of the epiglottis, providing resilience to return it to its initial position after swallowing. A Lingual (ante-rior) view. B Left lateral view. C Laryngeal (posterior) view. Fig. 37.19 Thyroid cartilage Left oblique view. Fig. 37.20 Cricoid cartilage A Anterior view. B Left lateral view. C Posterior view. Fig. 37.21 Arytenoid and corniculate cartilages Right cartilages. A Right lateral view. B Medial view. C Posterior view. Laryngeal prominence Cricotracheal lig. Cricothyroid joint Superior horn Thyrohyoid lig. Epiglottis Hyoid bone Foramen for superior laryngeal a. and v. and internal laryngeal n. Cricothyroid lig. Thyroid cartilage Cricoid cartilage Median thyrohyoid lig. Thyrohyoid membrane Thyroid cartilage Vocal lig. Median cricothyroid lig. Cricotracheal lig. Vocal process Cricoid cartilage Cricoarytenoid joint Corniculate cartilage Vestibular lig. Arytenoid cartilage Corniculate cartilage Thyroepiglottic lig. Inferior horn Cricoarytenoid lig. Superior horn Thyrohyoid membrane Foramen for superior laryngeal a. and v. and internal laryngeal n. Greater horn Lesser horn Epiglottic cartilage Cricothyroid joint Thyroid cartilage Vocal process Corniculate cartilage Lamina of cricoid cartilage Cricoarytenoid lig. Muscular process Arch of cricoid cartilage Vocal lig. Colliculus Conus elasticus Median cricothyroid lig. 37 Neck 527 Fig. 37.22 Structure of the larynx The larynx is suspended from the hyoid bone, primarily by the thyro-hyoid membrane. The hyoid bone provides the sites for attachment of the suprahyoid and infrahyoid muscles. A Left anterior oblique view. B Sagittal section, viewed from the left medial aspect. The arytenoid cartilage alters the position of the vocal folds during phonation. C Posterior view. Arrows indicate the directions of movement in the various joints. D Superior view. Crico-thyroid Straight part Oblique part Lateral crico-arytenoid Posterior cricoarytenoid Thyro-arytenoid Corniculate tubercle Cuneiform tubercle Aryepiglottic fold Thyroarytenoid muscle, thyroepiglottic part Lateral crico-arytenoid Articular facet for thyroid cartilage Posterior cricoarytenoid Arytenoid cartilage, muscular process Vocalis Arytenoid cartilage, vocal process Conus elasticus Middle crico-arytenoid lig. Epiglottis Posterior cricoarytenoid Transverse arytenoid Oblique arytenoid Thyro-arytenoid Cuneiform tubercle Aryepiglottic fold Oblique arytenoid F G H D S A Head & Neck 528 Larynx: Muscles & Levels Fig. 37.23 Laryngeal muscles The laryngeal muscles move the laryngeal cartilages relative to one an-other, affecting the tension and/or position of the vocal folds. Muscles that move the larynx as a whole (infra- and suprahyoid muscles) are described on p. 516. A Intrinsic laryngeal muscles, left lateral oblique view. B Intrinsic laryngeal muscles, left lateral view. Removed: Thyroid cartilage (left half). Revealed: Epiglottis and thyroarytenoid muscle. C Left lateral view with the epiglottis removed. D Posterior view. C Closed rima glottidis. B Open rima glottidis. A Laryngeal muscles, superior view. Table 37.9 Actions of the laryngeal muscles Muscle Action Effect on rima glottidis ① Cricothyroid m. Tightens the vocal folds None ② Vocalis m. ③ Thyroarytenoid m. Adducts the vocal folds Closes ④ Transverse arytenoid m. ⑤ Posterior cricoarytenoid m. Abducts the vocal folds Opens ⑥ Lateral cricoarytenoid m. Adducts the vocal folds Closes The cricothyroid is innervated by the external laryngeal n. All other intrinsic laryngeal mm. are innervated by the recurrent laryngeal n. Rima vestibuli Laryngeal ventricle Rima glottidis Glands Quadrangular membrane Vocalis Thyroid cartilage Laryngeal saccule Epiglottic cartilage Conus elasticus Vocal lig. Thyroarytenoid Vestibular lig. Epiglottis Vestibular fold Vocal fold Ventricle Cricoid cartilage Corniculate tubercle Cuneiform tubercle Aryepiglottic fold Piriform recess Lingual tonsil Hyoid bone Hyoepiglottic lig. Thyrohyoid lig. Median cricothyroid lig. Tracheal cartilage Vocal fold Vestibular fold Epiglottis Cricoid cartilage Membranous wall of trachea Esophagus Cricoid cartilage Thyroid cartilage III II I 37 Neck 529 Posterior view. Fig. 37.24 Cavity of the larynx A Posterior view with the larynx splayed open. B Midsagittal section viewed from the left side. Fig. 37.25 Vestibular and vocal folds Coronal section, superior view. Table 37.10 Levels of the larynx Level Space Extent I Supraglottic space (laryngeal vestibule) Laryngeal inlet (aditus laryngis) to vestibular folds II Transglottic space (intermediate laryngeal cavity) Vestibular folds across laryngeal ventricle (lateral evagination of mucosa) to vocal folds III Subglottic space (infraglottic cavity) Vocal folds to inferior border of cricoid cartilage Trachea Left lobe, thyroid gland Isthmus of thyroid gland Right lobe, thyroid gland Pyramidal lobe, thyroid gland Cricothyroid Median cricothyroid lig. Thyroid cartilage Inferior thyroid a. Parathyroid glands, inferior pair Parathyroid glands, superior pair Superior thyroid a. Prevertebral layer Buccopharyngeal fascia (continuous with pretracheal layer) Parathyroid glands Common carotid a. Vagus n. Carotid sheath Internal jugular v. Esophagus Thyroid gland Trachea Investing (superficial) layer Pretracheal visceral layer Platysma Sternocleido-mastoid Pretracheal muscular layer Retropharyngeal space Head & Neck 530 Neurovasculature of the Larynx, Thyroid & Parathyroids Fig. 37.26 Thyroid and parathyroid glands A Thyroid gland, anterior view. B Thyroid and parathyroid glands, posterior view. C Transverse section of neck at level of C6, superior view. Topographical relations of the thyroid and parathyroid glands. See p. 533 for coverage of the layers of the deep cervical fascia, which are shown here. Investing (superficial) layer Muscular pretracheal layer Visceral pretracheal layer Carotid sheath Prevertebral layer Superior laryngeal v. Superior thyroid v. Internal jugular v. Middle thyroid vv. Facial v. Inferior laryngeal v. Thyroid venous plexus Inferior thyroid v. Left brachio-cephalic v. Subclavian v. Median cricothyroid lig. Left recurrent laryngeal n. Esophagus Inferior thyroid a. Middle thyroid v. External laryngeal n. Inferior pharyngeal constrictor Superior laryngeal a. and v. Internal laryngeal n. Cricothyroid Thyrohyoid Thyrohyoid membrane Hyoid bone Thyroid gland Superior laryngeal n. Epiglottis Left recurrent laryngeal n. Inferior thyroid a. Middle thyroid v. Esophagus Posterior cricoarytenoid Galen’s anastomosis Superior laryngeal a. and v. Internal laryngeal n. Hyoid bone Median thyrohyoid lig. Tracheal brs. Thyro-arytenoid Lateral cricoarytenoid Median crico-thyroid lig. Cricothyroid Superior laryngeal a. Right vagus n. (CN X) Common carotid a. Cricothyroid br. Inferior laryngeal a. Inferior thyroid a. Thyrocervical trunk Right recurrent laryngeal n. Recurrent laryngeal nn. External laryngeal n. Left vagus n. (CN X) Aortic arch Left recurrent laryngeal n. Superior thyroid a. Internal laryngeal n. Left subclavian a. Superior laryngeal n. 37 Neck 531 Fig. 37.27 Arteries and nerves of the larynx Anterior view. Removed: Thyroid gland (right half). Fig. 37.28 Veins of the larynx Left lateral view. Note: The inferior thyroid vein generally drains into the left brachiocephalic vein. Fig. 37.29 Neurovasculature of the larynx Left lateral view. A Superficial layer. B Deep layer. Removed: Cricothyroid muscle and left lamina of thyroid cartilage. Retracted: Pharyngeal mucosa. F ② ① ③ Omoclavicular (subclavian) triangle Lesser supra-clavicular fossa Muscular triangle Omohyoid, inferior belly Sternocleido-mastoid Carotid triangle Occipital triangle Hyoid bone Submental triangle Submandibular (digastric) triangle Digastric, anterior and posterior bellies Omohyoid, superior belly Trapezius Submental triangle Carotid triangle Sternohyoid Lesser supra-clavicular fossa Clavicle Trapezius Occipital triangle Sternocleido-mastoid Digastric, posterior belly Digastric, anterior belly Submandibular triangle Omoclavicular (subclavian) triangle Omohyoid, superior belly Omohyoid, inferior belly Head & Neck 532 Topography of the Neck: Regions & Fascia A Right anterior oblique view. B Left posterior oblique view. Fig. 37.30 Cervical regions A Anterior view. B Left lateral view. Table 37.11 Regions of the neck Region Divisions Contents ① Anterior cervical region (triangle) Submandibular (digastric) triangle Submandibular gland and l.n., hypoglossal n. (CN XII), facial a. and v. Submental triangle Submental l.n. Muscular triangle Sternothyroid and sternohyoid mm., thyroid and parathyroid glands Carotid triangle Carotid bifurcation, carotid body, hypoglossal (CN XII) and vagus (CN X) nn. ② Sternocleidomastoid region Sternocleidomastoid, common carotid a., internal jugular v., vagus n. (CN X), jugular l.n. ③ Lateral cervical region (posterior triangle) Omoclavicular (subclavian) triangle Subclavian a., subscapular a., supraclavicular l.n. Occipital triangle Accessory n. (CN XI), trunks of brachial plexus, transverse cervical a., cervical plexus (posterior branches) ④ Posterior cervical region Nuchal mm., vertebral a., cervical plexus The sternocleidomastoid region also contains the lesser supraclavicular fossa. ③ ④ ② ① Nuchal lig. Spinal cord ① Sternohyoid Trapezius Parotid gland Sternocleidomastoid Carotid sheath Prevertebral layer Pretracheal muscular layer Investing (superficial) layer Pretracheal visceral layer Mandible Clavicle ① ③ ⑤ ② ④ Omohyoid 37 Neck 533 A Transverse section at level of C5 vertebra. B Midsagittal section, left lateral view. Fig. 37.31 Deep cervical fascial layers Anterior view. Table 37.12 Deep cervical fascia The deep cervical fascia is divided into four layers that enclose the structures of the neck. Layer Type of fascia Description ① Investing (superficial) layer Muscular Envelopes entire neck; splits to enclose sternocleidomastoid and trapezius muscles Pretracheal layer ② Muscular Encloses infrahyoid muscles ③ Visceral Surrounds thyroid gland, larynx, trachea, pharynx, and esophagus ④ Prevertebral layer Muscular Surrounds cervical vertebral column and associated muscles ⑤ Carotid sheath Neurovascular Encloses common carotid artery, internal jugular vein, and vagus nerve ③ Visceral pretracheal fascia Carotid sheath ⑤ Retropharyngeal space Prevertebral fascia ④ ② Muscular pretracheal fascia ① Superficial layer of deep cervical fascia Buccopharyngeal fascia Prevertebral layer (deep nuchal fascia) Superficial layer of deep cervical (superficial nuchal) fascia Parotid gland External jugular v. Great auricular n. Transverse cervical n. Supraclavicular nn. Pretracheal layer of deep cervical fascia Sternocleidomastoid, sternal head Jugular venous arch Anterior jugular v. Platysma Mandible Investing layer of deep cervical fascia Facial n. (CN VII), cervical br. Thyroid cartilage Sternohyoid (cut) Sternothyroid Sternocleido-mastoid Omohyoid, superior belly (cut) External jugular v. Thyrohyoid Internal jugular v. Median thyrohyoid lig. Superior thyroid a. Right common carotid a. Superior laryngeal a. External laryngeal n. Hypoglossal n. (CN XII) Internal laryngeal n. Thyrohyoid br. (C1 via CN XII) Cricothyroid Head & Neck 534 Topography of the Anterior Cervical Region Fig. 37.32 Anterior cervical triangle Anterior view. A Superficial layer. Removed: Subcutaneous platysma (right side) and investing layer of deep cervical fascia (center). B Deep layer. Removed: Pretracheal lamina (middle layer of cervical fascia). Cuts: Sternohyoid, sterno-thyroid, and thyrohyoid (right side); sternohyoid (left side). Accessory n. (CN XI) Internal jugular v. Thyrocervical trunk Subclavian v. Inferior laryngeal n. Inferior thyroid v. Suprascapular n. Brachial plexus Left common carotid a. Trapezius Phrenic n. Cricothyroid Superior thyroid a. Thyroid cartilage Superior laryngeal a. Vagus n. (CN X) Subclavian a. Thyrocervical trunk Suprascapular a. Transverse cervical a. Inferior thyroid a. Ascending cervical a. External laryngeal n. Internal jugular v. Vagus n. (CN X) Internal laryngeal n. Left recurrent laryngeal n. C8, anterior root Middle cervical ganglion Sympathetic trunk Anterior scalene Inferior thyroid a. Transverse cervical a. Internal thoracic a. Subclavian a. and v. Suprascapular a. Thoracic duct Brachial plexus Phrenic n. Trapezius Accessory n. (CN XI) Internal jugular v. Common carotid a. Thyroid cartilage Median thyrohyoid lig. Vagus n. (CN X) T1, anterior root Stellate ganglion Left common carotid a. External jugular v. External laryngeal n. Cricothyroid Vertebral a. Ascending cervical a. Thyrocervical trunk 37 Neck 535 C Deep anterior cervical region. D Root of the neck. Head & Neck 536 Topography of the Anterior & Lateral Cervical Regions Fig. 37.33 Deep anterior cervical region The deep midline viscera of the anterior cervical region are the larynx and thyroid gland. The two lateral neurovascular pathways primarily supply these organs. Mandible Hyoid Digastric, anterior belly Mylohyoid Sternocleido-mastoid Parotid gland Superior thyroid a. Sternohyoid Thyrohyoid Middle scalene Superior thyroid v. Trapezius Accessory n. Anterior scalene Brachial plexus Transverse cervical a. Phrenic n. Vagus n. Middle thyroid v. Left recurrent laryngeal n. Left common carotid a. Inferior thyroid vv. Thyroid gland Clavicle Brachiocephalic trunk Subclavian a. Thyrocervical trunk Pyramidal lobe Phrenic n. Right common carotid a. Internal jugular v. Cricothyroid Thyroid cartilage Thyrohyid membrane (median thyrohyoid lig.) Parotid gland Superior cervical ganglion Internal carotid a. Anterior scalene Middle scalene Internal jugular v. Superficial cervical a. Phrenic n. Ansa cervicalis Brachial plexus Omohyoid, inferior belly Vagus n. (CN X) Sternocleidomastoid Sternothyroid Inferior thyroid a. Thyroid gland Common carotid a. Sternohyoid Carotid bifurcation Superior thyroid a. Carotid body External carotid a. Facial a. and v. Hypoglossal n. (CN XII) Accessory n. (CN XI) Sympathetic trunk Fig. 37.34 Carotid triangle Right lateral view. Internal jugular and facial veins removed. Digastric, posterior belly Internal carotid a. External carotid a. Facial a. Hypoglossal n. (CN XII) Lingual a. Facial n. (CN VII) marginal mandibular br. Submandibular gland Superior laryngeal n. Hyoid bone Thyrohyoid br. (C1 via CN XII) Superior thyroid a. Thyrohyoid Sternothyroid Omohyoid, superior belly Internal jugular v. (cut) Superior cervical ganglion Occipital a. Vagus n. Superior root of ansa cervicalis (descendens hypoglossus) Carotid body Common carotid a. Sternocleidomastoid Investing layer of deep cerical fascia Ansa cervicalis 37 Neck 537 Fig. 37.35 Deep lateral cervical region Right lateral view with sternocleidomastoid windowed. Superficial cervical l.n. Accessory n. (CN XI) Trapezius Superficial cervical a. Superficial cervical v. Transverse cervical n. Sternocleido-mastoid Great auricular n. Supra-clavicular nn. Pretracheal layer of deep cervical fascia External jugular v. Lesser occipital n. Erb’s point Superficial (investing) layer of deep cervical fascia Prevertebral layer of deep cervical fascia Sternocleido-mastoid, posterior border Great auricular n. Lateral supra-clavicular nn. Trapezius, anterior border Erb’s point Intermediate supra-clavicular nn. Medial supra-clavicular nn. Clavicle External jugular v. Superficial (investing) layer of deep cervical fascia Transverse cervical and CN VII anastomosis Parotid gland Transverse cervical n. Lesser occipital n. Masseter Facial n. (CN VII), cervical br. Head & Neck 538 Topography of the Lateral Cervical Region Fig. 37.36 Lateral cervical region Right lateral view. The contents of the deep lateral cervical region are found in Fig. 37.34. A Subcutaneous layer. B Subfascial layer. Removed: Superficial (investing) layer of deep cervical fascia. External jugular v. Lateral supra-clavicular n. Trapezius Superficial cervical a. and v. Right subclavian v. Prevertebral layer of deep cervical fascia Sternocleido-mastoid Parotid gland Accessory n. (CN XI) Transverse cervical n. Lesser occipital n. Great auricular n. Omohyoid, inferior belly Intermediate supraclavicular n. Trapezius Posterior scalene Middle scalene Anterior scalene Omohyoid, inferior belly Superficial cervical a. Brachial plexus Sternocleido-mastoid Suprascapular a. Accessory n. (CN XI) Right subclavian v. Phrenic n. Splenius capitis Levator scapulae 37 Neck 539 C Deep layer. Removed: Pretracheal layer of deep cervical fascia. Revealed: Omohyoid, omoclavicular (subclavian) triangle. D Deepest layer. Removed: Prevertebral layer of deep cervical fascia. Revealed: Muscular floor of posterior triangle, brachial plexus, and phrenic nerve. Posterior cutaneous br. (cervical nn., posterior rami) Occipital l.n . Trapezius Splenius capitis Sternocleido-mastoid Great auricular n. Lesser occipital n. Semispinalis capitis Occipital a. and v. Accessory n. (CN XI) 3rd occipital n. Greater occipital n. Lesser occipital n. Head & Neck 540 Topography of the Posterior Cervical Region Fig. 37.37 Occipital and posterior cervical regions Posterior view. Subcutaneous layer (left), subfascial layer (right). The occiput is technically a region of the head, but it is included here due to the continuity of the vessels and nerves from the neck. Removed on right side: Investing layer of deep cervical fascia. Occipital a. Vertebral a. Rectus capitis posterior minor Greater occipital n. Rectus capitis posterior major Obliquus capitis inferior 3rd occipital n. Cervical posterior intertransversarius Longissimus capitis Great auricular n. Obliquus capitis superior Occipital a. Semispinalis capitis Sternocleido-mastoid Splenius capitis Semispinalis capitis Transverse process of atlas (C1) Spinous process of axis (C2) Splenius capitis Suboccipital n. Spinous process of C3 37 Neck 541 Fig. 37.38 Suboccipital triangle Right side, posterior view, windowed. The suboccipital triangle is bounded by the suboccipital muscles (rectus capitis posterior major and obliquus capitis superior and inferior) and contains the vertebral artery. The left and right vertebral arteries pass through the atlanto-occipital membrane and combine to form the basilar artery. Head & Neck 542 Frontal bone Coronal suture Parietal bone Lambdoid suture Zygomatic bone, temporal process Temporal bone, zygomatic process Mastoid foramen Tympanomastoid fissure Mastoid process Postglenoid tubercle Styloid process Squamous suture Zygomatic arch Zygomatic bone Oblique line Mandible, ramus Mandible, body Mental foramen Maxilla, zygomatic process Infraorbital foramen Nasal bone Lacrimal bone Ethmoid bone Sphenofrontal suture Sphenoid bone, greater wing Supraorbital foramen Sphenoparietal suture Sphenosquamous suture Mental protuberance Anterior nasal spine Pterion Asterion Articular tubercle (articular eminence) External acoustic meatus Glabella Zygomatic bone, frontal process 38 Bones of the Head Anterior & Lateral Skull Fig. 38.1 Lateral skull Left lateral view. Table 38.1 The skull is subdivided into the neurocranium (gray) and viscerocranium (orange). The neurocranium protects the brain, while the viscerocranium houses and protects the facial regions. Neurocranium Viscerocranium • Ethmoid bone (cribriform plate) • Frontal bone • Occipital bone • Parietal bone • Sphenoid bone • Temporal bone (petrous and squamous parts) • Ethmoid bone • Hyoid bone • Inferior nasal concha • Lacrimal bone • Mandible • Maxilla • Nasal bone • Palatine bone • Sphenoid bone (pterygoid process) • Temporal bone • Vomer Most of the ethmoid bone is in the viscerocranium; most of the sphenoid bone is in the neurocranium. The temporal bone is divided between the two. Bones of the skull Supraorbital foramen Supraorbital margin Frontal incisure (notch) Infraorbital margin Ethmoid bone, middle nasal concha Sphenoid bone, lesser wing Nasal bone Piriform (anterior nasal) aperture Anterior nasal spine Mental foramen Mental protuberance Mandible, body Mental tubercles Infraorbital foramen Maxilla, zygomatic process Sphenoid bone, greater wing Zygomatic bone, frontal process Temporal bone Sphenoid bone, greater wing Parietal bone Maxilla, frontal process Frontal bone Superciliary arch Inferior nasal concha Vomer Ethmoid bone, perpendicular plate Oblique line Mandible, ramus Orbit Intermaxillary suture Maxilla, alveolar process Nasion Glabella 38 Bones of the Head 543 Fig. 38.2 Anterior skull Anterior view. Clinical box 38.1 The framelike construction of the facial skeleton leads to characteristic patterns for fracture lines (classified as Le Fort I, II, and III fractures). Fractures of the face A Le Fort I. B Le Fort II. C Le Fort III. Supreme nuchal line Superior nuchal line Inferior nuchal line Mandible, ramus Incisive foramen Palatine bone Maxilla, palatine process Sphenoid bone, pterygoid process Occipital condyle Temporal bone, styloid process Mastoid foramina Temporal bone, mastoid process Lambdoid suture Parietal bone Sagittal suture Occipital bone External occipital protuberance (inion) Vomer Temporal bone, petrous part Temporal bone, squamous part Submandibular fossa Mandible, body Parietal foramina Lambda Mastoid notch Mylohyoid groove Mandibular foramen Mylohyoid line Genial (mental) spines Digastric fossa Median nuchal line (external occipital crest) Asterion Parietal eminence Clinical box 38.2 In the neonate, there are areas between still-growing cranial bones not occupied by bone: the fontanelles. While these regions close at different times, they have clinical implications. The posterior fontanelle provides a reference point for describing the position of the fetal head during childbirth, and the anterior fontanelle provides a potential access site for drawing cerebrospinal fluid in infants (e.g., in suspected meningitis). Cranial Fontanelles Sphenoidal fontanelle Sphenosquamosal suture Mastoid fontanelle Lamdoid suture Squamous suture Posterior fontanelle Anterior fontanelle Coronal suture Frontal suture Coronal suture Anterior fontanelle Posterior fontanelle Sagittal suture A B Head & Neck 544 Posterior Skull & Calvaria Fig. 38.3 Posterior skull Posterior view. Emissary v. Inner table Diploë Outer table Scalp Dural sinus Diploic vv. Calvaria Dura mater Frontal crest Groove for superior sagittal sinus Grooves for middle meningeal a. Granular foveolae (for arachnoid granulations) Parietal foramen Frontal bone Frontal sinus Parietal bone Parietal foramen Lambdoid suture Occipital bone Parietal bone Coronal suture Frontal bone Sagittal suture 38 Bones of the Head 545 Fig. 38.4 Calvaria A External calvaria, superior view. B Internal calvaria, inferior view. The interior of the calvaria is marked by grooves for the meningeal arteries, dural venous sinuses, and arachnoid granulations (see pp. 590–591). Fig. 38.5 Structure of the calvaria Cross section. Choana Median palatine suture Transverse palatine suture Palatine bone Greater palatine foramen Lesser palatine foramen Foramen ovale Foramen lacerum Foramen spinosum Carotid canal Jugular foramen Stylomastoid foramen Pharyngeal tubercle Hypoglossal canal Foramen magnum Inferior nuchal line Superior nuchal line Occipital condyle Mastoid foramen Mastoid process Styloid process Fossa of pterygoid canal Mandibular fossa Articular tubercle Zygomatic arch Zygomatic bone, temporal surface Inferior orbital fissure Zygomatic process Palatine process Incisive foramen Medial plate Lateral plate Vomer Parietal bone Condylar canal Mastoid notch (for digastric belly) Supreme nuchal line External occipital protuberance (inion) Pterygoid process Palatovaginal (pharyngeal) canal Petrotympanic fissure External occipital crest Hamulus (of medial pterygoid plate) Maxilla Head & Neck 546 Base of the Skull Fig. 38.6 Base of the skull: Exterior Inferior view. Revealed: Foramina and canals for blood vessels (see p. 582) and cranial nerves. Note: This view allows visual access into the posterior region of the nasal cavity. Posterior cranial fossa Middle cranial fossa Anterior cranial fossa Foramen magnum Dorsum sellae Petrous ridge (crest), temporal bone Jugum sphenoidale Posterior cranial fossa Middle cranial fossa Anterior cranial fossa Lesser wing of sphenoid bone Foramen magnum Optic canal Anterior clinoid process Foramen ovale Foramen spinosum Internal acoustic meatus Foramen magnum Cerebral fossa Cerebellar fossa Jugular foramen Temporal bone, petrous part Groove for lesser petrosal nerve Posterior clinoid process Sphenoid bone, greater wing Sphenoid bone, hypophyseal fossa Sphenoid bone, lesser wing Ethmoid bone, cribriform plate Frontal sinus Frontal crest Groove for transverse sinus Groove for sigmoid sinus Hypoglossal canal Clivus Foramen lacerum Hiatus of facial canal Ethmoid bone, crista galli Frontal bone Petrooccipital fissure Internal occipital protuberance Internal occipital crest Chiasmatic groove Dorsum sellae 38 Bones of the Head 547 Fig. 38.8 Base of the skull: Interior Superior view. Fig. 38.7 Cranial fossae The interior of the skull base consists of three successive fossae that become progressively deeper in the frontal-to-occipital direction. A Midsagittal section, left lateral view. B Superior view of opened skull. Head & Neck 548 Neurovascular Pathways Exiting or Entering the Cranial Cavity Fig. 38.9 Summary of the neurovascular structures exiting or entering the cranial cavity A Cranial cavity (interior of skull base), left side, superior view. B Exterior of skull base, left side, inferior view Greater palatine foramen Greater palatine n. and a. Lesser palatine foramina Lesser palatine n. and a. Incisive canal Nasopalatine n., sphenopalatine a. Mastoid foramen Emissary v. Hypoglossal canal Hypoglossal n., venous plexus of hypoglossal canal Condylar canal Condylar emissary v. Foramen spinosum Middle meningeal a., meningeal br. of mandibular n. (CN V3) Carotid canal Internal carotid a., internal carotid sym-pathetic plexus Petrotympanic fissure Anterior tympanic a., chorda tympani Stylomastoid foramen Facial n., stylomastoid a. Jugular foramen Posterior meningeal a. Inferior petrosal sinus Accessory n. Vagus n. Glossopharyngeal n. Internal jugular v. Cribriform plate Olfactory n., anterior and posterior ethmoidal aa. Internal acoustic meatus Facial n. Vestibulocochlear n. Labyrinthine a. and v. Foramen rotundum Maxillary n. (CN V2) Mandibular n. (CN V3), lesser petrosal n. accessory meningeal a. Carotid canal Internal carotid a., internal carotid sym-pathetic plexus Foramen spinosum Middle meningeal a., meningeal br. of mandibular n. (CN V3) Hiatus of canal for lesser petrosal n. Lesser petrosal n., superior tympanic a. Hiatus of canal for greater petrosal n. Greater petrosal n. Jugular foramen Posterior meningeal a. Inferior petrosal sinus Accessory n. Vagus n. Glossopharyn-geal n. Sigmoid sinus Foramen magnum Vertebral a. Accessory n. Medulla oblongata Posterior spinal a. Anterior spinal a. Spinal v. Optic canal Optic n., ophthalmic a. Superior orbital fissure Nasociliary n. Trochlear n. Frontal n. Lacrimal n. Superior oph-thalmic v. Oculomotor n. Abducent n. Foramen ovale Foramen lacerum Deep petrosal n., greater petrosal n. Olfactory bulb Anterior cranial fossa Optic n. (CN II) Internal carotid a. Oculomotor n. (CN III) Trochlear n. (CN IV) Facial and vestibulocochlear nn. (CN VII, CN VIII) Glossopharyngeal n. (CN IX) Vagus n. (CN X) Accessory n. (CN XI) Hypoglossal n. (CN XII) Tentorium cerebelli (cut) Tentorium cerebelli Inferior sagittal sinus Trigeminal n. (CN V) Abducent n. (CN VI) Lateral dural wall of cavernous sinus Infundibular stalk Diaphragma sella Olfactory tract Olfactory fibers (filia olfactoria) (CN I) Superior sagittal sinus Middle cranial fossa Posterior cranial fossa 38 Bones of the Head 549 Fig. 38.10 Cranial nerves exiting the cranial cavity Cranial cavity (interior of skull base), right side, superior view. Removed: Brain and tentorium cerebelli. The ends of the cranial nerves have been cut to reveal the fissures, fossae, or dural cave where they pass through the cranial fossa. Orbital plate Ethmoid air cells Crista galli Perpendicular plate Middle concha Ethmoid air cells Orbital plate Cribriform plate Crista galli Perpendicular plate Ethmoid infundibulum Ethmoid bulla Perpendicular plate Middle concha Uncinate process Crista galli Superior concha Crista galli Perpen-dicular plate Middle concha Posterior ethmoid foramen Orbital plate (lamina papyracea) Anterior ethmoid foramen Ethmoid air cells Head & Neck 550 Ethmoid & Sphenoid Bones Fig. 38.11 Ethmoid bone The ethmoid bone is the central bone of the nose and paranasal air sinuses (see pp. 616–619). A Anterior view. B Superior view. C Posterior view. D Left lateral view. The structurally complex ethmoid and sphenoid bones are shown here in isolation. The other bones of the skull are shown in their respective regions: orbit (see pp. 602–603), nasal cavity (see pp. 616–617), oral cavity (see pp. 636–637), and ear (see pp. 624–625). Pterygoid canal Pterygoid process Medial plate Lateral plate Foramen rotundum Temporal surface Orbital surface Superior orbital fissure Lesser wing Aperture of sphenoid sinus Sphenoid crest Pterygoid hamulus Anterior clinoid process Optic canal Pterygoid canal Foramen rotundum Medial plate Lateral plate Greater wing, cerebral surface Superior orbital fissure Lesser wing Posterior clinoid process Dorsum sellae Cancellous trabeculae Pterygoid process Pterygoid notch Aperture of sphenoid sinus Sphenoid crest Lesser wing Superior orbital fissure Foramen rotundum Temporal surface Greater wing Body Pterygoid fossa Pterygoid hamulus Foramen ovale Foramen spinosum Pterygoid process Lateral plate Medial plate Greater wing Foramen spinosum Foramen ovale Tuberculum sellae Posterior clinoid process Dorsum sellae Anterior clinoid process Foramen rotundum Optic canal Jugum sphenoidale Lesser wing Superior orbital fissure Greater wing Chiasmatic groove Hypophyseal fossa (sella turcica) 38 Bones of the Head 551 Fig. 38.12 Sphenoid bone The sphenoid bone is the most structurally complex bone in the human body. A Anterior view. B Superior view. C Posterior view. D Inferior view. Note: The vomer sits below the sphenoid crest (see p. 636). Occipitofrontalis, frontal belly (frontalis) Corrugator supercilii Orbicularis oculi Levator labii superioris alaeque nasi (O) Levator labii superioris (O) Zygomaticus minor (O) Zygomaticus major (O) Levator anguli oris (O) Buccinator Masseter (muscle of mastication) Orbicularis oris Depressor labii inferioris (O) Depressor anguli oris (O) Mentalis Depressor labii inferioris Depressor anguli oris Platysma Risorius Levator anguli oris Zygomaticus major Zygomaticus minor Levator labii superioris Nasalis Levator labii superioris alaeque nasi Procerus Epicranial aponeurosis (galea aponeurotica) Risorius (I) Head & Neck 552 39 Muscles of the Skull & Face Muscles of Facial Expression & of Mastication The muscles of the skull and face are divided into two groups. The muscles of facial expression make up the superficial muscle layer in the face. The muscles of mastication are responsible for the movement of the mandible during mastication (chewing). Fig. 39.1 Muscles of facial expression A Anterior view. Muscle origins (O) and inser-tions (I) indicated on left side of face. Occipitofrontalis, frontal belly (frontalis) Anterior auricular muscle Temporo-parietalis Epicranial aponeurosis Superior auricular muscle Posterior auricular muscle Occipitofrontalis, occipital belly (occipitalis) Platysma Depressor anguli oris Mentalis Depressor labii inferioris Risorius Zygomaticus major Zygomaticus minor Orbicularis oris Levator labii superioris alaeque nasi Levator labii superioris Nasalis Orbicularis oculi Masseter (cut) Medial pterygoid Lateral pterygoid Temporalis (cut) Lateral lig. of TMJ Joint capsule of TMJ Capsule of temporo-mandibular joint Styloid process, temporal bone Superficial layer Temporalis Masseter Deep layer 39 Muscles of the Skull & Face 553 B Left lateral view. Fig. 39.2 Muscles of mastication Left lateral view. A Superficial layer. B Deep layer. Removed: Mandible (coronoid process) and lower temporalis. Genioglossus Geniohyoid Mylohyoid Medial pterygoid Lateral pterygoid Temporalis Buccinator Suprahyoid mm. Digastric, anterior belly Orbicularis oculi Nasalis Trapezius Sternocleidomastoid Longissimus capitis Splenius capitis Rectus capitis posterior minor Rectus capitis posterior major Obliquus capitis superior Semispinalis capitis Platysma Depressor anguli oris Depressor labii inferioris Orbicularis oris Mentalis Buccinator Zygomaticus major Orbicularis oris Transverse part Alar part Zygomaticus minor Levator anguli oris Levator labii superioris alaeque nasi Orbital part Lacrimal part Corrugator supercilii Occipitofrontalis, occipital belly Mm. of facial expression: Facial n. (CN VII) Nuchal and intrinsic back mm.: Posterior rami of cervical nn. Sternocleidomastoid and trapezius: Acces-sory n. (CN XI) Depressor septi nasi Lateral pterygoid Masseter Temporalis Mm. of mastication: Trigeminal n., mandibu-lar division (CN V3) Medial pterygoid (see Fig. 39.4) Levator labii superioris Head & Neck 554 Muscle Origins & Insertions on the Skull Fig. 39.3 Lateral skull: Origins and insertions Left lateral view. Muscle origins are shown in red, insertions in blue. Note: There are generally no bony insertions for the muscles of facial expression. These muscles insert into skin and other muscles of facial expression. Fig. 39.4 Mandible: Origins and insertions Medial view of right hemimandible (inner surface). Muscle origins are shown in red, insertions in blue. Digastric, posterior belly Styloglossus Trapezius Sternocleidomastoid Stylohyoid Lingual mm.: Hypo-glossal n. (CN XII) Hyoglossus (see Fig. 40.25) Genioglossus (see Fig. 40.25) Tensor veli palatini Stylopharyngeus Levator veli palatini Pharyngeal mm.: Glossopharyngeal n. (CN IX) and vagus n. (CN X) Middle pharyngeal constrictor (not shown) Sternocleidomastoid and trapezius: Acces-sory n. (CN XI) Lateral pterygoid Masseter Temporalis Mm. of mastication: Trigeminal n., mandib-ular division (CN V3) Medial pterygoid Splenius capitis Rectus capitis posterior minor Rectus capitis posterior major Obliquus capitis superior Semispinalis capitis Nuchal and intrinsic back mm.: Posterior rami of cervical nn. Rectus capitis lateralis Rectus capitis anterior Longus capitis Prevertebral mm.: Ventral cervical n. rami and cervical plexus Longissimus capitis Geniohyoid Mylohyoid Stylohyoid Thyrohyoid Sternohyoid Omohyoid Stylohyoid Geniohyoid Thyrohyoid Omohyoid Sternohyoid Mylohyoid 39 Muscles of the Skull & Face 555 Fig. 39.5 Skull base: Origins and insertions Inferior view of external skull. Muscle origins are shown in red, insertions in blue. Fig. 39.6 Hyoid bone: Origins and insertions The larynx is suspended from the hyoid bone, primarily by the thyrohyoid membrane. The hyoid bone is the site for attachment for the suprahyoid and infrahyoid muscles. Muscle insertions are shown in blue. A Anterior view. B Oblique left lateral view. ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ Head & Neck 556 Muscle Facts (I) The muscles of facial expression originate on bone and/or fascia and insert into the subcutaneous tissue of the face. This allows them to produce their effects by pulling on the skin. Fig. 39.7 Occipitofrontalis Anterior view. Fig. 39.8 Muscles of the palpebral fissure and nose Anterior view. A Orbicularis oculi. B Nasalis. C Levator labii superioris alaeque nasi. Fig. 39.9 Muscles of the ear Left lateral view. Table 39.1 Muscles of facial expression: Forehead, nose, and ear Muscle Origin Insertion Main action(s) Calvaria ① Occipitofrontalis (frontal belly) Epicranial aponeurosis Skin and subcutaneous tissue of eyebrows and forehead Elevates eyebrows, wrinkles skin of forehead Palpebral fissure and nose ② Procerus Nasal bone, lateral nasal cartilage (upper part) Skin of lower forehead between eyebrows Pulls medial angle of eyebrows inferiorly, producing transverse wrinkles over bridge of nose ③ Orbicularis oculi Medial orbital margin, medial palpebral ligament, lacrimal bone Skin around margin of orbit, superior and inferior tarsal plates Acts as orbital sphincter (closes eyelids) • Palpebral portion gently closes • Orbital portion tightly closes (as in winking) ④ Nasalis Maxilla (superior region of canine ridge) Nasal cartilages Flares nostrils by drawing ala (side) of nose toward nasal septum ⑤ Levator labii superioris alaeque nasi Maxilla (frontal process) Alar cartilage of nose and upper lip Elevates upper lip, opens nostril Ear ⑥ Anterior auricular muscle Temporal fascia (anterior portion) Helix of the ear Pulls ear superiorly and anteriorly ⑦ Superior auricular muscle Epicranial aponeurosis on side of head Upper portion of auricle Elevates ear ⑧ Posterior auricular muscle Mastoid process Convexity of concha of ear Pulls ear superiorly and posteriorly There are no bony insertions for the muscles of facial expression. All muscles of facial expression are innervated by the facial nerve (CN VII) via temporal, zygomatic, buccal, mandibular, or cervical branches arising from the parotid plexus (see pp. 568–569). S A F D H G J K L 39 Muscles of the Skull & Face 557 Fig. 39.10 Muscles of the mouth A Zygomaticus major and minor, left lateral view. B Levator labii superioris and depressor labii inferioris, left lateral view. C Levator and depressor anguli oris, left lateral view. D Buccinator, left lateral view. E Orbicularis oris, anterior view. F Mentalis, anterior view. Table 39.2 Muscles of facial expression: Mouth and neck Muscle Origin Insertion Main action(s) Mouth ① Zygomaticus major Zygomatic bone (lateral surface, posterior part) Skin at corner of the mouth Pulls corner of mouth superiorly and laterally ② Zygomaticus minor Upper lip just medial to corner of the mouth Pulls upper lip superiorly Levator labii superioris alaeque nasi (see Fig. 39.8C) Maxilla (frontal process) Alar cartilage of nose and upper lip Elevates upper lip, opens nostril ③ Levator labii superioris Maxilla (frontal process) and infraorbital region Skin of upper lip, alar cartilages of nose Elevates upper lip, dilates nostril, raises angle of the mouth ④ Depressor labii inferioris Mandible (anterior portion of oblique line) Lower lip at midline; blends with muscle from opposite side Pulls lower lip inferiorly and laterally ⑤ Levator anguli oris Maxilla (below infraorbital foramen) Skin at corner of the mouth Raises angle of mouth, helps form nasolabial furrow ⑥ Depressor anguli oris Mandible (oblique line below canine, premolar, and first molar teeth) Skin at corner of the mouth; blends with orbicularis oris Pulls angle of mouth inferiorly and laterally ⑦ Buccinator Mandible, alveolar processes of maxilla and mandible, pterygo-mandibular raphe Angle of mouth, orbicularis oris Presses cheek against molar teeth, working with tongue to keep food between occlusal surfaces and out of oral vestibule; expels air from oral cavity/resists distension when blowing Unilateral: Draws mouth to one side ⑧ Orbicularis oris Deep surface of skin Superiorly: maxilla (median plane) Inferiorly: mandible Mucous membrane of lips Acts as oral sphincter • Compresses and protrudes lips (e.g., when whistling, sucking, and kissing) • Resists distension (when blowing) Risorius (see pp. 552–553) Fascia over masseter Skin of corner of the mouth Retracts corner of mouth as in grimacing ⑨ Mentalis Mandible (incisive fossa) Skin of chin Elevates and protrudes lower lip Neck Platysma (see pp. 552–553) Skin over lower neck and upper lateral thorax Mandible (inferior border), skin over lower face, angle of mouth Depresses and wrinkles skin of lower face and mouth; tenses skin of neck; aids in forced depression of the mandible There are no bony insertions for the muscles of facial expression. All muscles of facial expression are innervated by the facial nerve (CN VII) via temporal, zygomatic, buccal, mandibular, or cervical branches arising from its parotid plexus. A S Coronoid process Temporomandibular joint (TMJ) capsule Lateral pterygoid Masseter (cut) Lateral (temporo-mandibular) lig. Temporalis Zygomatic arch (cut) Frontal bone Parietal bone Zygomatic arch External acoustic meatus Mastoid process Styloid process Masseter (deep layer) Masseter (superficial layer) Temporalis Head & Neck 558 Muscle Facts (II) The muscles of mastication are located at various depths in the parotid and infratemporal regions of the face. They attach to the mandible and receive their motor innervation from the mandibular division of the trigeminal nerve (CN V3). The muscles of the oral floor that aid in opening the mouth are found on in Table 37.3 on p. 516. Table 39.3 Muscles of mastication: Masseter and temporalis Muscle Origin Insertion Innervation Action ① Masseter Superficial layer: zygomatic arch (anterior two thirds) Mandibular angle (masseteric tuberosity) Mandibular n. (CN V3) via masseteric n. Elevates (entire muscle) and protrudes (superficial fibers) the mandible Deep layer: zygomatic arch (posterior one third) ② Temporalis Temporal fossa (inferior temporal line) Coronoid process of mandible (apex and medial surface) Mandibular n. (CN V3) via deep temporal nn. Vertical fibers: Elevate mandible Horizontal fibers: Retract (retrude) mandible Unilateral: Lateral movement of mandible (chewing) Fig. 39.11 Masseter muscle Left lateral view. A Schematic. B Masseter with temporalis muscle. Fig. 39.12 Temporalis muscle Left lateral view. A Schematic. B Temporalis muscle. Removed: Masseter and zygomatic arch. D F Styloid process Coronoid process (cut) Articular disk Condylar head Zygomatic arch (cut) Inferior head Superior head Lateral pterygoid H G Pterygoid process, lateral plate Medial pterygoid (superficial head) Mandibular angle Medial pterygoid (deep head) D F G H Articular disk Head of mandible Superficial layer Medial pterygoid Deep layer Inferior head Superior head Temporalis Masseter Lateral pterygoid 39 Muscles of the Skull & Face 559 Table 39.4 Muscles of mastication: Pterygoid muscles Muscle Origin Insertion Innervation Action Lateral pterygoid ③ Superior head Greater wing of sphenoid bone (infratemporal crest) Temporomandibular joint (articular disk) Mandibular n. (CN V3) via lateral pterygoid n. Bilateral: Protrudes mandible (pulls articular disk forward) Unilateral: Lateral movements of mandible (chewing) ④ Inferior head Lateral pterygoid plate (lateral surface) Mandible (condylar process) Medial pterygoid ⑤ Superficial head Maxilla (tuberosity) Pterygoid tuberosity on medial surface of the mandibular angle Mandibular n. (CN V3) via medial pterygoid n. Bilateral: Elevates mandible with masseter; contributes to protrusion. Unilateral: small grinding movements. ⑥ Deep head Medial surface of lateral pterygoid plate and pterygoid fossa Fig. 39.13 Lateral pterygoid muscle Left lateral view. A Schematic. B Left lateral pterygoid muscle. Removed: Coronoid process and part of ramus of mandible. Fig. 39.14 Medial pterygoid muscle Left lateral view. A Schematic. B Left medial pterygoid muscle. Removed: Coro-noid process of mandible. Fig. 39.15 Masticatory muscle sling Oblique posterior view. A Schematic. B Revealed: Muscular sling formed by the masseter and medial pterygoid muscles that elevate the mandible. V1 V2 V3 I Olfactory n. II Optic n. III Oculomotor n. IV Trochlear n. V Trigeminal n. VI Abducent n. VII Facial n. VIII Vestibulo-cochlear n. IX Glossopharyngeal n. X Vagus n. XI Accessory n. XII Hypoglossal n. Head & Neck 560 40 Cranial Nerves Cranial Nerves: Overview Fig. 40.1 Cranial nerves Inferior (basal) view. The 12 pairs of cranial nerves (CN) are numbered according to the order of their emergence from the brainstem. Note: The sensory and motor fibers of the cranial nerves enter and exit the brainstem at the same sites (in contrast to spinal nerves, whose sensory and motor fibers enter and leave through posterior and anterior roots, respectively). For fiber color code, see Table 40.1. Nucleus of hypoglossal n. (CN XII) Dorsal motor (vagal) nucleus Nucleus ambiguus Salivatory nuclei Facial nucleus (CN VII) Nucleus of abducent n. (CN VI) Nucleus of trochlear n. (CN IV) Oculomotor n. nuclei (CN III) Spinal nucleus of accessory n. (CN XI) Nucleus of solitary tract Spinal nucleus of trigeminal n. (CN V) CN X CN IX CN V Trigeminal n. nuclei (CN V) Efferent (motor) nuclei Afferent (sensory) nuclei CN VIII CN VI CN VII Superior salivatory nucleus Inferior salivatory nucleus (CN IX) Nucleus ambiguus Nucleus of solitary tract Nucleus of hypoglossal n. (CN XII) Dorsal vagal nucleus (CN X) Spinal nucleus of trigeminal n. (CN V) Nucleus of abducent n. (CN VI) Motor nucleus Mesencephalic nucleus Nucleus of trochlear n. (CN IV) Nucleus of oculomotor n. Visceral oculomotor nucleus Spinal nucleus of accessory n. (CN XI) CN III Principal (partial) sensory nucleus CN V Facial nucleus CN VII 40 Cranial Nerves 561 The cranial nerves contain both afferent (sensory) and efferent (motor) axons that belong to either the somatic or the autonomic (visceral) nervous system (see pp. 694–695). The somatic fibers allow interaction with the environment, whereas the visceral fibers regulate the auto-nomic activity of internal organs. In addition to the general fiber types, the cranial nerves may contain special fiber types associated with par-ticular structures (e.g., auditory apparatus and taste buds). The cranial nerve fibers originate or terminate at specific nuclei, which are similarly classified as either general or special, somatic or visceral, and afferent or efferent. Fig. 40.2 Cranial nerve nuclei The sensory and motor fibers of cranial nerves III to XII originate and terminate in the brainstem at specific nuclei. A Posterior view with the cerebellum removed. B Midsagittal section, left lateral view. Table 40.1 Classification of cranial nerve fibers and nuclei This color coding is used in subsequent chapters to indicate fiber and nuclei classifications. Fiber type Example Fiber type Example General somatic efferent (somatomotor function) Innervate skeletal muscles General somatic afferent (somatic sensation) Conduct impulses from skin, skeletal muscle spindles General visceral efferent (visceromotor function) Innervate smooth muscle of the viscera, intraocular muscles, heart, salivary glands, etc. Special somatic afferent Conduct impulses from retina, auditory and vestibular apparatuses Special visceral efferent Innervate skeletal muscles derived from branchial arches General visceral afferent (visceral sensation) Conduct impulses from viscera, blood vessels Special visceral afferent Conduct impulses from taste buds, olfactory mucosa Table 40.2 Cranial nerves Cranial nerve Origin Functional fiber types CN I: Olfactory n. Telencephalon CN II: Optic n. Diencephalon CN III: Oculomotor n. Mesencephalon CN IV: Trochlear n. CN V: Trigeminal n. Pons CN VI: Abducent n. CN VII: Facial n. CN VIII: Vestibulocochlear n. Medulla oblongata CN IX: Glossopharyngeal n. CN X: Vagus n. CN XI: Accessory n. CN XII: Hypoglossal n. The olfactory and optic nerves are extensions of the brain rather than true nerves; they are therefore not associated with nuclei in the brainstem. Olfactory trigone Medial and lateral olfactory stria Ambient gyrus Semilunar gyrus Diagonal stria Olfactory tract Olfactory bulb Anterior perforated substance Prepiriform area Amygdala Medial olfactory stria Olfactory bulb Olfactory fibers Olfactory mucosa Prepiriform area Lateral olfactory stria Dorsal longitudinal fasciculus Reticular formation Tegmental nucleus Interpeduncular nucleus Habenular nuclei Longitu-dinal striae Medullary stria of thalamus Uncus, with amygdala below Head & Neck 562 Olfactory bulb (second-order sensory neurons) Olfactory fibers (CN I, first-order sensory neurons) Olfactory tract Superior concha Frontal sinus Nasal septum Cribriform plate, ethmoid bone Nasal septum (cut) Middle concha CN I & II: Olfactory & Optic Nerves The olfactory and optic nerves are not true peripheral nerves but extensions (tracts) of the telencephalon and diencephalon, respec-tively. They are therefore not associated with cranial nerve nuclei in the brainstem. Fig. 40.3 Olfactory nerve (CN I) Fiber bundles in the olfactory mucosa pass from the nasal cavity through the cribriform plate of the ethmoid bone into the anterior cranial fossa, where they synapse in the olfactory bulb. Axons from second-order afferent neurons in the olfactory bulb pass through the olfactory tract and medial or lateral olfactory stria, terminating in the cerebral cortex of the prepiriform area, in the amygdala, or in neighbor-ing areas. A Olfactory bulb and tract, inferior view. Note: The amygdala and prepiriform area are deep to the basal surface of the brain. C Olfactory fibers. Portion of left nasal sep-tum and lateral wall of right nasal cavity, left lateral view. B Course of the olfactory nerve. Parasagittal section, viewed from left side. Optic n. (CN II) Optic chiasm Meyer’s loop Upper visual field Striate area Lower visual field Lateral genicu-late body Optic tract Lateral ventricle Optic radia-tion Occipital pole Optic radiation Lateral geniculate body Medial geniculate body Optic tract Optic chiasm Optic n. (CN II) Lateral geniculate body Optic tract Optic chiasm Optic n. (CN II) Mesencephalon Thalamus Ophthalmic n. (CN V1) Optic tract Optic chiasm Optic canal Optic n. (CN II) passing through optic canal Superior orbital fissure 40 Cranial Nerves 563 Fig. 40.4 Optic nerve (CN II) The optic nerve passes from the eyeball through the optic canal into the middle cranial fossa. The two optic nerves join below the base of the diencephalon to form the optic chiasm, before dividing into the two optic tracts. Each of these tracts divides into a lateral and medial root. Many retinal cell ganglion axons cross the midline to the contralateral side of the brain in the optic chiasm. A Optic nerve in the geniculate visual pathway, left lateral view. D Optic nerve in the left orbit, lateral view. The optic nerve exits the orbit via the optic canal. Note: The other cranial nerves (III, IV, V1, and VI) entering the orbit do so via the superior orbital fissure. C Course of the optic nerve, inferior (basal) view. B Termination of the optic tract, left postero-lateral view of the brainstem. The optic nerve contains the axons of retinal ganglion cells, which terminate mainly in the lateral geniculate body of the diencephalon and in the mesen-cephalon (superior colliculus). Visceral oculomotor nucleus Abducent n. (CN VI) Trochlear n. (CN IV) Oculomotor n. (CN III) Cerebral peduncles of mesencephalon Medulla oblongata Nucleus of abducent n. Nucleus of trochlear n. Nucleus of oculomotor n. Pons Nucleus of oculo-motor n. Visceral oculomotor nucleus Cerebral crus of cerebral peduncle Cerebral aqueduct Tectum Central gray substance Red nucleus Substantia nigra Head & Neck 564 CN III, IV & VI: Oculomotor, Trochlear & Abducent Nerves Cranial nerves III, IV, and VI innervate the extraocular muscles (see p. 605). Of the three, only the oculomotor nerve (CN III) contains both somatic and visceral efferent fibers; it is also the only cranial nerve of the extraocular muscles to innervate multiple extra- and intraocular muscles. B Oculomotor nerve nuclei. Transverse section, superior view. A Emergence of the cranial nerves of the extraocular muscles. Anterior view of the brainstem. Table 40.3 Cranial nerves of the extraocular muscles Course Fibers Nuclei Function Effects of nerve injury Oculomotor nerve (CN III) Runs anteriorly from mesencephalon Somatic efferent Oculomotor nucleus Innervates: • Levator palpebrae superioris • Superior, medial, and inferior rectus • Inferior oblique Complete oculomotor palsy (paralysis of extra- and intraocular muscles): • Ptosis (drooping of eyelid) • Downward and lateral gaze deviation • Diplopia (double vision) • Mydriasis (pupil dilation) • Accommodation difficulties (ciliary paralysis) Visceral efferent Visceral oculomotor (Edinger-Westphal) nucleus Synapse with neurons in ciliary ganglia. Innervates: • Pupillary sphincter • Ciliary muscle Trochlear nerve (CN IV) Emerges from posterior surface of brainstem near midline, courses anteriorly around the cerebral peduncle Somatic efferent Nucleus of the trochlear n. Innervates: • Superior oblique • Diplopia • Affected eye is higher and deviated medially (dominance of inferior oblique) Abducent nerve (CN VI) Follows a long extradural path Somatic efferent Nucleus of the abducent n. Innervates: • Lateral rectus • Diplopia • Medial strabismus (sp.) (due to unopposed action of medial rectus) All three nerves enter the orbit through the superior orbital fissure; CN III and CN VI pass through the common tendinous ring of the extraocular muscles. The abducent nerve follows an extradural course; abducent nerve palsy may therefore develop in association with meningitis and subarachnoid hemorrhage. Fig. 40.5 Nuclei of the oculomotor, trochlear, and abducent nerves The trochlear nerve (CN IV) is the only cranial nerve in which all the fibers cross to the opposite side. It is also the only cranial nerve to emerge from the dorsal side of the brainstem and, consequently, has the longest intradural (intracranial) course of any cranial nerve. Abducent n. (CN VI) Oculomotor n. (CN III) Trochlear n. (CN IV) Frontal n. Lacrimal n. Superior ophthalmic v. Inferior oblique Superior oblique Levator palpe-brae superioris Superior rectus Lateral rectus Inferior rectus Medial rectus Optic n. (CN II) Supraorbital n. (cut) Abducent n. (CN VI) Trochlear n. (CN IV) Oculomotor n. (CN III) Optic n. (CN II) Lacrimal gland Levator palpe-brae superioris Superior oblique Lateral rectus Inferior rectus Medial rectus Superior rectus 40 Cranial Nerves 565 Lateral rectus (cut) Inferior oblique CN III, inferior division Levator palpebrae superioris Superior rectus Superior oblique Ciliary ganglion CN VI CN IV CN III Sympathetic root (postganglionic fibers from superior cervical ganglion via internal carotid plexus) Common tendinous ring Lateral rectus (cut), CN VI Trochlea Short ciliary nerves Mesencephalon Pons Pontomedullary junction Internal carotid artery and plexus Parasympathetic root (preganglionic fibers from CN III) Inferior rectus Medial rectus Fig. 40.6 Course of the nerves inner-vating the extraocular muscles Right orbit. B Anterior view. CN II exits the orbit via the optic canal, which lies medial to the supe-rior orbital fissure (site of emergence of CN III, IV, and VI). C Superior view of the opened orbit. Note the relationship between the optic canal and the superior orbital fissure. A Lateral view. Note: The oculomotor nerve supplies parasympathetic innervation to the intraocular muscles and somatic motor innervation to most of the extraocular muscles (also the levator palpebrae superioris). Its parasympathetic fibers synapse in the ciliary ganglion. Oculomotor nerve palsy may affect exclusively the parasympathetic or somatic fibers, or both concurrently. Motor nucleus Principal (pontine) sensory nucleus Mesencephalic nucleus Spinal nucleus CN V2 CN V1 Trigeminal ganglion Trigeminal n. (CN V) CN V3 Ophthalmic division (CN V1) Maxillary division (CN V2) Mandibular division (CN V3) Trigeminal ganglion Motor nucleus Principal sensory nucleus Mesencephalic nucleus Trigeminal n. (CN V) Pons 4th ventricle Head & Neck 566 CN V: Trigeminal Nerve The trigeminal nerve, the sensory nerve of the head, has three somatic afferent nuclei: the mesencephalic nucleus, which receives propriocep-tive fibers from the muscles of mastication; the principal (pontine) sensory nucleus, which chiefly mediates touch; and the spinal nucleus, which mediates pain and temperature sensation. The motor nucleus supplies motor innervation to the muscles of mastication. Fig. 40.7 Trigeminal nerve nuclei B Cross section through the pons, superior view. A Anterior view of the brainstem. Fig. 40.8 Divisions of the trigeminal nerve (CN V) Right lateral view. Table 40.4 Trigeminal nerve (CN V) Course Fibers Nuclei Function Effects of nerve injury Exits from the middle cranial fossa. Ophthalmic division (CN V1): Enters orbit through superior orbital fissure Maxillary division (CN V2): Enters pterygopalatine fossa through foramen rotundum Mandibular division (CN V3): Passes through foramen ovale into infratemporal fossa Somatic afferent • Principal (pontine) sensory nucleus of the trigeminal n. • Mesencephalic nucleus of the trigeminal n. • Spinal nucleus of the trigeminal n. Innervates: • Facial skin (A) • Nasopharyngeal mucosa (B) • Tongue (anterior two thirds) (C) Involved in the corneal reflex (reflex closure of eyelid) • Sensory loss (traumatic nerve lesions) • Herpes zoster ophthalmicus (varicella-zoster virus); herpes zoster of the face Special visceral efferent Motor nucleus of the trigeminal n. Innervates (via CN V3): • Muscles of mastication (temporalis, masseter, medial and lateral pterygoids (D)) • Oral floor muscles (mylohyoid, anterior digastric) • Tensor tympani • Tensor veli palatini Visceral efferent pathway • Lacrimal n. (CN V1) conveys parasympathetic fibers from CN VII along the zygomatic n. (CN V2) to the lacrimal gland • Lingual n. (CN V3) conveys parasympathetic fibers from CN VII (via the chorda tympani) to the submandibular and sublingual glands • Auriculotemporal n. (CN V3) conveys parasympathetic fibers from CN IX to the parotid gland Visceral afferent pathway Gustatory (taste) fibers from CN VII (via chorda tympani) travel with the lingual n. (CN V3) to the anterior two thirds of the tongue Fibers of certain cranial nerves adhere to divisions or branches of the trigeminal nerve, by which they travel to their destination. All three divisions contribute to dural innervation in the anterior and middle cranial fossae. D C B A Auriculotemporal n. Masseteric n. Lingual n. Inferior alveolar n. (in mandibular canal) Mental n. (in mental foramen) Deep temporal nn. Mandibular division (CN V3) Trigeminal ganglion Medial pterygoid nn. Buccal n. Inferior dental branches Meningeal branch Foramen ovale Infraorbital foramen Mandibular foramen Mylohyoid n. Meningeal branch Pterygopalatine ganglion Posterior superior alveolar nn. Middle superior alveolar n. Anterior superior alveolar branches Infraorbital n. Zygomatic n. (with communicating branch) Ganglionic branches to pterygopalatine ganglion Maxillary division (CN V2) Foramen rotundum Inferior orbital fissure Ciliary ganglion Long ciliary nn. Communicating branch to zygomatic n. Lacrimal n. (with gland) Infratrochlear n. Supratrochlear n. Supraorbital n. Anterior ethmoidal n. Posterior ethmoidal n. Frontal n. Nasociliary n. Recurrent meningeal branch Ophthalmic division (CN V1) Short ciliary nn. Nasociliary (sensory) root to ciliary ganglion Superior orbital fissure 40 Cranial Nerves 567 Fig. 40.9 Course of the trigeminal nerve divisions Right lateral view. A Ophthalmic division (CN V1). Partially opened right orbit. B Maxillary division (CN V2). Partially opened right maxillary sinus with the zygomatic arch removed. C Mandibular division (CN V3). Partially opened mandible with the zygomatic arch removed. Note: The mylohyoid nerve branches from the inferior alveolar nerve just before the mandibular foramen. Nervus intermedius Superior salivatory nucleus Facial nucleus Abducent nucleus Stylomastoid foramen Geniculate ganglion Pons Nucleus of solitary tract Facial nucleus Nucleus of solitary tract Superior salivatory nucleus Abducent nucleus Internal genu of facial n. Posterior auricular n. Stape-dial n. Facial n. (CN VII) Geniculate ganglion Chorda tympani Greater petrosal n. CN V3 CN V2 CN V1 Trigeminal ganglion Stylohyoid Pterygo-palatine ganglion Tympanic membrane Hiatus of canal for greater petrosal n. Petro-tympanic fissure Lingual n. Digastric, posterior belly Stylo-mastoid foramen Facial canal Internal acoustic meatus Stylo-mastoid foramen Posterior auricular n. Greater petrosal n. Internal acoustic meatus Chorda tympani Stapedial n. Geniculate ganglion Parotid plexus Temporal branches Zygomatic branches Buccal branches Marginal mandibular branch Cervical branch Parotid plexus Posterior auricular n. Facial n. Head & Neck 568 CN VII: Facial Nerve The facial nerve mainly conveys special visceral efferent (branchiogenic) fibers from the facial nerve nucleus to the muscles of facial expression. The other visceral efferent (parasympathetic) fibers from the superior salivatory nucleus are grouped with the visceral afferent (gustatory) fibers to form the nervus intermedius. Fig. 40.10 Facial nerve nuclei A Anterior view of the brainstem. B Cross section through the pons, superior view. Fig. 40.11 Branches of the facial nerve Right lateral view. A Facial nerve in the temporal bone. B Branches. C Parotid plexus. Submandibular ganglion Via communicating br. to lacrimal n. Greater petrosal n. Pterygopalatine ganglion Nucleus of the solitary tract Geniculate ganglion Trigeminal n. Facial n. Lingual n. Parotid gland Sublingual gland Submandibular gland Nasal glands Lacrimal gland Maxillary division Postganglionic sympathetic and parasympathetic fibers Pterygoid canal with n. of pterygoid canal Glandular brs. Chorda tympani Stylomastoid foramen Internal carotid a. with internal carotid plexus Mandibular division Superior saliva-tory nucleus Deep petrosal n. Trigeminal ganglion Taste buds of soft palate 40 Cranial Nerves 569 Fig. 40.12 Course of the facial nerve Right lateral view. This figure shows the distribution of all the fiber types in Table 40.5. Visceral efferent (parasympathetic) and special visceral afferent (taste) fibers shown in blue and green respectively. Postganglionic sympathetic fibers are shown in black. Table 40.5 Facial nerve (CN VII) Course Fibers Nuclei Function Effects of nerve injury Emerges in the cerebellopontine angle between the pons and olive; passes through the internal acoustic meatus into the temporal bone (petrous part), where it divides into: • Greater petrosal n. • Stapedial n. • Chorda tympani Certain special visceral efferent fibers pass through the stylomastoid foramen to the skull base, forming the intraparotid plexus Special visceral efferent Facial nucleus Innervate: • Muscles of facial expression • Stylohyoid • Digastric (posterior belly) • Stapedius Peripheral facial nerve injury: paralysis of muscles of facial expression on affected side Associated disturbances of taste, lacrimation, salivation, hyperacusis, etc. Visceral efferent (para-sympathetic) Superior salivatory nucleus Synapse with neurons in the pterygopalatine or submandibular ganglion. Innervate: • Lacrimal gland • Small glands of nasal mucosa, hard and soft palate • Submandibular gland • Sublingual gland • Small salivary glands of tongue (dorsum) Special visceral afferent Nucleus of the solitary tract Peripheral processes of fibers from geniculate ganglion form the chorda tympani (gustatory fibers from tongue) Somatic afferent Sensory fibers from the auricle, skin of the auditory canal, and outer surface of the tympanic membrane travel via CN VII to the principal sensory nucleus of the trigeminal n. Grouped to form nervus intermedius, which aggregates with the visceral efferent fibers from the facial n. nucleus. Flocculus of cerebellum Inferior vestibular nucleus Medial vestibular nucleus Lateral vestibular nucleus Superior vestibular nucleus Vestibulo-cochlear n. (CN VIII) Semi-circular canals Vestibular ganglion Vestibular root Direct fibers to cerebellum Medial vestibular nucleus Lateral vestibular nucleus Superior vestibular nucleus Posterior cochlear nucleus Anterior cochlear nucleus Vestibulo-cochlear n. (CN VIII) Cochlear root Cochlea with spiral ganglia Anterior cochlear nucleus Posterior cochlear nucleus Head & Neck 570 CN VIII: Vestibulocochlear Nerve The vestibulochochlear nerve is a special somatic afferent nerve that consists of two roots. The vestibular root transmits impulses from the vestibular apparatus; the cochlear root transmits impulses from the auditory apparatus. Fig. 40.13 Vestibulocochlear nerve: Vestibular part A Anterior view of the medulla oblon-gata and pons with cerebellum. B Cross section through the upper medulla oblongata. Fig. 40.14 Vestibulocochlear nerve: Cochlear part A Anterior view of the medulla oblon-gata and pons. B Cross section through the upper medulla oblongata. Table 40.6 Vestibulocochlear nerve (CN VIII) Part Course Fibers Nuclei Function Effects of nerve injury Vestibular part Pass from the inner ear through the internal acoustic meatus to the cerebellopontine angle, where they enter the brain Special somatic afferent Superior, lateral, medial, and inferior vestibular nuclei Peripheral processes from the semicircular canals, saccule, and utricle pass to the vestibular ganglion and then to the four vestibular nuclei Dizziness Cochlear part Anterior and posterior cochlear nuclei Peripheral processes beginning at the hair cells of the organ of Corti pass to the spiral ganglion and then to the two cochlear nuclei Hearing loss Internal carotid a. Vestibular n. Posterior ampullary n. Geniculate ganglion Greater petrosal n. Transverse crest Facial n. (CN VII) Cochlear n. Sacculo-ampullary n. Utriculo-ampullary n. Nervus intermedius (from CN VII) CN VIII Cochlear root Spiral ganglia Vestibular root Saccular n. Utricular n. Posterior ampullary n. Lateral ampullary n. Anterior ampullary n. Inferior part Superior part Vestibulo-cochlear n. (CN VIII) Vestibular ganglion Cochlear duct Saccule Utricle Semicircular ducts Cochlea Internal carotid plexus Lesser petrosal n. Chorda tympani Tympanic plexus Greater petrosal n. Tympanic n. Geniculate ganglion Facial n. (CN VII) Sigmoid sinus (ghosted) Anterior semi-circular canal Lateral semi-circular canal Posterior semi-circular canal Mastoid air cells Facial n. (in facial canal) Round window Internal jugular v. Pharyngotympanic (auditory) tube Internal carotid a. Semicanal of tensor tympani Oval window Posterior wall of tympanic cavity Anterior wall of tympanic cavity Roof of tympanic cavity (tegmen tympani) Vestibular root (CN VIII) Cochlear root (CN VIII) 40 Cranial Nerves 571 Fig. 40.15 Vestibular and cochlear (spiral) ganglia Note: The vestibular and cochlear roots are still separate structures in the petrous part of the temporal bone. Fig. 40.16 Vestibulocochlear nerve in the temporal bone A Medial wall of the tympanic cavity, oblique sagittal section. B Cranial nerves in the internal acoustic meatus. Posterior oblique view of the right meatus. Inferior part Jugular foramen Pharyngeal brs. Br. to carotid sinus Br. to stylo-pharyngeus Tympanic n. Inferior ganglion Superior ganglion Superior part Nucleus ambiguus Inferior saliva-tory nucleus Spinal nucleus of trigeminal n. Nucleus of the solitary tract Nucleus ambiguus Inferior part Inferior salivatory nucleus Superior part Glosso-pharyngeal n. Nucleus of the solitary tract ③ Carotid sinus ② ④ ⑤ Glossopharyngeal n. (CN IX) ⑥ Inferior ganglion Superior ganglion Vagus n. (CN X) ① Br. to carotid sinus Pharyngeal brs. Pharyngeal plexus Vagus n. (CN X) Stylopharyngeus Head & Neck 572 CN IX: Glossopharyngeal Nerve Fig. 40.17 Glossopharyngeal nerve nuclei A Anterior view of the medulla oblongata. B Cross section through the medulla oblongata, superior view. Not shown: Nuclei of the trigeminal nerve. Fig. 40.18 Course of the glossopharyngeal nerve Left lateral view. Note: Fibers from the vagus nerve (CN X) combine with fibers from the glossopharyngeal nerve (CN IX) to form the pharyngeal plexus and supply the carotid sinus. Table 40.7 Glossopharyngeal nerve branches ① Tympanic n. ② Br. to carotid sinus ③ Br. to stylopharyngeus muscle ④ Tonsillar brs. ⑤ Lingual brs. ⑥ Pharyngeal brs. Promontory of labyrinthine wall Tubarian br. Internal carotid plexus Inferior ganglion Tympanic canaliculus with tympanic n. Tympanic plexus Lesser petrosal n. Carotico-tympanic n. Glossopharyngeal n. (CN IX) Superior ganglion Pharyngotympanic (auditory) tube Glossopharyngeal n. (CN IX) Tympanic n. Trigeminal n. (CN V) Tympanic plexus Mandibular division (CN V3) Lesser petrosal n. Otic ganglion Auriculo-temporal n. Postganglionic parasym-pathetic fibers (run a short distance with the auriculotemporal n.) Parotid gland 40 Cranial Nerves 573 A B C D E F Table 40.8 Glossopharyngeal nerve (CN IX) Course Fibers Nuclei Function Effects of nerve injury Emerges from the medulla oblongata; leaves cranial cavity through the jugular foramen Visceral efferent (parasympathetic) Inferior salivatory nucleus Parasympathetic presynaptic fibers are sent to the otic ganglion; postsynaptic fibers are distributed to • Parotid gland (A) • Buccal gland • Labial gland Isolated lesions of CN IX are rare. Lesions are generally accompanied by lesions of CN X and CN XI (cranial part), as all three emerge jointly from the jugular foramen and are susceptible to injury in basal skull fractures. Special visceral efferent (branchiogenic) Nucleus ambiguus Innervate: • Constrictor muscles of the pharynx (pharyngeal branches join with the vagus nerve to form the pharyngeal plexus) • Stylopharyngeus Visceral afferent Nucleus of the solitary tract (inferior part) Receive sensory information from • Chemoreceptors in the carotid body (B) • Pressure receptors in the carotid sinus Special visceral afferent Nucleus of the solitary tract (superior part) Receives sensory information from the posterior third of the tongue (via the inferior ganglion) (C) Somatic afferent Spinal nucleus of trigeminal nerve Peripheral processes of the intracranial superior ganglion or the extracranial inferior ganglion arise from • Tongue, soft palate, pharyngeal mucosa, and tonsils (D, E) • Mucosa of the tympanic cavity, internal surface of the tympanic membrane, pharyngotympanic tube (tympanic plexus) (F) • Skin of the external ear and auditory canal (blends with the vagus n.) Fig. 40.19 Glossopharyngeal nerve in the tympanic cavity Left anterolateral view. The tympanic nerve contains visceral efferent (presynaptic parasympathetic) fibers for the otic ganglion, as well as somatic afferent fibers for the tympanic cavity and pharyngotympanic tube. It joins with sympathetic fibers from the internal carotid plexus (via the caroticotympanic nerve) to form the tympanic plexus. Fig. 40.20 Visceral efferent (parasympathetic) fibers of CN IX Spinal nucleus of trigeminal n. Dorsal vagal nucleus Superior laryngeal n. Pharyngeal br. Inferior ganglion Nucleus ambiguus Jugular foramen Superior ganglion Superior part Inferior part Nucleus of the solitary tract Spinal nucleus of trigeminal n. Dorsal vagal nucleus Nucleus ambiguus Olive Superior part Inferior part Nucleus of the solitary tract Head & Neck 574 CN X: Vagus Nerve Fig. 40.21 Vagus nerve nuclei A Anterior view of the medulla oblongata. B Cross section through the medulla oblongata, superior view. A B C D E F G Table 40.9 Vagus nerve (CN X) Course Fibers Nuclei Function Effects of nerve injury Emerges from the medulla oblongata; leaves the cranial cavity through the jugular foramen. CN X has the most extensive distribution of all the cranial nerves (vagus = “vagabond”), consisting of cranial, cervical, thoracic (see p. 87), and abdominal (see p. 215) parts. Special visceral efferent (branchio-genic) Nucleus ambiguus Innervate: • Pharyngeal muscles (via pharyngeal plexus with CN IX) • Muscles of the soft palate • Laryngeal muscles (superior laryngeal n. supplies the cricothyroid; inferior laryngeal n. supplies all other laryngeal muscles) The recurrent laryngeal n. supplies visceromotor innervation to the only muscle abducting the vocal cords, the posterior cricoarytenoid. Unilateral destruction of this nerve leads to hoarseness; bilateral destruction leads to respiratory distress (dyspnea). Visceral efferent (parasympa-thetic) Dorsal vagal nucleus Synapse in prevertebral or intramural ganglia. Innervate smooth muscle and glands of • Thoracic viscera (A) • Abdominal viscera (A) Somatic afferent Spinal nucleus of trigeminal nerve Superior (jugular) ganglion receives peripheral fibers from • Dura in posterior cranial fossa (C) • Skin of ear (D), external auditory canal (E) Special visceral afferent Nucleus of solitary tract (superior part) Inferior nodose ganglion receives peripheral processes from • Taste buds on the epiglottis and root of the tongue (F) Visceral afferent Nucleus of solitary tract (inferior part) Inferior ganglion receives peripheral processes from • Mucosa of lower pharynx at its esophageal junction (G) • Laryngeal mucosa above (superior laryngeal n.) and below (inferior laryngeal n.) the vocal fold (G) • Pressure receptors in the aortic arch (B) • Chemoreceptors in the para-aortic body (B) • Thoracic and abdominal viscera (A) Vagus n. (CN X) Left inferior laryngeal n. External br. (external laryngeal n.) Internal br. (internal laryngeal n.) ① Subclavian a. Cricothyroid ④ Brachio-cephalic trunk Right inferior laryngeal n. ② 3R 3L Internal br. (sensory) External br. (motor) Vagus n. (CN X) Cricothyroid ① ② 40 Cranial Nerves 575 Fig. 40.22 Course of the vagus nerve The vagus nerve gives off four major branches in the neck. The inferior laryngeal nerves are the terminal branches of the recurrent laryngeal nerves. Note: The left recurrent laryngeal nerve hooks around the aortic arch, while the right nerve hooks around the subclavian artery. A Branches of the vagus nerve in the neck. Anterior view. B Innervation of the pharyngeal and laryngeal muscles. Left lateral view. Table 40.10 Vagus nerve branches in the neck ① Pharyngeal brs. ② Superior laryngeal n. 3R Right recurrent laryngeal n. 3L Left recurrent laryngeal n. ④ Cervical cardiac brs. Trapezius Cortico-bulbar fibers Cranial root Foramen magnum Accessory n. (CN XI) Vagus n. (CN X) Nucleus ambiguus Sternocleido-mastoid Jugular foramen Spinal root Spinal nucleus of accessory n. Head & Neck 576 CN XI & XII: Accessory & Hypoglossal Nerves Fig. 40.23 Accessory nerve Posterior view of the brainstem with the cer-ebellum removed. Note: For didactic reasons, the muscles are displayed from the right side. Fig. 40.24 Accessory nerve lesions Lesion of the right accessory nerve. A Trapezius paralysis, posterior view. See Table 40.11 (below) for clini-cal correlation explanation. B Sternocleidomastoid paralysis, right anterolateral view. See Table 40.11 (below) for clinical correlation explanation. Table 40.11 Accessory nerve (CN XI) Course Fibers Nuclei Function Effects of nerve injury The spinal root emerges from the spinal cord (at the level of C1–C5/6), passes superiorly, and enters the skull through the foramen magnum, where it joins with the cranial root arising from the medulla oblongata. Both roots leave the skull through the jugular foramen. Within the jugular foramen, fibers from the cranial root pass to the vagus n. (internal branch). The spinal portion descends to the nuchal region as the external branch. Special visceral efferent Nucleus ambiguus (caudal part) Join CN X and are distributed with the recurrent laryngeal n. Innervate: • All laryngeal muscles (except cricothyroid) Trapezius paralysis: drooping of shoulder on affected side and difficulty raising arm above horizontal plane. This paralysis is a concern during neck operations (e.g., lymph node biopsies). An injury of the accessory n. will not result in complete trapezius paralysis (the muscle is also innervated by segments C3 and C4). Sternocleidomastoid paralysis: torticollis (wry neck, i.e., difficulty turning head). Unilateral lesions cause flaccid paralysis (the muscle is supplied exclusively by the accessory n.). Bilateral lesions make it difficult to hold the head upright. Somatic efferent Spinal nucleus of accessory n. Form the external branch of the accessory n. Innervate: • Trapezius • Sternocleidomastoid See text at top of page, and Table 40.2, regarding new data on cranial fibers of CN XI. The traditional “cranial root” of the accessory nerve (CN XI), with its cell bodies found in the nucleus ambiguus, is now considered a part of the vagus nerve (CN X) that travels with the spinal root of CN XI for a short distance before splitting off. The cranial fibers are therefore part of CN X distributed via the vagus nerve after traveling briefly with the spinal root of CN XI. The spinal root fibers, arising from the spinal nucleus of the accessory nerve are now considered to be the accessory nerve, continue on as the accessory nerve (CN XI). Foramen magnum C1 spinal n. Hypoglossal n. (CN XII) Hypoglossal canal Nucleus of the hypo-glossal n. Paralyzed genioglossus Tongue Hypoglossal trigone (in rhomboid fossa) Nucleus of the hypoglossal n. Olive Hypo-glossal n. Hyoglossus Cortico-bulbar fibers Styloglossus Hypo-glossal canal Nucleus of the hypo-glossal n. Precentral gyrus Vagus n. C1 Genioglossus Thyrohyoid 40 Cranial Nerves 577 A Anterior view. B Cross section through the medulla oblongata. A Normal genioglossus muscles. B Unilateral nuclear or peripheral lesion. Table 40.12 Hypoglossal nerve (CN XII) Course Fibers Nuclei Function Effects of nerve injury Emerges from the medulla oblongata, leaves the cranial cavity through the hypoglossal canal, and descends laterally to the vagus nerve. CN XII enters the root of the tongue above the hyoid bone. Somatic efferent Nucleus of the hypo-glossal n. Innervates: • Intrinsic and extrinsic muscles of the tongue (except the palatoglossus, supplied by CN X) Central hypoglossal paralysis (supranuclear): tongue deviates away from the side of the lesion. Nuclear or peripheral paralysis: tongue deviates toward the affected side (due to preponderance of muscle on healthy side) Flaccid paralysis: both nuclei injured; tongue cannot be protruded. Fig. 40.25 Hypoglossal nerve Posterior view of the brainstem with the cerebellum removed. Note: C1, which innervates the thyrohyoid and geniohyoid, runs briefly with the hypoglossal nerve. Fig. 40.26 Hypoglossal nerve nuclei Note: The nucleus of the hypoglossal nerve is innervated by cortical neurons from the contralateral side. Fig. 40.27 Hypoglossal nerve lesions Superior view. See Table 40.12 below for clinical correlation explanation. Head & Neck 578 Autonomic Innervation Fig. 40.28 Parasympathetic nervous system (cranial part): Overview There are four parasympathetic nuclei in the brainstem. The visceral efferent fibers of these nuclei travel along particular cra-nial nn., listed below. • Visceral oculomotor (Edinger–Westphal) nucleus: oculomotor n. (CN III) • Superior salivatory nucleus: facial n. (CN VII) • Inferior salivatory nucleus: glossopharyn-geal n. (CN IX) • Dorsal vagal nucleus: vagus n. (CN X) The preganglionic parasympathetic fibers often travel with multiple cranial nn. to reach their target organs. The vagus n. supplies all of the thoracic and abdominal organs as far as a point near the left colic flexure. Note: The sympathetic fibers to the head travel along the arteries to their target organs. Table 40.13 Parasympathetic ganglia in the head Nucleus Path of presynaptic fibers Ganglion Postsynaptic fibers Target organs Edinger-Westphal nucleus Oculomotor n. (CN III) Ciliary ganglion Short ciliary nn. (CN V1) Ciliary muscle (accommodation) Pupillary sphincter (miosis) Superior salivatory nucleus Nervus intermedius (CN VII root) → greater petrosal n. → n. of pterygoid canal Pterygopalatine ganglion • Maxillary n. (CN V2) → zygomatic n. → anastomosis → lacrimal n. (CN V1) • Orbital branches • Posterior superior nasal brs. • Nasopalatine nn. • Greater and lesser palatine nn. • Lacrimal gland • Glands of nasal cavity and paranasal sinuses • Glands of gingiva • Glands of hard and soft palate • Glands of pharynx Nervus intermedius (CN VII root) → chorda tympani → lingual n. (CN V3) Submandibular ganglion Glandular branches Submandibular gland Sublingual gland Inferior salivatory nucleus Glossopharyngeal n. (CN IX) → tympanic n. → lesser petrosal n. Otic ganglion Auriculotemporal n. (CN V3) Parotid gland Dorsal motor (vagal) nucleus Vagus n. (X) Ganglia near organs Fine fibers in organs, not individually named Thoracic and abdominal viscera → = is continuous with Otic ganglion Submandibular ganglion Pterygo-palatine ganglion Dorsal motor nucleus of the vagus Inferior salivatory nucleus Superior salivatory nucleus III VII IX X Visceral occulomotor (Edinger-Westphal) nucleus Abdominal ganglia Thoracic ganglia Ciliary ganglion Parasympathetic preganglionic fibers Parasympathetic postganglionic fibers 40 Cranial Nerves 579 Fig. 40.29 Sympathetic innervation of the head Sympathetic preganglionic neurons of the head originate in the lateral horn of the spinal cord (TI–T3). They exit into the sympathetic trunk and ascend to synapse in the superior cervical ganglion. Postganglionic neu-rons then travel with arterial plexuses. Postganglionic fibers that travel with the carotid plexus (on the internal carotid artery) join with the na-sociliary nerves (of CN V1) and then the long ciliary nerves to reach the dilator pupillae muscle (pupillary dilation); other postganglionic fibers travel through the ciliary ganglion (without synapsing) to reach the cili-ary muscle to participate in accomodation. Still other postganglionic fi-bers from the carotid plexus leave with the deep petrosal nerve, which joins with the greater petrosal nerve (CN VII), to form the nerve of the pterygoid canal (vidian nerve). This nerve travels to the pterygopalatine ganglion where it distributes fibers via branches of the maxillary nerve to the glands of the nasal cavity, maxillary sinus, hard and soft palate, gin-giva, and pharynx, and to sweat glands and blood vessels in the head. Postganglionic fibers from the superior cervical ganglion that travel with the facial artery plexus pass through the submandibular ganglion (with-out synapsing) to the submandibular and sublingual glands. Other post-ganglionic fibers travel with the middle meningeal plexus, through the otic ganglion (without synapsing), to the parotid gland. Table 40.14 Sympathetic fibers in the head Nucleus Path of presynaptic fibers Ganglion Postsynaptic fibers Target organs Lateral horn of spinal cord (TI–L2) Enter sympathetic trunk and ascend to superior cervical ganglion Superior cervical ganglion ICA plexus → nasociliary n. (CN V1) → long ciliary nn. (CN V1) Dilator pupillae muscle (mydriasis) Postganglionic fibers → ciliary ganglion→ short ciliary nn. (limited number of fibers) Ciliary muscle (sparse sympathetic fibers contributing to accommodation) ICA plexus → deep petrosal n. → n. of pterygoid canal → pterygopalatine ganglion → branches of maxillary n. (CN V2) Glands of nasal cavity Sweat glands Blood vessels Facial a. plexus → submandibular ganglion Submandibular gland Sublingual gland External carotid a. plexus Parotid gland passes through without synpasing; → = is continuous with ICA, internal carotid a. Pupillary dilation Accommodation Blood vessels Sweat glands Vasomotor innervation Vasomotor innervation Long ciliary n. (CN V1) External carotid a. plexus Ciliary ganglion Pterygoid canal N. of pterygoid canal Deep petrosal n. (CN VII) Pterygopalatine ganglion Facial a. plexus Internal carotid a. plexus Nasociliary n. Superior cervical ganglion (CN V1) Sympathetic postganglionic Mandibular division (CN V3, exits via foramen ovale) Deep temporal nn. (to temporalis) Buccal n. N. to lateral pterygoid and lateral pterygoid Lingual n. N. to medial pterygoid and medial pterygoid Inferior alveolar n. N. to masseter and masseter Parotid brs. Auriculo-temporal n. Meningeal br. Trigeminal n. (CN V) Trigeminal ganglion Maxillary division (CN V2) Ophthalmic division (CN V1) Buccinator Temporal brs. Zygomatic brs. Buccal brs. Marginal mandibular br. Cervical br. Parotid plexus Posterior auricular n. Facial n. (CN VII) Head & Neck 580 41 Neurovasculature of the Skull & Face Innervation of the Face Fig. 41.1 Motor innervation of the face Left lateral view. Five branches of the facial nerve (CN VII) provide motor innervation to the muscles of facial expression. The mandibular division of the trigeminal nerve (CN V3) supplies motor innervation to the muscles of mastication. B Motor innervation of the muscles of mastication (). A Motor innervation of the muscles of facial expression. Trigeminal ganglion Ophthalmic division (CN V1) Maxillary division (CN V2) Supraorbital n. Supratrochlear n. Infraorbital n. Buccal n. Mental n. Mylohyoid n. Lingual n. Inferior alveolar n. Masseteric n. Mandibular division (CN V3) Auriculotemporal n. Pterygopalatine ganglion Mental n. (from CN V3) Infraorbital n. (from CN V2) Supraorbital n. (from CN V 1) Mandibular division Transverse cervical n. Supraclavic-ular nn. Spinal nn., posterior rami Great auricular n. (C2, C3), anterior rami Lesser occipital n. (C2) anterior ramus Greater occi-pital n. (C2), posterior ramus Maxillary division Ophthalmic division Trigeminal n. (CN V) 41 Neurovasculature of the Skull & Face 581 Fig. 41.2 Sensory innervation of the face C Divisions of the trigeminal nerve, left lateral view. Indicates sensory nn. A Sensory branches of the trigeminal nerve, anterior view. The sensory branches of the three divisions emerge from the supraorbital, infra-orbital, and mental foramina, respectively. B Cutaneous innervation of the head and neck, left lateral view. The occiput and nuchal regions are supplied by the posterior rami (blue) of the spinal nerves (the greater occipital nerve is the posterior ramus of C2). Vertebral a. Common carotid a. Internal carotid a. Facial a. External carotid a. Ophthalmic a. Subclavian a. Angular a. External carotid a. Superior thyroid a. Internal carotid a. Ophthalmic a. Basilar a. Internal carotid a. Subclavian a. Posterior communicating a. Carotid bifurcation Supraorbital a. Supra-trochlear a. Dorsal nasal a. Posterior ethmoidal a. Vertebral a. Cavernous part Petrous part Cerebral part Meningeal br. Inferior hypophyseal a. Carotico-tympanic aa. A. of pterygoid canal Trigeminal ganglion br. Marginal tentorial br. Basal tentorial br. Neural br. Cavernous sinus br. Superior hypophyseal a. Posterior communicating a. Cervical part Ophthalmic a. Anterior choroidal a. Middle cerebral a. Anterior cerebral a. Head & Neck 582 Arteries of the Head & Neck The head and neck are supplied by branches of the common carotid artery. The common carotid splits at the carotid bifurcation into two branches: the internal and external carotid arteries. The internal carotid chiefly supplies the brain (p. 688), although its branches anastomose with the external carotid in the orbit and nasal septum. The external carotid is the major supplier of structures of the head and neck. Fig. 41.3 Internal carotid artery Left lateral view. The most important extra-cerebral branch of the internal carotid artery is the ophthalmic artery, which supplies the upper nasal septum (p. 620) and the orbit (p. 608). See pp. 688–689 for the arteries of the brain. A Schematic. B Parts and branches of the internal carotid artery. C Course of the internal carotid artery. Medial branch Posterior branches Anterior branches Terminal branches 41 Neurovasculature of the Skull & Face 583 Angular a. Superior labial a. Inferior labial a. Facial a. Maxillary a. Ascending pharyngeal a. Facial a. Lingual a. Superior thyroid a. Left subclavian a. Thyrocervical a. Left common carotid a. Vertebral a. Carotid bifurcation with carotid body Internal carotid a. Occipital a. Posterior auricular a. External carotid a. Superior laryngeal a. Superficial temporal a. Fig. 41.4 External carotid artery: Overview Left lateral view. B Course of the external carotid artery. Table 41.1 Branches of the external carotid artery Group Artery Anterior (p. 584) Superior thyroid a. Lingual a. Facial a. Medial (p. 584) Ascending pharyngeal a. Posterior (p. 585) Occipital a. Posterior auricular a. Terminal (p. 585) Maxillary a. Superficial temporal a. A Schematic of the external carotid artery. Clinical box 41.1 The carotid artery is often affected by atherosclerosis, a hardening of arterial walls due to plaque formation. The examiner can determine the status of the arteries using ultrasound. Note: The absence of atherosclerosis in the carotid artery does not preclude coronary heart disease or atherosclerotic changes in other locations. Carotid artery atherosclerosis A Common carotid artery with “normal” flow. B Calcified plaque in the carotid bulb. Superior thyroid a. Internal carotid a. Lingual a. Facial a. Ascending pharyngeal a. Ophthalmic a. Angular a. Head & Neck 584 Superficial temporal a. Angular a. Superior labial a. Inferior labial a. Maxillary a. Internal carotid a. Left common carotid a. Superior thyroid a. Submental a. Lingual a. Ascending palatine a. Ascending pharyngeal a. Dorsal nasal a. Mental a. Infraorbital a. Facial a. Tonsillar a. Glandular branches Branch of ophthalmic a. External Carotid Artery: Anterior, Medial & Posterior Branches Fig. 41.5 Anterior and medial branches Left lateral view. The arteries of the anterior aspect supply the anterior structures of the head and neck, including the orbit (p. 606), ear (p. 632), larynx (p. 530), pharynx (p. 654), and oral cavity. Note: The angular artery anastomoses with the dorsal nasal artery of the internal carotid (via the ophthalmic artery). A Arteries of the anterior and medial branches. The copious blood supply to the face makes facial injuries bleed profusely but heal quickly. There are extensive anastomoses between branches of the external carotid artery and between the external carotid artery and branches of the ophthalmic artery. B Course of the anterior and medial branches. Vertebral a. Left common carotid a. External carotid a. Occipital a. Posterior auricular a. Superficial temporal a. 41 Neurovasculature of the Skull & Face 585 Superficial temporal a. Maxillary a. Internal carotid a. Posterior auricular a. Occipital a. Descending branch Occipital branches Left common carotid a. Superior thyroid a. External carotid a. Lingual a. Ascending pharyngeal a. Posterior branch Occipital a. Facial a. Fig. 41.6 Posterior branches Left lateral view. The posterior branches of the external carotid artery supply the ear (p. 632), posterior skull (p. 594), and posterior neck muscles (p. 541). A Arteries of the posterior branch. B Course of the posterior branches. Table 41.2 Anterior, medial, and posterior branches of the external carotid artery Branch Artery Divisions and distribution Anterior brs. Superior thyroid a. Glandular br. (to thyroid gland); superior laryngeal a.; sternocleidomastoid br. Lingual a. Dorsal lingual brs. (to base of tongue, palatoglossal arch, tonsil, soft palate and epiglottis); sublingual a. (to sublingual gland, tongue, oral floor, oral cavity); sublingual br. to the sublingual gland; deep lingual a. Facial a. Ascending palatine a. (to pharyngeal wall, soft palate, pharyngotympanic tube); tonsillar branch (to palatine tonsils); submental a. (to oral floor, submandibular gland); labial aa.; angular a. (to nasal root) Medial br. Ascending pharyngeal a. Pharyngeal brs.; inferior tympanic a. (to mucosa of inner ear); posterior meningeal a. Posterior brs. Occipital a. Occipital brs.; descending br. (to posterior neck muscles) Posterior auricular a. Stylomastoid a. (to facial n. in facial canal); posterior tympanic a.; auricular br.; occipital br.; parotid br. For terminal brs., see Table 41.3 (p. 586). Maxillary a. Left common carotid a. Superficial temporal a. External carotid a. Frontal br. Zygomatico-orbital a. Transverse facial a. Maxillary a. External carotid a. Superficial temporal a. Middle temporal a. Parietal bone br. Head & Neck 586 External Carotid Artery: Terminal Branches The terminal branches of the external carotid artery consist of two major arteries: superficial temporal and maxillary. The superficial temporal artery supplies the lateral skull. The maxillary artery is a major artery for internal structures of the face. Fig. 41.7 Superficial temporal artery Left lateral view. Inflammation of the superficial temporal artery due to temporal arteritis can cause severe headaches. The course of the frontal branch of the artery can often be seen superficially under the skin of elderly patients. A Arteries of the terminal branch. B Course of the superficial temporal artery. Table 41.3 Terminal branches of the external carotid artery Branch Artery Divisions and distribution External carotid a. Superficial temporal a. Transverse facial a. (to soft tissues below the zygomatic arch); frontal brs.; parietal brs.; zygomatico-orbital a. (to lateral orbital wall) Maxillary a. Mandibular part Inferior alveolar a. (to mandible, teeth, gingiva); middle meningeal a.; deep auricular a. (to temporomandibular joint, external auditory canal); anterior tympanic a. Pterygoid part Masseteric a.; deep temporal brs.; pterygoid brs.; buccal a. Pterygopalatine part Posterosuperior alveolar a. (to maxillary molars, maxillary sinus, gingiva); infraorbital a. (to maxillary alveoli) Descending palatine a. Greater palatine a. (to hard palate) Lesser palatine a. (to soft palate, palatine tonsil, pharyngeal wall) Sphenopalatine a. Lateral posterior nasal aa. (to lateral wall of nasal cavity, conchae) Posterior septal brs. (to nasal septum) Parts not shown here. See Fig 41.27 (p. 599) and Table 41.8 (p. 601). Mental br. Pterygoid br. Inferior alveolar a. Mylohyoid br. Maxillary a. Middle meningeal a. Masseteric a. Buccal a. Deep tem-poral aa. Spheno-palatine a. Infra-orbital a. Anterior and posterior superior alveolar aa. Posterior superior alveolar a. Buccal a. Middle meningeal a. Deep auricular a. Anterior tympanic a. Inferior alveolar a. Anastomotic br. with lacrimal a. Parietal br. Frontal br. Middle meningeal a. Petrous br. Epidural hematoma Arachnoid Fracture Dura mater Ruptured middle meningeal a. Calvaria Greater palatine a. Lesser palatine a. Descending palatine a. Spheno-palatine a. Posterior septal brs. Lateral posterior nasal aa. 41 Neurovasculature of the Skull & Face 587 Fig. 41.8 Maxillary artery Left lateral view. The maxillary artery consists of three parts: mandibu-lar (blue), pterygoid (green), and pterygopalatine (yellow). A Divisions of the maxillary artery. B Course of the maxillary artery. Clinical box 41.2 Middle meningeal artery The middle meningeal artery supplies the meninges and overlying calvaria. Rupture of the artery (generally due to head trauma) results in an epidural hematoma. Sphenopalatine artery The sphenopalatine artery supplies the wall of the nasal cavity. Excessive nasopharyngeal bleeding from the branches of the sphenopalatine artery may necessitate ligation of the maxillary artery in the pterygopalatine fossa. C Lateral wall of right nasal cavity, medial view. A Right middle meningeal artery, medial view of opened skull. B Epidural hematoma. Schematic coronal section. Pterygoid plexus (deep temporal vv.) Superior and inferior ophthalmic vv. Angular v. Submental v. Facial v. Anterior jugular v. Left brachio-cephalic v. Sub-clavian v. Supra-scapular v. External jugular v. Internal jugular v. Superior thyroid v. Retromandibular v. Occipital v. Posterior auricular v. Maxillary v. Superficial temporal v. Supratrochlear v. Supraorbital v. Inferior labial v. Confluence of the sinuses Superior sagittal sinus Facial v. Left brachio-cephalic v. Subclavian v. Suprascapular v. Anterior Internal External Transverse sinus Sigmoid sinus Superficial temporal v. Jugular vv. Cavernous sinus Head & Neck 588 Veins of the Head & Neck Fig. 41.9 Veins of the head and neck Left lateral view. The veins of the head and neck drain into the brachio-cephalic vein. Note: The left and right brachiocephalic veins are not symmetrical. A Principal veins of the head and neck. B Superficial veins of the head and neck. Note: The course of the veins is highly variable. Table 41.4 Principal superficial veins Vein Region drained Location Internal jugular v. Interior of skull (including brain) Within carotid sheath External jugular v. Superficial head Within superficial cervical fascia Anterior jugular v. Neck, portions of head Sigmoid sinus Superior and inferior petrosal sinuses Superficial temporal v. Superior ophthal-mic v. Angular v. Deep facial v. External palatine v. Facial v. Internal jugular v. Retromandibular v. Pterygoid plexus Deep temporal vv. Maxillary v. Cavernous sinus Retromandibular v. posterior division Retromandibular v. anterior division External vertebral venous plexus Venous plexus around foramen magnum Transverse sinus Confluence of the sinuses Superior sagittal sinus Parietal emissary v. Sigmoid sinus Internal jugular v. Mastoid emissary v. Condylar emissary v. Occipital v. Occipital emissary v. External occipital protruberance 41 Neurovasculature of the Skull & Face 589 Fig. 41.10 Deep veins of the head Left lateral view. Removed: Upper ramus, condylar and coronoid pro-cesses of mandible. The pterygoid plexus is a venous network situated between the mandibular ramus and the muscles of mastication. The cav-ernous sinus connects branches of the facial vein to the sigmoid sinuses. Fig. 41.11 Veins of the occiput Posterior view. The superficial veins of the occiput communicate with the dural venous sinuses via emissary veins that drain to diploic veins (calvaria, p. 545). Note: The external vertebral venous plexus traverses the entire length of the spine (p. 45). Table 41.5 Venous anastomoses The extensive venous anastomoses in this region provide routes for the spread of infections. Extracranial vein Connecting vein Venous sinus Angular v. Superior and inferior ophthalmic vv. Cavernous sinus Vv. of palatine tonsil Pterygoid plexus; inferior ophthalmic v. Superficial temporal v. Parietal emissary vv. Superior sagittal sinus Occipital v. Occipital emissary v. Transverse sinus, confluence of the sinuses Posterior auricular v. Mastoid emissary v. Sigmoid sinus External vertebral venous plexus Condylar emissary v. Deep spread of bacterial infection from the facial region may result in cavernous sinus thrombosis. Arachnoid mater Superior sagittal sinus Dura mater (cut) Outer table Cranial bone Diploë Inner table Arachnoid granulations (arachnoid villi) Lateral lacuna (opened) Pia mater (on cerebral surface) Middle cerebral a. (branches) Superior cerebral vv. Bridging vv. Confluence of the sinuses Head & Neck 590 Meninges The brain and spinal cord are covered by membranes called meninges. The meninges are composed of three layers: dura mater (dura), arach-noid mater (arachnoid membrane), and pia mater. The subarachnoid space, located between the arachnoid mater and pia mater, contains cerebrospinal fluid (CSF, see p. 684). See p. 40 for the coverings of the spinal cord. Fig. 41.12 Layers of the meninges See pp. 686–687 for the veins of the brain. Fig. 41.13 Dural folds (septa) Left anterior oblique view. Two layers of meningeal dura come together, after separating from the periosteal dura during formation of a dural (venous) sinus, to form a dural fold or septum. These include the falx cerebri (separating right and left cerebral hemispheres); the tentorium cerebelli (supporting the cerebrum to keep it from crushing the un-derlying cerebellum); the falx cerebelli (not shown, separat-ing right and left cerebellar lobes under the tentorium); and the diaphragma sellae (forming the roof over the hypophy-seal fossa and invaginated by the hypophysis). Falx cerebri Diaphragma sellae Optic n. Crista galli Internal carotid a. Tentorial notch Tentorium cerebelli Ostia of bridging vv. Epidural hematoma Neurothelium Arachnoid Arachnoid trabeculae Cerebral cortex Pia mater Cerebral a. Subarachnoid space Cerebral v. Dura mater Cranial bone Subdural hemorrhage Diploic vv. A Coronal section through the meninges, anterior view. B Superior view of opened cranium. Left side: Dura mater (outer layer) cut to reveal arachnoid (middle layer). Right side: Dura mater and arachnoid removed to reveal pia mater (inner layer) lining the surface of the brain. Note: Arachnoid granulations, sites for reabsorption of cerebrospinal fluid into the venous blood, are protrusions of the arachnoid layer of the meninges into the venous sinus system. CN V1, V2, and V3 (meningeal brs.) CN V1 and V2 (tentorial brs.) Tentorium cerebelli CN X (meningeal brs.) CN V3 (meningeal br.) Ist and 2nd cervical nn. (meningeal brs.) Anterior and posterior ethmoidal nn. (meningeal brs.) Cribriform plate 41 Neurovasculature of the Skull & Face 591 Subarachnoid space Dura mater Ruptured middle meningeal a. Calvaria Bridging v. Inferior sagittal sinus Superior sagittal sinus Falx cerebri Sphenoid sinus Ruptured aneurysm Clinical box 41.3 Bleeding between the bony calvarium and the soft tissue of the brain (extracerebral hemorrhage) exerts pressure on the brain. A rise of intracranial pressure may damage brain tissue both at the bleeding site and in more remote brain areas. Three types of intracranial hemorrhage are distinguished based on the relationship to the dura mater. See pp. 688–689 for the arteries and pp. 686–687 for the veins of the brain. Extracerebral hemorrhages A Epidural hematoma (above the dura). B Subdural hematoma (below the dura). C Subarachnoid hemorrhage. Middle meningeal a. (frontal br.) Middle meningeal a. (via foramen spinosum) Vertebral a. (brs.) Occipital a. (mastoid br.) Middle meningeal a. (parietal br.) Fig. 41.14 Arteries of the dura mater Midsagittal section, left lateral view. See pp. 688–689 for the arteries of the brain. Fig. 41.15 Innervation of the dura mater Superior view. Removed: Tentorium cerebelli (right side). Inferior petrosal sinus Confluence of the sinuses Great cerebral v. Superior petrosal sinus Tentorium cerebelli Straight sinus Occipital sinus Anterior intercavernous sinus ① ③ ⑫ ⑨ ⑧ ⑩ ⑦ ⑪ ⑥ ④ ⑤ Venous plexus of foramen ovale Middle meningeal v. Basilar plexus Sigmoid sinus Inferior cerebral vv. Transverse sinus Superior sagittal sinus Marginal sinus Jugular foramen Petrosquamous sinus Posterior intercavernous sinus Cavernous sinus Sphenoparietal sinus Superior ophthalmic v. Galea aponeurotica Periosteal layer Scalp Outer table Diploë Inner table Lateral lacuna with arachnoid granulations Falx cerebri Bridging v. Superior cerebral vv. Scalp vv. Granular foveola Emissary v. Superior sagittal sinus Dura mater Meningeal layer Pia mater (on cerebral surface) Superior cerebral vv. Middle cerebral a. (brs.) Bridging vv. Dura mater Outer table Diploë Inner table Lateral lacuna (closed) Superior sagittal sinus Lateral lacuna (open) Arachnoid granulations (arachnoid villi) Confluence of the sinuses Cranial bone Falx cerebri Tentorium cerebelli G S D F A Ö K J L H Ä Y Table 41.6 Principal dural sinuses Upper group Lower group ① Superior sagittal sinus ⑦ Cavernous sinus ② Inferior sagittal sinus ⑧ Anterior inter-cavernous sinus ③ Straight sinus ⑨ Posterior inter-cavernous sinus ④ Confluence of the sinuses ⑩ Sphenoparietal sinus ⑤ Transverse sinus ⑪ Superior petrosal sinus ⑥ Sigmoid sinus ⑫ Inferior petrosal sinus The occipital sinus is also included in the upper group (see Fig. 49.1, p. 686). Head & Neck 592 Dural Sinuses The dura mater is composed of two layers that separate in the region of a venous sinus into an outer periosteal layer, which lines the calvaria and an inner meningeal layer, which forms the unattached boundaries of the sinus. In the region of a sinus, the two meningeal dural layers come together after forming the sinus to create a dural fold, or septa (see Fig. 41.13, p. 590). The network of venous sinuses collect blood from the scalp, the calvaria, and the brain and eventually drain into the internal jugular vein at the jugular foramen. Fig. 41.16 Formation of a dural sinus Fig. 41.17 Dural sinuses in the cranial cavity Superior view of opened cranial cavity. Dural sinus system ghosted in blue. Removed: Tentorium cerebelli (right side). A Structure of a dural sinus. Superior sagittal sinus, coronal section, anterior view. B Superior sagittal sinus in situ. Superior view of opened cranial cavity. The roof of the sinus (the periosteal layer of the dura attached to the calvaria) is removed. Left side: Areas of dura mater removed to show arachnoid granulations (protrusions of the arachnoid layer of the meninges) in the sinus. Right side: Dura mater and arachnoid layers removed to reveal pia mater adhering to the cerebral cortex. Ophthalmic a. Internal carotid a. Oculomotor n. (CN III) Abducent n. (CN VI) Trochlear n. (CN IV) Sensory root Motor root Trigeminal ganglion Cavernous sinus Internal carotid a. Optic chiasm (optic n., CN II) Middle cranial fossa Trigeminal nerve (CN V) A Superior view of the right anterior and middle cranial fossae. Removed: Lateral dural wall and roof of the cavern-ous sinus. The trigeminal ganglion is cut and retracted laterally following removal of its dural covering Ophthalamic a. Internal carotid a. Carotid siphon Ophthalmic n. (CN V1) Trochlear n. (CN IV) Oculomotor n. (CN III) Maxillary n. (CN V2) Trigeminal ganglion Anterior clinoid process Posterior clinoid process Hypophyseal fossa Clivus Dorello's canal Gruber's lig. Abducent n. (CN VI) Trigeminal n. (CN V) B Topography of the extradural course of the abducent nerve along the clivus and in the left cavernous sinus. Left lateral view. Note the long extradural path the abducent nerve follows along the clivus. It runs within the subarachnoid space, pierces the dura mater, passes under Gruber’s ligament through Dorello’s canal and enters the cavernous sinus at the tip of the petrous temporal bone (at the junction of the middle and posterior cranial fossae). It courses through the cavernous sinus lateral to the internal carotid artery to reach the orbit through the superior orbital fissure. 41 Neurovasculature of the Skull & Face 593 Fig. 41.18 Cavernous sinus and cranial nerves Fig. 41.19 Cavernous sinus, coronal section through middle cranial fossa Anterior view. The right and left cavernous sinuses connect via the intercavernous sinuses that pass around the hypophysis, which sits in the hypophyseal fossa after invaginating the diaphragma sellae. On each side, this coronal section cuts through the internal carotid artery twice due to the presence of the carotid siphon, a 180 degree bend in the cavernous part of the artery. Of the five cranial nerves, or their divisions, associated with the sinus only the abducent nerve (CN VI) is not embedded in the lateral dural wall. Optic n. Hypophysis Sphenoid sinus Internal carotid a. Oculomotor n. (CN III) Trochlear n. (CN IV) Abducent n. (CN VI) Ophthalmic n. (CN V1) Cavernous sinus Maxillary n. (CN V2) Facial n., marginal man-dibular br. Facial n., buccal brs. Facial n., zygomatic brs. Facial n., temporal brs. Mental n. (in mental foramen) Inferior alveolar a., mental br. Facial a. and v. Masseter Parotid duct Parotid gland Transverse facial a. Infraorbital a. and n. (in infraorbital foramen) Superficial temporal a. and v. Auriculotemporal n. Angular a. and v. Dorsal nasal a. Supraorbital n., medial and lateral brs. Supratrochlear n. Superficial temporal a. and v., auriculotemporal n. Zygomaticus major Inferior labial a. Superior labial a. Head & Neck 594 Topography of the Superficial Face Fig. 41.20 Superficial neurovasculature of the face Anterior view. Removed: Skin and fatty subcutaneous tissue; muscles of facial expression (left side). Zygomatico-orbital a. Supraorbital n. (CN V1) Supratrochlear n. (CN V1) Angular v. Infratrochlear n. (CN V1) Transverse facial a. Infraorbital n. (CN V2) Facial v. Mental n. (CN V3) Parotid duct Buccinator Masseter Brs. of parotid plexus of facial n. External jugular v. Retromandibular v., posterior division Great auricular n. (from cervical plexus [C2–C3]) Sternocleido-mastoid Parotid gland Lesser occipital n. (from cervical plexus [C2]) Posterior auricular v. Greater occipital n. (C2, posterior ramus) Occipital a. Superficial temporal a. and v. Auriculotemporal n. (CN V3) Superficial temporal a., parietal br. Superficial temporal a., frontal br. External nasal n. (CN V1) 41 Neurovasculature of the Skull & Face 595 Fig. 41.21 Superficial neurovasculature of the head Left lateral view. Supratrochlear n. (CN V1) Infratrochlear n. (CN V1) Infraorbital n. (CN V2) Mental n. (CN V3) Parotid duct Temporal brs. of parotid plexus (CN VII) Zygomatic brs. of parotid plexus (CN VII) Buccal brs. of parotid plexus (CN VII) Marginal mandibular br. of parotid plexus (CN VII) Cervical br. of parotid plexus (CN VII) Masseter Intraparotid plexus of the facial n. (CN VII) N. to digastric, posterior belly (CN VII) Posterior auricular n. (CN VII) Sternocleidomastoid Lesser occipital n. (cervical plexus [C2]) Greater occipital n. (posterior ramus of C2) Occipital a. Auriculotemporal n. (CN V3) Superficial temporal a., frontal and parietal brs. Inferior (cervicofacial) trunk External jugular v. Great auricular n. (cervical plexus [C2–C3]) N. to stylohyoid (CN VII) Superior (temporofacial) trunk Supraorbital n. (CN V1) External nasal n. (CN V1) Head & Neck 596 Topography of the Parotid Region & Temporal Fossa Fig. 41.22 Parotid region Left lateral view. Removed: Parotid gland, sternocleidomastoid, and veins of the head. Revealed: Parotid bed and carotid triangle. Parotid duct (cut) Zygomatic arch Temporalis Masseter Superior cervical ganglion Hypoglossal n. Facial n. Coronoid process Submandibular gland, superficial part Temporomandibular joint capsule 41 Neurovasculature of the Skull & Face 597 Fig. 41.24 Temporal fossa Left lateral view. Removed: Sternocleidomas-toid and masseter. Revealed: Temporal fossa and temporomandibular joint (p. 638). Fig. 41.23 Temporal fossa Left lateral view. The temporal fossa is located on the lateral aspect of the skull. It communicates with the infratemporal fossa inferiorly (medial to the zygomatic arch). The main component of the fossa is the large temporalis muscle. Temporal fossa (shaded) Frontal bone, zygomatic process Zygomatic bone, frontal process Zygomatic bone Coronoid process (in temporal fossa) Zygomatic arch (cut) Inferior temporal line Superior temporal line Supramastoid crest Infratemporal fossa (deep to ramus of mandible) Head & Neck 598 Topography of the Infratemporal Fossa Fig. 41.26 Infratemporal fossa: Superficial dissection Left lateral view. Removed: Ramus of mandible. Note: The mylohyoid nerve (see Fig. 45.15 and 45.17A) branches from the inferior alveolar nerve just before the mandibular foramen. Fig. 41.25 Bony boundaries of Infratemoral fossa Oblique external view of base of the skull. Temporal bone, squamous part Mandibular fossa Articular eminence External acoustic meatus Foramen spinosum Foramen ovale Lateral pterygoid plate Medial pterygoid plate Occipital condyle Foramen magnum Maxilla, palatine process Pterygomaxillary fissure Palatine bar, maxillary process Temporal bone, zygomatic process Sphenopalatine foramen Inferior orbital fissure Infratemporal surface of maxilla Pterygoid hamulus Palatine bone, pyramidal process Maxillary tuberosity Deep temporal nn. Temporalis (cut) Maxillary a. Superior alveolar nn. posterior superior alveolar br. (CN V2) Medial pterygoid, superﬁcial and deep heads Lingual n. Masseter (cut) Buccal n. and a. Facial a. and v. Superﬁcial temporal a. and v. Deep temporal aa. Lateral pterygoid, superior and inferior heads Inferior alveolar n. Ramus of mandible (cut) Sternocleidomastoid Retromandibular v., posterior division Facial n. (CN VII) Auriculotemporal n. Posterior superior alveolar a. Mandibular canal 41 Neurovasculature of the Skull & Face 599 Fig. 41.27 Infratemporal fossa: Deep dissection Left lateral view. Removed: Lateral pterygoid muscle (both heads). Revealed: Deep infratemporal fossa and mandibular nerve as it enters the mandibular canal via the foramen ovale in the roof of the fossa. Deep temporal n. Lateral pterygoid (cut) Temporalis (cut) Sphenopalatine a. Posterior superior alveolar a. Buccinator Buccal a. and n. Facial a. and v. Lingual n. Medial pterygoid, deep head Inferior alveolar a. and n. Masseter Facial n. Mylohyoid n. Middle meningeal a. Superficial temporal a. and v. Auriculotemporal n. Medial pterygoid, superficial head Maxillary a. Mandibular n. (CN V3) Sphenomandibular lig. Head & Neck 600 Neurovasculature of the Infratemporal Fossa Mandibular division (CN V3) Deep temporal nn. Buccal n. Lateral pterygoid n. Lingual n. Inferior alveolar n. Medial pterygoid n. Masseteric n. Parotid brs. Auriculo-temporal n. Meningeal br. N. of tensor tympani (with muscle) Facial n. Stylomastoid foramen Auriculo-temporal n. Lesser petrosal n. Communicating br. to auriculo-temporal n. Chorda tympani Mylohyoid n. Lingual n. Otic ganglion N. of tensor veli palatini (with muscle) Mandibular division (CN V3) Foramen ovale Inferior alveolar n. Mylohyoid n. Medial pterygoid n. Fig. 41.28 Mandibular nerve (CN V3) in the infratemporal fossa A Left lateral view. B Left medial view. Table 41.7 Nerves of the infratemporal fossa Nerve Nerve Fibers Distribution Muscular Branches (CN V3) Branchial motor Muscles of mastication; mylohyoid; tensor tympani; tensor veli palatini, anterior belly of digastric Auriculotemzporal (CN V3) General sensory Auricle, temporal region, and temporomandibular joint Visceral motor from glossopharyngeal n. (CN IX) Parotid gland Inferior alveolar (CN V3) General sensory Mandibular teeth; mental branch supplies skin of lower lip and chin Lingual (CN V3) General sensory Anterior two thirds of tongue, floor of mouth Buccal (CN V3) General sensory Skin and mucous membrane of cheek Meningeal (CN V3) General sensory Dura of middle cranial fossa Chorda tympani (CN VII) Special sensory taste Anterior two thirds of tongue Visceral motor Submandibular and sublingual glands via submandibular ganglion and lingual n (CN V3) Zygomatic process (cut) Greater palatine a. Lesser palatine a. Pterygoid process, lateral plate Maxillary a. Inferior orbital fissure ⑪ ⑩ ① ③ ④ ⑤ ⑥ ⑦ ② ⑨ ⑫ ⑬ ⑧ Pterygomaxillary fissure 41 Neurovasculature of the Skull & Face 601 Fig. 41.29 Arteries in the infratemporal fossa Left lateral view into area. The maxillary artery passes either superficial or deep to the lateral pterygoid in the infratemporal fossa (see Fig. 41.27, p. 599) and passes medially into the pterygopalatine fossa through the pterygo maxillary fissure. Table 41.8 Branches of the maxillary artery Part Artery Distribution Mandibular part (between the origin and the first circle around artery in Fig. 41.29) ① Inferior alveolar a. Mandible, teeth, gingiva ② Anterior tympanic a. Tympanic cavity ③ Deep auricular a. Temporomandibular joint, external auditory canal ④ Middle meningeal a. Calvaria, dura, anterior and middle cranial fossae Pterygoid part (between the first and second circles around the artery) ⑤ Masseteric a. Masseter m. ⑥ Deep temporal aa. Temporalis m. ⑦ Pterygoid brs. Pterygoid mm. ⑧ Buccal a. Buccal mucosa Pterygopalatine part (between the second and third circles around the artery) ⑨ Descending palatine a. Greater palatine a. Hard palate Lesser palatine a. Soft palate, palatine tonsil, pharyngeal wall ⑩ Posterior superior alveolar a. Maxillary molars, maxillary sinus, gingiva ⑪ Infraorbital a. Maxillary alveoli ⑫ A. of pterygoid canal ⑬ Sphenopalatine a. Lateral posterior nasal aa. Lateral wall of nasal cavity, choanae Posterior septal brs. Nasal septum Inferior orbital fissure Infraorbital groove Superior orbital fissure Ethmoid bone, orbital plate Infraorbital foramen Maxilla, orbital surface Lacrimal bone Maxilla, frontal process Nasal bone Anterior ethmoidal foramen Posterior ethmoidal foramen Optic canal (sphenoid bone) Frontal incisure Supraorbital foramen Frontal bone, orbital surface Zygomatico-orbital foramen Zygomatic bone Maxilla, frontal process Maxilla, anterior lacrimal crest Fossa of lacrimal sac (with opening for nasolacrimal duct) Infraorbital foramen Infraorbital canal Maxillary sinus Maxilla, orbital surface Maxillary hiatus Inferior orbital fissure Pterygopalatine fossa Foramen rotundum Superior orbital fissure Sphenoid, optic canal Anterior and posterior ethmoidal foramina Frontal bone, orbital surface Ethmoid bone Lacrimal bone Lacrimal bone, posterior lacrimal crest Head & Neck 602 42 Orbit & Eye Bones of the Orbit Fig. 42.1 Bones of the orbit A Anterior view. B Lateral view of right orbit. Table 42.1 Openings in the orbit for neurovascular structures Opening Nerves Vessels Optic canal Optic n. (CN II) Ophthalmic a. Superior orbital fissure Oculomotor n. (CN III) Trochlear n. (CN IV) Abducent n. (CN VI) Trigeminal n., ophthalmic division (CN V1) • Lacrimal n. • Frontal n. • Nasociliary n. Superior ophthalmic v. Inferior orbital fissure Infraorbital n. (CN V2) Zygomatic n. (CN V2) Infraorbital a. and v., inferior ophthalmic v. Infraorbital canal Infraorbital n. (CN V2), a., and v. Supraorbital foramen Supraorbital n. (lateral br.) Supraorbital a. Frontal incisure Supraorbital n. (medial br.) Supratrochlear a. Anterior ethmoidal foramen Anterior ethmoidal n., a., and v. Posterior ethmoidal foramen Posterior ethmoidal n., a., and v. The nasolacrimal canal transmits the nasolacrimal duct. Inferior orbital fissure Sphenoid bone, lesser wing Superior orbital fissure Sphenoid bone, greater wing Palatine bone, pyramidal process Infraorbital canal Maxilla, orbital surface Zygomatico-orbital foramen Zygomatic bone, orbital surface Frontal bone, orbital surface Frontal sinus Maxillary sinus Ethmoid bone, perpendicular plate Ethmoid bone, superior nasal concha Ethmoid bone, middle nasal concha Inferior nasal concha Maxilla, palatine process Superior orbital fissure Maxillary sinus Infraorbital canal Inferior orbital fissure Zygomatic bone, orbital surface Sphenoid bone, greater wing Sphenoid bone, lesser wing Frontal bone, orbital surface Optic canal Ethmoid bone, orbital plate (lamina papyracea) Frontal sinus Ethmoid bone, crista galli Ethmoid bone Vomer Orbital floor 42 Orbit & Eye 603 C Medial view of right orbit. D Coronal section, anterior view. Table 42.2 Structures surrounding the orbit Direction Bordering structure Superior Frontal sinus Anterior cranial fossa Medial Ethmoid sinus Inferior Maxillary sinus Certain deeper structures also have a clinically important relationship to the orbit: Sphenoid sinus Hypophysis (pituitary) Middle cranial fossa Cavernous sinus Optic chiasm Pterygopalatine fossa Lateral rectus Inferior rectus Inferior oblique Medial rectus Superior oblique Superior rectus Optic n. (CN II, in optic canal) Common tendinous ring Medial rectus Inferior rectus Superior rectus Lateral rectus Superior oblique Tendon of superior oblique Trochlea Inferior oblique (origin) Levator palpebrae superioris Inferior oblique (insertion) Head & Neck 604 Muscles of the Orbit Fig. 42.2 Extraocular muscles The eyeball is moved by six extrinsic muscles: four rectus (superior, inferior, medial, and lateral) and two oblique (superior and inferior). A Right eye, anterior view. B Right eye, superior view of opened orbit. A Starting with the eyes directed anteriorly, movement to any of the cardinal direc-tions of gaze (arrows) requires activation of two extraocular muscles, each of which is innervated by a different cranial nerve, thus testing the function of those pairs of muscles. B Starting with the eyes adducted or abducted, elevating or lowering the eyes activates only the oblique or the rectus muscles, respectively, allowing for testing of the function of individual muscles. Fig. 42.3 Testing the extraocular muscles Inferior oblique Inferior oblique Superior rectus Lateral rectus Medial rectus Superior oblique Inferior rectus Adduct Abduct Depress Elevate Abduct Lateral rectus Superior oblique Inferior oblique Superior rectus Inferior rectus Adduct Abduct Superior oblique 42 Orbit & Eye 605 A Superior rectus. B Medial rectus. C Inferior rectus. D Lateral rectus. E Superior oblique. F Inferior oblique. Table 42.3 Extraocular muscles Action (see Fig. 42.4) Muscle Origin Insertion Vertical axis (red) Horizontal axis (black) Anteroposterior axis (blue) Innervation Superior rectus Common tendinous ring (common annular tendon) Sclera of the eye Elevates Adducts Rotates medially Oculomotor n. (CN III), superior branch Medial rectus — Adducts — Oculomotor n. (CN III), inferior branch Inferior rectus Depresses Adducts Rotates laterally Lateral rectus — Abducts — Abducent n. (CN VI) Superior oblique Sphenoid bone+ Depresses Abducts Rotates medially Trochlear n. (CN IV) Inferior oblique Medial orbital margin Elevates Abducts Rotates laterally Oculomotor n. (CN III), inferior branch Starting from gaze directed anteriorly + The tendon of the superior oblique passes through a tendinous loop (trochlea) attached to the superomedial orbital margin. Fig. 42.4 Actions of the extraocular muscles Superior view of opened orbit. Vertical axis, red circle; horizontal axis, black; anteroposterior (visual/optical) axis, blue. Clinical box 42.1 Oculomotor palsies may result from a lesion involving an eye muscle or its associated cranial nerve (at the nucleus or along the course of the nerve). If one extraocular muscle is weak or paralyzed, deviation of the eye will be noted. Impairment of the coordinated actions of the extraocular muscles may cause the visual axis of one eye to deviate from its normal position. The patient will therefore perceive a double image (diplopia). Oculomotor palsies Orbital axes Visual (optical) axis Lateral rectus Superior rectus 23° D Normal visual and orbital axes. A Abducent nerve palsy. Disabled: Lateral rectus. B Trochlear nerve palsy. Disabled: Superior oblique. C Complete oculomotor palsy. Disabled: Superior, inferior, and medial recti and inferior oblique. Lacrimal a. Anterior ethmoidal a. Posterior ethmoidal a. Optic n. (CN II) Ophthalmic a. Internal carotid a. (in cavernous sinus) Middle meningeal a. (from maxillary a.) Anastomotic br. Central retinal a. Long posterior ciliary aa. Short posterior ciliary aa. Supraorbital a. Medial palpebral a. Supra-trochlear a. Dorsal nasal a. Lacrimal v. Ophthalmic v. Facial v. Infraorbital v. Inferior ophthalmic v. Angular v. Supra-trochlear v. Dorsal nasal v. Superior ophthalmic v. Cavernous sinus Danger triangle Head & Neck 606 Neurovasculature of the Orbit Fig. 42.6 Arteries of the orbit Superior view of the right orbit. Opened: Optic canal and orbital roof. Fig. 42.5 Veins of the orbit Lateral view of the right orbit. Removed: Lateral orbital wall. Opened: Maxillary sinus. Clinical box 42.2 Gravity allows venous blood from the danger triangle region of the face (see figure) to drain to the cavernous sinus via the valveless ophthalmic veins. Squeezing a pimple or boil in this facial region can result in infectious thrombi being forced into the venous system and passing back into the cavernous sinus. Cavernous sinus syndrome (CIS) is diagnosed by the loss of eyeball movement due to the various cranial nerves associated with the cavernous sinus becoming infected. The abducent nerve (CN VI) is bathed in blood within the sinus, the first ocular movement to be affected is lateral deviation of the eyeball. The oculomotor (CN III) and trochlear (CN IV) nerves, embedded in the dural lateral wall of the sinus are also eventually affected as the infection penetrates the dura. The eyeball becomes frozen in the orbit as all nerves activating the extraocular mm. become infected. CN V1 is also in the lateral dural wall so a tingling/parasthesia is felt in the sensory region covered (forehead). Occasionally CN V2 may also be involved and this parasthesia may also extend to the skin of the face below the orbit. The intercavernous sinuses allow the infection to spread to the cavernous sinus on the opposite side. If left untreated, death can result however cavernous sinus septic thrombophlebitis mortality has decreased from 100% to 20% with the of improvements in diagnosis and treatment. Cavernous sinus syndrome Naso-ciliary n. Oculomotor n., superior br. Optic n. (CN II) Ophthalmic division (CN V1) Internal carotid a. with internal carotid plexus Oculomotor n. (CN III) Trochlear n. (CN IV) Trigeminal n. (CN V) Abducent n. (CN VI) Trigeminal ganglion Mandibular division (CN V3) Maxillary division (CN V2) Oculomotor n., inferior br. Nasociliary (sensory) root Sympathetic root Parasym-pathetic root Ciliary ganglion Short ciliary nn. Long ciliary nn. Supratroch-lear n. Supraorbital n. Lacrimal n. (with gland) Frontal n. 42 Orbit & Eye 607 Fig. 42.7 Innervation of the orbit Lateral view of the right orbit. Removed: Temporal bony wall. Fig. 42.8 A course of the cranial nerves through the cavernous sinus toward the orbit Sella turcica with partially opened cavernous sinus on the right side, cranial view. The trigeminal ganglia are displayed on both sides. The right ganglion is pulled laterally from its normal position (thereby exposing the trigemi-nal cave = Meckel’s cave) to show the cavernous sinus and the internal carotid artery, which passes through the sinus. Note the abducent nerve also traverses the cavernous sinus and runs lateral to the carotid artery. All other nerves (oculomotor, trochlear and the three branches of the trigeminal) run rostrally and caudally in the lateral dural wall of the cavernous sinus. Most cases of intracavernous carotid aneurysm only involve the abducent nerve. The space-occupying aneurysm compresses the nerve, causing a loss of function. In cases with sudden onset of isolated abducent nerve palsy, carotid aneurysm should always be considered as a possible cause. In contrast, isolated trochlear nerve palsy is rare. More often, the trochlear nerve is one of multiple nerves affected, e.g. in cases of cavernous sinus thrombosis which involves all nerves traveling through the cavernous sinus, often affecting also both branches of the trigeminal nerve. Cavernous sinus Optic n. (CN I) Internal carotid a. Oculomotor n. (CN III) Trochlear n. (CN IV) Trigeminal n. (minor portion) Trigeminal n. (major portion) Facial n. (CN VII) Abducent n. (CN VI) Clivus Trigeminal n. (CN V) Trigeminal ganglion Trigeminal cave Mandibular n. (CN V3) Maxillary n. (CN V2) Trochlear n. (CN IV) Oculomotor n. (CN V1) Opthalmic n. (CN V1) Superior ophthalmic a. and v. Dorsal nasal a. and v. Supraorbital a. and n. Orbital septum Infraorbital a. and n. Facial a. and v. Angular a. and v. Lacrimal sac Infra-trochlear n. Supratrochlear n. Levator palpebrae superioris Superior tarsal m. Lacrimal gland, orbital part Lacrimal gland, palpebral part Lateral palpebral lig. Superior and inferior tarsus Medial palpebral lig. Superior ophthalmic v. Oculomotor n. (CN III), superior br. Lateral rectus Nasociliary n. Inferior orbital fissure Abducent n. (CN VI) Inferior ophthalmic v. Oculomotor n. (CN III), inferior br. Superior orbital fissure Ophthalmic a. Common tendinous ring Superior oblique Optic n. (CN II, in optic canal) Levator palpebrae superioris Trochlear n. (CN IV) Frontal n. Lacrimal n. Superior rectus Inferior rectus Medial rectus Head & Neck 608 Topography of the Orbit Fig. 42.9 Neurovascular structures of the orbit Anterior view. Right side: Orbicularis oculi removed. Left side: Orbital septum partially removed. Fig. 42.10 Passage of neurovascular structures through the orbit Anterior view. Removed: Orbital contents. Note: The optic nerve and ophthalmic artery travel in the optic canal. The remaining structures pass through the superior orbital fissure. Abducent n. (CN VI) Levator palpebrae superioris Frontal n. Lacrimal a. and n. (with gland) Inferior ophthalmic v. Cribriform plate Supratrochlear a. and n. Supraorbital a. Trochlear n. (CN IV) Superior ophthalmic v. Superior rectus Infratrochlear n. Nasociliary n. Posterior ethmoidal a. and n. Anterior ethmoidal a. and n. Supraorbital aa. and nn. Lateral rectus Levator palpebrae superioris Superior rectus Ciliary ganglion Superior oblique Lacrimal a. and n. Optic n. (CN II) Superior ophthalmic v. Nasociliary n. Medial rectus Inferior ophthalmic v. Long ciliary nn. Lateral rectus Short posterior ciliary aa., short ciliary nn. Nasociliary n. Abducent n. (CN VI) Trochlear n. (CN IV) Oculomotor n. (CN III) Lacrimal gland 42 Orbit & Eye 609 Fig. 42.11 Neurovascular contents of the orbit Superior view. Removed: Bony roof of orbit, peritorbita, and retro-orbital fat. A Upper level. B Middle level. Reflected: Levator palpebrae superioris and superior rectus. Revealed: Optic nerve. Eyeball Orbital septum Infraorbital n. Inferior oblique Maxillary sinus Orbital floor Inferior rectus Optic n. (with dural sheath) Superior rectus Bulbar fascia (Tenon’s capsule) Episcleral space Adipose tissue of the orbit Levator palpebrae superioris Orbital roof Periorbita Sclera Orbital septum Iris Orbital roof Orbital septum Orbicularis oculi, orbital part Upper eyelid Superior tarsus (with tarsal glands) Ciliary and sebaceous glands Inferior tarsus Lower eyelid Orbicularis oculi, palpebral part Inferior tarsal m. Infraorbital n. Sclera Retina Ciliary body Lens Cornea Superior tarsal m. Superior rectus Superior conjunctival fornix Levator palpebrae superioris Periorbita Head & Neck 610 Orbit & Eyelid Fig. 42.12 Topography of the orbit Sagittal section through the right orbit, medial view. Fig. 42.13 Eyelids and conjuctiva Sagittal section through the anterior orbital cavity. Levator palpebrae superioris Orbital septum Lacrimal gland, orbital part Lacrimal gland, palpebral part Lower eyelid Upper eyelid Infraorbital foramen Superior and inferior puncta Inferior nasal concha Nasolacrimal duct Lacrimal sac Medial palpebral lig. Superior and inferior lacrimal canaliculi Lacrimal caruncle 42 Orbit & Eye 611 Fig. 42.14 Lacrimal apparatus Right eye, anterior view. Removed: Orbital septum (partial). Divided: Levator palpebrae superioris (tendon of insertion). Perimenopausal women are frequently subject to chronically dry eyes (keratoconjunctivitis sicca), due to insufficient tear production by the lacrimal gland. Acute inflammation of the lacrimal gland (due to bacteria) is less common and characterized by intense inflammation and extreme tenderness to palpation. The upper eyelid shows a characteristic S-curve. Clinical box 42.3 Lacrimal drainage Lens Iris Scleral venous sinus (canal of Schlemm) Pigment epithelium of the ciliary body Posterior chamber Medial rectus Optic disk Lamina cribrosa Central retinal a. Optic n. (CN II) Fovea centralis Lateral rectus Vitreous body Ora serrata Zonular fibers Ciliary body, ciliary m. Corneoscleral limbus Chamber angle Cornea Anterior chamber Sclera Choroid Retina Ocular conjunctiva Hyaloid fossa Macula lutea Optical axis Orbital axis 23° Head & Neck 612 Eyeball Fig. 42.15 Structure of the eyeball Transverse section through right eyeball, superior view. Note: The orbital axis (running along the optic nerve through the optic disk) deviates from the optical axis (running through the center of the eye to the fovea centralis) by 23 degrees. Temporal Nasal Physiological cup Central retinal a. and v. (sites of entry and emergence) Optic disk (blind spot) Fovea centralis Macula lutea (yellow spot) Optic disk Central retinal a. Central retinal v. Macula lutea Long posterior ciliary aa. Lesser arterial circle of iris Greater arterial circle of iris Anterior conjunctival a. Arterial circle of Zinn (and von Haller) Short posterior ciliary aa. Pial vascular plexus Vorticose v. Anterior ciliary aa. Scleral venous sinus (canal of Schlemm) Choroid (choroido-capillary layer) Optic n. (CN II) Central retinal a. and v. 42 Orbit & Eye 613 Fig. 42.16 Blood vessels of the eyeball Transverse section through the right eyeball at the level of the optic nerve, superior view. The arteries of the eye arise from the ophthal-mic artery, a terminal branch of the internal carotid artery. Blood is drained by four to eight vorticose veins that open into the superior and inferior ophthalmic veins. Clinical box 42.4 A Retina of left eyeball, anterior view, schematic. B Normal optic fundus in the ophthalmoscopic examination. C High intracranial pressure; the edges of the optic disk appear less sharp. Optic fundus The optic fundus is the only place in the body where capillaries can be examined directly. Examination of the optic fundus permits observation of vascular changes that may be caused by high blood pressure or diabetes. Examination of the optic disk is important in determining intracranial pressure and diagnosing multiple sclerosis. Cornea Pupillary sphincter Iris stroma Pigmented iris epithelium (two layers) Greater arterial circle of iris Pupillary dilator Lesser arterial circle of iris Scleral venous sinus (canal of Schlemm) Conjunctiva Episcleral vv. Sclera Ciliary body Chamber angle Posterior chamber Iris Anterior chamber Cornea Trabecular meshwork Cornea Posterior chamber Scleral venous sinus (canal of Schlemm) Ciliary body Chamber angle Pupil Lens Zonular fibers Ciliary m. Sclera Ocular conjunctiva Pupillary dilator Pupillary sphincter Iris Anterior chamber Head & Neck 614 Cornea, Iris & Lens Fig. 42.17 Cornea, iris, and lens Transverse section through the anterior segment of the eye. Anterosuperior view. Fig. 42.18 Iris Transverse section through the anterior segment of the eye. Anterosuperior view. Clinical box 42.5 Aqueous humor produced in the posterior chamber passes through the pupil into the anterior chamber. It seeps through the spaces of the trabecular meshwork into the scleral venous sinus (canal of Schlemm) before passing into the episcleral veins. Obstruction of aqueous humor drainage causes an increase in intraocular pressure (glaucoma), which constricts the optic nerve in the lamina cribrosa. This constriction eventually leads to blindness. The most common glaucoma (approximately 90% of cases) is chronic (open-angle) glaucoma. The more rare acute glaucoma is characterized by red eye, strong headache and/or eye pain, nausea, dilated episcleral veins, and edema of the cornea. Glaucoma A Normal drainage. C Acute (angle-closure) glau-coma. The chamber angle is obstructed by iris tissue. Aque-ous fluid cannot drain into the anterior chamber, which pushes portions of the iris upward, blocking the chamber angle. B Chronic (open-angle) glaucoma. Drainage through the trabecular meshwork is impaired. Lens Fovea centralis Far vision Near vision Incident light rays Retina Lens Incident light rays Nearsightedness (myopia) Farsightedness (hyperopia) Normal vision Fovea centralis Iris Choroid Sclera Zonular fibers Ciliary processes Ciliary m. Ciliary body, pars plana Ciliary body, pars plicata Ora serrata Retina, optical part Lens 42 Orbit & Eye 615 A Normal pupil size. B Maximum constriction (miosis). C Maximum dilation (mydriasis). A Normal dynamics of the lens. B Abnormal lens dynamics. Fig. 42.19 Pupil Pupil size is regulated by two intraocular muscles of the iris: the pupillary sphincter, which narrows the pupil (parasympathetic innervation), and the pupillary dilator, which enlarges it (sympathetic innervation). Fig. 42.20 Lens and ciliary body Posterior view. The curvature of the lens is regulated by the muscle fibers of the annular ciliary body. Fig. 42.21 Light refraction by the lens Transverse section, superior view. In the normal (emmetropic) eye, light rays are refracted by the lens (and cornea) to a focal point on the retinal surface (fovea centralis). Tensing of the zonular fibers, with ciliary muscle relaxation, flattens the lens in response to parallel rays arriving from a distant source (far vision). Contraction of the ciliary muscle, with zonular fiber relaxation, causes the lens to assume a more rounded shape (near vision). Nasion Frontal process of maxilla Minor alar cartilages Major alar cartilage Lateral nasal cartilage Nasal bone Naris Anterior nasal spine Medial crus Lateral crus Nasal ala Septal cartilage Major alar cartilage Anterior cranial fossa Oral cavity Hypophyseal fossa Major alar cartilage, medial crus Posterior process Palatine bone, horizontal plate Vomer Sphenoid crest Sphenoid sinus Cribriform plate Ethmoid bone, perpendicular plate Crista galli Frontal sinus Nasal bone Septal cartilage Maxilla, palatine process Choana Nasal crest Incisive canal Frontal bone Head & Neck 616 43 Nasal Cavity & Nose Bones of the Nasal Cavity A Left lateral view. B Inferior view. A Left side of nasal septum in left nasal cavity. Parasagittal section. Fig. 43.1 Skeleton of the nose The skeleton of the nose is composed of an upper bony portion and a lower cartilaginous portion. The proximal portions of the nostrils (alae) are composed of connective tissue with small embedded pieces of cartilage. Fig. 43.2 Bones of the nasal cavity The left and right nasal cavities are flanked by lateral walls and separated by the nasal septum. Air enters the nasal cavity through the anterior nasal aperture and travels through three passages: the superior, middle, and inferior meatuses (arrows). These passages are separated by the superior, middle, and inferior conchae. Air leaves the nose through the choanae, entering the nasopharynx. Frontal sinus Crista galli Anterior cranial fossa Maxilla, palatine process Inferior concha Lacrimal bone Maxilla, frontal process Middle concha (ethmoid bone) Lateral plate Palatine bone, horizontal plate Sphenoid bone, lesser wing Hypophyseal fossa Medial plate Superior concha (ethmoid bone) Anterior nasal aperture Superior meatus Middle meatus Inferior meatus Choana Pterygoid process Middle cranial fossa Sphenoethmoidal recess Sphenoid sinus Cribriform plate Sphenoid sinus Sphenopalatine foramen Middle concha (cut) Maxilla, palatine process Inferior concha (cut) Palatine bone, perpendicular plate Superior concha (cut) Lacrimal bone Ethmoid bulla Orifices of posterior ethmoid sinus Uncinate process Maxillary hiatus Inferior meatus 43 Nasal Cavity & Nose 617 B Right lateral wall of the right nasal cavity. Sagittal section, medial view. Removed: Nasal septum. Note: The superior and middle conchae are parts of the ethmoid bone, whereas the inferior nasal conchae is a separate bone. C Right lateral wall of the right nasal cavity with the conchae removed. Sagittal section, medial view. Revealed: Paranasal sinuses (p. 618). Maxillary sinus Ethmoid sinus Frontal sinus Sphenoid sinus Age 60+ Age 20 Age 12 Age 8 Age 4 Age 1 Age 1 Age 4 Age 8 Age 12 Age 20 Inferior concha (cut) Superior concha (cut) Middle meatus Hiatus semilunaris Middle concha (cut) Inferior meatus Superior meatus Ethmoid bulla Spheno-ethmoidal recess Nasal septum Nasal cavity Orbit Maxillary sinus Inferior concha Middle concha Ethmoid sinus Frontal sinus Superior concha Head & Neck 618 Paranasal Air Sinuses Fig. 43.3 Location of the paranasal sinuses The paranasal sinuses (frontal, ethmoid, maxillary, and sphenoid) are air-filled cavities that reduce the weight of the skull. A Anterior view. B Left lateral view. C Pneumatization (the formation of air-filled cells and cavities) of the sinuses with age. The frontal (yellow) and maxillary (orange) sinuses develop gradually over the course of cranial growth. Fig. 43.4 Paranasal sinuses Arrows indicate the flow of mucosal secretions from the sinuses and the nasolacrimal duct into the nasal cavity (see Table 43.1). A Openings of the paranasal sinuses and nasolacrimal duct. Sagittal section, medial view of the right nasal cavity. B Paranasal sinuses and osteomeatal unit in the left nasal cavity. Coro-nal section, anterior view. Table 43.1 Nasal passages into which sinuses empty Sinuses/duct Nasal passage Via Sphenoid sinus (blue) Sphenoethmoidal recess Direct Ethmoid sinus (green) Posterior cells Superior meatus Direct Anterior and middle cells Middle meatus Ethmoid bulla Frontal sinus (yellow) Middle meatus Frontonasal duct into hiatus semilunaris Maxillary sinus (orange) Middle meatus Hiatus semilunaris Nasolacrimal duct (red) Inferior meatus Direct Superior orbital fissure (to middle cranial fossa) Ethmoid sinus Maxillary sinus Frontal sinus Anterior cranial fossa Parietal bone Temporal bone Vomer Inferior concha Zygomatic bone Sphenoid bone, greater wing Sphenoid bone, lesser wing Frontal bone Ethmoid bone Cribriform plate Palatine process of maxilla Vomer Ostium of maxillary sinus Maxillary sinus Uncinate process Middle meatus Superior meatus Orbit Frontal sinus Crista galli Orbital plate Superior concha Middle concha Inferior concha Inferior meatus Middle ethmoid sinus Perpendicular plate Ethmoid sinuses Deviated septum Maxillary sinus Inferior conchae 43 Nasal Cavity & Nose 619 Fig. 43.5 Bony structure of the paranasal sinuses Coronal section, anterior view. A Bones of the paranasal sinuses. B Ethmoid bone (red) in the paranasal sinuses. Clinical box 43.1 The normal position of the nasal septum creates two roughly symmetrical nasal cavities. Extreme lateral deviation of the septum may result in obstruction of the nasal passages. This may be corrected by removing portions of the cartilage (septoplasty). Sinusitis When the mucosa in the ethmoid sinuses becomes swollen due to inflammation (sinusitis), it blocks the flow of secretions from the frontal and maxillary sinuses in the osteomeatal unit (see Fig. 43.4). This may cause microorganisms to become trapped, causing secondary inflammations. In patients with chronic sinusitis, the narrow sites can be surgically widened to establish more effective drainage routes. Deviated septum C MRI through the paranasal sinuses. Anterior ethmoidal a. Anterior septal brs. (from anterior ethmoidal a.) Medial nasal br. Medial superior posterior nasal brs. (CN V2) Nasopalatine n. Posterior septal brs. (from spheno-palatine a.) Olfactory fibers (CN I) Olfactory bulb (CN I) Hypophyseal fossa Sphenoid sinus Frontal sinus Choana Torus tubarius Axis (C2) Pharyngeal orifice of pharyngotympanic (auditory) tube Anterior ethmoidal a. Anterior septal brs. Kiessel-bach’s area Posterior septal brs. Spheno-palatine a. Maxillary a. External carotid a. Internal carotid a. Ophthalmic a. Posterior ethmoidal a. Ophthalmic a. Posterior ethmoidal a. Anterior ethmoidal a. Lateral posterior nasal aa. Greater palatine a. Descending palatine a. Spheno-palatine a. Uvula Limen nasi Inferior meatus Nasal vestibule Inferior concha Superior meatus Salpingo-pharyngeal fold Pharyngeal tonsil Sphenoid sinus Sphenoethmoid recess Middle meatus Superior concha Middle concha Cribriform plate Head & Neck 620 Neurovasculature of the Nasal Cavity Fig. 43.6 Nasal septum and lateral wall A Nerves and arteries of the left side of the nasal septum. Fig. 43.7 Arteries of the nasal cavity Note: The venous drainage of the nasal cavity is into the anterior facial and ophthalmic veins. A Arteries of the left side of the nasal septum. B Arteries of the right lateral nasal wall. B Mucosa of the right lateral nasal wall. Sagittal section. Anterior ethmoidal a. Inferior posterior nasal br., lateral posterior nasal aa. Greater palatine n. and a. Lesser palatine a. and n. Pterygopalatine ganglion Trigeminal (CN V) ganglion Internal carotid a. Maxillary n. (V2) Olfactory fibers, posterior ethmoidal a. Olfactory bulb (CN I) Descending palatine a., greater and lesser palatine nn. Internal carotid plexus Greater petrosal n. Deep petrosal n. N. of the pterygoid canal Anterior ethmoidal n. (CN V1) Medial nasal brs. Medial superior posterior nasal brs. (CN V2) Nasopalatine n. (CN V2) Sphenopalatine foramen Pterygopalatine ganglion (in pterygopalatine fossa) CN V3 Trigeminal ganglion CN V1 CN V2 Olfactory bulb with fibers (CN I) Anterior ethmoidal n. (CN V1) External nasal br. Lateral nasal brs. Internal nasal brs. Inferior posterior nasal br. Greater palatine n. Lesser palatine nn. Pterygo-palatine ganglion Lateral superior posterior nasal brs. Olfactory fibers (CN I) A Lateral nasal wall with middle and inferior conchae removed to show anatomy of underlying meatuses. 43 Nasal Cavity & Nose 621 Ethmoid bulla Pharyngeal orifice of pharyngotympanic (auditory) tube Inferior concha (cut) Middle concha (cut) Inferior meatus Spheno-ethmoidal recess Sphenoid sinus Superior concha Frontal sinus Opening of frontonasal duct Semilunar hiatus Opening of nasolacrimal duct Nosebleeds Vascular supply to the nasal cavity arises from both the internal and external carotid arteries. The anterior part of the nasal septum contains a very vascularized region referred to as Kiesselbach’s area. This area is the most common site of significant nosebleeds. Clinical box 43.2 Fig. 43.8 Lateral nasal wall B Nerves and arteries of the right lateral nasal wall. Sagittal section. Removed: Sphenopalatine foramen. Fig. 43.9 Nerves of the nasal cavity Left lateral view. A Nerves of the left side of the nasal septum. B Nerves of the right lateral nasal wall. Head & Neck 622 Pterygopalatine Fossa The pterygopalatine fossa is a small pyramidal space just inferior to the apex of the orbit. It is continuous with the infratemporal fossa laterally through the pterygomaxillary fissure. The pterygopalatine fossa is a crossroad for neurovascular structures traveling between the middle cranial fossa, orbit, nasal cavity, and oral cavity. Fig. 43.10 Bony boundaries of pterygopalatine fossa A Left lateral view. The lateral approach through the infratemporal fossa via the pterygomaxillary fissure. Table 43.2 Communications of the Pterygopalatine Fossa Communication Direction Via Transmitted structures Middle cranial fossa Posterosuperiorly Foramen rotundum • Maxillary n. (CN V2) Middle cranial fossa Posteriorly in anterior wall of foramen lacerum Pterygoid (vidian) canal • N. of pterygoid canal, formed from: ◦ ◦ Greater petrosal n. (preganglionic parasympathetic fibers from CN VII) ◦ ◦ Deep petrosal n. (postganglionic sympathetic fibers from internal carotid plexus) • A. of pterygoid canal • Vv. of pterygoid canal Orbit Anterosuperiorly Inferior orbital fissure • Branches of maxillary n. (CN V2) ◦ ◦ Infraorbital n. ◦ ◦ Zygomatic n. • Infraorbital a. and vv. • Communicating vv. between inferior ophthalmic v. and pterygoid plexus of vv. Nasal cavity Medially Sphenopalatine foramen • Nasopalatine (sp) n. (CN V2), lateral and medial superior posterior nasal branches • Sphenopalatine a. and vv. Oral cavity Inferiorly Greater palatine canal (foramen) • Greater (descending) palatine n. (CN V2) and a. • Branches that emerge through lesser palatine canals: ◦ ◦ Lesser palatine nn. (CN V2) and aa. Nasopharynx Inferoposteriorly Palatovaginal (pharyngeal) canal • Pharyngeal branches of maxillary n. (CN V2), and pharyngeal a. Infratemporal fossa Laterally Pterygomaxillary fissure • Maxillary a., pterygopalatine (third) part • Posterior superior alveolar n., a., and v. Sphenoid, greater wing Temporal bone, squamous portion Maxilla, tuberosity Width of pterygomaxillary fissure Lateral plate, pterygoid process Sphenopalatine foramen Foramen rotundum (from middle cranial fossa) Pterygopalatine fossa (via pterygomaxillary fissure) Greater palatine canal (to oral cavity) Sphenopalatine foramen (to nasal cavity) Inferior orbital fissure (to orbit) Palatovaginal (pharyngeal) canal (from nasopharynx) Pterygoid canal (from middle cranial fossa) Maxilla Sphenoid Palatine Foramen rotundum Palatovaginal (pharyngeal) canal Pterygoid canal B Left lateral view. This color-coded version shows the location of the role of palatine bone. 43 Nasal Cavity & Nose 623 Communicating br. between CN V2 and pterygopalatine ganglion ⑤ N. of pterygoid canal entering ganglion from behind Pterygopalatine ganglion ⑧ Posterior superior/ inferior lateral nasal br. of CN V2 Nasopalatine n. Medial/lateral pterygopalatine fossa boundaries ④ Maxillary n. (CN V2) in foramen rotundum Lacrimal n. ① Infraorbital n. Zygomatico-temporal n. Inferior orbital fissure ② Zygomatic n. Zygomaticofacial n. Infraorbital nn. in foramen Posterior superior alveolar n. Descending palatine nn. ⑥, ⑦ Gr./Ls. palatine nn. Dental/gingival br. of superior alveolar nn. Superior orbital fissure The maxillary division of the trigeminal nerve (CN V2, see Fig. 40.9, p. 567) passes from the middle cranial fossa through the foramen rotun dum into the pterygopalatine fossa. The parasympathetic ptery-gopalatine ganglion receives postganglionic fibers from the greater petrosal nerve (the parasympathetic root of the nervus intermedius branch of the facial nerve). The preganglionic fibers of the pterygopala-tine ganglion synapse with ganglion cells that innervate the lacrimal, small palatal, and small nasal glands. The sympathetic fibers of the deep petrosal nerve (sympathetic root) and sensory fibers of the maxil-lary nerve (sensory root) pass through the pterygopalatine ganglion without synapsing. The pterygopalatine structures can be seen from the medial view in Fig. 43.8B, p. 621. Fig. 43.11 Nerves in the pterygopalatine fossa Left lateral view. For simplicity in a small, structurally compressed area, numbers are used to identify the nerves. The key to these numbers is found in Table 43.3 (below). Fig. 43.12 Coronal view of the pterygopalatine fossa Table 43.3 Nerves of the pterygopalatine fossa Origin of structures Passageway Transmitted nerves Orbit Inferior orbital fissure ① Infraorbital n. ② Zygomatic n. ③ Orbital brs. (from CN V2) Middle cranial fossa Foramen rotundum ④ Maxillary n. (CN V2) Base of skull Pterygoid (Vidian) canal ⑤ N. of pterygoid canal Palate Greater palatine canal ⑥ Greater palatine n. Lesser palatine canals ⑦ Lesser palatine nn. Nasal cavity Sphenopalatine foramen ⑧ Medial and lateral posterior superior and posterior inferior nasal brs. (from nasopalatine n., CN V2) ① Posterior superior alveolar n. Pharyngeal n. Pterygo-palatine ganglion Ganglionic br. ⑦ ② ③ ④ ⑧ ⑤ ⑥ Pterygomaxillary fissure Petrotympanic fissure Styloid process External acoustic meatus Mastoid process Mastoid foramen Temporal surface Mandibular fossa Zygomatic process Articular tubercle Tympanomastoid fissure Petrous apex Arterial groove Internal acoustic meatus Zygomatic process Styloid process Groove for sigmoid sinus Mastoid foramen Carotid canal Jugular fossa Mastoid foramen Mastoid notch Mastoid process Stylomastoid foramen External acoustic meatus Styloid process Mandibular fossa Articular tubercle Zygomatic process Head & Neck 624 44 Temporal Bone & Ear Temporal Bone Fig. 44.1 Temporal bone Left bone. The temporal bone consists of three major parts: squamous, petrous, and tympanic (see Fig. 44.2). A Left lateral view. B Inferior view. C Medial view. Tympanic part Petrous part Squamous part Styloid process Tympanic part Petrous part Squamous part Tympanic membrane Pharyngotympanic (auditory) tube Internal carotid a. Internal jugular v. Mastoid air cells Chorda tympani Mastoid process Facial n. (CN VII) Facial n. (CN VII), vestibulocochlear n. (CN VIII) Temporal bone, petrous part Internal acoustic meatus Posterior semicircular canal Lateral semicircular canal Anterior semicircular canal Cochlea 45° 90° 45° Mastoid process External acoustic meatus Temporal bone, squa-mous part Posterior semicircular canal Lateral semicircular canal Vestibule Canthomeatal plane Cochlea Anterior semicircular canal 30° 44 Temporal Bone & Ear 625 Fig. 44.2 Parts of the temporal bone A Left lateral view. B Inferior view. Clinical box 44.1 The mastoid process contains mastoid air cells that communicate with the middle ear; the middle ear in turn communicates with the nasopharynx via the pharyngotympanic (auditory) tube (A). Bacteria may use this pathway to move from the nasopharynx into the middle ear. In severe cases, bacteria may pass from the mastoid air cells into the cranial cavity, causing meningitis. Structures in the temporal bone The petrous portion of the temporal bone contains the middle and inner ear as well as the tympanic membrane. The bony semicircular canals are oriented at an approximately 45-degree angle from the coronal, transverse, and sagittal planes (B). Irrigation of the auditory canal with warm (44°C) or cool (30°C) water can induce a thermal current in the endolymph of the semicircular canal, causing the patient to manifest vestibular nystagmus (jerky eye movements, vestibulo-ocular reflex). This caloric testing is important in the diagnosis of unexplained vertigo. The patient must be oriented so that the semicircular canal of interest lies in the vertical plane (C). A B C Temporal bone, tympanic part Cartilagi-nous part Tympanic membrane Stapes Incus Malleus Sebaceous and cerumen glands Bony part External auditory canal Middle ear Tympanic membrane Head of mandible Vestibule Tympanic membrane Tympanic cavity Pharyngo-tympanic (auditory) tube Tensor tympani Stapes Cochlea Cochlear root Vestibular root Malleus Temporal bone, petrous part Anterior semicircular canal Lateral semicircular canal Posterior semicircular canal Incus External auditory canal Styloid process Vestibulo-cochlear n. (CN VIII) Head & Neck 626 External Ear & Auditory Canal The auditory apparatus is divided into three main parts: external, middle, and inner ear. The external and middle ear are part of the sound conduction apparatus, and the inner ear is the actual organ of hearing (see p. 634). The inner ear also contains the vestibular apparatus, the organ of balance (see p. 634). Fig. 44.3 Ear: Overview Coronal section through right ear, anterior view. Clinical box 44.2 Curvature of the external auditory canal The external auditory canal is most curved in its cartilaginous portion. When an otoscope is being inserted, the auricle should be pulled backward and upward so the speculum can be introduced into a straightened canal. Fig. 44.4 External auditory canal Coronal section through right ear, anterior view. The tympanic mem-brane separates the external auditory canal from the tympanic cavity (middle ear). The outer third of the auditory canal is cartilaginous, and the inner two thirds are osseous (tympanic part of temporal bone). A Insertion of otoscope. B Anterior view. C Transverse section. Helix Scaphoid fossa Antihelix Earlobe Intertragic incisure Tragus External auditory canal Concha Cymba conchae Triangular fossa Crura of antihelix Antitragus Superior auricular (posterior part of temporoparietal) Posterior auricular Antitragus Temporo-parietal Tragus External auditory canal Helicis minor Helicis major Superior auricular Anterior auricular External auditory canal Transverse muscle of the auricle Posterior auricular Oblique muscle of the auricle Posterior auricular Perforating brs. Posterior auricular a. Anastomotic arcades External carotid a. Facial n. (CN VII) Lesser occipital nn. and great auricular n. (cervical plexus) Vagus n. (CN X) and glossopharyngeal n. (CN IX) Auriculotemporal n. (trigeminal n., CN V) Facial n. Perforating brs. Superficial temporal a. Anterior auricular aa. Posterior auricular a. External carotid a. Maxillary a. Transverse facial a. Frontal br. Parietal br. Posterior auricular a. 44 Temporal Bone & Ear 627 Fig. 44.5 Structure of the auricle The auricle of the ear encloses a cartilaginous framework that forms a funnel-shaped receptor for acoustic vibrations. The muscles of the auricle are considered muscles of facial expression, although they are vestigial in humans. A Right auricle, right lateral view. B Cartilage and muscles of the right auricle, right lateral view. C Cartilage and muscles of the right auricle, medial view of posterior surface. Fig. 44.6 Arteries of the auricle A Right auricle, lateral view. Fig. 44.7 Innervation of the auricle A Right auricle, lateral view. B Right auricle, posterior view. B Right auricle, posterior view. Auricle External auditory canal Internal carotid a. Cochlea Facial n. Cochlear n. Vestibular n. Anterior semi-circular canal Endolymphatic sac Cochlear aqueduct Posterior semicircular canal Lateral semi-circular canal Sigmoid sinus Mastoid cells Incus Vestibule Malleus Pharyngotympanic (auditory) tube Tympanic cavity Sphenoid sinus Pharyngeal tonsil Superior concha Pharyngeal orifice Levator veli palatini Membranous lamina Salpingo-pharyngeus Tympanic membrane Pharyngotympanic tube, bony part Internal carotid a. Tensor veli palatini Cartilaginous part Pharyngotympanic (auditory) tube Internal jugular v. Sigmoid sinus Oral cavity Cribriform plate Internal carotid a. Hypophysis Hard palate Uvula Head & Neck 628 Middle Ear: Tympanic Cavity Fig. 44.8 Middle ear Right petrous bone, superior view. The tym-panic cavity of the middle ear communicates anteriorly with the pharynx via the pharyngo-tympanic (auditory) tube and posteriorly with the mastoid air cells. Fig. 44.9 Tympanic cavity and pharyngo-tympanic tube Medial view of opened tympanic cavity. Table 44.1 Boundaries of the tympanic cavity During chronic suppurative otitis media (inflammation of the middle ear), pathogenic bacteria may spread to adjacent regions. Direction Wall Anatomical boundary Neighboring structures Infection Anterior Carotid Opening to pharyngotympanic tube Carotid canal Lateral Membranous Tympanic membrane External ear Superior Tegmental Tegmen tympani Middle cranial fossa Meningitis, cerebral abscess (especially of temporal lobe) Medial Labyrinthine Promontory overlying basal turn of cochlea Inner ear CSF space (via petrous apex) Abducent paralysis, trigeminal nerve irritation, visual disturbances (Gradenigo’s syndrome) Inferior Jugular Temporal bone, tympanic part Bulb of jugular v. Sigmoid sinus Sinus thrombosis Posterior Mastoid Aditus to mastoid antrum Air cells of mastoid process Mastoiditis Facial n. canal Facial paralysis CSF, cerebrospinal fluid. Malleus Chorda tympani Tympanic membrane Tensor tympani Prominence of lateral semicircular canal Tendon of stapedius Tympanic n. (from CN IX) via tympanic canaliculus Tympanic plexus Promontory Stapes Prominence of facial canal Facial n. (CN VII) in facial canal Lesser petrosal n. (from tympanic plexus) Aditus (inlet) to mastoid antrum Incus Internal carotid plexus Lesser petrosal n. Chorda tympani Tympanic plexus Greater petrosal n. Tympanic n. entering tympanic canaliculus Geniculate ganglion Facial n. (CN VII) Sigmoid sinus, (ghosted) Anterior semi-circular canal Facial n. in facial canal Lateral semi-circular canal Posterior semi-circular canal Mastoid air cells Round window Internal jugular v. Pharyngotympanic (auditory) tube Internal carotid a. Semicanal of tensor tympani Oval window Anterior wall of tympanic cavity Promontory Roof of tympanic cavity (tegmen tympani) Cochlear n. (CN VIII) Vestibular n. (CN VIII) Opening for tendon of tensor tympani External auditory canal Tympanic membrane Hypo-tympanum Meso-tympanum Tendon of tensor tympani Stapes Epitympanum Malleus Incus Pharyngo-tympanic (auditory) tube 44 Temporal Bone & Ear 629 Fig. 44.10 Tympanic cavity A Levels of the tympanic cavity. Anterior view. The tympanic cavity is divided into three levels: epi-, meso-, and hypotympanum. B Coronal section, anterior view with the anterior wall removed. C Anatomical relationships of the tympanic cavity. Oblique sagittal section showing the medial wall. Tympanic cavity Oval window (with anular stapedial lig.) Stapes Incus Malleus Tympanic membrane Incudo-stapedial joint Incudomalleolar joint Posterior crus Base Anterior crus Short process Body Anterior process Head Handle Neck Malleus Incus Stapes Long process Handle Lateral process Neck Articular surface for incus Anterior process Head Lateral process Neck Articular surface for malleus Short process Body Lenticular process Long process Short process Body Base Posterior crus Anterior crus Neck Head Malleolar prominence Posterior malleolar fold Incus Stapes Temporal bone, tympanic part Cone of light Umbo Malleolar stria Pars tensa Anterior malleolar fold Pars flaccida Tympanic incisure I II III IV Stapedius tendon Malleolar stria Chorda tympani Superior malleolar fold Tympanic membrane Malleolar prominence Superior recess of tympanic membrane Lateral lig. of malleus Malleus Incus Umbo Head & Neck 630 Middle Ear: Ossicular Chain & Tympanic Membrane Fig. 44.11 Auditory ossicles Left ear. The ossicular chain consists of three small bones that establish an articular connection between the tympanic membrane and the oval window. A Auditory ossicles in the middle ear. Anterior view. B Bones of the ossicular chain. Medial view of the left ossicular chain. Fig. 44.12 Malleus (“hammer”) Left ear. A Posterior view. B Anterior view. Fig. 44.13 Incus (“anvil”) Left ear. A Medial view. B Anterolateral view. Fig. 44.14 Stapes (“stirrup”) Left ear. A Superior view. B Medial view. Fig. 44.15 Tympanic membrane Right tympanic membrane. The tympanic membrane is divided into four quadrantsquadrants (I–IV). A Lateral view of the right tympanic membranewith quadrants indicated. B Mucosal lining of the tympanic cavity. Posterolateral view with the tympanic membrane partially removed. Pyramidal eminence Incudo-stapedial joint Petrotympanic fissure Anterior process of malleus Stapedial membrane Facial n. (CN VII) Tympanic membrane Stylomastoid a. Stapedius Posterior tympanic a. Tendon of tensor tympani Anterior tympanic a. Chorda tympani Anterior lig. of malleus Tensor tympani Internal carotid a. Malleus Superior ligs. of the incus and malleus Incudo-malleolar joint Incus Posterior lig. of incus Anular stapedial lig. Chorda tympani Tympanic membrane Round window Oval window Anular stape-dial lig. Stapes Incus Malleus Basilar membrane Stapedius tendon Pyramidal eminence Oval window with anular stapedial lig. Stapes Incus Malleus Oval window Axis of movement 44 Temporal Bone & Ear 631 Fig. 44.16 Ossicular chain in the tympanic cavity Lateral view of the right ear. Revealed: Ligaments of the ossicular chain and muscles of the middle ear (stapedius and tensor tympani). A Vibration of the tympanic membrane causes a rocking movement in the ossicular chain. The mechanical advantage of the lever action of the ossicular chain amplifies the sound waves by a factor of 1.3. B The stapes in its normal posi-tion lies in the plane of the oval window. Clinical box 44.3 Sound waves funneled into the external auditory canal set the tympanic membrane into vibration. The ossicular chain transmits the vibrations to the oval window, which communicates them to the fluid column of the inner ear. Sound waves in fluid meet with higher impedance; they must therefore be amplified in the middle ear. The difference in surface area between the tympanic membrane and the oval window increases the sound pressure 17-fold. A total amplification factor of 22 is achieved through the lever action of the ossicular chain. If the ossicular chain fails to transform the sound pressure between the tympanic membrane and the footplate of the stapes, the patient will experience conductive hearing loss of magnitude 20 dB. Ossicular chain in hearing C Rocking of the ossicular chain causes the stapes to tilt. The movement of the stapes base against the membrane of the oval window (stapedial membrane) induces corresponding waves in the fluid column of the inner ear. D Propagation of sound waves by the ossicular chain. Internal carotid a. Occipital a. Posterior auricular a. Middle meningeal a. Ascending pharyngeal a. A S D F G H J Maxillary a. External carotid a. Superior tympanic a. Tensor tympani Anterior tympanic a. Pharyngotympanic (auditory) tube Tympanic membrane Handle of malleus Deep auricular a. Incudostapedial joint (stapes removed) Inferior tympanic a. Stylomastoid a. Posterior tympanic a. Chorda tympani Stapedial br., stylomastoid a. Facial n. (CN VII) Tegmen tympani Incus Mastoid antrum Head & Neck 632 Arteries of the Middle Ear Fig. 44.17 Arteries of the middle ear: Ossicular chain and tympanic membrane Medial view of the right tympanic membrane. With inflammation, the arteries of the tympanic membrane may become so dilated that their course can be observed (as shown here). Table 44.2 Principal arteries of the middle ear Origin Artery Distribution Internal carotid a. ① Caroticotympanic aa. Tympanic cavity (anterior wall), pharyngotympanic (auditory) tube External carotid a. Ascending pharyngeal a. (medial br.) ② Inferior tympanic a. Tympanic cavity (floor), promontory Maxillary a. (terminal br.) ③ Deep auricular a. Tympanic cavity (floor), tympanic membrane ④ Anterior tympanic a. Tympanic membrane, mastoid antrum, malleus, incus Middle meningeal a. ⑤ Superior tympanic a. Tympanic cavity (roof), tensor tympani, stapes Posterior auricular a. (posterior br.) Stylomastoid a. ⑥ Stylomastoid a. Tympanic cavity (posterior wall), mastoid air cells, stapedius m., stapes ⑦ Posterior tympanic a. Chorda tympani, tympanic membrane, malleus Carotico-tympanic aa. Tympanic n. Stylomastoid a., posterior tympanic br. Stapedial br. Mastoid a. Stylomastoid a., posterior tympanic br. Deep auricular a. Inferior tympanic a. Tubal a. Pharyngotympanic (auditory) tube Internal carotid a. Tensor tympani Lesser petrosal n. Superior tympanic a. Greater petrosal n. Superficial petrosal a. Facial n. (CN VII) Internal auditory a. Anterior crural a. Subarcuate a. Posterior crural a. Superficial petrosal a., descending br. Superficial petrosal a., ascending br. Stylomastoid a. Facial n. (CN VII) Anterior semicircular canal Promontory Stapedius Clinical box 44.4 Otitis media is an infection of the middle ear that occurs commonly in children often following an upper respiratory tract infection. Fluid that accumulates in the middle ear can temporarily diminish hearing and inflammation of the lining of the tympanic cavity can block the pharyngotympanic tube. The stapedius muscle protects the delicate inner ear by modifying the vibrations of very loud sounds as they are transmitted through the middle ear to the stapes. Paralysis of the muscle resulting from a lesion of the facial nerve causes an extreme sensitivity to sound, a condition known as hyperacusis. Otitis Media Hyperacusis 44 Temporal Bone & Ear 633 Fig. 44.18 Arteries of the middle ear: Tympanic cavity Right petrous bone, anterior view. Removed: Malleus, incus, portions of chorda tympani, and anterior tympanic artery. Greater petrosal n. Cochlea Cochlear n. Facial n. Vestibular n. Internal acoustic meatus Temporal bone, petrous part Semicircular canals Chorda tympani Tympanic cavity Geniculate ganglion Modiolus Lesser petrosal n. Helico-trema Cochlear duct Spiral ganglion Scala tympani Scala vestibuli Lateral ampullary n. Lateral semicir-cular canal Posterior semicir-cular canal Posterior ampullary n. Ductus reuniens Saccule (with saccular n.) Utricle (with utricular n.) Inferior part Superior part Anterior ampullary n. Endo-lymphatic sac Endolymphatic duct Anterior semi-circular duct Anterior semi-circular canal Vestibular ganglion (CN VIII) Ductus reuniens Saccule Utricle Endolymphatic duct Semicircular canals Scala tympani Scala vestibuli Cochlear canal Cochlear duct Round window Oval window Cochlear aqueduct Head & Neck 634 Inner Ear Fig. 44.19 Vestibular apparatus Right lateral view. A Schematic. Ampullary crests and maculae of utricle and saccule shown in red. B Structure of the vestibular apparatus. Fig. 44.20 Auditory apparatus The cochlear labyrinth and its bony shell form the cochlea, which con-tains the sensory epithelium of the auditory apparatus (organ of Corti). A Schematic. B Compartments of the cochlear canal, cross section. C Location of the cochlea. Superior view of the petrous part of the temporal bone with the cochlea sectioned transversely. The bony canal of the cochlea (spiral canal) makes 2.5 turns around its bony axis (modiolus). The inner ear consists of the vestibular apparatus (for balance) and the auditory apparatus (for hearing). Both are formed by a membranous labyrinth filled with endolymph floating within a bony labyrinth filled with perilymph and embedded in the petrous part of the temporal bone. Vestibular n. (CN VIII) Vestibular ganglion Cochlear n. (CN VIII) Nervus intermedius Vestibular a. V. of vestibular aqueduct V. of round window V. of cochlear aqueduct Vestibulo-cochlear a. Cochlear a. proper Common cochlear a. Facial n. (CN VII) Internal auditory a. and vv. Modiolus Spiral ganglion of cochlea Dura mater Endolymphatic sac Vestibular aqueduct Lateral semicircular duct Common crus Posterior semicircular duct Posterior ampulla Oval window Round window Posterior ampullary n. Nervus intermedius Vestibular ganglion, inferior part Vestibulocochlear n. (CN VIII), cochlear part Facial n. (CN VII) Vestibular ganglion, superior part Vestibulo-cochlear n. (CN VIII), vestibular part Cochlear com-municating br. Saccular n. Utricular n. Anterior ampullary n. Lateral ampullary n. Anterior semi-circular duct 44 Temporal Bone & Ear 635 Fig. 44.22 Blood vessels of the inner ear Right anterior view. The labyrinth receives its blood supply from the internal auditory artery, a branch of the anteroinferior cerebellar artery (see p. 688). Fig. 44.21 Innervation of the membranous labyrinth Right ear, anterior view. The vestibulocochlear nerve (CN VIII; see p. 570) transmits afferent impulses from the inner ear to the brainstem through the internal acoustic meatus. The vestibulocochlear nerve is divided into the vestibular and cochlear nerves. Note: The sensory or-gans in the semicircular canals respond to angular acceleration, and the macular organs respond to horizontal and vertical linear acceleration. Incisive fossa Inferior orbital fissure Choana Foramen for pterygoid plexus Posterior nasal spine Vomer Pterygoid fossa Foramen ovale Pterygoid process, lateral plate Pterygoid process, medial plate Pyramidal process Lesser palatine foramen Greater palatine foramen Palatine process of maxilla Median palatine suture Transverse palatine suture Pterygoid canal Greater palatine canal Pyramidal process Lateral plate Perpendicular plate Posterior nasal spine Transverse palatine suture Palatine process of maxilla Maxillary sinus Nasal crest Anterior nasal spine Palatine bone Pterygoid process Medial plate Palatine bone Septum of sphenoid sinus Optic canal Anterior clinoid process Superior orbital fissure Sphenoid, lesser wing Inferior orbital fissure Choana Median palatine suture Incisive foramen Palatine process of maxilla Medial plate Lateral plate Pterygoid fossa Inferior concha Ethmoid bone, perpendicular plate Middle concha Ostium of sphenoid sinus Vomer Pterygoid process A Inferior view. Head & Neck 636 45 Oral Cavity & Pharynx Bones of the Oral Cavity The floor of the nasal cavity (the maxilla and palatine bone) forms the roof of the oral cavity, the hard palate. The two horizontal processes of the maxilla (the palatine processes) grow to gether during development, eventually fusing at the median palatine suture. Failure to fuse results in a cleft palate. Fig. 45.1 Hard palate B Superior view. Removed: Maxilla (upper part). C Oblique posterior view. Greater horn Lesser horn Body Greater horn Lesser horn Greater horn Lesser horn Body Ramus of mandible Oblique line Alveoli (tooth sockets) Mental tubercles Mental foramen Alveolar process External oblique ridge Coronoid process Mental protuberance Head (condyle) of mandible Internal oblique ridge Pterygoid fovea Body of mandible Neck of mandible Mylohyoid line Mandibular foramen Coronoid process Head (condyle) of mandible Lingula Mylohyoid groove Sublingual fossa Internal oblique ridge Digastric fossa Superior and inferior mental spines (genial tubercles) Submandib ular fossa Angle Oblique line External oblique ridge Mental foramen Alveolar process Mental protuberance Mental tubercle Mandibular foramen Mandibular notch Coronoid process Ramus of mandible Pterygoid fovea Head (condyle) of mandible Condylar process Lingula Body of mandible 45 Oral Cavity & Pharynx 637 A Anterior view. B Posterior view. C Oblique left lateral view. A Anterior view. B Posterior view. C Oblique left lateral view. Fig. 45.2 Mandible The mandible (jaw) is connected to the viscero cranium at the temporomandibular joint (p. 638). Fig. 45.3 Hyoid bone The hyoid bone is suspended in the neck by muscles between the floor of the mouth and the larynx. Although not listed among the cranial bones, the hyoid bone gives attachment to the muscles of the oral floor. The greater horn and body of the hyoid are palpable in the neck. Mandibular fossa Head of mandible Articular disk Articular tubercle Mylohyoid groove Mandibular foramen Lingula Neck of mandible Neck of mandible Coronoid process Pterygoid fovea Head of mandible Petrotympanic fissure Articular tubercle Mandibular fossa External acoustic meatus (to external auditory canal) Zygomatic process, temporal bone Styloid process, temporal bone Mastoid process, temporal bone Stylomandibular lig. Joint capsule Lateral lig. Pterygoid process, medial plate Pterygoid process, lateral plate Stylomandibular lig. Spheno mandibular lig. Pterygospinous lig. Head & Neck 638 Temporomandibular Joint Fig. 45.4 Temporomandibular joint The head of the mandible articulates with the mandibular fossa in the temporomandibular joint. A Sagittally sectioned temporoman dibular joint, left lateral view. B Head of mandible, anterior view. C Head of mandible, posterior view. D Mandibular fossa of the temporo mandibular joint, inferior view. Fig. 45.5 Ligaments of the temporomandibular joint A Lateral view of the left temporomandibular joint. B Medial view of the right temporomandibular joint. Lateral pterygoid, inferior head Head of mandible Joint capsule Articular disk Articular tubercle Lateral pterygoid, superior head Mandibular fossa Joint capsule Articular disk Articular tubercle Mandibular fossa Mandibular n. (CN V3) Masseteric n. Deep temporal n. Auriculotemporal n. 15° 15° 45 Oral Cavity & Pharynx 639 Fig. 45.6 Movement of the temporomandibular joint Left lateral view. During the first 15 degrees of mandibular depression (opening of the mouth), the head of the mandible remains in the man dibular fossa. Past 15 degrees, the head of the mandible glides forward onto the articular tubercle. A Mouth closed. B Mouth opened to 15 degrees. C Mouth opened past 15 degrees. Clinical box 45.1 Dislocation of the temporomandibular joint Dislocation may occur if the head of the mandible slides past the articular tubercle. The mandible then becomes locked in a protruded position, a condition reduced by pressing on the mandibular row of teeth. Fig. 45.7 Innervation of the temporo mandibular joint capsule Superior view. Root Neck Crown Crown Neck Root Enamel Dentin Pulp chamber Gingival margin Alveolar bone Cementum Apex of root Interalveolar septum Molars Premolars Canine Incisors Incisive suture Interalveolar septum Incisive fossa Molars Premolars Canine Incisors Median palatine suture Transverse palatine suture Labial Mesial Lingual Distal Mesial Distal Buccal Distal Mesial Distal Mesial Palatal Labial Buccal Head & Neck 640 Teeth Fig. 45.8 Structure of a tooth Each tooth consists of hard tissue (enamel, dentin, cementum) and soft tissue (dental pulp) arranged into a crown, neck (cervix), and root. A Principal parts of a tooth (molar). Fig. 45.9 Permanent teeth Each half of the maxilla and mandible contains a set of three anterior teeth (two incisors, one canine) and five posterior (postcanine) teeth (two premolars, three molars). A Maxillary teeth. Inferior view of the maxilla. B Mandibular teeth. Superior view of the mandible. Fig. 45.10 Tooth surfaces The top of the tooth is known as the occlusal surface. B Histology of a tooth (mandibular incisor). 31 30 29 28 27 26 25 18 19 20 21 22 23 24 15 14 13 12 11 10 9 2 3 4 5 6 7 8 1 16 32 17 T S R Q P O N M L K J I H G F E D C B A Mandibular canal Articular tubercle Nasal septum Maxillary sinus Mandibular angle Bite guide of scanner 25 16 17 Condylar process Mandibular fossa Orbit 26 27 28 29 30 31 Impacted third molar (wisdom tooth) 1 32 45 Oral Cavity & Pharynx 641 Not fully erupted. Fig. 45.11 Coding of the teeth In the United States, the 32 permanent teeth are numbered sequen tially (not assigned to quadrants). The 20 deciduous (baby) teeth are coded A to J (upper arch), and K to T (lower arch), in a similar clock wise fashion. The third upper right molar is 1; the second upper right premolar is A. Fig. 45.12 Dental panoramic tomogram The dental panoramic tomogram (DPT) is a survey radiograph that allows preliminary assessment of the temporomandibular joints, maxil- lary sinuses, maxillomandibular bone, and dental status (carious lesions, location of wisdom teeth, etc.). DPT courtesy of Dr. U. J. Rother, Director of the Department of Diagnostic Radiology, Center for Dentistry and Oromaxillo-facial Surgery, Eppendorf University Medical Center, Hamburg, Germany. S D F G 1b 1a Mylohyoid Digastric, anterior belly Hyoglossus Hyoid bone Infrahyoid muscles Connective tissue sling Digastric, intermediate tendon Stylohyoid Digastric, posterior belly Mastoid process Styloid process Sublingual fold Genioglossus Geniohyoid Mylohyoid Hyoid bone Oral mucosa Sublingual papilla Stylohyoid Hyoglossus 1b S D F G 1a Mylohyoid raphe Head & Neck 642 Oral Cavity Muscle Facts Fig. 45.13 Muscles of the oral floor See pp. 516–517 for the infrahyoid muscles. A Suprahyoid muscles, left lateral view. C Suprahyoid muscles, superior view. B Left lateral view. D Superior view of the mandible and hyoid bone. Musculus uvulae Oropharynx (isthmus) Levator veli palatini Tensor veli palatini Uvula Pterygoid hamulus Palatine aponeurosis Hard palate Occipital condyles Opening of carotid canal Lateral pterygoid plate Inferior orbital fissure 45 Oral Cavity & Pharynx 643 Table 45.1 Suprahyoid muscles Fig. 45.14 Muscles of the soft palate Inferior view. The soft palate forms the pos terior boundary of the oral cavity, separating it from the oropharynx. Muscle Origin Insertion Innervation Action ① Digastric 1a Anterior belly Mandible (digastric fossa) Hyoid bone (body) Via an intermediate tendon with a fibrous loop Mylohyoid n. (from CN V3) Elevates hyoid bone (during swallowing), assists in opening mandible 1b Posterior belly Temporal bone (mastoid notch, medial to mastoid process) Facial n. (CN VII) ② Stylohyoid Temporal bone (styloid process) Via a split tendon ③ Mylohyoid Mandible (mylohyoid line) Via median tendon of insertion (mylohyoid raphe) Mylohyoid n. (from CN V3) Tightens and elevates oral floor, draws hyoid bone forward (during swallowing), assists in opening mandible and moving it side to side (mastication) ④ Geniohyoid Mandible (inferior mental spine) Body of hyoid bone Anterior ramus of C1 via hypoglossal n. (CN XII) Draws hyoid bone forward (during swallowing), assists in opening mandible ⑤ Hyoglossus Hyoid bone (superior border of greater cornu) Sides of tongue Hypoglossal n. (CN XII) Depresses and retracts the tongue Table 45.2 Muscles of the soft palate Muscle Origin Insertion Innervation Action Tensor veli palatini Medial pterygoid plate (scaphoid fossa); sphenoid bone (spine); cartilage of pharyngotympanic tube Palatine aponeurosis Medial pterygoid n. (CN V3) Tightens soft palate; opens inlet to pharyngotympanic (auditory) tube (during swallowing, yawning) Levator veli palatini Cartilage of pharyngotympanic tube; temporal bone (petrous part) Vagus n. via pharyngeal plexus Raises soft palate to horizontal position Musculus uvulae Uvula (mucosa) Palatine aponeurosis; posterior nasal spine Shortens and raises uvula Palatoglossus Tongue (side) Palatine aponeurosis Elevates tongue (posterior portion); pulls soft palate onto tongue Palatopharyngeus Tightens soft palate; during swallowing pulls pharyngeal walls superiorly, anteriorly, and medially For the palatoglossus, see Figs. 45.19, p. 646 and 45.24, p. 648; and for the palatopharyngeus, see Figs. 45.24, p. 648 and 45.29C, p. 653. Masseteric n. Mylohyoid n. Medial pterygoid nn. Lingual n. Inferior alveolar n. (in mandibular canal) Mental n. (and foramen) Pterygo palatine ganglion Mandibular division (CN V3, via foramen ovale) Trigeminal n. (CN V) Buccal n. Inferior dental brs. Zygomatic n. Infraorbital n. (and foramen) Maxillary division (CN V2, via foramen rotundum) Posterior Anterior Middle Superior alveolar nn. Auriculo temporal n. Superior labial brs. Anterior and middle superior alveolar brs. Posterior superior alveolar brs. Buccal n. Lesser palatine n. Greater palatine n. Nasopalatine n. Greater palatine n. and a. Greater palatine foramen Lesser palatine foramen Lesser palatine n. and a. Vomer Pterygoid process Median palatine suture Incisive foramen Nasopalatine n. Posterior septal brs. Head & Neck 644 Innervation of the Oral Cavity A Sensory innervation. Note: The buccal nerve is a branch of the man dibular division (CN V3). B Nerves and arteries. Fig. 45.15 Trigeminal nerve in the oral cavity Right lateral view. Fig. 45.16 Neurovasculature of the hard palate Inferior view. The hard palate receives sensory innervation primarily from terminal branches of the maxillary division of the trigeminal nerve (CN V2). The arteries of the hard palate arise from the maxillary artery. Trigeminal ganglion Mandibular division (CN V3) Chorda tympani Stylohyoid br. (with muscle) Digastric br. (with posterior belly) Mastoid process Stylomastoid foramen Mastoid cells Tympanic plexus Geniculate ganglion Glossopharyngeal n. (CN IX) Lingual n. Facial n. (CN VII) Digastric, anterior belly Mylohyoid Submandibular ganglion Mylohyoid n. Lingual n. Chorda tympani (CN VII) Inferior alveolar n. Mandibular division (CN V3) Trigeminal ganglion Submandibular ganglion Lingual n. Hypoglossal n. (CN XII) C1 spinal n., anterior ramus Inferior root of ansa cervicalis (descendens cervicalis) Ansa cervicalis Genioglossus Geniohyoid Geniohyoid br. (C1) Superior root of ansa cervicalis (descendens hypoglossus) 45 Oral Cavity & Pharynx 645 Fig. 45.17 Innervation of the oral floor muscles A Mylohyoid nerve (CN V3). Left lateral view with the left half of the mandible removed. B Facial nerve (CN VII). Sagittal section through the right petrous bone at the level of the mastoid process, medial view. C Anterior rami of the C1 spinal nerve, left lateral view. The muscles of the oral floor have a complex nerve supply with con tributions from the trigeminal nerve (CN V3), facial nerve (CN VII), and C1 spinal nerve via the hypoglossal nerve (CN XII). Palatopharyngeal fold Palatine tonsil Palato glossal fold Lingual tonsil Terminal sulcus Median furrow Dorsum of tongue Apex of tongue Body of tongue Root of tongue Foramen cecum Epiglottis Dorsum of tongue Genio glossus Genio hyoid Hyoid bone Hyoglossus Styloglossus Styloid process Palato glossus Mandible Apex of tongue Lingual aponeurosis Superior longi tudinal m. Inferior longi tudinal m. Hyoglossus Genioglossus Geniohyoid Mylohyoid Sublingual gland Transverse m. Vertical m. Lingual mucosa Lingual septum Digastric, anterior belly Vagus n. (CN X) Glosso pharyngeal n. (CN IX) Facial n. (CN VII, via chorda tympani) Lingual n. (CN V3) Glosso pharyngeal n. (CN IX) Vagus n. (CN X) Taste Somatic sensation Head & Neck 646 Tongue The dorsum of the tongue is covered by a highly specialized mucosa that supports its sensory functions (taste and fine tactile discrimina tion). The tongue is endowed with a very powerful muscular body to support its motor properties during mastication, swallowing, and speaking. A Left lateral view. B Coronal section, anterior view. Fig. 45.18 Structure of the tongue Superior view. The V-shaped sulcus terminalis divides the tongue into an anterior 2/3rds (oral, presulcal) and a posterior 1/3rd (pharyngeal, postsulcal). Fig. 45.19 Muscles of the tongue The extrinsic lingual muscles (genioglossus, hyoglossus, palatoglossus, and styloglossus) have bony attachments and move the tongue as a whole. The intrinsic lingual muscles (superior and inferior longitudinal muscles, transverse muscle, and vertical muscle) have no bony attach ments and alter the shape of the tongue. Fig. 45.20 Somatosensory and taste innervation of the tongue Superior view. Deep lingual a. Sublingual a. Mandible Submental a. and v. Hyoid bone Submandibular ganglion Lingual a. (from external carotid a.) Hypoglossal n. (CN XII) Glosso pharyngeal n. (CN IX) Styloid process Lingual n. (CN V3) Lingual v. (to internal jugular v.) Anterior lingual glands Apex of tongue Submandibular duct Lingual n. Deep lingual a. and v. Frenulum Sublingual fold Sublingual papilla Mandibular n. (CN V3) Medial pterygoid Styloglossus Submandibular duct Sublingual gland Sublingual papilla Lingual n. Genioglossus Otic ganglion Hypoglossal n. (CN XII) Lingual a. Submandibular ganglion Submandibular gland, deep part External carotid a. Hypoglossal n. (CN XII) Mylohyoid Tongue 45 Oral Cavity & Pharynx 647 A Inferior surface of the tongue. B Left lateral view. Fig. 45.21 Neurovasculature of the tongue The lingual muscles receive somatomotor innervation from the hypoglossal nerve (CN XII), with the exception of the palatoglossus (supplied by the vagus nerve, CN X). Fig. 45.22 Floor of mouth with tongue pulled from midline Right mandible, medial view. The oral cavity is generally dissected on a hemisected head. To see the relationship of structures and the base of the tongue/floor of the mouth, the tongue is pulled out of the plane of the dissection and an incision is made in the mucosa of the region. Genioglossus Geniohyoid Hyoid bone Thyrohyoid lig. Thyroid gland Vocal fold Vestibular fold Palatoglossal fold Uvula Salpingopharyngeal fold Soft palate Atlas (C1) Dens of axis (C2) Pharyngeal orifice of pharyngotympanic (auditory) tube Pharyngeal tonsil Torus tubarius with lymphatic tissue (tubal tonsils) Palatine tonsil Lingual tonsil Right choana Cricoid cartilage Epiglottis Ventricle Airway Foodway Laryngo pharynx Oro pharynx Naso pharynx A Open oral cavity. B Oral cavity with mucosa removed from the roof and walls. Soft palate Palatoglossal arch Palatopharyngeal arch Uvula Palatine tonsil Tongue Pterygomandibular fold Oral vestibule Lingual n. Medial pterygoid Buccal n. Palatine tonsil Palatopharyngeus Palatoglossus Pterygomandibular raphe Head & Neck 648 Topography of the Oral Cavity & Salivary Glands The oral cavity is located below the nasal cavity and anterior to the pharynx. It is bounded by the hard and soft palates, the tongue and muscles of the oral floor, and the uvula. Fig. 45.23 Divisions of the oral cavity Midsagittal section, left lateral view. A Organization of the oral cavity. B Boundaries of the oral cavity. Fig. 45.24 Oral cavity topography Right side, anterior view. Table 45.3 Divisions of the oral cavity Part Anterior boundary Posterior boundary Oral vestibule Lips/cheek Dental arches Oral cavity proper Dental arches Palatoglossal arch Fauces (throat) Palatoglossal arch Palatopharyngeal arch Buccinator Parotid duct Masseter Facial a. and v. Submandibular gland, super ficial part Sternocleido mastoid Parotid gland Accessory parotid gland Sublingual fold Genioglossus Submandibular duct Mylohyoid Hyoid bone Lingual a. Submandib ular gland, deep part Geniohyoid Sublingual gland Oral mucosa Sublingual papilla Stylohyoid Hyoglossus Lingual n. Inferior alveolar n. Mylohyoid n. Submandib ular gland, superficial part Parotid gland, superficial part Parotid plexus Parotid gland, deep part Sternocleido mastoid Facial n. (CN VII) Superficial temporal a. and v. 45 Oral Cavity & Pharynx 649 The three large, paired salivary glands are the parotid, submandibular, and sublingual glands. The parotid gland is a purely serous (watery) salivary gland. The sublingual gland is predominantly mucous; the submandibular gland is a mixed seromucous gland. Fig. 45.25 Salivary glands A Parotid gland, left lateral view. Note: The parotid duct penetrates the buccinator muscle to open opposite the second upper molar. B Facial nerve in the parotid gland, left lateral view. The branching of the facial nerve into the parotid plexus (see p. 568) separates the parotid gland into a superficial part and a deep part. C Submandibular and sublingual glands, superior view with tongue removed. Tonsillar fossa Uvula Palatine tonsil Palato pharyngeal arch Palato glossal arch Soft palate Choana Nasal septum Torus tubarius with lymphatic tissue (tubal tonsils) Soft palate Uvula Salpingo pharyngeal fold Dens of axis (C2) Pharyngeal orifice of pharyngo tympanic tube Pharyngeal tonsil Roof of pharynx Nasal conchae Soft palate Lymphatic tissue of lateral bands along salpingo pharyngeal fold Epiglottis Lingual tonsil Palatine tonsil Uvula Tubal tonsil (extension of pharyngeal tonsil) Pharyngeal tonsil Roof of pharynx Head & Neck 650 Tonsils & Pharynx Fig. 45.26 Tonsils A Palatine tonsils, anterior view. B Pharyngeal tonsils. Sagittal section through the roof of the pharynx. C Waldeyer’s ring. Posterior view of the opened pharynx. Table 45.4 Structures in Waldeyer’s ring Tonsil # Pharyngeal tonsil 1 Tubal tonsils 2 Palatine tonsils 2 Lingual tonsil 1 Lateral bands 2 Nasal septum Soft palate Thyroid gland Corniculate tubercle Cuneiform tubercle Aryepiglottic fold Epiglottis Root of tongue (lingual tonsil) Medial pterygoid Uvula Oral cavity opening via faucial isthmus into oropharynx Masseter Inferior nasal concha Digastric muscle, posterior belly Stylohyoid Middle nasal concha Nasal cavity opening via choana into nasopharynx Pharynx (cut) Laryngeal inlet opening into laryngopharynx Piriform recess Pharyngeal tonsil Sigmoid sinus Palatopharyngeal arch Enlarged palatine tonsil Choana Enlarged pharyngeal tonsil 45 Oral Cavity & Pharynx 651 Fig. 45.27 Pharyngeal mucosa Posterior view of the opened pharynx. The anterior portion of the muscular tube contains three openings: choanae (to the nasal cavity), faucial isthmus (to the oral cavity), and aditus (to the laryngeal inlet). Clinical box 45.2 Abnormal enlargement of the palatine tonsils due to severe viral or bacterial infection can result in obstruction of the oropharynx, causing difficulty swallowing. Tonsil infections Particularly well developed in young children, the pharyngeal tonsil begins to regress at 6 to 7 years of age. Abnormal enlargement is common, with the tonsil bulging into the nasopharynx and obstructing air passages, forcing the child to “mouth breathe.” Superior pharyngeal constrictor Digastric muscle, anterior belly Sternohyoid (cut) Thyrohyoid Esophagus Inferior pharyngeal constrictor Middle pharyngeal constrictor Hyoglossus Mylohyoid Pterygo mandibular raphe Stylopharyngeus Tensor veli palatini Levator veli palatini Digastric muscle, posterior belly Stylohyoid Styloglossus Buccinator Cricothyroid I2 I1 M2 M1 S4 S3 S2 S1 Superior pharyngeal constrictor Middle pharyngeal constrictor Inferior pharyngeal constrictor Head & Neck 652 Pharyngeal Muscles Fig. 45.28 Pharyngeal muscles: Left lateral view The pharyngeal musculature consists of the pharyngeal constrictors and the relatively weak pharyngeal elevators. A Pharyngeal muscles in situ. B Subdivisions of the pharyngeal constrictors. Table 45.5 Pharyngeal constrictors Superior pharyngeal constrictor S1 Pterygopharyngeal part S2 Buccopharyngeal part S3 Mylopharyngeal part S4 Glossopharyngeal part Middle pharyngeal constrictor M1 Chondropharyngeal part M2 Ceratopharyngeal part Inferior pharyngeal constrictor I1 Thyropharyngeal part I2 Cricopharyngeal part Stylohyoid Stylopharyngeus Inferior pharyngeal constrictor Middle pharyngeal constrictor Medial pterygoid Masseter Digastric muscle, posterior belly Superior pharyngeal constrictor Accessory muscle bundle Pharyngobasilar fascia Pharyngeal raphe Hyoid bone, greater horn Esophagus Tensor veli palatini Circular muscle fibers of esophagus Posterior cricoarytenoid Transverse arytenoid Middle pharyngeal constrictor Digastric, posterior belly Stylohyoid Levator veli palatini Medial pterygoid Stylopharyngeus Oblique arytenoid Styloid process Salpingo pharyngeus Palato pharyngeus Superior pharyngeal constrictor Musculus uvulae Pharyngeal elevators Angle of mandible Masseter Palato pharyngeus Superior pharyngeal constrictor Salpingo pharyngeus Levator veli palatini Pharyngeal tonsil Musculus uvulae Tubal orifice Pharyngotympanic tube, cartilaginous part Pterygoid process, medial plate Pterygoid hamulus Tensor veli palatini 45 Oral Cavity & Pharynx 653 Fig. 45.29 Pharyngeal muscles: Posterior view A Muscles of the posterior pharynx. B Muscles of the soft palate and pharyngo tympanic tube. The muscles of the fauces form the posterior boundary of the oral cavity. Cut on right side: Levator veli palatini and salpingopharyngeus. C Muscles in the opened pharynx. Pharyngobasilar fascia Occipital a. Superior pharyngeal constrictor Middle pharyngeal constrictor Thyroid gland Carotid body Internal carotid a. Superior laryngeal n. CN IX Sternocleido mastoid CN XII CN XI Superior cervical ganglion Sigmoid sinus CN XII Internal jugular v. Pharyngeal venous plexus Pharyngeal raphe Stylopharyngeus External carotid a. Ascending pharyngeal a. Sympathetic trunk Superior thyroid a. CN X Inferior pharyngeal constrictor Common carotid a. Head & Neck 654 Neurovasculature of the Pharynx Fig. 45.30 Neurovasculature in the parapharyngeal space Posterior view. Removed: Vertebral column and posterior structures. Fig. 45.31 Fasciae and potential tissue spaces in the head Transverse section at the level of the tonsillar fossa, superior view. Fascial boundaries are key to outlining pathways for the spread of infection. Potential spaces in the head, shown on this figure, become true spaces when they are infiltrated by products of infection. These spaces are defined by bones, muscles and fascia and initially confine an infection but eventually allow it to spread through communications between spaces. Retropharyngeal space Parotid space Buccal space Pterygomandibular space Submasseteric space Peritonsillar space Parapharyngeal space Inferior nasal concha Middle nasal concha CN IX Musculus uvulae Palatopharyngeus CN XII Superior laryngeal n. Sympathetic trunk Superior laryngeal a. and internal laryngeal n. Inferior laryngeal v. Cuneiform tubercle Corniculate tubercle Epiglottis Right recurrent laryngeal n. External jugular v. Inferior thyroid a. Common carotid a. Middle cervical ganglion Internal jugular v. Transverse part Oblique part CN X Sternocleidomastoid CN XI Occipital a. Salpingopharyngeus CN VII CN IX, X, XI CN VII, CN VIII, nervus intermedius CN V CN VI CN III CN X Left subclavian a. Brachiocephalic trunk CN X, right Inferior laryngeal n. (terminal br. of recurrent laryngeal n.) Posterior cricoarytenoid Superior cervical ganglion Choanae Left recurrent laryngeal n. Stellate ganglion Arytenoid CN X, left 45 Oral Cavity & Pharynx 655 Fig. 45.32 Neurovasculature of the opened pharynx Posterior view. CN III, oculomotor n.; CN V, trigeminal n.; CN VI, abducent n.; CN VII, facial n.; CN VIII, vestibulocochlear n.; CN IX, glossopharyngeal n.; CN X, vagus n.; CN XI, accessory n.; CN XII, hypoglossal n.. See Chapter 39 for the cranial nerves. Head & Neck 656 46 Sectional & Radiographic Anatomy Sectional Anatomy of the Head & Neck (I) Palatine process of the maxilla Orbital plate of ethmoid bone Buccinator Inferior nasal concha Middle nasal meatus and concha Greater palatine a. Oral cavity Tongue Digastric, anterior belly Anterior cranial fossa Oral vestibule First lower molar First upper molar Vomer Orbicularis oculi Inferior oblique Inferior rectus Periorbital fat Vitreous body Levator palpebrae superioris Ethmoid sinuses Inferior nasal meatus Maxillary sinus Infraorbital n. (from CN V2) in infraorbital groove Inferior alveolar a., n., and v. in mandibular canal Mylohyoid Genioglossus Geniohyoid Cartilaginous nasal septum Platysma Medial rectus Frontal lobe of cerebrum Fig. 46.1 Coronal section through the anterior orbital margin Anterior view. This section shows four regions of the head: the oral cav-ity, the nasal cavity and sinuses, the orbit, and the anterior cranial fossa. Muscles of the oral floor, the apex of the tongue, the hard palate, the neurovascular structures in the mandibular canal, and the first molar are all seen in the region of the oral cavity. This section reinforces the clinical implications of the relationship of the maxillary sinus with the maxillary teeth and the floor of the orbit and with the maxillary nerve in the infraorbital groove. The medial wall of the orbit shares a thin bony wall (orbital plate) with the ethmoid air cells (sinus). The section is enough anterior so that the lateral bony walls of the orbit are not included due to the lateral curvature of the skull. 46 Sectional & Radiographic Anatomy 657 Digastric, anterior belly Soft palate Inferior rectus Lateral rectus Optic n. (CN II) Superior rectus Superior oblique Frontal lobe of cerebrum Medial rectus Superior sagittal sinus Inferior alveolar n., a., and v. in mandibular canal Genioglossus Mylohyoid Buccal fat pad Maxillary sinus Zygomatic arch Nasal septum Olfactory n. (CN I) Falx cerebri Infraorbital n. (from CN V2) Buccinator Body of mandible Medial pterygoid Mandibular ramus Nasal cavity Masseter Tongue Geniohyoid Temporalis Ethmoid sinuses Coronoid process Lingual n., deep lingual a. and v. Hyoglossus Fig. 46.2 Coronal section through the orbital apex Anterior view. In this more posterior section than that of Fig. 46.1, the soft palate now separates the oral and nasal cavities. The buccal fat pad is also visible. The section is slightly angled, producing an apparent discontinuity in the mandibular ramus on the left side. Head & Neck 658 Sectional Anatomy of the Head & Neck (II) Fig. 46.3 Coronal section through the pituitary Anterior view. Parietal lobe Temporal lobe Maxillary n. (CN V2) Abducent n. (CN VI) Ophthalmic n. (CN V1) Trochlear n. (CN IV) Oculomotor n. (CN III) Superior sagittal sinus Middle cranial fossa Anterior cerebral a. Sphenoid sinus Nasopharynx Uvula Palatine tonsil Oropharynx Epiglottis Laryngopharynx Lingual n. Medial pterygoid Mandible, ramus Masseter Lateral pterygoid Inferior alveolar n. Zygomatic process, temporal bone Hypophysis in hypophyseal fossa Internal carotid a. Optic n. (CN II) Putamen Internal capsule Caudate nucleus, head Corpus callosum Lateral ventricle Falx cerebri Septum of sphenoid sinus Cavernous sinus Temporalis Lingual n. Inferior alveolar n. Mandibular n. (CN V3) 46 Sectional & Radiographic Anatomy 659 Fig. 46.4 Midsagittal section through the nasal septum Left lateral view. Olfactory bulb (CN I) Anterior cranial fossa Frontal sinus Clivus Nasal septum Hard palate, maxilla (palatine process) Soft palate Uvula Mylohyoid Geniohyoid Hyoid bone Laryngeal cartilage Epiglottis Choana C3 vertebra Axis (C2), dens Median atlantoaxial joint Nasopharynx Nuchal lig. Atlas (C1), anterior and posterior arches Transverse sinus Transverse lig. of atlas Foramen magnum Hypophysis Sphenoid sinus Corpus callosum Mandible Laryngopharynx Vallecula Oropharynx Head & Neck 660 Sectional Anatomy of the Head & Neck (III) Vertebral a. Oculomotor n. (CN III) Optic n. (CN II) Frontal sinus Ethmoid sinus Sphenoid sinus Middle nasal concha Inferior nasal concha Maxilla Superior labial vestibule Palatine process, palatine sulcus Longus capitis Palato-pharyngeus Mandible Digastric, anterior belly Lingual n. and deep lingual vv. Mylohyoid Tongue Hyoid bone Epiglottic cartilage and vallecula Thyroid cartilage C7 spinal n. C6 spinal n. Spinalis cervicis C5 spinal n. C4 spinal n. Laryngo-pharynx C3 spinal n. C2 spinal n. Splenius capitis Obliquus capitis inferior Rectus capitis posterior major Semispinalis capitis Vertebral a. Rectus capitis posterior minor Oral cavity Pharyngo-tympanic (auditory) tube Cerebellum Ponto-cerebellar cistern Posterior thalamic nuclei Uncus Medial segment of globus pallidus Internal capsule Caudate nucleus, head Lateral ventricle Inferior labial vestibule Tentorium cerebelli Fig. 46.5 Sagittal section through the medial orbital wall Left lateral view. This section passes through the inferior and middle conchae of the lateral nasal wall. Three of the four paranasal air sinuses (ethmoid, sphenoid, and frontal) are seen in this section and in relation to the nasal cavity into which they drain. In the region of the cervical spine, the vertebral artery is cut at multiple levels. The spinal nerves have been cut just prior to their lateral exit through the intervertebral foramina. 46 Sectional & Radiographic Anatomy 661 Frontal sinus Procerus Superior rectus Vitreous body Lateral rectus Optic n. (CN II) Inferior rectus CN V2 in pterygo-palatine fossa Sphenoid sinus Ethmoid sinus Maxillary sinus Lateral pterygoid Pharyngo-tympanic (auditory) tube Medial pterygoid Orbicularis oris Levator veli palatini Mylohyoid Genioglossus Digastric, anterior belly Tongue Palatine tonsil Palato-pharyngeus Submandibular gland Hyoid bone, greater cornu Thyroid cartilage, left lamina Inferior pharyngeal constrictor Hyoid bone, lesser cornu Retropharyngeal space C3 spinal n. Trapezius Splenius capitis Semispinalis capitis Rectus capitis posterior major Vertebral a. Condylar emissary v. Hypoglossal n. (CN XII) Transverse sinus Posterior meningeal a. CN IX, X, and XI in jugular foramen Internal carotid a. Trigeminal ganglion (CN V) Dentate gyrus Putamen Internal capsule External capsule Claustrum Extreme capsule Amygdala Obliquus capitis inferior Greater occipital n. (C2) Internal carotid a. Fig. 46.6 Sagittal section through the inner third of the orbit Left lateral view. This section passes through the maxillary, frontal, and sphenoid sinuses and a single ethmoidal air cell. The pharyngeal and masticatory muscles are revealed grouped around the cartilaginous part of the pharnygotympanic (auditory) tube. The palatine tonsil of the oral cavity and medial portion of the submandibular gland below the floor of the mouth are also seen in this section. Head & Neck 662 Sectional Anatomy of the Head & Neck (IV) Fig. 46.7 Transverse section through the optic nerve and pituitary Inferior view. Dorsum sellae Optic n. (CN II) Inferior sagittal sinus Superior sagittal sinus Falx cerebri Cerebellum, vermis Pons Interpeduncular fossa Basilar a. Internal carotid a. Lateral rectus Lacrimal gland Medial rectus Vitreous body Lens Ethmoid sinuses Nasal cavity Nasal septum Tentorium cerebelli Lateral ventricle, occipital horn Hypophysis Optic canal Oculo-motor n. (CN III) Temporalis Cavernous sinus 46 Sectional & Radiographic Anatomy 663 Trapezius Glosso-pharyngeal n. (CN IX) Dens of axis (C2) Soft palate (including tensor and levator veli palatini) Atlas (C1) Internal carotid a. Facial n. (CN VII) in parotid gland Accessory n. (CN XI) Hypoglossal n. (CN XII) Vagus n. (CN X) Median atlantoaxial joint Transverse lig. of atlas Occipital bone Spinal cord Posterior condylar emissary v. Semispinalis capitis Vertebral a. Splenius capitis Occipital a. Internal jugular v. Maxillary a. Inferior alveolar n. Lateral pterygoid Mandibular ramus Lingual n. Masseter Medial pterygoid Buccinator Levator anguli oris Mucoperiosteum of hard palate Maxilla Lateral pterygoid plate Fig. 46.8 Transverse section of head through the median atlantoaxial joint Superior view. This section passes through the soft palate and muco-periosteum of the hard palate. The articulation of the odontoid process (dens of C2) with the axis (C1) at the median atlantoaxial joint is shown, as well as the carotid sheath, containing the vertical neuro- vascular elements of the neck. The vertebral artery is sectioned as it prepares to enter the foramen magnum and fuse with its opposite to form the basilar artery. Head & Neck 664 Sectional Anatomy of the Head & Neck (V) Omohyoid Platysma Superior thyroid v. External jugular v. Accessory n. (CN XI), external branch Common carotid a., internal jugular v., and vagus n. (CN X) in carotid sheath Longissimus capitis Longissimus cervicis Levator scapulae Trapezius Splenius cervicis Splenius capitis Semispinalis cervicis C6 spinal n. Vertebral a. C5 spinal n. C4 spinal n. Sternocleido-mastoid Piriform recess Thyroid cartilage Laryngeal vestibule Epiglottic cartilage Longus colli Thyrohyoid Arytenoid cartilage Oropharynx C5 vertebra C6 vertebral body Spinous process of C7 Fig. 46.9 Transverse section of the neck Transverse section at the level of the C5 vertebral body. Inferior view. The internal and external jugular veins are separated by the sternoclei-domastoid. The accessory nerve (CN XI) is just medial to this muscle as it prepares to innervate it from behind. The elongated spinous process of the C7 vertebra (vertebra prominens) is also visible in the section due to the lordotic curvature of the neck. 46 Sectional & Radiographic Anatomy 665 Fig. 46.10 Transverse section at the level of the C6 vertebral body Inferior view. Sternohyoid Thyrohyoid Superior thyroid v. Thyroid gland Longus colli C4 spinal n. C6 spinal n. and C6 vertebra C7 spinal n. and C7 vertebra Laryngopharynx Semispinalis cervicis Splenius cervicis Posterior scalene Middle scalene Vertebral v. Vertebral a. Anterior scalene with C5 spinal n. Common carotid a., internal jugular v., and vagus n. (CN X) in carotid sheath Sternocleido-mastoid Arytenoid cartilage Omohyoid Thyroid cartilage Trapezius Levator scapulae T1, vertebral arch Superior thyroid v. Superior thyroid a. Thyroid gland Sternocleidomastoid Internal jugular v., vagus n. (CN X), and common carotid a. External jugular v. C6 spinal n. C7 spinal n. C8 spinal n. Intervertebral disk T1 vertebra and spinal n. Vertebral a. and v. Second rib Posterior scalene Middle scalene Esophagus Thyrocervical trunk Sternothyroid Sternohyoid Transverse process of T2 Phrenic n. with anterior scalene Inferior thyroid a. Cricoid cartilage Fig. 46.11 Transverse section of the neck Transverse section at the level of the C7/T1 vertebral junction. Inferior view. This section reveals the roots of spinal nerves C6 to C8 of the bra-chial plexus passing between the anterior and middle scalene muscles. The phrenic nerve is on the anterior surface of the anterior scalene and the components of the carotid sheath (internal jugular vein, common carotid artery, and vagus nerve) lie in the interval between this muscle, the sternocleidomastoid, and the thyroid gland. Head & Neck 666 Radiographic Anatomy of the Head & Neck (I) Fig. 46.12 Radiograph of the skull Anteroposterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radio-graphic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 46.13 Coronal MRI through the eyeball Anterior view. Superior sagittal sinus Roof of orbit Falx cerebri with superior frontal gyrus Levator palpebrae superioris, superior rectus, and supraorbital n. Lacrimal gland Lateral rectus Inferior rectus and inferior oblique Infraorbital a., v., and n. Maxillary sinus Maxilla, alveolar process Buccinator Lingual n., deep lingual a. and v. Genioglossus Mandibular tooth Ethmoid sinus Superior oblique with superior ophthalmic v. Eyeball Periorbital fat Medial rectus with ophthalmic a. Zygomatic bone Middle and inferior nasal conchae Nasal septum Tongue Mandible Mental protuberance Angle, mandible Maxilla Maxillary sinus Nasal septum and inferior nasal concha Ethmoidal air cells Roof of orbit Frontal sinus 46 Sectional & Radiographic Anatomy 667 Fig. 46.14 Radiograph of the skull Left lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 46.15 Midsagittal MRI through the nasal septum Left lateral view. Boxed area represents the location of the ventricular system, thalamus, and pons. A more detail labeled version of this area can be seen in Fig. 51.5, p. 700. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Manidble Hard palate Maxillary sinus Ethmoidal sinus Clivus Sphenoid sinus Anterior clinoid process Hypophyseal fossa (sella turcica) Frontal sinus Coronal suture Dorsum sellae Oral cavity Fourth ventricle Nasal bone Frontal sinus Hard palate Nasopharynx C2/C3 intervertebral disk Dens of axis (C2) and anterior arch of atlas (C1) Rectus capitis posterior minor Nuchal lig. Basilar a. Confluence of the sinuses Straight sinus Ethmoid sinus and sphenoid sinus Tongue Semispinalis capitis Corpus callosum Optic n. (CN II) Septum pellucidum Superior sagittal sinus Mandible, body Uvula Oropharynx Hypophysis Head & Neck 668 Radiographic Anatomy of the Head & Neck (II) Fig. 46.16 Radiograph of the skull Inferosuperior oblique view (Waters view). (Reproduced from Moeller TB, Reif E. Pocket Atlas of Radio-graphic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Fig. 46.17 Radiograph of the mandible Left lateral view. (Repro-duced from Moeller TB, Reif E. Pocket Atlas of Radiographic Anatomy, 3rd ed. New York, NY: Thieme; 2010.) Frontal sinus Nasal bone Orbit Nasal septum Infraorbital foramen Maxillary sinus Sphenoid sinus Mandible Oropharynx Tongue Sphenoid sinus, median septum Temporomandibular joint (TMJ) Mandible, condyle Mandible Mandibular foramen Mental protuberance Apical foramen of tooth Mental foramen Mandibular canal Mandible, angle Coronoid process, mandible Mandibular notch 46 Sectional & Radiographic Anatomy 669 Fig. 46.18 Transverse MRI through the orbit and nasolacrimal duct Inferior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Fig. 46.19 Transverse MRI through the neck Inferior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Temporalis Medial pterygoid between medial and lateral pterygoid plates Levator labii superioris Longus capitis Mastoid air cells Levator and tensor veli palatini Pharyngeal recess Maxilla with infraorbital canal Vertebral a. Internal jugular v. with CN IX, X, and XI Mandibular condyle Mandibular and auriculotemporal nn. (from CN V3) Lateral pterygoid Masseter Maxillary sinus Orbicularis oris Cerebellum, posterior lobe Sigmoid sinus Internal carotid a. Medulla oblongata Nasolacrimal duct Middle nasal concha Nasal septum Nasal bone Falx cerebri around superior sagittal sinus Occipital bone Fourth ventricle Platysma Thyroid gland Sternothyroid Vertebral a. and v. C7 spinal n. root Sternocleidomastoid with external jugular v. Cricoid cartilage Splenius capitis Spinalis cervicis Levator scapulae Common carotid a. Scalene mm. Internal jugular v. Larynx Anterior jugular vv. Sternohyoid and thyrohyoid Esophagus Semispinalis cervicis Thyroid cartilage C6 vertebral body, C7 posterior arch C7 spinous process Multifidus Trapezius Head & Neck 670 Radiographic Anatomy of the Head & Neck (III) Fig. 46.20 CT scan of temporo- mandibular joint (TMJ) Coronal section. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculo- skeletal System. New York, NY: Thieme; 2009.) Fig. 46.21 CT scan of temporo- mandibular joint (TMJ) Sagittal section, mouth closed. (Reproduced from Moeller TB, Reif E. Atlas of Sectional Anatomy: The Musculo- skeletal System. New York, NY: Thieme; 2009.) Temporalis Temporal bone Zygomatic process Inferior synovial cavity Lateral joint capsule Mandible, ramus Masseter Temporal lobe Articular disk Internal carotid a., petrous part Mandible, head Medial pterygoid Cerebrum, temporal lobe Inferior synovial cavity Retrodiskal region Mandible, head External acoustic meatus Mandible, neck Superior synovial cavity Articular disk Articular tubercle Temporalis Lateral pterygoid, superior head Lateral pterygoid, inferior head Mandible, ramus Inferior alveolar n. in mandibular canal 46 Sectional & Radiographic Anatomy 671 Fig. 46.23 Dural venous sinus system of the head Right lateral view. Lateral internal carotid arteriogram, venous phase. Fig. 46.22 Cranial MR angiography Cranial view. In this angiogram note that the right posterior cerebral a. arises from the internal carotid artery instead of the basilar artery—a variant. The normal configuration is seen on the left side. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Anteromedial frontal br. of anterior cerebral a. Anterior communicating a. Anterior cerebral a. Left posterior communicating a. Left posterior cerebral a. Temporal a. Ophthalmic a. Internal carotid a. Middle cerebral a. Right posterior cerebral a. Superior cerebellar a. Basilar a. Parieto-occipital a. Superior sagittal sinus Bridging vv. Inferior sagittal sinus Internal cerebral v. Great cerebral v. (of Galen) Confluence of sinuses Transverse sinus Superior petrosal sinus Cavernous sinus Sigmoid sinus Inferior petrosal sinus Occipital sinus Internal jugular v. 47 Brain Nervous System: Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 Nervous System: Development . . . . . . . . . . . . . . . . . . . . . . . 676 Brain, Macroscopic Organization . . . . . . . . . . . . . . . . . . . . . . 678 Diencephalon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 Brainstem & Cerebellum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 Ventricles & CSF Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 48 Blood Vessels of the Brain Veins & Venous Sinuses of the Brain . . . . . . . . . . . . . . . . . . . 686 Arteries of the Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 49 Functional Systems Anatomy & Organization of the Spinal Cord. . . . . . . . . . . . . . 690 Sensory & Motor Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 50 Autonomic Nervous System Autonomic Nervous System (I): Overview . . . . . . . . . . . . . . . 694 Autonomic Nervous System (II) . . . . . . . . . . . . . . . . . . . . . . . 696 51 Sectional & Radiographic Anatomy Sectional Anatomy of the Nervous System . . . . . . . . . . . . . . 698 Radiographic Anatomy of the Nervous System. . . . . . . . . . . 700 Brain & Nervous System Brain & Nervous System 674 47 Brain Nervous System: Overview CNS both white and gray matter A B C Brain Spinal cord Telencephalon (cerebral hemispheres) Diencephalon Cerebellum Brainstem Midbrain (mesencephalon) Pons Medulla oblongata Segments (only functionally, not morphologically displayable) Cranial/oral Parietal/ dorsal superior Frontal/cranial/ oral/rostral anterior Basal/ ventral inferior Ventral Caudal Dorsal Occipital/ caudal posterior ① ② Fig. 47.1 Morphology of the Central Nervous System (CNS) A and B Right side of the brain, medial view; C Section of the spinal cord, ventral view. A general morphological overview of the entire ner-vous system is necessary to help with understanding the material that follows. The CNS is divided into the brain and the spinal cord with the brain subdivided into the following regions: • Cerebral hemispheres (telencephalon or endbrain) • Interbrain (diencephalon) • Cerebellum • Brain stem composed of the midbrain (mesencephalon), pons (bridge) and medulla oblongata In contrast, the other part of the CNS, the spinal cord appears morpho-logically rather as one homogenous structure. In terms of its functions, however, the spinal cord can also be divided into segments. The division of gray and white matter is clearly visible: • Gray matter: centrally located, butterfly-shaped structure • White matter: substance that surrounds the “butterfly” Fig. 47.2 Axes of the nervous system and directional terms The same planes, axes and directional terms apply for both the entire body and the PNS. However, with the CNS, one differentiates between two axes: • Axis No. 1: Meynert axis: It corresponds to the axes of the body and is used to designate locations in the spinal cord, brainstem (truncus encephali) and cerebellum. • Axis No. 2: Forel axis. It turns horizontally through the diencephalon and telencephalon and forms an 80° angle to axis 1. As a result, the diencephalon and telencephalon lie “face down”. Note: In order to avoid topographical misunderstandings, the following directional terms for axis No. 2 (Forel axis) are used: • basal instead of ventral • parietal instead of dorsal • frontal and oral/rostral respectively instead of cranial • occipital instead of caudal 47 Brain 675 675 PNS A B C Ganglia Gray matter Nerves Spinal nerves Ganglion of cranial nerve Autonomic ganglia (these ganglia can be functionally and topographically further divised, see pp. 280–281) White matter ① anterior root ② posterior root Cranial nerves “False cranial nerves” (cranial nerves I and II) “True cranial nerves” (cranial nerves III-XII) Splanchnic nerves (can be functionally and topographically further divised, see pp. 280–281) Spinal ganglion (dorsal root ganglion) ① ② Fig. 47.3 Morphology of the peripheral nervous system A Segment of the spinal cord, ventral view; B Base of the brain; C view of sympathetic ganglia and nerves located anterior to the sacrum. The nerves and ganglia forming the peripheral nervous system are gen-erally named for the part of the CNS with which they communicate: • Spinal nerves (connect the periphery of the body with the spinal cord. Usually 31 or 32 pairs. Spinal nerves (except those related to verte-bral levels T1 to T11 or T12) generally have their anterior rami form plexuses for reasons of functionality. • Cranial nerves (connect the periphery of the body to the brain). 12 pairs. Nerve cells found within ganglia (in the PNS) can be classified based on their affiliation with a particular functional division of the nervous system: • Sensory neurons can be found within either division of the nervous system. In the PNS, sensory neurons are found within the sensory (dorsal root) ganglia on the posterior (dorsal) root of the spinal nerve. In the CNS, sensory neurons are found within the sensory nuclei asso-ciated with the appropriate cranial nerves that contain sensory fibers. • Ganglia of the autonomic nervous system contain postganglionic sympathetic and parasympathetic neurons that control the organs of the body. Autonomic ganglia are associated with splanchnic nerves that take vasomotor fibers to the viscera. The autonomic nervous system also demonstrates characteristic plexus formation. Note: The distinction of sensory nerves in the CNS applies except for a few special cases. For instance, cranial nerves I (olfactory) and II (optic) are not true nerves but parts of the telencephalon or diencephalon, which clearly makes them part of the CNS. For historical reasons, they have been called “nerves”, which is systematically false. These “bogus” cranial nerves (colored red on the brain in the figure above) are often contrasted with the 10 true cranial nerves (colored yellow on this figure), which are clearly part of the PNS. In the interest of clarity, further details are located within each region throughout the atlas. White matter Gray matter Brain Spinal cord Cauda equina Cranial nn. Spinal nn. White matter Cerebral cortex Basal ganglia Gray matter Brain & Nervous System 676 Nervous System: Development Fig. 47.4 Central and peripheral nervous systems The CNS consists of the brain and spinal cord, which constitute a functional unit. The PNS consists of the nerves emerging from the brain and spinal cord (cranial and spinal nerves, respectively). Nerves emerging from the spinal canal after the end of the spinal cord, form the cauda equina (see p. 41) Fig. 47.5 Gray and white matter in the CNS Nerve cell bodies appear gray in gross inspection, whereas nerve cell processes (axons) and their insulating myelin sheaths appear white. A Coronal section through the brain. B Transverse section through the spinal cord. Table 47.1 Development of the brain Primary vesicle Region Structure Neural tube Prosencephalon (forebrain) Telencephalon (cerebrum) Cerebral cortex, white matter, and basal ganglia Diencephalon Epithalamus (pineal), dorsal thalamus, subthalamus, and hypothalamus Mesencephalon (midbrain) Tectum, tegmentum, and cerebral peduncles Rhombencephalon (hindbrain) Metencephalon Cerebellum Cerebellar cortex, nuclei, and peduncles Pons Nuclei and fiber tracts Myelencephalon Medulla oblongata The mesencephalon, pons, and medulla oblongata are collectively known as the brainstem. Medulla oblongata Pons Insula Eye Medulla oblongata Insula Pons Precentral gyrus Central sulcus Postcentral gyrus Occipital lobe Cerebellum Medulla oblongata Pons Temporal lobe Lateral sulcus Frontal lobe Parietal lobe Cingulate gyrus Pineal gland Hypothalamus Occipital lobe Cerebellum Medulla oblongata Pons Corpus callosum Hypophysis Thalamus Mesencephalon Cranial flexure Optic cup Pons Medulla oblongata Cervical flexure Mammillary tubercle Olfactory bulb Hypophysis primordium Telodien-cephalic sulcus Frontal lobe Longitudinal cerebral fissure Optic chiasm Optic n. (CN II) Hypophysis Pons Medulla oblongata Cervical spinal cord Cerebellum Temporal lobe 47 Brain 677 677 Fig. 47.6 Embryonic development of the brain Left lateral view. A Start of 2nd month. B End of 2nd month. C 3rd month of development. D 7th month. Fig. 47.7 Adult brain See Fig. 47.10 for lobes of the cerebrum. CN, cranial nerve. A Left lateral view. B Basal view. C Right hemisphere, midsagittal section. Superior frontal gyrus Middle frontal gyrus Superior frontal sulcus Inferior frontal sulcus Opercular part Triangular part Orbital part Inferior frontal gyrus Frontal pole Anterior ramus Ascending ramus Posterior ramus Lateral sulcus (of Sylvius) Temporal pole Superior temporal gyrus Superior temporal sulcus Middle temporal gyri Inferior temporal sulcus Inferior temporal gyrus Preoccipital notch Transverse occipital sulcus Calcarine sulcus Occipital pole Parieto-occipital sulcus Precentral gyrus Central sulcus Postcentral gyrus Superior parietal lobule Intraparietal sulcus Inferior parietal lobule Supramarginal gyrus Angular gyrus Brain & Nervous System 678 Brain, Macroscopic Organization Frontal operculum Temporal operculum Insular lobe (= insula) Parietal operculum Fig. 47.8 Cerebrum Left lateral view. The cerebrum is part of the anterior subdivision of the embryonic forebrain (telencephalon)—the part of the adult forebrain that includes the cerebral hemispheres and associated structures. The surface anatomy of the cerebrum can be divided macroscopically into 4 lobes: frontal, parietal, temporal, and occipital. The surface contours of the cerebrum are defined by convolutions (gyri) and depressions (sulci). Fig. 47.9 Insular lobe Lateral view of the retracted left cerebral hemisphere. Part of the cerebral cortex sinks below the surface during development form-ing the insula (or insular lobe). Those portions of the cerebral cortex that overlie this deeper cortical region are called opercula (“little lids”). Central sulcus Lateral sulcus Insula Fornix Septum pellucidum Cingulate gyrus Corpus callosum Parieto-occipital sulcus Occipital pole Frontal pole Olfactory n. (CN I) Optic n. (CN II) Hypophysis Mammillary body Mesencephalon Longitudinal cerebral fissure 47 Brain 679 679 Frontal lobe Parietal lobe Temporal lobe Occipital lobe Insular lobe (insula) Limbic lobe (limbus) Fig. 47.10 Lobes in the cerebral hemispheres The isocortex also may be functionally divided into association areas (lobes). A Left hemisphere, lateral view. B Lateral view of the retracted left cerebral hemisphere. C Right hemisphere, medial view. D Basal view with the brainstem removed. Fig. 47.11 Midsagittal section of the brain showing the medial surface of the right hemisphere The brain has been split along the longitudinal cerebral fissure. Interventricular foramen Sulcus of corpus callosum Cingulate gyrus Medial frontal gyrus Cingulate sulcus Paracentral sulcus Paracentral lobule Central sulcus Marginal sulcus Corpus callosum Precuneus Choroid plexus Parieto-occipital sulcus Cuneus Calcarine sulcus Septum pellucidum Interthalamic adhesion Third ventricle Subcallosal gyrus Hypothalamic sulcus (ventral diencephalic sulcus) Thalamus 3rd ventricle Infundibular recess Tuber cinereum Hypothalamus Supraoptic recess Preoptic area Cerebral aqueduct Quadrigeminal plate Pineal (epiphysis) Stria medullaris thalami Interthalamic adhesion Choroid plexus Fornix Corpus callosum Anterior commissure Infundibulum Anterior lobe (adenohypophysis) Posterior lobe (neurohypophysis) Cerebral peduncle Tegmentum Cerebellum Mesencephalon Mammillary body Hypophysis The diencephalon is the posterior subdivision of the forebrain—the part of the adult forebrain that includes the thalamus and associated structures. Brain & Nervous System 680 Diencephalon v. Fig. 47.12 Diencephalon Right hemisphere, midsagittal section, medial view. The major components of the diencephalon are the thalamus, hypothalamus, and hypophysis (anterior lobe). The diencephalon is located below the corpus callosum, part of the cerebrum, and above the midbrain. The thalamus makes up four-fifths of the diencephalon but the only parts that can be seen externally are the hypothala-mus (seen on the basal aspect of the brain) and portions of the epithalamus. In the adult brain the diencephalon is involved in endocrine functioning and autonomic coordination of the pineal, neurohypophysis, and hy-pothalamus. It also acts as a relay station for sensory information and somatic motor control via the thalamus. Fig. 47.13 Arrangement of the diencephalon around the third ventricle Oblique transverse section through the telecephalon with the corpus callosum, fornix, and choroid plexus removed, poste-rior view. This figure clearly illustrates that the lateral wall of the third ventricle forms the medial boundary of the diencephalon. Optic nerve Infundibulum Mammillary body Optic tract Cerebral peduncle Superior colliculus Inferior colliculus Brachium of inferior colliculus Pineal Lateral geniculate body Pulvinar Thalamus Corpus callosum Quadri-geminal plate Cerebellum Fig. 47.14 The diencephalon and brainstem Left lateral view. The cerebral hemispheres have been removed from around the thala-mus. The cerebellum has also been removed. The parts of the diencephalon visible in this dissection are the thalamus, the lateral genic-ulate body, and the optic tract. The latter two are components of the visual pathway. This dissection illustrates the role the diencepha-lon plays in linking the underlying brainstem to the overlying cerebral hemispheres. Mammillary body Cerebral peduncle Substantia nigra Red nucleus Cerebral aqueduct Lateral geniculate body Optic tract Hypothalamus Optic nerve Optic chiasm Infundibulum Tuber cinereum Fig. 47.15 Location of the diencephalon in the adult brain Basal view of the brain (brainstem has been sectioned at the level of the pons). The structures that can be identified in this view represent those parts of the diencephalon situated on the basal surface of the brain. This view also demonstrates how the optic tract winds around the cerebral peduncles. The expansion of the telencephalon during development limits the number of structures of the diencephalon visible on the under surface of the brain. They are: • Optic nerve • Optic chiasm • Optic tract • Tuber cinerum with the infundibulum • Mammilary bodies • Lateral geniculate body • Neurohypophysis 47 Brain 681 681 Thalamus Lateral geniculate body Pulvinar Quadrigeminal plate Primary fissure Horizontal fissure Posterior lobe Posterolateral fissure Tonsil Medulla oblongata Flocculus Pons Cerebral peduncle Mammillary body Infundibulum Optic n. (CN II) Anterior lobe 4th ventricle Pyramid Choroid plexus Nodule Horizontal fissure Primary fissure Central lobule Pineal Prebiventral fissure Superior medullary velum Lingula Corpus callosum Fornix Choroid plexus Anterior commissure Hypothalamus Optic chiasm Infundibulum Adenohypophysis Neurohypophysis Tectal Posterior lobe Quadrangular lobule Simple lobule Superior semilunar lobule Folium of vermis Primary fissure Culmen Vermis Anterior lobe Horizontal fissure Inferior semilunar lobule Median part Lateral parts Middle cerebellar peduncle Trigeminal n. (CN V) Vestibulocochlear n. (CN VIII) Facial n. (CN VII) Central tegmental tract Olive Inferior cerebellar peduncle Superior cerebellar peduncle Anterior spino-cerebellar tract Superior medullary velum Nodule Pyramid of vermis Tonsil Vallecula Peduncle of flocculus Flocculus Middle cerebellar peduncle Inferior cerebellar peduncle Superior cerebellar peduncle Lingula Central lobule Flocculo-nodular lobe Horizontal fissure Uvula vermis 4th ventricle Intermediate parts The stalk-like region of the brain connecting the cerebral hemispheres to the cerebellum and spinal cord consists of the diencephalon (thala-mus and associated structures) and the brainstem—composed of the mesencephalon or midbrain, pons and medulla oblongata moving sequentially caudal. Fiber bundles pass through this region from the spinal cord on their way to and from the cerebrum; thick fiber bundles pass contralaterally from the cerebrum into the cerebellar hemispheres; and 10 of the 12 cranial nerves are associated with the brainstem. Brain & Nervous System 682 Brainstem & Cerebellum Fig. 47.16 Diencephalon, brainstem, and cerebellum Left lateral view. A Isolated structures. B Midsagittal section. Fig. 47.17 Cerebellum A Superior view. B Anterior view. Fig. 47.18 Cerebellar peduncles Tracts of afferent (sensory) or efferent (mo-tor) axons enter or leave the cerebellum through cerebellar peduncles. Afferent axons originate in the spinal cord, vestibular organs, inferior olive, and pons. Efferent axons origi-nate in the cerebellar nuclei. Dien-cephalon Hypophysis Mesen-cephalon Pons Rhomboid fossa Medulla oblongata Cerebellum 4th ventricle Cerebral aqueduct Pyramid of medulla oblongata Anterior median fissure Decussation of pyramids C1 spinal n., anterior root Pons Cerebral peduncle Oculomotor n. (CN III) Interpeduncular fossa Hypoglossal n. (CN XII) Accessory n. (CN XI) Vagus n. (CN X) Glossopharyngeal n. (CN IX) Vestibulocochlear n. (CN VIII) Nervus intermedius Facial n. (CN VII) Abducent n. (CN VI) Trigeminal n. (CN V) Olive Brachium of inferior colliculus Cerebral peduncle Trochlear n. (CN IV) Pons CN V, motor root CN V, sensory root CN VI Nervus intermedius CN VII CN XII Olive CN XI C1 spinal n., ventral root Anterolateral sulcus Posterolateral sulcus CN X CN IX Lateral aperture Middle cerebellar peduncle Inferior cerebellar peduncle CN VIII Superior cerebellar peduncle Inferior colliculus Superior colliculus Tectal plate Superior cerebellar peduncle Rhomboid fossa Middle cerebellar peduncle Inferior cerebellar peduncle Vestibular area Striae medullaris CN XII, trigone CN X, trigone Taenia cinerea Facial colliculus CN V Medial eminence Superior medullary velum CN IV Superior and inferior colliculi Brachium of inferior colliculus Brachium of superior colliculus Pineal Tubercle of nucleus cuneatus Tubercle of nucleus gracilis 47 Brain 683 683 Fig. 47.19 Brainstem The brainstem is the site of emergence and entry of the 10 pairs of true cranial nerves (CN III–XII). See pp. 560–561 for an overview of the cranial nerves and their nuclei. A Levels of the brainstem. B Anterior view. C Left lateral view. D Posterior view. Superior sagittal sinus Arachnoid granulations Inter-hemispheric cistern Choroid plexus (lateral ventricle) Choroid plexus (3rd ventricle) Ambient cistern Confluence of sinuses Vermian cistern Cerebral aqueduct Choroid plexus (4th ventricle) Cerebellomedullary cistern (cisterna magna) Median aperture Pontomedullary cistern Interpeduncular cistern Chiasmatic cistern Basal cistern Inter-ventricular foramen Cistern of lamina terminalis Straight sinus Central canal of the spinal cord Vertebral venous plexus Endoneural space Spinal n. Spinal cord Subarachnoid space Ventricle Subarachnoid space Vein or venous sinus Choroid plexus CSF flow Brain & Nervous System 684 Ventricles & CSF Spaces Fig. 47.20 Circulation of cerebrospinal fluid (CSF) The brain and spinal cord are suspended in CSF. Produced continually in the choroid plexus, CSF occupies the subarachnoid space and ventricles of the brain and drains through arachnoid granulations into the dural venous sinus system (primarily the superior sagittal sinus) of the cranial cavity. Smaller amounts drain along proximal portions of the spinal nerves into venous plexuses or lymphatic pathways. Lateral recess 4th ventricle Left lateral ventricle 3rd ventricle Cerebral aqueduct Right lateral ventricle Anterior horn Inferior horn Collateral trigone Posterior horn Lateral ventricle, left Lateral ventricle Anterior horn Collateral trigone Posterior horn Central canal Lateral recess Inferior horn Cerebral aqueduct 3rd ventricle 4th ventricle Lateral ventricle Interventricular foramen Lateral ventricle (anterior horn) Corpus callosum Interventricular foramen Fornix Interthalamic adhesion Suprapineal recess Pineal gland Collateral trigone Lateral ventricle (posterior horn) Central canal Median aperture of 4th ventricle Lateral recess, ends as a lateral aperture of 4th ventricle 4th ventricle Lateral ventricle (inferior horn) Cerebral aqueduct Hypophysis Infundibular recess Optic chiasm Supraoptic recess 3rd ventricle Pineal recess Lateral ventricle (central part) Fornix Interventricular foramen Septum pellucidum Corpus callosum Infundibulum Anterior commissure Pons Hypothalamus Cerebral peduncle (crus cerebri) Medulla oblongata Quadrigeminal plate 47 Brain 685 685 Fig. 47.21 Ventricular system The ventricular system is a continuation of the central spinal canal into the brain. Cast specimens are used to demonstrate the connections between the four ventricular cavities. A Superior view. B Lateral ventricles in transverse section. C Left lateral ventricle in parasagittal section. D Left lateral view. Fig. 47.22 Ventricular system in situ Left lateral view. A 3rd and 4th ventricles in midsagittal section. B Ventricular system with neighboring structures. B Right hemisphere, medial view. Bridging vv. Superior anastomotic v. Superficial middle cerebral v. Superior and inferior petrosal sinuses Petrosal v. Internal jugular v. Sigmoid sinus Occipital sinus Transverse sinus Confluence of sinuses Medial and lateral superior cerebellar vv. Inferior anastomotic v. Superior cerebral vv. Superior sagittal sinus Thalamostriate v. Superior sagittal sinus Anterior v. of septum pellucidum Anterior cerebral v. Internal cerebral v. Basilar v. Superior cerebellar v. Posteromedian medullary v. Occipital sinus Transverse sinus Confluence of sinuses Straight sinus Great cerebral v. Internal occipital v. Inferior sagittal sinus Superior cerebral vv. Additional information on the venous sinus system and dural folds of the cranial cavity can be found on pp. 590–593. Brain & Nervous System 686 48 Blood Vessels of the Brain Veins & Venous Sinuses of the Brain Fig. 48.1 Superficial cerebral veins A Left hemisphere, lateral view. Superficial middle cerebral v. Internal cerebral v. Inferior choroidal v. Peduncular v. Deep middle cerebral v. Anterior cerebral v. Posterior venous confluence Basilar v. Anterior communi-cating v. Great cerebral v. Superior cerebellar vv. Superior petrosal v. Posteromedian medullary v. Transverse medullary vv. Transverse pontine vv. Anterolateral and anteromedian pontine v. Basilar v. Interpeduncular v. Pontomesencephalic v. Trigeminal n. (CN V) 48 Blood Vessels of the Brain 687 Fig. 48.2 Basal cerebral venous system Basal (inferior) view. Fig. 48.3 Veins of the brainstem Basal (inferior) view. Cerebral part External carotid a. Aortic arch Cervical part Petrous part Basilar a. Vertebral a. Left subclavian a. Posterior cerebral a. Posterior communicating a. Common carotid a. Internal carotid a. Vertebral a. Posteroinferior cerebellar a. Labyrinthine a. CN VI Anteroinferior cerebellar a. Basilar a. CN V Superior cerebellar a. CN III Posterior cerebral a. Medial occipital a. (P4) Lateral occipital a. (P3) Postcommunicating part (A2) Precommunicating part (A1) Internal carotid a. Posterior communicating a. Anterior inferior cerebellar a. Posterior inferior cerebellar a. Vertebral a. Anterior spinal a. Pontine aa. Anterior choroidal a. Anterior communicating a. Postcommunicating part (P2) Precommunicating part (P1) Insular part (M2) Sphenoidal part (M1) Basilar a. Superior cerebellar a. Posterior cerebral a. Anterior cerebral a. Middle cerebral a. Brain & Nervous System 688 Arteries of the Brain Fig. 48.4 Internal carotid artery Left lateral view. See p. 582 for details of the internal carotid artery. Fig. 48.5 Arteries of the brainstem and cerebellum Left lateral view. Fig. 48.6 Arteries of the brain Basal (inferior) view. Posterior parietal a. A. of postcentral sulcus A. of central sulcus A. of precentral sulcus Prefrontal a. Lateral frontobasal a. Temporo-occipital br. Anterior, middle, and posterior temporal brs. Posterior parietal a., angular gyral br. Aa. of precentral, central, and postcentral sulci Middle cerebral a. Lateral frontobasal a. Temporo-occipital br. Anterior, middle, and posterior temporal brs. Parieto-occipital br. Dorsal callosal br. Medial occipital a. (P4) Middle and posterior temporal brs. Anterior temporal a. (P3) Posterior cerebral a. Parieto-occipital brs. Pericallosal a. Precuneal brs. Cingular br. Calloso-marginal a. Polar frontal a. Anterior cerebral a. Thalamus Internal capsule Hippo-campus Globus pallidus Putamen Claustrum Insula Caudate nucleus Lateral ventricle Corpus callosum Cortical margin Anterior cerebral a. Middle cerebral a. Posterior cerebral a. Corpus callosum Optic chiasm Thalamus Cerebral aqueduct Pineal (epiphysis) Lateral ventricle 3rd ventricle Septum pellucidum Cortical margin Anterior commissure 48 Blood Vessels of the Brain 689 Fig. 48.7 Cerebral arteries A Middle cerebral artery. Left hemisphere, lateral view. B Middle cerebral artery. Left lateral view with the lateral sulcus retracted. C Anterior and posterior cerebral arteries. Right hemisphere, medial view. Fig. 48.8 Cerebral arteries: Distribution areas The central gray and white matter have a complex blood supply (yellow) that includes the anterior choroidal artery. A Left hemisphere, lateral view. B Right hemisphere, medial view. Sensory nuclei Motor nuclei Anterior horn (motor) Lateral horn (visceromotor) Posterior horn (sensory) Columns Nuclei Anterior root Plexus Spinal cord segment Nuclear column Peripheral n. Multisegmental muscle innervation Monosegmental muscle innervation Brain & Nervous System 690 49 Functional Systems Anatomy & Organization of the Spinal Cord Gray commissure Central canal Posterior rootlets Sensory ganglion Posterior root Spinal nerve Anterior root Anterior rootlets Anterior medial fissure Anterior column Posterior column Anterolateral sulcus Anterior gray horn Intermediolateral gray horn Lateral column Posterior median sulcus Posterior intermediate sulcus Posterior gray horn Fig. 49.1 Anatomy of a spinal cord segment Three dimensional representation, oblique anterior view from upper left. The gray matter of the spinal cord is found internally, surrounding the central canal in an H-shaped, or butterfly-like, configuration. This is the reverse of what was seen in the brain where the gray matter was on the external aspect in a cortical configuration. The primary function of the spinal cord is to conduct impulses to and from the brain. To fa-cilitate this, both gray and white matter are organized into longitudinal groupings. Fig. 49.2 Organization of the gray matter Left oblique anterosuperior view. The gray matter of the spinal cord is divided into three columns (horns). • Anterior column (horn): contains motor neurons • Lateral column (horn): contains sympathetic or parasympathetic (visceromotor) neurons in selected regions • Posterior column (horn): contains sensory neurons Sensory (blue) and motor (red) neurons within these columns are clustered in nuclei according to function. Fig. 49.3 Innervation of muscles Motor neurons that innervate specific muscles are arranged into vertical columns in the anterior horn of gray matter, the columns themselves can be called nuclei, in a fashion similar to that seen in brainstem motor nuclei. Most muscles (intersegmental muscles) receive innervation from numerous motor nuclei spanning several spi-nal cord segments. Monosegmental (or indicator) muscles have their motor neurons located entirely within a single spinal cord segment. Sulcomarginal fasciculus Lateral fasciculus proprius Longitudinal fasciculus of posterior column Interfascicular fasciculus (only in cervical cord) Septomarginal fasciculus (only in thoracic cord) Philippe-Gombault triangle (only in sacral cord) α-motor neuron Inter-calated cell Association cell Commissural cell Projection neuron Spinal ganglion α-motor neuron Lower motor neuron Motor interneuron Upper motor neuron (in the motor cortex) Interneuron Neuron in the sensory cortex Tertiary sensory neuron Secondary sensory neuron Primary sensory neuron Posterior funiculus Anterior funiculus Descending tracts (motor) Ascending tracts (sensory) Lateral funiculus Funiculi Tracts Fig. 49.4 Organization of the white matter Left oblique anterosuperior view. The gray matter columns partition the white matter analogously into anterior, lateral, and posterior columns or funiculi. The white matter of the spinal cord contains ascending and descending tracts which are the CNS equivalent of peripheral nerves. Fig. 49.5 Overview of sensorimotor integration Schematic illustrates the pathway of incoming primary sensory neuron impulses, the axon of which ascends to synapse with the secondary and tertiary sensory neurons in the brainstem and cerebrum ending in a synapse on a neuron in the sensory cortex. An interneuron links this with an upper motor neuron in the motor cortex which then descends through the white matter funiculi of the spinal cord to a motor neuron, which then synapses with a lower motor neuron, the axon of which passes out the spinal nerve to the effector organ. Fig. 49.6 Principle intrinsic fascicles of the spinal cord (shaded yellow) Left oblique anterosuperior view. The majority of muscles have a multi segmental mode of innervaton that necessitates axons to ascend/descend multiple spinal cord segments to coordinate spinal reflexes. The neurons of these axons originate from interneurons in the gray matter forming intrinsic reflex pathways of the spinal cord. These axons are collected into intrinsic fascicles which are arranged chiefly around the gray matter. These bundles make up the intrinsic circuits of the spinal cord. Fig. 49.7 Intrinsic circuits of the spinal cord Sensory neurons are shown in blue, motor neurons in red. The neurons of the spinal reflex circuits are in black. These chains of interneurons, which are entirely contained within the spinal cord, comprise the intrinsic circuits of the cord. The axons of these intrinsic circuits pass to adjacent segments in intrinsic fascicles located along the edge of the gray matter. 49 Functional Systems 691 A S D F G H Sensory cortex (postcentral gyrus) 3rd neurons Thalamus 2nd neuron Medial lemniscus ② ② ① ③ ④ ⑤ 2nd neurons Nucleus gracilis Nucleus cuneatus Accessory nucleus cuneatus Cuneocere-bellar fibers α-motor neuron Sensory ganglion of spinal n. (with 1st neurons) Unconscious proprioception Position sense, conscious proprioception, vibration, touch Pain, temperature Pressure, touch Anterolateral system (spino-thalamic tracts) Brain & Nervous System 692 Sensory & Motor Pathways Fig. 49.8 Sensory pathways (ascending tracts) Table 49.1 Sensory pathways (ascending tracts) of the spinal cord Tract Location Function Neurons ① Anterior spino- thalamic tract Anterior funiculus Pathway for crude touch and pressure sensation 1st afferent neurons located in spinal ganglia; contain 2nd neurons and cross in the anterior commissure ② Lateral spino- thalamic tract Anterior and lateral funiculi Pathway for pain, temperature, tickle, itch, and sexual sensation ③ Anterior spino- cerebellar tract Lateral funiculus Pathway for unconscious coordination of motor activities (unconscious proprioception, automatic processes, e.g., jogging, riding a bike) to the cerebellum Projection (2nd) neurons receive proprioceptive signals from 1st afferent fibers originating at the 1st neurons of spinal ganglia ④ Posterior spino- cerebellar tract ⑤ Fasciculus cuneatus Posterior funiculus Pathway for position sense (conscious proprioception) and fine cutaneous sensation (touch, vibration, fine pressure sense, two-point discrimination) Conveys information from upper limb (not present below T3) Cell bodies of 1st neuron located in spinal ganglion; pass uncrossed to the dorsal column nuclei ⑥ Fasciculus gracilis Conveys information from lower limb The fasciculi cuneatus and gracilis convey information from the upper and lower limbs, respectively. At this spinal cord level, only the fasciculus cuneatus is present. A S D F G H Corticospinal tracts (pyramidal tract) Descending tracts from brainstem (extrapyramidal motor system) Precentral gyrus (primary motor cortex) Postcentral gyrus (primary somato-sensory cortex) Corticonuclear fibers Corticospinal fibers CN VII CN XII Pyramidal decussation Supplementary motor cortex, premotor cortex Corticospinal tracts Tegmental nucleus Substantia nigra Red nucleus Pyramidal tract Pyramid α-motor neuron (with interneurons) Spinal n., anterior root Spinal n., posterior root Inferior olive Ventral intermedius nucleus From cerebellum ② ① ③ ⑤ ⑦ ⑥ Leg Arm Face ④ ② ① 49 Functional Systems 693 Fig. 49.9 Motor pathways (descending tracts) Table 49.2 Descending tracts of the spinal cord Tract Function Corticospinal tract (pyramidal tract) ① Anterior corticospinal tract Most important pathway for voluntary motor function Originates in the motor cortex Corticonuclear fibers to motor nuclei of cranial nerves Corticospinal fibers to motor cells in anterior horn of the spinal cord Corticoreticular fibers to nuclei of the reticular formation ② Lateral corticospinal tract Descending tracts from the brainstem (Extrapyramidal motor system) ③ Rubrospinal tract Pathway for automatic and learned motor processes (e.g., walking, running, cycling) ④ Reticulospinal tract ⑤ Vestibulospinal tract ⑥ Tectospinal tract ⑦ Olivospinal tract Brain & Nervous System 694 50 Autonomic Nervous System Autonomic Nervous System (I): Overview Fig. 50.1 Autonomic nervous system The autonomic nervous system is the part of the peripheral nervous system that innervates smooth muscle, cardiac muscle, and glands. It is subdivided into the sympathetic (red) and the parasympathetic (blue) nervous systems, which often act in antagonistic fashion to regulate blood flow, secretions, and organ function. Both the sympathetic and parasympathetic nervous systems have a two-neuron pathway, which is under central nervous system control via an upper motor neuron with its cell body in the hypothalamus. In the sympathetic system, the preganglionic neuron synapses within the ganglia of the sympathetic trunk (paired, one on each side of vertebral column) or on one of the unpaired prevertebral ganglia located at the base of the artery for which the ganglion was named (celiac, superior and inferior mesen-teric). Sympathetic postganglionic neurons then either reenter spinal nerves via gray rami communicans and are distributed to their target structure or they reach their target structure by travelling with arteries. Except in the head, parasympathetic preganglionic neurons synapse in ganglia in the wall of the target organ. Short postganglioinc parasym-pathetic neurons then innervate the organ. In the head there are four parasympathetic ganglia: ciliary, pterygopalatine, submandibular, and otic, which are associated with cranial nerves III, VII, and IX, respec-tively. These four ganglia are responsible for distributing fibers to smooth muscle within the eye and to the salivary glands and glands of the nasal cavity, paranasal sinuses, hard and soft palate, and pharynx. Both sympathetic and parasympathetic preganglionic neurons secrete acetylcholine, which acts upon nicotinic receptors in the ganglia. Sympathetic postganglionic neurons secrete norepinephrine, which acts upon adrenoceptors (α or β) in target tissues. Parasympathetic postganglionic neurons secrete acetylcholine, which acts upon muscarinic receptors in target tissues. T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 L1 L2 L3 L4 L5 S2 Superior cervical ganglion Sympathetic trunk CN III CN VII CN IX CN X Cranial part: brainstem with para-sympathetic nuclei Parasympathetic ganglia (in the head) Pelvic splanchnic nn. Parts of the colon, rectum Stellate ganglion Superior mesenteric ganglion Inferior mesenteric ganglion Parasympathetic ganglia (within the walls of organs) Genitalia Bladder S5 S4 S3 Celiac ganglion Greater splanchnic n. Intestine Kidney Eye Lacrimal and salivary glands Cranial vessels Heart Lung Stomach Liver Pancreas Sympathetic nervous system Parasympathetic nervous system Stellate ganglion - inferior cervical ganglion and T1 sympathetic ganglion Sacral part: sacral cord with para-sympathetic nuclei Inferior hypogastric plexus A B S1 Minimal sympathetic preganglionic fibers traveling through the sacral splanchnic nerves will synapse in the ganglia located in the inferior hypogastric plexus. 50 Autonomic Nervous System 695 Table 50.1 Parasympathetic pathways Neuron Location of cell body Upper motor neuron Hypothalamus: The cell bodies of parasympathetic upper motor neurons are located in the hypothalamus. Their axons descend via white matter tracts to synapse with the lower motor neuron in the brainstem and sacral spinal cord (S2–S4). Preganglionic neuron (lower motor neuron) The parasympathetic nervous system is divided into two parts (cranial and sacral), based on the location of the preganglionic parasympathetic neurons. Brainstem cranial nerve nuclei: The axons of these secondary neurons leave the CNS as the motor root of cranial nn. III, VII, IX, and X. Spinal cord (S2–S4): The axons of the sacral parasympathetics originate from the S2–S4 spinal segments in a region of the spinal cord gray matter similar to that of the lateral horns where the sympathetic division arises from. These axons initially travel through the S2–S4 anterior roots before con-tinuing within the S2–S4 anterior rami. The axons then pass through pelvic splanchnic nerves that arise from the S2–S4 anterior rami before reaching the inferior hypogastric plexus, which then distributes the axons to the pelvic and hindgut structures. Postganglionic neuron Cranial nerve parasympathetic ganglia: The parasympathetic cranial nn. of the head each have at least one ganglion: • CN III: Ciliary ganglion • CN VII: Pterygopalatine ganglion and submandibular ganglion • CN IX: Otic ganglion • CN X: Small unnamed (intramural) ganglia close to target structures Distribution of postganglionic fibers Parasympathetic fibers course with other fiber types to their targets. In the head, the postganglionic fibers from the pterygopalatine ganglion (CN VII) and otic ganglion (CN IX) are distributed via branches of the trigeminal n. (CN V). Postganglionic fibers from the ciliary ganglion (CN III) course with sympathetic and sensory fibers in the short ciliary nn. (preganglionic fibers travel with the somatomotor fibers of CN III). In the thorax, abdomen, and pelvis, preganglionic parasympathetic fibers from CN X and the pelvic splanchnic nn. combine with postganglionic sympathetic fibers to form plexuses (e.g., cardiac, pulmonary, esophageal). Table 50.2 Sympathetic pathways Neuron Location of cell body Upper motor neuron Hypothalamus: The cell bodies of sympathetic upper motor neurons are located in the hypothalamus. Their axons descend via white matter tracts to synapse with the lower motor neuron in the lateral horn of the spinal cord (T1–L2). Preganglionic neuron (lower motor neuron) Intermediolateral gray horn of spinal cord (T1–L2): The lateral horn is the middle portion of the gray matter of the spinal cord, situated between the anterior and posterior horns. It contains exclusively autonomic (sympathetic) neurons. The axons of these neurons leave the CNS as the motor root of the spinal nn. and enter the paravertebral ganglia via the white rami communicans (myelinated). Preganglionic neurons in paravertebral ganglia All preganglionic sympathetic neurons enter the sympathetic chain. There they may synapse in a chain ganglion or ascend or descend to synapse. Preganglionic sympathetic neurons synapse in one of two places, yielding two types of sympathetic ganglia. Synapse in the paravertebral ganglia Pass without synapsing through the sympathetic ganglia. These fibers travel in the thoracic, lumbar, and sacral splanchnic nn. to synapse in the prevertebral ganglia. Postganglionic neuron Paravertebral ganglia: These ganglia form the sympathetic nerve trunks that flank the spinal cord. Postganglionic axons leave the sympathetic trunk via the gray rami communicans (unmyelinated). Prevertebral ganglia: Associated with peripheral plexuses, which spread along the abdominal aorta. There are three primary prevertebral ganglia: • Celiac ganglion • Superior mesenteric ganglion • Inferior mesenteric ganglion Distribution of postganglionic fibers Postganglionic fibers are distributed in two ways: 1. Spinal nerves: Postganglionic neurons may re-enter the spinal nn. via the gray rami communicans. These sympathetic neurons induce constriction of blood vessels of the skin and dilate the blood vessels of skeletal muscles, sweat glands, and arrector pili (muscle fibers attached to hair follicles, “goose bumps”). 2. Arteries and ducts: Nerve plexuses may form along existing structures. Postganglionic sympathetic fibers may travel with arteries to target structures. Viscera are innervated by this method (e.g., sympathetic innervation concerning vasoconstriction, bronchial dilatation, glandular secretions, pupillary dilatation, smooth muscle contraction). Brain & Nervous System 696 Autonomic Nervous System (II) Posterior root Sensory (spinal) ganglion Spinal n. Posterior ramus Anterior ramus Anterior root Spinal Cord Level L2 Spinal Cord Level L3 Somatic sensory (afferent) Somatic motor (efferent) Fig. 50.2 Typical spinal nerve All spinal nerves arising from the spinal cord contain somatic sensory (or afferent, from body wall) and somatic motor (or efferent, to body wall) fibers. Sensory fibers come from the posterior (back) region via the posterior ramus and anterolateral regions of the body wall via the anterior ramus of the spinal nerve. The somatic sensory fibers approach the spinal cord via the posterior root. The cell bodies for these fibers lie in the sensory (spinal/dorsal root) ganglion. They synapse with sensory neurons in the posterior horn of gray matter within the spinal cord sending the majority to the brain for interpretation. Somatic motor fibers have their neurons in the anterior horn of gray matter and send their fibers to the spinal n. via the anterior root. This pattern of somatic innervation occurs in all spinal nerves from C1 through S5, whether they are involved in a plexus or not. Fig. 50.3 ANS Circuitry Body wall dermatomes also require sympathetic fibers to contract smooth muscle and cause glands in the dermatome to secrete. Preganglionic sympathetic fibers (purple) arise from cell bodies in the intermediolateral gray horn of the spinal cord. They exit the spinal cord via the outgoing/efferent (anterior) root—along with the somatic motor (efferent) fibers—and enter the spinal nerve. The smooth muscle of the body wall requires innervation by postganglionic sympathetic fibers so the preganglionic fiber looks for the closest synapse site—the para- vertebral sympathetic ganglia—found in a chain-like arrangement on either side of the vertebral column. Each ganglion is connected to the spinal n. by communicating branches—the rami communicans. The white ramus communicans is found most lateral and conveys the pre- ganglionic (myelinated = white) sympathetic fiber to the ganglion. Once in the paravertebral ganglion one of two things can happen: a) The preganglionic sympathetic fiber can synapse in the ganglion and the postganglionic sympathetic fiber (orange) passes along the gray ramus communicans (unmyelinated) back to the spinal n. Now postganglionic sympathetic fibers can be distributed to structures in the dermatome via the anterior and posterior rami—along with somatic motor and sensory fibers. b) The preganglionic fiber can run up or down the sympathetic trunk to synapse in an upper or lower paravertebral ganglion. This is especially important as the source of sympathetic innervation is limited to spinal cord levels T1 to L2. This figure depicts sympathetic innervation from the last spinal cord segment to contain it (L2) descending along the sympathetic trunk to the paravertebral ganglion at L3. It synapses here and the postganglionic sympathetic fiber exits into the spinal nerve of L3. Note that there is only a gray ramus communicans at this level as white rami communicans are input fibers (T1–L2), while the gray are output fibers above and below T1 and L2. Therefore, there are more gray rami than white rami. Both anterior and posterior rami now contain postganglionic sympathetic fibers distributed to the dermatome of L3 along with the typical somatic sensory and motor fibers of each vertebral level. Now that the body wall has been supplied with postganglionic sympathetic innervation, we’ll turn our attention to the viscera. In the Posterior ramus Anterior ramus White ramus communicans Paravertebral (sympathetic) ganglion Spinal n. - L3 Gray ramus communicans Sympathetic trunk Wall of small intestine Intramural (terminal) ganglion Spinal cord level L2 Anterior root Splanchnic n. Prevertebral (collateral) ganglion Spinal n. - L2 Sensory (spinal) ganglion Posterior root Brain stem Vagus n. (CN X) (preganglionic parasympathetic) Gray ramus communicans Somatic afferent (sensory) Somatic efferent (motor) Sympathetic, preganglionic Sympathetic, postganglionic Parasympathetic, preganglionic Parasympathetic, postganglionic Visceral afferent (sensory) 50 Autonomic Nervous System 697 third option for preganglionic sympathetic fibers entering a preverte-bral ganglion, the fibers pass through the paravertebral ganglion at that level without synapsing and pass into a splanchnic n. to synapse in one of 3 primary prevertebral (or collateral) ganglia found in the abdomen along the anterior surface of the aorta at the base of one of the three main visceral branches (celiac a., superior mesenteric a., and inferior mesenteric a.). The postganglionic sympathetic fibers (orange) are distributed by following arterial branches to the viscera where they decrease the activity of the glands and peristalsis, constrict sphincters, and vasoconstrict the blood vessels. The body wall does not receive any parasympathetic innervation. Dilation of the blood vessel walls occurs as the postganglionic sympa-thetics stop firing to cause vasoconstriction. However, the intricate control of movement of the wall of the intestine, or secretion of the glands within its wall, does require the antagonistic input of the para- sympathetic division. Parasympathetic innervation to the viscera of the thorax and much of the abdomen (to the mid transverse colon) is supplied by the vagus n. (dark blue). The vagus n. sends branches to the various sympathetic prevertebral ganglia of the abdomen but they do not synapse there. They pass through, following the branches of the blood vessel to the wall of the organs supplied. There they synapse in tiny parasympathetic (intramural) ganglion within the wall of the organ. The postganglionic parasympathetic fibers (light blue) are therefore extremely short. The remainder of the abdominal and pelvic viscera receive their parasympathetic supply in a similar fashion but from preganglionic parasympathetic fibers from spinal cord levels S2-4. Viscera also exhibit pain, relayed back to the CNS as visceral afferents (dark green). Note that the visceral afferent fibers follow the pathway of the sympathetic pre- and postganglionic fibers back from the viscera. They pass through the prevertebral ganglion (without synapsing), and then back along the splanchnic n. and through the paravertebral ganglion (again, without synapsing). From there they travel along the white ramus communicans (as visceral afferent fibers, they are also myelinated) and follow the posterior root back to the sensory ganglion where the cell body is found interspersed amongst those for the body wall. They finally synapse in the posterior horn of gray matter in the spinal cord amongst the somatic afferents also synapsing there. This is the basis for referred pain as the brain finds it difficult to distinguish visceral pain from somatic pain as the latter outnumber the former very significantly. Therefore, pain from internal organs is often referred to sites on the body wall. Fig. 51.1 Sagittal section through the midline of the brain Fornix Interventricular foramen Septum pellucidum Optic chiasm Pineal Fourth ventricle Nodule Uvula Superior medullary velum Lingula Primary fissure Anterior lobe of cerebellum Corpus callosum, splenium Corpus callosum, trunk Corpus callosum, genu Inferior medullary velum Third ventricle Calcarine sulcus Parieto-occipital sulcus Infundibulum Hypophysis Anterior commissure Cingulate gyrus Pons Hypothalamus Cerebral peduncle (crus cerebri) Central canal Medulla oblongata Cerebral aqueduct Tectal plate Fornix, crus Posterior commissure Hippocampus Lateral geniculate body Medial geniculate body Longitudinal cerebral fissure Dentate gyrus Insula Caudate nucleus, tail Corpus callosum, trunk Lateral ventricle, central part Caudate nucleus, body Third ventricle Internal capsule, posterior limb Choroid plexus of lateral ventricle Thalamic nuclei Fimbria of hippocampus Choroid plexus of lateral ventricle Superior cerebellar peduncle Middle cerebellar peduncle Medulla oblongata Flocculus Horizontal fissure Anterior lobe of cerebellum Fig. 51.2 Frontal section through the brain I Brain & Nervous System 698 51 Sectional & Radiographic Anatomy Sectional Anatomy of the Nervous System Fig. 51.3 Frontal section through the brain II Anterior lobe of cerebellum Central gray matter Quadrigeminal plate, superior colliculus Rhomboid fossa Choroid plexus of fourth ventricle Lateral ventricle, central part Fornix, crus Longitudinal cerebral fissure Corpus callosum, trunk Choroid plexus of lateral ventricle Caudate nucleus, tail Internal cerebral veins Hippocampus Insula Posterior lobe of cerebellum Cerebral aqueduct Cerebellar tonsil Choroid plexus of lateral ventricle Caudate nucleus, body Pineal Thalamus, pulvinar Middle cerebellar peduncle Cerebral peduncle (crus cerebri) Mesencephalon Cerebral aqueduct Lateral geniculate body Vermis of cerebellum Hippocampus Striate area Lateral ventricle, posterior horn Insula Choroid plexus of lateral ventricle Quadrigeminal plate, superior colliculus Medial geniculate body Amygdala Red nucleus Substantia nigra Optic chiasm Medial geniculate body Optic tract Third ventricle, optic recess Fig. 51.4 Transverse section of the brain through the upper region of the brainstem 51 Sectional & Radiographic Anatomy 699 Fig. 51.5 MRI of the brain Midsagittal section, left lateral view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Fig. 51.6 MRI of the brain Transverse (axial) section through the cerebral hemispheres, inferior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Frontal lobe Corpus callosum (body) Fornix Septum pellicidum Corpus callosum (genu) Interthalamic adhesion Third ventricle Anterior cerebral a. Basilar a. Hypophysis Sphenoid sinus Internal cerebral v. Choroid plexus Parietal lobe Corpus callosum (splenium) Cerebellar vv. Straight sinus Midbrain Fourth ventricle Pons Cerebellum Medulla oblongata Falx cerebri Caudate nucleus Corpus callosum Putamen Thalamus Third ventricle Choroid plexus Great cerebral v. Frontal bone Superior sagittal sinus Frontal lobe Frontal horn of lateral ventricle Parietal lobe Temporal lobe Corpus callosum Posterior horn of lateral ventricle Posterior horn of lateral ventricle Superior sagittal sinus Occipital bone Additional radiological images of the blood supply to the brain can be found on p. 671. Brain & Nervous System 700 Radiographic Anatomy of the Nervous System Fig. 51.7 MRI of the brain Coronal section through the ventricular system. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Fig. 51.8 MRI of the neck Coronal section through the cervical spinal cord, anterior view. (Reproduced from Moeller TB, Reif E. Pocket Atlas of Sectional Anatomy, Vol 1, 4th ed. New York, NY: Thieme; 2014.) Interhemispheric fissure Corpus callosum Lateral ventricle Putamen Hypothalamus Basilar a. Anterior inferior cerebellar a. Vertebral a. Septum pellucidum Caudate nucleus (body) Thalamus Third ventricle Posterior cerebral a. Superior cerebellar a. Temporal lobe Internal carotid a. Atlas (C1) Parotid gland Internal jugular v. Spinal n. roots C4, C5, C6 Middle scalene Spinal n. root C8 Spinal cord Sternocleidomastoid Intervertebral foramen Vertebral a. Dens of C2 Medulla oblongata Temporal lobe 51 Sectional & Radiographic Anatomy 701 Index Index 704 Index A Abdomen. See also individual organs arteries of, 212t autonomic plexuses of, 212–217 bones of, 142–143 female, 196 lymphatic drainage of, 85, 202–209, 212t regions of, 140t retroperitoneal organs of, 159 transverse plane through, 140t walls of anterior, 148–149, 186–187 bony framework of, 142–143 layers of, 146 lymphatics of, 204–205 mesenteric sites of attachment to, 164 muscles of, 25, 144–149, 148t anterior, 148, 148t, 149 anterolateral, 144–145, 148, 149 posterior, 146–147, 148, 148t, 149 nerves of, 210 veins of, 194–195 Abdominal press, 64t Abduction at carpometacarpal joint of the thumb, 347 of extraocular muscles, 604, 605t at metacarpophalangeal joint, 346 Accommodation, in pupillary dilation, 579 Acetabulum, 233, 237, 238, 413, 415, 417 fossa of, 231 labrum of, 413, 415, 417, 473 lunate surface of, 231 margin of, 232 MRI of, 288 notch of, 231 radiograph of, 231 rim of, 231 roof of, radiograph of, 504 Acromion, 295, 297, 298, 299, 302, 303, 304, 305, 306, 307, 313, 316, 317, 318, 319, 320, 321, 381, 516 MRI of, 401 radiograph of, 400 Acute abdominal pain, 156 Adduction at carpometacarpal joint of the thumb, 347 of extraocular muscles, 604, 605t at metacarpophalangeal joint, 346 Adenohypophysis, 680, 682 Adhesion, interthalamic, 679, 680 MRI of, 700 Aditus (inlet), to the mastoid antrum, 629 Adventitia esophageal, 166 with visceral pelvic fascia, 252 Airway, 648 respiratory portion of, 121 Ala (wing) of ilium, 51, 232 computed tomography (CT) of, 223 nasal, 616 of sacrum, 7, 12, 13, 232, 233 of sphenoid bone, 542, 543 greater, 603, 617, 619 lesser, 547, 603, 617, 619, 622, 636 Alveolar bone, 640 Alveoli, 121 capillary bed of, 126 conditions of, 121 epithelial lining of, 121 pulmonary, 121, 126 respiratory compromise and, 121 tooth sockets, 637 Ampulla of ductus deferens, 156, 253, 266 hepatopancreatic, 178 posterior, 635 rectal, 249 in defection, 283 MRI of, 290 urethral, 264 of uterine tube, 257 Amygdala, 562, 661, 699 Anal region, 3 Anastomoses abdominal arterial, 187 between arteries of the large intestine, 193 Galen’s, 531 between inferior phrenic and suprarenal veins, 185 with medial brachial cutaneous nerve, 374 of mesenteric artery, 187 portocaval, 195, 199 between superior mesenteric and inferior pancreatic arteries, 191 transverse cervical, 538 venous, 589t Anconeus, 312, 314, 315, 323, 323t, 327, 334, 389 MRI of, 399 Angina pectoris, 101 Angle acromial, 299 aperture, sagittal (anterior), 413 costal, 56, 57, 59 epigastric, 122 of eyeball chamber, 612, 614, 614 inferior, 295, 297, 319, 320, 321 of scapula, 297, 299 radiograph of, 400 lateral, of scapula, 299 of mandible, 512, 513, 559, 641, 653, 666 sternal, 54, 56, 58, 80 subpubic, 234t superior, 295, 297, 317, 320, 513 of clavicle, 303 of scapula, 299 radiograph of, 400 Ankle. See also Joints (individual), talocrural bones of, 452–453 ligaments of, 460–461 mortise of, 410, 456 MRI of, 509 neurovasculature, 495 retinacula of, 467 tendon sheaths of, 467 ANS. See Autonomic nervous system (ANS) Ansa cervicalis, 607 inferior root of (descendens cervicalis), 607 superior root of (descendens hypoglossus), 607 Anterolateral system, 692 Antetarsus (phalanges), 452 Antihelix, 512, 627 crus of, 627 Antitragus, 512, 627 Antrum gastric, 167 mastoid, 632 aditus (inlet) to, 629 pyloric, 166, 167 Anulus fibrosis, 14, 17, 22, 98 disk herniation and, 15 Anus, 3, 229, 242, 243, 249, 284, 285. See also Sphincters, anal anterior border of, 229 canal of, 248–249 cleft of, 239 sphincters of external, 239, 240, 241t, 242, 243, 248, 249 subcutaneous part of, 249 Aorta, 89, 92, 106, 174, 208 abdominal, 5, 25, 36, 38, 67, 68, 80, 92, 104, 105, 109, 110, 147, 156, 159, 165, 169, 175, 180, 181, 182, 185, 187, 188, 190, Note: Clinical applications, imaging, sectional anatomy, and surface anatomy are found under these main headings, broken out by region. Italicized page numbers represent clinical applications. Tabular material is indicated by a “t” following the page number. Index 705 Artery/arteries (individual) 193, 194, 196, 197, 220, 221, 250, 268, 269, 270, 273, 274, 276, 277, 278, 472, 473 branches of, 186t computed tomography (CT) of, 224 ascending, 5, 79, 80, 81, 81t, 84, 88, 92, 94, 95, 96, 97, 99, 103, 124, 126, 131, 132 angiography of, 135 computed tomography (CT) of, 79, 134, 136 MRI of, 133, 135 descending, 79, 81, 81t, 91, 116 angiography of, 135 thoracic, 36, 44, 67, 68, 68t, 73, 80, 81, 81, 81t, 84, 86, 91, 106, 108, 109, 109t, 113, 115, 116, 126, 129, 130, 186, 364 branches of, 126 Aortic dissection, 81 Aperture in brainstem lateral, 683, 685 median, 684 piriform (anterior nasal), 543, 617 of sphenoid sinus, 551 thoracic inferior, 56 superior, 56 Apex of heart, 92, 93, 96, 97, 99, 100, 102, 124, 131, 132 of lung, 117 of orbit, 657 of patella, 435 of prostate gland, 266 of root of tooth, 649 of sacrum, 12 of tongue, 646, 647 of urinary bladder, 251, 252, 253, 267 Aponeurosis bicipital, 310, 322, 332, 386, 387, 396 dorsal, 467 epicranial, 552, 553 of flexor digitorum longus, 499 gluteal, 24 lingual, 646 oblique, external, 144, 146, 150, 151, 153, 488, 489, 492 palatine, 643 palmar, 295, 378, 390, 391, 392, 396 plantar, 457, 458, 463, 464, 466, 469, 485, 498, 499, 509 MRI of, 509 Appendicitis, 156 Appendix/appendices, 161, 172 of epididymis, 265 epiploic, 170 fibrous, of liver, 176, 177 of testis, 265 vermiform, 172, 245 vesicular, 257 Aqueduct cerebral, 564, 680, 681, 683, 684, 685, 689, 698, 699 cochlear, 628, 634 vestibular, 635 veins of, 635 Arachnoid mater, 40, 41, 587, 590 inner layer of, 38 Arcade anastomotic, 627 Riolan’s, 193 scapular, 365 Arch aortic, 36, 78, 80, 81t, 86, 88t, 89, 91, 92, 93, 96, 97, 102, 104, 105, 106, 108, 109, 110, 111, 116, 124, 125, 131, 132, 531, 688 angiography of, 135 with aortic plexus, 103 MRI of, 133 with thoracic plexus, 103 coracoacromial, 303, 305, 306, 307 costal, 56 margin of, 309 of cricoid cartilage, 527 dorsal metacarpal, 365 of forefoot, anterior, 462 iliopectineal, 152, 268, 489, 489t jugular venous, 521, 534 longitudinal, stabilizers of, 463 metatarsal, 462 palatoglossal, 648, 650 palatopharyngeal, 648, 650, 651 palmar venous deep, 364, 365, 366, 391, 393 metacarpal, 365 superficial, 364, 365, 366, 390, 391, 393 plantar deep, 472, 499 medial, 499 pubic, 232, 233 superciliary, 543 tendinous of levator ani, 238, 251, 258, 259 of soleus, 450, 494 transverse, stabilizers of, 462 venous dorsal, 474 plantar, 474 posterior, 474, 475 vertebral, 7, 9, 14, 19, 21, 22, 25, 665 of atlas (C1) anterior, 9, 21, 49, 66, 659 posterior, 8, 9, 18, 19, 21, 27, 659 fractures of, 9 joints of, 16t lumbar, 7, 11 MRI of, 669 pedicles of, 20, 23 zygomatic, 513, 542, 546, 558, 559, 597, 657 Area(s) intercondylar anterior, 433, 443 posterior, 433 Kiesselbach’s, 620 striate, 563, 699 vestibular, 683 Areola, 74, 75 Arm. See also Forearm; Wrist arteries of, 364–365 bones of, 296–297 motor pathways of, 693 MRI of, 398 muscles of, 386, 396–397 anterior, 308–311, 322, 322t posterior, 312–315, 323, 323t nerves of, 380 regions of anterior, 294 brachial, 386 cubital, 387 anterior, 294 posterior, 294 posterior, 294 windowed dissection of, 396 Artery/arteries (individual) alveolar anterior, 587 inferior, 599, 601, 601t, 656, 657 branches of, 594 posterior, 599 posterior superior, 587, 601, 601t angular, 582, 584, 594, 608 arcuate, 184, 189, 472, 497 auditory, internal, 635 auricular, 627 deep, 587, 601, 601t, 633 internal, 633 posterior, 583, 583, 585, 585t, 632 axillary, 68, 68t, 72, 74, 186, 364, 365, 372, 373, 375, 376, 377, 383, 384, 384t, 385, 386, 520, 584, 620, 632 MRI of, 398, 401 basilar, 44, 582, 662, 667, 688 MR angiography of, 671 MRI of, 701, 700 brachial, 76, 364, 365, 384, 385, 386, 388, 389, 397 deep MRI of, 398 deep, MRI of, 398 MRI of, 398, 399 Index 706 Artery/arteries (individual) (continued) Artery/arteries (individual) (continued) bronchial, 80, 126 buccal, 587, 598, 599, 601, 601t callosomarginal, 689 cardiac, 100–101, 100t caroticotympanic, 582, 633 carotid atheroscleosis of, 583 common, 25, 36, 66, 68, 78, 80, 81, 81t, 84, 85, 92, 96, 103, 108, 109, 111, 125, 126, 130, 365, 383, 520, 530, 531, 535, 537, 582, 654, 655, 664, 665, 688 angiography of, 135 branches of, 109 left, 364, 535, 536, 583, 583, 584, 586 MRI of, 669 normal blood flow in, 583 right, 534, 536 external, 520, 537, 582, 583, 620, 627, 632, 632t, 647, 654, 688 branches of, 583, 583, 584, 585t, 586, 586t plexus of, 579 internal, 520, 537, 565, 571, 582, 583, 584, 590, 593, 606, 607, 620, 621, 628, 629, 631, 632, 632t, 633, 654, 658, 661, 662, 688 computed tomography (CT) of, 670 course of, 582 with internal carotid plexus, 607 MR angiography of, 671 parts and branches of, 582, 688 petrous part of, 670 carpal, dorsal, 365 cecal anterior, 172, 192, 193, 200, 201 posterior, 192, 193, 200 cerebellar anterior, 688 anterior inferior MRI of, 701 anteroinferior, 688 middle, 689 posterior inferior, 688 superior, 688 MR angiography of, 671 MRI of, 701 cerebral, 590, 689 anterior, 582, 658, 689 MR angiography of, 671 MRI of, 700 parts and branches of, 671, 688 branches of, 688, 689 distribution areas of, 689 middle, 582, 590 MR angiography of, 671 posterior, 688, 689 branches of, 671 MRI of, 701 parts and branches of, 688 right, MR angiography of, 671 cervical ascending, 44, 365, 383, 520, 535 deep, 44, 365, 520 superficial, 365, 537, 538, 539 transverse, 365, 383, 520, 535, 536 choroidal, anterior, 582, 688 ciliary anterior, 613 long posterior, 606, 613 short posterior, 606, 609 circumflex, 98 femoral lateral, 472, 473, 492, 505 medial, 472, 473, 491, 492, 493 MRI of, 502, 505 humeral anterior, 364, 365 MRI of, 401 posterior, 364, 365, 369, 381, 401 iliac deep, 152, 186 superficial, 488, 492 right, angiography of, 134 scapular, 381, 383, 384, 385 MRI of, 398 clitoral deep, 275, 285 dorsal, 261, 275, 285 cochlear common, 635 proper, 635 colic left, 165, 169, 181, 183, 187, 193, 201, 219 middle, 156, 160, 170, 187, 192, 193, 200, 219 right, 19, 187, 192, 193, 200, 201, 218, 219 collateral middle, 364 radial, 364 right, 389, 3889 ulnar inferior, 364, 386, 388 superior, 364, 386, 387, 388 communicating anterior, 688 MR angiography of, 671 posterior, 582, 688 conjunctival, 613 coronary in aortic dissection, 81 branches of, 100, 100t, 101 computed tomography (CT) of, 134 distribution of, 101 disturbed blood flow in, 101 dominance patterns, 101 left branches of, 100, 100t, 134 computed tomography of, 134 opening of, 99 right, 81t, 98, 100, 101 angiography of, 134 branches of, 100, 100t, 101 cremasteric, 271 crural anterior, 633 posterior, 633 cystic, 176, 190, 191 branches of, 177 deep of arm, 364, 365, 381 MRI of, 398 of thigh, 472, 473, 492, 505 perforating arteries, 502 digital dorsal, 365, 392, 395, 497 MRI of, 405 palmar, 364, 365, 392, 393 common, 364, 365, 392, 393, 472 digitopalmar branches, 392 proper, 365, 392 of thumb, 392 plantar common, 472 proper, 472, 498, 499 dorsalis pedis, 497 epigastric, 68, 72 inferior, 151, 152, 153, 160, 164, 170, 186, 195, 250, 259, 271, 472 branches of, 268 with lateral umbilical fold, 245 obturator branch of, 188 superficial, 186, 188, 472, 492 superior left, 186, 188 right,186, 197 ethmoidal anterior, 606, 609, 620 posterior, 582, 606, 609, 620 facial, 537, 582, 583, 583, 584, 585, 585t, 594, 598, 599, 608, 649 plexus of, 579 transverse, 586, 594, 595 femoral, 150, 151, 154, 186, 188, 196, 271, 275, 277, 286, 287, 472, 473, 486, 488, 489, 492, 501 in adductor canal, 492 deep, MRI of, 502 MRI of, 288, 502, 505 segments of, 473 Index 707 Artery/arteries (individual) fibular, 472, 473, 494, 496 branches, 472, 494, 497 MRI of, 503 frontal, polar, 689 frontobasal, lateral, 689 gastric, 109 left, 109, 109t, 169, 181, 185, 186, 187, 188, 191, 192, 198, 200, 201 posterior, 191 right, 186, 187, 190, 192, 200, 201 short, 191 gastroduodenal, 186, 187, 190, 191, 192, 200 gastroomental left, 187, 190, 191, 198, 199 right, 187, 190, 191, 198, 199 genicular descending, 472, 492 inferior, 472 lateral inferior, 472, 495 lateral superior, 472, 495 medial inferior, 495 medial superior, 472, 495 middle, 472, 495 gluteal inferior, 268, 269t, 472, 490, 491, 492, 493 right, 270 superior, 268, 269t, 472, 491, 492, 493, left250 right, 274 hepatic, 165, 169, 176, 186, 187 common, 67, 109, 162, 179, 181, 190, 191, 192, 198, 199 left, 177, 199, 220 198 right, 176, 190 proper, 176, 181, 183, 185, 186, 190, 191, 192, 198, 199, 200, 201 branches of, 177 hypophyseal inferior, 582 superior, 582 ileal, 187, 192, 200, 201 ileocolic, 187, 192, 193, 200, 218, 223 colic branch, 192, 193, 200 ileal branch, 188, 192, 193, 196, 200 iliac, 252 circumflex deep, 152, 186, 271, 472 superficial, 186 common, 36, 104, 194, 196, 197, 204, 211, 273, 276, 472, 473 computed tomography (CT) of, 223, 224 external, 36 internal, 36, 104 left, 156, 186, 196, 201, 252, 269, 270, 273 MRI of, 48, 51, 291 right, 165, 186, 188, 193, 242, 243, 250, 268, 269, 272, 274 external, 152, 165, 186, 188, 244, 247, 248, 269, 271, 278, 472, 473, 492 left, 251, 269, 273 right, 261, 268, 269, 270, 272 internal, 188, 211, 261, 269, 271, 273, 276, 472, 473 anterior division of, 272 anterior trunk of, 250, 268 branches of, 268, 268t, 269, 269t left, 250, 273 posterior trunk of, 268 right, 250, 268, 269, 270, 274 MRI of, 288 iliolumbar, 188, 268, 269t right, 270, 272 infraorbital, 584, 587, 594, 601, 601t, 608, 666 infratrochlear, 609 intercostal, 68, 71, 72, 75, 90, 91, 113, 220 anterior, 36, 68, 68t branches of, 68 highest, 520 posterior, 36, 44, 68, 68t, 73, 80, 86, 89, 90, 91, 108, 109, 126, 186 branches of, 36, 44, 68, 68t, 126 supreme, 365 interlobar, of kidney, 184, 189 interosseous anterior, 364, 365, 388, 389, 393 MRI of, 399 common, 364, 365 posterior, 364, 365, 388, 389 dorsal, 365 recurrent, 365, 389 passage through interosseous membrane, 389 interventricular angiography of, 134 anterior, 93, 98 jejunal, 187, 192, 200, 201, 218, 221 labial inferior, 583, 584, 594 superior, 584, 594 labyrinthine, 688 lacrimal, 606, 609 laryngeal inferior, 531 superior, 520, 531, 534, 535, 583, 655 foramen for, 527 lingual, 583, 583, 584, 585, 585t, 649 deep, 647, 657, 666 lumbar, 44, 68, 80, 188 malleolar anterior, 472497 lateral, 472, 497 medial, 472 marginal, 192, 193 left, 100 right, MRI of, 134 masseteric, 587, 601, 601t maxillary, 583, 583, 586, 586t, 587, 598, 599, 601 branches of, 601, 601t meningeal middle, 587, 587, 591, 599, 601, 601t, 632 groove for, 545 ruptured, 591 posterior, 661 mental, 584 mesenteric, 270 inferior, 183, 187, 188, 193, 196, 197, 198, 250, 270, 273, 274, 276 superior, 156, 165, 168, 169, 179, 180, 181, 183, 185, 186, 187, 188, 191, 192, 193, 197, 199, 200, 201, 216, 220, 221, 250 anastomoses of, 191 computed tomography (CT) of, 222, 223, 224 occlusion of, 192 metacarpal dorsal, 395 palmar, 393 metatarsal dorsal, 472, 497 plantar, 472, 498, 499 musculophrenic, 66, 67, 67t, 68t, 186 nasal dorsal, 582, 584, 594, 606, 608 lateral posterior, 587, 620 obturator, 188, 259, 268, 269t, 270, 272, 273, 274, 286, 473 MRI of, 505 occipital, 537, 540, 541, 583, 583, 585, 585t, 595, 596, 632, 654, 655, 663 branches of, 585, 585t, 591 lateral, 688 medial, 688 middle, 689 posterior, 689 ophthalmic, 582, 584, 593, 620, 666 MR angiography of, 671 superior, 608 ovarian, 183, 185, 186, 188, 195, 196, 197, 199, 256, 259 branches of, 197 left, 273 in ovarian suspensory ligament, 251 right, 272 with suspensory ligament of ovary, 252 Index 708 Artery/arteries (individual) (continued) Artery/arteries (individual) (continued) palatine ascending, 584 descending, 587, 601, 601t, 620, 621 greater, 587, 601, 601t, 620, 621, 644, 656 lesser, 601, 601t, 621 palpebral, medial, 606 pancreatic dorsal, 187, 191 great, 187, 191 inferior, of tail, 187 pancreaticoduodenal, 187, 200 anterior/posterior, 187, 190 inferior, 187, 191, 192, 194 anterior, 192 posterior, 192 superior, 199 anterior, 190, 191, 192, 199 parietal, posterior, 689 parieto-occipital, MR angiography of, 671 pedal, dorsal, 472, 473, 499 penile deep, 265 branches of, 264 dorsal, 265, 269, 270, 271, 275, 284 deep, 270 perforating 1st, 493 2nd, 493 3rd, 493 pericallosal, 689 pericardiocophrenic, 66, 67, 67t, 73, 78, 89, 90, 91, 94, 115 perineal, 261, 270, 272, 275 petrosal greater, 633 lesser, 633 superficial, 633 branches of, 633 pharyngeal, ascending, 520, 583, 583, 584, 585, 585t, 632 phrenic inferior, 66, 67, 67t, 80, 185, 186, 188, 197, 250 superior, 66, 67, 67t, 73, 81t plantar deep, 497 lateral, 472, 495, 498, 499 medial, 472, 473, 495, 499 branches, 472, 495, 498 MRI of, 509 pontine, 688 popliteal, 472, 473, 474, 493, 494, 495 MRI of, 503, 507 posterolateral, angiography of, 134 prefrontal, 689 princeps pollicis, 365 pudendal, 246 external, 188, 271, 275, 472, 488, 492 internal, 248, 268, 269t, 270, 272, 274, 275, 284, 285, 287, 491 pulmonary, 90, 92, 104, 111, 124–125, 125t branches of, 125, 125t, 126 left, 11, 78, 81, 91, 93, 95, 96, 97, 106, 116, 124, 125, 125t MRI of, 135 opening of, 99 perfused, 105 with pulmonary plexus, 103 right, 88, 89, 90, 95, 96, 97, 106, 116, 117, 124, 125, 125t, 133, 135 computed tomography (CT) of, 133 MRI of, 133 radial, 364, 365, 387, 388, 389, 390, 391, 393, 394, 395, 397 dorsal carpal branch of, 389 MRI of, 399, 403, 404 recurrent, 364, 387 superficial palmar branch, 390, 392, 393 superficial palmar branch of, 364 rectal inferior, 187, 269, 275, 284, 285 left, 270, 272, 274 middle, 187, 188, 259, 268, 269t, 273 left, 270, 272, 274 right, 270, 272 superior, 187, 193, 201, 219, 270, 272, 274, 276 renal, 184, 186, 189, 201 computed tomography (CT) of, 222 left, 156, 183, 186, 188, 192, 196, 197, 199, 250 anterior branch of, 189 main trunk of, 189 posterior branch of, 189 right, 183, 184, 185, 197, 220 computed tomography (CT) of, 224 ureteral branch of, 189 variants of, 189 retinal, central, 606, 612, 613, 613 physiological, 613 sacral lateral, 36, 188, 268, 269t median, 36, 188, 196, 250, 268, 270, 272, 273, 274 scapular, 365 circumflex, 365, 381, 383, 384, 385 sciatic, 490, 493 scrotal anterior, 275 posterior, 270 segmental, 184 anterior inferior, 189 anterior superior, 189 inferior, 189 posterior, 189 superior, 189 segmentary medullary, 44 anterior, 44 greater anterior, 44 posterior, 44 sigmoid, 187, 193, 201, 219, 274 sphenopalatine, 587, 587, 599, 601, 601t, 620 spinal, 44 anterior, 40, 44, A688 posterior, 44 splenic, 44, 67, 109, 156, 162, 165, 169, 179, 180, 181, 183, 187, 190, 191, 198, 199, 200, 201, 220 computed tomography (CT) of, 222, 224 pancreatic branches of, 187, 191 stylomastoid, 631, 632 branches of, 632, 633 subarcuate, 633 subclavian, 36, 44, 66, 68, 68t, 74, 76, 80, 81, 81t, 86, 89, 90, 91, 92, 96, 103, 106, 108, 109, 111, 116, 125, 126, 130, 364, 365, 369, 535, 536, 575, 582 angiography of, 135 branches of, 365 groove for, 59 left, 364, 520, 531, 655, 688 subcostal, 186 sublingual, 647 submental, 584, 647 subscapular, 364, 383, 385 MRI of, 401 sulcal, 44 supraduodenal (variant), 191 supraorbital, 582, 606, 608, 609 suprarenal inferior, 184, 185, 189 left, 183, 197, 250 right, 183, 197 middle, 184 left, 183, 186, 250 right, 197 obliterated, 153, 160, 164, 170 superior, 184, 185 left, 183, 186, 188 right, 67, 183, 186, 197, 222 suprascapular, 364, 365, 380, 381, 383, 520, 535, 539 MRI of, 401 supratrochlear, 582, 606, 609 sural, 472 tarsal, lateral, 472, 497 temporal, 598 anterior, 689 Index 709 Autonomic nervous system (ANS) deep, 587, 598, 601, 601t middle, 586 MR angiography of, 671 superficial, 583, 583, 584, 585, 586, 586t, 594, 596, 598, 599, 649 branches of, 595 testicular, 152, 153, 154, 155, 183, 185, 186, 188, 197, 199, 250, 265, 271, 277 computed tomography (CT) of, 223 thoracic internal, 36, 66, 68t, 73, 78, 80, 81t, 109, 114, 115, 130, 186, 220, 364, 365, 520, 535, 667 lateral, 68, 72, 74, 186, 364, 383, 384, 385 superior, 186, 364, 384, 385 thoracoacromial, 364, 385 thoracodorsal, 186, 364, 365, 383, 384, 385 thyrocervical, 583 thyroid inferior, 365, 383, 530, 535, 537, 655, 665 superior, 383, 520, 531, 534, 535, 536, 537, 539, 582, 583, 583, 584, 585, 585t, 654, 665 tibial, 501 anterior, 472, 473, 496, 497 MRI of, 503 posterior, 456, 472, 473, 494, 495, 501 MRI of, 503 recurrent anterior, 472 posterior, 495 tonsillar, 584 tubal, 633 tympanic anterior, 587, 601, 601t, 631, 632 inferior, 633 posterior, 631, 632 superior, 633 ulnar, 295, 364, 365, 387, 388, 389, 390, 391, 392, 393, 397 branches of, 365 deep branch, 390, 393 dorsal carpal branch, 389, 395 MRI of, 398, 399, 403, 404 recurrent, 364 superficial branch, 391 umbilical, 104, 269t, 270, 271 adult remnant of, 105, 105t obliterated, 153, 160, 164, 170, 244, 245, 273 occluded part of, 251 patent part of, 268, 273 right, 188, 272 urethral, 265 uterine, 188, 259, 261, 269, 273 left, 272, 273 relations of, 273 right, 272 tubal branch of, 273 vaginal, 269, 272 vertebral, 17, 36, 40, 44, 46, 68, 81t, 109, 364, 365, 451, 520, 582, 583, 585, 591, 660, 661, 663, 664, 665, 688 angiography of, 135 groove for, 8, 9, 18 left, angiography of, 135 MRI of, 669, 701 in posterior atlanto-occipital membrane, 27 right, 80 in transverse formamen, 17 uncinate process proximity to, 17 vesical inferior, 188, 268, 269t, 279, 273, 286 superior, 268, 269, 270, 272, 273 in lateral vesicular ligament, 259 vestibulocochlear, 635 zygomatico-orbital, 586 Artery/arteries (regional) of abdomen, 186–187 of arm, 364–365 of auricle, 627 of back, 36. See also Cord, spinal, arteries of of brain, 688–689 of brainstem, 688 of breast (female), 74 of central sulcus, 689 of diaphragm, 67, 67t of ductus deferens, 268, 269, 271 of dura mater, 591 of forearm, 365 of hand, 365 of head, 582–583 of heart, 100–101, 100t of large intestine, 186–187 of larynx, 531 of limb (lower), 472–473 of limb (upper), 364–365 of lung, 124–125, 125t, 126 of middle ear, 632, 632t, 633 of nasal cavity, 620–621 of neck, 520, 582–583 of orbit, 606 of pancreas, 187, 191 of pelvis in female, 272–273 in male, 270–271 of penis, 275 of postcentral sulcus, 689 of precentral sulcus, 689 of rectum, 274 of round ligament, 150 of sole of foot, 472 of thoracic cavity, 80 of thoracic wall, 68, 68t of trunk, 36 of vestibular bulb, 261, 275, 285 Arthrosis, uncovertebral, 17 Ascites, 158 Aspiration, of foreign bodies, 120 Asterion, 542 Atherosclerosis, 101 of carotid artery, 583 Atlas (C1), 4, 6, 8, 17, 18, 19, 20, 26, 30, 31, 40, 59, 63, 320, 370, 372, 519, 659, 663 arch of anterior, 21, 49 posterior, 8, 18, 19, 21 49 fractures of, 9 lateral mass of, 18, 19 ligaments of cruciform, 19 transverse, 18, 19, 21 MRI of, 701 transverse process of, 26, 27, 30, 31, 46, 370, 513, 519, 541 tubercles of, posterior, 18, 21, 30, 519 imaging of, 21 Atrial septal defects, 105 Atrium/atria, of heart left, 88, 89, 92, 95, 102, 104, 105, 110, 130, 131, 132 computed tomography (CT) of, 134, 136, 137 right, 89, 92, 96, 97, 100, 102, 104, 105, 110, 124, 130, 131, 132 computed tomography (CT) of, 137 MRI of, 135 radiograph of, 132 Auditory apparatus, 634 Auricle of ear, 628 arteries of, 627 innervation of, 627 structure of, 627 of heart left, 93, 94, 96, 97 atrial appendage of, 100 computed tomography (CT) of, 134 right, 93, 96, 97 atrial appendage of, 100 computed tomography (CT) of, 134, 136 Auscultation sites. cardiac, 99 Autonomic nervous system (ANS) of anal sphincters, 283 of bronchial tree, 127 circuitry of, 696–697 of esophagus, 108 overview of, 212–213, 694–695 Index 710 Autonomic nervous system (ANS) (continued) Autonomic nervous system (ANS) (continued) parasympathetic, 578, 578t, 694–695 of abdomen, 212, 212t cardiac, 103 effect on organs, 694 pathways, 695t of pelvis, 212, 212t peripheral, 87t in thorax, 86, 87, 87t of pleura, 114 referred pain in, 214–215 sympathetic, 87, 694–695 of abdomen, 212, 212t, 213 effect on organs, 694 of genital organs in female, 281 in male, 280 pathways, 695t peripheral, 87t of thorax, 87, 87t of urinary organs, 282 typical spinal nerves of, 696 Axilla, 384–385 radial nerve compression in, 373 walls of, 384–385, 384t Axillary region, 54, 294 Axis (C2), 6, 8, 17, 18, 20, 26, 30, 31, 63, 320, 620, 650 body of, 21 dens of, 5, 6, 18, 21, 59, 411, 659, 663, 667 MRI of, 701 fractures, 9 radiograph, 8, 49 spinous process of, 9, 18, 27, 30, 46, 519, 541 Axis/axes of abduction/adduction, 347, 427 of the nervous system No. 1 (Meynert), 674 No. 2 (Forel), directional terms for, 674 optical, 612 orbital, 605 of pronation/supination, 330 of radioulnar joint pronation/supination, 330 uterine, longitudinal, 256 vaginal, longitudinal, 256 visual (optical), 605 Axons, preganglionic motor, 87t Azygos system, 45, 69, 82, 83, 83t, 194, 199 B Back. See also Vertebral column arteries of, 36. See also Cord, spinal, arteries of bones of. See Ribs; Vertebra/vertebrae; Vertebral column joints of. See Vertebral column ligaments of. See Vertebra/vertebrae; Vertebral column muscles of, 2 extrinsic, 28 superficial, 24 intrinsic, 313, 515t, 554, 555 deep, 28, 29, 32, 34t, 35, 518–519, 518t deep segmental, 34, 34t deep transversospinalis, 34–35, 34t intermediate, 28, 32–33, 32t superficial, 24, 25, 26–35, 28, 32t, 65 in nuchal region, 26–27, 313, 515, 515t, 554, 555 paravertebral, 31, 31t short nuchal, 30, 30t thoracolumbar fascia and, 24 nerves of, 38–39, 46–47 neurovasculature of of back, 36–37, 46–47 of nuchal region, 36 palpable structures of, 2 radiographic anatomy of, 48–51 reference lines of, 3 regions of, 3 surface anatomy of, 2–3 veins of, 37. See also Cord, spinal veins of Balance, sense of, 692 Ballottable patella sign, of knee effusion, 443 Bands, lateral, 650 Bifurcation aortic, 68, 186, 193 carotid, 537, 582, 583 tracheal, 120 Biopsy, percutaneous, of the liver, 111 Birth canal. See Canal(s), pelvic Bladder, urinary, 5, 152, 156, 182, 242, 243, 244, 245, 246, 250, 258, 259, 260, 264, 266, 271, 273, 278, 282, 286 apex of, 251, 252, 253, 267 autonomic nervous system of, 212t, 215 body of, 252, 253 computed tomography (CT) of, 224 in female, 252 fundus of, 252, 253 innervation of, 694 lymphatic drainage of, 276 in male, 5, 253 MRI of, 225, 288, 289, 290, 291, 505 neck of, 252, 266, 267 trigone of, 252, 253 uvula of, 252 with vesical plexus, 280 visceral pelvic fascia on, 242, 243 visceral peritoneum on, 242, 243 Bleeding. See also Hematoma: Hemorrhage from nose, 585 Blood flow. See also Circulation, of blood disturbed, in coronary arteries, 101 Blood pressure. See also Hypertension optic fundus examination and, 577 Body/bodies calcaneal, 459 carotid, 537, 583, 654 ciliary, 612, 614, 614, 615 of clitoris, 262, 263, 277 of epididymis, 265 geniculate lateral, 563, 681, 682, 698, 699 medial, 563, 698, 699 of the incus, 630 mammillary, 679, 680, 681, 682 of mandible, 542, 543, 637, 657, 667 of penis, 264 perineal, 242, 243, 244, 284, 285 of pubis, MRI of, 288 of sphenoid bone, 551 of stapes, 630 of sternum, 297, 306, 308 of tongue, 646 of urinary bladder, 252, 253 of uterus, 252, 256, 257, 260 MRI of, 288, 289 vertebral. See Vertebra/vertebrae, body vitreous, 612, 656, 661, 662 Bones. See also individual bones of abdomen, 142–143 of ankle, 452–453 of foot, 452–453 of hand, 342–344, 342t of head, 542–543, 542t of hip, 410–411 of knee, 432–433 of limb (upper), 296–297, 296–299 of nasal cavity, 616 of oral cavity, 636 of orbit, 602–603 of skull, 542–543, 542t of thigh, 410–411 of wrist, 342–345, 342t Brachial region, of arm, 386 Brain, 674. See also individual regions of brain arteries of, 688–689 development of, 676t embryonic development of, 677 macroscopic organization of, 678–679 venous sinuses of, 686–687 veins of, 686–687 Brainstem, 674, 681, 682–683, 697 arteries of, 688 parasympathetic neurons of, 695t parasympathetic nuclei of, 578, 578t, 694 preganglionic fibers of, 87t veins of, 687 Breast (female), 74–75 arteries of, 74 Index 711 Canal(s) cancer of, 77 lymph nodes and lymphatics of, 76, 76t mammography of, 77 nerves of, 74 structures of, 75 Breast cancer, lymph nodes in, 367 Bronchiole, 121, 126 terminal, 121 Bronchus/bronchi. See also Tree, bronchial foreign body in, 120 lobar bronchogram of, 133 computed tomography (CT) of, 136 inferior, 89, 90, 106, 116, 117, 120, 126, 127, 136 left, 120 right, 120 middle, 90, 117, 120, 127 superior, 90, 117, 120, 127 main, 111 bronchogram of, 133 left, 80, 81, 91, 92, 108, 109, 111, 113, 120, 124, 126, 127, 129, 131, 132 computed tomography (CT) of, 79, 136 MRI of, 133 origin of, 88 right, 79, 80, 89, 108, 111, 113, 120, 124, 125, 129, 133 computed tomography (CT) of, 79, 136 MRI of, 133 segmental subsegmental, 118, 118t Bulb carotid, calcified plaque in, 583 duodenal, 168 olfactory, 549, 562, 620, 621, 659, 677 penile, 245, 255, 264 MRI of, 290 veins of, 269 vestibular, 246, 254, 262, 263, 275, 285 artery of, 261, 275, 285 venous plexus of, 275 Bulbourethral glands, 266, 267, 284 Bulla, ethmoid, 550, 617, 618, 621 Bundle accessory muscle, 653 anteromedial, 475 atrioventricular (of His), 98, 102 interatrial, 102 intermodal, 102 anterior, 102 middle, 102 posterior, 102 neurovascular intercostal, 71 of the rectum, 259 posterolateral, 475 Bundle branches, 102 Bursa(e) acromial, subcutaneous, 307 of calcaneal tendon, 457 gastrocnemius-semimembranous (Baker’s cyst), 437 iliopectineal, 413, 489 infrapatellar, 442, 443 olecranon, 327 omental, 156, 157, 158, 159, 160, 166, 180, 181, 220 boundaries and walls of, 162–163, 163t location of, 163 recess of inferior, 163 splenic, 163, 174 superior, 163 vestibule of, 162, 163, 173 periarticular, 437 prepatellar, 435, 443 semimembranosus, 437, 495 subacromial, 306–307 MRI of, 401 subcoracoid, 304 subcutaneous, 495 subdeltoid, 306, 307, 383 subtendinous of gastrocnemius, 495 lateral, 437 medial, 437 of subscapularis, 304, 306, 307, 383 trochanteric, 413, 415, 491, 493 Buttocks, regions of, 3 C Caesarean section, 232 Calcaneus, 410, 445, 446, 448, 450, 452, 454, 455, 456, 458, 459, 460, 461, 462, 468, 471 MRI of, 509 Calvaria, 545, 587 diploë, 545 inner table of, 545 muscles of, 556t outer table of, 545 structure of, 545 Calyx major, 184, 189 computed tomography (CT) of, 224 minor, 184, 189 Canaliculus lacrimal inferior, 611 superior, 611 tympanic, 573, 629 Canal(s) adductor, 472, 473, 474, 492 anal, 248–249, 287 MRI of, 290 muscularis mucosae of, 274 regions of, 249t auditory external, 5, 411, 626, 627, 629 curvature of, 626 parts of, 626 carotid, 546, 548, 624 opening of, 643 central, 685, 698 cervical, 257 cervical axis in, 256 MRI of, 288, 289 cochlear, 634 condylar, 546, 548 Dorello’s, 593 facial, 547, 568, 571, 629 for greater petrosal nerve, 568 hypoglossal, 21, 546, 547, 548, 577 incisive, 548, 616 infraorbital, 602, 602t, 603 MRI of, 669 inguinal, 150, 151, 151t, 255, 489t gender differences in, 150 opening of, 151 structures of, 151, 151t malleolar, 485 mandibular, 567, 598, 644, 656, 657 radiograph of, 668 nasolacrimal, 602t obturator, 237, 238, 258, 268t optic, 547, 548, 551, 563, 602, 603, 604, 608, 636, 662 osseofibrous, 25 palatine, 622, 636 palatovaginal (pharyngeal), 546, 622 pelvic, female narrowest diameter of, 235 true conjugate of, 235 pterygoid, 551, 569, 579, 622 artery of, 601 fossa of, 546 nerve of, 579, 621, 623 pudendal (Alcock’s), 268t, 491 pyloric, 166, 167 sacral, 7, 7t, 12, 13, 45, 232, 233, 237 MRI of, 51 of Schlemm, 612, 613, 614 semicircular, 570, 634 anterior, 571, 625, 626, 628, 629, 633, 634 lateral, 571, 625, 626, 628, 629, 634 prominence of, 629 posterior, 571, 625, 626, 628, 629, 634 of spinal cord, central, 684 supraorbital, 602, 602t vertebral, 5, 14, 22, 23, 40 cauda equina in, 41 computed tomography (CT) of, 223 Index 712 Canal(s) (continued) Canal(s) (continued) MRI of, 48 spinal cord in, 220 Cancellous trabecule, 551 Cancer. See also Carcinoma of breast, 77 of colon, 173 metastatic, 77, 195 Canines (teeth), 640 Capillary bed, 126 cancer metastatic to, 195 Capitate, 295, 342, 342t, 343, 344, 345, 346, 347, 350, 361, 391, 404 MRI of, 404 radiograph of, 343 Capitulum, 300, 301, 326, 329, 329, 403 radiograph of, 402 Capsule of brain external, 661 extreme, 661 internal, 660, 661, 689 fat, perirenal, 182, 183 fibrous, 25 renal, 182, 184, 189 internal, 698 of joints, 306 articular, of wrist, 345 of atlantooccipital joint, 19, 29 distal interphalangeal, 349 of elbow joint, 329 of glenohumeral joint, 305, 307 of hip joint, 416 interphalangeal distal, 349 proximal, 349 of knee joint, 436–437, 442 of metatarsophalangeal joints, 460, 461, 471 distal, 349 of shoulder joint, 381 of temporomandibular joint, 553, 558, 597, 638, 639 innervation of, 639 lateral, 670 Tenon’s, 610 of zygapophyseal joint, 21, 22 prostatic, 266 splenic, 212t Carcinoma of breast, 77 of lung, 129 prostatic, 267 Cardia, gastric, 107, 166, 167 Cardiac impulse, 102 Cardiac tamponade, 95 Cardiac valves, 98–99 aortic, 88, 95, 98, 98t, 99, 131 auscultation site for, 98t, 99 computed tomography (CT) of, 137 MRI of, 131, 135 atrioventricular, 97, 99 auscultation site for, 98t, 99 cusp of, 97 posterior, 98 septal, 98 right (tricuspid), 97 computed tomography (CT) of, 137 MRI of, 135 with right atrioventricular orifice, 97 left (bicuspid), 97, 98, 98t auscultation of, 98t, 99 of coronary sinus, 97 of inferior vena cava, 97 pulmonary, 97, 98, 98t, 99, 100 semilunar, 98, 99 Carina at tracheal bifurcation, 120 urethral, 260 Carpal bones, 296, 342t, 344–345 distal row of, 345 proximal row of, 345 Carpal region, 294 Carpal tunnel syndrome, 391 Cartilage alar, 616 arytenoid, 120, 526, 527, 528, 664, 665 of auricle, 627 corniculate, 526, 527 costal, 56, 57, 61, 63, 303, 309 cricoid, 106, 107, 120, 513, 526, 527, 529, 665 MRI of, 669 epiglottic, 526, 527, 660, 664 of hyaline end plate, 14 laryngeal, 526, 659 nasal, lateral, 616 septal, 616 thyroid, 54, 78, 80, 88, 89, 103, 120, 127, 512, 517, 526, 527, 528, 529, 530, 534, 535, 536, 660, 661, 664, 665 MRI of, 669 tracheal, 120, 529 triradiate, of coxal bone, 231 Caruncle, lacrimal, 611 Cauda equina, 5, 15, 40, 43, 482, 676 computed tomgraphy (CT) of, 223 MRI of, 48 in vertebral canal, 41, 223 Cauda equina lesion, 42 Cave, trigeminal, 607 Cavernous sinus syndrome, 606 Cavity abdominopelvic divisions of, 156–157 mesentery of, 156, 158 organs of, 156, 159, 159t peritoneum of parietal, 158 visceral, 158 cranial cranial nerves exiting, 549 dural sinuses in, 592 neurovascular pathways exiting or entering, 548–549 glenohumeral joint, 304 glenoid, 297, 299, 303, 304, 305, 306, 307, 319, 383 MRI of, 401 radiograph of, 400 of knee joint, 442 laryngeal, 529 intermediate, 529t nasal, 618, 622t, 623t, 657, 662 arteries of, 620–621 bones of, 616 lateral wall of, 620, 621 nerves of, 621 neurovasculature of, 620–621 opening into choana, 651 oral, 616, 622t, 628, 636, 656, 660, 667 bones of, 636 boundaries of, 648 divisions of, 648, 648t floor of, 647 muscles of, 642, 645 innervation of, 644 muscles of, 557, 557t, 642, 645 opening of, 639, 651 topography of, 648–649 pelvic, 234, 234t, 260 divisions of, 157 pericardial, 88, 92, 95 cardiac tamponade and, 95 posterior, 94 peritoneal, 157, 159, 246 dissection of, 160 infracolic, 159 organs of, 159, 159t posterior wall of, 165 pleural, 55, 89, 128–129 lymphatic drainage of, 128–129 movement in respiration, 123 in pneumothorax, 123 positive pressure in, 123 right, 89 pulmonary, 78, 78t, 112–113 boundaries and reference lines of, 112, 112t Index 713 Colon scrotal, 153 synovial of wrist, 391 of temporomandibular gland inferior, computed tomography (CT) of, 670 superior, computed tomography (CT) of, 670 thoracic arteries of, 80 heart location in, 92 joints of, 60–61 lymphatics of, 84–85, 110 nerves of, 86–87, 87t veins of, 82–83 of tunica vaginalis, 155 tympanic, 626, 628, 629, 634 anterior wall of, 571, 629 boundaries of, 628t glossopharyngeal nerve in, 573 ossicular chain in, 631 posterior wall of, 571 roof of, 571, 629 uterine, 256, 257 MRI of, 289 Cecum, 161, 164, 170, 171, 172, 173, 244, 245 lymphatic drainage, 203, 209 radiograph of, 225 Cell(s) air ethmoid, 550, 666 mastoid, 571, 625, 628, 629, 645 MRI of, 669 alveolar macrophages, 121 association, 691 capillary endothelial, 121 commissural, 691 intercalcalated, 691 Cementum, of teeth, 640 Center of gravity, 5, 411 Central nervous system. See also Brain; Cord, spinal morphology of, 674 Cerebellum, 570, 660, 662, 674, 677, 680, 681, 682, 683 flocculus of, 570 lateral part of, 682 median part of, 682 MRI of, 700 posterior lobe of, MRI of, 669 vermis of, 699 Cerebral hemispheres cortical margins of, 689 lobes of, 679 right, medial surface of, 679 Cerebrospinal fluid (CSF) circulation of, 684 cisterns for, 21, 684 in dural sac, 15 extraction of sample of, 41 Cerebrum computed tomography (CT) of, 670 frontal lobe of, 656, 657 macroscopic organization of, 678 Cerumen glands, 626 Cervix, of uterus, 247, 252, 254, 258, 259, 286 lymphatic drainage of, 277 MRI of, 288 supravaginal part of, 256, 257, 260 uterine isthmus of, 256, 257 with uterine os, 261 vaginal part of, 256, 257, 260 Chambers, of eyeball anterior, 612, 614 posterior, 612, 614 Chest tube, insertion of, 72 Chiasm crural, 447, 451 optic, 563, 593, 603t, 677, 681, 682, 685, 689, 698, 699 plantar, 447, 451 Childbirth complications of, 232 epistotomy during, 263 Choanae, 546, 616, 617, 620, 636, 655, 659 right, 648 in tonsil infections, 651 Chorda tympani, 568, 569, 571, 600, 600t, 625, 629, 630, 631, 634, 645, 646 Choroid, 612, 615 Ciliary glands, 610 Circle arterial, of the iris greater, 613, 614 lesser, 613, 614 of Zinn (and von Haller), 613 Circulation, of blood, 92 lower body, 92 portal, 92 postnatal, 105 prenatal, 104 pulmonary, 92, 126 upper body, 92 Circumference, articular, 327, 331 Cirrhosis, hepatic, 175 Cistern ambient, 684 basal, 684 cerebellomedullary (cisterna magna), 21, 684 chiasmatic, 684 interhemispheric, 684 interpeduncular, 684 of lamina terminalis, 684 lumbar, 21. See also Sac, dural pontocerebellar, 660 pontomedullary, 684 vermian, 684 Cisterna chyli, 84, 85, 202, 203, 204, 207, 208, 279f Claustrum, 661, 689 Clavicle, 24, 59, 64, 76, 91, 130, 294, 295, 296, 297, 298, 302, 303, 304, 305, 307, 308, 309, 313, 316, 318, 319, 320, 321, 365, 381, 382, 383, 512, 513, 516, 532, 533, 536 acromial end of, 298, 303 borders of, 303 head of, 54 notch of, 56 radiograph of, 400 right, 298 shaft of, 298 sternal end of, 298, 303 Cleft, anal, 239 Clitoris, 254, 254t body of, 262, 263, 277 crus of, 246, 254, 261, 262, 263, 276, 285 glans of, 229, 254, 262, 277, 285 lymphatic drainage of, 277 prepuce of, 229, 262, 263 Clivus, 547, 593, 607, 659, 667 Coccyx, 4, 5, 6, 12–13, 36, 229, 232, 233, 234, 235, 236, 237, 238, 239, 248, 286, 411 MRI of, 48, 288, 289 Cochlea, 625, 626, 628 canals of, 596 Colitis, 173 Colliculus, 526, 527 facial, 683 inferior, 680, 681, 683 brachium of, 681, 683 seminal, 247, 253, 266 superior, 680, 681, 683, 699 brachium of, 683 Colon ascending, 161, 162, 164, 167, 170, 171, 172, 173, 174, 218 computed tomography (CT) of, 193 lymphatic drainage of, 208 MRI of, 51, 225 radiograph of, 225 site of attachment of, 165, 183 carcinoma of, 173 descending, 161, 162, 164, 165, 167, 169, 171, 172, 173, 174, 181, 193, 210, 220 computed tomography (CT) of, 223 Index 714 Colon (continued) Colon (continued) lymphatic drainage of, 203, 209 MRI of, 51, 225 radiograph of, 225 site of attchment of, 165 innervation of, 694 sigmoid, 161, 172, 173, 193, 242, 244, 245, 248, 261 lymphatic drainage of, 203 MRI of, 288, 289, 291 radiograph of, 225 transverse, 156, 159, 160, 161, 164, 169, 170, 171, 172, 173, 174, 193, 219, 220, 221 computed tomography (CT) of, 222, 224 lymphatic drainage of, 209, 209 MRI of, 225 radiograph of, 225 in ulcerative colitis, 173 Colonoscopy in colon carcinoma, 173 in ulcerative colitis, 173 Column(s) anal, 249 anterior, 260, 690 fecal, 283 lateral, 690 nuclear, 690 posterior, 690 renal, 184 vertebral. See Vertebral columns Commissure anterior, 698 of brain anterior, 680, 682, 685, 689 gray, 690 of labia, 262 of lip, 512 posterior, 698 of vagina, posterior, 229 Compartments infracolic, 161 of lower limb, 408 of wrist, 350–351 Compartment syndrome, 497 Concha cymba, of ear, 627 nasal inferior, 543, 603, 611, 617, 618, 619, 620, 621, 636, 651, 656, 660, 666 middle, 543, 550, 562, 603, 617, 618, 619, 620, 621, 636, 651, 655, 656, 660, 666 MRI of, 669 superior, 603 superior, 550, 562, 617, 618, 619, 620, 621, 628 Conduction system, cardiac, 98, 102 Condyle femoral lateral, 412, 413, 432, 433, 440, 441 MRI of, 507 radiograph of, 506 humeral, 300 medial, 336t, 412, 432, 433, 440, 441, 442, 450 mandibular MRI of, 669 radiograph of, 668 occipital, 18, 544, 546, 643 tibial lateral, 408, 410, 445, 448, 449, 496 MRI of, 507 radiograph of, 506 medial, 408, 410, 412, 432, 440, 441, 442, 445, 450, 496 MRI of, 507 radiograph of, 506 Cone, of light, 630 Confluence posterior venous, 687 of the sinuses, 588, 589, 592t, 667, 684, 686 MR angiography of, 671 Conjunctiva, ocular, 610, 612, 614, 614 Connection, intertendinous, 334, 358 dorsal digital expansion of, 341 Connective tissue of cardiac skeleton, 98 fibromuscular, anterior, MRI of, 291 interlobar, 75 of levator ani muscle, gaps in, 239 sling of, 642 subpleural, 126 Contractures, Dupuytren’s, 354 Conus, tendons of, 98 Conus arteriosus, 97 computed tomography (CT) of, 136, 137 Conus elasticus, 527, 528, 529 Conus lesions, 42 Conus medullaris, 5, 40, 41 developmental stages of, 41 MRI of, 48 Cord of brachial plexus, 383 lateral, 368, 368t, 369, 376, 384, 384t, 385, 386 branches of, 385 long (terminal) branches of, 374t short branches of, 374, 374t, 375 medial, 368, 368t, 369, 376, 377, 384t, 385, 386 branches of, 385 long (terminal) branches of, 374t short branches of, 374, 374t posterior, 368, 368t, 369, 371, 384t, 385 axillary nerves of, 372, 372t long (terminal) branches of, 371t radial nerves of, 373, 373t short branches of, 371, 371t oblique, 330 sacral, 694 spermatic, 144, 145, 150, 151, 154–155, 270, 284, 287, 479, 488, 492 contents of, 155t with cremaster muscle, 151 with internal spermatic fascia, 151 MRI of, 290 spinal, 5, 40, 67, 75, 113, 130, 221, 533, 663, 674, 676 arteries of, 40, 44 ascending pathways of, 692, 692t axons of, 695t central canal of, 684 cervical, 17, 42, 677 developmental stages of, 41 ganglion fibers, 87t in situ, 40 intrinsic circuits of, 691 lateral horn of, 579t level L2, 696, 697 level L3, 696, 697 lumbar, 42 meningeal layers, 40 morphology of, 674 MRI of, 21, 133, 701 nerves of, 40–43 organization of, 690 regions of, 42 segments of, 690 segments of, and spinal nerves, 42–43 thoracic MRI of, 48 T1 segment, 127 T2 segment, 108 T6 segment, 108 in vertebral canal, 181 umbilical, 105 veins of, 97, 99 Cornea, 610, 612, 614, 614 Cornua coccygeal, 12 of hyoid bone, 661 sacral, 12 Corpus callosum, 658, 659, 667, 677, 679, 680, 681, 682, 685, 689 body of MRI of, 700 genu of, 698 MRI of, 700 MRI of, 700, 701 Index 715 Diaphragm (thoracic) splenium of, 698 MRI of, 700 sulcus of, 679 trunk of, 698, 699 Corpus cavernosum, 253, 262, 264, 265, 267, 284 MRI of, 290, 291 tunica albuginea of, 265 Corpus spongiosum, 253, 255, 264, 284 MRI of, 290, 291 Cortex cerebral, basal, 676 premotor, 693 primary motor, 693 renal, 184 computed tomography (CT) of, 224 sensory, 692 of tibia, 506 Coxal bone, 230, 410 gluteal surface of, 231 pelvic ligament attachmment to, 237 relation to vertebral column, 411 triradiate cartilage of, 231 Cranial bones, 590 diploë of, 590, 592 inner table of, 590, 592 outer table of, 590, 592 Cranial nerves, 560–561, 593, 676. See also individual cranial nerves under Nerves (individual) classification of cranial fibers and nuclei, 561, 561t “false,” 675 ganglion of, 675 parasympathetic nuclei, 695t through the cavernous sinus, 607 “true,” 675 Cranial vessels, innervation of, 694 Crease of distal interphalangeal joint, 295 of metacarpophalangeal joint, 295 of proximal interphalangeal joint, 295 thenar (“life line”), 295 of wrist distal, 295 middle, 295 proximal, 295 Crest frontal, 545, 547 of greater tubercle, 300, 317, 318 iliac, 2, 3, 3t, 24, 28, 29, 47, 142, 143, 147, 182, 230, 231, 232, 233, 236, 321, 371, 408, 409, 410, 414, 416, 418, 421, 422, 425, 427, 428, 429, 479, 500 computed tomography (CT) of, 223 intermediate line of, 149 MRI of, 289 intertrochanteric, 414, 416, 427 radiograph of, 504 interureteric, 253 lacrimal, posterior, 602 of lesser tubercle of the humerus, 300, 301, 317, 321 nasal, 616, 636 occipital external, 546 internal, 547 petrous, 547 pubic, 232 of rib head, 59, 61 sacral lateral, 12, 13 median, 7, 12, 13, 232, 233 sphenoid, 551, 616 supracondylar, lateral, 339 supramastoid, 597 supraventricular, 97 terminal, 97 transverse, 571 trochanteric, 412 urethral, 264 Crista galli, 547, 550, 590, 603, 616, 617, 619 Crohn’s disease, 171 Crus cerebri, 698, 699 Crus/crura of alar cartilage lateral, 616 medial, 616 of the antihelix, 627 cerebral, of cerebral peduncle, 564 cerebri, 685 of clitoris, 246, 254, 261, 262, 263, 276, 285 common, of labyrinth canal, 635 of diaphragm left, 64, 65 right, 64, 65 of heart, 96 of inguinal ring lateral, 150, 151 medial, 150, 151 of penis, 245, 246, 264 of the stapes anterior, 630 posterior, 630 of superficial inguinal ring lateral, 489 medial, 489 CT. See Imaging Cubital region, 387 anterior, 294 posterior, 294 Cuboid, 448, 452, 453, 455, 458, 460, 461, 462 radiograph of, 508 tuberosity of, 451, 453 Culmen, 682 Cuneiform, 457, 462, 468 intermediate, 452, 453, 455, 462, 468 MRI of, 509 radiograph, 508 lateral, 452, 453, 455, 462 MRI of, 509 radiograph, 508 medial, 448, 451, 452, 453, 455, 458, 461, 462, 463, 471, 499 MRI of, 509 radiograph, 508 Cuneus, 679 Cup optic, 677 physiological, 613 Curvature of external auditory canal, 626 of stomach greater, 162, 166, 167 lesser, 166, 167 of uterus, 256 Cusp, of cardiac valves, 98, 99 Cymba conchae, 627 Cyst, gastrocnemius-semimembranous (Baker’s), 437 D Decussation, pyramidal, 693 Defecation, mechanism of, 283 Deltoid region, 3, 294 Dens, of axis (C2), 5, 8, 9, 17, 18, 21, 59, 411, 659, 663, 667 apex, MRI of, 21 apical ligament of, 18, 19, 21 axis of, radiograph of, 49 facet for, 9 MRI of, 48, 701 posterior articular surface of, 19 Dentin, of teeth, 640 Dentition. See Teeth Dermatomes of anterior trunk, 210 of back, 15, 39, 42 of body wall, 696 of limb (lower), 487 of limb (upper), 379 of nuchal region, 524 of thoracic wall, 71 Diaphragm (pelvic), 246, 246t, 248 fascia of, 239, 245, 246, 251, 261 inferior, 246, 248 superior, 246, 248, 249 muscles of, 240, 240t Diaphragm (thoracic), 5, 60, 64–65, 71, 73, 78, 79, 80, 83, 84, 85, 88, 89, 90, 91, 92, 93, 108, 109, 110, 111, 113, 116, 123, 128, 131, Index 716 Diaphragm (thoracic) (continued) Diaphragm (thoracic) (continued) 146, 152, 162, 164, 167, 169, 174, 181, 182, 185, 204, 206 apertures of, 147 arteries of, 67, 67t attachment to liver, 88, 95, 175 central tendon of, 64, 65, 67, 73, 88, 95, 115, 116, 146, 147 attachment to fibrous pericardium, 89, 94, 95 computed tomography (CT) of, 222 crus of left, 147, 168, 222 computed tomography (CT) of, 222 right, 147, 168 computed tomography (CT) of, 222 dome of, 132 left, 64 right, 64 hepatic surface of, 165, 169, 181, 183 hiatus (opening) of, esophageal, 107 movement in respiration, 60, 123 nerves of, 66, 67 neurovasculature of, 66–67 parts of costal, 64, 65, 65t, 67, 82, 115, 131, 146, 147, 220 superior, 164 lumbar, 64, 64t, 67, 147, 211, 222 sternal, 64, 64t, 65 Diaphragma sella, 549 Diastole, ventricular, 98, 131 Diencephalon, 561t, 674, 680–681, 682, 683 arrangement around third ventricle, 680 location in adult brain, 681 Digestive organs, lymphatic drainage of, 203 Diploë, 590, 592 Disk articular, 559, 638, 639 computed tomography (CT) of, 670 articular (ulnocarpal), 303, 346 intervertebral, 4, 5, 6, 14–15, 17, 20, 21, 22, 23, 56, 61, 237 C2/C3, 667 gaps in ligamentous reinforcement of, 23 herniation of, 15 L3/4, 48 L4/5, 237 L 4/5, 237 MRI of, 15, 21, 48, 133 radiograph of, 49, 50 relation to vertebral column, 14 split, 17 structure of, 14 surface of, 14 optic, 612 blind spot of, 613 Dislocations of acromioclavicular joint, 303 of elbow joint, 376 of hip joint, 415 of temporomandibular joint, 639 Diverticulum/diverticula duodenal, endoscopic appearance of, 169 epiphrenic, 107 esophageal, 107 hypopharyngeal (haryngo-esophageal), 107 parabronchial, 107 Zenker’s, 107 Dorsum of foot, 408 muscles of intrinsic, 468, 468t superficial intrinsic, 469, 469t topography anterior compartment, 496, 496t lateral compartment, 496, 496t of hand, 294, 358–359, 394–395 cutaneous innervation, 394 dorsal digital expansion of, 359 lymphatic drainage of, 367 muscles of, 358 neurovascular structures, 395 tendons of, 358 of penis, 271 of tongue, 646 Dorsum sellae, 547, 551, 662, 667 “Drawer sign,” 435 Duchenne’s limp, 483 Duct(s) alveolar, 121 bile, 165, 168, 176, 178, 181, 185, 190 common, 179, 220 extrahepatic, 178 hepatic location of, 178 lymphatic drainage of, 206 obstruction (in gallstones), 179 cochlear, 571, 634 cystic, 176, 178, 179 ejaculatory, 253, 255, 267 openings of, 253 orifices of, 266 endolymphatic, 634 frontonasal, opening of, 621 hepatic, 183 branches of, 177 common, 178, 179 left, 176, 178, 179 right, 178, 179 lactiferous, 75, 77 lymphatic, 84, 85, 88t nasolacrimal, 602, 611, 618t MRI of, 669 opening of, 621 pancreatic, 178, 179, 180 accessory, 168, 178, 180 main, 168 sphincter of, 178 parotid, 594, 595, 596, 597, 649 semicircular, 571 anterior, 634, 635 lateral, 635 posterior, 635 submandibular, 647, 649 terminal lobar unit (TDLU), 75, 77 thoracic, 84, 85, 91, 106, 110, 113, 129, 130, 202, 203, 207, 208, 279f, 535 Ductules efferent, 265 prostatic, 264 Ductus arteriosus, 104, 105 adult remnant of, 105t Ductus deferens, 152, 153, 154, 155, 245, 250, 251, 255, 255t, 265, 286 ampulla of, 156, 253, 266 artery of, 268, 269 with deferential plexus, 280 lymphatic drainage of, 277 MRI of, 291 right, 242, 251, 271 artery of, 270 Ductus reuniens, 634 Ductus venosus, 104 adult remnant of, 105t Duodenum, 159, 162, 163, 165, 166, 167, 174, 175, 179, 181, 182, 190, 193, 220 computed tomography (CT) of, 222, 224 location of, 168 lymphatic drainage of, 203, 207, 208 muscular coat of circular layer of, 168 longitudinal layer of, 168 papillary, endoscopy of, 167 parts of ascending part, 165, 168, 169, 178, 179, 181 computed tomography (CT) of, 222 descending part, 162, 165, 178, 180, 221 horizontal, 156, 164, 165, 168, 169, 171, 178, 181, 183, 224 superior, 165, 166, 168, 169, 171, 178, 180, 183 veins of, 198, 221 wall of, 178 sphincter system in, 178 Dural sinus, 545 Dura mater, 40, 41, 545, 587, 590, 635 arteries of, 591 innervation of, 591 layers of, 38, 40, 592 Index 717 Falx cerebri sinuses of. See Sinus, dural (venous) spinal, 15 Dysplasia, of the hip joint, 415 E Ear external, 626 inner, 634 blood vessels of, 635 middle, 626 arteries of, 632, 632t, 633 muscles of, 556, 556t Earlobe, 627 ECG (electrocardiogram), 102 Effusion of knee, ballottable patella sign of, 443 pleural, 72 Embolism, pulmonary, 125 Eminence articular, 542, 598 hypothenar, 294, 391 iliopubic, 142, 233, 489t intercondylar, 433 radiograph of, 506 medial, 683 parietal, 544 pyramidal, 631 in conductive hearing loss, 631 radial carpal, 350 thenar, 295, 391 ulnar carpal, 350 Enamel, of teeth, 640 Endometrium MRI of, 289 of the uterus, 256 Endoscopic retrograde cholangiopancreatography (ERCP) of duodenal papillary region, 169 Endoscopy, of stomach, 166 End plate, vertebral, 50 Enema, double-contrast barium, radiograph of, 225 Epicardium, 93 Epicondyle femoral lateral, 434, 435, 438 radiograph of, 506 medial radiograph of, 506 humeral, 376 in elbow injuries, 327 lateral, 295, 300, 309, 322, 323, 326, 327, 328, 329, 341, 403, 412, 434, 438, 449 arterial network of, 389 common head of flexors, 335 MRI of, 507 radiograph of, 402 medial, 295, 300, 301, 308, 309, 322, 323, 326, 327, 328, 329, 332, 337, 341, 377, 387, 396, 403, 410, 412, 434, 438, 450 common head of flexors, 334, 337, 396 femoral, 434, 435, 438 radiograph of, 402 tibial, lateral, 434, 435 Epicondylitis, lateral, 338 Epididymis, 153, 154, 255, 255t, 265, 271, 278, 280 appendix of, 265 body of, 155, 265 head of, 155, 265 MRI of, 290 in situ, 265 lymphatic drainage of, 277 tail of, 265 MRI of, 290 Epigastrum, 54 Epiglottis, 526, 527, 529, 531, 646, 648, 650, 651, 655, 658, 659 stalk of, 526 Epithelium, pigment, 612 Epitypanum, 629 Epoöphoron, 257 Erb’s point, 538 ERCP, 169 Esophagus, 5, 25, 38, 65, 67, 73, 79, 80, 81, 88t, 92, 106–109, 110, 111, 115, 116, 130, 131, 147, 156, 166, 167, 168, 169, 171, 174, 179, 183, 185, 188, 196, 204, 529, 530, 531, 652, 653, 665 adventia of, 166 arteries of, 109, 109t circular muscles fibers of, 653 computed tomography (CT) of, 136, 137 constrictions of, 106 in situ, 107 location of, 106 lymphatic drainage of, 110, 111 MRI of, 133, 669 nerves of, 108 neurovasculature of, 108–109 parts of abdominal, 106, 107, 108 cervical, 79, 86, 88, 89, 106, 107, 108 thoracic, 86, 87, 88, 89, 106, 107, 108, 116 structure of, 107 varices of, 199 veins of, 109, 109t walls of, 107 Ethmoid bone, 542, 542t, 543, 547, 550, 562, 602, 603, 616, 617, 619 orbital plate of, 656 in the paranasal sinuses, 619 perpendicular plate of, 636 Expiration lung contraction in, 123 pleural contraction in, 122 rib cage movement in, 60 thoracic changes in, 122 Eye, 677 cornea, 610, 612, 614 innervation of, 694 iris, 610, 612, 614, 614, 615 greater arterial circle of, 613 lesser arterial circle of, 613 lens, 610, 612, 614, 662 light refraction by, 615 retina, 610, 612 sclera, 610, 612, 614, 615 Eyeball, 610, 666 blood vessels of, 613 left, retina of, 613 MRI of, 666 muscles (extraocular) of, 604–605 actions of, 605, 605t testing of, 604 structure of, 612 Eyelids, 610, 611 F Face. See also Facial expression fractures of, 543 innervation of motor, 580 sensory, 581 motor pathways of, 693 neurovasculature of, 580–581 superficial, 594 Facet(s) articular for arytenoid cartilage, 526 for thyroid cartilage, 526, 528 for dens, 9 lateral, 435 medial, 435 radiograph of, 9, 50 of vertebra articular anterior, 8 inferior, 8, 9, 10, 12, 14, 16, 18, 22, 23, 49, 51 costal, 4, 16 inferior, 7, 10 superior, 7, 10 Facial expression, muscles of, 552–553, 554 of forehead, nose, and ear, 556, 556t motor innervation of, 580 of mouth and neck, 557, 557t occipitofrontalis, 556 Falx cerebri, 590, 592, 657, 658, 662, 666 MRI of, 669, 700 Index 718 Fascia Fascia abdominal, superficial, 488 deep layer of, 267 antebrachial, 354, 392 axillary, 382 biceps, 385 biceps brachii, 387 brachial, 382 buccopharyngeal, 530, 533 bulbar, 610 clavipectoral, 382 cremasteric, 150, 153, 154, 155, 265 dartos, 155, 155t deep cervical, 25, 533, 533t investing (superficial), 25, 88 layers of, 530, 533, 533t, 534, 537, 538, 539 pretracheal muscular layer, 25, 88 pretracheal visceral layer, 25 prevertebral layer, 25 of diaphragm (pelvic), 239, 246 inferior, 248 superior, 248, 249 endothoracic, 71, 72 gluteal, 490, 493 investing superior, 153 of the leg, deep, 496 nuchal deep, 24, 25, 28, 516, 533 superficial, 25 obturator, 238, 239, 261 pectoral, 75 pelvic, 245, 251, 258, 261 in female, 259 parietal, 246, 259 of pelvic floor, 239 visceral, 242, 243, 252, 253, 259 penile, 253 deep, 265, 275 superficial (Colles’), 154, 244, 265, 275 perineal, 239 superficial (Colles’), 239, 244, 246, 247 pharyngobasilar, 653, 654 plantar, deep layer, 499 pretracheal muscular, 533 visceral, 533 prevertebral, 533 psoas, 25 rectoprostatic, 242, 267, 287 rectovesical (Denonvillier), MRI of, 290 renal anterior layer, 25, 182 posterior layer, 25 retrorenal layer, 182 spermatic, 153, 154 external, 155, 155t, 265, 275 internal, 151, 154, 155t, 265, 271 superficial, 387 deep layer of, 265 thoracic, superficial, 75, 382 thoracolumbar, 312, 321, 371, 427 middle layer, 25 posterior layer, 25, 28, 29, 47, 313 of tissue spaces in the head, 654 transversalis, 25, 145, 146, 151, 152 Fascia lata, 151, 479, 486, 488, 490, 492, 493 Fasciculus cuneatus, 692, 692t Fasciculus/fascicles cardiac anterior, 102 middle, 102 posterior, 102 interfascicular, 691 intrinsic, of the spinal cord, 691 longitudinal, 18, 19, 21, 354, 691 dorsal, 562 proprius, lateral, 691 septomarginal, 691 sulcomarginal, 691 transverse, 354, 464 Fasciculus gracilis, 692, 692t Fat epidural, 15 periorbital, 656, 666 perirenal, 183 preperitoneal, 146 Fat body, anterior, of elbow, 403 Fat capsule, perirenal, 182, 183, 220, 250 Fat pad buccal, 657 of elbow (cubital) joint, 327 infrapatellar, 442, 443 infrapatellar, MRI of, 507 of inguinal region, 150 pararenal, 184 retrosternal, 88, 90 Fauces (throat), 648t Femur, 2, 410, 412–413, 428, 429, 434, 436, 437, 438, 442, 443, 444, 448, 449, 450, 500, 501 fractures of, 415 head of, 247, 286, 412, 413, 414, 415, 417, 473 arteries, 473 fovea on, 417 ligament of, 286, 417 MRI of, 288, 289, 291, 505 necrosis of, 473 radiograph of, 504 rotation of, 413 MRI of, 288, 502 neck, 410, 411, 412, 413, 415, 417, 473 anterior superior, radiograph of, 504 radiograph of, 504 patellar surface, 413, 435, 438, 440, 441, 442 radiograph of, 506 shaft of, 412, 415 Fetal circulatory structures, derivatives of, 105t Fibers afferent (sensory), 71 nuclei of, 561 circular muscle, of esophagus, 653 corticobulbar, 576, 577 corticonuclear, 693t corticoreticular, 693t corticospinal, 693t cuneocerebellar, 692 efferent (motor), 71 nuclei of, 561 elastic, 121 intercrural, 150, 151, 489 olfactory, 549, 562, 620 parasympathetic postganglionic, 103, 127, 212, 214, 215, 280, 281, 282, 569, 573, 697 preganglionic, 87t, 103, 212, 214, 215, 280, 281, 282, 578, 697 prerectal, 238 somatic afferent (sensory), 561, 561t somatic efferent (motor), 561, 561t somatic motor (efferent), 696–697 somatic sensory (afferent), 566t, 569, 573t, 574t, 696–697 special visceral afferent, 573t, 574t special visceral efferent (branchiogenic), 573t, 574t sympathetic of head, 579 postganglionic, 103, 212, 214, 215, 280, 281, 282, 565, 579, 579t, 697 distribution of, 695t preganglionic, 103, 127, 212, 214, 215, 280, 281, 282, 565, 697 visceral afferent (sensory), 566t, 574t, 697 special, 569 visceral efferent (parasympathetic), 566t, 573t, 574t special, 569 zonular, 612, 614, 615 Fibula, 410, 427, 428, 429, 430, 433, 434, 435, 436, 437, 438, 440, 441, 445, 451, 454, 457, 460, 495, 497, 500, 501 articular surfaces of, 456 fractures of, 433 head of, 408, 410, 418, 425, 431, 432, 433, 434, 435, 436, 439, 440, 441, 444, 445, 448, 449, 450, 451, 484, 496, 497 MRI of, 503, 507 radiograph, 506 MRI of, 503, 509 neck of, 432, 434, 484 Index 719 Foramen/foramina posterior surface of, 451 radiograph of, 506, 508 shaft of, 432 Filia olfactoria, 549 Filum terminale, 41 Fimbriae at abdominal ostium, 257 of hippocampus, 689, 698 Fingers. See also Phalanges; Thumb ligaments of, 352–353 anterior (palmar), 353 during extension and flexion, 352 nails of, 353 neurovasculature of, 392 Fingertip, longitudinal section of, 353 Fissure cerebral, 698 longitudinal, 677, 679, 698, 699 horizontal, 682, 698 interhemispheric,MRI of, 701 of lung horizontal, 113, 116, 117, 118, 119, 130 oblique, 116, 117, 118, 119, 130 medial, anterior, 690 median, anterior, 683 orbital inferior, 546, 567, 598, 601, 602, 603, 608, 622, 623, 636, 643 superior, 548, 551, 563, 567, 603, 608, 619, 623, 636 palpebral, muscles of, 556, 556t petrooccipital, 547 petrotympanic, 546, 548, 624, 631, 638 posterolateral, 682 prebiventral, 682 primary, 682 pterygomaxillary, 598, 601, 623 width of, 622 tympanomastoid, 542 Fistula, ileorectal, 171 Flexure cervical, 677 colic left (splenic), 160, 161, 164, 172, 173, 181, 193, 220 radiograph of, 225 right (hepatic), 162, 164, 167, 169, 171, 172, 173, 179, 181, 182 radiograph of, 225 cranial, 677 duodenal inferior, 168 superior, 168 duodenojejunal, 164, 168, 170, 171, 173, 221 perineal, 248 sacral, 248 Flocculus, 682, 698 of cerebellum, 570 peduncle of, 682 Floor of oral cavity, 642, 645, 647 orbital, 610 pelvic in situ, 239 muscles of, 238–240, 240t neurovascular pathways of, 268t outermost layer of, 240 Fluid accumulation ascitic, 158 cardiac tamponade and, 95, 158 Fold alar, 442 aryepiglottic, 528, 529, 651 circular, of intestine, 168, 170 endoscopic appearance of, 169 radiograph of, 225 dural, 590 gastric (rugae), 107 interureteral, 252 longitudinal, 252 malleolar anterior, 630 posterior, 630 superior, 630 palatoglossal, 646 palatopharyngeal, 646 pterygomandibular, 648 rectal, transverse inferior, 249 middle, 249 superior, 249 rectouterine, 244, 248, 273 uterosacral ligament in, 257 rugal, 166 salpingopharyngeal, 620, 648, 650 semilunar, 172 sublingual, 642, 647, 649 umbilical lateral, 152–153, 160, 164, 170 with inferior epigastric artery and vein, 245 medial, 251 median, 152, 160, 164, 170 with obliterated umbilical artery, 244, 245 with obliterated urachus, 245 uterosacral, with uterosacral ligament, 251 vesical, transverse, 244, 245, 251 vestibular, 529, 648 vocal, 529, 648 Folium, of vermis, 682 Follicle Graafian, bulge from, 256 lymphatic, 170 MRI of, 288 Fontanelles, cranial, 544 Foodway, 648 Foot bones of, 452–453 dorsum of, 408, 496, 496t functional position, 453 joints of, 454–455, 454–457 distal articular surfaces, 455 interphalangeal, 408 proximal articular surfaces, 455 ligaments of, 460–461 MRI of, 509 muscles of intrinsic, 468, 468t of sole of foot, 464–465, 470–471, 470t superficial intrinsic, 469, 469t neurovasculature of, 496–499 sole of, 409 arteries, 472 muscles of, 464–465, 470–471, 470t topography, 498–499 veins, 474 Foramen/foramina apical, of tooth, radiograph of, 668 ethmoid anterior, 550, 602, 602t posterior, 550, 602, 602t incisive, 546, 636, 644 infraorbital, 513, 542, 567, 594, 602, 611, 644 MRI of, 668 interventricular, 679, 684, 685, 698 intervertebral, 4, 10, 11, 15, 21, 23, 40, 71 disk herniation and, 15 inferior vertebral notch of, 10 MRI of, 48, 701 radiograph of, 50 jugular, 546, 548, 572, 574, 576, 592, 661 lacerum, 546, 547, 548 magnum, 40, 546, 547, 548, 567, 659 venous plexus around, 589 mandibular, 567, 637, 638 radiograph of, 668 mastoid, 542, 544, 546, 548, 624 mental, 513, 542, 567, 594, 637, 644 radiograph of, 668 nutrient, 23 obturator, 230, 231, 232, 233, 238 radiograph of, 504 omental (epiploic), 156, 164, 171 boundaries of, 163t ovale, 97, 546 547, 548, 567, 580, 598, 600, 636 adult remnant of, 105t closed, 105 Index 720 Foramen/foramina (continued) Foramen/foramina (continued) limbus of, 97 open, 105 venous plexus of, 592 palatine greater, 546, 548, 636, 644 lesser, 548, 636, 644 parietal, 544, 545 for the pterygoid plexus, 636 rotundum, 551, 567, 602, 633, 623, 644 sacral anterior, 6, 12, 13, 233 MRI of, 51 posterior, 233, 482 scapular, 299 sciatic anterior, 237 greater, 236, 237, 268t, 491t lesser, 236, 237 sphenopalatine, 598, 617, 621, 622 spinosum, 546, 547, 548, 551, 591, 598 stylomastoid, 546, 548, 568, 569, 600, 624, 645 for superior laryngeal artery, vein, and nerve, 527 supraorbital, 542, 543, 602 inferior, 602, 602t superior, 602, 602t transverse, 7, 7t, 16, 20 posterior, 9 vertebral artery in, 17 vertebral, 7, 7t, 16, 17, 18, 57 cervical, 7t, 9 lumbar, 7, 7t, 11 relation to intervertebral disks, 14 thoracic, 7t zygomatic, 543 zygomaticoorbital, 602, 603 Forearm, 324–325 anterior region of, 294 arteries of, 365 bones of, 296–297 MRI of, 399 muscles of, 396–397 anterior, 332–333, 336–337, 336t, 388 deep, 336t intermediate, 336t superficial, 336t flexors, 327 posterior, 334–335, 340t, 389 deep, 340t, 341 radialis muscles, 338–339, 338t superficial, 340t, 341 posterior region of, 294 windowed dissection, 396 Forebrain. See Prosencephalon Forefoot, 452, 508 anterior arch, 462 range of motion, 456 Foregut, nerves of, 216 Foreign body, aspiration of, 120 Fornix, 679, 680, 682, 685, 698, 699 conjunctival, superior, 610 MRI of, 700 vaginal anterior, 252, 256, 260 posterior, 252, 256, 260 Fossa(e) acetabular, 231, 415, 417, 473, 489 axillary, 294 cerebellar, 547 cerebral, 547 coronoid, 300, 326, 327, 329 radiograph of, 402 cranial, 547 anterior, 547, 549, 603t, 616, 617, 619, 656, 659 middle, 547, 549, 593, 603t, 617, 622t, 623t, 658 posterior, 547, 549 cubital, 366, 386, 387 digastric, 637 hyaloid, 612 hypophyseal, 547, 551, 593, 616, 620, 658, 667 iliac, 149, 230, 232 incisive, 636, 650 infraclavicular, 54, 382 infraspinous, 299, 304, 317t infratemporal, 597, 622t bony boundaries of, 599 nerves of, 600, 600t neurovasculature of, 600 superficial dissection of, 599 topography of, 598–599 inguinal lateral, 152, 245 medial, 152, 153 intercondylar, radiograph of, 506 interpeduncular, 662, 683 intraclavicular, 294 ischioanal, 246, 248, 287, 491 anterior recess of, 244, 245, 247 jugular, 512, 624 of lacrimal sac, 602 malleolar, lateral, 432, 433 mandibular, 546, 598, 624, 638, 639, 641 navicular, 264, 267 olecranon, 301, 326, 327, 328 radiograph of, 402 paravesical, 244, 247 popliteal, 422, 485 pterygoid, 551, 636 pterygopalatine, 602, 622–623 bony boundaries of, 622 communications of, 622t nerves of, 623, 623t radial, 300, 326, 329 rhomboid, 683, 699 scaphoid, 627 splenic, 183 sublingual, 637 submandibular, 637 subscapular, 299 supraclavicular, 54 greater, 512 lesser, 512, 532 suprascapular, 298 supraspinous, 59, 317t supravesical, 152, 244 temporal, 597 tonsillar, 650 triangular, 627 trochanteric, 412 Fovea articular, 324, 325, 327, 330, 331 of femoral head, 412 radiograph of, 504 pterygoid, 637, 638 Fovea centralis, 612, 613, 615 Foveolae, granular, 545, 592 Fractures cervical, 9 of distal radius (Colles’), 331 of elbow joint, 376 of face, 543 of femur, 415 fibular, 433 of humeral neck, 372 of humerus, 301 lumbar compression, 11 scaphoid, 343 of skull, 587 Frenulum of clitoris, 262 of tongue, 647 Frontal bone, 512, 513, 542, 542t, 543, 545, 547, 558, 597, 603, 616 MRI of, 700 orbital surface of, 602, 603 Fundus of gallbladder, 178 optic, 613 of stomach, 107, 109, 166, 167 computed tomography (CT) of, 224 of urinary bladder, 252, 253, 267 of uterus, 244, 247, 251, 252, 257, 260, 261, 273 lymphatic drainage of, 277 Index 721 Gland(s) Funiculus, 691 lateral, 691 posterior, 691 Furrow, median, 646 G Galea aponeurotica, 552, 592 Gallbladder, 160, 162, 163, 164, 167, 170, 174, 175, 176, 177, 178, 190, 220, 221 autonomic nervous system of, 214 body of, 178, 179 computed tomography (CT) of, 222 fundus of, 178 infundibulum of, 178 location of, 178 lymphatic drainage of, 206 neck of, 178, 179 referred pain in, 214 Gallstones, 179 Ganglion/ganglia abdominal, 578 aorticorenal, 213t, 215, 217, 218 autonomic, 675 basal, 676 celiac, 212, 213t, 216, 217, 281, 694 cervical inferior, 103 middle, 86, 87, 103, 108, 127, 535, 655 superior, 87, 103, 212, 537, 565, 579, 579t, 597, 655, 694 cervicothoracic, 127 ciliary, 565, 567, 578, 578t, 579, 607, 609 cochlear (spiral), 571, 634, 635, 675 geniculate, 568, 569, 571, 629, 634, 645 impar, 213 inferior, 572, 573, 574 intramural (terminal), 697 lumbar, 280, 282 mesenteric inferior, 212, 213t, 215, 217, 219, 281, 282, 694 superior, 212, 213t, 214, 215, 217, 218, 280, 281, 694 otic, 573, 578, 578t, 600, 647 parasympathetic, 578, 579, 694 paravertebral, 38, 42, 695t, 697 pelvic, 213t, 215 prevertebral, 212 pterygopalatine, 567, 568, 567, 569, 578, 578t, 579, 581, 623, 644 renal, 215, 280, 281 sacral, 217, 282 sensory (spinal), 690, 696, 697 spinal, 17, 38, 40, 41, 42, 43, 71, 73, 691 MRI of, 48 stellate, 87, 103, 535, 655, 694 submandibular, 578, 578t, 645 superior, 572, 573, 574 sympathetic, 38, 42, 43, 71, 87t thoracic, 90, 103, 108, 127, 578 trigeminal, 567, 569, 581, 593, 607, 621, 645, 661 vestibular, 570, 635 inferior part of, 571, 635 parts of, 634 superior part of, 571, 635 Gastritis, 167 Genitalia. See also individual genital organs in female autonomic innervation of, 281 coronal section of, 261 external, 254t, 262–263 blood vessels of, 275 veins of, 275 internal, 254t lymph nodes of, 279 overview of, 254t innervation of, 694 in male arteries of, 275 autonomic innervation of, 280 blood vessels of, 275 external, 255t blood vessels of, 275 internal, 255t lymphatic drainage of, 277 lymph nodes of, 278, 279 neurovasculature of, 284 overview of, 255t veins of, 275 overview of, 254–255 in female, 254t in male, 255t Genu of corpus callosum, 698 MRI of, 700 internal, of facial nerve, 568 Girdle pelvic, 230, 246, 410 shoulder, 296, 298 joints of, 302 in situ, 298 Glabella, 543 Gland(s) accessory sex, 255t bulbourethral, 253, 255, 255t, 264, 266, 267, 284 ciliary, 610 lacrimal, 565, 567, 569, 607, 609, 662, 666 orbital part of, 608, 611 palpebral part of, 608, 611 lingual, 647 mammary, 74 nasal, 569 parathyroid, 530 inferior pair, 530 neurovasculature of, 530 superior pair, 530 parotid, 533, 534, 536, 538, 539, 569, 573, 594, 595, 663 accessory, 649 deep part of, 649 MRI of, 701 superfical part of, 649 pineal, 677, 680, 682, 683, 685, 689, 698, 699 pituitary, 658 proctodeal, 274 prostate, 152, 156, 242, 245, 246, 247, 253, 255t, 266, 270, 282, 287 apex of, 266 base of, 266 carcinoma of, 267 central zone of, MRI of, 291 divisions of anatomical, 266 clinical, 266 in situ, 267 lymphatic drainage of, 277 MRI of, 291 peripheral zone of, MRI of, 291 with prostatic plexus, 280 transitional zone of, MRI of, 291 salivary, 649 sebaceous, 610, 626 seminal, 212t, 242, 255t, 266, 267, 270, 280, 286, 287 lymphatic drainage of, 277 MRI of, 290, 291 sex, accessory, 266–267 sublingual, 569, 646, 647, 649 submandibular, 512, 537, 569, 649, 661 superficial part of, 597 suprarenal, 174, 185 autonomic nervous system of, 212t left, 162, 179, 181, 182, 183, 185 location of, 182 lymphatic drainage of, 205 right, 165, 175, 181, 182, 183, 184, 185, 198, 220, 222, 250 in retroperitoneum, 183 tarsal, 610 thyroid, 531, 536, 537, 648, 651, 654, 665 lobes left, 530 pyramidal, 530 right, 530 MRI of, 669 neurovasculature of, 530 Index 722 Gland(s) (continued) Gland(s) (continued) urethral, orifices of, 252, 264 vestibular greater (Bartholin’s), 254, 254t, 262, 263, 285 lesser, 254t of vestibular and vocal folds, 529 Glans of clitoris, 229, 262, 277, 285 of penis, 155, 229, 264, 265, 267 255 corona of, 264, 275 Glaucoma, 614 Globus pallidus, 689 medial segment of, 660 Gluteal region, 3, 418, 421, 422, 424, 491 deep, 490 innervation cutaneous of, 482 muscles of, 409, 426t deep, 427 lateral, 425 posterior, 424 neurovasculature, 490 topography, 490–491 Gonads. See also Genitalia lymphatic drainage of, 205 Granulations, arachnoid, 545, 590, 592, 684 Gravity center of, 5, 411 line of, 5, 411 Gray matter, 42, 674, 675, 676 central, 699 organization of, 690 Groove arterial, 624 capitulotrochlear, 301, 326, 327, 329 chiasmatic, 547, 551 costal, 11, 71, 72, 73 deltopectoral, 54, 294, 366, 382 for fibularis longus tendon, 453 for flexor hallucis longus tendon, 459 for inferior vena cava, 177 infraorbital, 602, 656 intertubercular, 300, 301, 303, 304, 306, 311, 317, 318, 321, 375 tendon sheath in, 307 for lesser petrosal nerve, 547 for middle meningeal artery, 545 mylohyoid, 637, 638 for posterior tibialis tendon, 432 radial, 300, 315, 323, 373, 381 for sigmoid sinus, 547, 624 for subclavian artery, 519 for subclavius muscle, 298 for superior sagittal sinus, 545 for transverse sinus, 547 ulnar, 301, 326, 328, 341, 386 Growth plate, radiograph of, 508 Gubernaculum, 150, 154 Gutter, paracolic, 165 Gyrus ambient, 562 angular, 678 cingulate, 677, 679, 698 dentate, 661, 698 frontal medial, 679 middle, 678 superior, 666, 678 postcentral, 677, 678, 692 precentral, 577, 677, 678, 693 semilunar, 562 subcallosal, 679 supramarginal, 678 temporal inferior, 678 middle, 678 superior, 678 H Habenula, 680 Hallux, of interphalangeal joints, 454 Hamate, 342, 342t, 344, 345, 346, 350, 391, 404 hook of, 295, 337, 343, 345, 350, 361, 363, 391 MRI of, 405 radiograph of, 343 MRI of, 404 Hamulus, pterygoid, 546, 551, 598, 643, 653 Hand. See also Fingers; Thumb arteries of, 365 bones of, 296–297, 342–344, 342t carpal region of, 390–391 dorsal, 342, 346 dorsum of, 294, 358–359, 394–395 cutaneous innervation, 394 dorsal digital expansion of, 359 lymphatic drainage of, 367 muscles of, 358 neurovascular structures, 395 tendons of, 358 functional position of, 349 joints of, 346–347 ligaments of, 348–349 lymphatic drainage of, 367 muscles of insertions of, 357 intrinsic deep layer of, 356–357 hypothenar group, 360t, 361 metacarpal group, 360 middle layer of, 354–355, 356–357 superficial layer of, 354–355 thenar group, 360, 360t, 361 metacarpal, 362–363, 362t origins of, 357 palm of, 294, 295, 343, 392–393 MRI of, 405 neurovascular structures deep, 393 superficial, 392 “Hangman’s fracture,” 9 Haustra, colonic, 172 radiograph of, 225 Head, 512–513. See also Brain; Cavity, oral; Ear; Eye; Larynx; Nose; Orbit; Pharynx arteries of, 582–583 nerves of parasympathetic, 578, 578t sympathetic, 212 neurovasculature of, superficial, 595 organs of. See individually named organs palpable bony prominences, 513 regions of, 512 surface anatomy of, 512–513 tissue spaces of, 654 veins of, 588–589 deep, 589, 589t superficial, 8 Hearing, ossicular chain in, 631 Heart, 5, 92, 93, 131 arteries of, 100–101, 100t blood circulation in, 92 chambers of, 97 computed tomography (CT) of, 134, 224 conduction system in, 98, 102 diaphragmatic surface of, 94 electrocardiogram of, 102 functions and relations of, 92–93, 95 in situ, 93 innervation of, 694 lymphatic drainage of, 110 MRI of, 131, 135 radiograph of, 132 skeletal structure of, 98 surfaces of, 96 topographical relations of, 92 valves of. See Cardiac valves veins of, 100, 100t, 101 ventricular outflow tract of left, 131, 135 right, 131 Heart sounds, auscultation sites for, 99 Helicobacter pylori infection, 167 Helicotrema, 634 Helix, 512, 627 Hematoma epidural, 587, 590, 592 subdural, 592 Hemorrhage extracerebral, 591 Index 723 Imaging subarachnoid, 591 subdural, 590 Hernia/herniation diaphragmatic, 147 inguinal, 152–153, 153t direct, 153t femoral, 153t indirect, 153t intervertebral disk, 15 Hiatus adductor, 419, 420, 429, 472, 473, 474, 493 aortic, 64, 65, 80, 82, 84, 147, 188, 222 basilic, 366, 378 of canal for greater petrosal nerve, 548, 568 of canal for lesser petrosal nerve, 548 distal, 391 esophageal, 64, 65, 67, 82, 107, 147, 188, 222 as herniation site, 147, 147 of facial canal, 547 intervertebral disk, 15 maxillary, 617 proximal, 391 rectal, 238 sacral, 12, 40, 41, 232, 233, 237 semilunaris, 618, 621 triceps, 381, 381t urogenital, 238 Hilum of kidney, 182, 184 computed tomography (CT) of, 222 of lung, 117 of spleen, 180 Hindbrain. See Rhombencephalon Hindfoot, 452 range of motion, 456 Hindgut, nerves of, 215 Hippocampus, 689, 698, 699 fimbria of, 698 Hip. See Joints (individual), hip Hook, of hamate, 295, 337, 343, 345, 350, 361, 363, 391 MRI of, 405 radiograph of, 343 Horns anterior (motor), 690 of gray matter anterior, 690 lateral (visceromotor), 690 intermediolateral, 690 posterior, 690 of hyoid, 526, 527, 637 of lateral brain ventricles anterior, 685 frontal, MRI of, 700 inferior, 685 occipital, 662 posterior, 699 of thyroid cartilage, 526, 527 Humerus, 296, 300–301, 304, 306, 307, 318, 321, 326, 327, 328, 329, 339, 397, 403 borders of, 300, 301, 326 capitulum of, 327 condyle of, 300 epicondyle of. See Epicondyle, of humerus fractures of, 301 greater tubercle of, 300 head of, 300, 301, 303, 304, 383, 384t MRI of, 401 radiograph of, 400 lesser tubercle of, 300, 301 ligaments of, 306 MRI of, 398, 403 neck of anatomical, 300, 301, 304, 305 fractures of, 372 surgical, 300, 301 radiograph of, 402 shaft of, 300, 301, 316, 317, 322, 323 MRI of, 398 trochlea of, 301 Hydrocephalus, 232 Hyoid bone, 526, 527, 529, 531, 532, 536, 537, 637, 646, 647, 648, 649, 659, 660, 661 body of, 517, 526, 637 horns of greater, 526, 537, 653 lesser, 526, 637 muscle origins and insertions of, 555 Hyperacusis, 633 Hyperopia, 615 Hypertension, renal artery, 189 Hypophysis, 593, 628, 659, 662, 667, 677, 679, 683, 698 anterior lobe (adenohypophysis), 680, 682 in hypophyseal fossa, 658 MRI of, 700 posterior lobe (neurohypophysis), 680, 682 Hypothalamus, 677, 680, 682, 685, 698 MRI of, 701 parasympathetic and sympathetic neurons of, 695t Hypotympanum, 629 I Ileum, 156, 160, 168, 245 location of, 170 lymphatic drainage of, 203, 208 MRI of, 51, 225 radiograph of, 225 terminal, 164, 171, 172, 173 Crohn’s disease and, 171 MRI of, 171 Ilium, 237, 248, 258, 261, 321, 415 auricular surface of, 143 body of, 230, 231 gluteal surface of, 236 margins, 415 MRI of, 505 radiograph of, 225, 231 Imaging of abdomen computed tomography (CT) of coronal section through the kidneys, 224 of sagittal section through the aorta, 224 of transverse section through L2, 222 of transverse section through L3, 223 of transverse section through L4, 223 of transverse section through L5, 223 of transverse section through T12, 222 MRI of hepatic cirrhosis, 171 of the intestines, 225 of thickened wall of terminal section of ileum, 171 cholangiopancreatography, 179 radiography of double contrast barium enema, 225 of ileorectal fistula, 171 of intravenous pyelogram, 224 of large intestine in colitis, 173 of back MRI of lateral view of cervical spine, 8, 49 of lateral view of lumbar spine, 49 of left lateral view of normal umbar spine, 11 of left lateral view of osteoporotic lumbar spine, 11 of midsagittal T2-weighted view of cervical spine, 21 of oblique view of cervical spine, 49 of oblique view of sacrum, 51 of parasagittal view of lumbar spine, 48 of “whiplash” injury in cervical spine, 9 radiography of anteroposterior view fof thoracic spine, 49 dental panoramic tomogram, 641 of gallbladder and bile ducts MR cholangiopancreatography of, 179 ultrasonography, of bile duct obstruction, 179 of head and neck arteriography, of dural venous sinus system of the head, 671 computed tomography (CT) scan, of temporomandibular joint, 670 dental panoramic tomogram of, 641 Index 724 Imaging (continued) Imaging (continued) MR angiography, cranial, 671 MRI coronal section through cervical spinal cord, 701 coronal section through ventricular system, 701 midsagittal section of brain, 700 midsagittal through the nasal septum, 667 of neck, 701 through the eyeball, 666 transverse section through cerebral hemispheres, 700 transverse through the neck, 668 transverse through the orbit and nasolacrimal duct, 668 radiography of the mandible, 668 of the skull, 668 of limb (lower) MRI of Baker’s cyst, 437 of hip joints, 505 of the leg, 503 of the right ankle, 509 of the right foot, 509 of right hip joint, 505 of the thigh, 502 of transverse section of knee joint, 507 radiography of the ankle, 508 anterior-posterior view of the forefoot, 508 of knee in flexion, 506 of right hip joint, 504 of right hip joint (Lauenstein view), 504 of right knee joint, 506 ultrasonography, of hip dysplasia and dislocation, 415 of limb (upper) MRI coronal section of shoulder, 401 coronal section through carpal tunnel, 405 coronal section through humeroulnar and humeroradial joints, 403 coronal section through palm, 405 of hand, 405 of right wrist, 404 sagittal section of right shoulder, 401 of sagittal section of shoulder, 401 sagittal section through humeroulnar and humeroradial joints, 403 sagittal section through humeroulnar joint, 403 transverse section of arm, 398 transverse section of forearm, 399 transverse section of right shoulder, 401 of wrist, 404 radiography of elbow, 402 of hand, 404 of right shoulder, 400 of scapula, 400 of wrist, 343 ultrasonography, of anterior region of left shoulder, 400 of pelvis and perineum of child’s acetabulum, 231 MRI of coronal section of female pelvis, 289 of pelvis (male), 290–291 of prostate gland, 291 of sagittal section of female pelvis, 289 of testes, 290 of transverse section of female pelvis, 288 of thorax, 131–133 angiography, of aortic arch, 135 arteriography, of anterior view of pulmonary arteries and veins, 125 bronchogram anterioposterior view of lung, 133 of anterior-medial basal segment of lung, 133 of basal segment of lung, 133 computed tomography (CT), of transverse section of thorax, 79, 136–137 MRI of coronal view of thorax, 133 of sagittal section of heart, 131 of transverse section of heart, 131 radiography of anterior view of heart, 132 of lateral view of heart, 132 of left anterior oblique view, of right coronary artery, 134 of left lateral chest radiograph, 132 of posterior chest radiograph, 132 of right anterior oblique view, of left coronary artery, 134 Impressions aortic, 117 cardiac, 117 colic, 174 for costoclavicular ligament, 298 duodenal, 174 gastric, 174 renal, 174 suprarenal, 174 Incisors, 640 mandibular, 640 Incisure frontal, 543, 602, 602t intertragic, 627 Incus, 625, 626, 628, 629, 630, 631, 632 articular surface for, 630 body of, 630 in conductive hearing loss, 631 ligaments of, 631 process of lenticular, 630 long, 630 short, 630 Inflection points, of vertebral column, 5, 411 Infrascapular region, 3 Infundibulum, 97, 680, 681, 682, 685, 698 ethmoid, 550 of right outflow ventricular tract, 131 of uterine tube, 257 Inguinal region, 150–151 herniation of, 152–153, 153t in male, 150, 488 dissection of, 151 muscles of, 150 neurovasculature of, 150 structures in, 150, 489t topography, 488–489 Inion, 544, 546 Inlet esophageal, 88, 106 laryngeal, 529, 529t, 651 pelvic, 233, 234, 234t, 246t in female, diameters of, 235 plane of, 235 thoracic, 78, 79, 88, 122, 130 Innervation, cutaneous of anterior trunk, 210 of back, 15 of breast, 74 of dorsum of hand, 394 of neck, 624–625 of gluteal region, 482 of inguinal region, 478 of limb (lower), 487 of limb (upper), 379 of nuchal region, 624 of thoracic wall, 70 Inspiration deep, 60 lung expansion in, 123 in normal versus pneumothorax, 123 pleural cavity expansion in, 123 quiet, 60 rib cage movement in, 60 thoracic volume changes in, 123 Insula, 677, 679, 689, 698, 699 Interneurons, 693 Interscapular region, 3 Index 725 Joints (individual) Intersections, tendinous, 145, 149 Intestines innervation of, 694 large, 172. See also Anus; Appendix/ appendices; Cecum; Colon; Rectum anastomoses between arteries of, 193 in situ, 173 lymphatic drainage of, 209, 209 mesentery of, 172 nerves of, 219 radiograph of, 225 small, 174. See also Duodenum; Ileum; Jejunum computed tomography (CT) of, 223 convoluted, 161 mesentery of, 158, 159, 160 nerves of, 218 radiograph of, 225 referred pain in, 215 wall of, 697 Intraperitonal organs, sites of attachment of, 164–165 Iris, 610, 612, 614, 614, 615 greater arterial circle of, 613 lesser arterial circle of, 613 Ischemia, mesenteric, 192 Ischium, 248, 413 body of, 231 MRI of, 505 radiograph of, 231, 504 Isocortex, 679 Isthmus faucial, 651 of the oropharynx, 643 prostatic, 266 of thyroid gland, 530 uterine, 256, 257 J Jejunum, 168, 169, 170, 179, 180, 181, 221, 256, 260 computed tomography (CT) of, 222, 223, 224 in situ, 170 location of, 170 lymphatic drainage of, 203, 208 MRI of, 225 radiograph of, 225 Joints (individual) acromioclavicular, 59, 296, 298 MRI of, 401 radiograph of, 400 atlantoaxial, 16t, 19 lateral (capsule), 19, 20 median, 18, 659, 663 atlantooccipital lateral, 18 median, 16t, 19, 20 calcaneocuboid, 454, 455, 508 calcaneofibular, MRI of, 509 carpometacarpal, 346 MRI of, 405 of thumb, 346, 347, 350 movement of, 347 costotransverse, 56 costovertebral, 61 craniovertebral, 16t, 18–19, 30, 30t cricoarytenoid, 527 cricothyroid, 527 cuneocuboid, 454 cuneonavicular, 454 elbow (cubital), 296, 324–329, 326–327 arterial network of, 389 capsule of, 329 injuries to, 329 fracture/dislocation, 376 injury assessment in, 327 ligaments of, 328–329, 328t skeletal elements of, 327 soft-tissue elements of, 327 femopatellar, 435, 438 glenohumeral, 300, 304–305 capsule of, 305, 307 cavity of, 304 ligaments of, 305 hip, 410, 411, 414–415 bones of, 410–411 capsule of, 416 synovial membrane of, 417 weakness in, 417 dysplasia and dislocation, 415 ligaments of, 416 muscles of, 426–427, 426t anterior, 418, 420 lateral, 425 medial, 421 posterior, 422, 424 right, 414 humeroradial, 326, 328t radiograph of, 402 humeroulnar, 327, 328t radiograph of, 402 incudomalleolar, 630, 631 incudostapedial, 632 intercuneiform, 454 intermetatarsal, 454 interosseous, dorsal 1st, 346 4th, 346 interphalangeal, 295, 340, 340t, 346 distal, 340, 340t, 346, 352, 353, 454 collateral ligaments of, 348, 353 crease of, 295 joint capsule of, 349 radiograph, 508 of foot, 408 of the hallux, 454 middle, radiograph of, 404 proximal, 340, 340t, 346, 352, 454 collateral ligaments of, 348, 353 crease of, 295 joint capsule of, 349 radiograph, 508 radiograph of, 404 of thumb, 346 intertarsal, radiograph of, 508 intervertebral, 14, 16t knee, 410, 411, 434–435 bones of, 432–433 capsule of, 436–437 effusion of, 443 in flexion, 441, 441 ligaments, 436–437, 436t, 438 MRI of, 507 right, 434 metacarpophalangeal, 295, 340, 340t, 346, 352, 359, 392, 408 abduction and adduction at, 346 collateral ligaments of, 348 crease of, 295 flexion and extension of, 352 joint capsule of, 349 MRI of, 405 of thumb, 346 metatarsophalangeal, 454, 455 1st, 454 capsule of, 471 of great toe, 462 radiograph, 508 midcarpal, 346 articular surfaces of, 345 movement at, 348 patellofemoral, radiograph of, 506 radiocarpal, 346 articular surfaces of, 345 movement at, 348 radioulnar, 330–331 distal, 325, 331, 346 pronation/semipronation of, 331 rotation of, 331 MRI of, 399 pronation of, 330 proximal, 325, 326, 327, 328t, 330, 331 radiograph of, 402 supination of, 330 sacrococcygeal, 12 sacroiliac, 232, 411 ligaments of, 237 MRI of, 51 radiograph of, 13, 504 symphyseal surface of, 237 sternoclavicular, 59, 297, 298, 513 Index 726 Ligaments (individual) (continued) Joints (individual) (continued) sternocostal, 60 subtalar (talocalcaneal), 450, 454, 456, 457, 458–459 anterior compartment of, 457, 458 MRI of, 509 posterior compartment of, 457, 458 radiograph, 508 talocalcaneonavicular, 457, 508 talocrural (ankle), 410, 450, 454, 456, 457 ligaments of, 460–461, 460t MRI of, 509 radiograph, 508 talofibular, MRI of, 509 talonavicular, 454, 455 MRI of, 509 radiograph, 508 tarsal, transverse, 454 tarsometatarsal, 454 temporomandibular, 638 capsule of, 553, 558, 597, 639 innervation of, 639 computed tomography (CT) of, 670 dislocation of, 639 ligaments of, 638 movement of, 639 radiograph of, 668 tibiofibular, 432, 434, 439 MRI of, 507 transverse tarsal, 454 uncovertebral, 8, 16t, 17 zygopophyseal, 8, 10, 11, 16, 16t, 20 capsule of, 21, 22 MRI of, 48 Joints (of region or structure) of foot, 454–455, 454–457 distal articular surfaces, 455 proximal articular surfaces, 455 of hand, 346–347 of neck, 515, 515t of shoulder, 296–297, 302–303 acromioclavicular, 302 injuries of, 303, 303 glenohumeral, 302, 304–305 scapulothoracic, 302 sternoclavicular, 297, 298, 302 sternocostal, 303 of shoulder girdle, 297 of thigh, muscles of, 425 of thumb, 336t, 340t, 342, 343, 346, 347, 350, 360t of vertebral column, 16–19, 16t of wrist, 346–347 Jugum sphenoidale, 547, 551 Junction anorectal, 249 cervicothoracic, 3, 4 line of gravity through, 5 craniocervical, 4 gastroesophageal (Z line), 107 jugulofacial subclavian venous, 522 lumbosacral, 4 pontomedullary, 565 thoracolumbar, line of gravity through, 5 of triradiate cartilage, 231 ureteropelvic, 250 K Keratoconjunctivitis sicca, 611 Kidney, 25, 182–185, 183 autonomic nervous system effects on, 212t, 215 borders of, 184 innervation of, 694 left, 163, 165, 166, 169, 174, 179, 180, 181, 185, 220, 250 computed tomography (CT) of, 222, 224 location of, 182 lymphatic drainage of, 205 poles of inferior, 184 computed tomography (CT) of, 224 right computed tomography (CT) of, 224 superior, 162, 222 relations of, 182 renal hypertension and, 189 in retroperitoneum, 183 right, 162, 165, 167, 169, 175, 182, 185, 198, 221, 250, 254 computed tomography (CT) of, 222, 223, 224 in renal bed, 182 structure of, 184 suprarenal glands in, 183 Kidney stones, 251 Kiesselbach’s area, 585 Knee. See Joints (individual), knee Köhler’s teardrop figure, 504 Kyphosis, 4, 5 L Labium majus/labia majora, 229, 254, 254t Labium minus/labia minora, 229, 254, 254t, 260, 261, 262, 285 MRI of, 289 Labrum acetabular, 413, 415, 417, 473 glenoid, 307, 383, 401 iliocecal, 172 iliocolic, 172 Labyrinth, membranous, innervation of, 635 Lacrimal apparatus, 611 Lacrimal bone, 542, 602, 617 Lacrimal drainage, 611 Lacuna, lateral, 590, 592 Lacuna, of pelvic floor, 268t Lamina of cervical vertebrae, 9, 17 radiograph of, 49 of cricoid cartilage, 527 left, 661 of lumbar spine, MRI of, 48 of thoracic vertebra, 7, 10 of thoracolumbar spine, 23 of thyroid cartilage, 526 of vertebral arch, 7, 20 Lamina affixa, 680 Lamina cribosa, 612 Lamina papyracea, 550, 603 Lamina terminalis, cistern of, 684 Laryngopharynx, 648, 658, 659, 660, 665 Larynx arteries of, 531 cartilage of, 526 levels of, 529, 529t MRI of, 669 muscles of, 528–529, 528t actions of, 528t neurovasculature of, 531 structure of, 527 veins of, 531 “Lateral elbow syndrome,” 338 Leg bones of, 432–433 motor pathways of, 693 muscles of anterior compartment, 444, 449, 449t lateral compartment, 445, 448, 448t posterior compartment, 446–447, 450–451, 450t, 451 neurovasculature of, 497 anterior, 497 deep, 494 superficial, 494 topography of anterior compartment, 496, 496t lateral compartment, 496, 496t posterior compartment, 494–495, 496, 496t Lemniscus, medial, 692 Lens, 610, 612, 614, 615, 662 light refraction by, 615 Ligaments (individual) acromioclavicular, 302, 303, 304, 305, 307 alar, of dens, 18, 19 of ankle, 460–461 anococcygeal, 238, 239, 240 Index 727 Ligaments (individual) anular, 120, 329, 330, 352, 353, 354, 359, 464 MRI of, 399 of radius, 327, 328, 329, 331 apical, of dens, 18, 19, 21 arcuate lateral, 211 medial, 64, 65, 211 median, 64, 65, 187, 211 popliteal, 437 pubic, 251, 284 bifurcate, 458, 460, 461 broad, 256, 273 germinal epithelial covering of, 256 peritoneal covering of, 256 of uterus, 244, 251, 273 calcaneocuboid plantar, 463 calcaneocuboid, dorsal, 458 calcaneofibular, 460 calcaneonavicular, plantar, 457, 458, 461, 463, 466, 471, 509 cardinal (transverse cervical), 247, 258 carpal palmar, 350, 390, 391, 393 transverse, 350, 390, 391, 392, 396 carpometacarpal dorsal, 348 palmar, 349 cervical, transverse (cardinal), 247, 258, 259, 261 collateral, 352, 359 accessory, 352 of distal interphalangeal joints, 348, 353 femoral medial, 507 MRI of, 507 in flexion and extension, 441 lateral, 435, 436, 437, 439, 440, 442 fibular, 442 MRI of, 507 medial, 403, 435, 436, 436t, 437, 438, 439 MRI of, 507 of metacarpophalangeal joints, 348, 353 MRI of, 405 of proximal interphalangeal joints, 348, 353 radial, 327, 328, 329, 330 at wrist joint, 348 ulnar, 327, 328, 329, 330 anterior part of, 328 posterior part of, 328 transverse part of, 328 at wrist joint, 345, 346, 348 conoid, 303 coracoacromial, 302, 303, 304, 305, 306, 307 coracoclavicular, 303, 304, 305, 307, 380 dislocation of, 303 coracohumeral, 305 coronary, 176, 177 costoclavicular, impression for, 298 costotransverse lateral, 61 superior, 61 costoxiphoid, 61 cricoarytenoid, middle, 528 cricothyroid, 527 median, 527, 529, 530, 531 cricotracheal, 527 cruciate, 435, 440–441, 442 anterior, 439, 440, 442, 443 MRI of, 507 in flexion and extension, 441 posterior, 439, 440, 441 MRI of, 507 rupture of, 441 cruciform, 352, 353, 354, 464, 469 deltoid, 460, 460t, 461 MRI of, 509 denticulate, 40 extracapsular, 442 falciform, of liver, 152, 160, 167, 175, 176, 220 fundiform, of penis, 144, 154 gastrocolic, 162, 163, 165, 181 gastrosplenic, 162, 164, 171 glenohumeral inferior, 305 middle, 305 superior, 305 Gruber’s, 593 hepatoduodenal, 162, 163, 164, 165, 167, 169, 171, 173, 175, 181, 183 margin of, 216 hepatoesophageal, 163, 167 hepatogastric, 156, 162, 164, 165, 166, 167, 171, 175, 180, 181 hyoepiglottic, 529 iliofemoral, 416, 417, 419 ilioinguinal, 489t iliolumbar, 236, 416, 417, 419 inguinal, 142, 144, 145, 149, 150, 151, 204, 228, 236, 268, 271, 416, 418, 427, 473, 474, 475, 479, 481, 486, 488, 489, 492 deep, 488 reflected, 149, 151, 488, 489, 489t interarticular, 61 intercarpal dorsal, 348 palmar, 349 interclavicular, 303 interfoveolar, 151, 152 interspinous, 14, 20t, 21, 22 MRI of, 48 intertransverse, 18, 20, 20t, 22, 23 intracapsular, 442 ischiofemoral, 416, 417 lacunar, 150, 151, 488, 489, 489t lateral, of bladder, 258 lateral, of temporomandibular joint, 553 ligamenta flava, 21, 22, 23 longitudinal anterior, 20, 20t, 21, 22, 23, 63, 236, 416, 418 MRI of, 21 posterior MRI of, 21 meniscofemoral, posterior, 439, 440 metacarpal dorsal, 348 palmar, 349 transverse deep, 349, 352, 353, 354, 355, 359 superficial, 354 metatarsal dorsal, 460 transverse deep, 462 superficial, 464 nuchal, 18, 19, 21, 32t, 320, 512, 533, 659, 667 MRI of, 21, 48 ovarian, 257 proper, 256 MRI of, 288 palmar, 353, 356 palmar carpal, 295 palpebral lateral, 608 medial, 608 patellar, 418, 419, 421, 425, 435, 436, 436t, 438, 439, 440, 442, 443, 444, 445, 497 MRI of, 507 pectineal, 152, 236 pelvic, in female, 258–259 perineal, transverse, 284 phalangoglenoid, 352 phrenicocolic, 162 phrenicosplenic, 169 plamar, 349 plantar, 462, 466 long, 458, 461, 463, 465, 466, 471 popliteal, arcuate, 436t, 437, 495 popliteal, oblique, 431t, 436t, 437, 495 pterygospinous, 638 pubic, inferior, 238 pubofemoral, 416, 417 pubovesical, 258 pulmonary, 117 Index 728 Ligaments (individual) (continued) Ligaments (individual) (continued) radiocarpal dorsal, 348 palmar, 349 MRI of, 405 radioulnar dorsal, 330, 331, 348 palmar, 330, 331, 349 rectouterine, 258 round of liver, 152, 160, 164, 167, 170, 171, 174, 176, 177 MRI of, 288 of ovary, 254, 258 of uterus, 150, 243, 244, 247, 251, 252, 272, 273 sacrococcygeal, anterior, 237 sacroiliac anterior, 236, 237, 238, 416 MRI of, 51 interosseous, 237 posterior, 236, 237, 416 long, 236 MRI of, 51 short, 236 sacrospinous, 147, 236, 237, 238, 268, 286, 416, 421, 427, 491t sacrotuberous, 236, 237, 238, 268, 416, 423, 427, 431, 483, 485, 491, 491t, 493 sphenomandibular, 599, 638 splenorenal, 180 stapedial, anular, 630, 631 in conductive hearing loss, 631 sternoclavicular anterior, 302, 303 posterior, 302 sternocostal, radiate, 61 stylomandibular, 638 supraspinous, 20, 20t, 21, 22, 416 suspensory of breast (Cooper’s), 75 of duodenum, 168 of ovary, 254, 261 of penis, 253, 271, 275 of uterus, 244, 247, 251 with ovarian artery and vein, 252 syndesmotic, 461 talocalcaneal, interosseous, 454, 457, 458, 461, 509 talonavicular, dorsal, 460, 461 tarsal, dorsal, 460, 461 temporomandibular, lateral, 558 thyroepiglottic, 527 thyrohyoid, 527, 529, 648 median, 527, 531, 534, 535, 536 tibiofibular, 460, 460t anterior, 460, 460t, 461 posterior, 460, 460t, 461 transverse of atlas, 18, 19, 659, 663 of humerus, 304, 306 inferior, of scapula, 381t of knee, 438, 440 superior, of scapula, 299, 303, 304, 305, 306, 307, 370, 380, 381t trapezoid, 303 triangular, left, 175, 176, 177 ulnocarpal, palmar, 349 umbilical medial, 259 median, 250, 251, 252, 254, 255 uterosacral, 257, 258, 259, 286 with uterosacral fold, 251 of the uterus, 256 vesicouterine, 258 vesicular, lateral, 259 vestibular, 527, 529 vocal, 527, 529 Ligaments (regional) of fibular head, 438, 440 of fingers, 352–353 of foot, 460–461 of glenohumeral joint, 305 of hand, 348–349 of head of femur, 286, 413, 417, 473 of hip joint, 416 of incus posterior, 631 superior, 631 of knee joint, 436–437, 436t extrinsic, 436t intrinsic, 436t of malleus lateral, 630 superior, 631 of neck, 515, 515t of ovary, 243, 244, 252, 258 of pelvis, 236–237 in female, 258–259 of sacroiliac joint, 237 of temporomandibular joint, 638 of uterus, 251 of vena cava, 176 of wrist, 350–351 Ligamentum arteriosum, 89, 91, 93, 94 96, 97, 103, 105, 105t, 109, 125 Ligamentum flavum, 14, 15, 19, 20, 20t, 21, 22 Ligamentum teres hepatis. See Ligaments (individual), round, of liver Ligamentum venosum, 105, 105t, 177 Light, cone of, 630 Limb (lower). See also Ankle; Foot; Leg; Thigh arteries, 472–473 bones of, 410–411 dermatomes, 487 lymphatic drainage of, 85 lymphatics, 475 muscles of palpable, 409 nerves of cutaneous innervation, 487 lumbar plexus, 43, 208, 209 lumbosacral plexus, 478–481, 478t, 480t, 481t sacral plexus, 188, 210, 250, 268, 271, 281, 283, 476t, 477, 492 superficial, 486 neurovasculature of, 472–499 palpable bony prominences of, 408 regions of, 408 veins, 474–475 perforating veins, 475 Limb (upper). See also Arm; Hand arteries of, 364–365 bones of, 296–299 dermatomes of, 379 lymph nodes of, 367 muscles of, palpable, 294 nerves of brachial plexus, 368–369, 368t cutaneous innervation, 379 superficial cutaneous, 378 Limbus, 679 corneoscleral, 612 Limen nasi, 620 Linea alba, 144, 145, 146, 148, 149, 150, 488 Linea aspera, 414, 431 Linea terminalis, 157, 235, 480 Line(s) anocutaneous, 249 arcuate, 142, 143, 145, 146, 149, 152, 160, 170, 230, 232, 237, 238, 488 axillary anterior, 55 posterior, 55 epiphyseal, 415 gluteal anterior, 231 inferior, 231 posterior, 231, 427 of gravity, 5, 411 intercondylar, 412 intermediate, 232 intertrochanteric, 414, 417, 427, 430 “life line” (thenar crease), 295 Lisfranc’s joint line, 454 midaxillary, 55, 111, 111t midclavicular, 54, 75, 111, 112t Index 729 Lymphatic drainage mylohyoid, 517, 637 nuchal inferior, 30, 35, 519, 544, 546 median, 544 superior, 18, 26, 27, 29, 30, 33, 35, 313, 320, 519, 544, 546 supreme, 544, 546 oblique, 526, 542, 543, 637 parasternal, 55 paravertebral, 3t, 111, 111t pectineal, 230, 232, 233, 237, 274, 412, 414 scapular, 3t, 111, 111t semilunar, 140, 145 soleal, 432, 434, 451 sternal, 3t, 111, 111t supracondylar lateral, 412 medial, 412 temporal inferior, 597 superior, 597 transverse, of sacrum, 12 Z (gastroesophageal), 107 Lingual glands, 647 Lingula, 637, 638, 682, 698 of lung, 117, 133 Lip commissure of, 512 of iliac crest inner, 232 outer, 149, 232 of iliocecal labrum, 172 of uterine os, 260 Lithotomy position, 229 Liver, 71, 88, 104, 105, 113, 114, 131, 156, 159, 163, 174–175, 180, 182, 190, 221 attachment to diaphragm, 88, 95, 174 autonomic nervous system effects on, 212t, 214 bare area (diaphragmatic surface) of, 156, 158, 174, 175, 176, 177, 178 borders of, 176 cirrhosis of, 175 computed tomography (CT) of, 222 fibrous appendix of, 176, 177 hepatic surface of, 175 inferior surface of, 175 in situ, 173–174 innervation of, 694 ligaments of, 152, 160, 164, 167, 171, 174, 175, 176, 220 lobes of. See Lobes, of liver location of, 174, 207 lymphatic drainage of, 203, 206 MRI of, 133 relations of, 174 round ligament of, 152, 160, 164, 167, 170, 171, 174, 176, 177 visceral surface of, 176, 177 Lobectomy, of lung, 119 Lobes of brain anterior, 682 frontal, 677 MRI of, 700 occipital, 677 parietal, 677 posterior, 682 temporal, 677 computed tomography (CT) of, 670 MRI of, 700, 701 of cerebellum anterior, 682, 698, 699 flocculonodular, 682 posterior, 682, 699 MRI of, 669 of cerebral hemispheres, 679 frontal, 679 insular (insula), 678, 679 limbic (limbus), 679 occipital, 679 parietal, 658, 679 MRI of, 700 of cerebrum, frontal, 656, 657 of liver caudate, 163, 176, 177, 178 inferior, 113, 130 left, 160, 163, 164, 167, 171, 174, 175, 176, 178, 220 computed tomography (CT) of, 224 quadrate, 176 right, 160, 162, 163, 164, 167, 174, 177, 177t, 179, 220 computed tomography (CT) of, 224 of lung, 133 arteries and veins of, 125t of bronchus inferior, 114, 116, 117, 124, 126, 127 middle, 126, 127 superior, 126, 127 right, 160, 162, 163, 164, 167, 174, 177, 177t, 179, 220 computed tomography (CT) of, 224 inferior, 113, 116, 130 middle, 113, 116, 130 superior, 113, 116, 117, 124, 130 mammary, 74, 75 of prostate gland, 266 pyramidal, 536 temporal, 679 Lobule(s) central, 682 paracentral, 679 parietal inferior, 678 superior, 678 quadrangular, 682 semilunar inferior, 682 superior, 682 simple, 682 of mammary gland, 75, 77 testicular, 155, 265 Loop, Meyer’s, 563 Lordosis, 4, 5 Lower limb. See Limb (lower) Lumbar puncture, 41 Lumen alveolar, 121 capillary, 121 Lunate, 342, 342t, 343, 344, 345, 347, 350, 404 radiograph of, 343 surface of, 417 Lung(s) arteries of, 124–125, 125t, 126 borders, 55, 117 boundaries and reference lines, 111, 112, 112t computed tomography (CT) of, 136 fissures of horizontal, 117 oblique, 116, 117, 119 in situ, 116 innervation of, 694 left, 55, 78, 79, 116, 118, 124, 128, 129, 131 bronchopulmonary segmentation, 118, 118t, 119 computed tomography (CT) of, 136, 222 in left pulmonary cavity, 78 MRI of, 133 lobes of. See Lobes, of lung resection of, 119 respiratory mechanics of, 126 right, 55, 71, 78, 93, 112, 113, 114, 116, 118, 124, 128, 129, 130, 131, 182 MRI of, 131 in right pulmonary cavity, 78 segmentation of, 118, 118t, 119 segmentation of bronchopulmonary, 118–119, 118t, 119 neurovasculature of, 126–127 veins of, 125t 124–125, 126 Lunula of radius, 331 of semilunar valve, 99 Lymphatic drainage of abdomen, 202–209, 202t of breast, 76, 76t of digestive organs, 203 Index 730 Lymphatic drainage (continued) Lymphatic drainage (continued) of duodenum, 207 of esophagus, 110, 111 of genitalia in female, 277 in male, 277 of gonads, 205 of hand, 367 of heart, 110 of ileum, 203, 208 of intestines, 208, 209 of kidneys, 205 of limb (lower), 475 of liver, 203, 206, 207 of neck, 522 of oral floor, 522 of pancreas, 207 of pelvic organs, 279f of pelvis, 204–205 of pleural cavity, 128–129 of rectum, 209, 276 of spleen, 203, 207 of stomach, 203, 204, 206, 207 of thoracic cavity, 84–85, 110 of tongue, 522 of ureters, 205 of urethra, 276 of urinary bladder, 276 Lymph nodes (individual) aortic, lateral, 203, 205, 208, 277 left, 202t, 276t right, 202t, 203, 208, 276t appendicular, 203, 208 axillary, 202, 367 apical, 76 central, 76 humeral, 76 interpectoral, 76 levels of, 76t, 367 pectoral, 76 subscapular, 76 brachial, 76 brachiocephalic, 85, 87, 88, 90, 110 brochopulmonary, 85, 110 111, 128, 129 cancer metastases to, 195, 522 caval, lateral, 279f celiac, 110, 202, 203, 204, 206, 207, 208, 221 cervical, 76 deep, 129, 523, 523t inferior, 522 superior, 522 superficial, 523, 523t, 538 colic intermediate, 209 left, 208, 209 middle, 203, 208, 209 right, 203, 208, 209 cubital, 76 cystic, 203, 207 epicolic, 209 foraminal, 207 gastric left, 203, 206, 207 right, 203, 207 gastroomental left, 206, 207 right, 203, 206, 207 gluteal inferior, 205 superior, 205 hepatic, 203, 206, 207 ileocolic, 203, 208, 209 iliac common, 202, 202t, 204, 205, 276t, 278, 279, 475 external, 204, 205, 276, 276t, 277, 279, 279f, 475, 488 internal, 204, 276, 276t, 277, 279, 475 inguinal deep, 204, 205, 276, 276t, 277, 278, 279, 279f, 475 inferior, 488 superficial, 202, 205, 276, 276t, 277, 279f, 475, 488 horizontal group, 204, 278, 279 horizontal group of, 277 vertical group, 204, 279 vertical group of, 277 superolateral, 488 superomedial, 488 intercostal, 84, 110, 128 interiliac, 205, 279f intrapulmonary, 85, 128, 129 juxta-intestinal, 203, 208 lacunar intermediate, 204, 205, 279f lateral, 205, 279f medial, 205, 279 lateral jugular, 523 lumbar, 202t, 277, 475 intermediate, 205, 220, 278, 279, 279f lateral, 220 left, 208, 278 preaortic. See Lymph nodes, preaortic retroaortic. See Lymph nodes, retroaortic right, 278 mastoid, 523, 523t mesenteric inferior, 202t, 204, 208, 209, 276, 278, 279 intermediate, 204, 208 superior, 202t, 203, 204, 207, 208, 209 mesocolic, 203, 208 obturator, 279, 279f occipital, 523, 523t pancreatic, 206 inferior, 203, 207 superior, 203, 207 pancreaticoduodenal, 203, 207 inferior, 207 superior, 207 paracolic, 209 paraesophageal, 86, 111 paramammary, 85 pararectal, 279f parasternal, 76, 85, 128, 202 paratracheal, 11, 85, 110, 128, 129 parauterine, 279f paravaginal, 279f paravertebral, 85 parietal, 202t parotid deep, 523, 523t superficial, 523, 523t anterior, 523, 523t lateral, 523, 523t pelvic, visceral, 279f phrenic inferior, 110, 111, 128, 205, 206 superior, 85, 87, 91, 95, 110, 206 popliteal deep, 475, 495 superficial, 475 preaortic, 202t, 203, 205, 208, 220 precaval, 205, 279f prececal, 203, 208, 209 prevertebral, 110 prevesical, 279f promontory, 203, 206, 207, 278, 279, 279f pyloric, 203, 206, 207 rectal superior, 203, 208, 209 retroaortic, 202t, 204 retroauricular, 523, 523t retrocaval, 204, 205, 279f retropyloric, 203, 207 retrovesical, 279f Rosenmϋller’s, 480, 489 sacral, 204, 205, 276t, 277, 278, 279 sigmoid, 203, 208, 209 splenic, 203, 206, 207 subaortic, 205, 279f submandibular, 522, 523 submental, 522 subpyloric, 203, 206, 207 superolateral, 475 superomedial, 475 suprapyloric, 203, 206, 207 supratrochlear, 76 thoracic, 85 Index 731 Mesosalpinx tracheobronchial, 85, 87, 88, 110, 111, 128 inferior, 110, 111, 128, 129 superior, 128, 129 vesical, lateral, 279 Lymph nodes (regional) of abdomen, 202–209, 202t of biliary tract, 207 of breast, 76, 76t of genitalia female, 279 male, 278 of gonads, 205 of ileum, 208 of infracolic organs, 208–209 of inguinal region, 488 of jejunum, 208 of kidneys, 205 of large intestine, 208, 209 of limb (lower), 475 superficial, 488 of limb (upper), 367 of liver, 206 of mediastinum, 85, 87t, 110–111 of pelvis, 276, 276t horizontal group of, 276t vertical group of, 276t of stomach, 206 of supracolic organs, 206–207 of thoracic cavity, 84–85 of thoracic wall, 85 of ureter, 205 M Macrophages, alveolar, 121 Macula lutea, 612, 613, 613 Malleolus, 496 lateral, 408, 410, 432, 433, 445, 446, 448, 449, 450, 451, 454, 456, 460, 467, 474, 484, 485 radiograph, 508 medial, 408, 410, 432, 433, 444, 446, 449, 450, 451, 454, 456, 457, 460, 461, 462, 463, 467, 494, 495, 497 radiograph, 508 Malleus, 626, 628, 629, 630 anterior process of, 630 in conductive hearing loss, 631 handle of, 630, 632 head of, 630 lateral process of, 630 ligaments of, 631 neck of, 630 Mammography, of breast, 77 Mandible, 3, 66, 533, 534, 536, 542, 637, 646, 647, 659, 666, 667 angle of, 653, 666 body of, 542, 543, 637, 657, 667 head of, 517, 638, 639 computed tomography (CT) of, 670 head (condyle) of, 637 inferior border, 512 MRI of, 668 neck of, 637, 638 computed tomography (CT) of, 670 origins and insertions of, 554 radiograph of, 668 ramus of, 598, 637 Manubrium, 56, 58, 63, 88, 92, 130, 147, 297, 298, 303, 308, 309, 513 Margins acetabular, 232 infraorbital, 512, 543 orbital, anterior, 656 supraorbital, 512, 543 Mastication, muscles of, 553, 554, 555, 558–559, 558t, 559t motor innervation of, 580 Maxilla, 21, 513, 542, 543, 546, 602, 622, 659, 660, 663, 666 frontal process of, 616, 617 infratemporal surface of, 598 MRI of, 669 orbital surface of, 603 palatine process of, 619, 636, 656 Meatus acoustic external, 542, 558, 598, 624, 638 computed tomography (CT) of, 670 internal, 547, 548, 568, 625, 634 nasal inferior, 617, 656 middle, 618, 620, 656 superior, 617, 618, 619 Mediastinum, 78t, 116 anterior, 78, 79, 85, 88, 115, 132 divisions and contents of, 78–79, 78t, 88t, 89 inferior, 78, 78t, 90 anterior, 88t middle, 88t posterior, 88t lymphatic drainage of, 85, 110–111 middle, 79, 88 posterior, 79, 88 structures of, 90–91 superior, 78, 78t, 85, 88, 88t, 90 of testes, MRI of, 290 Medulla, renal, 184 computed tomography (CT) of, 222, 223 Medulla oblongata, 561t, 564, 570, 574, 674, 677, 682, 683, 685, 698 MRI of, 669, 700, 701 Membrane(s) atlantooccipital anterior, 19, 20, 21 posterior, 18, 20, 21, 27 basilar, in conductive hearing loss, 631 fibrous, 413, 417, 435, 442, 473 intercostal, external, 63 interosseous, 325, 337, 339, 346, 365, 389, 427, 431, 432, 436, 440, 445, 447, 448, 451, 460, 472, 473, 496, 500, 501 of forearm, 397 MRI of, 399, 503 obturator, 236, 268, 287, 417 perineal, 239, 240, 244, 245, 246, 252, 261, 263, 264, 285 petrotympanic, 568 quadrangular, 529 stapedial, 631 synovial, 304, 417, 435, 439, 473 intima, 442 reflection of, 417 subintima, 442 tectorial, 18, 19, 20, 21 thyrohyoid, 527, 531, 536 tympanic, 568, 625, 626, 628, 629, 630, 631, 632 in conductive hearig loss, 631 Meninges, layers of, 590 Meningocele, 40 Menisci, of knee, 439 injury to, 439 lateral, 438, 439, 440, 441, 442 anterior horn, MRI of, 507 intermediate portion, 507 posterior horn, MRI of, 507 medial, 438, 439, 441, 442 intermediate portion, 507 movement of, 439 Meralgia paresthetica, 478 Mesencephalon, 561t, 563, 565, 674, 677, 679, 680, 683, 699 cerebral peduncle of, 564 development of, 676t Mesentery, 156, 172 attachment to duodenal wall, 159, 164 of large intestine, 172 root of, 161, 165, 169, 181 of small intestine, 158, 159, 160, 171 Meshwork, trabecular, 614 Mesoappendix, 159, 165, 172 Mesocolon sigmoid, 161, 164, 165, 171, 172, 173, 242, 243, 245, 248 root of, 165 transverse, 156, 158, 159, 160, 162, 163, 164, 165, 170, 172, 173 root of, 164, 171, 181, 183, 191, 223 Mesometrium, 256, 257, 279 of broad ligament of uterus, 273 Mesosalpinx, 256, 257 Index 732 Mesotympanum Mesotympanum, 629 Mesovarium, 256, 257 Metacarpals, 295, 296, 342t, 352, 392, 404 1st, 297, 341, 342, 346, 394, 495 base of, 341 1st to 5th, 344, 345 2nd, 297, 339, 341, 363 base of, 337, 339 shaft of, 339 2nd through 5th, 363 3rd, 339, 352, 353, 359 base of, 339 transverse section of, 353 5th, 346, 361 base of, 337, 341 base of, 343, 347, 404 MRI of, 405 head of, 343, 347, 404 MRI of, 405 shaft of, 343, 347 MRI of, 405 Metastases, of tumors of breast cancer, 77 to lymph nodes, 209 venous drainage of, 195 Metatarsals, 410, 452 1st, 448, 450, 452, 453, 456, 458, 460, 461, 462 base, 452, 455 head, 452, 455, 462 shaft, 455 1st through 5th, 451, 455 2nd, 457, 499 base of, MRI of, 509 head, MRI of, 509 3rd, 499 base of, MRI of, 509 4th, base of, MRI of, 509 5th, 452, 453, 458, 461, 468 base, 455, 462, 508 head, 462 tuberosity of, 408, 410, 448, 451, 452, 453, 455, 456, 462, 464, 465, 467, 468, 471 5th, base of, MRI of, 509 Metatarsus, 452 Microdiscectomy, 15 Midbrain. See Mesencephalon Midfoot, 452 Midgut, nerves of, 215 Miosis, 615 Modiolus, 634, 635 Molars, 640, 656 impacted, 641 Mons pubis, 229, 262 Mortise, of ankle, 410, 432, 456, 457 ligaments of, 460t Motor cortex, supplementary, 693 Motor system, extrapyramidal, 693, 693t Mouth. See Cavity, oral Mucoperiosteum, of the hard palate, 663 Mucosa esophageal, 107 lingual, 646 olfactory, 562 oral, 642, 649 pharyngeal, 651 tracheal, 120 urethral, 252 Muscles (individual) abductor digiti minimi, 346, 354, 356, 357, 358, 360, 360t, 361, 390, 392, 393, 454, 464, 465, 466, 467, 469, 469t, 485, 498, 499 MRI of, 405, 509 abductor hallucis, 454, 456, 463, 464, 465, 466, 469, 469t, 472, 485, 498 MRI of, 509 abductor pollicis, 355 abductor pollicis brevis, 360, 360t, 361, 390, 393 abductor pollicis longus, 332, 334, 335, 340t, 356, 357, 358t, 374, 388, 389, 396 MRI of, 399 adductor brevis, 419, 420, 424, 428, 428t, 473, 480, 480t, 492, 500, 501 adductor hallucis, 463, 464, 466, 470t, 485, 495 MRI of, 509 oblique head, 465, 466, 471, 499 transverse head, 462, 465, 466, 471, 499 adductor longus, 418, 419, 420, 421, 424, 428, 428t, 473, 480, 480t, 492, 500, 501 MRI of, 502, 505 adductor magnus, 284, 285, 418, 419, 420, 422, 423, 424, 429, 429t, 472, 473, 474, 480, 480t, 490, 491, 492, 493, 500, 501 MRI of, 502, 505 tendinous insertion of, 420 tendinous part, 429, 429t adductor minimus, 420, 421 adductor pollicis, 354, 356, 357, 360, 360t, 392, 393 MRI of, 405 oblique head, 355, 356, 393 transverse head, 355, 356, 393 adductor pollicis brevis, 354, 355, 392 adductor pollicis longus, 355 articularis genus, 419, 420 arytenoid oblique, 653, 655 transverse, 528t, 653, 655 auricular anterior, 553, 556t posterior, 553, 556t superior, 553, 556t biceps brachii, 76, 294, 308, 322, 322t, 332, 333, 375, 382, 385, 386, 387, 388, 389, 396, 403 long head of, 306, 307, 308, 309, 310, 311, 322, 322t, 375, 384t, 396, 397 MRI of, 398, 401 radial tuberosity of, 311 short head of, 306, 307, 308, 309, 310, 311, 322, 322t, 375, 384t, 396, 397 MRI of, 398, 401 tendon of, 310 biceps femoris, 409, 419, 420, 431, 431t, 444, 446, 447, 494 long head of, 422, 423, 424, 425, 431, 431t, 445, 484, 485, 491, 493, 495, 496, 500, 501, 502 MRI of, 505, 507 short head of, 423, 424, 425, 431, 431t, 445, 484, 485, 493, 495, 496, 500, 501 MRI of, 502 brachialis, 309, 310, 311, 315, 322, 327, 332, 333, 386, 387, 388, 389, 397 MRI of, 398, 399, 403 brachioradialis, 14, 294, 311, 327, 333, 334, 335, 338t, 339, 373, 387, 388, 389, 393, 396, 397 MRI of, 398, 399, 403 buccinator, 552, 554, 557, 557t, 580, 595, 649, 652, 656, 657, 663, 666 bulbospongiosus, 156, 238, 240, 241t, 242, 246, 253, 262, 263, 264, 267, 284, 285 MRI of, 291 ciliary, 612, 614, 615 coccygeus, 238, 240, 240t, 268, 269 compressor urethrae, 240, 241, 252 coracobrachialis, 307, 308, 309, 310, 311, 318, 318t, 375, 383, 384t, 386, 396 MRI of, 398 corrugator cutis ani, 249 corrugator supercilii, 552, 554 cremaster, 144, 150, 151, 154, 155, 155t, 265 cricoarytenoid lateral, 528, 528t, 531 posterior, 528, 528t, 531, 655 cricothyroid, 528t, 530, 531, 534, 535, 536, 575, 652 oblique part, 528 straight part, 528 dartos, 153, 154, 155t deltoid, 2, 24, 54, 72, 294, 307, 308, 309, 310, 311, 312, 314, 315, 316–317, 380, 381, 382, 383, 384, 385, 386, 396 acromial part, 309, 314, 315, 316, 316t MRI of, 401 Index 733 Muscles (individual) clavicular part, 309, 314, 315, 316, 316t MRI of, 401 MRI of, 398, 401 scapular part MRI of, 401 spinal part, 314, 315, 316, 316t ultrasound of, 400 depressor anguli oris, 516, 552, 553, 554, 557, 557t depressor labii inferioris, 552, 553, 554, 557, 557t depressor septi nasi, 554 detrusor, 252, 253 digastric anterior belly of, 66, 516t, 517, 532, 536, 554, 642, 643t, 645, 646, 652, 656, 657, 661 posterior belly of, 516t, 517, 532, 537, 555, 642, 643t, 645, 651, 652, 653 nerve to, 596 digitorum, 334 dilator urethrae, 253 erector spinae, MRI of, 48 extensor carpi radialis, 388, 389 extensor carpi radialis brevis, 311, 312, 314, 315, 334, 335, 338t, 339, 357, 388, 389, 390, 397 MRI of, 399 extensor carpi radialis longus, 294, 311, 312, 314, 315, 334, 335, 338t, 339, 357, 388, 389, 390, 397, 403 MRI of, 399 extensor carpi ulnaris, 294, 312, 314, 334, 340t, 341, 357, 358, 358t, 388, 389, 397 common head of, 341 MRI of, 399 extensor digiti minimi, 334, 340t, 357, 358t, 397 common head of, 341 extensor digitorum, 294, 312, 314, 340t, 357, 358, 358t, 374, 389, 397, 403, 456 common head of, 341 dorsal venous network of, 294 MRI of, 399 extensor digitorum brevis, 444, 445, 467, 468, 468t extensor digitorum longus, 444, 445, 449, 449t, 467, 484, 496, 496t, 499, 501 MRI of, 503 extensor digitorum superficialis, MRI of, 399 extensor hallucis, 456 extensor hallucis brevis, 444, 445, 467, 468, 468t, 497, 499 MRI of, 503 extensor hallucis longus, 409, 444, 445, 449, 449t, 467, 484, 496, 496t, 497, 499, 501 MRI of, 503 extensor indicis, 334, 335, 340t, 341, 357, 358t, 389 MRI of, 399 extensor pollicis brevis, 334, 335, 341, 356, 357, 358, 358t, 374, 389, 397 superficial head of, 390 extensor pollicis longus, 335, 340t, 341, 358t, 374, 388, 389, 397 MRI of, 399 fibularis brevis, 444, 445, 446, 447, 448, 448t, 456, 466, 467, 484, 494, 496, 496t, 501 fibularis longus, 409, 425, 444, 445, 446, 447, 448, 448t, 456, 462, 464, 465, 467, 484, 494, 496, 496t, 497, 499, 501 MRI of, 503 fibularis tertius, 445, 449, 449t, 467, 496, 496t flexor carpi radialis, 294, 332, 336t, 337, 354, 357, 376, 387, 388, 390, 393, 396, 397, 403 MRI of, 399 flexor carpi ulnaris, 294, 312, 314, 332, 333, 334, 335, 336t, 337, 354, 357, 377, 387, 390, 393, 396, 397 MRI of, 399 flexor digiti minimi brevis, 354, 356, 357, 360, 360t, 390, 392, 393, 464, 465, 466, 470t, 471, 499 MRI of, 405 flexor digitorum brevis, 456, 463, 465, 466, 469, 469t, 471, 485, 498 MRI of, 509 flexor digitorum longus, 423, 424, 446, 447, 451, 451t, 456, 463, 464, 465, 466, 471, 494, 495, 496, 496t MRI of, 503 flexor digitorum profundus, 311, 314, 332, 335, 336t, 337, 376, 377, 397, 403 MRI of, 399, 403 flexor digitorum superficialis, 332, 336t, 388, 390, 393, 396, 397, 403 humeral-ulnar head of, 336, 337 humeroradial head of, 389 MRI of, 399 radial head of, 333, 336, 337, 389 ulnar head of, 333 flexor hallucis brevis, 463, 464, 465, 466, 470t, 499 of, 465 lateral head of, 465 flexor hallucis longus, 446, 447, 451, 451t, 456, 463, 464, 465, 466, 494, 495, 499, 501 flexor pollicis brevis, 354, 357, 360, 360t, 361 deep head of, 356 MRI of, 405 superficial head of, 392, 393 flexor pollicis longus, 332, 333, 336t, 355, 357, 376, 388, 389, 390, 393, 396, 397 MRI of, 399 superficial head of, 356 gastrocnemius, 409, 421, 425, 435, 450t, 485, 493, 495, 497, 500 lateral head of, 422, 423, 445, 446, 447, 450, 450t, 494, 495 MRI of, 503, 507 medial head of, 422, 423, 424, 444, 446, 447, 450, 450t, 494, 495, 501 MRI of, 503, 507 gemellus, 426t inferior, 422, 423, 424, 427, 483, 490, 491, 500 superior, 422, 423, 424, 427, 483, 490, 491, 500 genioglossus, 555, 577, 642, 645, 646, 647, 648, 649, 656, 657, 661, 666 geniohyoid, 64t, 516t, 517, 525, 554, 555, 642, 645, 646, 648, 649, 656, 657, 659 gluteus maximus, 2, 24, 28, 238, 284, 285, 286, 287, 409, 419, 420, 421, 422, 423, 424, 425, 426t, 427, 490, 493, 500 MRI of, 51, 288, 502, 505 gluteus medius, 2, 409, 419, 422, 423, 424, 425, 426t, 427, 483, 490, 493, 500 computed tomography (CT) of, 223, 224 MRI of, 51, 505 gluteus minimus, 247, 420, 423, 424, 426t, 427, 491, 493, 500 gracilis, 284, 285, 418, 419, 420, 421, 422, 428, 428t, 444, 446, 480, 480t, 490, 491, 492, 493, 494, 495, 500, 501 MRI of, 502, 503, 505 hyoglossus, 642, 643t, 646, 649, 652, 657 hypothenar, 388, 391 iliacus, 146, 147, 148t, 152, 211, 247, 250, 261, 271, 418, 419, 420, 421, 426t, 427, 479, 481, 489 computed tomography (CT) of, 223, 224 MRI of, 51, 289, 505 iliococcygeus, 238, 240, 240t iliocostalis, 28, 32, 32t, 515 iliocostalis cervicis, 32t, 33, 515 iliocostalis lumborum, 29, 32t, 33 iliocostalis thoracis, 29, 32t, 33 iliopsoas, 147, 149, 150, 152, 286, 418, 419, 420, 424, 426, 426t, 427, 481, 489, 492 MRI of, 288, 505 infrahyoid, 25, 516, 516t, 517, 525, 642 infraspinatus, 306, 313, 314, 315, 317, 317t, 370, 380, 381, 383 MRI of, 398, 401 intercostal, 62, 62t, 65, 66, 75, 91, 190 external, 28, 29, 62, 62t, 63, 71, 72, 73, 113, 144 Index 734 Muscles (individual) (continued) Muscles (individual) (continued) innermost, 62, 62t, 63, 71, 73, 86, 113 internal, 62, 62t, 63, 71, 73, 86, 113, 144, 147 MRI of, 401 interossei, 357, 377, 444, 454, 499 dorsal, 357, 362, 362t, 363, 395, 463, 465, 470t, 471, 497 1st, 35, 358, 393, 394, 466, 471 1st through 4th, 356, 466 2nd, 358, 359, 363, 466 3rd, 358, 359, 362, 362t, 363 4th, 358, 363, 464, 465, 466 MRI of, 405 MRI of, 405, 509 palmar, 357, 362, 362t 1st, 356, 363 2nd, 356, 363 3rd, 356 MRI of, 405 plantar, 463, 464, 465, 470t, 499 1st, 466 1st through 3rd, 466 2nd, 466 3rd, 464, 465, 466, 471 interspinales, 27, 29, 34t, 35 intertransversarius, cervical posterior, 541 intertransversii anteriores cervicis, 27 intertransversii cervicis, 34t intertransversii laterales lumborum, 29, 34t, 35 intertransversii mediales lumborum, 35 intertransversii posteriores cervicis, 34t, 35 ischiocavernosus, 238, 240, 241t, 242, 243, 246, 261, 263, 264, 284, 285 with crus of penis, 247 MRI of, 291 of larynx, 528–529, 528t latissimus dorsi, 2, 24, 25 47, 65, 73, 294, 307, 308, 309, 310, 311, 312, 313, 321, 321t, 371, 380, 382, 385, 386 aponeurotic origin of, 24, 28 costal part of, 321, 321t iliac part of, 321, 321t scapular part of, 321, 321t vertebral part of, 321, 321t levator anguli oris, 552, 557, 557t, 663 levator ani, 238, 240t, 242, 243, 244, 245, 247, 248, 252, 263, 274, 283, 284, 287 arch of, 238 with fascia of pelvic diaphragm, 261 gender-related structural differences in, 238 MRI of, 288, 289, 505 tendinous arch of, 251, 258, 259 levator costarum, 29 levator costarum breves, 29, 34t, 35 levator costarum longii, 29, 34t, 35 levator labii superioris, 553, 554, 557, 557t, 656 MRI of, 669 levator labii superioris alaeque nasi, 552, 553, 554, 556, 556t levator palpebrae superioris, 565, 604, 608, 609, 610, 656, 666 levator scapulae, 24, 25, 313, 314, 315, 320, 320t, 370, 516, 539, 664, 665 levator veli palatini, 555, 628, 643t, 652, 653, 661, 663 MRI of, 669 lingual, 555 longissimus, 28, 32, 515 longissimus capitis, 26, 27, 29, 32t, 33, 46, 515t, 541, 554, 555, 664 longissimus cervicis, 32t, 33, 664 longissimus thoracic, 29, 32t, 33 longitudinal inferior, 646 superior, 646 longus capitis, 31, 31t, 518, 519, 555, 660 MRI of, 669 longus colli, 25, 518, 664, 665 inferior oblique part, 519 oblique part of, 31, 31t superior oblique part of, 31, 31t vertical oblique part of, 31, 31t vertical part of, 519 lumbricals, 356, 362, 362t, 393, 463, 464, 465, 470t, 485, 499 1st, 363, 376 1st through 4th, 466, 471 2nd, 359, 363, 376 3rd, 363 4th, 363 MRI of, 405 masseter, 538, 552, 554, 555, 558, 558t, 594, 595, 596, 597, 598, 599, 649, 653, 657, 658, 663 computed tomography (CT) of, 670 deep layer of, 559 layers of, 553 MRI of, 669 superficial layer of, 559 mentalis, 552, 553, 554, 557, 557t multifidus, 29, 34, 34t, 35 MRI of, 48, 669 muscularis externa circular layer of, 248 longitudinal layer of, 248 musculus uvulae, 643, 643t, 653, 655 mylohyoid, 516t, 517, 536, 554, 555, 642, 643t, 645, 646, 647, 649, 652, 656, 657, 659, 660, 661 nasalis, 552, 553, 556, 556t alar part of, 554 transverse part of, 554 oblique arytenoid, 528 external, 2, 24, 28, 47, 65, 67, 72, 73, 144, 145, 146, 148, 148t, 149, 150, 164, 165, 167, 170, 174, 185, 308, 312, 313, 479, 488, 489 aponeurosis of, 144, 146, 150, 151, 153 attachment site of, 143 computed tomography (CT) of, 223 MRI of, 51 inferior, 565, 604, 605, 605t, 610, 656, 666 internal, 28, 29, 67, 71, 72, 145, 146, 148, 148t, 149, 150, 153, 165, 167, 170, 174, 185, 313, 479, 488 attachment site of, 143, 146 computed tomography (CT) of, 223 MRI of, 51 superior, 565, 604, 605, 605t, 608, 609, 657, 666 obliquus capitis inferior, 26, 27, 29, 30, 30t, 46, 518t, 519, 541, 660, 661 obliquus capitis superior, 26, 27, 29, 30, 30t, 46, 518t, 519, 541, 554, 555 obturator externus, 247, 287, 419, 420, 424, 429, 429t, 451, 540 MRI of, 288, 291, 505 obturator internus, 147, 238, 240, 240t, 244, 245, 246, 247, 248, 258, 261, 268, 286, 287, 421, 422, 423, 424, 426t, 427, 483, 490, 491, 493, 500 MRI of, 288, 289, 291, 505 occipitofrontalis frontal belly of, 553, 556t occipital belly of, 553, 554 omohyoid, 380, 516, 525, 533, 555, 664, 665 inferior belly of, 76, 382, 383, 512, 516, 517, 532, 537, 539 superior belly of, 516, 517, 532, 534, 537 opponens digiti minimi, 355, 356, 357, 360, 360t, 361, 393, 465, 466, 470t, 471, 499 MRI of, 405 opponens pollicis, 346, 354, 356, 360, 360t, 361, 390, 393 orbicularis oculi, 552, 553, 556, 556t, 656 lacrimal part of, 554 orbital part of, 554, 610 palpebral part of, 610 orbicularis oris, 552, 553, 554, 557, 557t, 661 MRI of, 669 palatoglossus, 643t, 646, 648 palatopharyngeus, 643t, 648, 653, 655, 660, 661 palmaris brevis, 354, 360, 360t, 390, 391, 396 palmaris longus, 332, 336t, 337, 376, 387, 388, 391, 393, 396, 397 papillary anterior, 97, 99, 102 Index 735 Muscles (individual) posterior, 97, 99 septal, 97 pectinate, 97 pectineus, 150, 151, 286, 419, 420, 424, 428, 428t, 488, 492 MRI of, 288, 502, 505 pectoralis major, 54, 72, 75, 76, 294, 309, 310, 311, 318, 318t, 374, 382, 383, 384, 384t, 385, 386, 396 abdominal part of, 308, 318, 318t clavicular part of, 308, 309, 318, 318t, 382 computed tomography (CT) of, 75 MRI of, 398, 401 sternocostal part of, 308, 309, 318, 318t, 382 pectoralis minor, 309, 310, 311, 319, 319t, 367, 374, 382, 383, 384, 385, 386 MRI of, 401 perineal deep transverse, 156, 238, 240, 240t, 244, 247, 255, 260, 261, 264, 266, 267 superficial transverse, 238, 240, 241t, 263, 275, 284, 285 pharyngeal constrictors inferior, 89, 531, 652, 652t, 654 middle, 555, 652, 652t, 653 posterior view of, 653 superior, 652, 652t, 653, 654 pharyngeal elevators, 653 piriformis, 147, 238, 240, 240t, 258, 269, 419, 420, 421, 423, 424, 426t, 472, 474, 483, 490, 491, 491t, 493, 500 plantaris, 422, 423, 424, 446, 447, 450, 450t, 493, 494, 495, 496, 496t, 500 platysma, 515t, 516, 516t, 530, 534, 552, 553, 557, 557t popliteus, 423, 424, 431, 437, 446, 447, 451, 451t, 494, 495 MRI of, 503 prevertebral, 31, 31t, 518, 518t, 555 procerus, 552, 556t, 661 pronator quadratus, 332, 333, 336t, 337, 354, 376, 388, 389, 390, 393 MRI of, 399 pronator teres, 310, 311, 332, 333, 335, 336t, 337, 388, 389, 396, 397, 403 humeral head of, 333, 376, 387, 388 MRI of, 399, 403 ulnar head of, 376, 387, 403 psoas major, 25, 64, 65, 67, 143, 147, 148, 149, 152, 211, 250, 259, 271, 418, 419, 420, 426t, 427, 479, 481, 489 computed tomography (CT) of, 222, 223, 224 MRI of, 51, 289, 505 psoas minor, 147, 211, 421, 427 attachment site of, 143 pterygoid, 558 lateral, 553, 554, 555, 559, 559t, 580, 599, 658, 661, 663 computed tomography (CT) of, 670 inferior head of, 559, 559t, 598, 670 lateral head of, 639 MRI of, 669 superior head of, 559, 559t, 598, 639, 670 medial, 553, 554, 555, 559, 559t, 580, 644, 647, 648, 651, 657, 658, 661, 663 computed tomography (CT) of, 670 deep head, 599 deep head of, 559, 559t MRI of, 669 superficial head of, 559, 559t medial, superior head of, 598 pubococcygeus, 238, 240t, 248 puborectalis, 238, 240t, 248, 274 pubovesical, 251 pupillary dilator, 614 pyramidalis, 147, 148t, 149 quadratus femoris, 247, 420, 422, 423, 424, 426t, 427, 483, 490, 491, 492, 493, 500, 501 quadratus lumborum, 25, 29, 64, 65, 67, 143, 147, 148t, 149, 211, 479 computed tomography (CT) of, 223 quadratus plantae, 456, 463, 465, 466, 470t, 471, 498, 499 MRI of, 509 quadriceps femoris, 228, 419, 430, 430t, 443 rectus inferior, 565, 604, 605, 605t, 610, 656, 666 lateral, 565, 604, 608, 609, 612, 657, 661, 662, 666 medial, 565, 604, 605, 605t, 608, 609, 612, 662, 666 superior, 565, 604, 605, 605t, 608, 609, 610, 657, 661, 666 rectus abdominalis, 65, 67, 72, 144, 145, 146, 148t, 149, 150, 151, 152, 153, 156, 160, 164, 170, 173, 242, 245, 286, 479, 488 lateral head of, 143 medial head of, 143 MRI of, 290 rectus capitis anterior, 31, 31t, 518, 518t, 519, 555 rectus capitis lateralis, 31, 31t, 518t, 519, 555 rectus capitis posterior major, 26, 27, 29, 30, 30t, 46, 518t, 519, 541, 554, 555, 661 rectus capitis posterior minor, 26, 27, 29, 30, 30t, 46, 518t, 519, 541, 554, 555, 660, 667 rectus femoris, 409, 418, 419, 420, 421, 425, 430, 445, 481, 492, 500, 501 MRI of, 288, 502 rhomboid major, 24, 28, 47, 313, 314, 315, 320, 320t, 370, 383 rhomboid minor, 24, 28, 313, 314, 315, 320, 320t, 370, 516 risorius, 552, 553, 557, 557t rotatores, 34, 35 rotatores breves, 34t, 35 rotatores longi, 34t, 35 rotatores thoracis longi, 29 salpingopharyngeus, 628, 653, 655 sartorius, 228, 409, 418, 419, 420, 421, 425, 430, 430t, 444, 481, 492, 500, 501 MRI of, 288, 502, 503, 505 scalene, 25, 59, 62, 62t, 66, 78, 80, 82, 89, 103, 106, 109, 383, 518 anterior, 365, 369, 372, 373, 375, 518, 519, 525, 535, 536, 537, 539, 665 middle, 62, 62t, 63, 80, 369, 372, 382, 518, 519, 525, 536, 537, 539, 665 MRI of, 701 MRI of, 669 posterior, 62, 62t, 63, 382, 518, 519, 539, 665 semimembranosus, 409, 419, 420, 421, 422, 423, 424, 431, 431t, 446, 485, 491, 493, 494, 495, 500, 501 MRI of, 502 semispinalis, 26, 34, 34t, 35, 515, 515t semispinalis capitis, 26, 27, 28, 34t, 35, 46, 312, 313, 540, 541, 554, 555, 660, 663, 667 semispinalis cervicis, 515, 664, 665 MRI of, 669 semispinalis thoracis, 34t, 35 semitendinosus, 409, 418, 419, 420, 421, 422, 423, 424, 431, 431t, 485, 491, 493, 494, 495, 500, 501 MRI of, 502, 503 serratus anterior, 24, 54, 73, 144, 302, 309, 310, 311, 313, 319, 319t, 370, 383, 384t, 385, 386 inferior part of, 319t intermediate part of, 319t MRI of, 401 superior part of, 319t tuberosity for, 59 serratus posterior, 28, 32, 32t, 33, 313 inferior, 24, 25, 28, 32t, 47 superior, 28, 32t, 33 superior posterior, 302 soleus, 15, 423, 424, 444, 445, 446, 447, 450, 485, 494, 495, 496, 497, 500, 501 innervation of, 15 MRI of, 503 tendinous arch, 494 spinalis, 28, 32, 32t spinalis cervicis, 29, 32, 32t MRI of, 669 Index 736 Muscles (individual) (continued) Muscles (individual) (continued) spinalis thoracis, 29, 32, 32t splenius, 32, 32t, 515 splenius capitis, 26, 27, 28, 29, 32t, 33, 47, 312, 313, 539, 540, 541, 554, 555, 660, 661, 663, 664 splenius cervicis, 26, 28, 29, 33, 313, 664, 665 stapedius, 569t, 629, 630, 631, 632, 633 sternocleidomastoid, 24, 25, 26, 27, 46, 54, 308, 309, 312, 313, 382, 515, 516, 516t, 530, 532, 533, 534, 536, 537, 539, 540, 541, 554, 555, 576, 595, 598, 649, 655, 664, 665 borders of, 538 clavicular head, 512, 516, 516t MRI of, 669, 701 sternal head, 512, 516, 516t, 534 sternohyoid, 516, 517, 525, 532, 533, 536, 537, 555, 665 medial MRI of, 669 sternothyroid, 516, 525, 534, 537, 665 medial MRI of, 669 styloglossus, 555, 577, 646, 647, 652 stylohyoid, 516t, 517, 555, 642, 643t, 649, 651, 652, 653 nerve to, 596 stylopharyngeus, 572, 652, 653 subclavius, 309, 310, 311, 319, 319t, 382, 383, 384 groove for, 298 MRI of, 401 nerve to, 368t, 369, 370, 370t subcostal, 62t, 63 subscapularis, 302, 306, 309, 310, 311, 317, 317t, 384t, 385, 386 MRI of, 398, 401 subtendinous bursa of, 307 supinator, 311, 314, 327, 332, 333, 334, 335, 340t, 341, 373, 387, 389, 403 humeral head of, 335 suprahyoid, 516, 516t, 517, 554, 642, 643 supraspinatus, 24, 305, 306, 309, 310, 311, 313, 315, 317, 317t, 370, 380, 381, 516 MRI of, 401 tarsal inferior, 610 superior, 608, 610 temporalis, 553, 554, 555, 558, 558t, 559, 597, 598, 599, 657, 658, 662 computed tomography (CT) of, 670 MRI of, 669 temporoparietalis, 553 tensor fasciae latae, 409, 418, 419, 422, 423, 424, 425, 426t, 483, 491, 492, 500 MRI of, 502, 505 tensor tympani, 626, 629, 631, 633 nerve of, 600 semicanal of, 571, 629 tensor veli palatini, 555, 628, 643t, 652, 653, 663 MRI of, 669 nerve of, 600 teres major, 2, 24, 294, 307, 308, 310, 311, 312, 313, 314, 315, 321, 321t, 371, 380, 381, 385, 386, 396 MRI of, 401 teres minor, 2, 306, 313, 314, 315, 317, 317t, 372, 380, 381 MRI of, 398 thenar, 388, 396 thyroarytenoid, 528, 528t, 529, 531 thyroepiglottic part, 528 thyrohyoid, 516, 517, 525, 531, 534, 536, 537, 555, 577, 652, 664, 665 tibialis anterior, 409, 421, 425, 444, 445, 447, 449, 449t, 456, 467, 484, 496, 496t, 499, 501 MRI of, 503 MRI of, 503 posterior, 423, 424, 446, 447, 451, 451t, 456, 462, 464, 465, 494, 495, 496, 496t, 501 MRI of, 503 transverse, 646 transversospinalis, 34, 34t, 35 transversus abdominis, 29, 64, 65, 67, 145, 148, 150, 151, 152, 153, 164, 165, 167, 170, 174, 185, 211, 479, 488 aponeurosis of, 145, 146, 223 attachment site of, 143 MRI of, 51 trapezius, 2, 24, 25, 26, 28, 47, 307, 308, 309, 310, 311, 313, 314, 315, 320, 320t, 380, 382, 512, 515, 516, 532, 533, 535, 536, 538, 539, 540, 554, 555, 576, 661, 663, 664, 665 ascending part of, 312, 320t, 380 borders of, 538 descending part of, 312, 320, 320t, 380, 516 MRI of, 669 transverse part of, 312, 320t, 380, 516 triceps brachii, 2, 24, 323, 323t, 327, 332, 334, 335, 373, 387, 388 lateral head of, 294, 312, 314, 315, 323, 323t, 381, 389, 397 MRI of, 398 long head of, 294, 305, 312, 314, 323, 323t, 381, 385, 386, 397 MRI of, 398 long head tendon of origin of, 323 medial head of, 314, 315, 323, 323t, 385, 386, 397 MRI of, 398 MRI of, 398, 403 tendon of insertion of, 323 triceps surae, 445, 447, 450, 450t, 467, 495, 496, 496t, 500 vastus intermedius, 419, 420, 423, 424, 430, 501 MRI of, 502 vastus lateralis, 409, 418, 419, 420, 423, 425, 430, 436, 444, 445, 492, 500, 501 MRI of, 502, 505, 507 vastus medialis, 409, 418, 419, 420, 421, 424, 430, 436, 444, 481, 492, 500, 501 MRI of, 505 vertical, 646 vocalis, 528, 528t, 529 zygomaticus major, 552, 553, 554, 557, 557t, 594 zygomaticus minor, 552, 553, 554, 557, 557t Muscles (regional) of abdominal wall anterior, 148, 148t anterolateral, 144–145, 148t deep posterior, 144, 148t lateral, 25 of arm anterior, 308–311 posterior, 312–315 of back, 2 extrinsic, 28 superficial, 24 intrinsic, 24, 25, 26–36, 65, 313, 554, 555 deep, 28, 29, 32, 34t, 35, 518–519, 518t deep segmental, 34, 34t deep transversospinalis, 34–35, 34t intermediate, 28, 32–33, 32t superficial, 24, 25, 26–35, 65 in nuchal region, 26–27, 313, 554, 555 paravertebral, 31, 31t short nuchal, 27, 30, 30t thoracolumbar fascia and, 24 of bronchial tree, 121 of esophagus, 107 of face, 552–553 of facial expression, 580 of foot, sole of, 464–465, 466 of forearm anterior, 388 anterior compartment, 332–333, 336–337, 336t Index 737 Nerves (named) deep, 336t extensors, common head of, 311, 314, 315 flexors, 327 common head of, 332, 333 superficial, common head of, 314, 315 innervation of, 336t intermediate, 336t posterior, 389 posterior compartment, 334–335 deep, 340t, 341 radialis muscles, 338–339, 338t superficial, 340t, 341 superficial, 336t radialis group, 373 of gluteal region, 409 deep, 427 lateral, 425 posterior, 424 weakness of, 483 of hand insertions of, 357 intrinsic deep layer of, 356–357 hypothenar group, 360t, 361 metacarpal group, 362–363, 362t middle layer of, 354–355, 356–357 superficial layer of, 354–355 thenar group, 360, 360t, 361 of hip, 419, 426–427, 426–427 426t anterior, 418, 420 medial, 421 posterior, 422, 424 of leg anterior compartment, 444, 449, 449t fibularis group, 495 lateral compartment, 444, 448, 448t posterior compartment, 446–447, 450–451, 450t, 451 of limb (lower), palpable, 409 of mastication, 553, 554, 555 motor innervation of, 580 of neck deep, craniovertebral joint, 30, 30t nuchal, 26, 515, 515t short nuchal, 27, 30, 30t superficial, 515, 515t, 516, 516t of oral cavity floor, 642 of orbit, 604–605 of pelvic floor, 238–241, 240t of perineum, 239, 240–241, 241t of pharynx, 652 of shoulder, 308–323 anterior muscles, 308–311 posterior muscles, 312–315 of rotator cuff, 317, 317t of skull, 552–553 origins and insertions of, 554–555 of soft palate, 643, 643t of suboccipital region, 27, 30, 518 of thigh anterior, 418, 420, 430, 430t medial, 421 deep layer of, 429, 429t superficial layer of, 428, 428t posterior, 422, 424, 431, 431t of thoracic walls, 60, 62–63, 62t of thorax, 54 of tongue, 646 Mydriasis, 615 Myelomeningocele, 40 Myocardial infarction, 81 types of, 101 Myocardium, 101, 103 Myometrium MRI of, 289 of the uterus, 256 Myopia, 615 N Nails, of fingers, 353 Naris, 616 Nasal bone, 513, 542, 543, 602, 616 MRI of, 668, 669 Nasion, 543, 616 Nasopharynx, 622t, 648, 651, 658, 659, 667 Navicular bone, 450, 452, 453, 455, 456, 457, 458, 460, 462, 463, 468 MRI of, 509 radiograph of, 508 Neck, 512–513. See also Larynx; Pharynx; Vertebrae/vertebra; Vertebral column arteries of, 520, 582–583 of humerus anatomical, 300, 301, 304, 305 surgical, 300, 301 lymphatics of, 522–523 MRI of, 669 muscles of, 515–516, 515t, 557t craniovertebral joint, 30, 30t facial expression, 557, 557t nuchal, 26, 27, 30, 30t, 515, 515t superficial, 515, 515t, 516, 516t nerves of, 38–39 of anterolateral region, 525 motor innervation, 525 parasympathetic innervation, 212 regions of, 512, 532, 532t anterior, 534–537 anterior cervical, 534–537 lateral cervical, 536–537, 538–539 occipital region, 540 posterior cervical, 540–541 surface anatomy of, 512–513 topography of, 532–533, 532–541 anterior cervical region, 534–535 veins of, 521 Nerves (named) abducent (CN VI), 549, 560, 564, 565, 593, 607, 608, 609, 655, 658, 683 effects of injury on, 564t nuclei of, 561, 561t, 564, 568 palsy of, 605 accessory (CN XI), 47, 380, 535, 536, 537, 538, 539, 540, 549, 554, 555, 560, 576, 654, 661, 663, 683 course of, 576t cranial root of, 576 fibers of, 576t function of, 576t lesions of, 576, 576t MRI of, 669 nuclei of, 561, 561t, 576t alveolar anterior, 644 anterior superior, alveolar branches, 567 branches of, 623 inferior, 580, 581, 598, 599, 600, 600t, 644, 645, 649, 656, 657, 658, 663 middle, 644 middle superior, 567 posterior, 644 posterior superior, 567, 623 superior, 598 ampullary anterior, 571, 634, 635 lateral, 571, 635 posterior, 571, 634, 635 anococcygeal, 284, 477 ansa cervicalis, 537 inferior root of (descendens cervicalis), 524, 525, 645 superior root of (descendens hypoglossus), 524, 525, 537, 645 antebrachial lateral, 375, 378, 387 medial, 368t, 378, 379, 386, 387 posterior, 373, 378 auricular great, 39, 46, 382, 524, 524t, 534, 538, 539, 540, 581, 595, 596 greater, 627 posterior, 568, 595 auriculotemporal, 567, 573, 580, 581, 594, 595, 596, 598, 599, 600, 600t, 627, 639, 644 MRI of, 669 Index 738 Nerves (named) (continued) Nerves (named) (continued) axillary, 39, 368, 368t, 369, 371, 371t, 372, 372t, 378, 379, 380, 381, 385 branches of, 372, 372t injuries to, 372 MRI of, 401 brachial inferior lateral, 373, 378, 380 medial, 368, 368t, 369, 374, 374t, 378, 379, 386 posterior, 373, 378, 380, 385, 386 superior lateral, 372, 378, 380 buccal, 567, 580, 581, 598, 599, 600, 600t, 644, 648 caroticotympanic, 573 cavernous, of penis, 280 cervical 1st, 591 2nd, 591 cardiac, 87, 103 transverse, 382, 524t, 525, 534, 538, 539, 581 ciliary long, 579, 607, 609 short, 565, 567, 607 clitoral, dorsal, 261, 285 clunial, 482, 487 inferior, 284, 285, 476, 482, 486, 493 middle, 39, 47, 284, 285, 482, 486, 493 superior, 284, 285, 482, 486, 493 coccygeal, 477 cochlear (CN VIII), 571, 628, 629, 635 cranial. See Cranial nerves cutaneous dorsal of foot, 494 intermediate, 484, 486, 496, 497, 499 lateral, 496, 499 medial, 496, 497 lateral of big toe, 484 of thigh, 476, 476t, 477, 482, 482t, 487, 492 of thigh, 476, 476t, 477, 487, 491, 493 digital dorsal, 373, 392, 394, 497 palmar, 392, 393 common, 376, 377, 379 dorsal branch, 392, 394 proper, 376, 377, 379 plantar common, 485, 498 proper, 485, 498 dorsal, of penis, 265, 275, 280, 284 ethmoidal anterior, 567, 609, 621 branches of, 591 posterior, 567, 609 facial (CN VII), 549, 554, 560, 568–569, 571, 597, 598, 599, 600, 625, 625, 627, 628, 629, 631, 632, 633, 634, 635, 645, 646, 655, 663, 682, 683, 688, 693 branches of, 534, 537, 538, 568, 569, 580, 594 course of, 569t effects of injury on, 569t in facial canal, 571, 629 functions of, 569t internal genu of, 568 nuclei of, 561, 561t, 569t parotid plexus of, 595 femoral, 150, 152, 210, 211, 286, 287, 476, 476t, 477, 478, 478t, 481, 481t, 486, 487, 489, 492 branches, 477, 481, 481t, 486 cutaneous distribution, 481 lateral, 210, 211 entrapment of, 478 MRI of, 502, 505 muscular compartment, 489 posterior, 284, 285 perineal branches of, 285 fibular common, 435, 476, 484, 487, 493, 494, 495, 496 MRI of, 502, 503 deep, 476, 484, 484t, 486, 496, 497, 499, 501 branches, 497 MRI of, 503 superficial, 484, 484t, 486, 496, 497 frontal, 567, 607, 609 genitofemoral, 185, 476, 476t, 477, 478, 478t, 479, 487, 489 branches of, 478, 478t, 488 femoral branch of, 210, 211 genital branch of, 150, 151, 211, 284, 285 glossopharyngeal (CN IX), 549, 555, 560, 572–573, 627, 645, 646, 654, 655, 661, 663, 683 branches of, 572t course of, 572, 573t effecs of injury on, 573t fibers of, 573t function of, 573t MRI of, 669 nuclei of, 561, 561t, 573t, 578 in tympanic cavity, 573 gluteal inferior, 476, 476t, 477, 482, 482t, 483, 490, 491 superior, 476, 476t, 477, 482, 482t, 483, 491, 493 hypogastric, 213, 281 left, 217, 218, 280, 281, 282 right, 217, 280, 281, 282 hypoglossal (CN XII), 524, 525, 534, 537, 549, 555, 560, 577, 597, 645, 647, 654, 655, 661, 663, 683, 693 course of, 577t effects of injury on, 577t fibers of, 577t function of, 577t lesions of, 577t nuclei of, 561, 561t, 577 iliohypogastric, 39, 70, 182, 183, 185, 476, 476t, 477, 478, 478t, 479, 482, 487 anterior cutaneous branch, 151, 210, 211 branches, 478, 478t, 486, 493 lateral cutaneous branch, 210, 211 ilioinguinal, 150, 151, 182, 183, 185, 210, 211, 275, 284, 285, 476, 476t, 477, 478, 478t, 479, 487, 488 infraorbital, 581, 594, 595, 596, 608, 610, 623, 644, 656, 657 MRI of, 666 infratrochlear, 567, 595, 596, 608 intercostal, 47, 66, 67, 70, 71, 72, 73, 74, 75, 86, 90, 91, 108, 210, 222, 378 branches of, 39, 70, 72, 73, 74, 210, 374, 379 course of, 71 intercostobrachial, 369, 374, 374t, 378, 379 interosseous anterior, MRI of, 399 posterior, 373, 389 of forearm, 397 labial anterior, 285 posterior, 285 lacrimal, 567, 609, 623 laryngeal external, 531, 535, 575 inferior, 535, 655 left, 575 right, 575 internal, 531, 534, 535, 575, 655 foramen for, 527 recurrent, 78, 87, 90, 108, 127, 531 left, 86, 88t, 89, 91, 94, 103, 108, 531, 535, 536 right, 86, 90, 103, 108, 531, 575t, 655 superior, 87, 103, 531, 537, 574, 575t, 654, 655 lingual, 537, 567, 568, 569, 580, 581, 598, 599, 600, 600t, 644, 645, 648, 649, 657, 658, 660, 663, 666 lumbar, 51, 280, 281 mandibular, 525, 599, 607, 639, 647, 658 in infratemporal fossa, 600 MRI of, 669 masseteric, 567, 580, 581, 600, 639, 644 maxillary, 593, 607, 621, 623, 658 Index 739 Nerves (named) median, 72, 368, 368t, 369, 374, 374t, 376, 383, 384, 385, 386, 387, 388, 389, 390, 391, 393, 394, 397 branches of, 379 exclusive area, 394 injuries to, 376 motor branches of, 376t MRI of, 398, 399, 403, 404 palmar branch of, 376, 378, 388, 392 recurrent branch, 390 roots of, 368t, 369, 376, 384 sensory branches of, 376t meningeal, 600t mental, 567, 581, 594, 595, 596, 644 musculocutaneous, 368, 368t, 369, 375, 379, 383, 384, 386, 387, 389 cutaneous distribution of, 375 motor branches of, 375, 375t sensory branches of, 375, 375t mylohyoid, 567, 581, 599, 600, 644, 645 nasal, external, 596 nasociliary, 567, 579, 607, 609 nasopalatine, 620, 621, 623, 644 branches of, 620 obturator, 211, 268, 273, 280, 281, 286, 476, 476t, 477, 478, 478t, 480, 481t, 486, 487 branches of, 477, 486, 492 MRI of, 505 obturator externus, 480, 480t occipital, 540 3rd, 39, 46, 47, 524t, 540, 541 greater, 39, 39t, 46, 524, 524t, 525, 540, 541, 595, 596, 661 lesser, 39, 39t, 46, 524, 524t, 525, 538, 539, 540, 595, 596, 627 oculomotor (CN III), 549, 560, 565, 593, 607, 609, 655, 658, 660, 662, 683, 688 effects of injury on, 564t inferior branch of, 607, 608 nuclei of, 561, 561t palsy of, 605 superior branch of, 608 visceral nucleus of, 564 olfactory (CN I), 560, 562, 657, 679 nuclei of, 561, 561t ophthalmic (CN V1), 524, 525, 563, 593, 607, 658 optic (CN II), 549, 560, 563, 590, 593, 604, 606, 607, 608, 609, 610, 612, 657, 658, 660, 661, 662, 667, 679, 681 nuclei of, 561, 561t palatine descending, 623 greater, 621, 623, 644 lesser, 621, 623, 644 palmar digital, common, 377 pectineus, 480, 480t, 481 pectoral lateral, 368, 368t, 369, 374, 374t, 382, 383 medial, 368, 368t, 369, 374, 374t, 382, 383, 384 perineal, 248, 284, 285 MRI of, 502 petrosal deep, 569, 579, 621 greater, 568, 569, 571, 621, 629, 634 hiatus of canal for, 548 hiatus of canal for, 568 lesser, 571, 573, 600, 629, 634 groove for, 547 hiatus of canal for, 548 pharyngeal, 623 phrenic, 64t, 66, 67, 73, 78, 88t, 89, 90, 91, 113, 130, 368, 369, 372, 383, 524, 525, 535, 536, 537, 539, 665 left, 66, 67, 78, 91, 93, 94, 95, 103 right, 66, 67, 78, 86, 90, 93, 103 plantar common, 498 lateral, 476, 485, 485t, 495, 498, 499 branches of, 498, 499 superficial branch of, 485 medial, 476, 485, 485t, 495, 498, 499 branches of, 498 MRI of, 509 pterygoid lateral, 600 medial, 567, 600 pudendal, 246, 248, 268, 280, 281, 283, 284, 285, 287, 476, 476t, 482, 482t, 491, 493 branches of, 284, 285, 490 quadriceps femoris, 481 radial, 368, 368t, 369, 371, 371t, 373, 379, 385, 392, 394, 397, 403 branches of, 368, 378, 387, 388 compression in the axilla, 373 deep branches of, 389 head, 403 motor branches of, 373, 373t, 385 MRI of, 398, 403 muscular branches of, 381, 389 in radial groove, 300, 373 sensory branches of, 373, 373t superficial branch of, 373, 388, 389, 391, 394 superficial palmar branch, 390 rectal, inferior, 280, 283, 284 saccular, 571, 634, 635 sacculoampullary, 571 sacral, 281, 282 saphenous, 476, 481, 486, 492 branches of, 486, 499 scapular, dorsal, 47, 368, 368t, 369, 370, 370t sciatic, 286, 287, 476, 476t, 477, 482, 482t, 483, 484, 485, 487, 490, 491, 493, 495, 500, 501 branches of, 484t MRI of, 288, 502, 505 scrotal, posterior, 280, 284 spinal, 17, 38, 40, 42–43, 47, 676, 690 anterior, 40, 45 anterior root of, 683, 693 branches of, 39, 380, 524 cervical, 369, 370, 371, 372, 577, 660, 664, 665, 669 groove for, 7, 8, 9 16, 17, 20 in intervertebral disk herniation, 15 posterior root, 693 root and rootlets of, 42, 683, 693 MRI of, 701 thoracic, 369 typical, 696 splanchnic, 67, 86, 90, 91, 108, 675, 697 greater, 67, 86, 90, 91, 108, 127, 214, 215, 216, 217, 218, 694 least, 215, 280, 281 lesser, 87, 215, 216, 217, 280, 281 lumbar, 212, 213, 215, 280, 281, 282 pelvic, 212, 215, 217, 280, 281, 282, 283, 694 sacral, 212, 213, 215, 280, 281, 282, 283 stapedial, 568 subcostal, 182, 185, 211, 476, 477 suboccipital, 39, 39t, 46, 524t, 541 greater, 39, 39t subscapular, 369 lower, 368t, 371, 371t, 384, 385 MRI of, 401 upper, 368t, 371, 371t, 385 supraclavicular, 210, 372, 378, 379, 380, 382, 524, 524t, 525, 534, 538, 581 intermediate, 538, 539 lateral, 538, 539 medial, 538 supraorbital, 565, 567, 581, 595, 596, 607, 608, 609, 666 branches of, 594 suprascapular, 368, 368t, 369, 370, 370t, 385, 535 lower, 385 MRI of, 401 in scapular notch, 380, 385 upper, 385 supratrochlear, 567, 581, 595, 596, 607, 608 sural, 476, 485, 486, 493, 496 lateral, 476, 484, 486, 493, 494, 496 medial, 485, 486, 493, 494, 496 MRI of, 503 temporal, deep, 567, 580, 598, 599, 600, 639 Index 740 Nerves (named) (continued) Nerves (named) (continued) thoracic internal, 72 long, 368, 368t, 369, 370t, 383, 384, 385 thoracodorsal, 368, 368t, 369, 371, 371t, 384, 385 tibial, 435, 476, 477, 484, 485, 485t, 486, 487, 493, 494, 495, 501 branches of, 485t, 494 cutaneous distribution of, 485 MRI of, 502, 503, 507 trigeminal (CN V), 549, 560, 566–567, 567t, 569, 580, 607, 644, 655, 682, 683, 687, 688 branches of, 567 course of, 566, 566t, 567 effects of nerve injury on, 566t fibers of, 566t mandibular division of, 554, 555, 566, 566t, 580, 581, 591, 600, 607, 621, 639, 644, 645 branches of, 591 course of, 567, 568 maxillary division of, 566, 566t, 573, 580, 581, 591, 607, 621, 644 branches of, 591, 623, 644 course of, 567, 568 motor root of, 593 nuclei of, 561, 561t, 566t, 573, 574t ophthalmic division of, 566, 566t, 573, 580, 581, 591, 607, 621 branches of, 591 course of, 567, 568 sensory root of, 593 trochlear (CN IV), 549, 560, 564, 565, 593, 594, 607, 609, 658 effects of injury on, 564t nuclei of, 561, 561t palsy of, 605 tympanic, 571, 572, 572t, 573, 629, 633 ulnar, 72, 295, 368t, 369, 374, 374t, 377, 379, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 397 deep branch of, 390, 391, 393 dorsal branch of, 377, 378, 394 dorsal carpal branch of, 395 motor branches of, 377t MRI of, 398, 399, 404 palmar branch of, 377, 378, 392 palsy of, 377 sensory branches of, 377t superficial branch of, 390, 391, 393 in ulnar tunnel, 388 utricular, 571, 634, 635 utriculoampullary, 571 vagus (CN X), 25, 78, 86, 87, 88t, 89, 90, 91, 93, 94, 95, 103, 106, 108, 109, 113, 127, 130, 530, 531, 535, 536, 537, 549, 572, 574–575, 576, 577, 591, 627, 646, 654, 655, 661, 663, 664, 665, 683, 697 branches of, 87t, 103, 572, 575, 575t, 591 course of, 575 dorsal nuclei of, 578 effects of nerve injury on, 574t fibers of, 574t function of, 574t MRI of, 669 nuclei of, 108, 127, 561, 561t, 574, 574t vestibular, 571, 628, 629, 634, 635 vestibulocochlear (CN VIII), 549, 560, 570–571, 625, 655, 682, 683 cochlear part of, 570, 570t, 635 cochlear root of, 571 course of, 570t effects of injury on, 570t fibers of, 570t function of, 570t nuclei of, 561, 561t in temporal bone, 571 vestibular part of, 570, 570t, 635 vestibular root of, 571 zygomatic, 567, 623, 644 zygomaticofacial, 623 zygomaticoorbital, 595 zygomaticotemporal, 623 Nerves (regional) of abdomen of abdominal walls, 210 of foregut, 216 of intestines, 218–219 of pelvis, 210–211 of spleen, 216 of back, 38–39, 46–47 cutaneous innervation of, 39 of breast, 74 cutaneous, superficial, 378 of diaphragm (thoracic), 66, 67 of foregut, 216 of infratemporal fossa, 600, 600t of limb (upper) brachial plexus course of, 369 structure of, 369 of limb (lower), superficial, 486–487 of lumbar plexus, 478–481, 478t, 480t of nasal cavity, 621 of pelvis, 210–211 of pleura, 114 of pterygoid canal, 569, 621 of pterygopalatine fossa, 623, 623t of spleen, 216 to subclavius muscle, 368t, 369, 370, 370t of thoracic cavity, 114 Nervous system autonomic (visceral). See Autonomic nervous system (ANS) central. See Central nervous system development of, 676–677 overview of, 674–675 peripheral development of, 676 morphology of, 675 sectional anatomy of, 698–699 Nervus intermedius, 568, 571, 635, 655, 683 Network (vascular) arterial, of elbow and lateral epicondyle, 389 carpal dorsal, 365, 395 palmar, 365 dorsal venous, 366, 378, 474 of foot, 474 peribronchial, 128 subpleural, 128 patellar vascular, 492 Neurocranium, 542t Neurohypophysis, 680, 682 Neurons first-order sensory, 562 α-motor, 691, 692, 693 postganglionic, 695t preganglionic, 695t projection, 691 upper motor, 695t Neurothelium, 590 Nipple, accessory, 74 Node atrioventricular (AV), 102, 103 sinoatrial (SA), 102, 103 Nodule of brain, 698 of choroid plexus, 682 Nosebleeds, 621 Nose. See also Cavity, nasal arteries of, 620–621 muscles of, 556, 556t septum of, 562, 618, 620, 650, 651, 657, 659, 662, 666 skeleton of, 616 Notch, 166 acetabular, 231 cardiac, 117 clavicular, 56, 58, 61 fibular, radiograph of, 508 frontal, 513, 543 intercondylar, 412, 413, 434, 440 jugular, 54, 56, 58, 512 mandibular, 637 mastoid, 544, 546, 624 pterygoid, 551 Index 741 Otitis media radial, 324, 325 of ulna, 327, 331 scapular, 59, 299, 303, 304, 305, 317, 370, 381, 381t, 385 suprascapular nerve in, 380 sciatic greater, 143, 232, 491t lesser, 232, 491t supraorbital, 513 suprascapular, 370 suprasternal, 512 tentorial, 590 thyroid inferior, 526 superior, 513, 526 trochlear, 324, 325, 327, 330, 331, 403 ulnar, 331 vertebral inferior, 10, 11 superior, 7, 10, 11, 14 Nucleus ambiguus, 561, 572, 574t, 576 Nucleus cuneatus, 692 accessory, 692 tubercle of, 683 Nucleus gracilis, 692 tubercle of, 683 Nucleus/nuclei caudate, 680, 689 body of, 698, 699 MRI of, 701 head of, 658, 660 MRI of, 700 tail of, 698, 699 cochlear anterior, 570 posterior, 570 of cranial nerves, 695t abducent (CN VI), 561, 561t, 564, 564t accessory (CN XI), 561, 561t, 651 facial (CN VII), 561, 561t, 568 glossopharyneal (CN IX), 561, 561t hyoglossal (CN XII), 561, 561t oculomotor (CN III), 561, 561t, 564, 564t, 565 olfactory (CN I), 561, 561t optic (CN II), 561, 561t, 565 trigeminal (CN V), 561, 561t trigeminal nerve (CN V), 572 trochlear (CN IV), 561, 561t, 564, 564t, 565 vagus (CN X), 561, 561t, 574t vestibulocochlear (CN VIII), 561, 561t, 570t facial, 568 habenular, 562 intermedius, ventral, 693 interpeduncular, 562 mesencephalic, 566 motor, 566, 690 parasympathetic, 694 principal (pontine) sensory, 566 red, 564, 681, 693, 699 salivatory inferior, 572, 578, 578t superior, 568, 569, 578, 578t of solitary tract, 561, 568, 569 spinal, 566 tegmental, 562, 693 thalamic, 698 posterior, 660 vagal, dorsal, 108, 212 vestibular inferior, 570 lateral, 570 medial, 570 posterior, 570 superior, 570 visceral oculomotor (Edinger-Westphal), 578, 578t Nucleus pulposus, 14, 15, 17, 22 in disk herniation, 15 MRI of, 48 “Nursemaid’s elbow,” 329 O Obturator internus, 238 Occipital bone, 28, 30t, 512, 513, 542t, 545, 663 basilar part of, 3t, 20, 21, 31 MRI of, 669, 700 Occiput, veins of, 589 Olecranon, 24, 294, 295, 296, 312, 314, 323, 324, 325, 326, 327, 328, 331, 334, 339, 341, 389 MRI of, 403 radiograph of, 402 Olfactory system, 549, 562 Olive, 574, 577, 683 inferior, 693 Omentum greater, 156, 158, 159, 160, 161, 162, 167, 170, 172, 173, 174, 181, 190, 193, 220 lesser, 156, 158, 159, 162, 166, 174, 181, 190 ligaments of, 162, 166, 175, 181 Opening of the carotid canal, 643 caval, 64, 65, 67, 82, 89, 108, 147 of ejaculatory ducts, 253 of frontonasal duct, 621 of nasal cavity, 651 of nasolacrimal duct, 621 of oral cavity, 651 saphenous, 271, 474, 486, 488 for tendon of tensor tympani, 629 Operaculum frontal, 678 parietal, 678 temporal, 678 Optic chiasm. See Chiasm, optic Optic radiation lower visual field of, 563 upper visual field of, 563 Oral floor, lymphatic drainage of, 522 Ora serrata, 612, 615 Orbit, 543, 602–603, 618, 619, 622t, 623t adipose tissue of, 610 arteries of, 606 bones of, 602–603 innervation of, 607 medial wall of, 660 MRI of, 668, 669 muscles of, 604–605 neurovasculature of, 606–607, 608–609 openings for neurovascular structures in, 602, 602t passage of neurovascular structures through, 608 roof of, 666 structures surrounding, 603, 603t topography of, 608–609, 610 veins of, 606 Orifice cardiac, 163, 164 of ejaculatory ducts, 266 ileocecal, 172 pharyngeal, 648 of pharyngotympanic (auditory) tube, 620, 621 of posterior ethmoid sinus, 617 pyloric, 168 tubal, 653 ureteral, 247, 253, 255, 286 urethral, 252, 264 external, 229, 252, 260, 262, 264, 285 internal, 25, 247 vaginal, 229, 260, 262, 285 Oropharynx, 643, 648, 658, 659, 664, 667 MRI of, 668 Os, uterine, 260, 261 anterior lip of, 260 posterior lip of, 260 Ossicles, auditory. See Ossicular chain Ossicular chain, 630 arteries of the middle ear and, 632 in tympanic cavity, 631 in hearing, 631 Ostium abdominal, 257 of maxillary sinus, 619 of sphenoid sinus, 636 Otitis media, 633 Index 742 Otoscopy Otoscopy, 626 Outlet, pelvic, 234t plane of, 234, 235 Ovary, 247, 254t, 273, 279, 281 free margin of, 256 left, 244, 251, 257, 261 ligaments of, 243, 244, 247, 252, 254, 257, 258, 261, 288 lymphatic drainage of, 277 medial surface of, 256 MRI of, 288 right, 252, 254, 272 P Pain acute abdominal, 156 kidney stone-related, 251 referred, 214–215 sense of, 692, 692t Palate, 623 hard, 628, 636, 643, 659, 667 mucoperiosteum of, 663 neurovasculature of, 644 soft, 648, 650, 651, 657, 659, 663 muscles of, 643, 643t taste buds of, 569 Palatine bone, 544, 546, 598, 603, 616, 622, 636 Palm, of hand, 295, 392–393 MRI of, 405 neurovascular structures deep, 393 superficial, 392 Palpable bony prominences of head, 513 of limb (lower), 408 of limb (upper), 295 of neck, 513 of thorax, 54 Palsy abducent, 605 oculomotor, 605 trochlear, 605 ulnar, 377 Pancreas, 159, 162, 163, 164, 166, 168, 169, 175, 180–181, 183, 190, 191, 220 arteries of, 163, 171, 174, 179, 187 autonomic nervous system effects on, 156, 212t body of, 165, 180, 181, 222 computed tomography (CT) of, 222, 224 head of, 165, 180, 221 innervation of, 694 location of, 180 lymphatic drainage of, 203, 206, 207 neck of, 156, 180, 185 tail of, 165, 180, 181, 185 artery of, 187, 191 transverse section of, 181 uncinate process of, 180, 181 veins of, 199 Papilla duodenal, 168, 169, 178, 179 ileal, 172 renal, 184 sublingual, 642, 647, 649 Paralysis hypoglossal, 577 sternocleidomastoid, 576t trapezius, 576t Parietal bone, 26, 513, 542, 542t, 543, 544, 545, 546, 558, 619 Parotid region, 596 Pars flaccida, 630 Pars intermedia, 262 Pars plicata, 615 Pars tensa, 630 Patches, Peyer’s, 170 Patella, 408, 410, 413, 418, 421, 425, 428, 429, 430, 434, 435, 438, 439, 442, 443, 444, 445, 448, 496, 497 apex of, 435 base of, 435 MRI of, 507 radiograph of, 506 surface of, 412 Pecten, anal, 249, 274 Pecten pubis, 232 Pectoral region, 54 Pedicle of cervical vertebrae, 7, 9, 50 of lumbar vertebrae, 15, 50 radiograph of, 49, 50 of thoracic vertebrae, 7, 10, 49 of vertebral arch, 7, 20, 23 Peduncle cerebellar, 682, 683 inferior, 682, 683 middle, 682, 683, 698, 699 superior, 682, 683, 698 cerebral, 680, 681, 682, 683, 685, 698, 699 cerebral crus of, 564 of mesencephalon, 564 of flocculus, 682 Pelvis, 246–247 arteries of, 212t, 269, 269t in female, 272–273 in male, 270–271 contents of in female, 243 in male, 242 divisions of, 246t false, 234, 234t, 246, 247t floor of in situ, 239 muscles of, 238–239, 240t gender-specific features of, 234t ligaments of, 236 deep pelvis and fascia, 258 in female, 258–259 lymphatic drainage of, 85, 204–205 lymphatics of, 277–278 lymph nodes of, 276, 276t in male, 233, 235 measurements of in female, 234–235 in male, 234–235 MRI of, in female, 288–289 nerves of, 217 neurovasculature of, 268t oblique section of, 247 palpable structures of, 228 peritoneal relationships in in female, 244 in male, 245 spaces of in female, 243 in male, 242 peritoneal, 244, 245 subperitoneal, 244, 245 surface anatomy of, 228–229 symphyseal surface of, 234 true, 234, 246, 246t veins of, 269, 269t in female, 272–273 in male, 270–271 Penis, 229, 255, 255t, 264–265, 278 arteries of, 275 body of, 264 bulb of, 245, 255, 264 MRI of, 290 corpus cavernosum of, 253, 267 corpus spongiosum of, 253 crus of, 245, 246, 247, 264 dorsum of, 271 glans of, 155, 229, 264, 265, 267 ligaments of, 144, 271, 275 nerves of, 280 root of, 264 suspensory ligament of, 253 veins of, 275 Pericardium, 67, 94–95 fibrous, 66, 89, 90, 93, 94, 95, 96, 103, 110, 111, 113, 114, 115, 116 attachment to diaphragm, 89, 94 serous, 113 parietal layer, 94, 95, 115, 116 visceral layer, 94, 95, 116 Perineal region, 229 anal triangle of, 229 urogenital triangle of, 229 Perineum body of, 239 Index 743 Plexus divisions of, 246t muscles of, 239, 240–241, 241t in female, 241 in male, 241 neurovasculature of in female, 285 in male, 284 regions of, 229 surgical gynecological, 229 Periorbita, 610 Peritoneum, 152, 158–159, 246 in situ, 239 parietal, 27, 107, 113, 146, 153, 160, 165, 167, 169, 175, 181, 182, 183, 220, 242, 244, 245, 248, 249, 251, 261 of abdominopelvic cavity, 158, 159 of rectum, 260 of rectum, 249 urogenital, 252, 253 visceral, 107, 113, 160, 181, 256 of abdominopelvic cavity, 159 of prostate gland, 267 of rectum, 242, 243 of urinary bladder, 242, 243, 260, 273 of uterus, 260, 273 Peritonitis, 158 Pes anserinus, 495, 497 Phalanx/phalanges, 295, 296, 297, 342t, 410, 452 base of, 347 distal, 346, 347, 353, 359 1st, 297, 347, 452, 453, 454, 461 base of, 337, 341 1st through 5th, 449 2nd, 342, 363 2nd through 5th, 363 4th, 296, 337 5th, 452, 453, 468 head of, 347 radiograph of, 404, 508 of second toe, 509 tuberosity of, 347 middle, 347, 353 2nd, 342 2nd through 5th, 337 4th, 296 5th, 343, 452, 453, 454, 468 head of, 343 radiograph, 508 of second toe, 509 shaft of, 343 proximal, 346, 347, 352 1st, 297, 361, 453, 454, 460, 461, 468 base of, 341, 453, 455, 462 head of, 452, 453 MRI of, 404, 405 shaft of, 452, 453 2nd, 342, 363 4th, 296 5th, 361, 452, 468 base of, 341, 404 MRI of, 405 of great toe, 462 radiograph, 508 of second toe, 509 shaft of, 347 Pharynx, 636, 651 neurovasculature of, 654–655 Philtrum, 512 Pia mater, 40, 590, 592 Pisiform, 295, 337, 342, 342t, 343, 344, 345, 346, 350, 363, 390, 391, 404 MRI of, 405 radiograph of, 343 Placenta, 104 Plane(s) of acetabular inlet, 413 canthomeatal, 625 of pelvic inlet, 234, 235 of pelvic outlet, 234, 235 sagittal, 413 subcostal, 54 transpyloric, 166 transumbilical, 228 Plateau, tibial, 410, 432, 439 radiograph, 506 Plate(s) cartilaginous, 121 cribiform, 542t, 547, 548, 550, 562, 591, 609, 616, 617, 619, 620 epiphyseal, 329 of ethmoidal bone, perpendicular, 636 lateral, 601, 636 medial, 636 orbital, 550, 602, 603, 619 of ethmoid bone, 656 of palatine bone horizontal, 616, 617 perpendicular, 543, 550, 603, 616, 617, 619, 636 of pterygoid process lateral, 598, 617, 622, 636, 643, 663 MRI of, 669 medial, 546, 598, 617, 638, 653 MRI of, 669 quadrigeminal, 680, 681, 682, 685, 699 tectal, 683, 698 Platysma, 656, 664 Pleura cavity of, 55 boundaries and reference lines of, 112 cervical (cupola), 55, 89, 106, 113, 116 costal part of, 65, 67, 71, 90, 91, 113, 114, 115 diaphragmatic part, 66, 71, 73, 89, 90, 91, 106, 113, 114, 116, 146, 174 innervation of, 114 mediastinal part, 66, 78, 89, 90, 91, 93, 94, 95, 106, 113, 114, 115, 116 parietal, 55, 66, 72, 112, 113 recesses of, 66, 115 subdivisions of, 114 visceral, 71, 72, 113, 114 Plexus aortic thoracic, 87, 103 nerve autonomic of genital organs, 280–281 of heart, 103 of intestines, 218–219 of urinary organs and rectum, 282–283 brachial, 25, 43, 78, 86, 89, 90, 91, 103, 106, 108, 368–369, 382, 383, 535, 536, 537, 539 course of, 369 infraclavicular part of, 368, 368t injuries to, 368 MRI of, 398, 401 posterior cord of, 371 structure of, 369 supraclavicular branches of, 370 supraclavicular part of, 368, 368t carotid common, 87 external, 87, 579 internal, 87, 565, 571, 573, 607, 621, 629 carotid, internal, 579 celiac, 213t, 214 cervical, 555, 595, 627 branches of, 380, 524, 525 coccygeal, 477 deferential, 280 enteric, 216 esophageal, 86, 87, 95, 106, 108, 109 facial artery, 579 gastric, 103, 216 anterior, 108, 216 posterior, 108, 109, 214 hepatic, 214, 216, 217 hypogastric inferior, 213t, 215, 217, 219, 281, 282, 286, 694 inferior right, 281 superior, 213t, 215, 217, 218, 280, 281, 282, 283 iliac, 213, 217, 282 intermesenteric, 280, 281, 282 lumbosacral, 476–477, 476t, 477 computed tomography (CT) of, 223 injuries to, 476 nerves of, 478–481, 478t, 480t, 481t Index 744 Plexus (continued) Plexus (continued) mesenteric inferior, 215, 219, 282 renal, 213t myenteric (Auerbach’s), 216 ovarian/testicular, 213t, 281 pampiniform, 265, 271 pancreatic, 214, 216 parotid, 568, 580, 595, 649 branches of, 568, 596 pharyngeal, 87, 572, 654 pial vascular, 613 posterior, 108, 109 pulmonary, 87, 103, 127 sacral, 43, 188, 210, 250, 268, 271, 281, 283, 476t, 477, 492 branches of, 283 nerves of, 482–483, 482t, 483–485, 484t splenic, 214, 216 submucosal (Meissner’s), 216 subserosal, 216 superior, 213t, 214, 215 suprarenal, 213t testicular, 154, 155, 216, 217, 218, 280, 282 tympanic, 571, 573, 629, 645 ureteral, 215, 217, 280, 281, 282 venous areolar, 69, 194 around foramen magnum, 589 basilar, 592 choroid, 679, 680, 682, 684 of lateral ventricle, 698, 699 MRI of, 700 external, 274 of foramen ovale, 592 hemorrhoidal, 249, 274 infraparotid, 596 pampiniform, 153, 154, 155 prostatic, 217, 280, 282 pterygoid, 521, 588, 589 foramen for, 636 rectal, 196, 274, 283 inferior, 215, 280 middle, 215, 217, 280, 281, 282 subcutaneous, 249 thyroid, 531 uterine, 196, 261, 272 uterovaginal, 281, 286 right, 281 vaginal, 261, 272 venous, 247 vesical, 196, 217, 280, 281, 282 vesicoprostatic, 286 of vestibular bulb, 276 visceral, 113 vertebral, 37, 45, 69, 87, 220, 684 anterior, 37 anterior external, 37, 45, 69 anterior internal, 37, 40, 41, 45, 69 external, 589, 589t Poles frontal, 678 occipital, 563, 678 temporal, 678 uterine, 256, 257 vascular, 256 Polythelia, 74 Pons, 561t, 565, 566, 568, 570, 662, 674, 677, 682, 683, 685, 698 MRI of, 700 Popliteal region, neurovasculature, 495 Position, sense of, 692, 692t Pouch perineal deep, 241, 246, 246t superficial, 246, 246t rectouterine (of Douglas), 243, 244, 251, 286 MRI of, 288, 289 rectovesical, 156, 157, 165, 242, 245, 267 superficial, 241 suprapatellar, 442, 443 during flexion, 443 vesicouterine, 243, 244, 251, 256, 260 Precuneus, 679 Pregnancy, ectopic, 156, 257 Premolars, 640 Prepiriform area, 562 Prepuce of clitoris, 229, 262, 263 of penis, 253 Pressure atmospheric, in pneumothorax, 123 sense of, 692, 692t Primordium, pituitary, 677 Process accessory, 11 lumbar, 7 alveolar, of mandible, 543, 637, 666 articular, of vertebrae, 4, 7, 7t inferior, 8, 9, 10, 11, 14, 16, 20, 23 superior, 7, 8, 9, 10, 11, 12, 13, 14, 16, 20, 22, 23, 48 MRI of, 48 radiograph of, 48 of calcaneal tuberosity lateral, 452, 459 medial, 452, 453, 459 caudate, 177 ciliary, 615 clinoid anterior, 547, 551, 593, 636, 667 posterior, 547, 551, 593 condylar, 641 coracoid, 295, 297, 298, 299, 302, 303, 304, 305, 306, 307, 309, 316, 317, 318, 319, 321, 322, 375, 385, 386 radiograph of, 400 coronoid, 324, 325, 326, 327, 330, 337, 403, 517, 558, 559, 597, 637, 638, 657 MRI of, 403 radiograph of, 402, 668 costal, of vertebrae, 6, 29 frontal, 543, 597, 602, 616, 617 of incus lenticular, 639 long, 639 short, 639 lumbar, 156 of malleus anterior, 630, 631 lateral, 630 mammillary, 11 MRI of, 48 mastoid, of temporal bone, 18 26, 27, 30, 31, 33, 512, 513, 519, 542, 544, 546, 624, 625, 638, 642, 645 maxillary, 598 medial, 471 muscular, of arytenoid cartilage, 526, 527, 528 palatine, of maxilla, 546, 616, 619, 636, 656, 659, 660 of posterior talus, 452, 453, 459 pterygoid, 544, 601, 636, 644 lateral plate of, 546, 551, 559, 617, 622, 636 medial plate of, 546, 551, 617, 636, 638, 653 pyramidal, 598, 603, 636 sacral, 13 spinous, of vertebrae, 3, 3t, 4, 5, 6, 7, 7t, 8, 9, 10, 11, 14, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 32t, 35, 56, 57, 61, 513 of axis (C2), 18, 27, 30, 46 519, 541 cervical, 320, 512, 541, 664 MRI of, 669 of intervertebral disk, 14 lumbar, 236, 237, 422, 425 MRI of, 48 as posterior landmark of back, 3t radiograph of, 49, 50 T1-T4, 320 T12, 320 thoracic, 371 of vertebrae prominens, 3t, 5, 33 styloid, 19, 542, 544, 546, 558, 559, 624, 646 of radius, 295, 324, 330, 331, 331, 339, 342, 343, 344, 345, 347, 404 of temporal bone, 18, 553, 625, 638 Index 745 Recess of ulna, 295, 324, 330, 342, 343, 344, 347, 404 superior articular, of sacrum, 233 temporal, of zygomatic bone, 542 transverse, of vertebrae, 6, 7, 7t, 8, 9, 10, 14, 16, 17, 18, 20, 22, 23, 35, 56, 57, 61 of atlas (C1), 27, 30, 31, 46, 370, 513, 519, 541 cervical, 8, 9, 27 contralateral, 50 costal facet on, 10 with groove for spinal nerve, 7, 9 ipsilateral, 50 lumbar, 11, 64 radiograph of, 49, 50 relation to intervertebral disk, 14 thoracic, 7, 10, 56, 64, 665 uncinate, 8, 9, 17, 156, 550, 617 vocal, of arytenoid cartilage, 526, 527, 528 xiphoid, 5, 54, 61, 63, 64, 88, 142, 144, 149, 309 zygomatic, 542, 543, 546, 597, 598, 601, 624, 638, 658 computed tomography (CT) of, 670 Prominence. See also Palpable bony prominences of facial canal, 629 laryngeal, 513, 526, 527 of lateral semicircular canal, 629 malleolar, 630 Promontory, 428 of labyrinthine wall, 573 pelvic, 232 sacral, 4, 5, 12, 13, 142, 233, 236, 237, 251, 418, 421, 427 computed tomography (CT) of, 224 MRI of, 48 radiograph of, 50 of tympanic cavity, 629, 633 Proprioception, 692 Prosencephalon, development of, 676t Protuberance mental, 512, 513, 542, 543, 637, 666 radiograph of, 668 occipital external, 18, 19, 20, 21, 26, 27, 30, 320, 512, 513, 544, 546, 589 internal, 20, 547 Pterion, 542 Pubic symphysis. See Symphysis, pubic Pubis, 248 body of, 230, 231 MRI of, 288 in female, 251, 258, 286 in male, 251 MRI of, 290, 291, 505 radiograph of, 231, 504 Pulvinar, 680, 681, 682, 699 Puncta inferior, 611 superior, 611 Pupil, 614, 615 regulation of size of, 615 Putamen, 658, 661, 689 MRI of, 700, 701 Pyramid(s), 693 decussation of, 683 of medulla oblongata, 683 renal (medullary), 184, 189 computed tomography (CT) of, 222, 223 of vermis, 682 R Radiography. See Imaging Radius, 296, 297, 323, 324–325, 326, 328, 331, 337, 339, 341, 342, 344, 347, 350, 357, 363, 375, 396, 397 borders of, 324, 325, 330 carpal articular surface of, 345 distal, fractures (Colles’) of, 331 head of, 295, 296, 324, 325, 326, 327, 329, 330 MRI of, 399 radiograph of, 402 lunula of, 331 MRI of, 405 neck of, 324, 330 shaft of, 324, 325 styloid process of, 295, 324, 325, 330, 331, 331, 339, 342, 343, 344, 345, 347, 404 Ramus communicates/rami communicates gray, 38, 40, 42, 43, 71, 90, 281, 697 white, 38, 40, 42, 43, 71, 90, 697 Ramus/rami of bones ischial, 230, 231, 232 MRI of, 289 ischiopubic, 229, 261 of mandible, 543, 598, 637, 658 computed tomography (CT) of, 670 of lateral sulcus anterior, 678 ascending, 678 posterior, 678 posterior (dorsal), 524t pubic, 262 inferior, 230, 231, 235, 242, 245, 246, 247, 261, 287 superior, 142, 143, 151, 230, 231, 232, 235, 242, 243, 428, 429 radiograph of, 504 of spinal nerves anterior, 39, 40, 43t, 71, 280, 281, 524t, 525, 581, 696, 697 of back, 38, 39t of lumbar nerves, 281 of neck, 39 of sacral nerves, 281 to sacral plexus, 482 of spinal cord, 38, 42, 43, 47 spinal cord segments and, 42 of thoracic spine, 43 cervical, 596 posterior, 554, 595 ventral, 555 posterior, 524, 525, 581, 696, 697 of brachial plexus, 369, 374 branches of, 380 dorsal, 482 superior MRI of, 505 Raphe anococcygeal, 238 mylohyoid, 517, 642 perineal, 229 pharyngeal, 107, 654 pterygomandibular, 648, 652, 653 Rays of foot lateral, 462 medial, 462 of kidney, medullary, 184 Recess. See also Mortise axillary, 304, 305, 306 costodiaphragmatic, 55, 71, 111, 115, 116, 117, 123, 130, 183, 222 costomedial, 113, 115, 130, 136 duodenal inferior, 161, 165, 169 superior, 161, 169 hepatorenal, 165 iliocecal, 165 inferior, 161 superior, 161, 165 infundibular, 680, 685 intersigmoidal, 161, 165 lateral, 685 of omental bursa inferior, 163, 174 splenic, 163, 174 superior, 163, 174 optic, 699 peritoneal, 160, 165 pharyngeal, MRI of, 669 pineal, 685 piriform, 529, 651, 664 pleural, 115 retrocecal, 161, 165 sacciform, 327, 328, 329 sphenoethmoidal, 617, 618, 620, 621 splenic, 163, 180 subhepatic, 165 Index 746 Recess (continued) Recess (continued) subphrenic, 165 subpopliteal, 437, 442 supraoptic, 680, 685 suprapineal, 685 of the tympanic membrane, superior, 630 Rectoanal inhibitory reflex, 283 Rectum, 156, 164, 165, 171 196, 172, 173, 179, 242, 243, 244, 245, 247, 248–249, 250, 252, 258, 260, 267, 271, 273, 278, 279, 283, 286 ampulla of, MRI of, 290 arteries of, 274 autonomic innervation of, 282–283 closure of, 248 computed tomography (CT) of, 224 in situ, 248 innervation of, 694 location of, 248 lymphatic drainage of, 276 MRI of, 288, 289, 291 with peritoneal covering, 251 peritoneal covering of, 249 regions of, 249t veins of, 274 visceral pelvic fascia on, 242, 243 visceral per 243, walls of, anterior, 260 Reference lines of back paravertebral, 3t posterior midline, 3t scapular, 3t of thorax, 55 Respiration diaphragm position during, 60, 123 mechanics of, 122–123 rib cage movement during, 60, 62 Respiratory compromise, conditions causing, 121 Rete calcaneal, 494 testis, 155, 265 MRI of, 290 Reticular formation, 562 Retina, 610, 612 Retinaculum extensor, 358, 359 inferior, 467, 495, 497 superior, 467, 495, 497 fibular inferior, 467 superior, 467 flexor, 295, 350, 354, 355, 356 376, 377, 388, 390, 391, 392, 393, 396, 494 MRI of, 404 medial, lateral, 430 patellar lateral, 430 lateral longitudinal, 436, 436t lateral transverse, 436, 436t medial longitudinal, 436 medial transverse, 436 Retrodiskal region, computed tomography (CT) of, 670 Retroperitoneum, structures of, 159t Rhombencephalon, development of, 676t Ribs, 383, 384t 1st, 56, 59, 63, 78, 80, 82, 86, 90, 91, 108, 122, 130, 297, 303, 307, 308, 319, 365, 369, 519 2nd, 59, 92, 130, 371, 519, 665 3rd, 130, 319 4th, 33, 142, 319 5th, 59, 149, 319 6th, 2, 57, 61, 142 7th, 2, 61 8th, 2, 33, 65, 71, 113, 142 9th, 2, 319, 370 10th, 2, 65, 142, 149, 180 12th, 2, 3, 3t, 29, 33, 56, 59, 65, 142, 182, 477 computed tomography (CT) of, 224 body (shaft) of, 57, 59 computed tomography (CT) of, 136 head of, 57, 59, 61 relationship to scapula, 299 structural elements of, 57t tubercle facet for, 61 types of, 57, 57t false, 57t floating, 57t true, 57t variations in size and shape, 59 Ridge mammary, of female breast, 74 marginal (epiphyseal ring), 14 oblique external, 637 internal, 637 petrous, 547 supracondylar lateral, 300, 326, 328 medial, 300, 301, 326 Rim, acetabular, 231 anterior, radiograph of, 504 bony, 414, 415 posterior, radiograph of, 504 superior, radiograph of, 504 Rima glottidis, 529 Rima vestibuli, 529 Ring cardiac lymphatic, 111, 206 femoral, 152, 489 inguinal deep, 145, 151, 151t, 152, 153, 244 superficial, 144, 149, 150, 151, 151t, 153, 154, 228, 275, 478, 479, 486, 489 tendinous, common, 565, 604, 608 venous, 45 Waldeyer’s, 650t Roof of acetabulum, 417, 430, 473 MRI of, 505 radiograph of, 504 of orbit, 610, 666 of tympanic cavity, 629 Root of brachial plexus, 369 of cochlea, 571, 626 of median nerve, 384 mesentery, 165 nasociliary (sensory), 567, 607 parasympathetic, 565, 607 of penis, 154, 264 of spinal nerves, 38, 669 anterior, 71, 482, 675, 683, 690, 696 posterior, 40, 675, 690, 696 sympathetic, 565, 607 of teeth, 640 of tongue, 646, 651 vestibular, 570, 571, 626 Rootlets anterior, 690 posterior, 690 Rotator cuff injury to, 307 muscles of, 317, 317t Rugae of stomach, 107 vaginal, 260 S Sac(s) alveolar, 121 dural, 15, 40, 41 developmental stages of, 41 MRI of, 48, 61 endolymphatic, 628, 634, 635 hernial, peritoneum of, 153t lacrimal, 608, 611 fossa of, 602 omental greater, 160 lesser. See Bursa(ae), omental pericardial, 73, 86, 92, 93, 95, 113 Saccule, 571, 634 laryngeal, 529 Sacrum, 2, 3t, 4, 6, 12, 36, 142, 230, 232, 237, 238, 248, 258, 321, 371, 408, 411, 421, 477 ala (wing) of, 7, 12, 13, 232, 233 Index 747 Sectional anatomy apex of, 12 base of, 7, 13, 233 body of, 7t double contrast barium enema of, 225 fused, 6 gender-specific features of, 234t hiatus of, 12, 40, 41 kyphosis of, 4 lateral part of, 7, 12, 13 MRI of, 51 MRI of, 48, 51, 288, 290 pelvic surface of, 233 process of, 13 promontory of, 4, 5, 7, 12, 13, 233 computed tomography (CT) of, 224 MRI of, 48 radiograph of, 50 radiograph of, 13, 225 sacral region of, 3 structural elements of, 7, 7t superior articular process of, 233 surface of, 12–13 anterior (pelvic), 13 articular, 12, 13 auricular, 12 pelvic, 13, 240 posterior, 13 transverse lines of, 12 vascular supply to, 36 Salivary glands, 649 Scala tympani, 634 Scala vestibuli, 634 Scalp, 545 Scaphoid, 342, 342t, 344, 345, 346, 347, 350, 361, 391, 404 fractures of, 343 MRI of, 405 radiograph of, 343 tubercle of, 295, 345, 350 Scapula, 3, 64, 296, 297, 299, 306, 307, 316, 321, 383, 384t, 513 borders of, 24, 297, 304, 306, 313, 317, 317t, 319, 320, 323, 370, 381 lateral, 297 computed tomography (CT) of, 136 inferior angle of, 2, 3, 3t, 297 ligaments of, 304, 306 margins of, MRI of, 400 MRI of, 398, 401 neck of, 299 MRI of, 400 neurovascular tracts of, 381, 381t spine of, 2, 24 superior angle of, 370 surfaces of anterior, 322 costal, 297, 303, 316 MRI of, 400 posterior, 297, 320, 323 Scapular region, 3, 294 Sclera, 610, 614, 615 Scrotum, 154–155, 229, 255t, 265, 267, 278, 284 lymphatic drainage of, 277 root of, posterior border of, 229 septum of, 156, 253 Sectional anatomy of abdomen, 220–221 coronal section of herniation through anterior abdominal wall, 152 of ileocecal orifice, 172 of posterior muscles of abdominal wall, 147 midsagittal section of abdominopelvic cavity in male, 156 of muscles of abdominal wall, 147 through abdominopelvic cavity in male, 158 sagittal section of inguinal region, 150, 151t of omental bursa, 156 of right kidney, 182 transverse section, 220 of anterior abdominal wall, 220 of colon and greater omentum, 173 of liver, 220, 221, 222 of omental bursa, 163 of pancreas, 180 of primary and secondary peritoneal organs, 159 through abdomen, at L1/L2 level, 182 through L1 vertebra, inferior view, 220 through L2 vertebra, inferior view, 221 through right testes, 155 through T12 vertebra, inferior view, 220 of back, 3 coronal section, of uncovertebral joints, 17 midsagittal section of cervical spine ligaments, 21 of lumbar disk herniation, 15 of spine (in adult male), 5 sagittal section, of intervertebral disk, 14 transverse section of cervical spinal cord, 40 of cervical vertebrae, 25 of lumbar vertebrae, 25 of sacral vertebrae, 13 of thoracolumbar fascia, 25 of fingers longitudinal section, of fingertip, 353 transverse section, of third metacarpal, 353 of head and neck coronal section of lymphatic drainage in tongue, 522 of muscles of tongue, 646 through middle cranial fossa, 593 through structures surrounding orbit, 603 through the anterior orbital margin, 656 through the orbital apex, 657 through the pituitary, 658 of tympanic cavity, 629 cross-section, of cochlear canal, 634 midsagittal section of cavity of the larynx, 529 through the nasal septum, 659 sagittal section of arytenoid cartilage, 527 through anterior orbital cavity, 610 through anterior segment of eye, 614 through the medial orbital wall, 660 through the median atlantoaxial joint, 663 transverse section at C6 level, superior view, 530 at the level of the C6 vertebral body, 665 at level of tonsillar fossa, 654 of the neck, 664, 665 through the optic nerve and pituitary, 662 of limb (lower) coronal section of hip joint, 415 of knee joint, 507 cross-section of neurovasculature of the foot, 499 through thigh and leg, 501 midsagittal, of right knee joint, 443 sagittal section of talocrural and subtalar joints, 457 transverse section of fibula, 433 of hip, 413 of patellofemoral joint, 435 windowed dissection of thigh and leg, 500 of limb (upper) coronal section of elbow joint, 327 of joints of foot, 456 of right hand, posterior (dorsal) view, 346 of subacromial and subdeltoid bursae, 307 cross-section through 3rd metacarpal head, 359 through arm and forearm, 397 through carpal tunnel, 391 Index 748 Sectional anatomy (continued) Sectional anatomy (continued) sagittal section of subacromial bursa and glenoid cavity, 306 through anterior shoulder wall, 382 through humeroradial and proximal radioulnar joints, 327 transverse section of carpal tunnel, 350 of radius, 325 through shoulder, 383 of ulna, 325 windowed dissection, 396 in neuroanatomy, 698–699 cross-section, through medulla oblongata, 570, 577 frontal section, through brain, 698, 699 midsagittal section of 3rd and 4th ventricles, 685 of brainstem and cerebellum, 682 of cranial nerve nuclei, 561 of right cerebral hemisphere, 677 parasagittal section, of left laterl ventricle, 685 sagittal section of brain through upper brainstem, 699 through midline of brain, 698 through the inner third of the orbit, 661 transverse section, through spinal cord, 676 of pelvis and perineum, 286–287 coronal section of pelvis (male), 290, 291 of prostate gland, 291 sagittal section, of pelvis (male), 290 transverse section of pelvis (female), 286 of pelvis (male), 286–287, 291 of prostate gland, 291 through bladder and seminal glands, 286 through bladder and uterine cervix, 286 through prostate gland and anal canal, 287 of thorax coronal section, of and through heart, 131 transverse section of midregion of thorax, 131 of parietal pleura, 113 of thoracic inlet, 130 thourhg the heart, 130, 131 Segmentation, of lungs arteries and veins of, 125t bronchopulmonary segments, 118–119, 118t Segmentectomy, of lung, 119 Sella(e) diaphragma, 590 turcica, 551, 667 Semicanal, of tensor tympani, 571, 629 Semilunar valves, of heart, 99 Sensorimotor integration, 691 Sensory pathways, of the spinal cord, 692, 692t Septal defects, 105 Septum pellucidum, 667, 679, 680, 685, 689, 698 anterior veins of, 686 MRI of, 700, 701 Septum/septa deviated, 619 fibrous, between pulmonary lobes, 126 interalveolar, 121, 640 interatrial, 97, 99, 131 intermuscular anterior, 496, 501 crural, anterior, 496 femoral lateral, 502 MRI of, 502 lateral, 381, 397, 501 medial, 397, 501 posterior, 501 transverse, 496, 501 interventricular, 97, 101, 102, 130 computed tomography (CT) of, 137 MRI of, 135 parts of membranous, 99 muscular, 99 trabaculae carneae of, 97, 99 lacrimal, 611 median, MRI of, 668 nasal, 562, 618, 620, 650, 651, 657, 659, 662, 666 cartilaginous, 656 MRI of, 667, 668, 669 orbital, 608, 610 penile, 265 plantar lateral, 464, 499 medial, 464, 499 rectovaginal, 244, 260 rectovesical, 253, 286 scrotal, 253, 267 of sphenoid sinus, 636, 658 vesicovaginal, 260 Sesamoid bones, 343 Sesamoids, 453, 455, 456, 469 lateral, 471 medial, 471 radiograph, 508 Sex glands, accessory, in males, 266–267 Sheath. See also Tendon sheath carotid, 25, 530, 533, 664, 665 common flexor tendon, 354 dural, 610 intertubercular tendon, 306, 307 rectus, 145, 150, 308 anterior layer of, 143, 144, 145, 146, 149, 153, 488 posterior layer of, 145, 146, 147, 149, 152 synovial, intertubercular, 304 of tendon. See Tendon sheath Shoulder anterior arteries of, 308 bones of, 296–299 bursae of subacromial, 306–307 MRI of, 401 subdeltoid, 306, 307, 383 deep dissection of, 383 joint capsule of, 381 superficial dissection of, 382 joints of, 302 acromioclavicular, 296, 298, 302 glenohumeral, 300, 302 scapulothoracic, 302 shoulder girdle, 296–297, 302 in situ, 298 sternoclavicular, 297, 298, 302 sternocostal, 303 muscles of, 308–323 anterior, 382–383 anterior muscles, 308–311 posterior muscles, 312–315 of rotator cuff, 317, 317t nerves of, 380. See also Plexus, nerve, brachial posterior, 380–381 “Shoulder separation,” 303 Sinus(es) anal, 249 aortic, 99 cavernous, 521, 588, 589, 592, 593, 606, 658, 662 cavernous sinus syndrome of, 606 lateral dural wall of, 549 MR angiography of, 671 cerebral, superior, 686 confluence of, 588, 589, 592t, 667, 684, 686 MR angiography of, 671 coronary, 94 valve of, 97 dorsal, 592, 592t dural (venous) in the cranial cavity, 592 formation of, 592 principal, 592 ethmoid, 603t, 618, 618t, 656, 657, 660, 661, 662, 667 middle, 619 posterior orifices of, 617 Index 749 Sphincters frontal, 545, 547, 603, 603t, 616, 618, 618t, 619, 620, 621, 659, 660, 661, 666, 667 MRI of, 668 intercavernous, 592 anterior, 592 marginal, 592 maxillary, 602, 603, 603t, 610, 618, 618t, 619, 636, 656, 657, 661, 666, 667 MRI of, 668, 669 ostium of, 619 occipital, 592, 686 internal, 686 MR angiography of, 671 paranasal, 618–619 air-related pneumatization of, 618 body structure of, 619 MR angiography of, 671 pericardial oblique, 94 transverse, 94, 95 petrosal inferior, 589, 592, 686 inferior, MR angiography of, 671 superior, 589, 592, 686 superior, MR angiography of, 671 petrosquamous, 592 renal, computed tomography of, 184 sagittal inferior, 549, 592t, 662, 686 MR angiography of, 671 MRI of, 700 superior, 37, 521, 549, 588, 589, 590, 592, 592t, 657, 658, 662, 666, 667, 684, 686 groove for, 545 MR angiography of, 671 MRI of, 669, 700 scleral venous, 614, 614 sigmoid, 37, 588, 589, 592, 592t, 628, 629, 651, 654 groove for, 547, 624 MR angiography of, 671 MRI of, 669 sphenoid, 21, 593, 603t, 616, 617, 620, 621, 658, 659, 660, 661, 667 aperture of, 551 MRI of, 668, 700 ostium of, 636 septum of, 636 sphenoparietal, 592 straight, 592, 592t, 667, 684, 686 MRI of, 700 transverse, 37, 521, 588, 589, 592, 592t, 659, 661, 686 groove for, 547 MR angiography of, 671 venous of the brain, 686–687 scleral, 612, 613 Sinusitis, 619 Sinus tarsi, 459 radiograph of, 508 Siphoid, carotid, 593 Skeleton cardiac, 98 of the nose, 616 pedal, divisions of, 452 thoracic, 55, 56–57 Skin of abdominal wall, 146 of cubital fossa, 387 pelvic, 246 penile, 265 perianal, 249, 249t scrotal, 153, 154, 155t of shoulder, 307 testicular, 155 Skull base of, 546–547, 623t interior, 547 muscle origins and insertions, 555 radiograph of, 49 bones of, 542–543, 542t of anterior skull, 543 neurocranium, 542t fractures of, 587 lateral, 542 muscles of, origins and insertions of, 554–555 neurovasculature of, 580 posterior, 544 radiograph of, 666–667, 668 Sleeve, dural, with spinal nerve, 15 Sling connective tissue, 642 crural, 147 of masticatory muscle, 559 of muscles of mastication, 559 Slip interosseus, 359 lumbrical, 359 Snuffbox, anatomic, 394 Sole, of foot. See Foot, sole of Somatomotor innervation, of anal sphincters, 283 Somatosensory innervation, of anal sphincters, 283 Spaces buccal, 654 endoneural, 684 epidural, 15, 40, 41 episcleral, 610 extraperitoneal, 157 infracolic, 165 infraglottic, 529t intercostal anterior, 85 posterior, 85 interscalene, 369, 383, 519 mesorectal, 259 paracolic, posterior, 223 parapharyngeal, 654 pararenal, 223 paravesical, 245 parotid, 654 pelvic in female, 233 in male, 232 perineal drainage of, 165 subcutaneous, 246t peritoneal, 157, 244, 245 peritonsillar, 654 pleural, 72, 123 chest tube insertion into, 72 in pneumothorax, 123 presacral, 259 pterygomandibular, 654 quadrangular, 381, 381t rectovaginal, 259 retro-inguinal, 489, 489t muscular compartment, 489, 489t vascular compartment, 489, 489t retroperitoneal, 157, 159, 259, 261 retropharyngeal, 530, 533, 654, 661 retropubic, 244, 245, 253, 259, 267 retrorectal, 244, 245, 259 retrovesical, 158, 267 subacromial, 302, 306 subarchnoid, 40, 590, 684 MRI of, 21 subdural, 40 subglottic, 529t submasseteric, 654 subperitoneal, 244, 245, 246 supraglottic, 529t tissue, of the head, 654 transglottic, 529t triangular, 381, 381t vesicovaginal, 259 Sphenoid bone, 542t, 551, 602 body of, 551 greater wing of, 542, 543, 547, 551, 603, 619 lesser wing, 547, 551 lesser wing of, 543, 603, 617, 619, 622 Sphincters anal defection mechanism of, 283 external, 239, 240, 241t, 242, 243, 252, 274, 284, 285 deep part of, 283 Index 750 Sphincters (continued) Sphincters (continued) MRI of, 289 subcutaneous part of, 249, 283 superficial part of, 249, 283 innervation of, 283 internal, 283 of bile duct, 178 lumbar, 159 pupillary, 614 pyloric, 166, 168 urethral external, 240, 241t, 245, 253 in female, 252 internal, 253 urethrogenital, 240 urethrovaginal, 240, 252, 260, 261 Spina bifida, 40 Spina occulta, 40 Spine (of region or structure) iliac anterior inferior, 142, 143, 144, 149 anterior superior, 418 posterior superior, 2, 3, 3t, 143 ischial, 142, 143 scapular, 2, 3, 24, 294, 295, 299, 304, 305, 312, 313, 316, 320, 323, 370, 380, 381 radiograph of, 400 Spine. See also Vertebrae/vertebra; Vertebral column cervical bones of, 8, 515, 515t injuries in, 9, 9 intrinsic muscles of, 26–27 joints of, 515, 515t ligaments of, 20–21, 515, 515t lordosis of, 4 radiograph of, 8, 49 uncovertebral joints of, 8, 16t, 17 curvatures of, 4, 5 development of, 4 iliac, 408 anterior, 236 anterior inferior, 230, 231, 232, 430, 431 anterior superior, 228, 230, 231, 232, 237, 408, 410, 414, 416, 421, 422, 425, 427, 430, 431, 479, 483, 484, 489t, 491, 491t, 492 radiograph of, 504 inferior, 232 posterior inferior, 231, 232, 233, 236, 414, 431 posterior superior, 230, 231, 232, 236, 237, 238, 408, 410, 414, 416, 425, 431, 491, 491t ischial, 229, 230, 231, 232, 233, 236, 237, 238, 286, 410, 414, 416, 427, 431, 489 line of gravity of, 5 lumbar, 4, 11 lordosis of, 4 MRI of, 48 osteoporotic, radiograph of, 11 process of, 3 lumbosacral, 40 mental inferior, 637 superior, 637 nasal anterior, 542, 543, 616, 636 posterior, 636 normal anatomical position of, 5 sacral. See Sacrum scapular, 296 thoracic, 4, 10 kyphosis of, 4 process of, 3, 3t radiograph of, 49 thoracolumbar, ligaments of, 22 Spleen, 162, 163, 165, 167, 171, 174, 175, 180, 190, 191, 198, 220 borders of, 164, 180, 181 computed tomography (CT) of, 222, 224 extremities of, 180 gastric surface of, 164, 181 in situ, 181 location of, 180 lymphatic drainage of, 203, 207 MRI of, 133 transverse section of, 181 Splenium, of corpus callosum, 698 MRI of, 700 Spondylolisthesis. traumatic, 9 Spondylophytes, 15, 17 Stalk, infundibular, 549 Stapedius, 631, 633 Stapes, 626, 629, 630 body of, 630 in conductive hearing loss, 631 crus of anterior, 630 posterior, 630 head of, 630 Stenosis, of coronary artery, 101 Sternum, 6, 57, 58, 61, 64, 67, 73, 79, 88, 92, 94, 95, 106, 122, 128, 130, 149, 156, 162, 169, 179, 309, 318 body of, 56, 58, 92 115, 131, 132, 142, 147, 149, 297, 306, 308 computed tomography (CT) of, 136 manubrium of. See Manubrium xiphoid process of. See Process, xiphoid Stigma, follicular, 256 Stomach, 5, 18, 86, 92, 93, 106, 108, 109, 110, 111, 159, 163, 166–167, 174, 180, 190, 220 autonomic nervous system effects on, 214 body of, 166, 167, 175, 221 with rugal folds, 166 cardia of, 166, 167 computed tomography (CT) of, 222, 224 curvature of greater, 162, 166, 167 lesser, 166, 167 fundus of, 166, 167 computed tomography (CT) of, 224 in situ, 167 innervation of, 694 location of, 166 lymphatic drainage of, 203, 207 muscular layers of, 166 posterior surface of, 162 pyloric part of, 164, 171, 220 referred pain in, 214 relations of, 166 veins of, 198 walls of anterior, 220 posterior, 220 Stria diagonal, 562 longitudinal, 562 malleolar, 630 medullary, of thalamus, 562 olfactory lateral, 562 medial, 562 Stria medullaris, 683 Stria medullaris thalami, 680 Striate area, 699 Stroma cervical, MRI of, 288 esophageal, 107 fibrous, 155 Submandibular glands, 512 superficial part of, 649 Submucosa duodenal, 168 of urethra, 252 Suboccipital region, 27 muscles of, 27, 30, 30t, 46 Substance anterior perforated, 562 gray, central, 564 Substantia nigra, 564, 693, 699 Sulcus/sulci anterolateral, 683, 690 calcaneal, 459 Index 751 Tendons calcarine, 678, 679, 698 central, 677, 678, 679 cingulate, 679 coronary, 96, 97 of corpus callosum, 679 frontal inferior, 678 opercular part of, 678 orbital part of, 678 triangular part of, 678 superior, 678 gluteal, 490 hypothalamic (ventral diencephalic), 680 interventricular anterior, 96 posterior, 96 intraparietal, 678 lateral (of Sylvius), 677, 678, 679 marginal, 679 occipital, transverse, 678 palatine, 660 paracentral, 679 parietooccipital, 678, 679 parieto-occipital, 698 plantar lateral, 498 medial, 498 posterior intermediate, 690 posterior median, 690 posterolateral, 683 precentral, artery of, 689 telodiencephalic, 677 temporal inferior, 678 superior, 678 terminal, 646 Supracapsular region, 3 Surface anatomy of abdomen, 140–141 of back, 2–3 of hand, 295 of head, 512–513 of kidney, 182, 184 of limb (lower), 408–409 of liver, 176 of neck, 512–513 of pelvis, 228–229 of stomach, 166 of thorax, 54 of upper limb, 294 of wrist, 295 Surfactant alveolar, 121 production of, 121 Sustentaculum tali, 153, 155, 156, 457, 458, 461, 462, 463, 471 Suture coronal, 542, 544, 545, 667 frontal, 544 incisive, 640 interalveolar, 640 intermaxillary, 543 lambdoid, 513, 542, 544, 545 palatine median, 546, 636, 644 transverse, 546, 636, 640 sagittal, 544, 544, 545 sphenofrontal, 542 sphenoparietal, 542 sphenosquamous, 542, 544 squamous, 542, 544 Symphysis pelvic, 251 pubic, 142, 147, 149, 228, 229, 230, 232, 233, 235, 236, 237, 238, 250, 252, 267, 287, 408, 411, 416, 418, 421, 489 in female, 258, 261 in male, 253 MRI of, 288, 289, 290 surface of, 230 Syndesmosis, tibiofibular, 432, 456, 461 fractures at, 433 Systole, ventricular, 98 T Taenia choroidea, 680 Taenia cinerea, 683 Taenia coli, 160, 164, 170, 171, 172, 242, 243, 248 Talus, 450, 451, 454, 456, 457, 458, 459, 460, 462, 463, 468 articular surfaces, 455, 459 body, 452, 453 head, 452, 453 MRI of, 509 neck, 452, 453 posterior process, 452, 453, 459 lateral tubercle, 453 medial tubercle, 453 radiograph, 508 superior trochlear surface, 455 surfaces of, 468 Tarsal bones, 452 Tarsal glands, 610 Tarsals, 410 Tarsus, 452 inferior, 608, 610 superior, 608, 610 Taste, tongue innervation and, 646 Taste buds, 569 Tectum, 564, 682 Teeth apical foramen of, radiograph of, 668 coding of, 641 dental panoramic tomogram of, 641 mandibular, 666 permanent, 640 sockets (alveoli) of, 637 structure of, 640 surfaces of, 640 Tegmentum, 680 Tegmen tympani, 571, 632 Telencephalon, 561t, 674 development of, 676t Temperature, sense of, 692, 692t Temporal bone, 18, 542, 542t, 543, 598, 619, 624, 638, 658 computed tomography (CT) of, 670 parts of, 625, 626, 630, 634 petrous, 542t, 544 squamous, 542t, 544, 622 structures of, 625 styloid process of, 553 vestibulocochlear nerve in, 571 Tendons abductor pollicis brevis, 358 abductor pollicis longus, 355, 356, 358, 394, 395 MRI of, 404 biceps brachii, 310, 322, 327, 332, 387, 389, 396, 403 long head, 304, 305, 383, 385 MRI of, 401 ultrasound of, 400 MRI of, 398, 403 biceps femoris, 445 brachialis, 322 brachioradialis, 358, 3339 MRI of, 399 calcaneal (Achilles’), 409, 445, 446, 450, 457, 463, 467, 494, 495, 500 bursa, 457 MRI of, 509 central (of diaphragm), 64, 65, 67, 73, 82, 106, 115, 116, 146, 147 attachment to fibrous pericardium, 89, 94, 95 conjoint, 143 conus, 98 digastric, intermediate, 642 extensor, dorsal compartments of, 358t extensor carpi radialis, MRI of, 399 extensor carpi radialis brevis, 334, 358, 389, 391, 394, 395 in lateral epicondylitis, 338 extensor carpi radialis longus, 334, 358, 391 MRI of, 399 extensor carpi ulnaris, 358, 391, 395 extensor digiti minimi, 358, 391, 395 Index 752 Tendons (continued) Tendons (continued) extensor digitorum, 294, 334, 353, 358, 359, 391, 394, 395, 409 dorsal digital expansion of, 334 dorsal venous network of, 294 extensor digitorum brevis, 467, 497 extensor digitorum longus, 449, 467, 497 MRI of, 503 extensor digitorum superficialis, 359 extensor hallucis brevis, 468, 497 extensor hallucis longus, 449, 467, 495, 497 MRI of, 503 extensor indicis, 358, 391, 394 extensor indicis longus, 358 extensor pollicis brevis, 358, 391, 394 extensor pollicis longus, 358, 389, 391, 394, 395 extensor retinaculum, 394, 395 extensors, superficial, common head of, 311 fibularis brevis, 448, 465, 497 fibularis longus, 448, 463, 465, 466, 471 groove for, 453 MRI of, 503, 509 tunnel for, 458 fibularis tertius, 449 flexor carpi radialis, 356 MRI of, 404 passage for, 350 flexor carpi ulnaris, 349, 356 flexor digitorum brevis, 464, 465, 498, 499 flexor digitorum longus, 451, 465, 471, 485, 498, 499 flexor digitorum profundus, 332, 333, 352, 353, 354, 355, 356, 357, 363, 389, 391, 392 MRI of, 404, 405 flexor digitorum superficialis, 352, 353, 354, 355, 356, 357, 388, 389, 391, 392 MRI of, 404, 405 flexor hallucis longus, 451, 464, 465, 485 groove for, 459 flexor pollicis longus, 332, 333, 354, 355, 356 MRI of, 405 flexor pollicis profundus, MRI of, 405 flexors, superficial, common head of, 310 gracilis, 428 MRI of, 503 iliopsoas, 473 intertubercular, sheath of, 306, 307 palmaris longus, 294, 354, 390, 392 pes anserinus, 418, 419, 421, 422, 431, 444 plantaris, 446, 450, 495 MRI of, 503 quadriceps, 435, 442, 443 MRI of, 507 quadriceps femoris, 418, 430, 438 rectus femoris, 436 MRI of, 502 sartorius, MRI of, 503 semimembranosus, 431, 495 semitendinosus, MRI of, 503 stapedius, 629, 630 in conductive hearing loss, 631 subscapularis, ultrasound of, 400 superior oblique, 604 supraspinatus, 307 MRI of, 401 rotator cuff injury of, 307 tensor tympani, 629, 631 tibialis anterior, 466, 497 posterior, 451, 465, 466, 471 groove for, 4 MRI of, 503 triceps brachii, 323 triceps surae, MRI of, 503 vastus intermedius, 436 Tendon sheath of ankle, 467 common flexor, 354 digital, of thumb, communications of, 355 dorsal carpal, 358 flexor, blood supply to, 392 flexor digitorum profundus, 359 flexor digitorum superficialis, 359 “Tennis elbow,” 338 Tentorium cerebelli, 549, 590, 591, 592, 660, 662 Testes, 153, 253, 253t, 255, 255t, 265, 278, 280 appendix of, 265 asymmetric venous drainage of, 271 coverings of, 154 in situ, 265 lymphatic drainage of, 277 mediastinum of, 155, 265 MRI of, 290 MRI of, 290 rete, 155, 265 MRI of, 290 Thalamus, 563, 677, 680, 681, 682, 689, 692, 699 medullary stria of, 562 MRI of, 700, 701 Thigh bones of, 410–411 crease of, lateral, 229 muscles of anterior, 418, 420, 430, 430t medial, 421 superficial layer of, 428, 428t posterior, 422, 424, 431, 431t regions of, 408 topography anterior, 492 medial, 492 posterior, 493 Thoracic cage, joints of, 60–61 Thorax. See also individual organs cavity of arteries of, 89 computed tomography (CT) of, 136–137 divisions of, 78 lymphatics of, 84–85, 110 nerves of, 86–87, 87t pleural cavities, 55 during respiration, 60 chest tube insertion in, 72 muscles, 54 nerves of parasympathetic nervous system, 87, 87t sympathetic nervous system, 87, 87t palpable structures of, 54 regions of, 54 segments of, 57, 57t skeletal structure of, 56–57 surface anatomy of, 54 walls of, 56–57 anterior, 85 arteries of, 68, 68t muscles of, 62–63, 62t, 70–71 nerves of, 60, 70 neurovasculature of, 72–73, 85 posterior, 85 veins of, 69 Thromboembolism, 125 Thumb carpometacarpal joint of, 346, 347, 350 movement of, 347 digital tendon sheath of, communications of, 355 interphalangeal joint of, 346 lymphatic drainage of, 367 metacarpophalangeal joint of, 346 palmar digital nerve, 392 Thymus, 78, 88t atrophy of, 89 retrosternal fat pad of, 88, 90 Tibia, 409, 410, 421, 427, 428, 429, 432, 435, 436, 437, 441, 443, 444, 454, 456, 457, 458, 460, 461, 467, 495, 496, 497, 500 borders of, 432 head of, 432 MRI of, 503, 509 radiograph, 506, 508 shaft, 432, 449 surfaces articular, 456 lateral, 432 Index 753 Trunk medial, 408, 432, 438 posterior, 451 Tissue. See also Connective tissue of abdomen wall fatty, 153 membranous, 146, 150, 153 adipose, of the orbit, 610 erectile, female, 263 lymphatic, 650 subcutaneous, 387 of shoulder, 307 superficial membranous later of, 155 Toe, great metatarsophalangeal joint of, 462 proximal phalanx of, 462 Tongue, 5, 577, 648, 656, 657, 660, 661, 666, 667 apex of, 646, 647 body of, 646 dorsum of, 646 lymphatic drainage of, 522 MRI of, 668 muscles of, 646 neurovasculature of, 647 root of, 646, 651 structure of, 646 Tonsil, 682 cerebellar, 699 lingual, 529, 646, 648, 650, 650t palatine, 648, 650, 650t, 658, 661 enlarged, 651 veins of, 589t pharyngeal, 620, 628, 648, 650, 650t, 651, 653 tubal, 648, 650, 650t Tonsil infections, 651 Torus tubarius, 620, 648 with lymphatic tissue, 650 Touch, sense of, 692, 692t Trabeculae, arachnoid, 590 Trachea, 5, 10, 25, 78, 80, 81, 86, 88t, 90, 92, 95, 103, 107, 108, 109, 110, 111, 115, 120, 123, 124, 126, 127, 128, 129, 132, 526, 530 bifurcation of, 129 conditions associated with, 120 membranous wall of, 120, 529 parts of cervical, 79, 88, 106, 120 thoracic, 79, 88, 106, 120 Tract ascending (sensory), 691, 692, 692t biliary, in situ, 179 corticospinal anterior, 693, 693t lateral, 693, 693t corticospinal (pyramidal), 693 693t descending (motor), 691 from brainstem, 693, 693t gastrointestinal, autonomic nervous system effects of, 212t iliopubic, 151, 152 iliotibial, 146, 409, 418, 419, 420, 422, 425, 426, 427, 444, 445, 446, 483, 492, 493, 496, 500, 501 MRI of, 502, 505 olfactory, 549, 562 olivospinal, 693t optic, 563, 681, 699 pinocerebellar, anterior, 682 pyramidal, 693 reticulospinal, 693t rubrospinal, 693t solitary, nucleus of, 561, 568, 569 inferior part of, 572, 574, 574t superior part of, 572, 574, 574t spinocerebellar anterior, 692, 692t posterior, 692, 692t spinothalamic, 692 anterior, 692, 692t lateral, 692, 692t tectospinal, 693t tegmental, central, 682 ventricular outflow left, 131, 135 right, 131 vestibulospinal, 693t Tragus, 512, 627 Trapezium, 337, 342, 342t, 344, 345, 346, 347, 350, 361, 391, 394 MRI of, 404, 405 radiograph of, 343 tubercle of, 295, 337, 343, 344, 345, 350 Trapezoid, 342, 342t, 343, 344, 345, 346, 347, 350, 363, 404 MRI of, 404 radiograph of, 343 Tree, bronchial, 118, 121 conditions associated with, 121 divisions of, 121 lymphatics of, 128 neurovasculature of, 126–127 in respiration, 123 respiratory portion of, 121 Triad, portal, 169, 181 Triangle anal, 229 carotid, 512, 532, 532t, 537 cervical anterior, 532, 532t, 534 posterior, 532 clavipectoral, 54, 294 “danger,” 606 Einthoven’s, 102 Hesselbach’s, 152, 153 iliolumbar (of Petit), 47 Killian’s, 107 lumbar, 3, 24 fibrous lumbar (of Grynfeltt), 47 lumbosacral (Bochdalek’s), 64, 65 muscular (omotracheal), 512, 532, 532t occipital, 532, 532t omoclavicular, 532, 532t Philippe-Gombault, 691 sternocostal, 65 submandibular, 512, 532, 532t submental, 512, 532, 532t suboccipital, 541 urogenital, 229, 246 Trigone collateral, 685 femoral, 408 fibrous, 98 hypoglossal, 577, 683 olfactory, 562 of urinary bladder, 252, 253 of vagus nerve, 683 Triquetrum, 295, 342, 342t, 343, 344, 345, 346, 350, 391, 404 radiograph of, 343 Trochanter greater, 2, 408, 410, 411, 412, 413, 415, 416, 417, 422, 424, 427, 429, 430, 473 MRI of, 505 radiograph of, 504 lesser, 410, 414, 416, 417, 427, 429, 430 radiograph of, 504 Trochlea, 300, 326, 565, 604 femoral, 413 humeral, 327 of humerus, 301 MRI of, 403 radiograph of, 402 of talus, radiograph, 508 Trunk arterial brachiocephalic, 36, 78, 80, 81, 81t, 84, 86, 90, 92, 93, 96, 103, 106, 108, 109, 116, 125, 126, 127, 130, 250, 364, 369, 536, 575, 655 celiac, 66, 67, 80, 84, 109, 110, 156, 162, 168, 179, 181, 186t, 187, 188, 190–191, 196, 197, 198, 199, 250 branches of, 187, 190 with celiac lymph nodes, 206 computed tomography (C) of, 224 distribution of, 187, 196 to pancreas, duodenum, and spleen, 187 cervicofacial, inferior, 596 Index 754 Trunk (continued) Trunk (continued) coronary left, 110 right, 110 of corpus callosum, 698, 699 costocervical, 36, 81, 365, 520 of internal iliac artery, 250 MRI of, 135 pulmonary, 81, 89, 92, 93, 94, 96, 97, 102, 103, 104, 105, 106, 116, 124, 125, 131, 132 computed tomography (CT) of, 134, 136 MRI of, 131 thyrocervical, 80, 81t, 109, 364, 365, 383, 520, 531, 535, 536, 665 left, angiography of, 135 right, angiography of, 135 lymphatic bronchomediastinal, 84, 128 intestinal, 202, 203, 204, 207, 208 jugular left, 85, 110, 129 right, 85, 129 lumbar left, 85, 129, 202, 204, 205, 208, 279f right, 85, 129, 202, 204, 279f lumbosacral, 280, 281, 477 subclavian left, 85, 129 right, 85, 129 nerve of anterior trunk cutaneous innervation of, 210 dermatomes of, 210 brachial plexus lower, 369 middle, 369 upper, 369 sympathetic, 42, 43, 67, 86, 87, 87t, 90, 91, 103, 108, 113, 130, 211, 212, 214, 216, 280, 283, 535, 537, 654, 694, 697 computed tomography (CT) of, 137 left, 115 lumbar ganglia of, 213, 280, 282 in mediastinum, 91 right, 115, 131 sacral ganglia of, 282 vagal, 87, 108 anterior, 95, 108, 130, 214, 216, 217, 218 anterior, hepatic branch of, 214, 216, 218 anterior, pyloric branch of, 216, 218 anterior, with esophageal plexus, 108 posterior, 85, 109, 214, 215, 216, 217, 218 posterior, celiac branch of, 214, 216 posterior, hepatic branch of, 214, 216 posterior, pyloric branch of, 214 Tube ovarian, 272 pharyngotympanic (auditory), 571, 573, 625, 626, 628, 629, 632, 633, 660, 661 parts of, 628 pharyngeal orifice of, 620, 621, 648 uterine, 243, 244, 247, 254, 254t, 256, 257, 258, 273, 281 ampulla of, 257 ectopic pregnancy and, 257 isthmus of, 255, 257, 257 left, 251, 257, 261 lymphatic drainage of, 277 proximal part of, 277 right, 252, 254 uterine part of, 257 Tuber cinereum, 680, 681 Tubercle adductor, 412, 420 articular, 542, 546, 624, 638, 639 of atlas (C1) anterior, 8 posterior, 8, 18, 30, 519 cervical anterior, 7, 8, 9 posterior, 9 conoid, 298 corniculate, 528, 529, 651 costal, 56, 57, 59, 61 criniculate, 655 cuneiform, 528, 529, 651, 655 dorsal, of radius, 324, 330, 331, 334, 335, 341, 344, 345, 357 greater, of humerus, 2, 54, 295, 300, 303, 304, 305, 309, 317, 318, 322, 323, 370 crest of, 317 MRI of, 401 ultrasound of, 400 iliac, 232, 233, 236 infraglenoid, 299, 304 intercondylar lateral, radiograph of, 506 medial MRI of, 507 radiograph of, 506 lesser, of humerus, 54, 295, 300, 303, 304, 306, 317, 318, 322, 328 crest of, 317, 321 ultrasound of, 400 mammillary, 677 medial, 463 mental, 543, 637 of nucleus cuneatus, 683 of nucleus gracilis, 683 pharyngeal, 546 postglenoid, 542 pubic, 142, 149, 150, 228, 230, 231, 232, 236, 238, 262, 408, 410, 414, 416 of ribs, facet for, 61 of scalene, 519 of scaphoid, 295, 345, 350 superior, of thryoid cartilage, 526 supraglenoid, 299, 304, 322 of talus posterior process lateral, 453, 459 medial, 453 of transverse process anterior, 16, 17, 20, 31 posterior, 16, 17, 20 of trapezium, 295, 337, 343, 344, 345, 350 Tuberculum sellae, 551 Tuberosity of 5th metatarsal, 408, 410, 448, 451, 452, 453, 455, 456, 462, 464, 465, 467, 468, 471 calcaneal, 408, 450, 451, 452, 453, 456, 459, 462, 464, 469 lateral process, 452, 459 medial process, 452, 453, 459 posterior, radiograph of, 508 of cuboid, 451, 453, 469 deltoid, 300, 316 of distal phalanges, 343, 347, 353 gluteal, 427, 431 greater, 317 humeroradial, 403 iliac, 142, 230, 233, 237 ischial, 2, 143, 229, 230, 231, 232, 235, 236, 237, 238, 239, 263, 284, 285, 287, 408, 411, 416, 427, 431, 489, 490, 500 gender-specific features of, 234t MRI of, 288 radiograph of, 504 maxillary, 598, 622 navicular, 408, 455 radial, 311, 322, 324, 325, 326, 329, 330, 337, 403 radiograph of, 402 radioulnar, proximal, 403 sacral, 12, 13, 237 tibial, 408, 410, 425, 429, 430, 432, 433, 434, 435, 436, 438, 441, 444, 449 MRI of, 503 radiograph of, 506 ulnar, 322, 324, 326, 329, 330, 337 Tunica albuginea, 155, 265, 275 MRI of, 290 Tunica dartos, 265 Tunica vaginalis cavity of, 155 parietal layer of, 154, 155, 155t, 265 visceral layer of, 154, 155, 155t, 265 Index 755 Veins (individual) Tunnel carpal, 295, 344, 350 apertures of, 391 bony boundaries of, 350 ligaments of, 350 MRI of, 405 roof of, 295 structures in, 391 walls of, 391 for fibularis longus tendon, 458 radial, 373, 387 tarsal, 495 ulnar, 295, 350, 388, 390, 391, 392 U Ulcerative colitis, 173 Ulcers, gastric, 156, 167 Ulna, 296, 297, 323, 325, 326, 327, 328, 329, 339, 341, 342, 344, 347, 350, 357, 363, 375, 396, 397 borders of, 325, 330, 341 head of, 324, 325, 330, 336, 343 MRI of, 399, 405 radiograph of, 402 shaft of, 295, 324, 325, 330 styloid process of, 295, 324, 330, 342, 343, 344, 347, 404 Umbilicus, 104, 105, 144, 145, 146, 152 computed tomography (CT) of, 233 Umbo, 630 Unconscious coordination, 692 Uncus, 660 Upper limb. See Limb (upper) Urachus, obliterated, 160, 164, 170, 244, 245 Ureters, 182, 183, 184, 196, 221, 248, 250–251, 266 anatomical constrictions of, 251 autonomic nervus system effects on, 215 distal, computed tomography (CT) of, 224 in female, 251, 259 in situ, 250 left, 165, 182, 183, 188, 189, 199, 269, 270, 272, 273 in female, 252 in male, 251, 253 lymphatic drainage of, 205 in male, 250, 251, 255 MRI of, 291 parts of abdominal, 250, 282 pelvic, 250, 282 relations of, 273 right, 165, 182, 183, 184, 185, 197, 242, 243, 247, 257, 270, 271, 272 computed tomography (CT) of, 223, 224 in female, 252 intramural part of, 252 in male, 251 passage through broad ligament of uterus, 251 upper, 215 Ureter(s), 165, 182, 183, 184, 185, 188, 189, 196, 197, 199, 221 lymphatic drainage of, 205 computed tomography (CT) of, 223 Urethra, 258, 260, 266, 267 female, 261 in female, 196, 252 lymphatic drainage of, 276 in male, 253 membranous part of, 247, 264, 266 MRI of, 288, 289, 290, 291 mucosa of, 252 preprostatic part of, 264 prostatic part of, 253, 264, 266, 287 spongy part of, 246, 253, 255, 264, 265, 266 submucosa of, 252 Uterus, 243, 244, 246, 254, 254t, 256–257, 258, 279, 281 autonomic nervous system effects on, 212t axis of, 256 body of, 252, 256, 257, 260 MRI of, 288, 289 cavity of, 257 MRI of, 289 cervix of, 247, 252, 254 embryonic remnants of, 257 fundus of, 244, 247, 251, 252, 257, 260, 261, 273 lymphatic drainage of, 277 junctional zone of, 289 ligaments of, 243, 244 broad, 251, 256 round, 247, 252, 258, 261 suspensory, 247, 251, 252 normal curvature and position of, 256 posterior surface of, 251, 256 visceral peritoneum on, 260 Utricle, 571, 634 prostatic, 253 Uvula, 628, 643, 648, 650, 651, 658, 659, 667, 698 of urinary bladder, 252 V Vagina, 196, 243, 244, 246, 252, 257, 260–261, 281 anterior, fascia over, 258 anterior wall of, 257, 260 axis of, longitudinal, 256 commissure of, posterior, 229 fornix of anterior, 252, 256 posterior, 252, 256 location of, 260 lymphatic drainage of, 205, 277 MRI of, 288, 289 orifice of, 229 relations of, 260 structure of, 260 upper portion of, 254t vestibule of, 246, 247, 260, 262 walls of anterior, 260 MRI of, 289 posterior, 260 Vallecula, 659, 660, 682 Valves anal, 249 of heart. See Cardiac valves of Kerckring, 168 Varices, esophageal, 199 Varicoceles, 271 Vasa recta, 192 Vault, plantar, 462 Veins (individual) alveolar, inferior, 657 anastomotic inferior, 686 superior, 686 angular, 521, 588, 589, 589t, 594, 606, 608 antebrachial, median, 366, 378, 387 anular, 595 appendicular, 195, 200 arcuate, 184 auditory, internal, 635 auricular, posterior, 521, 588, 589t axillary, 69, 72, 74, 194, 366, 383, 384, 384t, 385, 386 MRI of, 398, 401 azygos, 37, 45, 66, 67, 69, 73, 79, 82, 84, 88, 89, 90, 109, 111, 113, 115, 127, 139, 194, 195 arch of, 106 branches of, 83t computed tomography (CT) of, 136 basilar, 76, 686, 687 basilic, 378, 387 median, 366, 387 MRI of, 398, 399 basivertebral, 45 Boyd’s, 475 brachial, 366, 384, 385, 397 deep MRI of, 398 MRI of, 398, 399 brachiocephalic, 66, 69, 82, 83t left, 37, 45, 78, 82, 83, 84, 88, 89, 90, 93, 106, 109, 110, 116, 124, 127, 130, 521, 531, 588 right, 37, 78, 82, 83, 84, 90, 92, 93, 106, 109, 110, 124, 127, 130 Index 756 Veins (individual) (continued) Veins (individual) (continued) bridging, 590, 592, 686 MR angiography of, 671 ostia of, 590 bronchial, 127 cardiac, 100, 100t great, 100, 100t middle (posterior inerventricular), 100, 100t small, 100, 100t cecal, 200, 201 cephalic, 69, 72, 194, 294, 366, 378, 382, 383, 384, 387 accessory, 366, 378 in deltopectoral groove, 294 median, 366 MRI of, 398, 399 cerebellar MRI of, 700 superior, 686, 687 lateral, 686 medial, 686 cerebral, 590 anterior, 686, 687 great (of Galen), 592, 686, 687 MR angiography of, 671 MRI of, 700 inferior, 592 internal, 686, 687, 699 MR angiography of, 671 MRI of, 700 middle, 592, 686 deep, 687 superficial, 687 superficial, 686 superior, 590, 592 cervical deep, 521 superficial, 538, 539 choroidal, inferior, 687 clitoral deep, 261, 275 dorsal, deep, 272 colic, 195 left, 165, 169, 181, 183, 195, 201 middle, 160, 162, 170, 195, 198, 199, 200, 201 right, 195, 200, 201 communicating, anterior, 687 cremasteric, 271 Crockett’s, 475 cubital, median, 294, 366, 378, 387 deep, 387 MRI of, 399 cutaneous dorsal intermediate, 486 medial, 486 femoral anterior, 474 lateral, of the thigh, 486, 488, 489 posterior, of the thigh, 486, 490 superficial, of upper limb, 378 sural lateral, 486 medial, 486 cystic, 195 deep, of thigh, 474, 505 MRI of, 398 perforating veins, 502 digital dorsal, 366, 378 palmar, 366 plantar, 474 diploic, 545, 590 Dodd’s, 475 emissary, 37, 545, 592 condylar, 589, 661 posterior, 663 mastoid, 589 occipital, 589 parietal, 589 epigastric inferior, 151, 152, 153, 160, 164, 170, 194, 196, 250, 259, 271 with lateral umbilical fold, 245 superficial, 72, 195, 474, 486, 488 superior, 194, 195 episcleral, 614 esophageal, 195, 198 facial, 521, 523, 531, 537, 588, 594, 595, 598, 606, 608, 649 deep, 589 femoral, 69, 150, 151, 154, 158, 194, 196, 271, 286, 287, 475, 486, 488, 489, 492, 501 in adductor canal, 492 anterior, 474 anterior cutaneous, 488 circumflex medial, 474 MRI of, 502 deep MRI of, 502 MRI of, 288, 502, 505 in saphenous opening, 474 femoropopliteal, 474 fibular, 496 MRI of, 503 gastric, 109 left, 195, 198, 199, 201 right, 195, 198, 199, 200, 201 short, 195, 199 gastroomental left, 195, 198, 199 right, 195, 198, 199, 200 genicular, 474 gluteal inferior, 196, 490, 491 right, 270, 274 superior, 196, 493 right, 274 hemiazygos, 37, 45, 66, 67, 69, 82, 83, 84, 91, 109, 113, 115, 127, 130, 194, 195 accessory, 37, 45, 66, 69, 82, 83, 83t, 84, 91, 109, 127 hepatic, 165, 169, 177, 179, 181, 183, 195, 198, 199 branches of, 177 intermediate, 177 humeral circumflex posterior, MRI of, 401 ileal, 195, 200 ileocolic, 195, 200, 201 iliac, 252 circumflex deep, 152, 196, 271 superficial, 194, 474, 486, 488 common, 37, 45, 69, 156, 194, 195, 196, 201 computed tomography (CT) of, 224 left, 252, 270, 273 MRI of, 291 right, 165, 195, 201, 242, 243, 274 external, 37, 45, 69, 152, 196, 244, 247, 248, 271, 474, 475, 492 left, 251, 273 right, 261, 270, 272 internal, 37, 45, 196, 261, 271, 273, 475 MRI of, 288 superficial, 69 infraorbital, 606, 666 intercapitular, 366, 378 intercostal, 45, 69, 71, 72, 75, 90, 91, 113, 220 anterior, 37, 69, 83tt, 194 posterior, 37, 47, 66, 69, 73, 82, 83, 83t, 90, 91, 109, 194 superior left, 83, 91 right, 83t supreme, 82, 83t interlobar, of kidney, 184 internal anterior division of, 272 anterior trunk of, 250 left, 250, 273 right, 250, 270, 274 interosseous, anterior, 366 MRI of, 399 jejunal, 195, 200, 201 jugular anterior, 521, 534, 588, 588t MRI of, 669 Index 757 Veins (individual) external, 66, 69, 72, 83t, 109, 382, 383, 521, 534, 535, 538, 539, 588, 588t, 595, 596, 655, 664 left, 109 MRI of, 669 internal, 25, 37, 66, 69, 76, 78, 83, 84, 109, 110, 111, 124, 383, 521, 522, 523, 530, 531, 534, 535, 536, 537, 571, 588, 588t, 589, 625, 628, 629, 654, 655, 663, 664, 665, 686 junction with subclavian vein, 110 left, 82, 84, 85, 89, 92, 106, 109, 111, 124 MR angiography of, 671 MRI of, 669, 701 right, 37, 45, 69, 124, 129 labial, inferior, 588 labyrinthine, 688 lacrimal, 606 laryngeal external, 534 inferior, 531, 655 recurrent, left, 655 superior, 531 foramen for, 527 lingual, 521, 522, 666 deep, 647, 657, 660 lumbar, 37, 69, 82, 83, 194, 196 ascending, 37, 45, 67, 82, 83, 194, 196, 199 marginal left, 100 right, 100 maxillary, 521, 588, 589 medullary posteromedian, 686, 687 transverse, 687 meningeal, middle, 592 mesenteric, 270 inferior, 195, 199, 200, 201, 274 computed tomography (CT) of, 222, 224 superior, 168, 169, 179, 180, 181, 183, 185, 191, 195, 199, 200, 201, 220, 221 computed tomography (CT) of, 223 metacarpal, palmar, 366 metatarsal, plantar, 474 musculocutaneous, MRI of, 83t, 398 musculophrenic, 194 nasal dorsal, 606, 608 external, 595 obturator, 196, 273, 286 MRI of, 505 right, 270, 272, 274 occipital, 521, 540, 588, 589, 589t ophthalmic, 606 inferior, 521, 588, 606, 608, 609 superior, 521, 588, 589, 592, 606, 608, 609, 666 ovarian, 83, 256, 259 left, 196, 273 in ovarian suspensory ligament, 251 right, 183, 185, 195, 272 with suspensory ligament of ovary, 252 pancreatic, 195 pancreaticoduodenal, 195, 198, 199, 200 inferior, 195 posterior superior, 195 paraumbilical, 152, 195 peduncular, 687 penile, 275 bulb of, 269 dorsal deep, 265, 269, 271, 275, 284 superficial, 265, 267, 275 perforator, 366, 387 pericardiacophrenic, 67, 73, 78, 83t, 90, 91, 94, 115 pericardial, 83, 83t perineal, 194, 195, 270, 272, 276 periumbilical, 69, 194, 195 petrosal, 686 superior, 687 phrenic, 93 anastomosis with left suprarenal vein, 185, 196, 197 inferior, anastomosis with left suprarenal vein, 185 superior, 83t plantar lateral, 474, 498, 499 medial, 474 MRI of, 509 pontine anterolateral, 687 anteromedian, 687 pontomesencephalic, 687 popliteal, 493, 494, 495 MRI of, 503, 507 portal, 92, 104, 105, 165, 176, 177, 178, 181, 183, 185, 191, 192, 194, 195, 198, 201 anastomosis between umbilical vein, 104 branches of, 177 computed tomography (CT) of, 222 distribution of, 195 to pancreas and spleen, 199 to stomach and duodenum, 198 pudendal, 69, 246 external, 194, 271, 275, 474, 486, 488 internal, 196, 248, 270, 275, 284, 285, 287, 491 left, 270, 272 right, 274 pulmonary, 90, 92, 95, 103, 124–125, 125t, 131 branches of, 117 computed tomography (CT) of, 134, 136, 137 inferior, 124, 125, 125t left, 83, 89, 91, 92, 94, 96, 97, 104, 124 with pulmonary plexus, 103 right, 83, 89, 90, 94, 96, 97, 106, 116, 124, 133 computed tomography (CT) of, 137 inferior, 124, 125, 215t MRI of, 135 superior, 124, 125, 125t superior, 100, 124, 125, 125t radial, 366 MRI of, 399, 403, 404 radicular anterior, 45 posterior, 45 rectal of external venous plexus, 274 inferior, 195, 269, 284, 285 left, 270, 272 right, 274 middle, 195 left, 270, 272 right, 270, 272, 274 superior, 195, 201, 270, 272, 274 renal, 83, 156, 183, 184, 185, 192, 195, 196, 197, 199 computed tomography (CT) of, 222 left, 250 right, computed tomography (CT) of, 224 tributaries of, 197t retinal, central, 613, 613 physiological, 613 retromandibular, 521, 588, 589 anterior division of, 589 posterior division of, 595, 598 sacral lateral, 196 right, 270 median, 196, 250, 273, 274 saphenous, 494 accessory, 474, 486 great, 69, 194, 475, 486, 488, 494, 496 MRI of, 502, 503 small, 474, 486, 494, 495, 496 MRI of, 503 scapular, 592 circumflex, MRI of, 398 scrotal anterior, 275 posterior, 269, 270 Index 758 Veins (individual) (continued) Veins (individual) (continued) segmental, 184 sigmoid, 195, 201, 274 splenic, 156, 165, 180, 181, 185, 191, 195, 198, 199, 200, 201, 220 computed tomography (CT) of, 222, 224 subclavian, 66, 69, 74, 83, 83t, 90, 109, 111, 194, 195, 382, 383, 521, 531, 535, 588 left, 78, 82, 84, 85, 89, 91, 92, 106, 109, 110, 111, 116, 124 junction with internal jugular vein, 110 right, 37, 45, 82, 96, 97, 106, 116, 124, 133, 539 junction with internal jugular vein, 110 subcostal, 37, 45, 69, 194 submental, 588, 647 subscapular, 366 MRI of, 401 sulcal, 45 supraorbital, 588 suprarenal, 185 anastomosis with inferior phrenic vein, 185 left, 183, 196, 197, 199, 250 right, 183, 184, 196, 197, 250 suprascapular, 588 MRI of, 401 supratrochlear, 588, 606 sural, 494 temporal, 598 deep, 588, 589 superficial, 521, 588, 589, 589t, 594, 598, 599, 649 branches of, 595 testicular, 152, 250, 265, 271 left, 250 right, 83, 183, 185 computed tomography (CT) of, 223 thalamostriate, 680, 686 thoracic internal, 69, 72, 73, 78, 83t, 89, 114, 115, 130, 194, 195 right, 82 lateral, 74, 76 thoracoepigastric, 69, 72, 194 thymic, 83t thyroid inferior, 66, 82, 83, 83t, 109, 127, 531, 535, 536 middle, 531, 536 superior, 521, 531, 536, 588, 664, 665 tibial, 501 anterior, 496, 497 inferior, 474 MRI of, 503 posterior, 456, 474, 475, 501 MRI of, 503 transsphincteric, 274 ulnar, 366 MRI of, 398, 399, 403, 404 umbilical, 104 aastomosis between portal vein, 104 adult remnant of, 105t obliterated, 105 uterine, 196, 273 left, 272 right, 272 vertebral, 40, 521, 665 MRI of, 669 right, 45 vesical inferior, 196 in female, 196 left, 272 right, 272 superior, 273 left, 270, 272 right, 270 vorticose, 613 Veins (regional) of abdomen, 194–195 of back, 37. See also Cord, spinal, veins of basal cerebral venous system, 687 of brain, 686–687 of brainstem, 687 of esophagus, 109, 109t of head, 588–589 deep, 589, 589t superficial, 588, 588t of larynx, 531 of limb (lower), 474–475 perforating veins, 475 superfical, 488 superficial, 486 of limb (upper), 366 superficial cutaneous, 378 of lungs, 124–125, 125t of neck, 521 of orbit, 606 of palatine tonsil, 589t of pelvis in female, 272–273 in male, 270–271 of penis, 269 of rectum, 274 of round ligament, 150 of septum pellucidum, 686 of spinal cord, 45 of thoracic wall, 69 of vestibular aqueduct, 635 of vestibular bulb, 276 Velum, medullary inferior, 698 superior, 682, 683, 698 Vena cava inferior, 25, 37, 45, 48, 65, 67, 69, 73, 79, 82, 83 92, 94, 95, 97, 100, 104, 111, 124, 147, 163, 165, 166, 168, 169, 174, 175, 176, 178, 179, 181, 182, 183, 188, 192, 193, 195, 196, 197, 199, 200, 206, 211, 220, 221, 250, 269, 273, 274, 277, 475 in caval opening, 89 computed tomography (CT) of, 137, 224 groove for, 177 location of, 194 MRI of, 48 tributaries of, 194t valve of, 97 superior, 37, 45, 66, 69, 78, 79, 82, 83, 84, 89, 90, 92, 93, 94, 95 96, 97, 100, 102, 103, 104, 105, 109, 110, 111, 116, 124, 127, 131, 132, 194, 195 computed tomography (CT) of, 136 MRI of, 131, 135 thoracic tributaries of, 83t Venipuncture, 366 Ventricles of brain 3rd, 679, 680, 684, 685, 689, 698, 699 arrangement of diencephalon around, 680 MRI of, 700 4th, 566, 667, 682, 683, 698 MRI of, 669, 700 cerebrospinal fluid circulation and, 684 lateral, 563, 658, 660, 662, 680, 684, 689 central part, 698, 699 choroid plexus of, 698, 699 left, 685 MRI of, 701 right, 685 lateral aperture of, 685 of heart disastole in, 98 left, 89, 92, 93, 96, 97, 100, 101, 102, 104, 105, 110, 124, 130, 131, 132 computed tomography (CT) of, 137 right, 92, 93, 96, 97, 100, 101, 104, 105, 110, 124, 130, 131, 132 anterior veins of, 100 computed tomography (CT) of, 137 MRI of, 135 systole in, 98 of larynx, 529 of oral cavity, 648 Ventricular septal defects (VSDs), 105 Ventricular system, 685 in situ, 685 Vermis, 662, 682 of cerebellum, 699 Index 759 Zygomatic bone Vertebrae/vertebra body of, 6, 7, 7t, 8, 9, 10, 11, 14, 17, 22, 23, 56, 65, 147 computed tomography (CT) of, 224 joints of, 16t MRI of, 21 cervical, 4, 6, 7, 8, 17 arthrosis of, 17 C1. See Atlas (C1) C2. See Axis (C2) C3, 659 C4, 525 C5, 4, 6, 39, 368, 369, 664 C5-C7, anterior divisions of, 369 C6, 4, 6, 7t, 25, 369 body of, 664, 665, 669 C7. See Vertebra prominens C8, 369 anterior root, 535 C8-T1, anterior divisions of, 369 MRI of, 48, 49 pedicle of, 7, 9, 50 radiograph of, 48, 49 structural elements of, 7, 7t typical, 8 lumbar, 4, 6, 7t, 11, 67 L1, 4, 5, 6, 11, 40, 41, 477, 525 L2, 4, 6, 25, 222, 696, 697 L3, 4, 6, 223, 696, 697 L4, 4, 6, 194, 195, 220, 411, 416, 480, 481 MRI of, 48, 51, 289, 505 radiograph of, 48, 50 spinous process of, 236, 425 structural elements of, 7, 7t typical, 7t L5, 4, 6, 40, 147, 156, 223, 242, 243, 252, 268, 280, 281, 283, 416, 427, 477 spinous process of, 237 sacral. See sacrum thoracic body of, 7, 7t, 147 C5-T1, posterior divisions of, 369 MRI of, 38, 133 radiograph of, 48, 49 structural elements of, 7t, 47 T1, 4, 10, 368, 369, 665 anterior root, 535 T1-T12, 6 T2, transverse process of, 665 T3, body of, 374 T8, 65, 147 T12, 4, 6, 10, 40, 48, 65, 133, 147, 220, 222, 477 body of, 309 typical, 7t, 10 Vertebral column. See also Spine abnormal curves of, 5 bones of, 4, 6 developmental stages of, 41 inflection points of, 411 joints of, 16–19, 16t ligaments of, 20–22, 20t midsagittal section of, 4, 5 regions of, 3, 4 vertebrae comprising, 4 Vertebra prominens (C7), 2, 3, 3t, 6, 8, 21, 30, 40, 308, 369, 370, 513 MRI of, 21, 48 posterior arch of, 369 as posterior landmark in back, 3t radiograph of, 8 spinous process of, 3, 5, 33, 35 Vesicle, seminal, 280 Vestibular apparatus, 634 Vestibule, 254, 625, 626, 628 bulb of, 262, 263 labial inferior, 660 superior, 660 laryngeal, 529t, 664 nasal, 620 of omental bursa, 162, 163, 174 oral, 648, 648t, 656 of vagina, 246, 247, 260, 262 with labium minus, 260 Vibration, sense of, 692, 692t Villi, arachnoid, 590, 592 Vincula brevia, 359, 392 Vincula longum, 392 Vinculum longa, 359 Visual fields lower, 563 upper, 563 Volume of lung, respiratory changes in, 123 thoracic, respiratory changes in, 123 Vomer, 543, 544, 546, 603, 616, 619, 636, 644, 656 Vulva, 254t W “Watershed” zone, 202 Whiplash injury, 9 White matter, 674, 675, 676 organization of, 691 Window oval, 571, 629, 634, 635 in conductive hearig loss, 631 round, 571, 629, 634, 635 in conductive hearing loss, 631 Wing. See Ala (wing) Wrist bones of, 342–345, 342t carpal bones, 344–345 compartments of, 350–351 joints of, 346–347 ligaments of, 350–351 MRI of, 404 radiograph of, 343 Z Zinn, arterial circle of, 613 Zona orbicularis, 473 Zygomatic bone, 512, 542, 543, 597, 619, 666 frontal process of, 542 orbital surface of, 603 temporal process of, 542 temporal surface of, 546 MedOne Access the additional media content with this e-book via Thieme MedOne. Connect with us on social media Quick Access After you successfully register and activate your code, you can find your book and additional online media at medone.thieme.com/9781684202034 or with this QR code. Key Features • Over 2,000 full-color anatomy illustrations • “Labels-on, labels-off” function that makes studying easy and fun • Ability to zoom and custom size or crop images Access additional media now! With three easy steps, unlock access to the additional media for this title via MedOne, Thieme’s online platform. 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2565	https://www.sciencedirect.com/science/article/pii/S2210740124000433	Skip to article My account Sign in View PDF Clinics and Research in Hepatology and Gastroenterology Volume 48, Issue 5, May 2024, 102322 Review Pathophysiology and management of enteric hyperoxaluria Author links open overlay panel, , , , rights and content Under a Creative Commons license Open access Abstract Enteric hyperoxaluria is a metabolic disorder resulting from conditions associated with fatty acid malabsorption and characterized by an increased urinary output of oxalate. Oxalate is excessively absorbed in the gut and then excreted in urine where it forms calcium oxalate crystals, inducing kidney stones formation and crystalline nephropathies. Enteric hyperoxaluria is probably underdiagnosed and may silently damage kidney function of patients affected by bowel diseases. Moreover, the prevalence of enteric hyperoxaluria has increased because of the development of bariatric surgical procedures. Therapeutic options are based on the treatment of the underlying disease, limitation of oxalate intakes, increase in calcium salts intakes but also increase in urine volume and correction of hypocitraturia. There are few data regarding the natural evolution of kidney stone events and chronic kidney disease in these patients, and there is a need for new treatments limiting kidney injury by calcium oxalate crystallization. Keywords Oxalate Kidney Gut Enteric hyperoxaluria Microbiota Calcium Cited by (0) © 2024 The Authors. Published by Elsevier Masson SAS.
2566	https://www.vocabulary.com/dictionary/bombardment	Bombardment - Definition, Meaning & Synonyms \| Vocabulary.com Compete in the Vocabulary Bowl: 192 countries, 50 states, countless prizes. Register now!X SKIP TO CONTENT Log inSign up Dictionary Vocabulary Lists VocabTrainer™ Word Finder All Dictionary Vocabulary Lists Random Word Featured Dictionary Vocabulary Lists More results Any words Any words 2-letter words 3-letter words 4-letter words 5-letter words 6-letter words 7-letter words 8-letter words 9-letter words 10-letter words 11-letter words 12-letter words 13-letter words 14-letter words 15-letter words Starting with Starting with Ending with Including Containing in order Advanced Search Find Word Part of speech- [x] noun - [x] verb - [x] adjective - [x] adverb Syllable range Between and Restrict to Synonym Synonym Antonym of Definition contains Rhymes with Example of Ends with Parts of Type of Find Word Random Word bombardment Add to list Share Copy link /bɑmˈbɑrdmɪnt/ /bɒmˈbɑdmɪnt/ IPA guide Other forms: bombardments When a lot of bombs fall on a city or area, it is a bombardment. But a bombardment can also mean a pummeling with lots of other things: questions, spit balls, or e-mails. During the early stages of a war, the capital city of the country under attack may suffer a severe bombardment meant to bring on a quick surrender. If there are many casualties or the citizens of the country are treated brutally by the occupying force, military officials may find themselves under a bombardment of questions and accusations about how the war was handled. Definitions of bombardment noun an attack by dropping bombs synonyms:bombing see more see lesstypes:show 4 types... hide 4 types... bombing runthat part of the flight that begins with the approach to the target; includes target acquisition and ends with the release of the bombs area bombing, carpet bombing, saturation bombingan extensive and systematic bombing intended to devastate a large target dive-bombinga bombing run in which the bomber releases the bomb while flying straight toward the target loft bombing, toss bombinga bombing run in which the bomber approaches the target at a low altitude and pulls up just before releasing the bomb type of:attack, onrush, onset, onslaught(military) an offensive against an enemy (using weapons) noun the heavy fire of artillery to saturate an area rather than hit a specific target synonyms:barrage, barrage fire, battery, shelling see more see lesstype of:fire, firingthe act of firing weapons or artillery at an enemy noun the act (or an instance) of subjecting a body or substance to the impact of high-energy particles (as electrons or alpha rays) see more see lesstype of:radiationthe act of spreading outward from a central source noun the rapid and continuous delivery of linguistic communication (spoken or written) “a bombardment of mail complaining about his mistake” synonyms:barrage, onslaught, outpouring see more see lesstype of:language, linguistic communicationa systematic means of communicating by the use of sounds or conventional symbols Cite this entry Style: MLA MLA APA Chicago "Bombardment."Vocabulary.com Dictionary, Vocabulary.com, Accessed 29 Sep. 2025. Copy citation Try a free round of VocabTrainer, vocabulary learning that adapts to you! The Vocabulary Bowl Schools win state, national, and global awards Register now Examples from books and articles All sources All sources Fiction Arts / culture News Business Sports Science / Med Technology loading examples... Further, as a precautionary measure for those in the occupied territories, everyone living within a zone of twenty miles from the coast was warned to prepare for bombardments. The Diary of a Young Girl by Anne Frank The first bombardment showed us our mistake, and under it the world as they had taught it to us broke in pieces. All Quiet on the Western Front: A Novel by Erich Maria Remarque No laboratory in the world was as well equipped as theirs to discover artificial radioactivity, for none was as capable of such sustained bombardment. Big Science by Michael Hiltzik The destroyer Wadsworth had begun the shore bombardment right on schedule. Code Talker: A Novel About the Navajo Marines of World War Two by Joseph Bruchac < prev \| next > DISCLAIMER: These example sentences appear in various news sources and books to reflect the usage of the word ‘bombardment'. Views expressed in the examples do not represent the opinion of Vocabulary.com or its editors. Send us feedback Word Family bombardmentbombardmentscounterbombardment bombardbombardedbombardingbombardsbombardment the "bombard" family Vocabulary lists containing bombardment 100 SAT Words Beginning with "B" Find lists of SAT words organized by every letter of the alphabet here: A, B, C, D, E, F, G, H, I, J, K & L, M, N, O, P, Q, R, S, T, U, V, and W, X, Y & Z. This Week in Words: February 19 - 23, 2018 No time to scour the headlines or watch the news? No problem! We’ve rounded up the top words heard, read, debated, and discussed this week. This week was one of progress. Investigators looking into Russian interference in the '16 Presidential election indicted Russian entities accused of sowing discord in the American political system. Progress was made on scientific fronts as well. A pilot program designed to harness the energy of passing trains is underway, and a treatment to desensitize peanut allergy sufferers appears to offer real hope. And finally, progress was made in your cereal bowl. Lucky Charms announced that it will add a magical unicorn to its marshmallow lineup. We live in miraculous times. Take a look back at the week that was, vocabulary style. All Quiet on the Western Front Erich Maria Remarque Translated by A.W. Wheen, Remarque's masterpiece details the experiences of German soldiers during World War I. MORE VOCABULARY LISTS 2 million people are mastering new words.Master a word Sign up now (it’s free!) Whether you’re a teacher or a learner, Vocabulary.com can put you or your class on the path to systematic vocabulary improvement. 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2567	https://www.lehman.edu/faculty/anchordoqui/chapter06.pdf	Chapter 6 Circular Motion 6.1 Introduction ............................................................................................................. 1 6.2 Cylindrical Coordinate System .............................................................................. 2 6.2.1 Unit Vectors ...................................................................................................... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates ................................................................................................................ 4 Example 6.1 Area Element of Disk .......................................................................... 5 6.3 Circular Motion: Velocity and Angular Velocity ............................................ 6 6.3.1 Geometric Derivation of the Velocity for Circular Motion .......................... 8 6.4 Circular Motion: Tangential and Radial Acceleration ................................... 9 6.5 Period and Frequency for Uniform Circular Motion .................................... 11 6.5.1 Geometric Interpretation for Radial Acceleration for Uniform Circular Motion ...................................................................................................................... 12 6-1 Chapter 6 Circular Motion And the seasons they go round and round And the painted ponies go up and down We're captive on the carousel of time We can't return we can only look Behind from where we came And go round and round and round In the circle game 1 Joni Mitchell 6.1 Introduction Special cases often dominate our study of physics, and circular motion is certainly no exception. We see circular motion in many instances in the world; a bicycle rider on a circular track, a ball spun around by a string, and the rotation of a spinning wheel are just a few examples. Various planetary models described the motion of planets in circles before any understanding of gravitation. The motion of the moon around the earth is nearly circular. The motions of the planets around the sun are nearly circular. Our sun moves in nearly a circular orbit about the center of our galaxy, 50,000 light years from a massive black hole at the center of the galaxy. We shall describe the kinematics of circular motion, the position, velocity, and acceleration, as a special case of two-dimensional motion. We will see that unlike linear motion, where velocity and acceleration are directed along the line of motion, in circular motion the direction of velocity is always tangent to the circle. This means that as the object moves in a circle, the direction of the velocity is always changing. When we examine this motion, we shall see that the direction of change of the velocity is towards the center of the circle. This means that there is a non-zero component of the acceleration directed radially inward, which is called the centripetal acceleration. If our object is increasing its speed or slowing down, there is also a non-zero tangential acceleration in the direction of motion. But when the object is moving at a constant speed in a circle then only the centripetal acceleration is non-zero. In all of these instances, when an object is constrained to move in a circle, there must exist a force F  acting on the object directed towards the center. In 1666, twenty years before Newton published his Principia, he realized that the moon is always “falling” towards the center of the earth; otherwise, by the First Law, it would continue in some linear trajectory rather than follow a circular orbit. Therefore there must be a centripetal force, a radial force pointing inward, producing this centripetal acceleration. 1 Joni Mitchell, The Circle Game, Siquomb Publishing Company. 6-2 Because Newton’s Second Law m = F a   is a vector equality, it can be applied to the radial direction to yield Fr = mar . (6.1.1) 6.2 Cylindrical Coordinate System We first choose an origin and an axis we call the z -axis with unit vector ˆ k pointing in the increasing z-direction. The level surface of points such that z = zP define a plane. We shall choose coordinates for a point P in the plane z = zP as follows. One coordinate, r , measures the distance from the z -axis to the point P . The coordinate r ranges in value from 0 ≤r ≤∞. In Figure 6.1 we draw a few surfaces that have constant values of r . These level surfaces’ are circles. Figure 6.1 level surfaces for the coordinate r Our second coordinate measures an angular distance along the circle. We need to choose some reference point to define the angle coordinate. We choose a ‘reference ray’, a horizontal ray starting from the origin and extending to +∞ along the horizontal direction to the right. (In a typical Cartesian coordinate system, our ‘reference ray’ is the positive x-direction). We define the angle coordinate for the point P as follows. We draw a ray from the origin to the point P . We define the angle θ as the angle in the counterclockwise direction between our horizontal reference ray and the ray from the origin to the point P , (Figure 6.2). All the other points that lie on a ray from the origin to infinity passing through P have the same value as θ . For any arbitrary point, our angle coordinate θ can take on values from 0 ≤θ < 2π . In Figure 6.3 we depict otherlevel surfaces’, which are lines in the plane for the angle coordinate. The coordinates (r,θ) in the plane z = zP are called polar coordinates. 6-3 Figure 6.2 the angle coordinate Figure 6.3 Level surfaces for the angle coordinate 6.2.1 Unit Vectors We choose two unit vectors in the plane at the point P as follows. We choose ˆ r to point in the direction of increasing r , radially away from the z-axis. We choose ˆ θ to point in the direction of increasing θ . This unit vector points in the counterclockwise direction, tangent to the circle. Our complete coordinate system is shown in Figure 6.4. This coordinate system is called a ‘cylindrical coordinate system’. Essentially we have chosen two directions, radial and tangential in the plane and a perpendicular direction to the plane. If we are given polar coordinates (r,θ) of a point in the plane, the Cartesian coordinates (x, y) can be determined from the coordinate transformations x = rcosθ , (6.2.1) y = rsinθ . (6.2.2) Figure 6.4 Cylindrical coordinates Figure 6.5 Unit vectors at two different points in polar coordinates. 6-4 Conversely, if we are given the Cartesian coordinates (x, y), the polar coordinates (r,θ) can be determined from the coordinate transformations r = +(x2 + y2)1 2 , (6.2.3) θ = tan−1(y / x) . (6.2.4) Note that r ≥0 so we always need to take the positive square root. Note also that tanθ = tan(θ + π) . Suppose that 0 / 2 θ π ≤ ≤ , then x ≥0 and y ≥0 . Then the point (−x,−y) will correspond to the angle θ + π . The unit vectors also are related by the coordinate transformations ˆ r = cosθ ˆ i + sinθ ˆ j, (6.2.5) ˆ θ = −sinθ ˆ i + cosθ ˆ j. (6.2.6) Similarly ˆ i = cosθ ˆ r −sinθ ˆ θ , (6.2.7) ˆ j = sinθ ˆ r + cosθ ˆ θ . (6.2.8) One crucial difference between polar coordinates and Cartesian coordinates involves the choice of unit vectors. Suppose we consider a different point S in the plane. The unit vectors in Cartesian coordinates ˆ ˆ ( , ) S S i j at the point S have the same magnitude and point in the same direction as the unit vectors ˆ ˆ ( , ) P P i j at P . Any two vectors that are equal in magnitude and point in the same direction are equal; therefore ˆ ˆ ˆ ˆ , S P S P = = i i j j . (6.2.9) A Cartesian coordinate system is the unique coordinate system in which the set of unit vectors at different points in space are equal. In polar coordinates, the unit vectors at two different points are not equal because they point in different directions. We show this in Figure 6.5. 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates Consider a small infinitesimal displacement d s  between two points P1 and P2 (Figure 6.6). This vector can be decomposed into d  s = dr ˆ r + rdθ ˆ θ + dz ˆ k . (6.2.10) Consider an infinitesimal area element on the surface of a cylinder of radius r (Figure 6.7). The area of this element has magnitude 6-5 dA = rdθdz . (6.2.11) Area elements are actually vectors where the direction of the vector dA  points perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface. So for the surface of the cylinder, the infinitesimal area vector is d  A = rdθdz ˆ r . (6.2.12) Figure 6.6 Displacement vector d s  between two points Figure 6.7 Area element for a cylinder: normal vector ˆ r Example 6.1 Area Element of Disk Consider an infinitesimal area element on the surface of a disc (Figure 6.8) in the xy -plane. Figure 6.8 Area element for a disc: normal ˆ k Figure 6.9 Volume element 6-6 Solution: The area element is given by the vector ˆ d rd dr θ = A k  . (6.2.13) An infinitesimal volume element (Figure 6.9) is given by dV = rdθ dr dz . (6.2.14) The motion of objects moving in circles motivates the use of the cylindrical coordinate system. This is ideal, as the mathematical description of this motion makes use of the radial symmetry of the motion. Consider the central radial point and a vertical axis passing perpendicular to the plane of motion passing through that central point. Then any rotation about this vertical axis leaves circles invariant (unchanged), making this system ideal for use for analysis of circular motion exploiting of the radial symmetry of the motion. 6.3 Circular Motion: Velocity and Angular Velocity We can now begin our description of circular motion. In Figure 6.10 we sketch the position vector ( ) t r  of the object moving in a circular orbit of radius r . At time t , the particle is located at the point P with coordinates (r,θ(t)) and position vector given by  r(t) = r ˆ r(t) . (6.3.1) Figure 6.10 A circular orbit. Figure 6.11 Unit vectors At the point P , consider two sets of unit vectors ( ˆ r(t) , ˆ θ(t)) and ( ˆ i , ˆ j). In Figure 6.11 we see that a vector decomposition expression for ˆ r(t) and ˆ θ(t) in terms of ˆ i and ˆ j is given by ˆ r(t) = cosθ(t) ˆ i + sinθ(t) ˆ j, (6.3.2) 6-7 ˆ θ(t) = −sinθ(t) ˆ i + cosθ(t) ˆ j. (6.3.3) We can write the position vector as  r(t) = r ˆ r(t) = r(cosθ(t) ˆ i + sinθ(t) ˆ j). (6.3.4) The velocity is then  v(t) = d r(t) dt = r d dt (cosθ(t) ˆ i + sinθ(t) ˆ j) = r(−sinθ(t) dθ(t) dt ˆ i + cosθ(t) dθ(t) dt ˆ j), (6.3.5) where we used the chain rule to calculate that d dt cosθ(t) = −sinθ(t) dθ(t) dt , (6.3.6) d dt sinθ(t) = cosθ(t) dθ(t) dt . (6.3.7) We now rewrite Eq. (6.3.5) as  v(t) = r dθ(t) dt (−sinθ(t)ˆ i + cosθ(t) ˆ j). (6.3.8) Finally we substitute Eq. (6.3.3) into Eq. (6.3.8) and obtain an expression for the velocity of a particle in a circular orbit  v(t) = r dθ(t) dt ˆ θ(t) . (6.3.9) We denote the rate of change of angle with respect to time by the Greek letter ω , ω ≡dθ dt , (6.3.10) which can be positive (counterclockwise rotation in Figure 6.10), zero (no rotation), or negative (clockwise rotation in Figure 6.10). This is often called the angular speed but it is actually the z -component of a vector called the angular velocity vector.  ω = dθ dt ˆ k = ω ˆ k . (6.3.11) The SI units of angular velocity are [rad ⋅s−1]. Thus the velocity vector for circular motion is given by  v(t) = rω ˆ θ(t) ≡vθ ˆ θ(t), (6.3.12) 6-8 where the ˆ θ -component of the velocity is given by vθ = r dθ dt . (6.3.13) We shall call vθ the tangential component of the velocity. 6.3.1 Geometric Derivation of the Velocity for Circular Motion Consider a particle undergoing circular motion. At time t , the position of the particle is  r(t) . During the time interval Δt , the particle moves to the position  r(t + Δt) with a displacement Δ r . Figure 6.12 Displacement vector for circular motion The magnitude of the displacement, Δr , is represented by the length of the horizontal vector Δr  joining the heads of the displacement vectors in Figure 6.12 and is given by Δ r = 2r sin(Δθ / 2) . (6.3.14) When the angle θ Δ is small, we can approximate sin( / 2) / 2 θ θ Δ ≅Δ . (6.3.15) This is called the small angle approximation, where the angle θ Δ (and hence / 2 θ Δ ) is measured in radians. This fact follows from an infinite power series expansion for the sine function given by 3 5 1 1 sin 2 2 3! 2 5! 2 θ θ θ θ Δ Δ Δ Δ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + −⋅⋅⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ . (6.3.16) When the angle / 2 θ Δ is small, only the first term in the infinite series contributes, as successive terms in the expansion become much smaller. For example, when / 2 /30 0.1 θ π Δ = ≅ , corresponding to 6o, 3 4 ( / 2) /3! 1.9 10 θ − Δ ≅ × ; this term in the power series is three orders of magnitude smaller than the first and can be safely ignored for small angles. 6-9 Using the small angle approximation, the magnitude of the displacement is Δ r ≅r Δθ . (6.3.17) This result should not be too surprising since in the limit as Δθ approaches zero, the length of the chord approaches the arc length r Δθ . The magnitude of the velocity,  v ≡v , is then seen to be proportional to the rate of change of the magnitude of the angle with respect to time, v ≡ v = lim Δt→0 Δ r Δt = lim Δt→0 r Δθ Δt = r lim Δt→0 Δθ Δt = r dθ dt = r ω . (6.3.18) The direction of the velocity can be determined by considering that in the limit as Δt →0 (note that Δθ →0 ), the direction of the displacement Δ r approaches the direction of the tangent to the circle at the position of the particle at time t (Figure 6.13). Figure 6.13 Direction of the displacement approaches the direction of the tangent line Thus, in the limit Δt →0 , Δ r ⊥ r , and so the direction of the velocity  v(t) at time t is perpendicular to the position vector  r(t) and tangent to the circular orbit in the +ˆ θ -direction for the case shown in Figure 6.13. 6.4 Circular Motion: Tangential and Radial Acceleration When the motion of an object is described in polar coordinates, the acceleration has two components, the tangential component aθ , and the radial component, ar . We can write the acceleration vector as  a = ar ˆ r(t) + aθ ˆ θ(t). (6.4.1) 6-10 Keep in mind that as the object moves in a circle, the unit vectors ˆ r(t) and ˆ θ(t) change direction and hence are not constant in time. We will begin by calculating the tangential component of the acceleration for circular motion. Suppose that the tangential velocity vθ = rω is changing in magnitude due to the presence of some tangential force, where ω is the z -component of the angular velocity; we shall now consider that ω(t) is changing in time, (the magnitude of the velocity is changing in time). Recall that in polar coordinates the velocity vector Eq. (6.3.12) can be written as  v(t) = rω ˆ θ(t) . (6.4.2) We now use the product rule to determine the acceleration.  a(t) = d v(t) dt = r dω(t) dt ˆ θ(t) + rω(t) dˆ θ(t) dt . (6.4.3) Recall from Eq. (6.3.3) that ˆ θ(t) = −sinθ(t)ˆ i + cosθ(t) ˆ j. So we can rewrite Eq. (6.4.3) as  a(t) = r dω(t) dt ˆ θ(t) + rω(t) d dt (−sinθ(t)ˆ i + cosθ(t) ˆ j). (6.4.4) We again use the chain rule (Eqs. (6.3.6) and (6.3.7)) and find that  a(t) = r dω(t) dt ˆ θ(t) + rω(t) −cosθ(t) dθ(t) dt ˆ i −sinθ(t) dθ(t) dt ˆ j ⎛ ⎝ ⎜ ⎞ ⎠ ⎟. (6.4.5) Recall that ω ≡dθ / dt , and from Eq. (6.3.2), ˆ r(t) = cosθ(t) ˆ i + sinθ(t) ˆ j, therefore the acceleration becomes  a(t) = r dω(t) dt ˆ θ(t) −rω 2(t) ˆ r(t). (6.4.6) We denote the rate of change of ω with respect to time by the Greek letter α , α ≡dω dt , (6.4.7) which can be positive, zero, or negative. This is often called the angular acceleration but it is actually the z -component of a vector called the angular acceleration vector.  α = dω dt ˆ k = d 2θ dt2 ˆ k ≡α ˆ k . (6.4.8) 6-11 The SI units of angular acceleration are [rad ⋅s−2]. The tangential component of the acceleration is then aθ = rα . (6.4.9) The radial component of the acceleration is given by ar = −rω 2 < 0 . (6.4.10) Because ar < 0 , that radial vector component  ar(t) = −rω 2 ˆ r(t) is always directed towards the center of the circular orbit. 6.5 Period and Frequency for Uniform Circular Motion If the object is constrained to move in a circle and the total tangential force acting on the object is zero, total 0 F θ = . By Newton’s Second Law, the tangential acceleration is zero, 0 aθ = . (6.5.1) This means that the magnitude of the velocity (the speed) remains constant. This motion is known as uniform circular motion. The acceleration is then given by only the acceleration radial component vector  ar(t) = −rω 2(t) ˆ r(t) uniform circular motion . (6.5.2) Since the speed v = r ω is constant, the amount of time that the object takes to complete one circular orbit of radius r is also constant. This time interval, T , is called the period. In one period the object travels a distance s vT = equal to the circumference, s = 2πr ; thus s = 2πr = vT . (6.5.3) The period T is then given by T = 2πr v = 2πr r ω = 2π ω . (6.5.4) The frequency f is defined to be the reciprocal of the period, f = 1 T = ω 2π . (6.5.5) 6-12 The SI unit of frequency is the inverse second, which is defined as the hertz, 1 s [Hz] − ⎡ ⎤≡ ⎣ ⎦ . The magnitude of the radial component of the acceleration can be expressed in several equivalent forms since both the magnitudes of the velocity and angular velocity are related by v = r ω . Thus we have several alternative forms for the magnitude of the centripetal acceleration. The first is that in Equation (6.6.3). The second is in terms of the radius and the angular velocity, ar = rω 2 . (6.5.6) The third form expresses the magnitude of the centripetal acceleration in terms of the speed and radius, ar = v2 r . (6.5.7) Recall that the magnitude of the angular velocity is related to the frequency by 2 f ω π = , so we have a fourth alternate expression for the magnitude of the centripetal acceleration in terms of the radius and frequency, ar = 4π 2r f 2 . (6.5.8) A fifth form commonly encountered uses the fact that the frequency and period are related by f = 1/ T = ω / 2π . Thus we have the fourth expression for the centripetal acceleration in terms of radius and period, ar = 4π 2r T 2 . (6.5.9) Other forms, such as 4π 2r 2 f / T or 2πrω f , while valid, are uncommon. Often we decide which expression to use based on information that describes the orbit. A convenient measure might be the orbit’s radius. We may also independently know the period, or the frequency, or the angular velocity, or the speed. If we know one, we can calculate the other three but it is important to understand the meaning of each quantity. 6.5.1 Geometric Interpretation for Radial Acceleration for Uniform Circular Motion An object traveling in a circular orbit is always accelerating towards the center. Any radial inward acceleration is called centripetal acceleration. Recall that the direction of the velocity is always tangent to the circle. Therefore the direction of the velocity is 6-13 constantly changing because the object is moving in a circle, as can be seen in Figure 6.14. Because the velocity changes direction, the object has a nonzero acceleration. Figure 6.14 Direction of the velocity for circular motion. Figure 6.15 Change in velocity vector. 6-14 The calculation of the magnitude and direction of the acceleration is very similar to the calculation for the magnitude and direction of the velocity for circular motion, but the change in velocity vector, Δv , is more complicated to visualize. The change in velocity ( ) ( ) t t t Δ = + Δ − v v v    is depicted in Figure 6.15. The velocity vectors have been given a common point for the tails, so that the change in velocity, Δv , can be visualized. The length Δv  of the vertical vector can be calculated in exactly the same way as the displacement Δr . The magnitude of the change in velocity is 2 sin( / 2) v θ Δ = Δ v  . (6.6.1) We can use the small angle approximation ( ) sin / 2 / 2 θ θ Δ ≅Δ to approximate the magnitude of the change of velocity, v θ Δ ≅ Δ v  . (6.6.2) The magnitude of the radial acceleration is given by ar = lim Δt→0 Δ v Δt = lim Δt→0 v Δθ Δt = v lim Δt→0 Δθ Δt = v dθ dt = v ω . (6.6.3) The direction of the radial acceleration is determined by the same method as the direction of the velocity; in the limit Δθ →0, Δ ⊥ v v  , and so the direction of the acceleration radial component vector  ar(t) at time t is perpendicular to position vector ( ) t v  and directed inward, in the ˆ −r -direction.
2568	https://static1.squarespace.com/static/5fe101b108d85d5e817a934a/t/60e5a3958a9bbf37cec9d1da/1625662358035/The_Orthocentre_and_the_Pedal_Triangle_of_a_Triangle_Norra_Real.pdf	The Orthocentre and the Pedal Triangle of a Triangle Axel Hagerud, Rilind Hoti, Neo Dahlfors & Isak Fleig∗ Norra real gymnasieskola May 2021 Contents 1 Introduction 2 2 The location of the orthocentre 2 3 The pedal and orthic triangles 4 4 Inscribed quadrilaterals associated with the orthocentre 6 5 The reﬂections of the orthocentre 8 6 Conclusion 12 7 Exercises for the reader 12 ∗Minervagymnasium 1 1 Introduction In geometry one of the most common objects is the triangle. Being able to make further constructions from the information given is the key to solving all but the most basic problems you may encounter in geometry. This document is about the orthocentre and the pedal triangle of a triangle, which will be introduced shortly. The orthocentre is one of the most important points of a triangle. The pedal triangle, especially the orthic pedal triangle, is also frequently key to solving problems. 2 The location of the orthocentre To ﬁnd the orthocentre of a triangle ABC, let R, S, T be the orthogonal projections of the vertices A, B, C onto the lines generated by BC, CA, AB respectively. The orthocentre H is then the point where the lines AR, BS, CT intersect. A H T B C R S A H T B C R S Figure 1: H is the orthocentre of triangle ABC. Exercise 2.1. Investigare when the orthocentre lies inside, and when it lies outside the triangle. Does the orthocentre ever lie on the perimeter of the triangle? The very existence of an orthocentre relies on the fact that AR, BS, CT always intersect at a single point. Theorem 2.1. The altitudes of a triangle meet at a point, the orthocentre. Proof. To prove this, the fact that the perpendicular bisectors of a triangle meet at a point will be used. This can be shown as follows: 2 A D B E C O A B E C O D Figure 2: The perpendicular bisectors of a triangle meet at one point, the centre of the circumscribed circle. Here an acute and an obtuse triangle are shown. Lemma (The perpendicular bisectors of a triangle intersect). Consider the triangle ABC. The perpendicular bisectors of the sides AB and AC meet at a point O and intersect AB and AC at D and E respectively. Since \|AD\| = \|BD\| and ∠ADO = 90◦= ∠BDO, the triangles ADO and BDO are congruent by side-angle-side. This implies that \|AO\| = \|BO\|. Similarly, it can be shown that \|AO\| = \|CO\|, implying that \|BO\| = \|CO\|. Using simple trigonometry, this implies that O lies on the perpendicular bisector of side BC, meaning that the perpendicular bisectors of a triangle meet at one point. Given a triangle ABC, three additional triangles, congruent with ABC, can be constructed as in Figure 3: X Z B C Y A Figure 3: Triangles △ABZ, △ACY and △BCX congruent with triangle △ABC are constructed on the sides of △ABC. From the way the triangles are constructed it immediately follows that BC is parallel with Y Z, which can be noticed by the angles in Figure 3. Since the 3 altitude of triangle ABC through A is perpendicular to BC, it must also be perpendicular to Y Z. This, in addition to the fact that \|AY \| = \|AZ\| means that said altitude also is the perpendicular bisector of segment Y Z. Using identical arguments, it can be shown that all the altitudes of triangle ABC are the perpendicular bisectors of triangle XY Z. Since the perpendicular bisectors of a triangle are concurrent, it follows that the altitudes of any given triangle ABC always meet at one point. The proof above works without need for modiﬁcation for acute and obtuse triangles. The same result can also be achieved using Ceva’s Theorem or the Euler Line. 3 The pedal and orthic triangles A P T B C R S A H T B C R S Figure 4: There are many pedal triangles STR of the triangle ABC. The picture to the right shows a pedal triangle that is also the orthic triangle. A pedal triangle can be obtained by projecting a point onto the sides of a triangle and then connecting the projections, as in Figure 4 where △SRT is a pedal triangle of △ABC. Depending on the chosen point for projection, a pedal triangle will have diﬀerent properties. Both the medial triangle and the intouch triangle are pedal triangles, generated by projecting the centre of the circumscribed or inscribed circle of a triangle onto its sides. If the point being projected is the orthocentre, the pedal triangle becomes the orthic triangle. The orthic triangle can also be obtained by connecting the three points where the altitudes intersect with the lines generated by the sides of the triangle. The orthic triangle has some very interesting properties. Theorem 3.1. The orthocentre of the acute angled triangle △ABC is also the centre of the inscribed circle of the triangle △STR. Proof. The centre of the inscribed circle is the point where the internal angle bisectors meet. Thus, proving the theorem is equivalent to showing that the 4 angles ∠RST, ∠RTS and ∠SRT have the respective angular bisectors SH, TH and RH. Due to rotational symmetrical reasons this only needs to be proven for one of the three vertices mentioned above. To show that ∠TRH = ∠SRH ﬁrst construct a circle with the side AB as its diameter. The inverse of the inscribed angle theorem implies that the points R and S also lie on the arc of this circle. The inscribed angle theorem is now used to show that ∠ABS = ∠ARS. By constructing an additional circle with diameter AC the same reasoning can be used to show that ∠ART = ∠ACT. A H T B C R S Figure 5: The inscribed angle theorem is used to show that ∠ABS = ∠ARS and ∠ART = ∠ACT. The angles ∠BTH and ∠CSH are equal, both are 90◦. Since ∠BHT and ∠SHC are opposite angles this means that the triangles △BHT and △CHS share two angles. This implies that the triangles are similar, which means that △TBH = ∠SCH. B C T H S Figure 6: The triangles are similar. That implies that ∠TBH = ∠SCH. Returning to the conﬁguration in Figure 5 the above shown statement now 5 means that ∠TRH = ∠SRH. This proves that RH is the bisector of ∠TRS. This implies that H is the centre of the inscribed circle of the orthic triangle, since the bisectors of the three angles SH, TH and RH all meet in H, the orthocentre of △ABC. The proof is thus complete. Exercise 3.1. In Figure 5, show that the triangles ABC and AST are similar. Exercise 3.2. It is well known that the area of a triangle is base·height 2 . Show that this formula gives the same area irregardless of the choice of base of the triangle. 4 Inscribed quadrilaterals associated with the orthocentre A H T B C R S Figure 7: H is the orthocentre and △RST is the orthic triangle of △ABC. When studying the conﬁguration of the orthocentre H of a triangle ABC along with the vertices S, R and T of its orthic triangle as in Figure 7, one can observe a strong correlation between these points and inscribed quadrilaterals, which can be very useful for problem solving. In Figure 7, the quadrilateral ATHS is inscribed in a circle. This is because a quadrilateral is inscribed in a circle if and only if two opposite angles add up to 180◦, and ∠ATH + HSA = 90◦+ 90◦= 180◦. The quadrilateral BCST is also inscribed in a circle by the inverse of the inscribed angle theorem, since ∠BTC = ∠BSC = 90◦. Similarly, it can be shown that all of the quadrilaterals ATHS, BRHT, CSHR, BCST, CATR and ABRS 6 are inscribed in circles, and since both 90◦= 90◦and 90◦+ 90◦= 180◦, the circles will remain even if diﬀerent conﬁgurations permute the order of the points. An example will illustrate how these inscribed quadrilaterals may be used to solve problems. Example 4.1 (IMO Shortlist 2010 G1). Let ABC be an acute triangle with D, E, F the feet of the altitudes lying on BC, CA, AB respectively. One of the intersection points of the line EF and the circumcircle is P. The lines BP and DF meet at point Q. Prove that \|AP\| = \|AQ\|. C H B P D Q A E F Figure 8: The construction given in the problem, with the added orthocentre H of △ABC. The following solution holds only for the conﬁguration given in Figure 8. If the other intersection of EF and the circumcirle of △ABC is chosen as P, the proof is similar but minor changes are needed in the angle calculations. Solution. Note that ∠QPA = 180◦−∠BPA (Supplementary angles) = ∠BCA (Inscribed quadrilateral APBC) = ∠DCA (D on ray − − → CB) = 180◦−∠DFA (Inscribed quadrilateral AFDC) = ∠QFA (Supplementary angles) which implies that the quadrilateral AQPF is inscribed in a circle because of the inverse of the inscribed angle theorem (P, F lie on the same side of AQ). 7 But then ∠PQA = 180◦−∠PFA (Inscribed quadrilateral AQPF) = 180◦−∠BFE (Vertical angles) = ∠BCE (Inscribed quadrilateral BFEC) = ∠BCA (A on ray − − → CE) = 180◦−∠BPA (Inscribed quadrilateral APBC) = ∠QPA (Supplementary angles) gives that ∠PQA = ∠QPA. This means that △APQ is isosceles and it follows that segments AP and AQ are of equal length. 5 The reﬂections of the orthocentre One of the more unexpected, but surprisingly useful properties of the orthocentre, is it’s symmetry with respect to the sides of the triangle. As a quick reminder, the reﬂection of a point P over a line ℓis the point P ′ such that PP ′ is perpendicular to ℓand the distances from both points to ℓare equal. P P ′ ℓ Figure 9: P ′ is the reﬂection of P over the line ℓ. Theorem 5.1. In the triangle ABC with orthocentre H, the reﬂections of H over the lines AB, BC and CA lie on the circumscribed circle of the triangle ABC. 8 H A B C H A B C Figure 10: The reﬂections of the orthocentre over the sides of the triangle. The following proof assumes that the triangle is acute, as in the leftmost picture of Figure 10. The proof of the obtuse case can be obtained by minor changes in the angle calculations. Proof. Consider only the reﬂection of H over BC since the two other calculations will be identical in approach. Also, let D and E be the feet of the perpendiculars from A and B to BC and CA respectively, and let H′ be the reﬂection of H over BC. H A B C H′ D E Figure 11: H′ is the reﬂection of H over BC. By deﬁnitions, ∠HDB = 90◦= ∠BDH′ and segments HD and DH′ are 9 equal in length. Triangles HBD and H′BD are then congruent by side-angle-side, allowing for the following calculation: ∠H′BC = ∠H′BD (D on ray − − → BC) = ∠DBH (△H′BD ∼ = △HBD) = 90◦−∠BHD (Sum of angles in △HBD) = 90◦−∠EHA (Vertical angles) = 90◦−(90◦−∠HAE) = ∠HAE (Sum of angles in △EHA) = ∠H′AC (H′, C on rays − − → AH, − → AE). Thus ∠H′BC = ∠H′AC, and since B and A lie on the same side of H′C, the inverse of the inscribed angle theorem gives that A, B, H′ and C all lie on a circle. Since this circle passes through both A, B and C, it is the circumscribed circle of triangle ABC which concludes the proof. Exercise 5.1. It is possible to reﬂect objects over points as well as lines. A point P ′ is said to be the reﬂection of P over the point Q if Q is the midpoint of segment PP ′. Show that the reﬂections of the orthocentre over the midpoints of a triangle also lie on the circumscribed circle of the triangle. Example 5.1. (Polish MO Finals 2019) Let ABC be an acute triangle. The points X and Y lie on the segments AB and AC, respectively, and are such that \|AX\| = \|AY \| and the segment XY passes through the orthocentre of the triangle ABC. The lines tangent to the circumcircle of the triangle AXY at the points X and Y intersect at the point P. Prove that the points A, B, C, P are concyclic. A H X B C Y P Figure 12: The construction given in the problem statement. Before contemplating the solution, the reader should be aware of the following result. 10 Theorem 5.2 (Tangent-chord theorem). Let A, B and C be points on a circle, and let P be a point on the tangent to the circle at A such that B and P are on opposite sides of AC. Then ∠ABC = ∠PAC as in Figure 13. A C B P Figure 13: The tangent-chord theorem. It is a very useful tool in handling tangency conditions, but sadly often goes overlooked in the Swedish mathematics curriculum. Solution. Let R and Q be the intersections closest to A of PX and PY with the circumscribed circle of triangle ABC. A H X R B C P Y Q Figure 14: R and Q are deﬁned in the section above. Close inspection of Figure 14 suggests that Q and R are reﬂections of H over the lines AC and AB. This is indeed the case since the calculation ∠HY A = XY A (X on ray − − → Y X) = ∠Y XA (△AXY isosceles) = ∠QY A (Tangent-chord theorem) 11 gives that ∠HY A = ∠QY A, implying that the reﬂection of H over CA lies on ray − − → Y Q due to symmetry. The fact that the reﬂection of H over CA lies on the circumscribed circle of triangle ABC now implies that this reﬂection is Q, since it is the only point on ray − − → Y Q that lies on this circumscribed circle. In a similar way it can be established that R is the reﬂection of H over AB. Now note that ∠QPR = ∠Y PX (Y, X on rays − − → PQ, − → PR) = 180◦−∠PXY −∠XY P (Sum of angles in △PXY ) = 180◦−2∠XAY (Tangent-chord theorem) = 180◦−2∠BAC (B, C on rays − − → AX, − → AY ) implies that ∠QPR = 180◦−2∠BAC. From here, showing that ∠QCR = 180◦−2∠BAC would be enough to prove that A, B, P and C lie on a circle because of the inverse of the inscribed angle theorem. This is just straightforward angle chasing and the solution is thus complete. Exercise 5.2. Complete the angle chasing in the solution above by showing that ∠QCR = 180◦−2∠BAC. 6 Conclusion The results and examples discussed above have hopefully been both insightful and entertaining. The reader is highly encouraged to investigate the concepts of the Euler line and the Nine point circle, both of which are related to the orthocentre and the orthic triangle, but too extensive too do them justice in this short text. To conclude, a few practice problems are given for the reader to apply the ideas presented and build new problem solving strategies. 7 Exercises for the reader 1. Show that if H is the orthocentre of triangle ABC, then A is the orthocentre of triangle HBC. This is also called the orthocentric system, where four points on a plane always can form a triangle and its corresponding orthocentre. 2. Find four similar triangles with vertices among the points A, B, C, D, E, F, where △DEF is the orthic triangle of △ABC. 3. Given a triangle ABC and its orthocentre H, prove that the reﬂections of H over the midpoints of segments AB, BC and CA all lie on the circumscribed circle of triangle ABC. 4. In triangle ABC with orthocentre H and midpoints K, L and M of sides AB, BC and CA respectively, the point P is chosen on the circumscribed 12 circle of △ABC such that ∠HAP = 90◦. Show that the midpoint of segment HP lies on the circumscribed circle of △KLM. Extra challenging problem: Prove that the orthic triangle has the smallest perimeter among the triangles that can be inscribed in an acute triangle. This problem is also called Fagnano’s problem and is very diﬃcult. The reader is encouraged to ﬁrst try to prove the problem geometrically and then look up some famous solutions to the problem and attempt to complete those solutions. Some interesting geometric solutions have been carried out by Hermann Schwarz and Lip´ ot Fej´ er. 13
2569	https://math.stackexchange.com/questions/2161043/why-are-many-equations-not-solvable-by-applying-only-elementary-functions	algebra precalculus - Why are many equations not solvable by applying only Elementary functions? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Why are many equations not solvable by applying only Elementary functions? Ask Question Asked 8 years, 7 months ago Modified6 years, 2 months ago Viewed 590 times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. Why are many equations not solvable by applying only Elementary functions? Elementary functions: "In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots). The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms." (Wikipedia - Elementary function) Let us discuss this on the example equation x + e^x = 0, x real. I already know, the reason is that x and e^x are algebraically independent. (Is there a reputable reference for that?) But what is the the exact and complete reason of the non-solvability of this equation or of such equations only by Elementary functions? algebra-precalculus Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Feb 25, 2017 at 16:23 user405214user405214 79 3 3 bronze badges 1 7 To a certain extent, this is like asking "why are so many numbers irrational?".Chappers –Chappers 2017-02-25 16:26:36 +00:00 Commented Feb 25, 2017 at 16:26 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 0 Save this answer. Show activity on this post. There are just not enough elementary functions to cover even many common equations. But this problem is more theoretically, because it is easy to get solutions numerically with arbitary precision in most cases. A similar consequence is that many easy-looking functions do not have a closed-form-antiderivate. Ironically, the function e−x 2 e−x 2 , playing a very important role in probability theory, is such a function. Even polynomials with degree 5 5 or higher cannot be generally solved by radicals. An equation must be very special to have a closed-form-solution. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Feb 25, 2017 at 17:45 PeterPeter 86.8k 17 17 gold badges 87 87 silver badges 238 238 bronze badges 1 My question is: "But what is the the exact and complete reason of the non-solvability of this equation or of such equations only by Elementary functions?" Which very special form must an equation have to be solvable by applying only Elementary functions? (Hint: Liouvillian antiderivatives are not necessarily essential here, as J. F. Ritt showed.)user405214 –user405214 2017-02-25 18:19:39 +00:00 Commented Feb 25, 2017 at 18:19 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Let's take the definition of Elementary functions of Liouville and Ritt: Wikipedia: Elementary function. 1.) Let T 1(z)T 1(z), T 2(z)T 2(z) and T(z)T(z) denote mathematical terms which contain a variable z z. Each ordinary equation T 1(z)=T 2(z)T 1(z)=T 2(z) of one unknown z z can be transformed into an equation T(z)=c T(z)=c, with c c a constant. If the term T(z)T(z) contains only elementary functions, T T can be treated as an elementary function. If you want to solve the equation algebraically, you have to invert T T. The theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of elementary functions can have an inverse which is an elementary function. 2.) A method of proof for certain transcendental equations is given in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22. Two methods for simpler transcendental elementary equations are given in [Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50 and in [Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448. Both need the proof of Schanuel's conjecture what currently is an unsolved mathematical problem. Your example of an equation is treated there. Why are many equations not solvable by applying only elementary expressions? The proofs and proof methods show that the partial inverses cannot be represented by only elementary operations of algebraic numbers (Elementary numbers of Lin and Chow). Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 6, 2019 at 17:42 answered May 11, 2017 at 19:44 IV_IV_ 8,096 1 1 gold badge 24 24 silver badges 63 63 bronze badges Add a comment\| You must log in to answer this question. 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2570	https://www.studies.nawaz.org/posts/asymptotic-expansions/	Asymptotic Expansions Toggle navigationStudy Notes Tags Archives Tags About Asymptotic Expansions Posted by Beetle B. on Thu 25 September 2014 A decreasing sequence g k(N)g k(N) is called an asymptotic scale if g k+1(N)=o(g k(N))g k+1(N)=o(g k(N)). f(N)∼c 0 g 0(N)+c 1 g 1(N)+…f(N)∼c 0 g 0(N)+c 1 g 1(N)+… is called an asymptotic expansion of f f. Theorem Assume that a rational generating function f(z)/g(z)f(z)/g(z) with f(z)f(z) and g(z)g(z) relatively prime and g(0)=0 g(0)=0 has a unique pole 1/β 1/β of smallest modulus and that the multiplicity of β β is ν ν.Then [z n]f(z)g(z)∼C β n n ν−1[z n]f(z)g(z)∼C β n n ν−1 where C=ν(−β)ν f(1/β)g(ν)(1/β)C=ν(−β)ν f(1/β)g(ν)(1/β) A simple example: a n=5 a n−1−6 a n−2,a 0=0,a 1=1 a n=5 a n−1−6 a n−2,a 0=0,a 1=1 After making the recurrence valid for all n n (using Kronecker delta functions), and then converting to generating functions, we have: A(z)=5 z A(z)−6 z 2 A(z)+z=z(1−3 z)(1−2 z)A(z)=5 z A(z)−6 z 2 A(z)+z=z(1−3 z)(1−2 z) The pole of the smallest modulus is 1/3. Plugging into the expression above, we get that a n∼3 n a n∼3 n Well Known Expansions e x=1+x+x 2 2!+x 3 3!+O(x 4)e x=1+x+x 2 2!+x 3 3!+O(x 4) ln(1+x)=x−x 2 2+x 3 3+O(x 4)ln⁡(1+x)=x−x 2 2+x 3 3+O(x 4) (1+x)k=1+k x+(k 2)x 2+(k 3)x 3+O(x 4)(1+x)k=1+k x+(k 2)x 2+(k 3)x 3+O(x 4) 1 1−x=1+x+x 2+x 3+O(x 4)1 1−x=1+x+x 2+x 3+O(x 4) H N=ln N+γ+1 2 N−1 12 N 2+O(1 N 4)H N=ln⁡N+γ+1 2 N−1 12 N 2+O(1 N 4) ln N!=N ln N−N+ln 2 π N−−−−√+O(1 N)ln⁡N!=N ln⁡N−N+ln⁡2 π N+O(1 N) These are more accurate as x x approaches 0. Normally we’re interested as N N approaches ∞∞. We can simply substitute x=1/N x=1/N. Techniques Simplification Simply discard unneeded terms. If you have O(1/N)O(1/N), then ignore all 1/N 2 1/N 2 terms. Substitution In a known expansion, replace x x with something appropriate. Factoring Estimate the leading term, factor it out, and expand the rest: ln(N 2+N)=ln(N 2(1+1/N))=2 ln N+ln(1+1/N)ln⁡(N 2+N)=ln⁡(N 2(1+1/N))=2 ln⁡N+ln⁡(1+1/N) Now you can expand the last term… Multiplication Do term by term multiplication to some desired accuracy, then combine and simplify. (H N)2=(ln N+γ+O(1 N))(ln N+γ+O(1 N))(H N)2=(ln⁡N+γ+O(1 N))(ln⁡N+γ+O(1 N)) Division Expand, then factor the denominator, expand 1/(1−x)1/(1−x), then multiply. H N ln(N+1)=ln N+γ+O(1 N)ln N+O(1 N)H N ln⁡(N+1)=ln⁡N+γ+O(1 N)ln⁡N+O(1 N) Composition e H N e H N Expand H N H N and then apply the Taylor Series. Exp/Log Trick Sometimes it’s convenient to convert a function as follows: f(x)=exp(ln(f(x)))f(x)=exp⁡(ln⁡(f(x))) Then expand the ln ln function, and then use Taylor on the exponential. Asymptotics of Finite Sums Sometimes we need an approximation for ∑k=N k=0 f(k)∑k=0 k=N f(k) for large N N. Bounding the Tail If the series is rapidly decreasing, make the sum go to infinity: N!∑k=0 k=N(−1)k k!=N!e−1−R N N!∑k=0 k=N(−1)k k!=N!e−1−R N where R N=N!∑k>N(−1)k k!R N=N!∑k>N(−1)k k! By looking at the first few terms of R N R N, we can immediately see that it is bounded by O(1/N)O(1/N). Using The Tail If the series is rapidly increasing, the last term may dominate. ∑k=0 k=N k!=N!(1+1 N+∑k=0 N−2 k!N!)∑k=0 k=N k!=N!(1+1 N+∑k=0 N−2 k!N!) By looking at the last terms in the summation, we can see that the summation is bounded by O(1/N 2)O(1/N 2). Approximating with an Integral Just convert the sum to an integral. Try to find some justification,though… Euler-Maclaurin Summation Let f f be defined on [1,∞)[1,∞). Let its derivatives exist and let it be absolutely integrable.Then: ∑k=1 N f(k)=∫N 1 f(x)d x+1 2 f(N)+C f+1 12 f′(N)−1 720 f′′′(N)+…∑k=1 N f(k)=∫1 N f(x)d x+1 2 f(N)+C f+1 12 f′(N)−1 720 f‴(N)+… This series diverges, so take care with how many terms you take. Bivariate Asymptotics The function may have two variables (usually N N and k k). How you expand may depend on the relative values of each. Some well known ones: Normal (2 N N−k)∼e−k 2/N π N−−−√+O(1 N 3/2)(2 N N−k)∼e−k 2/N π N+O(1 N 3/2) This holds regardless of k k. (2 N N−k)∼e−k 2/N π N−−−√+(1+O(1 N)+O(k 4 N 3))(2 N N−k)∼e−k 2/N π N+(1+O(1 N)+O(k 4 N 3)) This holds when k k is near the“center”. Poisson (N k)(λ N)k(1−λ N)N−k∼λ k e−λ k!+o(1)(N k)(λ N)k(1−λ N)N−k∼λ k e−λ k!+o(1) Holds for all/most k k. (N k)(λ N)k(1−λ N)N−k∼λ k e−λ k!(1+O(1 N)+O(k N))(N k)(λ N)k(1−λ N)N−k∼λ k e−λ k!(1+O(1 N)+O(k N)) This holds when k k is near the“center”. Q N!(N−k)!N k∼e−k 2/(2 N)+O(1 N−−√)N!(N−k)!N k∼e−k 2/(2 N)+O(1 N) Holds for all/most k k. N!(N−k)!N k∼e−k 2/(2 N)(1+O(k N)+O(k 3 N 2))N!(N−k)!N k∼e−k 2/(2 N)(1+O(k N)+O(k 3 N 2)) This holds when k k is near the“center”. Estimating Sums with Bivariate Estimates If you have a sum of a bivariate function, you may need to utilize different estimates for different portions of the sum. Laplace Method Laplace’s method involves approximating the summand with a continuous function at the region(s) where the summand is greatest. For the rest of the range, bound the sum with the tail of your function. Then integrate your function. It does take some skill in picking the range that you define to be the tail. You need to pick a value where you’re fairly sure it is bounded above by your continuous function. tags : asymptotic expansion Previous Article: The Average Number of Leaves in a Binary Tree of N N Internal Nodes Next Article: Red Black Trees Blog powered by Pelican, which takes great advantage of Python. © Beetle B.
2571	https://www.mathplanet.com/education/algebra-1/exponents-and-exponential-functions/properties-of-exponents	Mathplanet A free service from Mattecentrum Menu Properties of exponents Do excercises Show all 3 exercises Properties of exponents I Properties of exponent II Properties of exponent III In earlier chapters we introduced powers. $$x^{3}=x\cdot x\cdot x$$ There are a couple of operations you can do on powers and we will introduce them now. We can multiply powers with the same base $$x^{4}\cdot x^{2}=\left (x\cdot x\cdot x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{6}$$ This is an example of the product of powers property tells us that when you multiply powers with the same base you just have to add the exponents. $$x^{a}\cdot x^{b}=x^{a+b}$$ We can raise a power to a power $$\left ( x^{2} \right )^{4}= \left (x\cdot x \right )\cdot \left (x\cdot x \right ) \cdot \left ( x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{8}$$ This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents. When you raise a product to a power you raise each factor with a power $$\left (xy \right )^{2}= \left ( xy \right )\cdot \left ( xy \right )= \left ( x\cdot x \right )\cdot \left ( y\cdot y \right )=x^{2}y^{2}$$ This is called the power of a product property $$\left (xy \right )^{a}= x^{a}y^{a}$$ As well as we could multiply powers we can divide powers. $$\frac{x^{4}}{x^{2}}=\frac{x\cdot x\cdot {\color{red} \not}{x}\cdot {\color{red} \not}{x}}{{\color{red} \not}{x}\cdot {\color{red} \not}{x}}=x^{2}$$ This is an example of the quotient of powers property and tells us that when you divide powers with the same base you just have to subtract the exponents. $$\frac{x^{a}}{x^{b}}=x^{a-b},\: \: x\neq 0$$ When you raise a quotient to a power you raise both the numerator and the denominator to the power. $$\left (\frac{x}{y} \right )^{2}=\frac{x}{y}\cdot \frac{x}{y}=\frac{x\cdot x}{y\cdot y}=\frac{x^{2}}{y^{2}}$$ This is called the power of a quotient power $$\left (\frac{x}{y} \right )^{a}=\frac{x^{a}}{y^{a}},\: \: y\neq 0$$ When you raise a number to a zero power you'll always get 1. $$1=\frac{x^{a}}{x^{a}}=x^{a-a}=x^{0}$$ $$x^{0}=1,\: \: x\neq 0$$ Negative exponents are the reciprocals of the positive exponents. $$x^{-a}=\frac{1}{x^{a}},\: \: x\neq 0$$ $$x^{a}=\frac{1}{x^{-a}},\: \: x\neq 0$$ The same properties of exponents apply for both positive and negative exponents. In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power $$\sqrt{x}=\sqrt{x}=x^{\frac{1}{2}}$$ Video lesson Simplify the following expression using the properties of exponents $$\frac{( 7^{5}) ^{10}\cdot 7^{200}}{\left ( 7^{-2} \right )^{30}}$$ Do excercises Show all 3 exercises Properties of exponents I Properties of exponent II Properties of exponent III More classes on this subject Algebra 1 Exponents and exponential functions: Exponential growth functions
2572	https://www.bbc.com/learningenglish/burmese/course/towards-advanced/unit-26/tab/grammar	BBC Learning English - Course: Towards Advanced / Unit 26 / Grammar Reference Homepage Accessibility links Skip to content Accessibility Help Sign in Notifications Home News Sport Weather iPlayer Sounds Bitesize CBeebies CBBC Food Home News Sport Business Innovation Culture Travel Earth Video Live More menu Search BBC Home News Sport Weather iPlayer Sounds Bitesize CBeebies CBBC Food Home News Sport Business Innovation Culture Travel Earth Video Live Close menu Learning English Inspiring language learning since 1943 ဗမာစာChange language Learn English How do I Learn English ဗမာစာ Change Language English فارسی 中文 ไทย ဗမာစာ 한국어 አማርኛ Afaan Oromoo ትግርኛ Learn English Search How do I Learn English 26 Unit 26: Towards Advanced Grammar, news, vocabulary and pronunciation Open unit selector Close unit selector Unit 26 Towards Advanced Select a unit 1 Towards advanced 2 Towards advanced 3 Towards advanced 4 Towards advanced 5 Towards advanced 6 Towards advanced 7 Towards advanced 8 Towards advanced 9 Towards Advanced 10 Towards Advanced 11 Towards Advanced 12 Towards Advanced 13 Towards Advanced 14 Towards Advanced 15 Towards Advanced 16 Towards Advanced 17 Towards Advanced 18 Towards Advanced 19 Towards Advanced 20 Towards Advanced 21 Towards Advanced 22 Towards Advanced 23 Towards Advanced 24 Towards Advanced 25 Towards Advanced 26 Towards Advanced 27 Towards Advanced 28 Towards Advanced 29 Towards Advanced 30 Towards Advanced Sessions Vocabulary reference Grammar reference Grammar Reference Session 1 - Masterclass Inversion 2 Inversion happens in English for emphasis, dramatic purpose or formality. In order to invert, the subject verb object order of a normal sentence is changed in some way. 1. Reduced Conditionals: Conditionals are sentences in English which express the result or possible result of a real or imagined action. The usually start with if: If you go to town, will you get me a cola? (1st conditional) If I were an animal, I would be a dog. (2nd conditional) If I had stayed longer, I would have learned a new language. (3rd conditional) In second and third conditionals we canremove the ifandinvert the subject and auxiliary verb.This is considered to be more formal and so more polite. In the case of the second conditional, if the verb is an action we use were and the infinitive. WereIan animal, I would be a dog. Were I togoon holiday, I would go to Jamaica. (If I went on holiday...) Had Istayed longer, I would have learned a new language. To invert a first conditional in this way, we need to use the word ‘should’. Should makes a first conditional more polite and more tentative. Then we remove the if and invert the subject and auxiliary verb as normal.So: If you should go to town, will you get me a cola? Should yougo to town, will you get me a cola? Negativesin these formsare not contracted. So: Should you notgo to town… Were I nota human… Had I notleft so early… 2. Adverbs of place or movement: Adverbs of place or movement usually come after the verb in a clause.When anadverb of place or movementis put at the beginning of a clause, thenthe whole verb phrase, andnot just the auxiliary verb, can be put before the subject. This is done for dramatic effect and is usually conveyed in a written style and even more so when introducing a new noun - such as in a story. So, for example: The spy came through the window. Through the windowcamethe spy (adverb of movement +complete verb phrase+ subject) 300 men would stand in the pass. In the passwould stand300 men. (adverb of place +complete verb phrase+ subject) This is common with shorter adverbs in speech, such as: here & there. If a pronoun is used instead of a noun, it must go before the verb. There sat my father. There he sat. On ran the racers. On they ran. I opened the box and out jumped a puppy! I opened the box and out it jumped. 3. Consequences of an adjective:We can usesoplusan adjective, then weinvertthe normal subject and auxiliary verb, and finally we use ‘that’ to emphasise how strongly something’s description affected us and what the consequence was. We can do the same thing with a noun using such. So beautiful was she that I fell in love immediately (so + adjective + inversion + that + consequence) Such a beautiful woman was she that I fell in love immediately. (such + noun + inversion + that + consequence) Practise Grammar View sessions in this unit Session Vocabulary Vocabulary Practise this vocabulary Download centre --------------- Latest course content About About BBC Learning English Courses Course site maps Learning FAQ Contact BBC Learning English Social TikTok YouTube Instagram Facebook Social media house rules Back to topRSS Explore the BBC Home News Sport Weather iPlayer Sounds Bitesize CBeebies CBBC Food Home News Sport Business Innovation Culture Travel Earth Video Live Terms of Use About the BBC Privacy Policy Privacy Policy Cookies Cookies Accessibility Help Parental Guidance Contact the BBC Make an editorial complaint BBC emails for you Advertise with us Copyright © 2025 BBC. The BBC is not responsible for the content of external sites. Read about our approach to external linking.
2573	https://shop.elsevier.com/books/henrys-clinical-diagnosis-and-management-by-laboratory-methods/mcpherson/978-0-323-67320-4	We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. For more information, see our Cookie Policy. View Cart My account Henry's Clinical Diagnosis and Management by Laboratory Methods 24th Edition - June 1, 2021 Newer edition is available Imprint: Elsevier Authors: Richard A. McPherson, Matthew R. Pincus Language: English Hardback ISBN: 9780323673204 9 7 8 - 0 - 3 2 3 - 6 7 3 2 0 - 4 For more than 100 years, Henry's Clinical Diagnosis and Management by Laboratory Methods has been recognized as the premier text in clinical laboratory medicine, widely used by… Read more Purchase options Info / Buy BACK-TO-SCHOOL Fuel your confidence! Up to 25% off learning resources Shop now For more than 100 years, Henry's Clinical Diagnosis and Management by Laboratory Methods has been recognized as the premier text in clinical laboratory medicine, widely used by both clinical pathologists and laboratory technicians. Leading experts in each testing discipline clearly explain procedures and how they are used both to formulate clinical diagnoses and to plan patient medical care and long-term management. Employing a multidisciplinary approach, it provides cutting-edge coverage of automation, informatics, molecular diagnostics, proteomics, laboratory management, and quality control, emphasizing new testing methodologies throughout. Remains the most comprehensive and authoritative text on every aspect of the clinical laboratory and the scientific foundation and clinical application of today's complete range of laboratory tests. 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Surgical Pathologists in training and practicing clinical pathologists Part 1: The Clinical Laboratory General Concepts and Administrative Issues Optimizing Laboratory Workflow and Performance Preanalysis Analysis: Principles of Instrumentation Mass Spectrometry and Applications Analysis: Clinical Laboratory Automation Point-of-Care Testing and Physician Office Laboratories Postanalysis: Medical Decision Making Interpreting Laboratory Results Laboratory Statistics Quality Control Clinical Laboratory Informatics Financial Management Ethics in Laboratory Medicine Part 2: Clinical Chemistry Evaluation of Renal Function, Water , Electrolytes and Acid-Base Function Biochemical Markers of Bone Metabolism Carbohydrates Lipids and Dyslipoproteinemia Cardiac Injury, Atherosclerosis, and Thrombotic Disease Specific Proteins Clinical Enzymology Evaluation of Liver Function Laboratory Diagnosis of Gastrointestinal and Pancreatic Disorders Toxicology and Therapeutic Drug Monitoring Evaluation of Endocrine Function Reproductive Function and Pregnancy Vitamins and Trace Elements Chemical Basis for Analyte Assays and Common Interferences Part 3: Urine and Other Body Fluids Basic Examination of Urine Cerebrospinal, Synovial, Serous Body Fluids, and Alternative Specimens Part 4: Hematology and Transfusion Medicine Basic Examination of Blood and Bone Marrow Hematopoiesis Erythrocytic Disorders Leukocytic Disorders The Flow Cytometric Evaluation for Hematopoietic Neoplasia Immunohematology Transfusion Medicine Hemapheresis Tissue Banking and Progenitor Cells Part 5: Hemostasis and Thrombosis Coagulation and Fibrinolysis Platelet Disorders and Von Willebrand Disease Laboratory Approach to Thrombotic Risk Antithrombotic Therapy Part 6: Immunology and Immunopathology Overview of the Immune System and Immunologic Disorders Immunoassays and Immunochemistry Laboratory Evaluation of the Cellular Immune System Laboratory Evaluation of Immunoglobulin Function and Humoral Immunity Mediators of Inflammation: Complement Mediators of Inflammation: Cytokines and Adhesion Molecules Human Leukocyte Antigen: The Major Histocompatibility Complex of Man The Major Histocompatibility Complex and Disease Immunodeficiency Disorders Clinical and Laboratory Evaluation of Systemic Autoimmune Rheumatic Diseases Vasculitis Organ-Specific Autoimmune Diseases Allergic Diseases Part 7: Medical Microbiology Medical Bacteriology In Vitro Testing of Antimicrobial Agents Mycobacteria Mycotic Diseases Spirochete Infections Chlamydial and Mycoplasmal Infections Rickettsiae and Other Related Intracellular Bacteria Viral Infections Medical Parasitology Specimen Collection and Handling for Diagnosis of Infectious Diseases Part 8: Molecular Pathology Introduction to Molecular Pathology Molecular Diagnostics: Basic Principles and Techniques Polymerase Chain Reaction and Other Nucleic Acid Amplification Technology Hybridization Array Technologies Applications of Cytogenetics in Modern Pathology Molecular Diagnosis of Genetic Diseases Molecular Genetics of Neuro-Psychiatric Disorders: Current Research and Perspectives Identity Testing: Use of DNA Analysis in Parentage, Forensic, and Missing Persons Testing Pharmacogenomics and Personalized Medicine Part 9: Clinical Pathology of Cancer Diagnosis and Management of Cancer Using Serologic and Other Body Fluid Markers Oncoproteins and Early Tumor Detection Molecular Diagnosis of Hematopoietic Neoplasms Molecular Genetic Pathology of Solid Tumors High-Throughput Genomic and Proteomic Technologies in the Post-Genomic Era Appendices Physiologic Solutions, Buffers, Acid-Base Indicators, Standard Reference Materials, and Temperature Conversions Desirable Weights, Body Surface Area, and Body Mass Index Approximations of Total Blood Volume Periodic Table of Elements SI Units Common Chimeric Genes Identified in Human Malignancies Disease/Organ Panels Edition: 24 Newer edition is available Published: June 1, 2021 Imprint: Elsevier Language: English Richard A. McPherson Affiliations and expertise Professor of Pathology, Emeritus, Virginia Commonwealth University, Richmond, Virginia Matthew R. Pincus Affiliations and expertise Professor, Department of Pathology, State University of New York Downstate Medical Center, Brooklyn, New York; Chief, Department of Pathology and Laboratory Medicine, New York Harbor VA Medical Center, New York, New York Related books
2574	https://www.nature.com/articles/s41588-020-0614-5	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. View all journals Search Log in Explore content About the journal Publish with us Sign up for alerts RSS feed Genomic diversifications of five Gossypium allopolyploid species and their impact on cotton improvement Download PDF Download PDF Article Open access Published: Genomic diversifications of five Gossypium allopolyploid species and their impact on cotton improvement Z. Jeffrey Chen ORCID: orcid.org/0000-0001-5006-80361,2 na1, Avinash Sreedasyam ORCID: orcid.org/0000-0001-7336-70123 na1, Atsumi Ando1 na1, Qingxin Song1,2 na1, Luis M. De Santiago ORCID: orcid.org/0000-0002-3796-48294 na1, Amanda M. Hulse-Kemp5, Mingquan Ding1,6, Wenxue Ye2, Ryan C. Kirkbride ORCID: orcid.org/0000-0001-9523-259X1, Jerry Jenkins ORCID: orcid.org/0000-0002-7943-39973, Christopher Plott3, John Lovell3, Yu-Ming Lin4, Robert Vaughn4, Bo Liu4, Sheron Simpson7, Brian E. Scheffler ORCID: orcid.org/0000-0003-1968-89527, Li Wen8, Christopher A. Saski8, Corrinne E. Grover ORCID: orcid.org/0000-0003-3878-54599, Guanjing Hu ORCID: orcid.org/0000-0001-8552-73949, Justin L. Conover ORCID: orcid.org/0000-0002-3558-60009, Joseph W. Carlson10, Shengqiang Shu ORCID: orcid.org/0000-0002-4336-899410, Lori B. Boston3, Melissa Williams3, Daniel G. Peterson11, Keith McGee12, Don C. Jones13, Jonathan F. Wendel ORCID: orcid.org/0000-0003-2258-50819, David M. Stelly ORCID: orcid.org/0000-0002-3468-41194, Jane Grimwood ORCID: orcid.org/0000-0002-8356-83253 & ¦ Jeremy Schmutz3,10 Nature Genetics volume 52, pages 525533 (2020)Cite this article 54k Accesses 360 Citations 336 Altmetric Metrics details Subjects Epigenomics Genome assembly algorithms Plant breeding Plant hybridization Sequence annotation Abstract Polyploidy is an evolutionary innovation for many animals and all flowering plants, but its impact on selection and domestication remains elusive. Here we analyze genome evolution and diversification for all five allopolyploid cotton species, including economically important Upland and Pima cottons. Although these polyploid genomes are conserved in gene content and synteny, they have diversified by subgenomic transposon exchanges that equilibrate genome size, evolutionary rate heterogeneities and positive selection between homoeologs within and among lineages. These differential evolutionary trajectories are accompanied by gene-family diversification and homoeolog expression divergence among polyploid lineages. Selection and domestication drive parallel gene expression similarities in fibers of two cultivated cottons, involving coexpression networks and N6-methyladenosine RNA modifications. Furthermore, polyploidy induces recombination suppression, which correlates with altered epigenetic landscapes and can be overcome by wild introgression. These genomic insights will empower efforts to manipulate genetic recombination and modify epigenetic landscapes and target genes for crop improvement. Similar content being viewed by others Convergence and divergence of diploid and tetraploid cotton genomes Article 29 October 2024 Post-polyploidization centromere evolution in cotton Article 03 March 2025 Regulatory controls of duplicated gene expression during fiber development in allotetraploid cotton Article Open access 16 October 2023 Main Polyploidy or whole-genome duplication provides genomic opportunities for evolutionary innovations in many animal groups and all flowering plants1,2,3,4,5, including most important crops such as wheat, cotton and canola or oilseed rape6,7,8. The common occurrence of polyploidy may suggest its advantage and potential for selection and adaptation2,3,9, through rapid genetic and genomic changes as observed in newly formed Brassica napus10, Tragopogon miscellus11 and polyploid wheat12, and/or largely epigenetic modifications as in Arabidopsis and cotton polyploids5,13. Cotton is a powerful model for revealing genomic insights into polyploidy3, providing a phylogenetically defined framework of polyploidization (~1.5million years ago (Ma))14, followed by natural diversification and crop domestication15. The evolutionary history of the polyploid cotton clade is longer than that of some other allopolyploids, such as hexaploid wheat (~8,000years)12, tetraploid canola (~7,500years)16 and tetraploid Tragopogon (~150years)11. Polyploidization between an A-genome African species (Gossypium arboreum (Ga)-like) and a D-genome American species (G. raimondii (Gr)-like) in the New World created a new allotetraploid or amphidiploid (AD-genome) cotton clade (Fig. 1a)14, which has diversified into five polyploid lineages, G. hirsutum (Gh) (AD)1, G. barbadense (Gb) (AD)2, G. tomentosum (Gt) (AD)3, G. mustelinum (Gm) (AD)4 and G. darwinii (Gd) (AD)5. G. ekmanianum and G. stephensii are recently characterized and closely related to Gh17. Gh and Gb were separately domesticated from perennial shrubs to become annualized Upland and Pima cottons15. To date, global cotton production provides income for ~100 million families across ~150 countries, with an annual economic impact of ~US$500billion worldwide6. However, cotton supply is reduced due to aridification, climate change and pest emergence. Future improvements in cotton and sustainability will involve use of the genomic resources and gene-editing tools becoming available in many crops9,18,19. Cotton genomes have been sequenced for the D-genome (Gr)20 and A-genome (Ga)21 diploids and two cultivated tetraploids22,23,24,25,26. These analyses have shown structural, genetic and gene expression variation related to fiber traits and stress responses in cultivated cottons, but the impact of polyploidy on selection and domestication among the wild and cultivated polyploid cotton species remains poorly understood6. Here we report high-quality genomes for all five allotetraploid species and show that despite wide geographic distribution and diversification, allotetraploid cotton genomes retained the syntenic gene content and genomic diversity relative to respective extant diploids. Evolutionary rate heterogeneities, gene loss and positively selected genes characterize the two subgenomes of each species but differ among polyploid lineages. Transposable elements (TEs) are dynamically exchanged between the two subgenomes, facilitating genome-size equilibration following allopolyploidy. Gene expression diversity in the fiber tissues involves selection, coexpression networks and N6-methyladenosine (m6A) RNA modifications. In cultivated polyploid cottons, recombination suppression correlates with DNA hypermethylation and weak chromatin interactions and can be overcome by wild introgression and possibly epigenetic remodeling. The results offer unique insights into polyploid genome evolution and provide valuable genomic resources for cotton research and improvement. Results Sequencing, assembly and annotation Sequencing of the five allotetraploid cotton genomes entailed using complementary whole-genome shotgun strategies, including sequencing by single-molecule real-time (PacBio SEQUEL and RSII, ~440Ã genome equivalent), Illumina (HiSeq and NovaSeq, ~286Ã) (Supplementary Dataset 1a) and chromatin conformation capture (Hi-C seq) (~326Ã) (Methods). Homozygous single nucleotide polymorphisms (SNPs) and insertions/deletions (indels) were also used to correct the consensus sequence (Supplementary Dataset 1b,c). The rate of anchored scaffolds is 97% in Gb and 99% or higher in the other 4 species. Scaffolds were oriented, ordered and assembled into 26 pseudo-chromosomes with very low (0.10.8%) gaps (Table 1 and Supplementary Dataset 1d). The assembled genomes range in size from 2.2 to 2.3 gigabase pairs (Gbp; Table 1), slightly smaller than the sum of the two A- and D-genome diploids (1.7/A+0.8/D2.5Gbp/AD)20,21. Nearly 73% of the assembled genomes are repeats and TEs (Supplementary Dataset 1e), predominantly in pericentromeric regions in Gm (Fig. 1b) and the other 4 species (Extended Data Fig. 1). The completeness and contiguity of these genomes compare favorably with Sanger-based sequences of sorghum27 and Brachypodium28. The euchromatic sequences of 5 polyploid genomes are complete (Supplementary Note), as supported by BUSCO scores (>97%) and 36,880 (>99%) primary transcripts from the Gr version 2 release20 (Supplementary Dataset 1b), with the number of protein-coding genes predicted to range from 74,561 (Gb) to 78,338 (Gt; Table 1), which are 3,0004,000 more than reported in Gh and Gb23. Although the A subgenome (1.7Gbp) is twice the size of the D subgenome (0.8Gbp)20,21, mirroring the ancestral state of their extant diploids, the two have similar numbers of protein-coding genes (ratio of D/A1.06; Supplementary Dataset 1f). As an indication of the improved contiguity (Supplementary Note), the contig length in the Gh genome increases 6.9-fold with a 7.7-fold reduction in fragmentation (6,733 versus 51,849), compared to the published sequences22. The improvement is substantial in the Gb genome with a 15.9-fold reduction in N50 contigs and a 23-fold increase in N50 contig length (from 77.6 to 1,800kilobase pairs (kb)). Moreover, most quality scores are 2-5-fold higher in the 3 wild polyploid species than in Gh and Gb (Table 1). Reciprocal 24-nucleotide masking and syntenic analyses show that our Gh and Gb assemblies have ~23- and 2.7-fold more unique sequences, respectively, than the published ones22 also with variable gap sizes (10200kb; Extended Data Fig. 2a). Some specific genes are present in our annotations and the published data, which are largely related to gene copy number variation (more decreases than increases). Other differences include inversions (132133megabase pairs (Mb)) with two large ones (A06 and D03) present in similar regions of both Gh and Gb22 (Extended Data Fig. 2b), which could result from errors and/or unresolved alternative haplotypes; these inversions were confirmed using Hi-C data (Extended Data Fig. 2c). Notably, the published Hai7124 strain22 is a Gb local strain that is different from Gb 3-79, and Gh TM-1 strains may vary; these can also contribute to the observed variation. Evolution within and between five polyploids Using the diploid20,21 and 5 polyploid cotton genomes, we estimated divergence at 5859Ma between Gossypium and its relative Theobroma cacao (Extended Data Fig. 3a and Supplementary Note), 4.75.2Ma between the extant diploids (Extended Data Fig. 3b), and 1.01.6Ma between polyploid and diploid clades. Genome-wide phylogenetic analysis (Extended Data Fig. 4a) supports a monophyletic origin for the five allotetraploid species29. Within the polyploid clade, the highest divergence (~0.63Ma) occurs between Gm and the other 4 species, with the most recent divergence (~0.20Ma) between Gb and Gd. This genomic diversification was accompanied by biogeographic radiation to the Galapagos Islands (Gd), the Hawaiian Islands (Gt), South America (northeastern Brazil) (Gm)30, Central and South America, the Caribbean, and the Pacific (Gh and Gb)31, with separate distribution and domestication of diploid cultivated cottons in southern Arabia, North Africa, western India and China32 (Extended Data Fig. 4b). Over the last 8,000years, Upland (Gh) and Pima (Gb) cottons were independently domesticated in northwest South America and the Yucatan Peninsula of Mexico, respectively, under strong human selection, leading to the modern annualized crops15. After whole-genome duplication, duplicate genes may be lost or diverge in functions33, but the pace of this process has rarely been studied in allopolyploids. Using 17,136 homoeolog pairs shared among all 5 allotetraploid species, we demonstrate that most (14,583, 85.5%) homoeolog pairs evolved at statistically indistinguishable rates throughout the polyploid clade relative to the diploids (Supplementary Dataset 2a), but those with rate shifts occur more commonly in the A (1,476, 8.5%) than in the D (845, 5%) subgenome. We further revealed that the D homoeologs generally acquire substitution mutations more quickly than the A homoeologs in most lineages, whereas the Gh and Gt lineages experience a greater rate of divergence in the A than in the D homoeologs (Supplementary Dataset 2b). This relative acceleration of A-homoeolog divergence is mirrored in lineage-specific rate tests; the Gh/Gt clade including Upland cotton has the fastest evolving A homoeologs and the slowest evolving D homoeologs among five polyploids. These results demonstrate pervasive lineage-specific rate heterogeneities between subgenomes and among different polyploid cottons. We examined patterns of gene loss and gain using 4,369 single-copy orthologs (SCOs), which are present in both diploids and in one or more allotetraploids (Extended Data Fig. 4c). Analysis of gene loss and gain among these basally shared homoeologs in the five polyploid lineages showed the highest level of net gene loss between the initial polyploidization and Gm, with threefold higher levels in the A subgenome (547 net gene losses) than in the D subgenome (149). Other polyploids have fewer gene losses with no subgenomic bias. Among the homoeologs shared by all five polyploid species (Fig. 2a), the number of genes under positive selection (Ka/Ks values>1) is the highest (3,2003,300) in Gm with the longest branch relative to others, and the lowest between Gb and Gd (~1,100), the most recently diverged polyploid clade (Supplementary Dataset 3). Across different polyploid lineages, 1020% more D homoeologs are under positive selection than A homoeologs, suggesting a concerted evolutionary impact on subgenomic functions in all polyploid species. Genomic diversity among five polyploids The two subgenomes in each of the five polyploid species are highly conserved at the chromosomal, gene content and nucleotide levels (Fig. 1b and Extended Data Fig. 1). The D subgenomes have fewer and smaller inversions than the A subgenomes (Fig. 1c), as reported for Gh25, except for a few small inversions in D10 of GtGm and GmGb and D12 of GdGtGm. This level of structural conservation is similar to some polyploids such as wheat7 and Arabidopsis suecica34, but is different from others such as B. napus10, peanut35 and T. miscellus11, which show rapid homoeologous shuffling. The genomic conservation is extended to gene order, collinearity and synteny (Fig. 1c). Among the annotated genes (74,56178,338), 56,870 orthologous groups or 65,300 genes (32,650 homoeologous pairs) (8488%) are shared among all 5 species (Fig. 2a and Supplementary Dataset 1f). The number of SNPs is in the range of 412 million (1.75.2 SNPskb1) or 0.190.53% among 5 polyploid genomes (Supplementary Dataset 4 and Supplementary Note). Gm has the highest SNP level (0.53%) relative to the other 4 species, with the lowest between the most recently diverged species Gb and Gd (~0.19%). Similar trends of indels range from ~5.55Mb (~0.76%) in GmGt to ~3.35Mb (~0.34%) in GbGd (Extended Data Fig. 1 and Supplementary Dataset 5). The level of overall variation of SNPs and indels among cotton species is low, comparable to natural variation (3.54.1 SNPskb1) between Brachypodium accessions28 but lower than that (~7.4 SNPskb1) for subspecies of rice36. SNPs are more frequent in pericentromeric regions, while indel distributions coincide with gene densities (Fig. 1b and Extended Data Fig. 1). TE exchanges between two subgenomes that equilibrate the genome-size variation The size difference between the Ga (~1.7Gbp) and Gr (~0.8Gbp)20,21 genomes is preserved in the respective A and D subgenomes of the 5 allotetraploid species (Fig. 3a). The A subgenome consists of a substantial amount of repetitive DNA in centromeric and pericentromeric regions (Fig. 3b). However, the A subgenome has 4.05.9% lower repetitive DNA content than the A-genome diploid (Ga), whereas the D subgenome has 1.52.9% higher content than the D-genome diploid (Gr) in Gh (Fig. 3c) and the other 4 species (Extended Data Fig. 5a). Consistently, the D subgenome has 1020% more long terminal repeat (LTR) TEs than the D-genome diploid, while the A subgenome has 311% fewer LTRs than the A-genome diploid. These changes in subgenomic TEs may account for slight genome downsizing (Table 1) and genome-size equilibration following allopolyploidy in all five species, suggesting that the evolutionary tape is replayed across polyploid lineages. Copia- and Gypsy-like TEs are the most abundant LTRs in the Gh genome25. Estimates indicate that divergence of 5.6% (Gt) to 15.5% (Gh) and 39.7% (Gb) LTRs occurred during polyploid diversification (<0.6Ma; Extended Data Fig. 5bf). Since polyploid formation, LTRs increased substantially in the D subgenome of all five polyploids (Fig. 3d). The results indicate activation of LTRs in the D subgenome following polyploidization or movement of LTRs from the A to D subgenome37. Indeed, some Copia- and Gypsy-like elements are present in the D subgenome but absent in the extant D-genome diploid (Extended Data Fig. 5g). Gene family diversification The domesticated (Gh and Gb) and wild (Gm, Gt and Gd) cotton species share 417 (403) and 464 (359) unique genes (orthogroups) in respective groups (Fig. 2a), and no species-specific orthogroups are identified, although they possess distinct phenotypic traits such as fiber length (Fig. 1a) and flower morphology (Fig. 2c,d). The unique genes in the two domesticated cottons are over-represented in biological processes such as microtubule-based movement and lipid biosynthetic process and transport in the domesticated cottons (Fig. 2e; P<0.05), reflecting the traits related to fiber development and cottonseed oil. Moreover, many of these genes are under positive selection and overlap regions of domestication traits including fiber yield and quality in Upland cotton38 (Supplementary Dataset 6). The unique genes in all three wild polyploid species, however, are enriched for pollination and reproduction (Fig. 2f), suggesting a role of these genes in reproductive adaptation in natural environments. Plants have evolved an intricate innate immune system to protect them from pathogens and pests through intracellular disease-resistance (R) proteins as a defense response39. Among the R genes (Methods and Supplementary Note), each species has its unique R genes with very few genes shared between species (Fig. 2b and Supplementary Dataset 7), despite 5 wild and cultivated species sharing a core R-gene set (271), suggesting extensive diversification of R genes during selection and domestication. This is in contrast to a shared set of unique genes (related to fiber and seed traits) between the two cultivated species and the other shared set (related to reproductive and adaptive traits) among the three wild species (Fig. 2a) Between the two subgenomes, the D subgenome has higher numbers of R genes (7.8%) than does the A subgenome (P=0.0126, Students t-test; Supplementary Dataset 7). Using the published data40, we found expression induction of ~96% of 291 and 384 predicted R genes in the A and D subgenomes, respectively, by bacterial blight pathogens; 19 in D and 7 in A are upregulated at significant levels (error corrected, FDR=0.05 and P<0.001, exact test), while a similar trend of R-gene expression is observed after the reniform nematode attack (Supplementary Dataset 8), suggesting a contribution of the D-genome species to disease-resistance traits. Gene expression diversity In the five allotetraploid species sequenced, gene expression diversity is dynamic and pervasive across developmental stages and between subgenomes (Supplementary Dataset 9). Principal component analysis shows clear separation of expression between developmental stages (PC1) and between subgenomes (PC3; Extended Data Fig. 6a), with more D homoeologs expressed than A homoeologs in most tissues examined (Extended Data Fig. 7), consistent with higher levels of tri-methylation of Lys4 on histone H3 (H3K4me3) in the former than in the latter41. Notably, expression correlates more closely with the subgenomic variation than with tissue types, except for fiber elongation and cellulose biosynthesis, where subgenomic expression patterns are more closely correlated between Upland and Pima cottons (Extended Data Fig. 6b). This may suggest that domestication drives parallel expression similarities of fiber-related genes in the two cultivated species. These differentially expressed genes in fibers may contribute to fiber development, as they show enrichment of GO groups in hydrolase and GTPase-binding activities (Extended Data Fig. 8a,b). Hydrolases are essential for plant cell wall development42, and Ras and Ran GTPases are implicated in the transition from primary to secondary wall synthesis in fibers43. Moreover, translation and ribosome biosynthesis pathway genes are enriched during fiber elongation in Upland cotton and during cellulose biosynthesis in Pima cotton, consistent with faster fiber development in Upland cotton and longer fiber duration in Pima cotton44. Expression networks and m6A RNA in fibers Gene expression diversity is also reflected by coexpression modules in fibers among four species (Supplementary Dataset 10 and Supplementary Note). These module-related genes show higher semantic similarities between domesticated cottons (GhGb) than with two wild species (Gt and Gm). The modules include supramolecular fiber organization genes in Upland cotton and brassinosteroid signaling genes in Pima cotton, which could affect fiber cell elongation45. The two wild species have different biological functions and transcription factors enriched in fiber-related gene modules (Supplementary Dataset 11), which may account for the fiber traits that are very different from those of the domesticated species (Fig. 1a). Transcriptional and post-transcriptional regulation, including the activity of small RNAs and DNA methylation, mediates fiber cell development46. Modification of m6A messenger RNA can stabilize mRNA and promote translation with a role in developmental regulation of plants and animals47. In Upland cotton, m6A peaks are found largely in the 5Ê¹ and 3Ê¹ untranscribed regions (Extended Data Fig. 8c) of 1,205 genes in developing fibers (Supplementary Dataset 12), at levels 7-fold more than in leaves (Extended Data Fig. 8d) (P<0.002, Students t-test), while the number of expressed genes is similar in both tissues. Notably, both m6A-modified mRNAs and transcriptome data in the fibers target the genes involved in translation, hydrolase activity and GTPase-binding activities (Extended Data Fig. 8a). These results indicate that mRNA stability and translational activities may determine fiber elongation and cellulose biosynthesis when cell cycles arrest in fiber cells. Recombination and epigenetic landscapes Polyploidy leads to low genetic recombination, as observed in B. napus48, which may comprise bottlenecks for breeding improvement. To determine the recombination landscapes in polyploid cottons, we genotyped 17,134 SNPs using the new Gh sequence and the CottonSNP63K array49 and identified a total of 1,739 low-recombination haplotype blocks (cold spots) in Upland cotton using whole-genome population-based linkage analysis50 (Methods and Supplementary Note). These blocks (average ~678.9kb with 8.4SNPs) span 1.18Gbp (~52%) of the genome, including ~58% and ~41% in the A and D subgenomes, respectively (Fig. 4a), and are dispersed among all chromosomes with large ones predominately near pericentromeric regions. Recombination is generally suppressed throughout haplotype blocks, in contrast to that in subtelomeric regions (Extended Data Fig. 9a). Chromosome A08 has 62 haplotype blocks, including an exceptionally large one (~72Mb) (Fig. 4b). Interestingly, interspecific hybridization between different tetraploids can increase recombination rates in these regions. For example, in the Gb Ã GhF2 population, recombination rates increased more than 46cMMb1 in the left region (2930Mb) and in two other regions in the same Gb Ã GhF2 population. Recombination rates were also increased in the Gm Ã GhBC1F1 population (Fig. 4b). Similar increases were observed in the homoeologous D08 low-recombination haplotype blocks in the Gb Ã GhF2 population. Moreover, these haplotype blocks of either parent segregated with expected ratios within the population of Gh Ã GmBC2F1 (Extended Data Fig. 9b) or Gh Ã GtBC3F1 (Extended Data Fig. 9c). These data suggest the stability and selection of these haplotype regions during domestication and breeding. Notably, genome-wide recombination cold spots (haplotype block) and hotspots (no haplotype block) correlated with the DNA methylation frequency at CG, CHG (H=A, T or C) and CHH sites in the cultivated allotetraploids Gh and Gb (Pearson r=0.994; Fig. 4c and Extended Data Fig. 10a,b), with higher methylation frequencies in the cold spots than in the hotspots (analysis of variance (ANOVA), P<1-10e). The data support the role of DNA methylation in altering recombination landscapes, as reported in Arabidopsis51,52. Consistent with this notion, DNA methylation changes that are induced in the interspecific hybrid (Ga Ã Gr) are also largely maintained in the five allotetraploid cotton species, creating hundreds and possibly thousands of epialleles, including the ones responsible for photoperiodic flowering and worldwide cultivation of cotton53. Moreover, recombination events in all three interspecific crosses (Gb Ã GhF2, Gm Ã GhBC1F1 and Gt Ã GhBC1F1) correlated negatively with the average numbers of strongly connecting sites (intensity>5) (P<8.842Ã1016) and their connection intensities (P<7.26Ã1012) of the Hi-C chromatin matrix (Pearson r=0.874; Extended Data Fig. 10c). Recombination hotspots have fewer but more intense chromatin interactions within short distances, while the cold spots tend to have more but weaker interactions in long distances (Extended Data Fig. 10c,d). For example, 2 hotspots and 9 cold spots in the A08 region (Extended Data Fig. 10d), including 7 cold spots spanning ~32Mb correlated with weak Hi-C intensities and DNA hypermethylation (Extended Data Fig. 10e). These data indicate that DNA hypermethylation and weak chromatin interactions interfere with recombination events in polyploid cottons. Discussion Despite wide geographic distribution and diversification, five allotetraploid cotton genomes have largely retained the gene content and genomic synteny relative to respective extant diploids. This level of genome stability is in contrast to rapid genomic changes observed in some newly formed allotetraploids such as B. napus10 and T. miscellus11. However, in cultivated canola, the two subgenomes are relatively undisrupted8, probably because the extant parental species existing today to make new tetraploids10 may be different from the ones that formed cultivated canola ~7,500 years ago16 and likely became extinct. In addition, all five cotton polyploid species have a monophyletic origin, which is similar to the origin of wild and domesticated tetraploid peanuts54, but different from recurrent formation of Tragopogon tetraploids55. Notably, since polyploid formation 11.5Ma, the evolution of 2 subgenomes in each of the 5 allotetraploid cotton species does not exhibit a simple asymmetrical pattern, as reported in Upland cotton25. Instead, the two subgenomes have diversified and experienced novel heterogeneous evolutionary trajectories, including partial equilibration of subgenome size mediated by differential TE exchanges, pervasive evolutionary rate shifts, and positive selection between homoeologs within and among lineages. These features present in all five allotetraploid species suggest that the evolutionary tape is replayed during polyploid diversification and speciation. Among the five allotetraploid genomes, no species-specific orthologs were identified, except for one set of the unique genes related to fiber and seed traits in the two domesticated cottons and another set of the unique genes for reproduction and adaptation in the three wild polyploid species. However, R-gene families have rapidly evolved in each allotetraploid and extensively diversified during selection and domestication. These genomic diversifications have been accompanied by dynamic and prevalent gene expression changes during growth and development between wild and cultivated polyploid species, including parallel gene expression, coexpression networks and m6A mRNA modifications in fibers of the cultivated species. Remarkably, polyploid cotton genomes show recombination suppression or haplotype blocks, which correlate with altered epigenetic landscapes and can be overcome by wild introgression and possibly epigenetic manipulation. This finding is contemporary to the discovery of the Ph1 locus that inhibits pairing of homoeologous chromosomes in polyploid wheat56,57. The recombination suppression may help maintain a repository of epigenes or epialleles that were generated by interspecific hybridization accompanied by polyploidization and could have shaped polyploid cotton evolution, selection and domestication53. These conceptual advances and genomic and epigenetic resources will help improve cotton fiber yield and quality as a sustainable alternative to petroleum-based synthetic fibers. Modifying epigenetic landscapes and using gene-editing tools may also overcome the limited genetic diversity within polyploid cottons. These principles may facilitate future efforts to concomitantly enhance the economic yield and sustainability of this global crop and possibly other polyploid crops. Methods Plant materials G. hirsutum L. acc. TM-1 (1008001.06), G. barbadense L. acc. 3-79 (1400233.01), G. tomentosum L. (7179.01,02,03), G. darwinii L. (AD5-32, no. 1808015.09) and G. mustelinum L. (1408120.09, 1408120.10, 1408121.01, 1408121.02, 1408121.03) were grown in a greenhouse in College Station at Texas A&M University. Young leaves were collected for preparation of high-molecular-weight DNA using a published method58. Total RNA was extracted from leaf, root, stem, square, cotyledon, hypocotyl, meristem, petal, stamen, exocarp, ovule (0, 3, 7, 14, 21 and 35days post anthesis (DPA)) and fiber (7, 14, 21 and 35DPA) tissues in Gh; from leaf, root, stem, square, cotyledon, flower, ovule (14DPA) and fiber (14DPA) tissues in Gb; from leaf, root, stem, square, cotyledon and fiber (14DPA) tissues in Gm; from leaf, root, stem, square, flower, ovule (0, 7, 14, 21 and 28DPA) and fiber (7, 14, 21 and 28 DPA) tissues in Gt; and from leaf, root and stem tissues in Gd. Two or three biological replicates were used for RNA-seq and m6A RNA-seq analyses. Genome sequencing and assembly Sequencing reads were collected using Illumina HiSeq and NovaSeq and PacBio SEQUEL and RSII platforms. We sequenced and assembled five Gossypium genomes using high-coverage (>74Ã) single-molecule real-time long-read sequencing (Pac Biosciences). A total of six Illumina libraries were sequenced using the HiSeq platform, and two libraries were sequenced using NovaSeq. Initially, all five species were assembled using MECAT59 and subsequently polished using long reads, as well as Illumina reads. Gb and Gh were polished using QUIVER60, while Gd, Gt and Gm were polished using ARROW60. Ten Hi-C libraries were sequenced for five cotton genomes (two for each species). The total amount of Illumina sequenced for all 5 species (Supplementary Dataset 1) is 4,361,212,302 reads for a total of 286.4Ã of high-quality Illumina bases. A total of 105,182,984 PacBio reads were sequenced for all 5 genomes with a total coverage of 439.61Ã. Chromosome integration of Gb and Gh leveraged a combination of published Gh synteny and Hi-C scaffolding. A total of 148,239 unique, non-repetitive, non-overlapping 1-kb sequences were extracted from the published Gh genome25 and aligned to the Gh and Gb MECAT assemblies. Misjoins in the MECAT assembly were identified, and the assembly was scaffolded with Hi-C data using the JUICER pipeline61. Small rearrangements to both genomes were made using the JUICEBOX interface62. Finally, a set of 5,275 clones (474.3Mb total sequence) were used to patch remaining gaps in the Gh assembly. A total of 626 gaps were patched resulting in 1,871,050base pairs (bp) being added to the assembly. Gd and Gm were integrated into chromosomes using Gb (3-79) synteny, whereas Gt was integrated using the Gh release assembly version 1 Final refinements to the Gt assembly were made using the JUICER/JUICEBOX pipeline61. In all five of the assemblies, care was taken to ensure that the telomere was properly oriented in the chromosomes, and the resulting sequence was screened for retained vector and/or contaminants. Genome annotation and gene prediction procedures are provided in the Supplementary Note. Dot plots (pairwise comparisons) were generated using Gepard version 1.30 (ref. 63). The input data consist of 2 FASTA files, as well as the appropriate flags (-seq1 FASTA_FILE_1 -seq2 FASTA_FILE_2 -matrix edna.mat -zoom 65000 -word 18 -lower 0 -upper 20 -greyscale 0 -format png), with the -zoom flag from 65,000 (D subgenome) to 119,000 (A subgenome). The edna.mat file is part of the Gepard version 1.30 release. As a rule of thumb, this factor is generated by dividing the number of bases of the input FASTA file by 1,000. The output from the Gepard command is a PNG image file. Procedures for the analysis of SNPs and indels are provided in the Supplementary Note. Comparative analysis with published assemblies Assessment of genome completeness We evaluated the genome assembly completeness by k-mer masking (24-nucleotide) reciprocally between Gh (TM-1)22 and Gh (TM-1, this study) and between Gb (Hai7124)22 and Gb (3-79, this study). The unmasked contiguous sequences of the unshared sequence were extracted into a FASTA file and analyzed using FASTA statistics. BBMap ( and Custom Python scripts (Supplementary Note) were used for this analysis. Genome comparisons using Hi-C data The Hi-C libraries IKCF (Gh) and ILDE (Gb) were aligned to published Gh and Gb reference genomes using BWA-MEM64."). Heatmaps were generated using the JUICER-pre command, and visualized using JUICEBOX62."). Inversions and rearrangements were further identified using JUICEBOX. Analysis of chromosomal collinearity, structural rearrangements and gene family composition between reference assemblies Published Gh and Gb assemblies22 were aligned to the assemblies generated in this study using Minimap2 (ref. 65) with the parameter setting -ax asm5 --eqx. The resulting alignments were used to identify structural rearrangements and local variations using SyRI66."). The gene copy numbers and gene families between assemblies were identified using OrthoFinder67.") based on all annotated protein-coding sequences. Analysis of evolutionary rate changes and gene gain and loss Evolutionary rate changes in subgenomes of allopolyploid cotton during diversification Rates of evolution for each subgenome of each species across the phylogeny were calculated using pairwise p-distances for the same 17,136 orthologs in all 5 polyploid species (Extended Data Fig. 4a). The distribution of p-distances between each species was compared for both subgenomes using a one-tailed Wilcoxon signed rank test and Bonferroni correction for multiple testing. Differences in evolutionary rates between the subgenomes within each species were evaluated using a modified relative rate test whereby p-distance distributions were compared for both subgenomes to determine which had the greater p-distance (that is, higher inferred rate). Differences in subgenome evolutionary rates among lineages were estimated using a modified relative rate test that again used the Wilcoxon signed rank test with the p-distances of 17,136 genes, here comparing p-distances between two species relative to an outgroup species. This test was repeated for all possible pairs of tip and outgroup combinations. We also summed the total number of differences contained within all orthologs between each pairwise set of species, excluding all sites in which any of the orthologs contained a gap sequence (Supplementary Dataset 2a). Chi-square tests were used to determine the significance of these total substitution counts (Supplementary Dataset 2b). Analysis of gene loss and gain after polyploid cotton formation A total of 32,622 groups of SCOs were identified between subgenomes of all 5 allopolyploids and the diploids Gr and Ga (Extended Data Fig. 4c). Of those, the 4,369 SCO groups that were present in both diploid species but absent in at least 1 allopolyploid subgenome were evaluated for gene losses specific to allopolyploids. The list of SCO groups was converted into a binary matrix of gene occurrence and mapped onto the inferred phylogeny of ten allopolyploid subgenomes (with five taxa each in the At- and Dt-subgenome clades, rooted by the respective diploid progenitors). Using a likelihoodbased mixture model assuming predominantly gene losses over gains and stochastic mapping implemented in GLOOME68, both the total number of gene gains and losses per branch and the associated probability of each event across the phylogeny were estimated. Identification of homoeologs under selection The homoeolog pairs of five species were used for estimating non-synonymous/synonymous (Ka/Ks) values. Every pair of the sequences were aligned using the MUSCLE alignment software69 and then transferred to the AXT format for identifying positively selected genes (Ka/Ks>1) using the KaKs calculator70. Positively selected genes in A and D homoeologs were compared pairwise among 5 species (Supplementary Dataset 2). Analyses of repetitive sequences and TEs Pairwise comparison of 18-nucleotide sequences between homoeologous chromosomes was performed by Gepard plots63. Analysis of the k-mer content of all of the genomes was conducted by LTR-harvest71 according to the manual. The whole-genome sequences were suffixed first and then indexed using the seed length 20. The frequency of individual 20-nucleotide sequences was estimated using in-house Perl scripts. This analysis was applied to the two diploid cotton species, Ga and Gr, and the five tetraploid allopolyploids, with the A or D subgenome examined separately. The software LTR-harvest71 and LTR-finder72 was used for identifying full-length LTR retrotransposons. The identification parameters were as follows. For LTR-harvest: overlaps best -seed 20 -minlenltr 100 -maxlenltr 2000 -mindistltr 3000 -maxdistltr 25000 -similar 85 -mintsd 4 -maxtsd 20 -motif tgca -motifmis 1 -vic 60 -xdrop 5 -mat 2 -mis -2 -ins -3 -del -3. For LTR-finder: -D 15000 -d 1000 -L 7000 -l 100 -p 20 -C -M 0.9. The two datasets were integrated to remove false positives using the LTR-retriever packages73. The insertion time was estimated using the formula T=Ks/2r, where Ks is the divergence rate and r (3.48Ã109) is the substitution rate in cotton17. Full-length TE sequences were extracted from each of the seven species and were used to build a TE database; the cd-hit software74 was applied to remove redundancies through self-sequence similarity tests, and sequences with identity>90% were grouped into the same cluster. A cluster present in only one species was defined as a species-specific TE cluster, and those present in more than one species were considered shared TE clusters. A total of 98,794 full-length LTRs were identified in all 7 cotton species and grouped into 20,583 clusters for analysis of their origins in Ga, Gr, and the A and D subgenomes in 5 allotetraploids. R-gene family and expression analysis in response to pathogen treatments We detected nucleotide-binding site, leucine-rich repeat (NBSLRR) motifs with the pfamscan tool75 that uses the hidden Markov model search tool (HMMER) version 3.2.1 (ref. 76) by searching primary protein-coding transcripts of each of the 5 allotetraploid cottons against the raw hidden Markov model for the NB-ARC-domain family downloaded from Pfam (PF00931). Identified NBSLRR protein-coding genes for each of the allotetraploid cottons were further analyzed for amino-terminal (TIR/coiled-coil/other) and other functional domains by searching them against the Pfam-A hidden Markov model with the PfamScan tool and HMMER version 3.1 (ref. 76) with default settings (Supplementary Note). Short-read sequencing data for bacterial blight were downloaded from the Sequence Read Archive from the NCBI Bioproject accession PRJNA395458 (ref. 40). Reniform nematode sequence data were downloaded from the NCBI Bioproject accession PRJNA269348. Sequence data were aligned to the 653 predicted R genes from the Gh version 2.0 (this study) with Bowtie2 version 2.3.4.1 and filtered for true-pair alignments. Fragments per kilobase million (FPKM) and read counts per million were determined with RSEM version 1.3.0. Differentially expressed R genes were determined with edgeR77 using false discovery rate (FDR)-corrected P values of 0.05. Of the 291 A-subgenome and 384 D-subgenome predicted R genes, we found FPKM expression profiles (>1) for at least 1 condition in 281 and 372 of the A- and D-subgenome predicted R genes, respectively. Similarly, in response to reniform nematode challenge in Gh, 274 of 291 A-subgenome and 370 of 384 D-subgenome predicted R genes were expressed at the FPKM level (>1) for at least 1 of the 4 conditions tested. RNA-seq library construction, sequencing and data normalization Total RNA was extracted from leaf, root, stem, square, flower, ovule and fiber samples from Gh, Gb, Gt, Gm and Gd species (2 replicates each for 124 samples; Supplementary Dataset 9), using PureLink Plant RNA Reagent (ThermoFisher). After DNase treatment, RNA-seq libraries were constructed using an NEBNext Ultra II RNA Library Kit (NEB), and 150-bp paired-end sequences were generated using an Illumina Hiseq 2500. Paired-end sequence data were quality trimmed (Q¥25) and reads shorter than 50bp after trimming were discarded. Sequences were then aligned to respective allotetraploid cotton genomes and counts of reads uniquely mapping to annotated genes were obtained using STAR (version 2.5.3a). Outliers among the biological replicates were verified on the basis of the Pearson correlation coefficient, r2¥0.85. Fragments per kilobase of exon per million (FPKM) fragments mapped values were calculated for each gene by normalizing the read count data to both the length of the gene and the total number of mapped reads in the sample and considered as the metric for estimating gene expression levels78. Normalized count data were obtained using the relative logarithm expression (RLE) method in DESeq2 (version 1.14.1)79. Genes with low expression were filtered out, by requiring ¥2 RLE-normalized counts in at least 2 samples for each gene. Additional data for RNA-seq expression in fiber (28DAP) tissue in both Gh and Gb were downloaded from the published data44 and processed as described above and in the Supplementary Note. Statistical analysis of differentially expressed genes To measure the gene expression differences between homoeologous genes in RNA-seq data, we used the DESeq2 package in R based on the negative binomial distribution (Supplementary Note). Only genes with log2[fold change]¥1, BenjaminiHochberg-adjusted P<0.05 were retained. The comparison of highly expressed homoeologous gene pairs between subgenomes in different tissues was carried out using a binomial test (P<0.05). GO enrichment was analyzed using topGO80, an R Bioconductor package with Fishers exact test; only GO terms with P<0.05 (FDR<0.05) were considered significant. Principal component analysis and correlation coefficient analysis To visualize subgenome and tissue expression relatedness, we used categorized gene expression values. These expression values were averaged across replicates and log2-transformed. Principal component analysis employed singular value decomposition via the prcomp function in R81. Categorized gene expression values were used in this analysis. Pearsons correlation coefficients were determined and hierarchical clustering was carried out using the Euclidian distance and complete linkage method. m6A RNA-seq data analysis m6A RNA-seq libraries were constructed using a modified protocol as previously described82. Briefly, total RNA was extracted from young leaf and fiber tissues at 7DPA (2 replicates each) from Gh by using PureLink Plant RNA Reagent (ThermoFisher). mRNA was collected from total RNA by the Oligotex mRNA mini kit (QIAGEN), fragmented and pulled down using an m6A antibody, followed by library construction using the NEBNext Ultra II RNA Library Kit (NEB) without polyA tail selection. Fragmented mRNA-seq libraries (control; input) and m6A RNA-seq libraries (IP) were sequenced using an Illumina Hiseq 2500 and 150-bp reads. Illumina reads were mapped to the Gh genome using Tophat 2.1.1 (ref. 83), and the uniquely mapped reads were used to identify m6A peaks with the Bioconductor package exomePeak84 (Supplementary Dataset 12). GO terms were extracted from the GeneAnnotation_info.txt file. Identified m6A peak genes were analyzed by the Bioconductor package topGO80 to identify significantly over-represented GO terms (P<0.0001). The location of RNA (5Ê¹UTR, CDS or 3Ê¹UTR) for each m6A RNA-seq read (both input and IP) was identified using the intersect function of Bedtools85. Single, double and triple asterisks indicate statistical significance levels of P<0.05, P<0.01 and P<0.001, respectively (Students t-test). We extracted the gene expression data for Gh leaf and fiber at 7DPA corresponding to m6A peak genes. All refers to the expression level of all identified homoeologous genes in the leaf and fiber samples, while peak corresponds to the expression level of the identified m6A peaks for the genes in leaf (161 genes) and fiber (1,205 genes) samples. Single, double and triple asterisks indicate statistical significance levels of P<0.05, P<0.01 and P<0.001, respectively (Students t-test). Fluorescence in situ hybridization of A and D homoeologous chromosomes Procedures for the preparation of metaphase chromosomes in Gh and fluorescence in situ hybridization were adopted from a published protocol86, with a modification that the cotton root tips were pretreated with cycloheximide (25ppm) for 3h at room temperature. The 25S rDNA fragment was obtained from Arabidopsis87 and originally provided by R. Hasterok from Poland. Synthetic oligonucleotides for forward and reverse plant telomeric sequences were PCR-amplified and products were labeled by nick translation to create probe to detect telomeres88. Genotyping and recombination rate analyses Genotyping data representing an improved cotton panel of 257 Gh accessions were acquired from a previously published diversity analysis49 utilizing the CottonSNP63K array89. The genotyping data in 2 segregating populations included 18 lines each representing 1 family of a Gh Ã GmBC2F1 population and 33 lines each representing 1 family of a Gh Ã GtBC3F1 population. SNPs with a minor allele frequency greater than 5% and that had less than 10% missing data were retained. Genotyping data were further filtered for homeo-SNPs that occur due to intragenomic sequence identity89. Array ID sequences were aligned to the Joint Genome Institute Gh version 2.0 sequence assembly using BLASTn90 (version 2.7.1+) with a minimum e-value cutoff of 1Ã1010. Homoeologous alignments were corrected for using previously published SNP segregation data89,91, as well as interspecific, bi-parental linkage mapping populations from their respective Gh Ã GmBC1F1 and Gh Ã GtBC1F1 initial mapping populations. Genotyping data were then imputed and phased using Beagle (version 4.1)92, and genotypes were converted to ABH format to distinguish genotypic parentage. It is notable that erroneous SNP calling is a common problem in polyploids and especially in the AD-genome allotetraploid cotton because of homoeologous and paralogous sequences. This issue has been addressed through several methods89,93,94. In this study, we used the published method89 to avoid erroneous genotype calling and to provide accurate chromosome-specific and homoeologous haplotype structure. Furthermore, we used a historical estimation of recombination95, as shown in the haplotype structure using confidence intervals, as well as in two segregating populations, which led to the accurate estimates of recombination rates between parental alleles using linkage disequilibrium analysis95. The haplotype block partitioning was conducted with PLINK50 (Supplementary Note). The recombination map for chromosome A08 of Gh was developed using 4 SNP-based genetic maps, including 3 of interspecific crosses between Gb Ã Gh (F2, n=195), Gt Ã Gh (BC1F1, n=85) and Gm Ã Gh (BC1F1, n=59) and 1 consensus map that was generated using 3 intraspecific populations91. All genetic maps were aligned to the Joint Genome Institute Gh version 2.0 sequence assembly using the previously stated methods. Recombination map visualization was estimated using the R package MareyMap96 using the nonlinear LOESS method97, and the number of surrounding markers used to fit a local polynomial was 7.5% of the total number of markers per chromosome. Final map plotting was conducted using the R package ggplot2 (ref. 98). Localized recombination rates for chromosomes A08 and D08 were estimated using a 1-Mb non-overlapping sliding window with a minimum of 4 SNPs per window as a linear regression threshold using MareyMap. DNA methylation analysis Methylome sequencing data were downloaded from a published report53. In brief, methylC-seq reads of all allopolyploid cottons were mapped to genome sequences of Gh and Gb, respectively, using Bismark with the parameters (--score_min L,0,-0.2 -X 1000 --no-mixed --no-discordant)99. Only the uniquely mapped reads were retained and used for further analysis. Reads mapped to the same site were collapsed into a single consensus molecule to reduce clonal bias. Cytosine counts were combined into 1,000-bp windows using methylKit 1.2.4 (ref. 100). The DNA methylation (CG, CHG and CHH) levels (percentage of methylated cytosines) and average Hi-C seq statistics (number of connections, intensity or interaction matrix, and distance) in each recombination spot were compared using custom Python scripts. The Pearson correlation coefficient (r) was estimated using singular value decomposition via the prcomp function in R81. Single, double and triple asterisks indicate statistical significance levels of P<0.001, P<1Ã105 and P<1Ã1010, respectively, using one-way ANOVA. Chromatin conformation capture (Hi-C) sequencing analysis Hi-C seq libraries were constructed using a previously described protocol101,102, with modifications. Briefly, young leaves from Gh, Gb, Gt, Gm and Gd (2 replicates each) and fiber samples from Gh were fixed in 1% formaldehyde, and nuclei were extracted. Fixed chromatin was digested with DpnII, filled in using biotin-14-dATP and ligated. The biotin-labeled DNA was extracted and pulled down to construct HiC-seq libraries. Sequencing of Hi-C seq libraries was performed using an Illumina Hiseq 2500 and 150-bp reads. Reads were mapped to respective genomes and analyzed by HiC-Pro103. The Hi-C read coverage is 205Ã for Gh, 45Ã for Gb, 36Ã for Gm, 22Ã for Gd and 17Ã for Gt. The Hi-C data were largely used to correct orientations and misalignments in the assemblies of contigs and scaffolds. For Gh, Hi-C data were used to generate chromatin connection heatmaps with the HiCPlotter ( Single, double and triple asterisks indicate statistical significance levels of P<0.001, P<1Ã105 and P<1Ã1010, respectively, using one-way ANOVA. Reporting Summary Further information on research design is available in the Nature Genetics Research Reporting Summary linked to this article. 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CAS PubMed Google Scholar 103. Servant, N. et al. HiC-Pro: an optimized and flexible pipeline for Hi-C data processing. Genome Biol. 16, 259 (2015). PubMed PubMed Central Google Scholar Download references Acknowledgements We thank J. R. Ecker, E. S. Dennis, T. Zhang, A. H. Paterson, R. G. Cantrell and C. L. Brubaker for their roles in coordinating the sequencing white paper and J. A. Udall for initial discussion of the cotton diversity project. We also thank Texas Advanced Computing Center, Iowa State University Research Information Technology Unit and the Bioinformatics Center at Nanjing Agricultural University for computational support and assistance. This work is supported by grants from the National Science Foundation (IOS1444552 and IOS1739092 to Z.J.C., IOS1826544 to J.F.W.), the US Department of Agriculture (6066-21310-005-00-D to B.E.S., NACA 58-6066-6-046 and NACA 58-6066-6-059 to D.G.P.) and Cotton Incorporated (14-371 to Z. J.C., 13-965 to J.S., 18-195 to J.F.W., 13-466TX, 13-636, 13-694 and 18-201 to D.M.S.). The work conducted by the US Department of Energy Joint Genome Institute is supported by the Office of Science of the US Department of Energy under contract no. DE-AC02-05CH11231 (S.Shu and J.W.C.). The work is also supported by grants from the National Natural Science Foundation of China (91631302 to Q.S. and Z.J.C.), Jiangsu Collaborative Innovation Center for Modern Crop Production (Q.S. and W.Y.) and the Natural Science Foundation of Zhejiang Province, China (LY17C060005 to M.D.). Author information Author notes These authors contributed equally: Z. Jeffrey Chen, Avinash Sreedasyam, Atsumi Ando, Qingxin Song, Luis M. De Santiago. Authors and Affiliations Department of Molecular Biosciences, The University of Texas at Austin, Austin, TX, USA Z. Jeffrey Chen, Atsumi Ando, Qingxin Song, Mingquan Ding & Ryan C. Kirkbride 2. State Key Laboratory for Crop Genetics and Germplasm Enhancement, Nanjing Agricultural University, Nanjing, China Z. Jeffrey Chen, Qingxin Song & Wenxue Ye 3. HudsonAlpha Institute for Biotechnology, Huntsville, AL, USA Avinash Sreedasyam, Jerry Jenkins, Christopher Plott, John Lovell, Lori B. Boston, Melissa Williams, Jane Grimwood & Jeremy Schmutz 4. Department of Soil and Crop Sciences, Texas A&M University System, College Station, TX, USA Luis M. De Santiago, Yu-Ming Lin, Robert Vaughn, Bo Liu & David M. Stelly 5. US Department of Agriculture-Agricultural Research Service, Genomics and Bioinformatics Research Unit, Raleigh, NC, USA Amanda M. Hulse-Kemp 6. College of Agriculture and Food Science, Zhejiang A&F University, Linan, China Mingquan Ding 7. US Department of Agriculture-Agricultural Research Service, Genomics and Bioinformatics Research Unit, Stoneville, MS, USA Sheron Simpson & Brian E. Scheffler 8. Department of Plant and Environmental Sciences, Clemson University, Clemson, SC, USA Li Wen & Christopher A. Saski 9. Department of Ecology, Evolution, and Organismal Biology, Iowa State University, Ames, IA, USA Corrinne E. Grover, Guanjing Hu, Justin L. Conover & Jonathan F. Wendel 10. The US Department of Energy Joint Genome Institute, Walnut Creek, CA, USA Joseph W. Carlson, Shengqiang Shu & Jeremy Schmutz Institute for Genomics, Biocomputing and Biotechnology and Department of Plant and Soil Sciences, Mississippi State University, Mississippi State, MS, USA Daniel G. Peterson 12. School of Agriculture and Applied Sciences, Alcorn State University, Lorman, MS, USA Keith McGee 13. Agriculture and Environmental Research, Cotton Incorporated, Cary, NC, USA Don C. Jones Authors Z. Jeffrey Chen View author publications Search author on:PubMed Google Scholar 2. Avinash Sreedasyam View author publications Search author on:PubMed Google Scholar 3. Atsumi Ando View author publications Search author on:PubMed Google Scholar 4. Qingxin Song View author publications Search author on:PubMed Google Scholar 5. Luis M. De Santiago View author publications Search author on:PubMed Google Scholar 6. Amanda M. Hulse-Kemp View author publications Search author on:PubMed Google Scholar 7. Mingquan Ding View author publications Search author on:PubMed Google Scholar 8. Wenxue Ye View author publications Search author on:PubMed Google Scholar 9. Ryan C. Kirkbride View author publications Search author on:PubMed Google Scholar 10. Jerry Jenkins View author publications Search author on:PubMed Google Scholar Christopher Plott View author publications Search author on:PubMed Google Scholar 12. John Lovell View author publications Search author on:PubMed Google Scholar 13. Yu-Ming Lin View author publications Search author on:PubMed Google Scholar 14. Robert Vaughn View author publications Search author on:PubMed Google Scholar 15. Bo Liu View author publications Search author on:PubMed Google Scholar 16. Sheron Simpson View author publications Search author on:PubMed Google Scholar 17. Brian E. Scheffler View author publications Search author on:PubMed Google Scholar 18. Li Wen View author publications Search author on:PubMed Google Scholar 19. Christopher A. Saski View author publications Search author on:PubMed Google Scholar 20. Corrinne E. Grover View author publications Search author on:PubMed Google Scholar 21. Guanjing Hu View author publications Search author on:PubMed Google Scholar 22. Justin L. Conover View author publications Search author on:PubMed Google Scholar 23. Joseph W. Carlson View author publications Search author on:PubMed Google Scholar 24. Shengqiang Shu View author publications Search author on:PubMed Google Scholar 25. Lori B. Boston View author publications Search author on:PubMed Google Scholar 26. Melissa Williams View author publications Search author on:PubMed Google Scholar 27. Daniel G. Peterson View author publications Search author on:PubMed Google Scholar 28. Keith McGee View author publications Search author on:PubMed Google Scholar 29. Don C. Jones View author publications Search author on:PubMed Google Scholar 30. Jonathan F. Wendel View author publications Search author on:PubMed Google Scholar 31. David M. Stelly View author publications Search author on:PubMed Google Scholar 32. Jane Grimwood View author publications Search author on:PubMed Google Scholar 33. Jeremy Schmutz View author publications Search author on:PubMed Google Scholar Contributions Z.J.C., J.G., D.M.S., B.E.S. and C.A.S. conceived and designed the project, A.S., A.A., Q.S., L.M.D.S., A.M.H.-K., M.D., J.J., R.C.K., Y.-M.L., C.P., J.L., B.L., C.E.G., G.H., J.L.C. and L.W. generated the data, B.E.S., D.G.P., D.C.J., K.M., R.V., S.Simpson, S.Shu, J.W.C., L.B.B., M.W. and W.Y. provided materials, reagents and technical support, Z.J.C., A.S., A.A., Q.S., L.M.D.S., A.M.H.-K., J.L., A.M.H.-K., C.E.G., G.H., J.L.C., D.M.S., C.A.S., J.G. and J.S. analyzed the data, and Z.J.C., J.G., J.S., A.S., A.A., L.M.D.S., A.M.H.-K., D.M.S., C.A.S. and J.F.W. wrote the paper. All authors have read and approved the paper. Corresponding authors Correspondence to Z. Jeffrey Chen or Jane Grimwood. Ethics declarations Competing interests Cotton Incorporated is a not-for-profit company working with cotton scientists, the textile industry and consumers. Additional information Publishers note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Extended data Extended Data Fig. 1 Sequencing features of four cotton allotetraploid species. ad, Chromosomal features and synteny of G. hirsutum (Gh) (a), G. barbadense (Gb) (b), G. tomentosum (Gt) (c), and G. darwinii (Gd) (d) genomes. Notes in the circos plots: (a) estimated lengths of 13A and 13 D homoeologous pseudochromosomes; (b) density distribution of annotated genes; (c) TE content (Gypsy, steel blue; Copia, grey; other repeats, orange); (d, e) stacked SNP (d) and INDEL (e) densities between species, respectively (see inset); (f) syntenic blocks between the homoeologous A and D chromosomes. The densities in plots in (b-e) are represented in 1Mb with overlapping 200-kb sliding windows. Source data Extended Data Fig. 2 Summary of completeness assessment and collinearity and similarity between G. hirsutum (Gh) and G. barbadense (Gb) genomes. a, Summary of genome completeness assessment by 24-mer reciprocal masking between the published22 and our assemblies of Gh and Gb genomes. b, Nucleotide alignment dot plots comparing the collinearity and similarity between the genomes of Gh (published22 vs. this study, left panel) and Gb (Hai712422 vs. 3-79 of this study, right panel). Plots show y axis (bottom to top) for chromosomes A01-A13 and D01-D1322 and x axis (left to tight) for chromosomes A01-13 and D01-D13 (this study). Boxed regions represent inversions and rearrangements assessed using Hi-C data. Minimum nucleotide alignment length = 1 Kb; color scale, mean percent identity per query. c, Hi-C interaction maps indicating rearrangements and inversions in the published Gh genome22 with several small rearrangements flanking a large 200-Kb gap in A02, a large inversion in A06, and rearrangements in D08. Source data Extended Data Fig. 3 Estimates of divergence time based on synonymous substitution rates (Ks). a, The divergence time is estimated to be 58-59 million years ago (Mya) between Theobroma cacao and Gossypium. Data shown using Ks bin size of 0.001. Divergence time [T=Ks/(2r)] was estimated using the synonymous substitution rate (r) of 3.48Ã109 synonymous substitutions per synonymous site per year17 and 10,562 single copy orthologs between subgenomes and species. Ks values >1 were removed to eliminate saturated synonymous sites. b, The synonymous substitution rate, Ks, distribution for orthologs (n=21,567), and estimates of divergence time between allotetraploid subgenomes and progenitor-like diploid genomes. Gh: G. hirsutum; Gb: G. barbadense; Ga: G. arboreum; Gr: G. raimondii; Gm: G. mustelium. Using a penalized-likelihood based on the concatenated nuclear tree (including branch lengths), the divergence between diploid-tetraploid clade is estimated to be 1-1.6 Mya. Source data Extended Data Fig. 4 Monophyletic origin and diversification of five allotetraploid species. a, The phylogeny of the polyploid species using 18,672 orthologous (37,344 homoeologous) genes and improved coalescence analysis. b, Geographic distribution and diversification of the five allotetraploid species G. hirsutum, G. barbadense, G. tomentosum, G. darwinii, and G. mustelium and their progenitor-like diploids, G. arboreum and G. raimondii. The world map was made using R scripts, and the distribution maps were redrawn based on published maps for Gd, Gt, and Gm30, Gh and Gb31, and diploid cultivated cottons32. c, Patterns of gene gain and loss using 4,369 single-copy orthologs (SCOs) (out of total 32,622), which are present in both diploids and in one or more allotetraploids. Numbers above and below each branch indicate number of gene gain (A-blue/D-red subgenome) or loss (A-green/D-purple subgenome), respectively. Source data Extended Data Fig. 5 Analysis of 20-nucleotide sequence distributions in subgenomes and Copia and Gypsy insertion time in five allotetraploid cotton species. a, Cumulative percentage (y axis) of 20-nucleotide sequences and their frequencies (x axis) is lower in the A subgenome than in the A (Ga) genome and higher in the D subgenome than in the D (Gr) genome in G. mustelium (Gm), G. tomentosum (Gt), G. barbadense (Gb), and G. darwinii (Gd) (from left to right). b-f, Number of Copia and Gypsy elements (y axis, left) relative to the estimated time of insertion (x axis) in G. hirsutum (b), G. barbadense (c), G. darwinii (d), G. tomentosum (e), and G. mustelinum (f). The right (y axis) shows cumulative % of Copia and Gypsy in the genome over divergence time (orange line). The number shown in each species indicates cumulative % of Copia and Gypsy at ~600 Kya. Note: Divergence time [T=Ks/(2r)] was estimated using the synonymous substitution rate (r) of 3.4Ã109 synonymous substitutions per synonymous site per year. g, Movement of TEs from the A subgenomes to the D subgenomes in allotetraploids. The number of each TE cluster (TC3-TC3060, top-bottom) is shown in the right. Color scale, TE density. Source data Extended Data Fig. 6 Gene expression diversity between subgenomes and among different developmental stages and five allotetraploid cotton species. a, Principal component analysis (PCA) of all genes during vegetative (leaf, stem, and root), reproductive (ovules at 0-35 DAP and square), fiber elongation (7, 14, and 21 DAP), and cellulose biosynthesis (28 and 35 DAP) stages, separating gene expression diversity among different developmental stages and between A and D subgenomes (marked by the dotted lines. b, Clustering analysis of 96 RNA-seq datasets with 2 biological replicates in fiber elongation (E), cellulose biosynthesis (C), vegetative (veg), and reproductive (rep) stages of cotton development. Source data Extended Data Fig. 7 Homeolog expression differences in four allotetraploid cotton species. a, Expression levels of homoeologs were compared among different tissues in each speces. The number of homoeologous genes that are more highly expressed (log2-fold change ¥1, Benjamini-Hochberg adjusted P<0.05; Wald test) in the A or D subgenome. Asterisks indicate P<0.05 (two-sided binomial test). b, Classification of homoeologous pairs by expression patterns. The downward arrow marks the fraction that shows differential expression in different tissues of four species. c-f, Number of homoeolog pairs (y axis) whose expression levels are A>D (pale blue), D>A (dark blue), sub- or neo-funcationalization in A (dark green) or in D (pale green) in G. hirsutum (c), G. tomentosum (d), G. barbadense (e), and G. mustelinum (f). Tissue types are shown in x axis. G. darwinii was not included in the analysis due to a small number of tissue types available for the study. Source data Extended Data Fig. 8 Gene Ontology (GO) analysis of differentially expressed genes and analysis of m6A mRNA modifications in Upland cotton. a, GO analysis of upregulated genes in two cultivated cottons and three wild relatives (>2-fold change, FPKM>5, and ANOVA p-value < 0.05) and m6A-associated genes in the leaf and fiber of Upland cotton. Color bars=-log10(p-value). b, GO analysis of upregulated genes (>2-fold change, FPKM>5, and ANOVA p-value < 0.05) in different tissues of G. hirsutum and G. barbadense. Color bars=-log10(p-value). c, Density of m6A marks in the genic region, 5Ê¹ and 3Ê¹ UTR of ethe xpressed genes in the fiber (red) and leaf (green). Students t-test was used to compare between m6A immuno-precipitated and fragmented (control) RNA reads with single () and triple () asterisks indicating statistical significance levels of P<0.05 and <0.001, respectively. d, Expression levels (y axis) of the genes with m6A peaks in the leaf (161 genes) and fiber (1,205 genes) (green), relative to all homoeologous genes (red). Students t-test was used to compare between m6A-associated genes and all homoeologous genes with double () and triple () asterisks indicating statistical significance levels of P<0.01 and <0.001, respectively. Source data Extended Data Fig. 9 Recombination rate distribution in G. hirsutum and inheritance of haplotype blocks in two breeding populations. a, Recombination rate distribution between A and D subgenomes. The recombination bins are based on overlapping 5-Mb windows. The dashed grey lines indicate 50% of individuals recombined in the window. The pale blue polygons link syntenic regions. The x axis is scaled independently for each homoeologous chromosome. b, Linkage disequilibrium heatmap of chromosome A08 of the G. hirsutumXG. mustelinum BC2F1 population. Genotypes of 18 lines each representative of one family, two parents, and F1 are shown using the CottonSNP63K array (top panel). Red, yellow, and blue colors show the genotypes homozygous for G. hirsutum, homozygous for G. mustelinum, and heterozygous for both species, respectively. Heatmap (bottom panel) consists of equidistant tiles that indicate linkage disequilibrium as determined by a normalized coefficient of linkage disequilibrium (D) between pairs of markers. Markers corresponding to SNP positions above the heatmap are congruent to the introgressed genotypes (x axis). c, Linkage disequilibrium heatmap of chromosome A08 of the G. hirsutumXG. tomentosum BC3F1 population. Genotypes of 33 lines each representative of one family, two parents, and F1 are shown using the CottonSNP63K array (top panel). Red, yellow, and blue colors show the genotypes homozygous for G. hirsutum, homozygous for G. tomentosum, and heterozygous for both species, respectively. Heatmap (bottom panel) consists of equidistant tiles that indicate linkage disequilibrium as determined by a normalized coefficient of linkage disequilibrium (D) between pairs of markers. Markers corresponding to SNP positions above the heatmap are congruent to the introgressed genotypes (x axis). Source data Extended Data Fig. 10 Correlation of DNA methylation levels and chromatin connecting sites and intensities with recombination cold (haplotype block) and hot (no block) spots. a, Average percentage (%) of CG (circle), CHG (triangle), and CHH (cross) methylation in the recombination hot (red) and cold (blue) spots between Gb (y axis) and Gh (x axis), with an enlarged image showing CHH methylation levels. Pearson correlation coefficient is 0.994. b, Average methylation percentage (y axis) of the recombination spots in different cross in CG, CHG, and CHH sites (x axis). Colors indicate recombination hot and cold spots in the three interspecific crosses GhXGbF2 (red and blue), GmXGhBC1F1 (pink and light blue), and GtXGhBC1F1 (white and black), respectively. ANOVA was used for statistical tests with ingle (), double (), and triple () asterisks indicating statistical significance levels of P-value<0.001, <1e-5, and <1e-10, respectively. c, Chromatin interaction matrices show correlation of chromatin connecting intensity (y axis, cutoff >5) with average chromatin connecting numbers (x axis, 20-Kb window) of recombination hot (red) and cold (blue) spots in the three interspecific crosses, GhXGbF2 (circles), GmXGhBC1F1 (triangles), GtXGhBC1F1 (squares). Pearson correlation coefficient is -0.874 with triple () asterisks indicating the statistical significance level of P-value<1e-10 (Students t-test). d, Comparison of Hi-C interaction matrix (log2-intensity) in chromosome A08 of the GbXGhF2 cross, consisting of recombination hot (red) and cold spots (blue). Locations for one hot spot and two cold spots are shown. e, Zoom-in images of two cold and one hot spots in Hi-C interaction matrix (log2 intensity) in chromosome A08, consisting of recombination hot (red) and cold spots (blue), with CG (black), CHG (blue), and CHH (red) methylation densities (100-kb sliding windows). Values at the top of the heatmap represent Hi-C window size (20-kb) and genomic locations (Mb). Gh: G. hirsutum; Gb: G. barbadense; Gt: G. tomentosum; Gm: G. mustelinum. Source data Supplementary information Supplementary Information Supplementary Note Reporting Summary Supplementary Data Twelve supplementary datasets. Source data Source Data Fig. 1 Sequence statistics, genomic features and syntenic relationships. Source Data Fig. 2 List of genes specific to domesticated cottons and wild species, respectively. Source Data Fig. 3 TE compositions among five species. Source Data Fig. 4 Low-recombination haplotype blocks, and their corresponding methylation data. Source Data Extended Data Fig. 1 Sequence statistics, genomic features and syntenic relationships. Source Data Extended Data Fig. 2 Copy number variants (CNVs) and structural variations in TM-1 and 3-79 relative to the published data. Source Data Extended Data Fig. 3 List of Sequence Read Archive files and SCOs for maximum likelihood and coalescent analyses. Source Data Extended Data Fig. 4 Single-copy orthologs for phylogenetic analysis and for gene loss and gain tests among five species. Source Data Extended Data Fig. 5 Statistics of TEs between A and D subgenomes among five species and their respective A and D extant diploids. Source Data Extended Data Fig. 6 RNA-seq gene expression data among different tissues and species. Source Data Extended Data Fig. 7 RNA-seq expression data for homoeologs in five species. Source Data Extended Data Fig. 8 GO analysis of the differentially expressed genes among five species and in the fiber and leaf with m6A RNA modifications in upland cotton. Source Data Extended Data Fig. 9 Genomic locations of low-recombination haplotype blocks. Source Data Extended Data Fig. 10 Comparative analysis for methylome-seq and Hi-C seq data with recombination hotspot and cold-spot distributions. Rights and permissions Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit Reprints and permissions About this article Cite this article Chen, Z.J., Sreedasyam, A., Ando, A. et al. Genomic diversifications of five Gossypium allopolyploid species and their impact on cotton improvement. Nat Genet 52, 525533 (2020). Download citation Received: Accepted: Published: Issue Date: DOI: Share this article Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 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2575	https://www.mathopenref.com/coordpointdist.html	Math Open Reference Home Contact About Subject Index Distance from a point to a line (Coordinate Geometry) The distance from a point to a line is the shortest distance between them - the length of a perpendicular line segment from the line to the point. Try this Adjust the sliders to change the line equation and drag the point C. Note the distance from the point to the line. You can also drag the origin point at (0,0). When we talk about the distance from a point to a line, we mean the shortest distance. If you draw a line segment that is perpendicular to the line and ends at the point, the length of that line segment is the distance we want. In the figure above, this is the distance from C to the line. There are many ways to calculate this distance. In this volume, four methods are described: Method 1. When the line is horizontal or vertical If you are lucky and the line is either exactly horizontal or vertical (parallel to the x or y axis), then the distance is very easy to calculate. See Distance from a point to a horizontal / vertical line Method 2. Using two line equations From the equation of the given line we find the equation of the line perpendicular to it that passes through the given point. (Does not work for vertical lines.) See Distance from a point to a line using line equations Method 3. Using trigonometry The distance is found using trigonometry on the angles formed. See Distance from a point to a line using trigonometry Method 4. By formula Given the equation of the line in slope - intercept form, and the coordinates of the point, a formula yields the distance between them. (Does not work for vertical lines.) See Distance from a point to a line using a formula Limitations In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off. For more see Teaching Notes Other Coordinate Geometry topics Introduction to coordinate geometry The coordinate plane The origin of the plane Axis definition Coordinates of a point Distance between two points Introduction to Lines in Coordinate Geometry Line (Coordinate Geometry) Ray (Coordinate Geometry) Segment (Coordinate Geometry) Midpoint Theorem Distance from a point to a line - When line is horizontal or vertical - Using two line equations - Using trigonometry - Using a formula Intersecting lines Cirumscribed rectangle (bounding box) Area of a triangle (formula method) Area of a triangle (box method) Centroid of a triangle Incenter of a triangle Area of a polygon Algorithm to find the area of a polygon Area of a polygon (calculator) Rectangle Definition and properties, diagonals Area and perimeter Square Definition and properties, diagonals Area and perimeter Trapezoid Definition and properties, altitude, median Area and perimeter Parallelogram Definition and properties, altitude, diagonals Print blank graph paper (C) 2011 Copyright Math Open Reference. All rights reserved
2576	https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/resources/ch9sourc_b_field/	Chapter 9 Sources of Magnetic Fields 9.1 Biot-Savart Law ....................................................................................................... 2 Interactive Simulation 9.1 : Magnetic Field of a Current Element .......................... 3 Example 9.1: Magnetic Field due to a Finite Straight Wire ...................................... 3 Example 9.2: Magnetic Field due to a Circular Current Loop .................................. 6 9.1.1 Magnetic Field of a Moving Point Charge ....................................................... 9 Animation 9.1 : Magnetic Field of a Moving Charge ............................................. 10 Animation 9.2 : Magnetic Field of Several Charges Moving in a Circle ................ 11 Interactive Simulation 9.2 : Magnetic Field of a Ring of Moving Charges .......... 11 9.2 Force Between Two Parallel Wires ....................................................................... 12 Animation 9.3 : Forces Between Current-Carrying Parallel Wires ......................... 13 9.3 Ampere’s Law ........................................................................................................ 13 Example 9.3: Field Inside and Outside a Current-Carrying Wire ............................ 16 Example 9.4: Magnetic Field Due to an Infinite Current Sheet .............................. 17 9.4 Solenoid ................................................................................................................. 19 Examaple 9.5: Toroid ............................................................................................... 22 9.5 Magnetic Field of a Dipole .................................................................................... 23 9.5.1 Earth’s Magnetic Field at MIT ....................................................................... 24 Animation 9.4 : A Bar Magnet in the Earth’s Magnetic Field ................................ 26 9.6 Magnetic Materials ................................................................................................ 27 9.6.1 Magnetization ................................................................................................. 27 9.6.2 Paramagnetism ................................................................................................ 30 9.6.3 Diamagnetism ................................................................................................. 31 9.6.4 Ferromagnetism .............................................................................................. 31 9.7 Summary ................................................................................................................ 32 9.8 Appendix 1: Magnetic Field off the Symmetry Axis of a Current Loop ............... 34 9.9 Appendix 2: Helmholtz Coils ................................................................................ 38 Animation 9.5 : Magnetic Field of the Helmholtz Coils ......................................... 40 Animation 9.6 : Magnetic Field of Two Coils Carrying Opposite Currents ........... 42 Animation 9.7 : Forces Between Coaxial Current-Carrying Wires ......................... 43 0Animation 9.8 : Magnet Oscillating Between Two Coils ....................................... 43 Animation 9.9 : Magnet Suspended Between Two Coils ........................................ 44 9.10 Problem-Solving Strategies ................................................................................. 45 9.10.1 Biot-Savart Law: ........................................................................................... 45 9.10.2 Ampere’s law: ............................................................................................... 47 9.11 Solved Problems .................................................................................................. 48 9.11.1 Magnetic Field of a Straight Wire ................................................................ 48 9.11.2 Current-Carrying Arc .................................................................................... 50 9.11.3 Rectangular Current Loop ............................................................................. 51 9.11.4 Hairpin-Shaped Current-Carrying Wire ........................................................ 53 9.11.5 Two Infinitely Long Wires ........................................................................... 54 9.11.6 Non-Uniform Current Density ...................................................................... 56 9.11.7 Thin Strip of Metal ........................................................................................ 58 9.11.8 Two Semi-Infinite Wires .............................................................................. 60 9.12 Conceptual Questions .......................................................................................... 61 9.13 Additional Problems ............................................................................................ 62 9.13.1 Application of Ampere's Law ....................................................................... 62 9.13.2 Magnetic Field of a Current Distribution from Ampere's Law ..................... 62 9.13.3 Cylinder with a Hole ..................................................................................... 63 9.13.4 The Magnetic Field Through a Solenoid ...................................................... 64 9.13.5 Rotating Disk ................................................................................................ 64 9.13.6 Four Long Conducting Wires ....................................................................... 64 9.13.7 Magnetic Force on a Current Loop ............................................................... 65 9.13.8 Magnetic Moment of an Orbital Electron ..................................................... 65 9.13.9 Ferromagnetism and Permanent Magnets ..................................................... 66 9.13.10 Charge in a Magnetic Field ......................................................................... 67 9.13.11 Permanent Magnets ..................................................................................... 67 9.13.12 Magnetic Field of a Solenoid ...................................................................... 67 9.13.13 Effect of Paramagnetism ............................................................................. 68 1Sources of Magnetic Fields 9.1 Biot-Savart Law Currents which arise due to the motion of charges are the source of magnetic fields. When charges move in a conducting wire and produce a current I, the magnetic field at any point P due to the current can be calculated by adding up the magnetic field contributions, G , from small segments of the wire ddB s G , (Figure 9.1.1). Figure 9.1.1 Magnetic field dB G at point P due to a current-carrying element I d s G . These segments can be thought of as a vector quantity having a magnitude of the length of the segment and pointing in the direction of the current flow. The infinitesimal current source can then be written as I d s G . Let r denote as the distance form the current source to the field point P, and the corresponding unit vector. The Biot-Savart law gives an expression for the magnetic field contribution, G , from the current source, ˆr dB Id s G , 02 ˆ4 I d d r μ π × = s rB GG (9.1.1) where 0 μ is a constant called the permeability of free space : (9.1.2) 70 4 10 T m/A μ π −= × ⋅ Notice that the expression is remarkably similar to the Coulomb’s law for the electric field due to a charge element dq : 20 1 ˆ4 dq d r πε =E r G (9.1.3) Adding up these contributions to find the magnetic field at the point P requires integrating over the current source, 202wire wire ˆ4 I dd r μ π × = =∫ ∫ s rB B GG G (9.1.4) The integral is a vector integral, which means that the expression for B is really three integrals, one for each component of B G G . The vector nature of this integral appears in the cross product G . Understanding how to evaluate this cross product and then perform the integral will be the key to learning how to use the Biot-Savart law. ˆI d ×s r Interactive Simulation 9.1 : Magnetic Field of a Current Element Figure 9.1.2 is an interactive ShockWave display that shows the magnetic field of a current element from Eq. (9.1.1). This interactive display allows you to move the position of the observer about the source current element to see how moving that position changes the value of the magnetic field at the position of the observer. Figure 9.1.2 Magnetic field of a current element. Example 9.1: Magnetic Field due to a Finite Straight Wire A thin, straight wire carrying a current I is placed along the x-axis, as shown in Figure 9.1.3. Evaluate the magnetic field at point P. Note that we have assumed that the leads to the ends of the wire make canceling contributions to the net magnetic field at the point .P Figure 9.1.3 A thin straight wire carrying a current I. 3Solution : This is a typical example involving the use of the Biot-Savart law. We solve the problem using the methodology summarized in Section 9.10. (1) Source point (coordinates denoted with a prime) Consider a differential element ˆ'd dx =s i G carrying current I in the x-direction. The location of this source is represented by ˆ' 'x=r i G . (2) Field point (coordinates denoted with a subscript “ P”) Since the field point P is located at ( , ) (0, )x y a= , the position vector describing P is .ˆ P a=r j G (3) Relative position vector The vector is a “relative” position vector which points from the source point to the field point. In this case, 'P= −r r r G G G ˆ 'a x= −r j ˆi G , and the magnitude 2\| \| 'r a= = +r 2 x G is the distance from between the source and P. The corresponding unit vector is given by 22 ˆ ˆ' ˆ ˆˆ sin cos ' a xr a x θ θ−= = = −+ r j ir j i G (4) The cross product ˆd ×s r G The cross product is given by ˆ ˆ ˆ ˆˆ ( ' ) ( cos sin ) ( 'sin )d dx dx θ θθ× = × − + =s r i i j k G (5) Write down the contribution to the magnetic field due to Id s G The expression is 0022 ˆ sin ˆ4 4 I Id dx d r r μ μ θ ππ × = = s rB k GG which shows that the magnetic field at P will point in the ˆ+k direction, or out of the page. (6) Simplify and carry out the integration 4The variables θ, x and r are not independent of each other. In order to complete the integration, let us rewrite the variables x and r in terms of θ. From Figure 9.1.3, we have 2 / sin csc cot csc r a a x a dx a θ θ d θ θ θ= =⎧⎪⎨ = ⇒ = − ⎪⎩ Upon substituting the above expressions, the differential contribution to the magnetic field is obtained as 2002 ( csc )sin sin 4 ( csc ) 4 I Ia ddB da a μμθ θ θ θ θπθπ−= = − Integrating over all angles subtended from 1 θ− to 2 θ (a negative sign is needed for 1 θ in order to take into consideration the portion of the length extended in the negative x axis from the origin), we obtain 21002 sin (cos cos )4 4 I IB da a θθ1 μ μ θ θθππ−= − = +∫ θ (9.1.5) The first term involving 2 θ accounts for the contribution from the portion along the + x axis, while the second term involving 1 θ contains the contribution from the portion along the axis. The two terms add! x− Let’s examine the following cases: (i) In the symmetric case where 2 1 θ θ= − , the field point P is located along the perpendicular bisector. If the length of the rod is 2L , then 21cos / 2 L L a θ = + and the magnetic field is 00122 cos 2 2 I I LB a a L a μ μθππ= = + (9.1.6) (ii) The infinite length limit L → ∞ This limit is obtained by choosing ( ,1 2 ) (0, 0) θ θ = . The magnetic field at a distance a away becomes 0 2 I B a μ π = (9.1.7) 5Note that in this limit, the system possesses cylindrical symmetry, and the magnetic field lines are circular, as shown in Figure 9.1.4. Figure 9.1.4 Magnetic field lines due to an infinite wire carrying current I. In fact, the direction of the magnetic field due to a long straight wire can be determined by the right-hand rule (Figure 9.1.5). Figure 9.1.5 Direction of the magnetic field due to an infinite straight wire If you direct your right thumb along the direction of the current in the wire, then the fingers of your right hand curl in the direction of the magnetic field. In cylindrical coordinates ( , , )r z ϕ where the unit vectors are related by ˆ ˆ ˆ× =r φ z , if the current flows in the + z-direction, then, using the Biot-Savart law, the magnetic field must point in the ϕ -direction. Example 9.2: Magnetic Field due to a Circular Current Loop A circular loop of radius R in the xy plane carries a steady current I, as shown in Figure 9.1.6. (a) What is the magnetic field at a point P on the axis of the loop, at a distance z from the center? (b) If we place a magnetic dipole ˆ z μ=μ k G at P, find the magnetic force experienced by the dipole. Is the force attractive or repulsive? What happens if the direction of the dipole is reversed, i.e., ˆ z μ= − μ k G 6Figure 9.1.6 Magnetic field due to a circular loop carrying a steady current. Solution: (a) This is another example that involves the application of the Biot-Savart law. Again let’s find the magnetic field by applying the same methodology used in Example 9.1. (1) Source point In Cartesian coordinates, the differential current element located at ˆ' (cos ' sin 'R ˆ) φ φ= +r i G j can be written as ˆ ˆ( '/ ') ' '( sin ' cos ' ) Id I d d d IRd φ φ φ φ φ= = − +s r i j G G . (2) Field point Since the field point P is on the axis of the loop at a distance z from the center, its position vector is given by ˆ P z=r k G . (3) Relative position vector 'P= −r r r G G G The relative position vector is given by ˆ ˆ ˆ' cos ' sin 'P R R φφ− = − − +r = r r i j k G G G z and its magnitude ( )22 2( cos ') sin 'r R R z R φφ= = − + − + = +r G 2 2 z (9.1.9) is the distance between the differential current element and P. Thus, the corresponding unit vector from Id s G to P can be written as 'ˆ \| ' PP r \| − = = − r rrr r r G GG G G 7 (9.1.8) (4) Simplifying the cross product The cross product can be simplified as ( ') Pd × −s r r G G G ( )ˆ ˆ ˆ ˆ ˆ( ') ' sin ' cos ' [ cos ' sin ' ]ˆ ˆ ˆ'[ cos ' sin ' ] P d R d R R zR d z z R φ φφφφφ φφ× − = − + × − − += + + s r r i j i j ki j k G G G (9.1.10) (5) Writing down dB G Using the Biot-Savart law, the contribution of the current element to the magnetic field at P is 000230223/ 2 ˆ ( '4 4 4 \|ˆ ˆ ˆcos ' sin ' '4 ( ) PP I I I d d dd r rIR z z R dR z 3 )' \| μ μμπππμφφ φπ × −× ×= = = −+ += + s r rs r s rB r ri j k G G GG G GG G G (9.1.11) (6) Carrying out the integration Using the result obtained above, the magnetic field at P is 20223/ 2 0 ˆ ˆ ˆcos ' sin ' '4 ( ) IR z z R dR z π μφφ φ π+ += +∫ i j kB G (9.1.12) The x and the y components of B can be readily shown to be zero: G 200223/ 2 223/ 2 0 2cos ' ' sin ' 004 ( ) 4 ( ) x IRz IRz B dR z R z π πμμφ φφππ= =+ +∫ = (9.1.13) 200223/ 2 223/ 2 0 2sin ' ' cos ' 004 ( ) 4 ( ) y IRz IRz B dR z R z π πμμφ φφππ= = − + +∫ = (9.1.14) On the other hand, the z component is 222200223/ 2 223/ 2 223/ 2 0 2'4 ( ) 4 ( ) 2( ) z IR IR IR B dR z R z R z π μμ πφππ= = =+ +∫ 0 μ+ (9.1.15) Thus, we see that along the symmetric axis, zB is the only non-vanishing component of the magnetic field. The conclusion can also be reached by using the symmetry arguments. 8The behavior of 0/zB B where 0 0 / 2 B I R μ= is the magnetic field strength at , as a function of 0z = /z R is shown in Figure 9.1.7: Figure 9.1.7 The ratio of the magnetic field, 0/zB B , as a function of /z R (b) If we place a magnetic dipole ˆ z μ=μ k G at the point P, as discussed in Chapter 8, due to the non-uniformity of the magnetic field, the dipole will experience a force given by ˆ( ) ( ) zB z z zdB B dz μ μ ⎛ ⎞= ∇ ⋅ = ∇ = ⎜ ⎟⎝ ⎠ F μ B G G k G (9.1.16) Upon differentiating Eq. (9.1.15) and substituting into Eq. (9.1.16), we obtain 20225/ 2 3 ˆ2( ) zB IR z R z μ μ = − + F k G (9.1.17) Thus, the dipole is attracted toward the current-carrying ring. On the other hand, if the direction of the dipole is reversed, ˆ z μ= − μ k G , the resulting force will be repulsive. 9.1.1 Magnetic Field of a Moving Point Charge Suppose we have an infinitesimal current element in the form of a cylinder of cross-sectional area A and length ds consisting of n charge carriers per unit volume, all moving at a common velocity v G along the axis of the cylinder. Let I be the current in the element, which we define as the amount of charge passing through any cross-section of the cylinder per unit time. From Chapter 6, we see that the current I can be written as n Aq I=v G (9.1.18) The total number of charge carriers in the current element is simply , so that using Eq. (9.1.1), the magnetic field d dN n A ds = B G due to the dN charge carriers is given by 90 0 02 2ˆ ˆ( \| \|) ( ) ( )4 4 4 nAq d n A ds q dN q d r r 2ˆ r μ μμπππ × ×= = = v s r v r v rB × GG GG G (9.1.19) where r is the distance between the charge and the field point P at which the field is being measured, the unit vector ˆ G points from the source of the field (the charge) to P.The differential length vector is defined to be parallel to v / r=r r d s G G . In case of a single charge, , the above equation becomes 1dN = 02 ˆ4 qr μ π × = v rB GG (9.1.20) Note, however, that since a point charge does not constitute a steady current, the above equation strictly speaking only holds in the non-relativistic limit where v , the speed of light, so that the effect of “retardation” can be ignored. c The result may be readily extended to a collection of N point charges, each moving with a different velocity. Let the ith charge be located at ( iq , , )i i ix y z and moving with velocity . Using the superposition principle, the magnetic field at P can be obtained as: iv G 03/ 2 2221 ˆ ˆ ˆ( ) ( ) ( )4 ( ) ( ) ( ) Niiiiiiiii x x y y z zqx x y y z z μπ= ⎡ ⎤− + − + −⎢ ⎥= × ⎢ ⎥⎡ ⎤− + − + −⎣ ⎦⎣ ⎦∑ i j kB v G G (9.1.21) Animation 9.1 : Magnetic Field of a Moving Charge Figure 9.1.8 shows one frame of the animations of the magnetic field of a moving positive and negative point charge, assuming the speed of the charge is small compared to the speed of light. Figure 9.1.8 The magnetic field of (a) a moving positive charge, and (b) a moving negative charge, when the speed of the charge is small compared to the speed of light. 10 Animation 9.2 : Magnetic Field of Several Charges Moving in a Circle Suppose we want to calculate the magnetic fields of a number of charges moving on the circumference of a circle with equal spacing between the charges. To calculate this field we have to add up vectorially the magnetic fields of each of charges using Eq. (9.1.19). Figure 9.1.9 The magnetic field of four charges moving in a circle. We show the magnetic field vector directions in only one plane. The bullet-like icons indicate the direction of the magnetic field at that point in the array spanning the plane. Figure 9.1.9 shows one frame of the animation when the number of moving charges is four. Other animations show the same situation for N =1, 2, and 8. When we get to eight charges, a characteristic pattern emerges--the magnetic dipole pattern. Far from the ring, the shape of the field lines is the same as the shape of the field lines for an electric dipole. Interactive Simulation 9.2 : Magnetic Field of a Ring of Moving Charges Figure 9.1.10 shows a ShockWave display of the vectoral addition process for the case where we have 30 charges moving on a circle. The display in Figure 9.1.10 shows an observation point fixed on the axis of the ring. As the addition proceeds, we also show the resultant up to that point (large arrow in the display). Figure 9.1.10 A ShockWave simulation of the use of the principle of superposition to find the magnetic field due to 30 moving charges moving in a circle at an observation point on the axis of the circle. 11 Figure 9.1.11 The magnetic field due to 30 charges moving in a circle at a given observation point. The position of the observation point can be varied to see how the magnetic field of the individual charges adds up to give the total field. In Figure 9.1.11, we show an interactive ShockWave display that is similar to that in Figure 9.1.10, but now we can interact with the display to move the position of the observer about in space. To get a feel for the total magnetic field, we also show a “iron filings” representation of the magnetic field due to these charges. We can move the observation point about in space to see how the total field at various points arises from the individual contributions of the magnetic field of to each moving charge. 9.2 Force Between Two Parallel Wires We have already seen that a current-carrying wire produces a magnetic field. In addition, when placed in a magnetic field, a wire carrying a current will experience a net force. Thus, we expect two current-carrying wires to exert force on each other. Consider two parallel wires separated by a distance a and carrying currents I1 and I2 in the + x-direction, as shown in Figure 9.2.1. Figure 9.2.1 Force between two parallel wires The magnetic force, , exerted on wire 1 by wire 2 may be computed as follows: Using the result from the previous example, the magnetic field lines due to I 12 F G 2 going in the + x-direction are circles concentric with wire 2, with the field 2B G pointing in the tangential 12 direction. Thus, at an arbitrary point P on wire 1, we have 2 0 2 ˆ( / 2 )I a μ π= − B j G , which points in the direction perpendicular to wire 1, as depicted in Figure 9.2.1. Therefore, ( ) 0 2 0 1 212 1 2 1 ˆ ˆ ˆ2 2 I I I l I I l a a μμππ⎛ ⎞= × = × − = − ⎜ ⎟⎝ ⎠ F B i j GG G l k (9.2.1) Clearly points toward wire 2. The conclusion we can draw from this simple calculation is that two parallel wires carrying currents in the same direction will attract each other. On the other hand, if the currents flow in opposite directions, the resultant force will be repulsive. 12 F G Animation 9.3 : Forces Between Current-Carrying Parallel Wires Figures 9.2.2 shows parallel wires carrying current in the same and in opposite directions. In the first case, the magnetic field configuration is such as to produce an attraction between the wires. In the second case the magnetic field configuration is such as to produce a repulsion between the wires. (a) (b) Figure 9.2.2 (a) The attraction between two wires carrying current in the same direction. The direction of current flow is represented by the motion of the orange spheres in the visualization. (b) The repulsion of two wires carrying current in opposite directions. 9.3 Ampere’s Law We have seen that moving charges or currents are the source of magnetism. This can be readily demonstrated by placing compass needles near a wire. As shown in Figure 9.3.1a, all compass needles point in the same direction in the absence of current. However, when , the needles will be deflected along the tangential direction of the circular path (Figure 9.3.1b). 0I ≠ 13 Figure 9.3.1 Deflection of compass needles near a current-carrying wire Let us now divide a circular path of radius r into a large number of small length vectors G , that point along the tangential direction with magnitude ˆs∆ ∆s = φ s∆ (Figure 9.3.2). Figure 9.3.2 Amperian loop In the limit , we obtain 0∆ →s GG ( )0022 Id B ds rr μ I π μπ⎛ ⎞⋅ = = =⎜ ⎟⎝ ⎠∫ ∫B s G Gv v (9.3.1) The result above is obtained by choosing a closed path, or an “Amperian loop” that follows one particular magnetic field line. Let’s consider a slightly more complicated Amperian loop, as that shown in Figure 9.3.3 Figure 9.3.3 An Amperian loop involving two field lines 14 The line integral of the magnetic field around the contour abcda is (9.3.2) 2211 0 ( ) 0 [ (2 )] abcda ab bc cd cd d d d dB r B r θπ θ⋅ = ⋅ + ⋅ + ⋅ + ⋅= + + + −∫ ∫ ∫ ∫ ∫B s B s B s B s B s G G G G GG G G Gv d G where the length of arc bc is 2r θ , and 1 (2 )r π θ− for arc da . The first and the third integrals vanish since the magnetic field is perpendicular to the paths of integration. With 10 / 2 1B I r μ π= and 2 0 / 2 2B I r μ π= , the above expression becomes 00002121 ( ) [ (2 )] (2 )2 2 2 2abcda I I I Id r rr r 0 I μ μμ μ θ π θθπ θ μπππ π⋅ = + − = + − =∫ B s G G v (9.3.3) We see that the same result is obtained whether the closed path involves one or two magnetic field lines. As shown in Example 9.1, in cylindrical coordinates ( , , )r z ϕ with current flowing in the +z-axis, the magnetic field is given by 0 ˆ( / 2 )I r μ π=B φ G . An arbitrary length element in the cylindrical coordinates can be written as ˆ ˆ ˆd dr r d dz ϕ= + +s r φ z G (9.3.4) which implies 0000closed path closed path closed path (2 )2 2 2 I I Id r d dr μμμ I ϕ ϕππππ⎛ ⎞⋅ = = = =⎜ ⎟⎝ ⎠∫ ∫ ∫B s G Gv v v μ (9.3.5) In other words, the line integral of d⋅∫ B s G G v around any closed Amperian loop is proportional to enc I , the current encircled by the loop. Figure 9.3.4 An Amperian loop of arbitrary shape. 15 The generalization to any closed loop of arbitrary shape (see for example, Figure 9.3.4) that involves many magnetic field lines is known as Ampere’s law: 0enc d I μ⋅∫ B s = G G v (9.3.6) Ampere’s law in magnetism is analogous to Gauss’s law in electrostatics. In order to apply them, the system must possess certain symmetry. In the case of an infinite wire, the system possesses cylindrical symmetry and Ampere’s law can be readily applied. However, when the length of the wire is finite, Biot-Savart law must be used instead. Biot-Savart Law 02ˆ4 I dr μ π × = ∫ s rB GG general current source ex: finite wire Ampere’s law 0 enc d I μ⋅∫ B s = G Gv current source has certain symmetry ex: infinite wire (cylindrical) Ampere’s law is applicable to the following current configurations: Infinitely long straight wires carrying a steady current I (Example 9.3) Infinitely large sheet of thickness b with a current density J (Example 9.4). Infinite solenoid (Section 9.4). Toroid (Example 9.5). We shall examine all four configurations in detail. Example 9.3: Field Inside and Outside a Current-Carrying Wire Consider a long straight wire of radius R carrying a current I of uniform current density, as shown in Figure 9.3.5. Find the magnetic field everywhere. Figure 9.3.5 Amperian loops for calculating the B G field of a conducting wire of radius R. 16 Solution: (i) Outside the wire where r , the Amperian loop (circle 1) completely encircles the current, i.e., R≥ enc I I= . Applying Ampere’s law yields ( ) 02d B ds B r I π μ⋅ = = =∫ ∫B s G Gv v which implies 0 2 I B r μ π = (ii) Inside the wire where r , the amount of current encircled by the Amperian loop (circle 2) is proportional to the area enclosed, i.e., R< 2enc 2 r I IR ππ⎛ ⎞= ⎜ ⎟⎝ ⎠ Thus, we have ( ) 200 2 22 2 Irrd B r I B R R μππ μ ππ⎛ ⎞⋅ = = ⇒ =⎜ ⎟⎝ ⎠∫ B s G Gv We see that the magnetic field is zero at the center of the wire and increases linearly with r until r=R . Outside the wire, the field falls off as 1/r . The qualitative behavior of the field is depicted in Figure 9.3.6 below: Figure 9.3.6 Magnetic field of a conducting wire of radius R carrying a steady current I . Example 9.4: Magnetic Field Due to an Infinite Current Sheet Consider an infinitely large sheet of thickness b lying in the xy plane with a uniform current density 0 ˆJ=J i G . Find the magnetic field everywhere. 17 Figure 9.3.7 An infinite sheet with current density 0 ˆJ=J i G . Solution: We may think of the current sheet as a set of parallel wires carrying currents in the + x-direction. From Figure 9.3.8, we see that magnetic field at a point P above the plane points in the −y-direction. The z-component vanishes after adding up the contributions from all wires. Similarly, we may show that the magnetic field at a point below the plane points in the + y-direction. Figure 9.3.8 Magnetic field of a current sheet We may now apply Ampere’s law to find the magnetic field due to the current sheet. The Amperian loops are shown in Figure 9.3.9. Figure 9.3.9 Amperian loops for the current sheets For the field outside, we integrate along path . The amount of current enclosed by is 1 C 1C 18 enc 0 ( )I d J b= ⋅ =∫∫ J A GG A (9.3.7) Applying Ampere’s law leads to 0enc 00 (2 ) ( )d B I J b μ μ⋅ = = =∫ B s G Gv A A or 0 0 / 2 B J b μ= . Note that the magnetic field outside the sheet is constant, independent of the distance from the sheet. Next we find the magnetic field inside the sheet. The amount of current enclosed by path is 2C enc 0 (2 \| \| )I d J z= ⋅ =∫∫ J A GG A (9.3.9) Applying Ampere’s law, we obtain 0enc 00 (2 ) (2 \| \| )d B I J z μ μ⋅ = = =∫ B s G Gv A A or 0 0 \| \|B J z μ= . At , the magnetic field vanishes, as required by symmetry. The results can be summarized using the unit-vector notation as 0z = 000000 ˆ, / 2 2ˆ, / 2 / 2 ˆ, / 2 2 J b z bJ z b z bJ b z b μμμ⎧− >⎪⎪⎪= − − < <⎨⎪⎪ < − ⎪⎩ jB jj G (9.3.11) Let’s now consider the limit where the sheet is infinitesimally thin, with . In this case, instead of current density G , we have surface current 0b → 0 ˆJ=J i ˆK=K i G , where 0K J b = .Note that the dimension of K is current/length. In this limit, the magnetic field becomes 00 ˆ, 02ˆ, 02 K zK z μμ⎧− >⎪⎪= ⎨ ⎪ < ⎪⎩ jBj G (9.3.12) 9.4 Solenoid A solenoid is a long coil of wire tightly wound in the helical form. Figure 9.4.1 shows the magnetic field lines of a solenoid carrying a steady current I. We see that if the turns are closely spaced, the resulting magnetic field inside the solenoid becomes fairly uniform, 19 (9.3.8) (9.3.10) provided that the length of the solenoid is much greater than its diameter. For an “ideal” solenoid, which is infinitely long with turns tightly packed, the magnetic field inside the solenoid is uniform and parallel to the axis, and vanishes outside the solenoid. Figure 9.4.1 Magnetic field lines of a solenoid We can use Ampere’s law to calculate the magnetic field strength inside an ideal solenoid. The cross-sectional view of an ideal solenoid is shown in Figure 9.4.2. To compute B G ,we consider a rectangular path of length l and width w and traverse the path in a counterclockwise manner. The line integral of B G along this loop is (9.4.1) 0 0 0 d d d dBl ⋅ ⋅ + ⋅ + ⋅ + ⋅= + + +∫ ∫ ∫ ∫ ∫1 2 3 4B s = B s B s B s B G G G G GG G G Gv d s G Figure 9.4.2 Amperian loop for calculating the magnetic field of an ideal solenoid. In the above, the contributions along sides 2 and 4 are zero because B G is perpendicular to . In addition, GG along side 1 because the magnetic field is non-zero only inside the solenoid. On the other hand, the total current enclosed by the Amperian loop is d s G =B 0 enc I NI = , where N is the total number of turns. Applying Ampere’s law yields 0 d Bl N μ⋅ = = I∫ B s G G v (9.4.2) or 20 00 NI B nI l μ μ= = (9.4.3) where represents the number of turns per unit length., In terms of the surface current, or current per unit length /n N l= K nI = , the magnetic field can also be written as, 0 B K μ= (9.4.4) What happens if the length of the solenoid is finite? To find the magnetic field due to a finite solenoid, we shall approximate the solenoid as consisting of a large number of circular loops stacking together. Using the result obtained in Example 9.2, the magnetic field at a point P on the z axis may be calculated as follows: Take a cross section of tightly packed loops located at z’ with a thickness ' , as shown in Figure 9.4.3 dz The amount of current flowing through is proportional to the thickness of the cross section and is given by , where ( ') ( / ) dI I ndz I N l dz = = ' /n N l= is the number of turns per unit length. Figure 9.4.3 Finite Solenoid The contribution to the magnetic field at P due to this subset of loops is 2200223/ 2 223/ 2 (2[( ') ] 2[( ') ] z R RdB dI nIdz z z R z z R μμ= =− + − + ') (9.4.5) Integrating over the entire length of the solenoid, we obtain 22/ 2 00223/ 2 222/ 2 02222/ 2 / 2 ' '2 [( ') ] 2 ( ') ( / 2) ( / 2) 2 ( / 2) ( / 2) lzlll nIR nIR dz z zB z z R R z z RnI l z l zz l R z l R μμμ−−−= =− + − +⎡ ⎤− += +⎢ ⎥⎢ − + + + ⎥⎣ ⎦∫ (9.4.6) 21 A plot of 0/zB B , where 0 0B nI μ= is the magnetic field of an infinite solenoid, as a function of /z R is shown in Figure 9.4.4 for 10 l R= and 20 l R= . Figure 9.4.4 Magnetic field of a finite solenoid for (a) 10 l R= , and (b) .20 l R= Notice that the value of the magnetic field in the region \| \| / 2 z l< is nearly uniform and approximately equal to 0B . Examaple 9.5: Toroid Consider a toroid which consists of N turns, as shown in Figure 9.4.5. Find the magnetic field everywhere. Figure 9.4.5 A toroid with N turns Solutions: One can think of a toroid as a solenoid wrapped around with its ends connected. Thus, the magnetic field is completely confined inside the toroid and the field points in the azimuthal direction (clockwise due to the way the current flows, as shown in Figure 9.4.5.) Applying Ampere’s law, we obtain 22 0(2 )d Bds B ds B r N π μ⋅ = = = =∫ ∫ ∫B s G Gv v v I or 0 2 NI B r μ π = (9.4.8) where r is the distance measured from the center of the toroid.. Unlike the magnetic field of a solenoid, the magnetic field inside the toroid is non-uniform and decreases as 1/ .r 9.5 Magnetic Field of a Dipole Let a magnetic dipole moment vector ˆ μ= −μ k G be placed at the origin ( e.g., center of the Earth) in the plane. What is the magnetic field at a point ( e.g ., MIT) a distance r away from the origin? yz Figure 9.5.1 Earth’s magnetic field components In Figure 9.5.1 we show the magnetic field at MIT due to the dipole. The y- and z-components of the magnetic field are given by 20033 3 sin cos , (3cos 1) 4 4 yz B Br r μ μ μ μθ θθ π π= − = − − (9.5.1) Readers are referred to Section 9.8 for the detail of the derivation. In spherical coordinates (r, θ, φ), the radial and the polar components of the magnetic field can be written as 03 2sin cos cos 4 ryz B B B r μ μ θ θ θ π = + = − (9.5.2) 23 (9.4.7) and 03 cos sin sin 4 yz B B B r θ μ μ θ θ θ π = − = − (9.5.3) respectively. Thus, the magnetic field at MIT due to the dipole becomes 03 ˆ ˆˆ (sin 2 cos )4 r B B r θ ˆ μ μ θ θπ= + = − +B θ r θ r G (9.5.4) Notice the similarity between the above expression and the electric field due to an electric dipole p G (see Solved Problem 2.13.6): 30 1 ˆ ˆ(sin 2 cos )4 pr θ θπε = +E θ r G The negative sign in Eq. (9.5.4) is due to the fact that the magnetic dipole points in the −z-direction. In general, the magnetic field due to a dipole moment μ G can be written as 03 ˆ ˆ 3( )4 r μ π ⋅ −= μ r r μB G GG (9.5.5) The ratio of the radial and the polar components is given by 0303 2 cos 4 2 cot sin 4 r B rBr θ μ μ θπ θ μ μ θπ−= =− (9.5.6) 9.5.1 Earth’s Magnetic Field at MIT The Earth’s field behaves as if there were a bar magnet in it. In Figure 9.5.2 an imaginary magnet is drawn inside the Earth oriented to produce a magnetic field like that of the Earth’s magnetic field. Note the South pole of such a magnet in the northern hemisphere in order to attract the North pole of a compass. It is most natural to represent the location of a point P on the surface of the Earth using the spherical coordinates ( , , )r θ φ , where r is the distance from the center of the Earth, θ is the polar angle from the z-axis, with 0 θ π≤ ≤ , and φ is the azimuthal angle in the xy plane, measured from the x-axis, with 0 2 φ π≤ ≤ (See Figure 9.5.3.) With the distance fixed at , the radius of the Earth, the point P is parameterized by the two angles Er r= θ and φ . 24 Figure 9.5.2 Magnetic field of the Earth In practice, a location on Earth is described by two numbers – latitude and longitude. How are they related to θ and φ ? The latitude of a point, denoted as δ , is a measure of the elevation from the plane of the equator. Thus, it is related to θ (commonly referred to as the colatitude) by 90 δ θ= ° − . Using this definition, the equator has latitude 0 , and the north and the south poles have latitude ° 90 ± ° , respectively. The longitude of a location is simply represented by the azimuthal angle φ in the spherical coordinates. Lines of constant longitude are generally referred to as meridians .The value of longitude depends on where the counting begins. For historical reasons, the meridian passing through the Royal Astronomical Observatory in Greenwich, UK, is chosen as the “prime meridian” with zero longitude. Figure 9.5.3 Locating a point P on the surface of the Earth using spherical coordinates. Let the z-axis be the Earth’s rotation axis, and the x-axis passes through the prime meridian. The corresponding magnetic dipole moment of the Earth can be written as 00000 ˆ ˆ ˆ(sin cos sin sin cos )ˆ ˆ ˆ( 0.062 0.18 0.98 ) EEE μ θ φ θ φθμ= + += − + − μ i ji j k G k 25 (9.5.7) where , and we have used 22 27.79 10 A m E μ = × ⋅ 0 0( , ) (169 ,109 ) θ φ = ° ° . The expression shows that Eμ G has non-vanishing components in all three directions in the Cartesian coordinates. On the other hand, the location of MIT is for the latitude and 71 for the longitude ( north of the equator, and 71 west of the prime meridian), which means that 42 N ° W° 42 ° ° 90 42 48 m θ = ° − ° = ° , and 360 71 289 m φ = ° − ° = ° . Thus, the position of MIT can be described by the vector (9.5.8) MIT ˆ ˆ ˆ(sin cos sin sin cos )ˆ ˆ ˆ(0.24 0.70 0.67 ) EmmmmmE rr θ φθ φθ= + += − + r i ji j k G k The angle between E−μ G and is given by MIT r G 11MIT MIT cos cos (0.80) 37 \| \|\| \| EME E θ − −⎛ ⎞− ⋅= =⎜ ⎟−⎝ ⎠ r μr μ = ° G G G G (9.5.9) Note that the polar angle θ is defined as 1 ˆˆcos ( ) θ − = ⋅r k , the inverse of cosine of the dot product between a unit vector ˆr for the position, and a unit vector ˆ+k in the positive z- direction, as indicated in Figure 9.6.1. Thus, if we measure the ratio of the radial to the polar component of the Earth’s magnetic field at MIT, the result would be 2 cot 37 2.65 rBB θ = ° ≈ (9.5.10) Note that the positive radial (vertical) direction is chosen to point outward and the positive polar (horizontal) direction points towards the equator. Animation 9.4 : Bar Magnet in the Earth’s Magnetic Field Figure 9.5.4 shows a bar magnet and compass placed on a table. The interaction between the magnetic field of the bar magnet and the magnetic field of the earth is illustrated by the field lines that extend out from the bar magnet. Field lines that emerge towards the edges of the magnet generally reconnect to the magnet near the opposite pole. However, field lines that emerge near the poles tend to wander off and reconnect to the magnetic field of the earth, which, in this case, is approximately a constant field coming at 60 degrees from the horizontal. Looking at the compass, one can see that a compass needle will always align itself in the direction of the local field. In this case, the local field is dominated by the bar magnet. Click and drag the mouse to rotate the scene. Control-click and drag to zoom in and out. 26 Figure 9.5.4 A bar magnet in Earth’s magnetic field 9.6 Magnetic Materials The introduction of material media into the study of magnetism has very different consequences as compared to the introduction of material media into the study of electrostatics. When we dealt with dielectric materials in electrostatics, their effect was always to reduce E G below what it would otherwise be, for a given amount of “free” electric charge. In contrast, when we deal with magnetic materials, their effect can be one of the following: (i) reduce B below what it would otherwise be, for the same amount of "free" electric current ( diamagnetic materials); G (ii) increase B a little above what it would otherwise be ( paramagnetic materials); G (iii) increase B a lot above what it would otherwise be ( ferromagnetic materials). G Below we discuss how these effects arise. 9.6.1 Magnetization Magnetic materials consist of many permanent or induced magnetic dipoles. One of the concepts crucial to the understanding of magnetic materials is the average magnetic field produced by many magnetic dipoles which are all aligned. Suppose we have a piece of material in the form of a long cylinder with area A and height L, and that it consists of N magnetic dipoles, each with magnetic dipole moment μ G , spread uniformly throughout the volume of the cylinder, as shown in Figure 9.6.1. 27 Figure 9.6.1 A cylinder with N magnetic dipole moments We also assume that all of the magnetic dipole moments μ G are aligned with the axis of the cylinder. In the absence of any external magnetic field, what is the average magnetic field due to these dipoles alone? To answer this question, we note that each magnetic dipole has its own magnetic field associated with it. Let’s define the magnetization vector M G to be the net magnetic dipole moment vector per unit volume: 1 iiV = ∑M μ G G (9.6.1) where V is the volume. In the case of our cylinder, where all the dipoles are aligned, the magnitude of G is simply M /M N AL μ= . Now, what is the average magnetic field produced by all the dipoles in the cylinder? Figure 9.6.2 (a) Top view of the cylinder containing magnetic dipole moments. (b) The equivalent current. Figure 9.6.2(a) depicts the small current loops associated with the dipole moments and the direction of the currents, as seen from above. We see that in the interior, currents flow in a given direction will be cancelled out by currents flowing in the opposite direction in neighboring loops. The only place where cancellation does not take place is near the edge of the cylinder where there are no adjacent loops further out. Thus, the average current in the interior of the cylinder vanishes, whereas the sides of the cylinder appear to carry a net current. The equivalent situation is shown in Figure 9.6.2(b), where there is an equivalent current eq I on the sides. 28 The functional form of eq I may be deduced by requiring that the magnetic dipole moment produced by eq I be the same as total magnetic dipole moment of the system. The condition gives eq I A N μ= (9.6.2) or eq NI A μ = (9.6.3) Next, let’s calculate the magnetic field produced by eq I . With eq I running on the sides, the equivalent configuration is identical to a solenoid carrying a surface current (or current per unit length) K . The two quantities are related by eq I NK L AL μ M= = = (9.6.4) Thus, we see that the surface current K is equal to the magnetization M , which is the average magnetic dipole moment per unit volume. The average magnetic field produced by the equivalent current system is given by (see Section 9.4) 00M B K M μ μ= = (9.6.5) Since the direction of this magnetic field is in the same direction as M , the above expression may be written in vector notation as G 0M μ=B M G G (9.6.6) This is exactly opposite from the situation with electric dipoles, in which the average electric field is anti-parallel to the direction of the electric dipoles themselves. The reason is that in the region interior to the current loop of the dipole, the magnetic field is in the same direction as the magnetic dipole vector. Therefore, it is not surprising that after a large-scale averaging, the average magnetic field also turns out to be parallel to the average magnetic dipole moment per unit volume. Notice that the magnetic field in Eq. (9.6.6) is the average field due to all the dipoles. A very different field is observed if we go close to any one of these little dipoles. Let’s now examine the properties of different magnetic materials 29 9.6.2 Paramagnetism The atoms or molecules comprising paramagnetic materials have a permanent magnetic dipole moment. Left to themselves, the permanent magnetic dipoles in a paramagnetic material never line up spontaneously. In the absence of any applied external magnetic field, they are randomly aligned. Thus, =M 0 GG and the average magnetic field MB G is also zero. However, when we place a paramagnetic material in an external field , the dipoles experience a torque 0 B G 0 = ×τ μ B GG G that tends to align μ G with 0B G , thereby producing a net magnetization parallel to M G 0 B G . Since MB G is parallel to 0B G , it will tend to enhance . The total magnetic field is the sum of these two fields: 0B G B G 00M μ= + = +B B B B M0 G G G G G (9.6.7) Note how different this is than in the case of dielectric materials. In both cases, the torque on the dipoles causes alignment of the dipole vector parallel to the external field. However, in the paramagnetic case, that alignment enhances the external magnetic field, whereas in the dielectric case it reduces the external electric field. In most paramagnetic substances, the magnetization M G is not only in the same direction as B G , but also linearly proportional to . This is plausible because without the external field there would be no alignment of dipoles and hence no magnetization M 0 0 B G 0 B G G . The linear relation between M G and G is expressed as 0B 00 m χ μ = BM G G (9.6.8) where m χ is a dimensionless quantity called the magnetic susceptibility . Eq. (10.7.7) can then be written as 0 (1 )m m χ κ= + =B B 0B G G G (9.6.9) where 1m m κ χ= + (9.6.10) is called the relative permeability of the material. For paramagnetic substances, ,or equivalently, 1m κ > 0m χ > , although m χ is usually on the order of 10 to . The magnetic permeability 6−3 10 − m μ of a material may also be defined as 0 (1 )m m 0m μ χ μ κ μ = + = (9.6.11) 30 Paramagnetic materials have 0m μ μ> . 9.6.3 Diamagnetism In the case of magnetic materials where there are no permanent magnetic dipoles, the presence of an external field G will induce magnetic dipole moments in the atoms or molecules. However, these induced magnetic dipoles are anti-parallel to , leading to a magnetization M and average field 0 B 0 B G G M B G anti-parallel to 0B G , and therefore a reduction in the total magnetic field strength. For diamagnetic materials, we can still define the magnetic permeability, as in equation (8-5), although now 1m κ < , or 0m χ < , although m χ is usually on the order of 510 − − to 910 − − . Diamagnetic materials have 0m μ μ< . 9.6.4 Ferromagnetism In ferromagnetic materials, there is a strong interaction between neighboring atomic dipole moments. Ferromagnetic materials are made up of small patches called domains ,as illustrated in Figure 9.6.3(a). An externally applied field 0B G will tend to line up those magnetic dipoles parallel to the external field, as shown in Figure 9.6.3(b). The strong interaction between neighboring atomic dipole moments causes a much stronger alignment of the magnetic dipoles than in paramagnetic materials. Figure 9.6.3 (a) Ferromagnetic domains. (b) Alignment of magnetic moments in the direction of the external field G .0B The enhancement of the applied external field can be considerable, with the total magnetic field inside a ferromagnet 10 or times greater than the applied field. The permeability of a ferromagnetic material is not a constant, since neither the total field or the magnetization M increases linearly with 34 10 m κ B G G 0 B G . In fact the relationship between and is not unique, but dependent on the previous history of the material. The M G 0 B G 31 phenomenon is known as hysteresis . The variation of M G as a function of the externally applied field G is shown in Figure 9.6.4. The loop abcdef is a hysteresis curve .0B Figure 9.6.4 A hysteresis curve. Moreover, in ferromagnets, the strong interaction between neighboring atomic dipole moments can keep those dipole moments aligned, even when the external magnet field is reduced to zero. And these aligned dipoles can thus produce a strong magnetic field, all by themselves, without the necessity of an external magnetic field. This is the origin of permanent magnets. To see how strong such magnets can be, consider the fact that magnetic dipole moments of atoms typically have magnitudes of the order of 23 210 A m − ⋅ .Typical atomic densities are atoms/m 3 . If all these dipole moments are aligned, then we would get a magnetization of order 29 10 23 229 36 (10 A m )(10 atoms/m ) 10 A/m M − ⋅∼ ∼ MThe magnetization corresponds to values of 0M μ=B G G of order 1 tesla, or 10,000 Gauss, just due to the atomic currents alone. This is how we get permanent magnets with fields of order 2200 Gauss. 9.7 Summary • Biot-Savart law states that the magnetic field dB G at a point due to a length element dG carrying a steady current I and located at rs G away is given by 02 ˆ4 I d d r μ π × = s rB GG where r = r G and is the permeability of free space. 70 4 10 T m/A μ π −= × ⋅ • The magnitude of the magnetic field at a distance r away from an infinitely long straight wire carrying a current I is 32 (9.6.12) 02 I B r μ π = • The magnitude of the magnetic force between two straight wires of length A carrying steady current of B F 1 and 2I I and separated by a distance r is 012 2 B I IF r μ π = A • Ampere’s law states that the line integral of d⋅B s G G around any closed loop is proportional to the total steady current passing through any surface that is bounded by the close loop: 0enc d I μ⋅ =∫ B s G G v • The magnetic field inside a toroid which has N closely spaced of wire carrying a current I is given by 0 2 NI B r μ π = where r is the distance from the center of the toroid. • The magnetic field inside a solenoid which has N closely spaced of wire carrying current I in a length of l is given by 00 N B I nl μμ= = I where n is the number of number of turns per unit length. • The properties of magnetic materials are as follows: Materials Magnetic susceptibility m χ Relative permeability 1m m κ χ= + Magnetic permeability 0mm μ κ μ = Diamagnetic 5 910 10 − −− −∼ 1m κ < 0m μ μ< Paramagnetic 5 310 10 − − ∼ 1m κ > 0m μ μ> Ferromagnetic 1m χ 1m κ 0m μ μ 33 9.8 Appendix 1: Magnetic Field off the Symmetry Axis of a Current Loop In Example 9.2 we calculated the magnetic field due to a circular loop of radius R lying in the xy plane and carrying a steady current I, at a point P along the axis of symmetry. Let’s see how the same technique can be extended to calculating the field at a point off the axis of symmetry in the yz plane. Figure 9.8.1 Calculating the magnetic field off the symmetry axis of a current loop. Again, as shown in Example 9.1, the differential current element is ˆ ˆ'( sin ' cos ' ) Id R d φ φφ= − +s i G j ˆ)and its position is described by ˆ' (cos ' sin 'R φ φ= +r i G jk . On the other hand, the field point P now lies in the yz plane with r jˆ ˆ P y z= + G , as shown in Figure 9.8.1. The corresponding relative position vector is ( )ˆ ˆ' cos ' sin 'P ˆR y R z φφ− = − + − +r = r r i j k G G G (9.8.1) with a magnitude ( )22 2 2 2 2( cos ') sin ' 2 sin r R y R z R y z yR φ φφ= = − + − + = + + −r G (9.8.2) and the unit vector 'ˆ \| ' PP r \| − = = − r rrr r r G GG G G pointing from Id s G to P. The cross product ˆd ×s r G can be simplified as (9.8.3) ( )( ) ˆ ˆ ˆ ˆ ˆˆ ' sin ' cos ' [ cos ' ( sin ') ]ˆ ˆ ˆ'[ cos ' sin ' sin ' ] d R d R y R zR d z z R y φ φφφφφ φφφ× = − + × − + − += + + − s r i j i j ki j k G 34 Using the Biot-Savart law, the contribution of the current element to the magnetic field at P is ( )( ) 0003/ 2 23222 ˆ ˆ ˆcos ' sin ' sin 'ˆ '4 4 4 2 sin ' z z R yI I IR d dd dr r R y z yR φφφμμμ φ πππφ+ + −× ×= = =+ + − i j ks r s rB G G GG (9.8.4) Thus, magnetic field at P is ( ) ( )( ) 203/ 2 0222 ˆ ˆ ˆcos ' sin ' sin '0, , '4 2 sin ' z z R yIR y z dR y z yR π φφφμ φ πφ+ + −=+ + −∫ i j kB G (9.8.5) The x-component of B can be readily shown to be zero G ( ) 203/ 2 0222 cos ' ' 04 2 sin ' x IRz dBR y z yR π μ φ φπφ= =+ + −∫ (9.8.6) by making a change of variable 2 2 2 2 sin w R y z yR ' φ= + + − , followed by astraightforward integration. One may also invoke symmetry arguments to verify that x B must vanish; namely, the contribution at ' φ is cancelled by the contribution at ' π φ− .On the other hand, the y and the z components of B G , ( ) 203/ 2 0222 sin ' '4 2 sin ' y IRz dBR y z yR π μ φ φπφ=+ + −∫ (9.8.7) and ( )( ) 203/ 2 0222 sin ' '4 2 sin ' z R y dIR BR y z yR π φ φμπφ−=+ + −∫ (9.8.8) involve elliptic integrals which can be evaluated numerically. In the limit , the field point P is located along the z-axis, and we recover the results obtained in Example 9.2: 0y = 200223/ 2 223/ 2 0 2sin ' ' cos ' 004 ( ) 4 ( ) y IRz IRz B dR z R z π πμμφ φφππ= = − + +∫ = (9.8.9) and 35 22 220 02 2 3/ 2 2 2 3/ 2 2 2 3/ 2 02'4 ( ) 4 ( ) 2( ) z IR IR IR B dR z R z R z π μμ πφππ= = =+ +∫ 0 μ+ (9.8.10) Now, let’s consider the “point-dipole” limit where 2 2 1/ 2 ( )R y z r+ = , i.e., the characteristic dimension of the current source is much smaller compared to the distance where the magnetic field is to be measured. In this limit, the denominator in the integrand can be expanded as ( ) 3/ 2 23/ 2 2 2 23 223 21 2 sin '2 sin ' 11 3 2 sin '1 2 R yR R y z yR r rR yR r r φφφ−− ⎡ ⎤−+ + − = +⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞−= − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦ … (9.8.11) This leads to 220320222200550 3 2 sin '1 s4 23 3sin ' '4 4 y I Rz R yR in ' 'B dr rI IR yz R yz dr r ππ μφ φ φπμμ πφ φ π π⎡ ⎤⎛ ⎞−≈ −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦= =∫∫ (9.8.12) and 2203203222203222032032222032 3 2 sin '1 ( sin ') 4 23 9 31 sin ' sin ' '4 2 23 324 232 higher order ter 4 z I R R yR B Rr rI R R R Ry 'y d R dr r r rI R R Ry Rr r rI R yr r ππ μφφ φπμ φ φ φπμπππμ ππ⎡ ⎤⎛ ⎞−≈ − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞= − − − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞= − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦= − +∫∫ ms ⎡ ⎤⎢ ⎥⎣ ⎦ (9.8.13) The quantity 2( )I R π may be identified as the magnetic dipole moment IA μ = , where 2 A R π= is the area of the loop. Using spherical coordinates where sin y r θ= and cos z r θ= , the above expressions may be rewritten as 2005 ( ) 3( sin )( cos ) 3 sin cos 4 4 y I R r rB r r μ πμ 3 θ θμ θππ= = θ (9.8.14) 36 and 22220003233 ( ) 3 sin 2 (2 3sin ) (3 4 4 4 z I R rB r r r r μμμπθμμθπππ⎛ ⎞= − = − =⎜ ⎟⎝ ⎠ 2 cos 1) θ − (9.8.15) Thus, we see that the magnetic field at a point r due to a current ring of radius R may be approximated by a small magnetic dipole moment placed at the origin (Figure 9.8.2). R Figure 9.8.2 Magnetic dipole moment ˆ μ=μ k G The magnetic field lines due to a current loop and a dipole moment (small bar magnet) are depicted in Figure 9.8.3. Figure 9.8.3 Magnetic field lines due to (a) a current loop, and (b) a small bar magnet. The magnetic field at P can also be written in spherical coordinates ˆˆrB B θ = +B r θ G (9.8.16) The spherical components rB and B θ are related to the Cartesian components yB and zB by sin cos , cos sin r y z y zB B B B B B θ θ θθ= + = − θ ˆ (9.8.17) In addition, we have, for the unit vectors, ˆ ˆ ˆˆˆ sin cos , cos sin θ θθ= + = −r j k θ j θ k (9.8.18) Using the above relations, the spherical components may be written as 37 ( ) 2203/ 2 022 cos '4 2 sin sin ' r IR dBR r rR π μθφπθ φ=+ −∫ (9.8.19) and ( ) ( )( ) 203/ 2 022 sin ' sin ', 4 2 sin sin ' r R dIR B rR r rR πθ φθ φμθ πθ φ−=+ −∫ (9.8.20) In the limit where R r , we obtain 222000330 cos 2 cos 2 cos '4 4 4 r IR IR B dr r π μθμμ 3 r π θμφπππ≈ = =∫ θ (9.8.21) and ( )( ) ( ) 203/ 2 02222222203202003303 sin ' sin '4 2 sin sin '3 3 3 sin sin 1 sin ' 3 sin sin ' '4 2 2 2( ) sin 2 sin 3 sin 4 4sin 4 r R dIR BR r rR IR R R R R r Rr r r rIR I RR Rr rr πθπ φθ φμπθ φμθ d θ φθ φπμμ π θπ θ π θππμ μ θ φ π −=+ −⎡ ⎤⎛ ⎞ ⎛ ⎞≈ − − + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣ ⎦≈ − + ==∫∫ (9.8.22) 9.9 Appendix 2: Helmholtz Coils Consider two N-turn circular coils of radius R, each perpendicular to the axis of symmetry, with their centers located at / 2 z l= ± . There is a steady current I flowing in the same direction around each coil, as shown in Figure 9.9.1. Let’s find the magnetic field B G on the axis at a distance z from the center of one coil. Figure 9.9.1 Helmholtz coils 38 Using the result shown in Example 9.2 for a single coil and applying the superposition principle, the magnetic field at (a point at a distance ( , 0) P z / 2 z l− away from one center and / 2 z l+ from the other) due to the two coils can be obtained as: 20top bottom 223/ 2 223/ 2 1 12 [( / 2) ] [( / 2) ] z NIR B B B z l R z l R μ ⎡ ⎤= + = +⎢ ⎥− + + +⎣ ⎦ (9.9.1) A plot of 0/zB B with 00 3/ 2 (5 / 4) NI B R μ = being the field strength at and 0z = l R= is depicted in Figure 9.9.2. Figure 9.9.2 Magnetic field as a function of /z R . Let’s analyze the properties of zB in more detail. Differentiating zB with respect to z, we obtain 20225/ 2 225/ 2 3( / 2) 3( / 2) ( ) 2 [( / 2) ] [( / 2) ] zz NIR dB z l z lB z dz z l R z l R μ ⎧ ⎫− +′ = = − −⎨ ⎬− + + +⎩ ⎭ (9.9.2) One may readily show that at the midpoint, 0z = , the derivative vanishes: 0 0 z dB dz = = (9.9.3) Straightforward differentiation yields 22202225/ 2 2225/ 2 227 / 2 3 15( / 2) ( ) 2 [( / 2) ] [( / 2) ]3 15( / 2) [( / 2) ] [( / 2) ] z N IR d B z lB z dz z l R z l Rz lz l R z l R μ ⎧ −′′ = = − +⎨ − + − +⎩⎫+− + ⎬+ + + + ⎭ 227 / 2 (9.9.4) 39 At the midpoint , the above expression simplifies to 0z = 22202225/ 2 202220227 / 2 6 15 (0) 2 [( / 2) ] 2[( / 2) ]6( )2 [( / 2) ] zz NI d B lB dz l R l RNI R ll R μμ= 27 / 2 ⎧ ⎫′′ = = − +⎨ ⎬+ +⎩ ⎭−= − + (9.9.5) Thus, the condition that the second derivative of zB vanishes at 0z = is . That is, the distance of separation between the two coils is equal to the radius of the coil. Aconfiguration with l is known as Helmholtz coils . l R= R= For small z, we may make a Taylor-series expansion of ( ) zB z about 0z = : 2 1( ) (0) (0) (0) ... 2! zzzz B z B B z B z′ ′′ = + + + (9.9.6) The fact that the first two derivatives vanish at 0z = indicates that the magnetic field is fairly uniform in the small z region. One may even show that the third derivative vanishes at as well. (0) zB′′′ 0z = Recall that the force experienced by a dipole in a magnetic field is (B = ∇ ⋅F μ )B G GG . If we place a magnetic dipole ˆ z μ=μ k G at 0z = , the magnetic force acting on the dipole is ˆ( ) zB z z zdB B dz μ μ ⎛ ⎞= ∇ = ⎜ ⎟⎝ ⎠ F G k (9.9.7) which is expected to be very small since the magnetic field is nearly uniform there. Animation 9.5 : Magnetic Field of the Helmholtz Coils The animation in Figure 9.9.3(a) shows the magnetic field of the Helmholtz coils. In this configuration the currents in the top and bottom coils flow in the same direction, with their dipole moments aligned. The magnetic fields from the two coils add up to create a net field that is nearly uniform at the center of the coils. Since the distance between the coils is equal to the radius of the coils and remains unchanged, the force of attraction between them creates a tension, and is illustrated by field lines stretching out to enclose both coils. When the distance between the coils is not fixed, as in the animation depicted in Figure 9.9.3(b), the two coils move toward each other due to their force of attraction. In this animation, the top loop has only half the current as the bottom loop. The field configuration is shown using the “iron filings” representation. 40 (a) (b) Figure 9.9.3 (a) Magnetic field of the Helmholtz coils where the distance between the coils is equal to the radius of the coil. (b) Two co-axial wire loops carrying current in the same sense are attracted to each other. Next, let’s consider the case where the currents in the loop flow in the opposite directions, as shown in Figure 9.9.4. Figure 9.9.4 Two circular loops carrying currents in the opposite directions. Again, by superposition principle, the magnetic field at a point with is (0, 0, )P z 0z > 2012223/ 2 223/ 2 1 12 [( / 2) ] [( / 2) ] zzz NIR B B B z l R z l R μ ⎡ ⎤= + = −⎢ ⎥− + + +⎣ ⎦ (9.9.8) A plot of 0/zB B with 0 0 / 2 B NI R μ= and l R= is depicted in Figure 9.9.5. 41 Figure 9.9.5 Magnetic field as a function of /z R .Differentiating zB with respect to z, we obtain 20225/ 2 225/ 2 3( / 2) 3( / 2) ( ) 2 [( / 2) ] [( / 2) ] zz NIR dB z l z lB z dz z l R z l R μ ⎧ ⎫− +′ = = − +⎨ ⎬− + + +⎩ ⎭ (9.9.9) At the midpoint, , we have 0z = 20225/ 2 3(0) 00 2 [( / 2) ] zz NIR dB lB zdz l R μ′ = == + ≠ (9.9.10) Thus, a magnetic dipole ˆ z μ=μ G k placed at 0z = will experience a net force: 20225/ 2 (0) 3ˆ ˆ( ) ( ) 2 [( / 2) ] zzBzzz NIR dB lB dz l R μ μ μ μ ⎛ ⎞= ∇ ⋅ = ∇ = =⎜ ⎟ +⎝ ⎠ F μ B k k G GG (9.9.11) For , the above expression simplifies to l R= 05 / 2 2 3 ˆ2(5 / 4) zB NI R μ μ=F k G (9.9.12) Animation 9.6 : Magnetic Field of Two Coils Carrying Opposite Currents The animation depicted in Figure 9.9.6 shows the magnetic field of two coils like the Helmholtz coils but with currents in the top and bottom coils flowing in the opposite directions. In this configuration, the magnetic dipole moments associated with each coil are anti-parallel. (a) (b) Figure 9.9.6 (a) Magnetic field due to coils carrying currents in the opposite directions. (b) Two co-axial wire loops carrying current in the opposite sense repel each other. The 42 field configurations here are shown using the “iron filings” representation. The bottom wire loop carries twice the amount of current as the top wire loop. At the center of the coils along the axis of symmetry, the magnetic field is zero. With the distance between the two coils fixed, the repulsive force results in a pressure between them. This is illustrated by field lines that are compressed along the central horizontal axis between the coils. Animation 9.7 : Forces Between Coaxial Current-Carrying Wires Figure 9.9.7 A magnet in the TeachSpin ™ Magnetic Force apparatus when the current in the top coil is counterclockwise as seen from the top. Figure 9.9.7 shows the force of repulsion between the magnetic field of a permanent magnet and the field of a current-carrying ring in the TeachSpin ™ Magnetic Force apparatus. The magnet is forced to have its North magnetic pole pointing downward, and the current in the top coil of the Magnetic Force apparatus is moving clockwise as seen from above. The net result is a repulsion of the magnet when the current in this direction is increased. The visualization shows the stresses transmitted by the fields to the magnet when the current in the upper coil is increased. Animation 9.8 : Magnet Oscillating Between Two Coils Figure 9.9.8 illustrates an animation in which the magnetic field of a permanent magnet suspended by a spring in the TeachSpin TM apparatus (see TeachSpin visualization), plus the magnetic field due to current in the two coils (here we see a "cutaway" cross-section of the apparatus). 43 Figure 9.9.8 Magnet oscillating between two coils The magnet is fixed so that its north pole points upward, and the current in the two coils is sinusoidal and 180 degrees out of phase. When the effective dipole moment of the top coil points upwards, the dipole moment of the bottom coil points downwards. Thus, the magnet is attracted to the upper coil and repelled by the lower coil, causing it to move upwards. When the conditions are reversed during the second half of the cycle, the magnet moves downwards. This process can also be described in terms of tension along, and pressure perpendicular to, the field lines of the resulting field. When the dipole moment of one of the coils is aligned with that of the magnet, there is a tension along the field lines as they attempt to "connect" the coil and magnet. Conversely, when their moments are anti-aligned, there is a pressure perpendicular to the field lines as they try to keep the coil and magnet apart. Animation 9.9 : Magnet Suspended Between Two Coils Figure 9.9.9 illustrates an animation in which the magnetic field of a permanent magnet suspended by a spring in the TeachSpin TM apparatus (see TeachSpin visualization), plus the magnetic field due to current in the two coils (here we see a "cutaway" cross-section of the apparatus). The magnet is fixed so that its north pole points upward, and the current in the two coils is sinusoidal and in phase. When the effective dipole moment of the top coil points upwards, the dipole moment of the bottom coil points upwards as well. Thus, the magnet the magnet is attracted to both coils, and as a result feels no net force (although it does feel a torque, not shown here since the direction of the magnet is fixed to point upwards). When the dipole moments are reversed during the second half of the cycle, the magnet is repelled by both coils, again resulting in no net force. This process can also be described in terms of tension along, and pressure perpendicular to, the field lines of the resulting field. When the dipole moment of the coils is aligned with that of the magnet, there is a tension along the field lines as they are "pulled" from both sides. Conversely, when their moments are anti-aligned, there is a pressure perpendicular to the field lines as they are "squeezed" from both sides. 44 Figure 9.9.9 Magnet suspended between two coils 9.10 Problem-Solving Strategies In this Chapter, we have seen how Biot-Savart and Ampere’s laws can be used to calculate magnetic field due to a current source. 9.10.1 Biot-Savart Law: The law states that the magnetic field at a point P due to a length element carrying a steady current I located at G away is given by ds G r 0023 ˆ4 4 I Id dd r r μ μππ × ×= = s r s rB G G GG The calculation of the magnetic field may be carried out as follows: (1) Source point : Choose an appropriate coordinate system and write down an expression for the differential current element I ds G , and the vector 'r G describing the position of I ds G .The magnitude is the distance between ' \| '\| r = r G I ds G and the origin. Variables with a “prime” are used for the source point. (2) Field point : The field point P is the point in space where the magnetic field due to the current distribution is to be calculated. Using the same coordinate system, write down the position vector Pr G for the field point P. The quantity \| \|P Pr = r G is the distance between the origin and P. (3) Relative position vector : The relative position between the source point and the field point is characterized by the relative position vector 'P= −r r r G G G . The corresponding unit vector is 'ˆ \| ' PP r \| − = = − r rrr r r G GG G G where is the distance between the source and the field point P.\| \| \| '\| Pr = = −r r r G G G (4) Calculate the cross product ˆd ×s r G or d ×s r G G . The resultant vector gives the direction of the magnetic field G , according to the Biot-Savart law. B (5) Substitute the expressions obtained to dB G and simplify as much as possible. 45 (6) Complete the integration to obtain Bif possible. The size or the geometry of the system is reflected in the integration limits. Change of variables sometimes may help to complete the integration. Below we illustrate how these steps are executed for a current-carrying wire of length L and a loop of radius R. Current distribution Finite wire of length L Circular loop of radius R Figure (1) Source point ˆ' 'ˆ( '/ ') ' ' xd d dx dx dx == = r is r i GG G ˆ ˆ' (cos ' sin ' ) ˆ ˆ( '/ ') ' '( sin ' cos ' ) Rd d d d Rd φ φ φ φ φ φ φ= += = − + r i js r i j G G G (2) Field point P ˆ P y=r j G ˆ P z=r k G (3) Relative position vector G G G 'P= −r r r 2222 ˆ ˆ'\| \| 'ˆ ˆ'ˆ' y xr xy xx y = −= = +−= + r j irj ir GG y 2 22 2ˆ ˆ ˆcos ' sin '\| \|ˆ ˆ ˆcos ' sin 'ˆ R Rr R zR RR z φφφφ= − − += = +− − += + r i jri jr zz kk G G (4) The cross product G ˆd ×s r 2 2ˆˆ y dx d y x ′ × = ′+ ks r G 22 ˆ ˆ ˆ'( cos ' sin ' )ˆ R d z z Rd R z φ φφ+ +× = + i js r G k (5) Rewrite dB G 02 2 3/ ˆ4 ( ) I y dx d y x μπ′= ′+ kB G 2 0223/ 2 ˆ ˆ ˆ'( cos ' sin ' )4 ( ) I R d z z Rd R z μ φ φφπ+ += + i jB kG (6) Integrate to get B G / 2 02 2 3/ / 2 02 200'4 ( ' ) 4 ( / 2) xyLzL BBIy dx B y xI Ly y L μπμπ−=== += +∫ 2 20223/ 2 020223/ 2 022200223/ 2 223/ 2 0 cos ' ' 04 ( )sin ' ' 04 ( )'4 ( ) 2( ) xyz IRz B dR zIRz B dR zIR IR B dR z R z πππ μ φ φπμφ φπμμφπ= =+= =+= =+ +∫∫∫ 46 9.10.2 Ampere’s law: Ampere’s law states that the line integral of d⋅B s G G around any closed loop is proportional to the total current passing through any surface that is bounded by the closed loop: 0enc d I μ⋅ =∫ B s G G v To apply Ampere’s law to calculate the magnetic field, we use the following procedure: (1) Draw an Amperian loop using symmetry arguments. (2) Find the current enclosed by the Amperian loop. (3) Calculate the line integral d⋅∫ B s G Gv around the closed loop. (4) Equate with d⋅∫ B s G Gv 0 enc I μ and solve for B G . Below we summarize how the methodology can be applied to calculate the magnetic field for an infinite wire, an ideal solenoid and a toroid. System Infinite wire Ideal solenoid Toroid Figure (1) Draw the Amperian loop (2) Find the current enclosed by the Amperian loop enc I I= enc I NI = enc I NI = (3) Calculate along the loop d⋅∫ B s G Gv (2 )d B r π⋅ =∫ B s G Gv d Bl⋅ =∫ B s G G v (2 )d B r π⋅ =∫ B s G G v 47 (4) Equate 0 enc I μ with to obtain d⋅∫ B s G Gv B G 02 I B r μ π = 00 NI B nI l μ μ= = 02 NI B r μ π = 9.11 Solved Problems 9.11.1 Magnetic Field of a Straight Wire Consider a straight wire of length L carrying a current I along the + x-direction, as shown in Figure 9.11.1 (ignore the return path of the current or the source for the current.) What is the magnetic field at an arbitrary point P on the xy -plane? Figure 9.11.1 A finite straight wire carrying a current I. Solution: The problem is very similar to Example 9.1. However, now the field point is an arbitrary point in the xy -plane. Once again we solve the problem using the methodology outlined in Section 9.10. (1) Source point From Figure 9.10.1, we see that the infinitesimal length dx ′ described by the position vector ˆ' 'x=r G i constitutes a current source ˆ( )I d Idx ′=s i G . (2) Field point As can be seen from Figure 9.10.1, the position vector for the field point P is ˆ ˆx y= +r i j G . (3) Relative position vector The relative position vector from the source to P is ˆ' ( ') P ˆx x y− = − +r = r r i j G G G , with 221 \| \| \| ' \| [( ) ]Pr x x′= = − = − +r r r G G G 2 y being the distance. The corresponding unit vector is 48 2 2 1ˆ ˆ' ( )ˆ \| ' \| [( ) ] PP 2 x x yr x x ′− − += = = ′− − + r rr ir r r y j G GGG G (4) Simplifying the cross product The cross product d can be simplified as ×s r G G ˆ ˆ ˆ ˆ( ' ) [( ') ] 'dx x x y y dx × − + =i i j k where we have used ˆ ˆ and ˆ ˆ× =i i 0 G ˆ× =i j k . (5) Writing down dB G Using the Biot-Savart law, the infinitesimal contribution due to Id s G is 0002322 ˆ ˆ4 4 4 [( ) I I Id d y dx d r r x x y 3 2 ] μ μμπππ ′× ×= = = ′− + s r s rB k G G GG (9.11.1) Thus, we see that the direction of the magnetic field is in the ˆ+k direction. (6) Carrying out the integration to obtain B G The total magnetic field at P can then be obtained by integrating over the entire length of the wire: / 2 / 2 00223 2 22/ 2 wire / 2 02222 ( )ˆ ˆ4 [( ) ] 4 ( )( / 2) ( / 2) ˆ4 ( / 2) ( / 2) LLLL Iy dx I x xd x x y y x x yI x L x Ly x L y x L y μμππμπ−−′ ′−= = = − ′− + ′− +⎡ ⎤− += − −⎢ ⎥⎢ − + + + ⎥⎣ ⎦∫ ∫B B kk G G k (9.11.2) Let’s consider the following limits: (i) 0x = In this case, the field point P is at ( , ) (0, )x y y= on the y axis. The magnetic field becomes 49 0 02 2 2 2 2 2/ 2 / 2 / 2 ˆ ˆ cos 4 2( / 2) ( / 2) ( / 2) I IL L Ly yL y L y L y μμ 0 ˆ2 Iy μ θ ππ⎡ ⎤− += − − = =⎢ ⎥⎢ − + + + ⎥ +⎣ ⎦ B k G π k k (9.11.3) in agreement with Eq. (9.1.6). (ii) Infinite length limit Consider the limit where ,L x y . This gives back the expected infinite-length result: 0 / 2 / 2 ˆ4 / 2 / 2 2 0 ˆI IL Ly L L y μππ− +⎡ ⎤= − − =⎢ ⎥⎣ ⎦ B G μ k k (9.11.4) If we use cylindrical coordinates with the wire pointing along the + z-axis then the magnetic field is given by the expression 0 ˆ2 I r μ π =B φ G (9.11.5) where is the tangential unit vector and the field point P is a distance r away from the wire. ˆφ 9.11.2 Current-Carrying Arc Consider the current-carrying loop formed of radial lines and segments of circles whose centers are at point P as shown below. Find the magnetic field B G at P. Figure 9.11.2 Current-carrying arc Solution: According to the Biot-Savart law, the magnitude of the magnetic field due to a differential current-carrying element I d s G is given by 50 0 0 02 2ˆ ' '4 4 4 dI I r d dB dr r μμμθ Ir θ πππ×= = = s r G (9.11.6) For the outer arc, we have 0outer 0 '4 4 0 I IB db b θ μ μ θθ π π= =∫ (9.11.7) The direction of is determined by the cross product outer B G ˆd ×s r G which points out of the page. Similarly, for the inner arc, we have 0inner 0 '4 4 0 I IB da a θ μ μ θθ π π= =∫ (9.11.8) For , points into the page. Thus, the total magnitude of magnetic field is inner B G ˆd ×s r G 0inner outer 1 1 (into page) 4 Ia b μ θπ⎛ ⎞= = −⎜ ⎟⎝ ⎠ B B + B G G G (9.11.9) 9.11.3 Rectangular Current Loop Determine the magnetic field (in terms of I, a and b) at the origin O due to the current loop shown in Figure 9.11.3 Figure 9.11.3 Rectangular current loop 51 For a finite wire carrying a current I, the contribution to the magnetic field at a point P is given by Eq. (9.1.5): ( )01 2cos cos 4 IB r μ θ θπ= + where 1 and 2 θ θ are the angles which parameterize the length of the wire. To obtain the magnetic field at O, we make use of the above formula. The contributions can be divided into three parts: (i) Consider the left segment of the wire which extends from ( , to . The angles which parameterize this segment give co ) ( , )x y a= − +∞ ( , )a d− + 1s 1 θ = ( 1 0 θ = ) and 22 cos /b b a θ = − + 2 . Therefore, ( )0 01 1 2 2 2cos cos 14 4 I I bB a a b a μμθ θππ⎛ ⎞= + = −⎜ +⎝ ⎠ ⎟ (9.11.10) The direction of is out of page, or 1B G ˆ+k . (ii) Next, we consider the segment which extends from ( , ) ( , )x y a b= − + to .Again, the (cosine of the) angles are given by ( , )a b+ + 122 cos aa b θ = + (9.11.11) 2122 cos cos aa b θ θ= = + (9.11.12) This leads to 02222222 4 2 I a aB b a b a b b a b μμππ⎛ ⎞= + =⎜ ⎟ 0 Ia +⎝ ⎠ (9.11.13) The direction of is into the page, or 2B G ˆ−k . (iii) The third segment of the wire runs from ( , ) ( , )x y a b= + + to ( , )a+ +∞ . One may readily show that it gives the same contribution as the first one: 31 B B= (9.11.14) 52 Solution: The direction of is again out of page, or 3B G ˆ+k . The magnetic field is ( ) 001231222222222022 ˆ ˆ2 12 2ˆ2 I Iba a b b a bI b a b b aab a b μμππμπ⎛ ⎞= + + = + = − −⎜ ⎟+⎝ ⎠= + − −+ B B B B B B k kk G G G G G G a (9.11.15) Note that in the limit , the horizontal segment is absent, and the two semi-infinite wires carrying currents in the opposite direction overlap each other and their contributions completely cancel. Thus, the magnetic field vanishes in this limit. 0a → 9.11.4 Hairpin-Shaped Current-Carrying Wire An infinitely long current-carrying wire is bent into a hairpin-like shape shown in Figure 9.11.4. Find the magnetic field at the point P which lies at the center of the half-circle. Figure 9.11.4 Hairpin-shaped current-carrying wire Solution: Again we break the wire into three parts: two semi-infinite plus a semi-circular segments. (i) Let P be located at the origin in the xy plane. The first semi-infinite segment then extends from ( , ) ( , )x y r= −∞ − to (0, )r− . The two angles which parameterize this segment are characterized by 1cos 1 θ = ( 1 0 θ = ) and 2 2cos 0 ( / 2) θ θ π= = . Therefore, its contribution to the magnetic field at P is ( )0 01 1 2cos cos (1 0) 4 4 0 4 I IB r r Ir μ μθ θ μ π ππ= + = + = (9.11.16) The direction of is out of page, or 1B G ˆ+k . 53 (ii) For the semi-circular arc of radius r, we make use of the Biot-Savart law: 02 ˆ4 I dr μ π × = ∫ s rB GG (9.11.17) and obtain 0220 4 4 0 I Ird B r r π μ μ θ π= =∫ (9.11.18) The direction of is out of page, or 2B G ˆ+k . (iii) The third segment of the wire runs from ( , ) (0, )x y r= + to ( , r)−∞ + . One may readily show that it gives the same contribution as the first one: 031 4 I B B r μ π = = (9.11.19) The direction of is again out of page, or 3B G ˆ+k . The total magnitude of the magnetic field is 00012312 ˆ ˆ2 (2 4 4 I I Ir r r ˆ2 ) μ μ μ π ππ= + + = + = + = +B B B B B B k k k G G G G G G (9.11.20) Notice that the contribution from the two semi-infinite wires is equal to that due to an infinite wire: 0131 ˆ2 2 I r μ π+ = =B B B k G G G (9.11.21) 9.11.5 Two Infinitely Long Wires Consider two infinitely long wires carrying currents are in the −x-direction. 54 Figure 9.11.5 Two infinitely long wires (a) Plot the magnetic field pattern in the yz -plane. (b) Find the distance d along the z-axis where the magnetic field is a maximum. Solutions: (a) The magnetic field lines are shown in Figure 9.11.6. Notice that the directions of both currents are into the page. Figure 9.11.6 Magnetic field lines of two wires carrying current in the same direction. (b) The magnetic field at (0, 0, z) due to wire 1 on the left is, using Ampere’s law: 00122 2 2 I IB r a z μ μπ π= = + (9.11.22) Since the current is flowing in the – x-direction, the magnetic field points in the direction of the cross product 1 ˆ ˆ ˆ ˆˆˆ( ) ( ) (cos sin ) sin cos ˆ θ θθ− × = − × + = −i r i j k j k θ (9.11.23) Thus, we have ( )01 2 2ˆ ˆsin cos 2 Ia z μ θ θπ= −+ B j G k (9.11.24) For wire 2 on the right, the magnetic field strength is the same as the left one: 1 2B B= .However, its direction is given by 2 ˆ ˆ ˆ ˆˆˆ( ) ( ) ( cos sin ) sin cos ˆ θ θθ− × = − × − + =i r i j k j + k θ (9.11.25) 55 Adding up the contributions from both wires, the z-components cancel (as required by symmetry), and we arrive at 0122222 sin ˆ( ) Ia za z 0 ˆIz μ θμππ+ = = ++ B = B B j j G G G (9.11.26) Figure 9.11.7 Superposition of magnetic fields due to two current sources To locate the maximum of B, we set /dB dz 0= and find ( ) 220022222222 1 2 0( ) I IdB z a zdz a z a z a z μμππ⎛ ⎞ −= − =⎜ ⎟+ +⎝ ⎠ + 2 = (9.11.27) which gives z a= (9.11.28) Thus, at z=a , the magnetic field strength is a maximum, with a magnitude 0max 2 I B a μ π = (9.11.29) 9.11.6 Non-Uniform Current Density Consider an infinitely long, cylindrical conductor of radius R carrying a current I with a non-uniform current density J r α= (9.11.30) where α is a constant. Find the magnetic field everywhere. 56 Figure 9.11.8 Non-uniform current density Solution: The problem can be solved by using the Ampere’s law: 0enc d I μ⋅ =∫ B s G G v (9.11.31) where the enclosed current Ienc is given by ( )( )enc ' 2 ' 'I d r r d α π= ⋅ = r∫ ∫J A GG (9.11.32) (a) For r , the enclosed current is R< 32enc 0 22 ' ' 3 r rI r dr παπα = =∫ (9.11.33) Applying Ampere’s law, the magnetic field at P1 is given by ( ) 30122 3 rB r μ πα π = (9.11.34) or 201 3 B r αμ = (9.11.35) The direction of the magnetic field 1B G is tangential to the Amperian loop which encloses the current. (b) For , the enclosed current is r R> 32enc 0 22 ' ' 3 R R I r dr πα πα = =∫ (9.11.36) which yields 57 ( ) 30222 3 R B r μ πα π = (9.11.37) Thus, the magnetic field at a point P2 outside the conductor is 302 3 R B r αμ = (9.11.38) A plot of B as a function of r is shown in Figure 9.11.9: Figure 9.11.9 The magnetic field as a function of distance away from the conductor 9.11.7 Thin Strip of Metal Consider an infinitely long, thin strip of metal of width w lying in the xy plane. The strip carries a current I along the + x-direction, as shown in Figure 9.11.10. Find the magnetic field at a point P which is in the plane of the strip and at a distance s away from it. Figure 9.11.10 Thin strip of metal 58 Consider a thin strip of width dr parallel to the direction of the current and at a distance r away from P, as shown in Figure 9.11.11. The amount of current carried by this differential element is dr dI I w ⎛= ⎜⎝ ⎠ ⎞⎟ (9.11.39) Using Ampere’s law, we see that the strip’s contribution to the magnetic field at P is given by 0enc 0 (2 ) ( )dB r I dI π μ μ= = (9.11.40) or 00 2 2 dI I dr dB r r w μ μπ π⎛= = ⎜⎝ ⎠ ⎞⎟ (9.11.41) Figure 9.11.11 A thin strip with thickness carrying a steady current dr I . Integrating this expression, we obtain 00 ln 2 2 sws I Idr s wB w r w s μμππ+ +⎛ ⎞ ⎛= =⎜ ⎟ ⎜⎝ ⎠ ⎝∫ ⎞⎟⎠ (9.11.42) Using the right-hand rule, the direction of the magnetic field can be shown to point in the +z-direction, or 0 ˆln 1 2 I ww s μπ⎛ ⎞= +⎜ ⎟⎝ ⎠ B k G (9.11.43) Notice that in the limit of vanishing width, ,w s ln(1 / ) /w s w s + ≈ , and the above expression becomes 0 ˆ2 I s μ π =B k G (9.11.44) which is the magnetic field due to an infinitely long thin straight wire. 59 Solution: 9.11.8 Two Semi-Infinite Wires A wire carrying current I runs down the y axis to the origin, thence out to infinity along the positive x axis. Show that the magnetic field in the quadrant with of the xy plane is given by ,x y > 0 02222 1 14 z I x yB x y y x y x x y μπ⎛ ⎞⎜= + + +⎜ + +⎝ ⎠ ⎟⎟ (9.11.45) Solution: Let ( , )P x y be a point in the first quadrant at a distance from a point (0 1r , ') y on the y-axis and distance from on the x-axis. 2r ( ', 0) x Figure 9.11.12 Two semi-infinite wires Using the Biot-Savart law, the magnetic field at P is given by 00011222212wire wire ˆ ˆˆ4 4 4y xI I Id ddd r r 2 r μ μμπππ × ×× = = = +∫ ∫ ∫ ∫ s r s rs rB B G GGG G (9.11.46) Let’s analyze each segment separately. (i) Along the y axis, consider a differential element 1 ˆ'd dy = − s j G which is located at a distance 1 ˆ ( ' ˆ)x y y= + −r i G j y from P. This yields (9.11.47) 1 1 ˆ ˆ ˆ ˆ( ' ) [ ( ') ] 'd dy x y y x d × = − × + − =s r j i j k G G 60 (ii) Similarly, along the x-axis, we have 2 ˆ'd dx =s i G and 2 ˆ( ') ˆx x y= − +r i j G which gives 22 ˆ'd y d × =s r kx G G (9.11.48) Thus, we see that the magnetic field at P points in the + z-direction. Using the above results and 2 21 ( ') r x y y= + − and ( )2 22r x x′= − + y , we obtain 00223/ 2 2200 '4 [ ( ') ] 4 [ ( ') ] z I Ix dy y dx B x y y y x x 3/ 2 ' μ μππ∞ ∞= ++ − + −∫ ∫ (9.11.49) The integrals can be readily evaluated using 223/ 2 220 1[ ( ) ] b ds ab a s b b a b ∞ = ++ − + ∫ (9.11.50) The final expression for the magnetic field is given by 02222 1 1 ˆ4 I y xx yx x y y x y μπ⎡ ⎤= + + +⎢⎢ + +⎣ ⎦ B k G ⎥⎥ (9.11.51) We may show that the result is consistent with Eq. (9.1.5) 9.12 Conceptual Questions Compare and contrast Biot-Savart law in magnetostatics with Coulomb’s law in electrostatics. If a current is passed through a spring, does the spring stretch or compress? Explain. How is the path of the integration of d⋅∫ B s G G v chosen when applying Ampere’s law? Two concentric, coplanar circular loops of different diameters carry steady currents in the same direction. Do the loops attract or repel each other? Explain. Suppose three infinitely long parallel wires are arranged in such a way that when looking at the cross section, they are at the corners of an equilateral triangle. Can currents be arranged (combination of flowing in or out of the page) so that all three wires (a) attract, and (b) repel each other? Explain. 61 9.13 Additional Problems 9.13.1 Application of Ampere's Law The simplest possible application of Ampere's law allows us to calculate the magnetic field in the vicinity of a single infinitely long wire. Adding more wires with differing currents will check your understanding of Ampere's law. (a) Calculate with Ampere's law the magnetic field, \| \| ( ) B r=B G , as a function of distance r from the wire, in the vicinity of an infinitely long straight wire that carries current I.Show with a sketch the integration path you choose and state explicitly how you use symmetry. What is the field at a distance of 10 mm from the wire if the current is 10 A? (b) Eight parallel wires cut the page perpendicularly at the points shown. A wire labeled with the integer k (k = 1, 2, ... , 8) bears the current 2 k times 0I (i.e., 02kI k I = ). For those with k = 1 to 4, the current flows up out of the page; for the rest, the current flows down into the page. Evaluate d⋅∫ B s G Gv along the closed path (see figure) in the direction indicated by the arrowhead. (Watch your signs!) Figure 9.13.1 Amperian loop (c) Can you use a single application of Ampere's Law to find the field at a point in the vicinity of the 8 wires? Why? How would you proceed to find the field at an arbitrary point P? 9.13.2 Magnetic Field of a Current Distribution from Ampere's Law Consider the cylindrical conductor with a hollow center and copper walls of thickness as shown in Figure 9.13.2. The radii of the inner and outer walls are a and b respectively, and the current I is uniformly spread over the cross section of the copper. b a− 62 (a) Calculate the magnitude of the magnetic field in the region outside the conductor, . (Hint: consider the entire conductor to be a single thin wire, construct an Amperian loop, and apply Ampere's Law.) What is the direction of r b> B G ? Figure 9.13.2 Hollow cylinder carrying a steady current I. (b) Calculate the magnetic field inside the inner radius, r < a. What is the direction of B G ? (c) Calculate the magnetic field within the inner conductor, a < r < b. What is the direction of G ?B (d) Plot the behavior of the magnitude of the magnetic field B(r) from r = 0 to . Is B(r) continuous at r = a and r = b? What about its slope? 4r = b (e) Now suppose that a very thin wire running down the center of the conductor carries the same current I in the opposite direction. Can you plot, roughly, the variation of B(r)without another detailed calculation? (Hint: remember that the vectors d G from different current elements can be added to obtain the total vector magnetic field.) B 9.13.3 Cylinder with a Hole A long copper rod of radius a has an off-center cylindrical hole through its entire length, as shown in Figure 9.13.3. The conductor carries a current I which is directed out of the page and is uniformly distributed throughout the cross section. Find the magnitude and direction of the magnetic field at the point P. 63 Figure 9.13.3 A cylindrical conductor with a hole. 9.13.4 The Magnetic Field Through a Solenoid A solenoid has 200 closely spaced turns so that, for most of its length, it may be considered to be an ideal solenoid. It has a length of 0.25 m, a diameter of 0.1 m, and carries a current of 0.30 A. (a) Sketch the solenoid, showing clearly the rotation direction of the windings, the current direction, and the magnetic field lines (inside and outside) with arrows to show their direction. What is the dominant direction of the magnetic field inside the solenoid? (b) Find the magnitude of the magnetic field inside the solenoid by constructing an Amperian loop and applying Ampere's law. (c) Does the magnetic field have a component in the direction of the wire in the loops making up the solenoid? If so, calculate its magnitude both inside and outside the solenoid, at radii 30 mm and 60 mm respectively, and show the directions on your sketch. 9.13.5 Rotating Disk A circular disk of radius R with uniform charge density σ rotates with an angular speed ω . Show that the magnetic field at the center of the disk is 0 12 B R μ σω = Hint: Consider a circular ring of radius r and thickness dr . Show that the current in this element is ( / 2 )dI dq r dr ω π ωσ = = . 9.13.6 Four Long Conducting Wires Four infinitely long parallel wires carrying equal current I are arranged in such a way that when looking at the cross section, they are at the corners of a square, as shown in Figure 9.13.5. Currents in A and D point out of the page, and into the page at B and C.What is the magnetic field at the center of the square? 64 Figure 9.13.5 Four parallel conducting wires 9.13.7 Magnetic Force on a Current Loop A rectangular loop of length l and width carries a steady current w 1I . The loop is then placed near an finitely long wire carrying a current 2I , as shown in Figure 9.13.6. What is the magnetic force experienced by the loop due to the magnetic field of the wire? Figure 9.13.6 Magnetic force on a current loop. 9.13.8 Magnetic Moment of an Orbital Electron We want to estimate the magnetic dipole moment associated with the motion of an electron as it orbits a proton. We use a “semi-classical” model to do this. Assume that the electron has speed v and orbits a proton (assumed to be very massive) located at the origin. The electron is moving in a right-handed sense with respect to the z-axis in a circle of radius r = 0.53 Å, as shown in Figure 9.13.7. Note that 1 Å = .10 10 m− Figure 9.13.7 65 (a) The inward force required to make the electron move in this circle is provided by the Coulomb attractive force between the electron and proton ( me is the mass of the electron). Using this fact, and the value of r we give above, find the speed of the electron in our “semi-classical” model. [Ans: .] 2 /em v r 6 2.18 10 m/s × (b) Given this speed, what is the orbital period T of the electron? [Ans: .] 16 1.52 10 s−× (c) What current is associated with this motion? Think of the electron as stretched out uniformly around the circumference of the circle. In a time T, the total amount of charge q that passes an observer at a point on the circle is just e [Ans: 1.05 mA. Big!] (d) What is the magnetic dipole moment associated with this orbital motion? Give the magnitude and direction. The magnitude of this dipole moment is one Bohr magneton, B μ . [Ans: along the −z axis.] 24 29.27 10 A m −× ⋅ (e) One of the reasons this model is “semi-classical” is because classically there is no reason for the radius of the orbit above to assume the specific value we have given. The value of r is determined from quantum mechanical considerations, to wit that the orbital angular momentum of the electron can only assume integral multiples of h/2 π, where is the Planck constant. What is the orbital angular momentum of the electron here, in units of 34 6.63 10 J/s h −= × / 2 h π ? 9.13.9 Ferromagnetism and Permanent Magnets A disk of iron has a height and a radius 1.00 mm h = 1.00 cm r = . The magnetic dipole moment of an atom of iron is 23 21.8 10 A m μ − = × ⋅ . The molar mass of iron is 55.85 g, and its density is 7.9 g/cm 3 . Assume that all the iron atoms in the disk have their dipole moments aligned with the axis of the disk. (a) What is the number density of the iron atoms? How many atoms are in this disk? [Ans: ; .] 28 38.5 10 atoms/m × 22 2.7 10 atoms × (b) What is the magnetization in this disk? [Ans: , parallel to axis.] M G 61.53 10 A/m × (c) What is the magnetic dipole moment of the disk? [Ans: 20.48 A m ⋅ .] (d) If we were to wrap one loop of wire around a circle of the same radius r, how much current would the wire have to carry to get the dipole moment in (c)? This is the “equivalent” surface current due to the atomic currents in the interior of the magnet. [Ans: 1525 A.] 66 9.13.10 Charge in a Magnetic Field A coil of radius R with its symmetric axis along the + x-direction carries a steady current I.A positive charge q moves with a velocity ˆv=v G j when it crosses the axis at a distance x from the center of the coil, as shown in Figure 9.13.8. Figure 9.13.8 Describe the subsequent motion of the charge. What is the instantaneous radius of curvature? 9.13.11 Permanent Magnets A magnet in the shape of a cylindrical rod has a length of 4.8 cm and a diameter of 1.1 cm. It has a uniform magnetization M of 5300 A/m, directed parallel to its axis. (a) Calculate the magnetic dipole moment of this magnet. (b) What is the axial field a distance of 1 meter from the center of this magnet, along its axis? [Ans: (a) , (b) , or .] 2 22.42 10 A m −× ⋅ 94.8 10 T−× 54.8 10 gauss −× 9.13.12 Magnetic Field of a Solenoid (a) A 3000-turn solenoid has a length of 60 cm and a diameter of 8 cm. If this solenoid carries a current of 5.0 A, find the magnitude of the magnetic field inside the solenoid by constructing an Amperian loop and applying Ampere's Law. How does this compare to the magnetic field of the earth (0.5 gauss). [Ans: 0.0314 T, or 314 gauss, or about 600 times the magnetic field of the earth]. We make a magnetic field in the following way: We have a long cylindrical shell of non-conducting material which carries a surface charge fixed in place (glued down) of , as shown in Figure 9.13.9 The cylinder is suspended in a manner such that it is free to revolve about its axis, without friction. Initially it is at rest. We come along and spin it up until the speed of the surface of the cylinder is . 2 C/m σ 0 v 67 Figure 9.13.9 (b) What is the surface current K on the walls of the cylinder, in A/m? [Ans: 0K v σ= .] (c) What is magnetic field inside the cylinder? [Ans. 0 0 0B K v μ μ σ = = , oriented along axis right-handed with respect to spin.] (d) What is the magnetic field outside of the cylinder? Assume that the cylinder is infinitely long. [Ans: 0]. 9.13.13 Effect of Paramagnetism A solenoid with 16 turns/cm carries a current of 1.3 A. (a) By how much does the magnetic field inside the solenoid increase when a close-fitting chromium rod is inserted? [Note: Chromium is a paramagnetic material with magnetic susceptibility .] 42.7 10 χ −= × (b) Find the magnitude of the magnetization M G of the rod. [Ans: (a) 0.86 μT; (b) 0.68 A/m.] 68
2577	https://mathematicscentre.com/taskcentre/178match.htm	Match Triangles Match Triangles Task 178 ... Years 4 - 8 Summary This is one of several tasks that begin with an easily accessed concrete/visual pattern, but lead on to considerable algebraic possibilities. Students are asked to make triangle chains, similar to those used in engineering for strengthening structures. The main challenge in the problem is: If I tell you the number of triangle sections, can you tell me the number of 'matches' I need to make it? This task is a partner for Task 154, 4 Arm Shapes and others. Using a suite of tasks like this means that algebra becomes concrete and visual, and it makes sense. This cameo includes an Investigation Guide and a From The Classroom section which explains why a class of primary students developed a PowerPoint to 'teach the other teachers'. Match Triangles also appears on the Picture Puzzles Pattern & Algebra B menu where the problem is presented using one screen, two learners, concrete materials and a challenge.#### Materials About 11 sticks Triangle dot paper Content basic number skills seeking & seeing patterns generalisation equivalent algebraic expressions symbolic representation substituting into equations solving equations graphing ordered pairs relationship to gradient and y intercept Iceberg A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.Answers are: No. of Triangles 1 2 3 4 5...10...100 Number of Matches 3 5 7 9 11...21...201 and students should keep this record in their journal. The iceberg begins with the last challenge: Explain your answer. There are at least four ways to do this and it is important to realise that the way the student 'sees' is the one that makes sense to them. The way we 'see' the generalisation may be different, but it is not more correct. Generalisation A To find the number of matches count two for each triangle, then and add one to close off the end. Generalisation B To find the number of matches start with one, then add two for every triangle. Generalisation C To find the number of matches multiply the number of triangles by three (3 matches to make a triangle), then subtract one less than the number of triangles (because there will be double matches at all the joins). Generalisation D To find the number of matches start with 3 (the first triangle), then add two for each of the remaining triangles. Encourage students to record their view and at least one other. Once the generalisation has been made orally, record in words as here. The written words are the genesis of symbolic representation as an equation: Generalisation A To find the number of matches (M =) count two for each triangle (2T), then and add one to close off the end (+ 1). Generalisation B To find the number of Matches (M =) start with one (1), then add two for every triangle (+ 2T). Generalisation C To find the number of Matches (M =) multiply the number of triangles by three (3T), then subtract one less than the number of triangles (- [T - 1]). Generalisation D To find the number of Matches (M =) start with 3 (3), then add two for each of the remaining triangles (+ 2[T - 1]). which become: Generalisation A ... M = 2T + 1 Generalisation B ... M = 1 + 2T Generalisation C ... M = 3T - (T - 1) Generalisation D ... M = 3 + 2(T - 1) These are all equivalent algebraic expressions and, by reference to the match triangle pattern, students will be able to tell you what each symbol means and why particular operations and numbers are there. Extend further with questions such as: Do these different ways of seeing the pattern give the same answers for 5, 17, 26 triangles? Suppose I tell you the number of matches I have. Can you tell me number of triangles in the chain I could build? Can I tell you any number for the number of matches? Discuss. Choose any five numbers for the triangles. Work out the number of matches in each case and make pairs of numbers like this (triangles, matches). If these pairs were plotted on a graph what would you expect to see? Plot them to check your hypothesis. If you joined up these dots with a pencil line, how could you measure the slope (gradient) of the line? Which number does it go through on the vertical axis? What happens if we change the match pattern? All tasks have three lives. The task form (above) is an invitation to work like a mathematician. To this can be added an Investigation Guide to lead students deeper into the iceberg of a task. This is the second life. The third life is as a whole class investigation to model the work of a mathematician. This third life is described below. Usually teachers prepare their own Investigation Guides but in this case two teachers have gone further. Jodi Wilson and Maria Antoniou, Mt. Eliza Secondary College, have prepared their own Guide and then offered it to colleagues through this cameo. We very much appreciate this form of sharing and encourage others to do the same. Match Triangles Investigation Guide There are also Guides like this included for the 20 tasks in the Maths With Attitude Pattern & Algebra Years 7 & 8 kit and for 10 tasks in the Maths With Attitude Pattern & Algebra Years 9 & 10 kit. Guides such as these can lead to students publishing a report of their investigation. See Recording & Publishing for examples of student reports in various forms and see Assessment for a rubric for assessing such reports that has also been submitted by Jodi & Maria. Whole Class Investigation Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.To convert this task to a whole class lesson with the purpose of addressing all the mathematics above while modelling what it means to work like a mathematician, you need plenty of pop sticks or an alternative such as straws cut to size. However, for a greater level of involvement, which is sensible management because it adds purpose to using the sticks before they are distributed, begin with each student quickly preparing a newspaper roll. Students bring this to a central floor space and a whole class model of the triangle chain is quickly constructed. Discussion and challenge begins here and the table top models are then used to explore and confirm hypotheses before returning to the public floorboard model for discussion and extension. (See From The Classroom below.) Many teachers report that an added advantage of lessons like these is that when the next lesson is a related 'toolbox lesson', perhaps from a text book, students often comment that the ...stuff in the book is simple. For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 164, Match Triangles. Is it in Maths With Attitude? Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. The Match Triangles task is an integral part of: MWA Pattern & Algebra Years 3 & 4 MWA Pattern & Algebra Years 7 & 8 The Match Triangles lesson is an integral part of: MWA Pattern & Algebra Years 7 & 8 Match Triangles task is also included in the Task Centre Kit for Aboriginal Students. From The Classroom St. Monica's Primary School Evatt _Sarah Collis Year 5_> Miss Collis called this an 'aha' moment and couldn't stop smiling and we felt successful. So say the Year 5 students of St. Monica's Primary School. Sarah writes: Match Triangles was our first whole class investigation and the other teachers were wondering how they could implement them into their maths program. So the students developed a short slide show explaining our steps and sharing our experience to demonstrate how engaged the children were and how much they enjoyed the challenge of going beyond what was on a piece of paper. Follow this link to Task Centre Home page.
2578	https://www.khanacademy.org/math/revision-term-1-tg-math-class-12/xb0b22d4b45602eb1:week-1/xb0b22d4b45602eb1:complex-numbers/e/dividing_complex_numbers	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
2579	https://www.youtube.com/watch?v=FLD5xrq0rvU	How to Prove the Sum of an Arithmetic Progression : ExamSolutions ExamSolutions 284000 subscribers 1436 likes Description 138097 views Posted: 7 Jul 2014 Tutorial on the proof of the sum of an arithmetic progression. Go to for the index, playlists and more maths videos on arithmetic progressions, series and other maths topics. THE BEST THANK YOU: 67 comments Transcript: hi there now you quite often asked to prove the sum of the first end terms of an arithmetic progression or an arithmetic sequence and that is the purpose of this video showing you how we go about setting this out and proving the sum of the first end terms in fact I'll show you two formulas that we can use so let's just just go back remind ourselves what an arithmetic progression is we take the first term which is a and then to that first term we add a value often called the common difference and we denote it by D so we're going to get the second term a plus d and to the second term we add D again and so we get for the third term a + 2D add D again fourth term becomes a + 3D so you can see for any term we end up with a + 1 less than that term time D so for the nth term it would be a + nus1 D and the N minus1 term the one before this would be a + n minus 2D so hoping you've got that idea so how do we go about working out what the sum of all of these n terms would be well we call it SN so we just push it over here SN the sum of the first n terms and in other words then we're just adding these terms up so we just write them out again but I'd encourage you to leave a little bit of space here rather than just writing a and then plus a plus d let's just move it across a little so we'll put it say about here and we'll have the next term the second term A+ D and then we're going to add this term a + 2D the third term and I'm going to again leave a bit of space and put a + 2D here and so it's going to be plus and so on and then as we come up here okay we'll just put plus a few more dots there I think plus and then we've got a + nus 2D for this one and then plus the nth term which will be a + n minus1 d and what I'd like to do is we'll just put these in Brackets to separate the terms okay just make it a little clearer I hope okay so do that now what I'm going to do next might seem a bit strange okay but what I'm going to do is reverse the terms around still going to be the same total so we're just going to call it SN again but we're going to start with this term so we'll just write that in as a + nus1 D all right and then we'll write this one in plus and then we've got a + nus 2 multipli by D and the one before this okay would be a + nus 3D and we'll put plus and so on and then for these last two terms would be on these last two terms okay so we would have underneath this one a plus d and then under this term would have the first term plus a let's bracket these off again because it's hard to read so we just bracket off all of those terms there so I hope you can see that part okay now here's the interesting bit we'll call this equation one and we'll call this equation two and what we do do is we add these two equations together 1 + 2 so what we end up with is SN plus another SN so that's clearly going to be two sn's and it's going to equal now when we look at adding these two terms up we've got a plus another a so it's going to be 2 a and then we've got plus n minus1 d so just put that in plus n minus 1 D now what happens when we add these two together in fact just before I do that what I'll do is I'll put that in Brackets just to separate it off okay so we're going to add these two terms together next and what happens we get a plus a to a again now if we expand this bracket here we get N D minus 2D D Well - 2D plus this D is min -1d so we end up with N D minus 1D that's the same as this N D minus 1D when you expand it so we get exactly the same plus nus1 d now maybe you can start to see what might happen now when we add these two together we we get a + a 2 a again and if we expand this bracket we got N D minus 3D and we add 2D to it so we end up with N D minus 1D in other words when factorized n-1 D again and this is going to carry on for every term that we pair up okay and that's the reason why we reverse the terms so we create this one common summation okay so this is going to be in all the terms we're going to get this so let's just say plus and so on all the way down when we add these two together we get our 2 A Plus and then you'll notice if you work that out you get n minus 1 D and for the last one here A and A again is 2 a plus easy to see in this one nus1 D so let's just put some brackets around that to create those terms and so what we've got here is n of these brackets so when it comes to summing these we've therefore got 2 SN equals n lots of 2 a now I'm going to put this in square brackets now okay do away with those red brackets there 2 a + n-1 D and all I've got to do now is if I divide both sides by two I therefore have the sum of the first n terms equals the number of terms n / 2 and that's all multiplied by twice the first term 2 a in other words plus the number of terms n minus one time the common difference D and there is one particular version for the sum of the first n terms now there is another formula that we can use and I'll show you how we can derive that it's very easy if we take the formula we've got here for SN the sum of the first n terms then we can see it's n / 2 but instead of writing 2 a what I'm going to do is just write a plus another a there's our 2 a and then we've got + n -1 D the common difference there now when we look at a + n -1 D what I notice is that this is up here here it's the last term the nth term in other words of our arithmetic progression or arithmetic sequence so I can change this to therefore the sum of the first n terms equals n /2 we'll put our square bracket again here and then we've got a plus and instead of a + nus1 D I'm just going to write that as being our last term just write it as L let's just say where L is the last term okay the last term or you could say it is the nth term okay let's just squeeze that in there the nth term so what we've got here then is two formulas which uh I would encourage you to learn you'll generally find them in your formula book but uh there they are anyway okay and as you can see the proofs aren't that difficult but uh it's just setting them out giving yourself plenty of room when you're trying to prove this leave these spaces at this end okay so that your terms can line up all right okay so there we go that brings us now to the end of this particular proof for
2580	http://materias.df.uba.ar/fiquia2013c2/files/2012/07/Tabla-de-datos.pdf	Contents The following is a directory of all tables in the text; those included in this Data section are marked with an asterisk. The remainder will be found on the pages indicated. Physical properties of selected materials Masses and natural abundances of selected nuclides 1.1 Pressure units (4) 1.2 The gas constant in various units (9) 1.3 The composition of dry air at sea level (11) 1.4 Second virial coeﬃcients 1.5 Critical constants of gases 1.6 van der Waals coeﬃcients 1.7 Selected equations of state (19) 2.1 Varieties of work (34) 2.2 Temperature variation of molar heat capacities 2.3 Standard enthalpies of fusion and vaporization at the transition temperature 2.4 Enthalpies of transition [notation] (51) 2.5 Thermodynamic data for organic compounds 2.6 Thermochemical properties of some fuels (53) 2.7 Thermodynamic data for inorganic compounds 2.7a Standard enthalpies of hydration at inﬁnite dilution 2.7b Standard ion hydration enthalpies 2.8 Expansion coeﬃcients and isothermal compressibilities 2.9 Inversion temperatures, normal freezing and boiling points, and Joule–Thomson coeﬃcients 3.1 Standard entropies (and temperatures) of phase transitions 3.2 Standard entropies of vaporization of liquids 3.3 Standard Third-Law entropies [see Tables 2.5 and 2.7] 3.4 Standard Gibbs energies of formation [see Tables 2.5 and 2.7] 3.5 The Maxwell relations (104) 3.6 The fugacity of nitrogen 5.1 Henry’s law constants for gases 5.2 Freezing point and boiling point constants 5.3 Standard states [summary of deﬁnitions] (158) 5.4 Ionic strength and molality (164) 5.5 Mean activity coeﬃcients in water 5.6 Relative permitivities (dielectric constants) Data section 7.1 Varieties of electrode (216) 7.2 Standard potentials 7.3 The electrochemical series of metals () 7.4 Acidity constants for aqueous solutions 8.1 The Schrödinger equation (255) 8.2 Constraints of the uncertainty principle (271) 9.1 The Hermite polynomials (293) 9.2 The error function 9.3 The spherical harmonics (302) 9.4 Properties of angular momentum (309) 10.1 Hydrogenic radial wavefunctions (324) 10.2 Eﬀective nuclear charge 10.3 Ionization energies 10.4 Electron aﬃnities 11.1 Some hybridization schemes (368) 11.2 Bond lengths 11.3 Bond dissociation enthalpies 11.4 Pauling and Mulliken electronegativities 11.5 Ab initio calculations and spectroscopic data (398) 12.1 The notation for point groups (408) 12.2 The C2v character table (415) 12.3 The C3v character table (416) 13.1 Moments of inertia [formulae] (440) 13.2 Properties of diatomic molecules 13.3 Typical vibrational wavenumbers 14.1 Colour, frequency, and energy of light 14.2 Ground and excited states of O2 (483) 14.3 Absorption characteristics of some groups and molecules 14.4 Characteristics of laser radiation and their chemical applications (500) 15.1 Nuclear spin and nuclear structure (515) 15.2 Nuclear spin properties 15.3 Hyperﬁne coupling constants for atoms 17.1 Rotational and vibrational temperatures [see also Table 13.2] (594) DATA SECTION 989 17.2 Symmetry numbers [see also Table 13.2] (596) 17.3 Contributions to the molecular partition function [formulae] (615) 17.4 Thermodynamic functions in terms of the partition function [formulae] (616) 17.5 Contributions to mean energies and heat capacities [formulae] (616) 18.1 Dipole moments, polarizabilities, and polarizability volumes 18.2 Partial charges in polypeptides (622) 18.3 Multipole interaction energies (630) 18.4 Lennard-Jones (12,6)-potential parameters 18.5 Surface tensions of liquids 19.1 Radius of gyration 19.2 Diﬀusion coeﬃcients in water 19.3 Frictional coeﬃcients and molecular geometry 19.4 Intrinsic viscosity 20.1 The seven crystal systems (699) 20.2 The crystal structures of some elements (716) 20.3 Ionic radii 20.4 Madelung constants (719) 20.5 Lattice enthalpies 20.6 Magnetic susceptibilities 21.1 Collision cross-sections 21.2 Transport properties of gases 21.3 Transport properties of perfect gases [summary of formulae] (758) 21.4 Viscosities of liquids 21.5 Limiting ionic conductivities in water 21.6 Ionic mobilities in water 21.7 Debye–Hückel–Onsager coeﬃcients for (1,1)-electrolytes 21.8 Diﬀusion coeﬃcients 22.1 Kinetic data for ﬁrst-order reactions 22.2 Kinetic data for second-order reactions 22.3 Integrated rate laws (803) 22.4 Arrhenius parameters 23.1 Photochemical processes (846) 23.2 Common photophysical processes (846) 23.3 Values of R0 for donor–acceptor pairs (852) 24.1 Arrhenius parameters for gas-phase reactions 24.2 Arrhenius parameters for reactions in solution [see Table 22.4] 24.3 Summary of uses of k [notation] (902) 25.1 Maximum observed enthalpies of physisorption 25.2 Enthalpies of chemisorption 25.3 Activation energies of catalysed reactions 25.4 Properties of catalysts (929) 25.5 Chemisorption abilities (930) 25.6 Exchange current densities and transfer coeﬃcients 25.7 Summary of acronyms (950) A1.1 The SI base units (960) A1.2 A selection of derived units (961) A1.3 Common SI preﬁxes (961) A1.4 Some common units (962) A3.1 Refractive indices relative to air Character tables The following tables reproduce and expand the data given in the short tables in the text, and follow their numbering. Standard states refer to a pressure of p7 = 1 bar. The general references are as follows: AIP: D.E. Gray (ed.), American Institute of Physics handbook. McGraw Hill, New York (1972). AS: M. Abramowitz and I.A. Stegun (ed.), Handbook of mathematical functions. Dover, New York (1963). E: J. Emsley, The elements. Oxford University Press (1991). HCP: D.R. Lide (ed.), Handbook of chemistry and physics. CRC Press, Boca Raton (2000). JL: A.M. James and M.P. Lord, Macmillan’s chemical and physical data. Macmillan, London (1992). KL: G.W.C. Kaye and T.H. Laby (ed.), Tables of physical and chemical constants. Longman, London (1973). LR: G.N. Lewis and M. Randall, resived by K.S. Pitzer and L. Brewer, Thermodynamics. McGraw-Hill, New York (1961). NBS: NBS tables of chemical thermodynamic properties, published as J. Phys. and Chem. Reference Data, 11, Supplement 2 (1982). RS: R.A. Robinson and R.H. Stokes, Electrolyte solutions. Butterworth, London (1959). TDOC: J.B. Pedley, J.D. Naylor, and S.P. Kirby, Thermochemical data of organic compounds. Chapman & Hall, London (1986). 990 DATA SECTION Physical properties of selected materials r/(g cm−3) Tf /K Tb/K r/(g cm−3) Tf /K Tb/K at 293 K† at 293 K† Elements Aluminium(s) 2.698 933.5 2740 Argon(g) 1.381 83.8 87.3 Boron(s) 2.340 2573 3931 Bromine(l) 3.123 265.9 331.9 Carbon(s, gr) 2.260 3700s Carbon(s, d) 3.513 Chlorine(g) 1.507 172.2 239.2 Copper(s) 8.960 1357 2840 Fluorine(g) 1.108 53.5 85.0 Gold(s) 19.320 1338 3080 Helium(g) 0.125 4.22 Hydrogen(g) 0.071 14.0 20.3 Iodine(s) 4.930 386.7 457.5 Iron(s) 7.874 1808 3023 Krypton(g) 2.413 116.6 120.8 Lead(s) 11.350 600.6 2013 Lithium(s) 0.534 453.7 1620 Magnesium(s) 1.738 922.0 1363 Mercury(l) 13.546 234.3 629.7 Neon(g) 1.207 24.5 27.1 Nitrogen(g) 0.880 63.3 77.4 Oxygen(g) 1.140 54.8 90.2 Phosphorus(s, wh) 1.820 317.3 553 Potassium(s) 0.862 336.8 1047 Silver(s) 10.500 1235 2485 Sodium(s) 0.971 371.0 1156 Sulfur(s, α) 2.070 386.0 717.8 Uranium(s) 18.950 1406 4018 Xenon(g) 2.939 161.3 166.1 Zinc(s) 7.133 692.7 1180 d: decomposes; s: sublimes; Data: AIP, E, HCP, KL. † For gases, at their boiling points. Inorganic compounds CaCO3(s, calcite) 2.71 1612 1171d CuSO4·5H2O(s) 2.284 383(–H2O) 423(–5H2O) HBr(g) 2.77 184.3 206.4 HCl(g) 1.187 159.0 191.1 HI(g) 2.85 222.4 237.8 H2O(l) 0.997 273.2 373.2 D2O(l) 1.104 277.0 374.6 NH3(g) 0.817 195.4 238.8 KBr(s) 2.750 1003 1708 KCl(s) 1.984 1049 1773s NaCl(s) 2.165 1074 1686 H2SO4(l) 1.841 283.5 611.2 Organic compounds Acetaldehyde, CH3CHO(l, g) 0.788 152 293 Acetic acid, CH3COOH(l) 1.049 289.8 391 Acetone, (CH3)2CO(l) 0.787 178 329 Aniline, C6H5NH2(l) 1.026 267 457 Anthracene, C14H10(s) 1.243 490 615 Benzene, C6H6(l) 0.879 278.6 353.2 Carbon tetrachloride, CCl4(l) 1.63 250 349.9 Chloroform, CHCl3(l) 1.499 209.6 334 Ethanol, C2H5OH(l) 0.789 156 351.4 Formaldehyde, HCHO(g) 181 254.0 Glucose, C6H12O6(s) 1.544 415 Methane, CH4(g) 90.6 111.6 Methanol, CH3OH(l) 0.791 179.2 337.6 Naphthalene, C10H8(s) 1.145 353.4 491 Octane, C8H18(l) 0.703 216.4 398.8 Phenol, C6H5OH(s) 1.073 314.1 455.0 Sucrose, C12H22O11(s) 1.588 457d DATA SECTION 991 Table 1.5 Critical constants of gases pc/atm Vc/(cm3 mol−1) Tc/K Zc TB/K Ar 48.00 75.25 150.72 0.292 411.5 Br2 102 135 584 0.287 C2H4 50.50 124 283.1 0.270 C2H6 48.20 148 305.4 0.285 C6H6 48.6 260 562.7 0.274 CH4 45.6 98.7 190.6 0.288 510.0 Cl2 76.1 124 417.2 0.276 CO2 72.85 94.0 304.2 0.274 714.8 F2 55 144 H2 12.8 64.99 33.23 0.305 110.0 H2O 218.3 55.3 647.4 0.227 HBr 84.0 363.0 HCl 81.5 81.0 324.7 0.248 He 2.26 57.76 5.21 0.305 22.64 HI 80.8 423.2 Kr 54.27 92.24 209.39 0.291 575.0 N2 33.54 90.10 126.3 0.292 327.2 Ne 26.86 41.74 44.44 0.307 122.1 NH3 111.3 72.5 405.5 0.242 O2 50.14 78.0 154.8 0.308 405.9 Xe 58.0 118.8 289.75 0.290 768.0 Data: AIP, KL. Masses and natural abundances of selected nuclides Nuclide m/u Abundance/% H 1H 1.0078 99.985 2H 2.0140 0.015 He 3He 3.0160 0.000 13 4He 4.0026 100 Li 6Li 6.0151 7.42 7Li 7.0160 92.58 B 10B 10.0129 19.78 11B 11.0093 80.22 C 12C 12 98.89 13C 13.0034 1.11 N 14N 14.0031 99.63 15N 15.0001 0.37 O 16O 15.9949 99.76 17O 16.9991 0.037 18O 17.9992 0.204 F 19F 18.9984 100 P 31P 30.9738 100 S 32S 31.9721 95.0 33S 32.9715 0.76 34S 33.9679 4.22 Cl 35Cl 34.9688 75.53 37Cl 36.9651 24.4 Br 79Br 78.9183 50.54 81Br 80.9163 49.46 I 127I 126.9045 100 Exact value. Table 1.4 Second virial coeﬃcients, B/(cm3 mol−1) 100 K 273 K 373 K 600 K Air −167.3 −13.5 3.4 19.0 Ar −187.0 −21.7 −4.2 11.9 CH4 −53.6 −21.2 8.1 CO2 −142 −72.2 −12.4 H2 −2.0 13.7 15.6 He 11.4 12.0 11.3 10.4 Kr −62.9 −28.7 1.7 N2 −160.0 −10.5 6.2 21.7 Ne −6.0 10.4 12.3 13.8 O2 −197.5 −22.0 −3.7 12.9 Xe −153.7 −81.7 −19.6 Data: AIP, JL. The values relate to the expansion in eqn 1.22 of Section 1.3b; convert to eqn 1.21 using B′ = B/RT. For Ar at 273 K, C = 1200 cm6 mol−1. 992 DATA SECTION Table 1.6 van der Waals coeﬃcients a/(atm dm6 mol−2) b/(10−2 dm3 mol−1) a/(atm dm6 mol−2) b/(10−2 dm3 mol−1) Ar 1.337 3.20 H2S 4.484 4.34 C2H4 4.552 5.82 He 0.0341 2.38 C2H6 5.507 6.51 Kr 5.125 1.06 C6H6 18.57 11.93 N2 1.352 3.87 CH4 2.273 4.31 Ne 0.205 1.67 Cl2 6.260 5.42 NH3 4.169 3.71 CO 1.453 3.95 O2 1.364 3.19 CO2 3.610 4.29 SO2 6.775 5.68 H2 0.2420 2.65 Xe 4.137 5.16 H2O 5.464 3.05 Data: HCP. Table 2.2 Temperature variation of molar heat capacities† a b/(10−3 K−1) c/(105 K2) Monatomic gases 20.78 0 0 Other gases Br2 37.32 0.50 −1.26 Cl2 37.03 0.67 −2.85 CO2 44.22 8.79 −8.62 F2 34.56 2.51 −3.51 H2 27.28 3.26 0.50 I2 37.40 0.59 −0.71 N2 28.58 3.77 −0.50 NH3 29.75 25.1 −1.55 O2 29.96 4.18 −1.67 Liquids (from melting to boiling) C10H8, naphthalene 79.5 0.4075 0 I2 80.33 0 0 H2O 75.29 0 0 Solids Al 20.67 12.38 0 C (graphite) 16.86 4.77 −8.54 C10H8, naphthalene −115.9 3.920 × 103 0 Cu 22.64 6.28 0 I2 40.12 49.79 0 NcCl 45.94 16.32 0 Pb 22.13 11.72 0.96 † For Cp,m/(J K−1 mol−1) = a + bT + c/T2. Source: LR. DATA SECTION 993 Table 2.5 Thermodynamic data for organic compounds (all values are for 298 K) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1 mol−1)† C 7 p,m/(J K−1 mol−1) DcH 7/(kJ mol−1) C(s) (graphite) 12.011 0 0 5.740 8.527 −393.51 C(s) (diamond) 12.011 +1.895 +2.900 2.377 6.113 −395.40 CO2(g) 44.040 −393.51 −394.36 213.74 37.11 Hydrocarbons CH4(g), methane 16.04 −74.81 −50.72 186.26 35.31 −890 CH3(g), methyl 15.04 +145.69 +147.92 194.2 38.70 C2H2(g), ethyne 26.04 +226.73 +209.20 200.94 43.93 −1300 C2H4(g), ethene 28.05 +52.26 +68.15 219.56 43.56 −1411 C2H6(g), ethane 30.07 −84.68 −32.82 229.60 52.63 −1560 C3H6(g), propene 42.08 +20.42 +62.78 267.05 63.89 −2058 C3H6(g), cyclopropane 42.08 +53.30 +104.45 237.55 55.94 −2091 C3H8(g), propane 44.10 −103.85 −23.49 269.91 73.5 −2220 C4H8(g), 1-butene 56.11 −0.13 +71.39 305.71 85.65 −2717 C4H8(g), cis-2-butene 56.11 −6.99 +65.95 300.94 78.91 −2710 C4H8(g), trans-2-butene 56.11 −11.17 +63.06 296.59 87.82 −2707 C4H10(g), butane 58.13 −126.15 −17.03 310.23 97.45 −2878 C5H12(g), pentane 72.15 −146.44 −8.20 348.40 120.2 −3537 C5H12(l) 72.15 −173.1 C6H6(l), benzene 78.12 +49.0 +124.3 173.3 136.1 −3268 Table 2.3 Standard enthalpies of fusion and vaporization at the transition temperature, ΔtrsH7/(kJ mol−1) Tf/K Fusion Tb/K Vaporization Tf /K Fusion Tb/K Vaporization Elements Ag 1234 11.30 2436 250.6 Ar 83.81 1.188 87.29 6.506 Br2 265.9 10.57 332.4 29.45 Cl2 172.1 6.41 239.1 20.41 F2 53.6 0.26 85.0 3.16 H2 13.96 0.117 20.38 0.916 He 3.5 0.021 4.22 0.084 Hg 234.3 2.292 629.7 59.30 I2 386.8 15.52 458.4 41.80 N2 63.15 0.719 77.35 5.586 Na 371.0 2.601 1156 98.01 O2 54.36 0.444 90.18 6.820 Xe 161 2.30 165 12.6 K 336.4 2.35 1031 80.23 Inorganic compounds CCl4 250.3 2.47 349.9 30.00 Data: AIP; s denotes sublimation. CO2 217.0 8.33 194.6 25.23 s CS2 161.2 4.39 319.4 26.74 H2O 273.15 6.008 373.15 40.656 44.016 at 298 K H2S 187.6 2.377 212.8 18.67 H2SO4 283.5 2.56 NH3 195.4 5.652 239.7 23.35 Organic compounds CH4 90.68 0.941 111.7 8.18 CCl4 250.3 2.5 350 30.0 C2H6 89.85 2.86 184.6 14.7 C6H6 278.61 10.59 353.2 30.8 C6H14 178 13.08 342.1 28.85 C10H8 354 18.80 490.9 51.51 CH3OH 175.2 3.16 337.2 35.27 37.99 at 298 K C2H5OH 158.7 4.60 352 43.5 994 DATA SECTION Hydrocarbons (Continued) C6H6(g) 78.12 +82.93 +129.72 269.31 81.67 −3302 C6H12(l), cyclohexane 84.16 −156 +26.8 204.4 156.5 −3920 C6H14(l), hexane 86.18 −198.7 204.3 −4163 C6H5CH3(g), methylbenzene (toluene) 92.14 +50.0 +122.0 320.7 103.6 −3953 C7H16(l), heptane 100.21 −224.4 +1.0 328.6 224.3 C8H18(l), octane 114.23 −249.9 +6.4 361.1 −5471 C8H18(l), iso-octane 114.23 −255.1 −5461 C10H8(s), naphthalene 128.18 +78.53 −5157 Alcohols and phenols CH3OH(l), methanol 32.04 −238.66 −166.27 126.8 81.6 −726 CH3OH(g) 32.04 −200.66 −161.96 239.81 43.89 −764 C2H5OH(l), ethanol 46.07 −277.69 −174.78 160.7 111.46 −1368 C2H5OH(g) 46.07 −235.10 −168.49 282.70 65.44 −1409 C6H5OH(s), phenol 94.12 −165.0 −50.9 146.0 −3054 Carboxylic acids, hydroxy acids, and esters HCOOH(l), formic 46.03 −424.72 −361.35 128.95 99.04 −255 CH3COOH(l), acetic 60.05 −484.5 −389.9 159.8 124.3 −875 CH3COOH(aq) 60.05 −485.76 −396.46 178.7 CH3CO2 −(aq) 59.05 −486.01 −369.31 +86.6 −6.3 (COOH)2(s), oxalic 90.04 −827.2 117 −254 C6H5COOH(s), benzoic 122.13 −385.1 −245.3 167.6 146.8 −3227 CH3CH(OH)COOH(s), lactic 90.08 −694.0 −1344 CH3COOC2H5(l), ethyl acetate 88.11 −479.0 −332.7 259.4 170.1 −2231 Alkanals and alkanones HCHO(g), methanal 30.03 −108.57 −102.53 218.77 35.40 −571 CH3CHO(l), ethanal 44.05 −192.30 −128.12 160.2 −1166 CH3CHO(g) 44.05 −166.19 −128.86 250.3 57.3 −1192 CH3COCH3(l), propanone 58.08 −248.1 −155.4 200.4 124.7 −1790 Sugars C6H12O6(s), α-d-glucose 180.16 −1274 −2808 C6H12O6(s), β-d-glucose 180.16 −1268 −910 212 C6H12O6(s), β-d-fructose 180.16 −1266 −2810 C12H22O11(s), sucrose 342.30 −2222 −1543 360.2 −5645 Nitrogen compounds CO(NH2)2(s), urea 60.06 −333.51 −197.33 104.60 93.14 −632 CH3NH2(g), methylamine 31.06 −22.97 +32.16 243.41 53.1 −1085 C6H5NH2(l), aniline 93.13 +31.1 −3393 CH2(NH2)COOH(s), glycine 75.07 −532.9 −373.4 103.5 99.2 −969 Data: NBS, TDOC. † Standard entropies of ions may be either positive or negative because the values are relative to the entropy of the hydrogen ion. Table 2.5 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1 mol−1)† C 7 p,m/(J K−1 mol−1) DcH 7/(kJ mol−1) DATA SECTION 995 Table 2.7 Thermodynamic data for elements and inorganic compounds (all values relate to 298 K) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Aluminium (aluminum) Al(s) 26.98 0 0 28.33 24.35 Al(l) 26.98 +10.56 +7.20 39.55 24.21 Al(g) 26.98 +326.4 +285.7 164.54 21.38 Al3+(g) 26.98 +5483.17 Al3+(aq) 26.98 −531 −485 −321.7 Al2O3(s, α) 101.96 −1675.7 −1582.3 50.92 79.04 AlCl3(s) 133.24 −704.2 −628.8 110.67 91.84 Argon Ar(g) 39.95 0 0 154.84 20.786 Antimony Sb(s) 121.75 0 0 45.69 25.23 SbH3(g) 124.77 +145.11 +147.75 232.78 41.05 Arsenic As(s, α) 74.92 0 0 35.1 24.64 As(g) 74.92 +302.5 +261.0 174.21 20.79 As4(g) 299.69 +143.9 +92.4 314 AsH3(g) 77.95 +66.44 +68.93 222.78 38.07 Barium Ba(s) 137.34 0 0 62.8 28.07 Ba(g) 137.34 +180 +146 170.24 20.79 Ba2+(aq) 137.34 −537.64 −560.77 +9.6 BaO(s) 153.34 −553.5 −525.1 70.43 47.78 BaCl2(s) 208.25 −858.6 −810.4 123.68 75.14 Beryllium Be(s) 9.01 0 0 9.50 16.44 Be(g) 9.01 +324.3 +286.6 136.27 20.79 Bismuth Bi(s) 208.98 0 0 56.74 25.52 Bi(g) 208.98 +207.1 +168.2 187.00 20.79 Bromine Br2(l) 159.82 0 0 152.23 75.689 Br2(g) 159.82 +30.907 +3.110 245.46 36.02 Br(g) 79.91 +111.88 +82.396 175.02 20.786 Br−(g) 79.91 −219.07 Br−(aq) 79.91 −121.55 −103.96 +82.4 −141.8 HBr(g) 90.92 −36.40 −53.45 198.70 29.142 Cadmium Cd(s, γ) 112.40 0 0 51.76 25.98 Cd(g) 112.40 +112.01 +77.41 167.75 20.79 Cd2+(aq) 112.40 −75.90 −77.612 −73.2 996 DATA SECTION Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Cadmium (Continued) CdO(s) 128.40 −258.2 −228.4 54.8 43.43 CdCO3(s) 172.41 −750.6 −669.4 92.5 Caesium (cesium) Cs(s) 132.91 0 0 85.23 32.17 Cs(g) 132.91 +76.06 +49.12 175.60 20.79 Cs+(aq) 132.91 −258.28 −292.02 +133.05 −10.5 Calcium Ca(s) 40.08 0 0 41.42 25.31 Ca(g) 40.08 +178.2 +144.3 154.88 20.786 Ca2+(aq) 40.08 −542.83 −553.58 −53.1 CaO(s) 56.08 −635.09 −604.03 39.75 42.80 CaCO3(s) (calcite) 100.09 −1206.9 −1128.8 92.9 81.88 CaCO3(s) (aragonite) 100.09 −1207.1 −1127.8 88.7 81.25 CaF2(s) 78.08 −1219.6 −1167.3 68.87 67.03 CaCl2(s) 110.99 −795.8 −748.1 104.6 72.59 CaBr2(s) 199.90 −682.8 −663.6 130 Carbon (for ‘organic’ compounds of carbon, see Table 2.5) C(s) (graphite) 12.011 0 0 5.740 8.527 C(s) (diamond) 12.011 +1.895 +2.900 2.377 6.113 C(g) 12.011 +716.68 +671.26 158.10 20.838 C2(g) 24.022 +831.90 +775.89 199.42 43.21 CO(g) 28.011 −110.53 −137.17 197.67 29.14 CO2(g) 44.010 −393.51 −394.36 213.74 37.11 CO2(aq) 44.010 −413.80 −385.98 117.6 H2CO3(aq) 62.03 −699.65 −623.08 187.4 HCO3 −(aq) 61.02 −691.99 −586.77 +91.2 CO3 2−(aq) 60.01 −677.14 −527.81 −56.9 CCl4(l) 153.82 −135.44 −65.21 216.40 131.75 CS2(l) 76.14 +89.70 +65.27 151.34 75.7 HCN(g) 27.03 +135.1 +124.7 201.78 35.86 HCN(l) 27.03 +108.87 +124.97 112.84 70.63 CN−(aq) 26.02 +150.6 +172.4 +94.1 Chlorine Cl2(g) 70.91 0 0 223.07 33.91 Cl(g) 35.45 +121.68 +105.68 165.20 21.840 Cl−(g) 34.45 −233.13 Cl−(aq) 35.45 −167.16 −131.23 +56.5 −136.4 HCl(g) 36.46 −92.31 −95.30 186.91 29.12 HCl(aq) 36.46 −167.16 −131.23 56.5 −136.4 Chromium Cr(s) 52.00 0 0 23.77 23.35 Cr(g) 52.00 +396.6 +351.8 174.50 20.79 DATA SECTION 997 Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Chromium (Continued) CrO4 2−(aq) 115.99 −881.15 −727.75 +50.21 Cr2O7 2−(aq) 215.99 −1490.3 −1301.1 +261.9 Copper Cu(s) 63.54 0 0 33.150 24.44 Cu(g) 63.54 +338.32 +298.58 166.38 20.79 Cu+(aq) 63.54 +71.67 +49.98 +40.6 Cu2+(aq) 63.54 +64.77 +65.49 −99.6 Cu2O(s) 143.08 −168.6 −146.0 93.14 63.64 CuO(s) 79.54 −157.3 −129.7 42.63 42.30 CuSO4(s) 159.60 −771.36 −661.8 109 100.0 CuSO4·H2O(s) 177.62 −1085.8 −918.11 146.0 134 CuSO4·5H2O(s) 249.68 −2279.7 −1879.7 300.4 280 Deuterium D2(g) 4.028 0 0 144.96 29.20 HD(g) 3.022 +0.318 −1.464 143.80 29.196 D2O(g) 20.028 −249.20 −234.54 198.34 34.27 D2O(l) 20.028 −294.60 −243.44 75.94 84.35 HDO(g) 19.022 −245.30 −233.11 199.51 33.81 HDO(l) 19.022 −289.89 −241.86 79.29 Fluorine F2(g) 38.00 0 0 202.78 31.30 F(g) 19.00 +78.99 +61.91 158.75 22.74 F−(aq) 19.00 −332.63 −278.79 −13.8 −106.7 HF(g) 20.01 −271.1 −273.2 173.78 29.13 Gold Au(s) 196.97 0 0 47.40 25.42 Au(g) 196.97 +366.1 +326.3 180.50 20.79 Helium He(g) 4.003 0 0 126.15 20.786 Hydrogen (see also deuterium) H2(g) 2.016 0 0 130.684 28.824 H(g) 1.008 +217.97 +203.25 114.71 20.784 H+(aq) 1.008 0 0 0 0 H+(g) 1.008 +1536.20 H2O(s) 18.015 37.99 H2O(l) 18.015 −285.83 −237.13 69.91 75.291 H2O(g) 18.015 −241.82 −228.57 188.83 33.58 H2O2(l) 34.015 −187.78 −120.35 109.6 89.1 Iodine I2(s) 253.81 0 0 116.135 54.44 I2(g) 253.81 +62.44 +19.33 260.69 36.90 998 DATA SECTION Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Iodine (Continued) I(g) 126.90 +106.84 +70.25 180.79 20.786 I−(aq) 126.90 −55.19 −51.57 +111.3 −142.3 HI(g) 127.91 +26.48 +1.70 206.59 29.158 Iron Fe(s) 55.85 0 0 27.28 25.10 Fe(g) 55.85 +416.3 +370.7 180.49 25.68 Fe2+(aq) 55.85 −89.1 −78.90 −137.7 Fe3+(aq) 55.85 −48.5 −4.7 −315.9 Fe3O4(s) (magnetite) 231.54 −1118.4 −1015.4 146.4 143.43 Fe2O3(s) (haematite) 159.69 −824.2 −742.2 87.40 103.85 FeS(s, α) 87.91 −100.0 −100.4 60.29 50.54 FeS2(s) 119.98 −178.2 −166.9 52.93 62.17 Krypton Kr(g) 83.80 0 0 164.08 20.786 Lead Pb(s) 207.19 0 0 64.81 26.44 Pb(g) 207.19 +195.0 +161.9 175.37 20.79 Pb2+(aq) 207.19 −1.7 −24.43 +10.5 PbO(s, yellow) 223.19 −217.32 −187.89 68.70 45.77 PbO(s, red) 223.19 −218.99 −188.93 66.5 45.81 PbO2(s) 239.19 −277.4 −217.33 68.6 64.64 Lithium Li(s) 6.94 0 0 29.12 24.77 Li(g) 6.94 +159.37 +126.66 138.77 20.79 Li+(aq) 6.94 −278.49 −293.31 +13.4 68.6 Magnesium Mg(s) 24.31 0 0 32.68 24.89 Mg(g) 24.31 +147.70 +113.10 148.65 20.786 Mg2+(aq) 24.31 −466.85 −454.8 −138.1 MgO(s) 40.31 −601.70 −569.43 26.94 37.15 MgCO3(s) 84.32 −1095.8 −1012.1 65.7 75.52 MgCl2(s) 95.22 −641.32 −591.79 89.62 71.38 Mercury Hg(l) 200.59 0 0 76.02 27.983 Hg(g) 200.59 +61.32 +31.82 174.96 20.786 Hg2+(aq) 200.59 +171.1 +164.40 −32.2 Hg2 2+(aq) 401.18 +172.4 +153.52 +84.5 HgO(s) 216.59 −90.83 −58.54 70.29 44.06 Hg2Cl2(s) 472.09 −265.22 −210.75 192.5 102 HgCl2(s) 271.50 −224.3 −178.6 146.0 HgS(s, black) 232.65 −53.6 −47.7 88.3 DATA SECTION 999 Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Neon Ne(g) 20.18 0 0 146.33 20.786 Nitrogen N2(g) 28.013 0 0 191.61 29.125 N(g) 14.007 +472.70 +455.56 153.30 20.786 NO(g) 30.01 +90.25 +86.55 210.76 29.844 N2O(g) 44.01 +82.05 +104.20 219.85 38.45 NO2(g) 46.01 +33.18 +51.31 240.06 37.20 N2O4(g) 92.1 +9.16 +97.89 304.29 77.28 N2O5(s) 108.01 −43.1 +113.9 178.2 143.1 N2O5(g) 108.01 +11.3 +115.1 355.7 84.5 HNO3(l) 63.01 −174.10 −80.71 155.60 109.87 HNO3(aq) 63.01 −207.36 −111.25 146.4 −86.6 NO3 −(aq) 62.01 −205.0 −108.74 +146.4 −86.6 NH3(g) 17.03 −46.11 −16.45 192.45 35.06 NH3(aq) 17.03 −80.29 −26.50 111.3 NH4 +(aq) 18.04 −132.51 −79.31 +113.4 79.9 NH2OH(s) 33.03 −114.2 HN3(l) 43.03 +264.0 +327.3 140.6 43.68 HN3(g) 43.03 +294.1 +328.1 238.97 98.87 N2H4(l) 32.05 +50.63 +149.43 121.21 139.3 NH4NO3(s) 80.04 −365.56 −183.87 151.08 84.1 NH4Cl(s) 53.49 −314.43 −202.87 94.6 Oxygen O2(g) 31.999 0 0 205.138 29.355 O(g) 15.999 +249.17 +231.73 161.06 21.912 O3(g) 47.998 +142.7 +163.2 238.93 39.20 OH−(aq) 17.007 −229.99 −157.24 −10.75 −148.5 Phosphorus P(s, wh) 30.97 0 0 41.09 23.840 P(g) 30.97 +314.64 +278.25 163.19 20.786 P2(g) 61.95 +144.3 +103.7 218.13 32.05 P4(g) 123.90 +58.91 +24.44 279.98 67.15 PH3(g) 34.00 +5.4 +13.4 210.23 37.11 PCl3(g) 137.33 −287.0 −267.8 311.78 71.84 PCl3(l) 137.33 −319.7 −272.3 217.1 PCl5(g) 208.24 −374.9 −305.0 364.6 112.8 PCl5(s) 208.24 −443.5 H3PO3(s) 82.00 −964.4 H3PO3(aq) 82.00 −964.8 H3PO4(s) 94.97 −1279.0 −1119.1 110.50 106.06 H3PO4(l) 94.97 −1266.9 H3PO4(aq) 94.97 −1277.4 −1018.7 −222 1000 DATA SECTION Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Phosphorus (Continued) PO4 3−(aq) 94.97 −1277.4 −1018.7 −221.8 P4O10(s) 283.89 −2984.0 −2697.0 228.86 211.71 P4O6(s) 219.89 −1640.1 Potassium K(s) 39.10 0 0 64.18 29.58 K(g) 39.10 +89.24 +60.59 160.336 20.786 K+(g) 39.10 +514.26 K+(aq) 39.10 −252.38 −283.27 +102.5 21.8 KOH(s) 56.11 −424.76 −379.08 78.9 64.9 KF(s) 58.10 −576.27 −537.75 66.57 49.04 KCl(s) 74.56 −436.75 −409.14 82.59 51.30 KBr(s) 119.01 −393.80 −380.66 95.90 52.30 Kl(s) 166.01 −327.90 −324.89 106.32 52.93 Silicon Si(s) 28.09 0 0 18.83 20.00 Si(g) 28.09 +455.6 +411.3 167.97 22.25 SiO2(s, α) 60.09 −910.94 −856.64 41.84 44.43 Silver Ag(s) 107.87 0 0 42.55 25.351 Ag(g) 107.87 +284.55 +245.65 173.00 20.79 Ag+(aq) 107.87 +105.58 +77.11 +72.68 21.8 AgBr(s) 187.78 −100.37 −96.90 107.1 52.38 AgCl(s) 143.32 −127.07 −109.79 96.2 50.79 Ag2O(s) 231.74 −31.05 −11.20 121.3 65.86 AgNO3(s) 169.88 −129.39 −33.41 140.92 93.05 Sodium Na(s) 22.99 0 0 51.21 28.24 Na(g) 22.99 +107.32 +76.76 153.71 20.79 Na+(aq) 22.99 −240.12 −261.91 59.0 46.4 NaOH(s) 40.00 −425.61 −379.49 64.46 59.54 NaCl(s) 58.44 −411.15 −384.14 72.13 50.50 NaBr(s) 102.90 −361.06 −348.98 86.82 51.38 NaI(s) 149.89 −287.78 −286.06 98.53 52.09 Sulfur S(s, α) (rhombic) 32.06 0 0 31.80 22.64 S(s, β) (monoclinic) 32.06 +0.33 +0.1 32.6 23.6 S(g) 32.06 +278.81 +238.25 167.82 23.673 S2(g) 64.13 +128.37 +79.30 228.18 32.47 S2−(aq) 32.06 +33.1 +85.8 −14.6 SO2(g) 64.06 −296.83 −300.19 248.22 39.87 SO3(g) 80.06 −395.72 −371.06 256.76 50.67 DATA SECTION 1001 Table 2.7a Standard enthalpies of hydration at inﬁnite dilution, ΔhydH7/(kJ mol−1) Li+ Na+ K+ Rb+ Cs+ F− −1026 −911 −828 −806 −782 Cl− −884 −783 −685 −664 −640 Br− −856 −742 −658 −637 −613 I− −815 −701 −617 −596 −572 Entries refer to X+(g) + Y−(g) →X+(aq) + Y−(aq). Data: Principally J.O’M. Bockris and A.K.N. Reddy, Modern electrochemistry, Vol. 1. Plenum Press, New York (1970). Table 2.7b Standard ion hydration enthalpies, ΔhydH 7/(kJ mol−1) at 298 K Cations H+ (−1090) Ag+ −464 Mg2+ −1920 Li+ −520 NH4 + −301 Ca2+ −1650 Na+ −405 Sr2+ −1480 K+ −321 Ba2+ −1360 Rb+ −300 Fe2+ −1950 Cs+ −277 Cu2+ −2100 Zn2+ −2050 Al3+ −4690 Fe3+ −4430 Anions OH− −460 F− −506 Cl− −364 Br− −337 I− −296 Entries refer to X±(g) →X±(aq) based on H+(g) →H+(aq); ΔH7 = −1090 kJ mol−1. Data: Principally J.O’M. Bockris and A.K.N. Reddy, Modern electrochemistry, Vol. 1. Plenum Press, New York (1970). Table 2.7 (Continued) M/(g mol−1) DfH7/(kJ mol−1) DfG7/(kJ mol−1) Sm 7 /(J K−1mol−1)† C 7 p,m/(J K−1mol−1) Sulfur (Continued) H2SO4(l) 98.08 −813.99 −690.00 156.90 138.9 H2SO4(aq) 98.08 −909.27 −744.53 20.1 −293 SO4 2−(aq) 96.06 −909.27 −744.53 +20.1 −293 HSO4 −(aq) 97.07 −887.34 −755.91 +131.8 −84 H2S(g) 34.08 −20.63 −33.56 205.79 34.23 H2S(aq) 34.08 −39.7 −27.83 121 HS−(aq) 33.072 −17.6 +12.08 +62.08 SF6(g) 146.05 −1209 −1105.3 291.82 97.28 Tin Sn(s, β) 118.69 0 0 51.55 26.99 Sn(g) 118.69 +302.1 +267.3 168.49 20.26 Sn2+(aq) 118.69 −8.8 −27.2 −17 SnO(s) 134.69 −285.8 −256.9 56.5 44.31 SnO2(s) 150.69 −580.7 −519.6 52.3 52.59 Xenon Xe(g) 131.30 0 0 169.68 20.786 Zinc Zn(s) 65.37 0 0 41.63 25.40 Zn(g) 65.37 +130.73 +95.14 160.98 20.79 Zn2+(aq) 65.37 −153.89 −147.06 −112.1 46 ZnO(s) 81.37 −348.28 −318.30 43.64 40.25 Source: NBS. † Standard entropies of ions may be either positive or negative because the values are relative to the entropy of the hydrogen ion. 1002 DATA SECTION Table 3.1 Standard entropies (and temperatures) of phase transitions, ΔtrsS7/(J K−1 mol−1) Fusion (at Tf) Vaporization (at Tb) Ar 14.17 (at 83.8 K) 74.53 (at 87.3 K) Br2 39.76 (at 265.9 K) 88.61 (at 332.4 K) C6H6 38.00 (at 278.6 K) 87.19 (at 353.2 K) CH3COOH 40.4 (at 289.8 K) 61.9 (at 391.4 K) CH3OH 18.03 (at 175.2 K) 104.6 (at 337.2 K) Cl2 37.22 (at 172.1 K) 85.38 (at 239.0 K) H2 8.38 (at 14.0 K) 44.96 (at 20.38 K) H2O 22.00 (at 273.2 K) 109.0 (at 373.2 K) H2S 12.67 (at 187.6 K) 87.75 (at 212.0 K) He 4.8 (at 1.8 K and 30 bar) 19.9 (at 4.22 K) N2 11.39 (at 63.2 K) 75.22 (at 77.4 K) NH3 28.93 (at 195.4 K) 97.41 (at 239.73 K) O2 8.17 (at 54.4 K) 75.63 (at 90.2 K) Data: AIP. Table 2.8 Expansion coeﬃcients, α, and isothermal compressibilities, κT a/(10−4 K−1) kT /(10−6 atm−1) Liquids Benzene 12.4 92.1 Carbon tetrachloride 12.4 90.5 Ethanol 11.2 76.8 Mercury 1.82 38.7 Water 2.1 49.6 Solids Copper 0.501 0.735 Diamond 0.030 0.187 Iron 0.354 0.589 Lead 0.861 2.21 The values refer to 20°C. Data: AIP(α), KL(κT). Table 2.9 Inversion temperatures, normal freezing and boiling points, and Joule–Thomson coeﬃcients at 1 atm and 298 K TI/K Tf/K Tb/K mJT/(K atm−1) Air 603 0.189 at 50°C Argon 723 83.8 87.3 Carbon dioxide 1500 194.7s 1.11 at 300 K Helium 40 4.22 −0.062 Hydrogen 202 14.0 20.3 −0.03 Krypton 1090 116.6 120.8 Methane 968 90.6 111.6 Neon 231 24.5 27.1 Nitrogen 621 63.3 77.4 0.27 Oxygen 764 54.8 90.2 0.31 s: sublimes. Data: AIP, JL, and M.W. Zemansky, Heat and thermodynamics. McGraw-Hill, New York (1957). DATA SECTION 1003 Table 3.6 The fugacity coeﬃcient of nitrogen at 273 K p/atm f p/atm f 1 0.999 55 300 1.0055 10 0.9956 400 1.062 50 0.9912 600 1.239 100 0.9703 800 1.495 150 0.9672 1000 1.839 200 0.9721 Data: LR. Table 5.1 Henry’s law constants for gases at 298 K, K/(kPa kg mol−1) Water Benzene CH4 7.55 × 104 44.4 × 103 CO2 30.1 × 103 8.90 × 102 H2 1.28 × 105 2.79 × 104 N2 1.56 × 105 1.87 × 104 O2 7.92 × 104 Data: converted from R.J. Silbey and R.A. Alberty, Physical chemistry. Wiley, New York (2001). Table 3.2 Standard entropies of vaporization of liquids at their normal boiling point DvapH 7/(kJ mol−1) qb/°C DvapS 7/(J K−1 mol−1) Benzene 30.8 80.1 +87.2 Carbon disulﬁde 26.74 46.25 +83.7 Carbon tetrachloride 30.00 76.7 +85.8 Cyclohexane 30.1 80.7 +85.1 Decane 38.75 174 +86.7 Dimethyl ether 21.51 −23 +86 Ethanol 38.6 78.3 +110.0 Hydrogen sulﬁde 18.7 −60.4 +87.9 Mercury 59.3 356.6 +94.2 Methane 8.18 −161.5 +73.2 Methanol 35.21 65.0 +104.1 Water 40.7 100.0 +109.1 Data: JL. Table 3.3 Standard Third-Law entropies at 298 K: see Tables 2.5 and 2.7 Table 3.4 Standard Gibbs energies of formation at 298 K: see Tables 2.5 and 2.7 1004 DATA SECTION Table 5.6 Relative permittivities (dielectric constants) at 293 K Nonpolar molecules Polar molecules Methane (at −173°C) 1.655 Water 78.54 (at 298 K) 80.10 Carbon tetrachloride 2.238 Ammonia 16.9 (at 298 K) 22.4 at −33°C Cyclohexane 2.024 Hydrogen sulﬁde 9.26 at −85°C 5.93 (at 283 K) Benzene 2.283 Methanol 33.0 Ethanol 25.3 Nitrobenzene 35.6 Data: HCP. Table 5.5 Mean activity coeﬃcients in water at 298 K b/b7 HCl KCl CaCl2 H2SO4 LaCl3 In2(SO4)3 0.001 0.966 0.966 0.888 0.830 0.790 0.005 0.929 0.927 0.789 0.639 0.636 0.16 0.01 0.905 0.902 0.732 0.544 0.560 0.11 0.05 0.830 0.816 0.584 0.340 0.388 0.035 0.10 0.798 0.770 0.524 0.266 0.356 0.025 0.50 0.769 0.652 0.510 0.155 0.303 0.014 1.00 0.811 0.607 0.725 0.131 0.387 2.00 1.011 0.577 1.554 0.125 0.954 Data: RS, HCP, and S. Glasstone, Introduction to electrochemistry. Van Nostrand (1942). Table 5.2 Freezing-point and boiling-point constants Kf /(K kg mol−1) Kb/(K kg mol−1) Acetic acid 3.90 3.07 Benzene 5.12 2.53 Camphor 40 Carbon disulﬁde 3.8 2.37 Carbon tetrachloride 30 4.95 Naphthalene 6.94 5.8 Phenol 7.27 3.04 Water 1.86 0.51 Data: KL. DATA SECTION 1005 Table 7.2 Standard potentials at 298 K. (a) In electrochemical order Reduction half-reaction E7/V Reduction half-reaction E7/V Strongly oxidizing Cu2+ + e−→Cu+ +0.16 H4XeO6+ 2H+ + 2e−→XeO3 + 3H2O +3.0 Sn4+ + 2e−→Sn2+ +0.15 F2 + 2e−→2F− +2.87 AgBr + e−→Ag + Br− +0.07 O3 + 2H+ + 2e−→O2 + H2O +2.07 Ti4+ + e−→Ti3+ 0.00 S2O8 2−+ 2e−→2SO4 2− +2.05 2H+ + 2e−→H2 0, by deﬁnition Ag2+ + e−→Ag+ +1.98 Fe3+ + 3e−→Fe −0.04 Co3+ + e−→Co2+ +1.81 O2 + H2O + 2e−→HO2 −+ OH− −0.08 H2O2 + 2H+ + 2e−→2H2O +1.78 Pb2+ + 2e−→Pb −0.13 Au+ + e−→Au +1.69 In+ + e−→In −0.14 Pb4+ + 2e−→Pb2+ +1.67 Sn2+ + 2e−→Sn −0.14 2HClO + 2H+ + 2e−→Cl2 + 2H2O +1.63 AgI + e−→Ag + I− −0.15 Ce4+ + e−→Ce3+ +1.61 Ni2+ + 2e−→Ni −0.23 2HBrO + 2H+ + 2e−→Br2 + 2H2O +1.60 Co2+ + 2e−→Co −0.28 MnO4 −+ 8H+ + 5e−→Mn2+ + 4H2O +1.51 In3+ + 3e−→In −0.34 Mn3+ + e−→Mn2+ +1.51 Tl+ + e−→Tl −0.34 Au3+ + 3e−→Au +1.40 PbSO4 + 2e−→Pb + SO4 2− −0.36 Cl2 + 2e−→2Cl− +1.36 Ti3+ + e−→Ti2+ −0.37 Cr2O7 2−+ 14H+ + 6e−→2Cr3+ + 7H2O +1.33 Cd2+ + 2e−→Cd −0.40 O3 + H2O + 2e−→O2 + 2OH− +1.24 In2+ + e−→In+ −0.40 O2 + 4H+ + 4e−→2H2O +1.23 Cr3+ + e−→Cr2+ −0.41 ClO4 −+ 2H+ + 2e−→ClO3 −+ H2O +1.23 Fe2+ + 2e−→Fe −0.44 MnO2 + 4H+ + 2e−→Mn2+ + 2H2O +1.23 In3+ + 2e−→In+ −0.44 Br2 + 2e−→2Br− +1.09 S + 2e−→S2− −0.48 Pu4+ + e−→Pu3+ +0.97 In3+ + e−→In2+ −0.49 NO3 −+ 4H+ + 3e−→NO + 2H2O +0.96 U4+ + e−→U3+ −0.61 2Hg2+ + 2e−→Hg2 2+ +0.92 Cr3+ + 3e−→Cr −0.74 ClO−+ H2O + 2e−→Cl−+ 2OH− +0.89 Zn2+ + 2e−→Zn −0.76 Hg2+ + 2e−→Hg +0.86 Cd(OH)2 + 2e−→Cd + 2OH− −0.81 NO3 −+ 2H+ + e−→NO2 + H2O +0.80 2H2O + 2e−→H2 + 2OH− −0.83 Ag+ + e−→Ag +0.80 Cr2+ + 2e−→Cr −0.91 Hg2 2+ + 2e−→2Hg +0.79 Mn2+ + 2e−→Mn −1.18 Fe3+ + e−→Fe2+ +0.77 V2+ + 2e−→V −1.19 BrO−+ H2O + 2e−→Br−+ 2OH− +0.76 Ti2+ + 2e−→Ti −1.63 Hg2SO4 + 2e−→2Hg + SO4 2− +0.62 Al3+ + 3e−→Al −1.66 MnO4 2−+ 2H2O + 2e−→MnO2 + 4OH− +0.60 U3+ + 3e−→U −1.79 MnO4 −+ e−→MnO4 2− +0.56 Sc3+ + 3e−→Sc −2.09 I2 + 2e−→2I− +0.54 Mg2+ + 2e−→Mg −2.36 CU+ + e−→Cu +0.52 Ce3+ + 3e−→Ce −2.48 I3 −+ 2e−→3I− +0.53 La3+ + 3e−→La −2.52 NiOOH + H2O + e−→Ni(OH)2 + OH− +0.49 Na+ + e−→Na −2.71 Ag2CrO4 + 2e−→2Ag + CrO4 2− +0.45 Ca2+ + 2e−→Ca −2.87 O2 + 2H2O + 4e−→4OH− +0.40 Sr2+ + 2e−→Sr −2.89 ClO4 −+ H2O + 2e−→ClO3 −+ 2OH− +0.36 Ba2+ + 2e−→Ba −2.91 [Fe(CN)6]3−+ e−→[Fe(CN)6]4− +0.36 Ra2+ + 2e−→Ra −2.92 Cu2+ + 2e−→Cu +0.34 Cs+ + e−→Cs −2.92 Hg2Cl2 + 2e−→2Hg + 2Cl− +0.27 Rb+ + e−→Rb −2.93 AgCl + e−→Ag + Cl− +0.22 K+ + e−→K −2.93 Bi3+ + 3e−→Bi +0.20 Li+ + e−→Li −3.05 1006 DATA SECTION Table 7.2 Standard potentials at 298 K. (b) In electrochemical order Reduction half-reaction E 7/V Reduction half-reaction E 7/V Ag+ + e−→Ag +0.80 I2 + 2e−→2I− +0.54 Ag2+ + e−→Ag+ +1.98 I− 3 + 2e−→3I− +0.53 AgBr + e−→Ag + Br− +0.0713 In+ + e−→In −0.14 AgCl + e−→Ag + Cl− +0.22 In2+ + e−→In+ −0.40 Ag2CrO4 + 2e−→2Ag + CrO4 2− +0.45 In3+ + 2e−→In+ −0.44 AgF + e−→Ag + F− +0.78 In3+ + 3e−→In −0.34 AgI + e−→Ag + I− −0.15 In3+ + e−→In2+ −0.49 Al3+ + 3e−→Al −1.66 K+ + e−→K −2.93 Au+ + e−→Au +1.69 La3+ + 3e−→La −2.52 Au3+ + 3e−→Au +1.40 Li+ + e−→Li −3.05 Ba2+ + 2e−→Ba +2.91 Mg2+ + 2e−→Mg −2.36 Be2+ + 2e−→Be −1.85 Mn2+ + 2e−→Mn −1.18 Bi3+ + 3e−→Bi +0.20 Mn3++ e−→Mn2+ +1.51 Br2 + 2e−→2Br− +1.09 MnO2 + 4H+ + 2e−→Mn2+ + 2H2O +1.23 BrO−+ H2O + 2e−→Br−+ 2OH− +0.76 MnO4 −+ 8H+ + 5e−→Mn2+ + 4H2O +1.51 Ca2+ + 2e−→Ca −2.87 MnO4 −+ e−→MnO4 2− +0.56 Cd(OH)2+ 2e−→Cd + 2OH− −0.81 MnO4 2−+ 2H2O + 2e−→MnO2 + 4OH− +0.60 Cd2+ + 2e−→Cd −0.40 Na+ + e−→Na −2.71 Ce3+ + 3e−→Ce −2.48 Ni2+ + 2e−→Ni −0.23 Ce4+ + e−→Ce3+ +1.61 NiOOH + H2O + e−→Ni(OH)2 + OH− +0.49 Cl2 + 2e−→2Cl− +1.36 NO3 −+ 2H+ + e−→NO2 + H2O −0.80 ClO−+ H2O + 2e−→Cl−+ 2OH− +0.89 NO3 −+ 4H+ + 3e−→NO + 2H2O +0.96 ClO4 −+ 2H+ + 2e−→ClO3 −+ H2O +1.23 NO3 −+ H2O + 2e−→NO2 −+ 2OH− +0.10 ClO4 −+ H2O + 2e−→ClO3 −+ 2OH− +0.36 O2 + 2H2O + 4e−→4OH− +0.40 Co2+ + 2e−→Co −0.28 O2 + 4H+ + 4e−→2H2O +1.23 Co3+ + e−→Co2+ +1.81 O2 +e−→O2 − −0.56 Cr2+ + 2e−→Cr −0.91 O2 + H2O + 2e−→HO2 −+ OH− −0.08 Cr2O7 2−+ 14H++ 6e−→2Cr3+ + 7H2O +1.33 O3 + 2H+ + 2e−→O2 + H2O +2.07 Cr3+ + 3e−→Cr −0.74 O3 + H2O + 2e−→O2 + 2OH− +1.24 Cr3+ + e−→Cr2+ −0.41 Pb2+ + 2e−→Pb −0.13 Cs+ + e−→Cs −2.92 Pb4+ + 2e−→Pb2+ +1.67 Cu+ + e−→Cu +0.52 PbSO4 + 2e−→Pb + SO4 2− −0.36 Cu2+ + 2e−→Cu +0.34 Pt2+ + 2e−→Pt +1.20 Cu2+ + e−→Cu+ +0.16 Pu4+ + e−→Pu3+ +0.97 F2 + 2e−→2F− +2.87 Ra2+ + 2e−→Ra −2.92 Fe2+ + 2e−→Fe −0.44 Rb+ + e−→Rb −2.93 Fe3+ + 3e−→Fe −0.04 S + 2e−→S2− −0.48 Fe3+ + e−→Fe2+ +0.77 S2O8 2−+ 2e−→2SO4 2− +2.05 [Fe(CN)6]3−+ e−→[Fe(CN)6]4− +0.36 Sc3+ + 3e−→Sc −2.09 2H+ + 2e−→H2 0, by deﬁnition Sn2+ + 2e−→Sn −0.14 2H2O + 2e−→H2 + 2OH− −0.83 Sn4+ + 2e−→Sn2+ +0.15 2HBrO + 2H+ + 2e−→Br2 + 2H2O +1.60 Sr2+ + 2e−→Sr −2.89 2HClO + 2H+ + 2e−→Cl2 + 2H2O +1.63 Ti2+ + 2e−→Ti −1.63 H2O2 + 2H+ + 2e−→2H2O +1.78 Ti3+ + e−→Ti2+ −0.37 H4XeO6 + 2H+ + 2e−→XeO3+3H2O +3.0 Ti4+ + e−→Ti3+ 0.00 Hg2 2+ + 2e−→2Hg +0.79 Tl+ + e−→Tl −0.34 Hg2Cl2 + 2e−→2Hg + 2Cl− +0.27 U3+ + 3e−→U −1.79 Hg2+ + 2e−→Hg +0.86 U4++ e−→U3+ −0.61 2Hg2+ + 2e−→Hg2 2+ +0.92 V2+ + 2e−→V −1.19 Hg2SO4 + 2e−→2Hg + SO4 2− +0.62 V3+ + e−→V2+ −0.26 Zn2+ + 2e−→Zn −0.76 DATA SECTION 1007 Table 7.4 Acidity constants for aqueous solutions at 298 K. (a) In order of acid strength Acid HA A− Ka pKa Hydriodic HI I− 1011 −11 Hydrobromic HBr Br− 109 −9 Hydrochloric HCl Cl− 107 −7 Sulfuric H2SO4 HSO4 − 102 −2 Perchloric HClO4 ClO4 − 4.0 × 101 −1.6 Hydronium ion H3O+ H2O 1 0.0 Oxalic (COOH)2 HOOCCO2 − 5.6 × 10−2 1.25 Sulfurous H2SO3 HSO3 − 1.4 × 10−2 1.85 Hydrogensulfate ion HSO4 − SO4 2− 1.0 × 10−2 1.99 Phosphoric H3PO4 H2PO4 − 6.9 × 10−3 2.16 Glycinium ion +NH3CH2COOH NH2CH2COOH 4.5 × 10−3 2.35 Hydroﬂuoric HF F− 6.3 × 10−4 3.20 Formic HCOOH HCO2 − 1.8 × 10−4 3.75 Hydrogenoxalate ion HOOCCO2 − C2O4 2− 1.5 × 10−5 3.81 Lactic CH3CH(OH)COOH CH3CH(OH)CO2 − 1.4 × 10−4 3.86 Acetic (ethanoic) CH3COOH CH3CO2 − 1.4 × 10−5 4.76 Butanoic CH3CH2CH2COOH CH3CH2CH2CO2 − 1.5 × 10−5 4.83 Propanoic CH3CH2COOH CH3CH2CO2 − 1.4 × 10−5 4.87 Anilinium ion C6H5NH3 + C6H5NH2 1.3 × 10−5 4.87 Pyridinium ion C5H5NH+ C6H5N 5.9 × 10−6 5.23 Carbonic H2CO3 HCO3 − 4.5 × 10−7 6.35 Hydrosulfuric H2S HS− 8.9 × 10−8 7.05 Dihydrogenphosphate ion H2PO4 − HPO4 2− 6.2 × 10−8 7.21 Hypochlorous HClO ClO− 4.0 × 10−8 7.40 Hydrazinium ion NH2NH3 + NH2NH2 8 × 10−9 8.1 Hypobromous HBrO BrO− 2.8 × 10−9 8.55 Hydrocyanic HCN CN− 6.2 × 10−10 9.21 Ammonium ion NH4 + NH3 5.6 × 10−10 9.25 Boric B(OH)3 B(OH)4 − 5.4 × 10−10 9.27 Trimethylammonium ion (CH3)3NH+ (CH3)3N 1.6 × 10−10 9.80 Phenol C6H5OH C6H5O− 1.0 × 10−10 9.99 Hydrogencarbonate ion HCO3 − CO3 2− 4.8 × 10−11 10.33 Hypoiodous HIO IO− 3 × 10−11 10.5 Ethylammonium ion CH3CH2NH3 + CH3CH2NH2 2.2 × 10−11 10.65 Methylammonium ion CH3NH3 + CH3NH2 2.2 × 10−11 10.66 Dimethylammonium ion (CH3)2NH2 + (CH3)2NH 1.9 × 10−11 10.73 Triethylammonium ion (CH3CH2)3NH+ (CH3CH2)3N 1.8 × 10−11 10.75 Diethylammonium ion (CH3CH2)2NH2 + (CH3CH2)2NH 1.4 × 10−11 10.84 Hydrogenarsenate ion HAsO4 2− AsO4 3− 5.1 × 10−12 11.29 Hydrogenphosphate ion HPO4 2− PO4 3− 4.8 × 10−13 12.32 Hydrogensulﬁde ion HS− S2− 1.0 × 10−19 19.00 At 293 K. 1008 DATA SECTION Table 7.4 Acidity constants for aqueous solutions at 298 K. (b) In alphabetical order Acid HA A− Ka pKa Acetic (ethanoic) CH3COOH CH3CO2 − 1.4 × 10−5 4.76 Ammonium ion NH4 + NH3 5.6 × 10−10 9.25 Anilinium ion C6H5NH3 + C6H5NH2 1.3 × 10−5 4.87 Boric B(OH)3 B(OH)4 − 5.4 × 10−10 9.27 Butanoic CH3CH2CH2COOH CH3CH2CH2CO2 − 1.5 × 10−5 4.83 Carbonic H2CO3 HCO3 − 4.5 × 10−7 6.35 Diethylammonium ion (CH3CH2)2NH2 + (CH3CH2)2NH 1.4 × 10−11 10.84 Dihydrogenphosphate ion H2PO4 − HPO4 2− 6.2 × 10−8 7.21 Dimethylammonium ion (CH3)2NH2 + (CH3)2NH 1.9 × 10−11 10.73 Ethylammonium ion CH3CH2NH3 + CH3CH2NH2 2.2 × 10−11 10.65 Formic HCOOH HCO2 − 1.8 × 10−4 3.75 Glycinium ion +NH3CH2COOH NH2CH2COOH 4.5 × 10−3 2.35 Hydrazinium ion NH2NH3 + NH2NH2 8 × 10−9 8.1 Hydriodic HI I− 1011 −11 Hydrobromic HBr Br− 109 −9 Hydrochloric HCl Cl− 107 −7 Hydrocyanic HCN CN− 6.2 × 10−10 9.21 Hydroﬂuoric HF F− 6.3 × 10−4 3.20 Hydrogenarsenate ion HAsO4 2− AsO4 3− 5.1 × 10−12 11.29 Hydrogencarbonate ion HCO3 − CO3 2− 4.8 × 10−11 10.33 Hydrogenoxalate ion HOOCCO2 − C2O4 2− 1.5 × 10−5 3.81 Hydrogenphosphate ion HPO4 2− PO4 3− 4.8 × 10−13 12.32 Hydrogensulfate ion HSO4 − SO4 2− 1.0 × 10−2 1.99 Hydrogensulﬁde ion HS− S2− 1.0 × 10−19 19.00 Hydronium ion H3O+ H2O 1 0.0 Hydrosulfuric H2S HS− 8.9 × 10−8 7.05 Hypobromous HBrO BrO− 2.8 × 10−9 8.55 Hypochlorous HClO ClO− 4.0 × 10−8 7.40 Hypoiodous HIO IO− 3 × 10−11 10.5 Lactic CH3CH(OH)COOH CH3CH(OH)CO2 − 1.4 × 10−4 3.86 Methylammonium ion CH3NH3 + CH3NH2 2.2 × 10−11 10.66 Oxalic (COOH)2 HOOCCO2 − 5.6 × 10−2 1.25 Perchloric HClO4 ClO4 − 4.0 × 101 −1.6 Phenol C6H5OH C6H5O− 1.0 × 10−10 9.99 Phosphoric H3PO4 H2PO4 − 6.9 × 10−3 2.16 Propanoic CH3CH2COOH CH3CH2CO2 − 1.4 × 10−5 4.87 Pyridinim ion C5H5NH+ C6H5N 5.9 × 10−6 5.23 Sulfuric H2SO4 HSO4 − 102 −2 Sulfurous H2SO3 HSO3 − 1.4 × 10−2 1.85 Triethylammonium ion (CH3CH2)3NH+ (CH3CH2)3N 1.8 × 10−11 10.75 Trimethylammonium ion (CH3)3NH+ (CH3)3N 1.6 × 10−10 9.80 At 293 K. DATA SECTION 1009 Table 9.2 The error function z erf z z erf z 0 0 0.45 0.475 48 0.01 0.011 28 0.50 0.520 50 0.02 0.022 56 0.55 0.563 32 0.03 0.033 84 0.60 0.603 86 0.04 0.045 11 0.65 0.642 03 0.05 0.056 37 0.70 0.677 80 0.06 0.067 62 0.75 0.711 16 0.07 0.078 86 0.80 0.742 10 0.08 0.090 08 0.85 0.770 67 0.09 0.101 28 0.90 0.796 91 0.10 0.112 46 0.95 0.820 89 0.15 0.168 00 1.00 0.842 70 0.20 0.222 70 1.20 0.910 31 0.25 0.276 32 1.40 0.952 28 0.30 0.328 63 1.60 0.976 35 0.35 0.379 38 1.80 0.989 09 0.40 0.428 39 2.00 0.995 32 Data: AS. Table 10.2 Screening constants for atoms; values of Zeﬀ= Z −σ for neutral ground-state atoms H He 1s 1 1.6875 Li Be B C N O F Ne 1s 2.6906 3.6848 4.6795 5.6727 6.6651 7.6579 8.6501 9.6421 2s 1.2792 1.9120 2.5762 3.2166 3.8474 4.4916 5.1276 5.7584 2p 2.4214 3.1358 3.8340 4.4532 5.1000 5.7584 Na Mg Al Si P S Cl Ar 1s 10.6259 11.6089 12.5910 13.5745 14.5578 15.5409 16.5239 17.5075 2s 6.5714 7.3920 8.3736 9.0200 9.8250 10.6288 11.4304 12.2304 2p 6.8018 7.8258 8.9634 9.9450 10.9612 11.9770 12.9932 14.0082 3s 2.5074 3.3075 4.1172 4.9032 5.6418 6.3669 7.0683 7.7568 3p 4.0656 4.2852 4.8864 5.4819 6.1161 6.7641 Data: E. Clementi and D.L. Raimondi, Atomic screening constants from SCF functions. IBM Res. Note NJ-27 (1963). J. chem. Phys. 38, 2686 (1963). 1010 DATA SECTION Table 10.4 Electron aﬃnities, Eea/(kJ mol−1) H He 72.8 −21 Li Be B C N O F Ne 59.8 ≤0 23 122.5 −7 141 322 −29 −844 Na Mg Al Si P S Cl Ar 52.9 ≤0 44 133.6 71.7 200.4 348.7 −35 −532 K Ca Ga Ge As Se Br Kr 48.3 2.37 36 116 77 195.0 324.5 −39 Rb Sr In Sn Sb Te I Xe 46.9 5.03 34 121 101 190.2 295.3 −41 Cs Ba Tl Pb Bi Po At Rn 45.5 13.95 30 35.2 101 186 270 −41 Data: E. Table 10.3 Ionization energies, I/(kJ mol−1) H He 1312.0 2372.3 5250.4 Li Be B C N O F Ne 513.3 899.4 800.6 1086.2 1402.3 1313.9 1681 2080.6 7298.0 1757.1 2427 2352 2856.1 3388.2 3374 3952.2 Na Mg Al Si P S Cl Ar 495.8 737.7 577.4 786.5 1011.7 999.6 1251.1 1520.4 4562.4 1450.7 1816.6 1577.1 1903.2 2251 2297 2665.2 2744.6 2912 K Ca Ga Ge As Se Br Kr 418.8 589.7 578.8 762.1 947.0 940.9 1139.9 1350.7 3051.4 1145 1979 1537 1798 2044 2104 2350 2963 2735 Rb Sr In Sn Sb Te I Xe 403.0 549.5 558.3 708.6 833.7 869.2 1008.4 1170.4 2632 1064.2 1820.6 1411.8 1794 1795 1845.9 2046 2704 2943.0 2443 Cs Ba Tl Pb Bi Po At Rn 375.5 502.8 589.3 715.5 703.2 812 930 1037 2420 965.1 1971.0 1450.4 1610 2878 3081.5 2466 Data: E. DATA SECTION 1011 Table 11.2 Bond lengths, Re/pm (a) Bond lengths in speciﬁc molecules Br2 228.3 Cl2 198.75 CO 112.81 F2 141.78 H2 + 106 H2 74.138 HBr 141.44 HCl 127.45 HF 91.680 HI 160.92 N2 109.76 O2 120.75 (b) Mean bond lengths from covalent radii H 37 C 77(1) N 74(1) O 66(1) F 64 67(2) 65(2) 57(2) 60(3) Si 118 P 110 S 104(1) Cl 99 95(2) Ge 122 As 121 Se 104 Br 114 Sb 141 Te 137 I 133 Values are for single bonds except where indicated otherwise (values in parentheses). The length of an A-B covalent bond (of given order) is the sum of the corresponding covalent radii. Table 11.3a Bond dissociation enthalpies, ΔH7(A-B)/(kJ mol−1) at 298 K Diatomic molecules H-H 436 F-F 155 Cl-Cl 242 Br-Br 193 I-I 151 O=O 497 C=O 1076 N.N 945 H-O 428 H-F 565 H-Cl 431 H-Br 366 H-I 299 Polyatomic molecules H-CH3 435 H-NH2 460 H-OH 492 H-C6H5 469 H3C-CH3 368 H2C=CH2 720 HC.CH 962 HO-CH3 377 Cl-CH3 352 Br-CH3 293 I-CH3 237 O=CO 531 HO-OH 213 O2N-NO2 54 Data: HCP, KL. 1012 DATA SECTION Table 11.4 Pauling (italics) and Mulliken electronegativities H He 2.20 3.06 Li Be B C N O F Ne 0.98 1.57 2.04 2.55 3.04 3.44 3.98 1.28 1.99 1.83 2.67 3.08 3.22 4.43 4.60 Na Mg Al Si P S Cl Ar 0.93 1.31 1.61 1.90 2.19 2.58 3.16 1.21 1.63 1.37 2.03 2.39 2.65 3.54 3.36 K Ca Ga Ge As Se Br Kr 0.82 1.00 1.81 2.01 2.18 2.55 2.96 3.0 1.03 1.30 1.34 1.95 2.26 2.51 3.24 2.98 Rb Sr In Sn Sb Te I Xe 0.82 0.95 1.78 1.96 2.05 2.10 2.66 2.6 0.99 1.21 1.30 1.83 2.06 2.34 2.88 2.59 Cs Ba Tl Pb Bi 0.79 0.89 2.04 2.33 2.02 Data: Pauling values: A.L. Allred, J. Inorg. Nucl. Chem. 17, 215 (1961); L.C. Allen and J.E. Huheey, ibid., 42, 1523 (1980). Mulliken values: L.C. Allen, J. Am. Chem. Soc. 111, 9003 (1989). The Mulliken values have been scaled to the range of the Pauling values. Table 11.3b Mean bond enthalpies, ΔH7(A-B)/(kJ mol−1) H C N O F Cl Br I S P Si H 436 C 412 348(i) 612(ii) 838(iii) 518(a) N 388 305(i) 163(i) 613(ii) 409(ii) 890(iii) 946(iii) O 463 360(i) 157 146(i) 743(ii) 497(ii) F 565 484 270 185 155 Cl 431 338 200 203 254 242 Br 366 276 219 193 I 299 238 210 178 151 S 338 259 496 250 212 264 P 322 201 Si 318 374 466 226 (i) Single bond, (ii) double bond, (iii) triple bond, (a) aromatic. Data: HCP and L. Pauling, The nature of the chemical bond. Cornell University Press (1960). DATA SECTION 1013 Table 13.2 Properties of diatomic molecules §0/cm−1 qV/K B/cm−1 qR/K r/pm k/(N m−1) D/(kJ mol−1) s 1H2 + 2321.8 3341 29.8 42.9 106 160 255.8 2 1H2 4400.39 6332 60.864 87.6 74.138 574.9 432.1 2 2H2 3118.46 4487 30.442 43.8 74.154 577.0 439.6 2 1H19F 4138.32 5955 20.956 30.2 91.680 965.7 564.4 1 1H35Cl 2990.95 4304 10.593 15.2 127.45 516.3 427.7 1 1H81Br 2648.98 3812 8.465 12.2 141.44 411.5 362.7 1 1H127I 2308.09 3321 6.511 9.37 160.92 313.8 294.9 1 14N2 2358.07 3393 1.9987 2.88 109.76 2293.8 941.7 2 16O2 1580.36 2274 1.4457 2.08 120.75 1176.8 493.5 2 19F2 891.8 1283 0.8828 1.27 141.78 445.1 154.4 2 35Cl2 559.71 805 0.2441 0.351 198.75 322.7 239.3 2 12C16O 2170.21 3122 1.9313 2.78 112.81 1903.17 1071.8 1 79Br81Br 323.2 465 0.0809 10.116 283.3 245.9 190.2 1 Data: AIP. Table 13.3 Typical vibrational wavenumbers, #/cm−1 C-H stretch 2850–2960 C-H bend 1340–1465 C-C stretch, bend 700–1250 C=C stretch 1620 –1680 C.C stretch 2100–2260 O-H stretch 3590–3650 H-bonds 3200–3570 C=O stretch 1640–1780 C.N stretch 2215–2275 N-H stretch 3200–3500 C-F stretch 1000–1400 C-Cl stretch 600–800 C-Br stretch 500–600 C-I stretch 500 CO3 2− 1410–1450 NO3 − 1350–1420 NO2 − 1230–1250 SO4 2− 1080–1130 Silicates 900–1100 Data: L.J. Bellamy, The infrared spectra of complex molecules and Advances in infrared group frequencies. Chapman and Hall. Table 14.1 Colour, frequency, and energy of light Colour l/nm n/(1014 Hz) §/(104 cm−1) E/eV E/(kJ mol−1) Infrared >1000 <3.00 <1.00 <1.24 <120 Red 700 4.28 1.43 1.77 171 Orange 620 4.84 1.61 2.00 193 Yellow 580 5.17 1.72 2.14 206 Green 530 5.66 1.89 2.34 226 Blue 470 6.38 2.13 2.64 254 Violet 420 7.14 2.38 2.95 285 Near ultraviolet 300 10.0 3.33 4.15 400 Far ultraviolet <200 >15.0 >5.00 >6.20 >598 Data: J.G. Calvert and J.N. Pitts, Photochemistry. Wiley, New York (1966). 1014 DATA SECTION Table 15.2 Nuclear spin properties Nuclide Natural Spin I Magnetic g-value g/(107 T−1 s−1) NMR frequency at abundance % moment m/mN 1 T, n/MHz 1n 1 – 2 −1.9130 −3.8260 −18.324 29.164 1H 99.9844 1 – 2 2.792 85 5.5857 26.752 42.576 2H 0.0156 1 0.857 44 0.857 45 4.1067 6.536 3H 1 – 2 2.978 96 −4.2553 −20.380 45.414 10B 19.6 3 1.8006 0.6002 2.875 4.575 11B 80.4 3 – 2 2.6886 1.7923 8.5841 13.663 13C 1.108 1 – 2 0.7024 1.4046 6.7272 10.708 14N 99.635 1 0.403 76 0.403 56 1.9328 3.078 17O 0.037 5 – 2 −1.893 79 −0.7572 −3.627 5.774 19F 100 1 – 2 2.628 87 5.2567 25.177 40.077 31P 100 1 – 2 1.1316 2.2634 10.840 17.251 33S 0.74 3 – 2 0.6438 0.4289 2.054 3.272 35Cl 75.4 3 – 2 0.8219 0.5479 2.624 4.176 37Cl 24.6 3 – 2 0.6841 0.4561 2.184 3.476 Radioactive. μ is the magnetic moment of the spin state with the largest value of mI: μ = gIμNI and μN is the nuclear magneton (see inside front cover). Data: KL and HCP. Table 14.3 Absorption characteristics of some groups and molecules Group §max/(104 cm−1) lmax/nm emax/(dm3 mol−1 cm−1) C=C (π ←π) 6.10 163 1.5 × 104 5.73 174 5.5 × 103 C=O (π←n) 3.7–3.5 270–290 10–20 -N=N-2.9 350 15 >3.9 <260 Strong -NO2 3.6 280 10 4.8 210 1.0 × 104 C6H5-3.9 255 200 5.0 200 6.3 × 103 5.5 180 1.0 × 105 [Cu(OH2)6]2+(aq) 1.2 810 10 [Cu(NH3)4]2+(aq) 1.7 600 50 H2O (π ←n) 6.0 167 7.0 × 103 DATA SECTION 1015 Table 15.3 Hyperﬁne coupling constants for atoms, a/mT Nuclide Spin Isotropic Anisotropic coupling coupling 1H 1 – 2 50.8(1s) 2H 1 7.8(1s) 13C 1 – 2 113.0(2s) 6.6(2p) 14N 1 55.2(2s) 4.8(2p) 19F 1 – 2 1720(2s) 108.4(2p) 31P 1 – 2 364(3s) 20.6(3p) 35Cl 3 – 2 168(3s) 10.0(3p) 37Cl 3 – 2 140(3s) 8.4(3p) Data: P.W. Atkins and M.C.R. Symons, The structure of inorganic radicals. Elsevier, Amsterdam (1967). Table 18.1 Dipole moments, polarizabilities, and polarizability volumes m/(10−30 C m) m/D a/(10−40 J−1 C2 m2) a′/(10−30 m3) Ar 0 0 1.66 1.85 C2H5OH 5.64 1.69 C6H5CH3 1.20 0.36 C6H6 0 0 10.4 11.6 CCl4 0 0 10.3 11.7 CH2Cl2 5.24 1.57 6.80 7.57 CH3Cl 6.24 1.87 4.53 5.04 CH3OH 5.70 1.71 3.23 3.59 CH4 0 0 2.60 2.89 CHCl3 3.37 1.01 8.50 9.46 CO 0.390 0.117 1.98 2.20 CO2 0 0 2.63 2.93 H2 0 0 0.819 0.911 H2O 6.17 1.85 1.48 1.65 HBr 2.67 0.80 3.61 4.01 HCl 3.60 1.08 2.63 2.93 He 0 0 0.20 0.22 HF 6.37 1.91 0.51 0.57 HI 1.40 0.42 5.45 6.06 N2 0 0 1.77 1.97 NH3 4.90 1.47 2.22 2.47 1,2-C6H4(CH3)2 2.07 0.62 Data: HCP and C.J.F. Böttcher and P. Bordewijk, Theory of electric polarization. Elsevier, Amsterdam (1978). 1016 DATA SECTION Table 18.4 Lennard-Jones (12,6)-potential parameters (e/k)/K r0/pm Ar 111.84 362.3 C2H2 209.11 463.5 C2H4 200.78 458.9 C2H6 216.12 478.2 C6H6 377.46 617.4 CCl4 378.86 624.1 Cl2 296.27 448.5 CO2 201.71 444.4 F2 104.29 357.1 Kr 154.87 389.5 N2 91.85 391.9 O2 113.27 365.4 Xe 213.96 426.0 Source: F. Cuadros, I. Cachadiña, and W. Ahamuda, Molec. Engineering, 6, 319 (1996). Table 18.5 Surface tensions of liquids at 293 K g/(mN m−1) Benzene 28.88 Carbon tetrachloride 27.0 Ethanol 22.8 Hexane 18.4 Mercury 472 Methanol 22.6 Water 72.75 72.0 at 25°C 58.0 at 100°C Data: KL. Table 19.2 Diﬀusion coeﬃcients of macromolecules in water at 20°C M/(kg mol−1) D/(10−10 m2 s−1) Sucrose 0.342 4.586 Ribonuclease 13.7 1.19 Lysozyme 14.1 1.04 Serum albumin 65 0.594 Haemoglobin 68 0.69 Urease 480 0.346 Collagen 345 0.069 Myosin 493 0.116 Data: C. Tanford, Physical chemistry of macromolecules. Wiley, New York (1961). Table 19.1 Radius of gyration of some macromolecules M/(kg mol−1) Rg/nm Serum albumin 66 2.98 Myosin 493 46.8 Polystyrene 3.2 × 103 50 (in poor solvent) DNA 4 × 103 117.0 Tobacco mosaic virus 3.9 × 104 92.4 Data: C. Tanford, Physical chemistry of macromolecules. Wiley, New York (1961). DATA SECTION 1017 Table 19.4 Intrinsic viscosity Macromolecule Solvent q/°C K/(10−3 cm3 g−1) a Polystyrene Benzene 25 9.5 0.74 Cyclohexane 34† 81 0.50 Polyisobutylene Benzene 23† 83 0.50 Cyclohexane 30 26 0.70 Amylose 0.33 m KCl(aq) 25† 113 0.50 Various Guanidine 7.16 0.66 proteins‡ hydrochloride + HSCH2CH2OH † The θ temperature. ‡ Use [η] = KNa; N is the number of amino acid residues. Data: K.E. Van Holde, Physical biochemistry. Prentice-Hall, Englewood Cliﬀs (1971). Table 20.3 Ionic radii (r/pm)† Li+(4) Be2+(4) B3+(4) N3− O2−(6) F−(6) 59 27 12 171 140 133 Na+(6) Mg2+(6) Al3+(6) P3− S2−(6) Cl−(6) 102 72 53 212 184 181 K+(6) Ca2+(6) Ga3+(6) As3−(6) Se2−(6) Br−(6) 138 100 62 222 198 196 Rb+(6) Sr2+(6) In3+(6) Te2−(6) I−(6) 149 116 79 221 220 Cs+(6) Ba2+(6) Tl3+(6) 167 136 88 d-block elements (high-spin ions) Sc3+(6) Ti4+(6) Cr3+(6) Mn3+(6) Fe2+(6) Co3+(6) Cu2+(6) Zn2+(6) 73 60 61 65 63 61 73 75 † Numbers in parentheses are the coordination numbers of the ions. Values for ions without a coordination number stated are estimates. Data: R.D. Shannon and C.T. Prewitt, Acta Cryst. B25, 925 (1969). Table 19.3 Frictional coeﬃcients and molecular geometry Major axis/Minor axis Prolate Oblate 2 1.04 1.04 3 1.11 1.10 4 1.18 1.17 5 1.25 1.22 6 1.31 1.28 7 1.38 1.33 8 1.43 1.37 9 1.49 1.42 10 1.54 1.46 50 2.95 2.38 100 4.07 2.97 Data: K.E. Van Holde, Physical biochemistry. Prentice-Hall, Englewood Cliﬀs (1971). Sphere; radius a, c = af0 Prolate ellipsoid; major axis 2a, minor axis 2b, c = (ab2)1/3 f = f0 Oblate ellipsoid; major axis 2a, minor axis 2b, c = (a2b)1/3 f = f0 Long rod; length l, radius a, c = (3a2/4)1/3 f = f0 In each case f0 = 6πηc with the appropriate value of c. 5 6 7 (1/2a)2/3 (3/2)1/3{2 ln(l/a)−0.11} 1 2 3 5 6 7 (a2/b2 −1)1/2 (a/b)2/3 arctan[(a2/b2 −1)1/2] 1 2 3 5 6 7 (1 −b2/a2)1/2 (b/a)2/3 ln{[1 + (1 −b2/a2)1/2]/(b/a)} 1 2 3 1018 DATA SECTION Table 20.5 Lattice enthalpies, ΔHL 7/(kJ mol−1) F Cl Br I Halides Li 1037 852 815 761 Na 926 787 752 705 K 821 717 689 649 Rb 789 695 668 632 Cs 750 676 654 620 Ag 969 912 900 886 Be 3017 Mg 2524 Ca 2255 Sr 2153 Oxides MgO 3850 CaO 3461 SrO 3283 BaO 3114 Sulﬁdes MgS 3406 CaS 3119 SrS 2974 BaS 2832 Entries refer to MX(s) →M+(g) + X−(g). Data: Principally D. Cubicciotti, J. Chem. Phys. 31, 1646 (1959). Table 20.6 Magnetic susceptibilities at 298 K c/10−6 cm/(10−4 cm3 mol−1) Water −90 −16.0 Benzene −7.2 −6.4 Cyclohexane −7.9 −8.5 Carbon tetrachloride −8.9 −8.4 NaCl(s) −13.9 −3.75 Cu(s) −96 −6.8 S(s) −12.9 −2.0 Hg(l) −28.5 −4.2 CuSO4·5H2O(s) +176 +192 MnSO4·4H2O(s) +2640 +2.79 × 103 NiSO4·7H2O(s) +416 +600 FeSO4(NH4)2SO4·6H2O(s) +755 +1.51 × 103 Al(s) +22 +2.2 Pt(s) +262 +22.8 Na(s) +7.3 +1.7 K(s) +5.6 +2.5 Data: KL and χm = χM/ρ. Table 21.1 Collision cross-sections, σ/nm2 Ar 0.36 C2H4 0.64 C6H6 0.88 CH4 0.46 Cl2 0.93 CO2 0.52 H2 0.27 He 0.21 N2 0.43 Ne 0.24 O2 0.40 SO2 0.58 Data: KL. Table 21.2 Transport properties of gases at 1 atm k/(J K−1 m−1 s−1) h/mP 273 K 273 K 293 K Air 0.0241 173 182 Ar 0.0163 210 223 C2H4 0.0164 97 103 CH4 0.0302 103 110 Cl2 0.079 123 132 CO2 0.0145 136 147 H2 0.1682 84 88 He 0.1442 187 196 Kr 0.0087 234 250 N2 0.0240 166 176 Ne 0.0465 298 313 O2 0.0245 195 204 Xe 0.0052 212 228 Data: KL. DATA SECTION 1019 Table 21.6 Ionic mobilities in water at 298 K, u/(10−8 m2 s−1 V−1) Cations Anions Ag+ 6.24 Br− 8.09 Ca2+ 6.17 CH3CO2 − 4.24 Cu2+ 5.56 Cl− 7.91 H+ 36.23 CO3 2− 7.46 K+ 7.62 F− 5.70 Li+ 4.01 [Fe(CN)6]3− 10.5 Na+ 5.19 [Fe(CN)6]4− 11.4 NH4 + 7.63 I− 7.96 [N(CH3)4]+ 4.65 NO3 − 7.40 Rb+ 7.92 OH− 20.64 Zn2+ 5.47 SO4 2− 8.29 Data: Principally Table 21.4 and u = λ/zF. Table 21.7 Debye–Hückel–Onsager coeﬃcients for (1,1)-electrolytes at 25°C Solvent A/(mS m2 mol−1/ B/(mol dm−3)−1/2 (mol dm−3)1/2) Acetone (propanone) 3.28 1.63 Acetonitrile 2.29 0.716 Ethanol 8.97 1.83 Methanol 15.61 0.923 Nitrobenzene 4.42 0.776 Nitromethane 111 0.708 Water 6.020 0.229 Data: J.O’M. Bockris and A.K.N. Reddy, Modern electrochemistry. Plenum, New York (1970). Table 21.5 Limiting ionic conductivities in water at 298 K, λ/(mS m2 mol−1) Cations Anions Ba2+ 12.72 Br− 7.81 Ca2+ 11.90 CH3CO2 − 4.09 Cs+ 7.72 Cl− 7.635 Cu2+ 10.72 ClO4 − 6.73 H+ 34.96 CO3 2− 13.86 K+ 7.350 (CO2)2 2− 14.82 Li+ 3.87 F− 5.54 Mg2+ 10.60 [Fe(CN)6]3− 30.27 Na+ 5.010 [Fe(CN)6]4− 44.20 [N(C2H5)4]+ 3.26 HCO2 − 5.46 [N(CH3)4]+ 4.49 I− 7.68 NH4 + 7.35 NO3 − 7.146 Rb+ 7.78 OH− 19.91 Sr2+ 11.89 SO4 2− 16.00 Zn2+ 10.56 Data: KL, RS. Table 21.4 Viscosities of liquids at 298 K, η/(10−3 kg m−1 s−1) Benzene 0.601 Carbon tetrachloride 0.880 Ethanol 1.06 Mercury 1.55 Methanol 0.553 Pentane 0.224 Sulfuric acid 27 Water† 0.891 † The viscosity of water over its entire liquid range is represented with less than 1 per cent error by the expression log(η20/η) = A/B, A = 1.370 23(t −20) + 8.36 × 10−4(t −20)2 B = 109 + t t = θ/°C Convert kg m−1 s−1 to centipoise (cP) by multiplying by 103 (so η ≈1 cP for water). Data: AIP, KL. 1020 DATA SECTION Table 21.8 Diﬀusion coeﬃcients at 25°C, D/(10−9 m2 s−1) Molecules in liquids Ions in water I2 in hexane 4.05 H2 in CCl4(l) 9.75 K+ 1.96 Br− 2.08 in benzene 2.13 N2 in CCl4(l) 3.42 H+ 9.31 Cl− 2.03 CCl4 in heptane 3.17 O2 in CCl4(l) 3.82 Li+ 1.03 F− 1.46 Glycine in water 1.055 Ar in CCl4(l) 3.63 Na+ 1.33 I− 2.05 Dextrose in water 0.673 CH4 in CCl4(l) 2.89 OH− 5.03 Sucrose in water 0.5216 H2O in water 2.26 CH3OH in water 1.58 C2H5OH in water 1.24 Data: AIP and (for the ions) λ = zuF in conjunction with Table 21.5. Table 22.1 Kinetic data for ﬁrst-order reactions Phase q/°C k/s−1 t1/2 2 N2O5 →4 NO2 + O2 g 25 3.38 × 10−5 5.70 h HNO3(l) 25 1.47 × 10−6 131 h Br2(l) 25 4.27 × 10−5 4.51 h C2H6 →2 CH3 g 700 5.36 × 10−4 21.6 min Cyclopropane →propene g 500 6.71 × 10−4 17.2 min CH3N2CH3 →C2H6 + N2 g 327 3.4 × 10−4 34 min Sucrose →glucose + fructose aq(H+) 25 6.0 × 10−5 3.2 h g: High pressure gas-phase limit. Data: Principally K.J. Laidler, Chemical kinetics. Harper & Row, New York (1987); M.J. Pilling and P.W. Seakins, Reaction kinetics. Oxford University Press (1995); J. Nicholas, Chemical kinetics. Harper & Row, New York (1976). See also JL. Table 22.2 Kinetic data for second-order reactions Phase q/°C k/(dm3 mol−1 s−1) 2 NOBr →2 NO + Br2 g 10 0.80 2 NO2 →2 NO + O2 g 300 0.54 H2 + I2 →2 HI g 400 2.42 × 10−2 D2 + HCl →DH + DCl g 600 0.141 2 I →I2 g 23 7 × 109 hexane 50 1.8 × 1010 CH3Cl + CH3O− methanol 20 2.29 × 10−6 CH3Br + CH3O− methanol 20 9.23 × 10−6 H+ + OH−→H2O water 25 1.35 × 1011 ice −10 8.6 × 1012 Data: Principally K.J. Laidler, Chemical kinetics. Harper & Row, New York (1987); M.J. Pilling and P.W. Seakins, Reaction kinetics. Oxford University Press (1995); J. Nicholas, Chemical kinetics. Harper & Row, New York (1976). DATA SECTION 1021 Table 22.4 Arrhenius parameters First-order reactions A/s−1 Ea/(kJ mol−1) Cyclopropane →propene 1.58 × 1015 272 CH3NC →CH3CN 3.98 × 1013 160 cis-CHD=CHD →trans-CHD=CHD 3.16 × 1012 256 Cyclobutane →2 C2H4 3.98 × 1013 261 C2H5I →C2H4 + HI 2.51 × 1017 209 C2H6 →2 CH3 2.51 × 107 384 2 N2O5 →4 NO2 + O2 4.94 × 1013 103 N2O →N2 + O 7.94 × 1011 250 C2H5 →C2H4 + H 1.0 × 1013 167 Second-order, gas-phase A/(dm3 mol−1 s−1) Ea/(kJ mol−1) O + N2 →NO + N 1 × 1011 315 OH + H2 →H2O + H 8 × 1010 42 Cl + H2 →HCl + H 8 × 1010 23 2 CH3 →C2H6 2 × 1010 ca. 0 NO + Cl2 →NOCl + Cl 4.0 × 109 85 SO + O2 →SO2 + O 3 × 108 27 CH3 + C2H6 →CH4 + C2H5 2 × 108 44 C6H5 + H2 →C6H6 + H 1 × 108 ca. 25 Second-order, solution A/(dm3 mol−1 s−1) Ea/(kJ mol−1) C2H5ONa + CH3I in ethanol 2.42 × 1011 81.6 C2H5Br + OH−in water 4.30 × 1011 89.5 C2H5I + C2H5O−in ethanol 1.49 × 1011 86.6 CH3I + C2H5O−in ethanol 2.42 × 1011 81.6 C2H5Br + OH−in ethanol 4.30 × 1011 89.5 CO2 + OH−in water 1.5 × 1010 38 CH3I + S2O3 2−in water 2.19 × 1012 78.7 Sucrose + H2O in acidic water 1.50 × 1015 107.9 (CH3)3CCl solvolysis in water 7.1× 1016 100 in methanol 2.3× 1013 107 in ethanol 3.0× 1013 112 in acetic acid 4.3× 1013 111 in chloroform 1.4× 104 45 C6H5NH2 + C6H5COCH2Br in benzene 91 34 Data: Principally J. Nicholas, Chemical kinetics. Harper & Row, New York (1976) and A.A. Frost and R.G. Pearson, Kinetics and mechanism. Wiley, New York (1961). 1022 DATA SECTION Table 24.1 Arrhenius parameters for gas-phase reactions A/(dm3 mol−1 s−1) Ea/(kJ mol−1) P Experiment Theory 2 NOCl →2 NO + Cl2 9.4 × 109 5.9 × 1010 102.0 0.16 2 NO2 →2 NO + O2 2.0 × 109 4.0 × 1010 111.0 5.0 × 10−2 2 ClO →Cl2 + O2 6.3 × 107 2.5 × 1010 0.0 2.5 × 10−3 H2 + C2H4 →C2H6 1.24 × 106 7.4 × 1011 180 1.7 × 10−6 K + Br2 →KBr + Br 1.0 × 1012 2.1 × 1011 0.0 4.8 Data: Principally M.J. Pilling and P.W. Seakins, Reaction kinetics. Oxford University Press (1995). Table 24.2 Arrhenius parameters for reactions in solution. See Table 22.4 Table 25.1 Maximum observed enthalpies of physisorption, ΔadH 7/(kJ mol−1) C2H2 −38 H2 −84 C2H4 −34 H2O −59 CH4 −21 N2 −21 Cl2 −36 NH3 −38 CO −25 O2 −21 CO2 −25 Data: D.O. Haywood and B.M.W. Trapnell, Chemisorption. Butterworth (1964). Table 25.2 Enthalpies of chemisorption, ΔadH7/(kJ mol−1) Adsorbate Adsorbent (substrate) Ti Ta Nb W Cr Mo Mn Fe Co Ni Rh Pt H2 −188 −188 −167 −71 −134 −117 N2 −586 −293 O2 −720 −494 −293 CO −640 −192 −176 CO2 −682 −703 −552 −456 −339 −372 −222 −225 −146 −184 NH3 −301 −188 −155 C2H4 −577 −427 −427 −285 −243 −209 Data: D.O. Haywood and B.M.W. Trapnell, Chemisorption. Butterworth (1964). DATA SECTION 1023 Table 25.3 Activation energies of catalysed reactions Catalyst Ea/(kJ mol−1) 2 HI →H2 + I2 None 184 Au(s) 105 Pt(s) 59 2 NH3 →N2 + 3 H2 None 350 W(s) 162 2 N2O →2 N2 + O2 None 245 Au(s) 121 Pt(s) 134 (C2H5)2O pyrolysis None 224 I2(g) 144 Data: G.C. Bond, Heterogeneous catalysis. Clarendon Press, Oxford (1986). Table 25.6 Exchange current densities and transfer coeﬃcients at 298 K Reaction Electrode j0/(A cm−2) a 2 H+ + 2 e−→H2 Pt 7.9 × 10−4 Cu 1 × 10−6 Ni 6.3 × 10−6 0.58 Hg 7.9 × 10−13 0.50 Pb 5.0 × 10−12 Fe3+ + e−→Fe2+ Pt 2.5 × 10−3 0.58 Ce4+ + e−→Ce3+ Pt 4.0 × 10−5 0.75 Data: Principally J.O’M. Bockris and A.K.N. Reddy, Modern electrochemistry. Plenum, New York (1970). Table A3.1 Refractive indices relative to air at 20°C 434 nm 589 nm 656 nm Benzene 1.5236 1.5012 1.4965 Carbon tetrachloride 1.4729 1.4676 1.4579 Carbon disulﬁde 1.6748 1.6276 1.6182 Ethanol 1.3700 1.3618 1.3605 KCl(s) 1.5050 1.4904 1.4973 Kl(s) 1.7035 1.6664 1.6581 Methanol 1.3362 1.3290 1.3277 Methylbenzene 1.5170 1.4955 1.4911 Water 1.3404 1.3330 1.3312 Data: AIP. C1 E h = 1 (1) A 1 Cs = Ch E σh h = 2 (m) A′ 1 1 x, y, Rz x2, y2, z2, xy A″ 1 −1 z, Rx, Ry yz, xz Ci = S2 E i h = 2 (⁄) Ag 1 1 Rx, Ry, Rz x2, y2, z2, xy, xz, yz Au 1 −1 x, y, z Character tables The groups C1, Cs, Ci 1024 DATA SECTION C4v, 4mm E C2 2C4 2σv 2σd h = 8 A1 1 1 1 1 1 z, z2, x2 + y2 A2 1 1 1 −1 1 Rz B1 1 1 −1 1 −1 x2 −y2 B2 1 1 −1 −1 1 xy E 2 −2 0 0 0 (x, y), (xz, yz) (Rx, Ry) C5v E 2C5 2C5 2 5σv h = 10, a = 72° A1 1 1 1 1 z, z2, x2 + y2 A2 1 1 1 −1 Rz E1 2 2 cos α 2 cos 2α 0 (x, y), (xz, yz) (Rx, Ry) E2 2 2 cos 2α 2 cos α 0 (xy, x2 −y2) C6v, 6mm E C2 2C3 2C6 3σd 3σv h = 12 A1 1 1 1 1 1 1 z, z2, x2 + y2 A2 1 1 1 1 −1 1 Rz B1 1 −1 1 −1 −1 1 B2 1 −1 1 −1 1 −1 E1 2 −2 −1 1 0 0 (x, y), (xz, yz) (Rx, Ry) E2 2 2 −1 −1 0 0 (xy, x2 −y2) C2v, 2mm E C2 σv σ′ v h = 4 A1 1 1 1 1 z, z2, x2, y2 A2 1 1 −1 −1 xy Rz B1 1 −1 1 −1 x, xz Ry B2 1 −1 −1 1 y, yz Rx C3v, 3m E 2C3 3σv h = 6 A1 1 1 1 z, z2, x2 + y2 A2 1 1 −1 Rz E 2 −1 0 (x, y), (xy, x2 −y2) (xz, yz) (Rx, Ry) The groups Cnv DATA SECTION 1025 C•v E 2Cφ† •σv h = • A1(Σ+) 1 1 1 z, z2, x2 + y2 A2(Σ−) 1 1 −1 Rz E1(Π) 2 2 cos φ 0 (x, y), (xz, yz) (Rx, Ry) E2(Δ) 2 2 cos 2φ 0 (xy, x2 −y2) † There is only one member of this class if φ = π. D4, 422 E C2 2C4 2C′ 2 2C2 ″ h = 8 A1 1 1 1 1 1 z2, x2 + y2 A2 1 1 1 −1 −1 z Rz B1 1 1 −1 1 −1 x2 −y2 B2 1 1 −1 −1 1 xy E 2 −2 0 0 0 (x, y), (xz, yz) (Rx, Ry) D3, 32 E 2C3 3C′ 2 h = 6 A1 1 1 1 z2, x2 + y2 A2 1 1 −1 z Rz E 2 −1 0 (x, y), (xz, yz), (xy, x2 −y2) (Rx, Ry) D2, 222 E C2 z C2 y C2 x h = 4 A1 1 1 1 1 x2, y2, z2 B1 1 1 −1 −1 z, xy Rz B2 1 −1 1 −1 y, xz Ry B3 1 −1 −1 1 x, yz Rx The groups Dn The groups Dnh D3h, %2m E σh 2C3 2S3 3C′ 2 3σv h = 12 A′ 1 1 1 1 1 1 1 z2, x2 + y2 A′ 2 1 1 1 1 −1 −1 Rz A1 ″ 1 −1 1 −1 1 −1 A2 ″ 1 −1 1 −1 −1 1 z E′ 2 2 −1 −1 0 0 (x, y), (xy, x2 −y2) E″ 2 −2 −1 1 0 0 (xz, yz) (Rx, Ry) 1026 DATA SECTION D•h E 2Cf •C′ 2 i 2iC• iC′ 2 h = • A1g(Σg +) 1 1 1 1 1 1 z2, x2 + y2 A1u(Σu +) 1 1 1 −1 −1 −1 z A2g(Σg −) 1 1 −1 1 1 −1 Rz A2u(Σu −) 1 1 −1 −1 1 1 E1g(Πg) 2 2 cos φ 0 2 −2 cos φ 0 (xz, yz) (Rx, Ry) E1u(Πu) 2 2 cos φ 0 −2 2 cos φ 0 (x, y) E2g(Δg) 2 2 cos 2φ 0 2 2 cos 2φ 0 (xy, x2 −y2) E2u(Δu) 2 2 cos 2φ 0 −2 −2 cos 2φ 0 D5h E 2C5 2C5 2 5C2 σh 2S5 2S5 3 5σv h = 20 a = 72° A′ 1 1 1 1 1 1 1 1 1 x2 + y2, z2 A′ 2 1 1 1 −1 1 1 1 −1 Rz E′ 1 2 2 cos α 2 cos 2α 0 2 2 cos α 2 cos 2α 0 (x, y) E′ 2 2 2 cos 2α 2 cos α 0 2 2 cos 2α 2 cos α 0 (x2 −y2, xy) A″ 1 1 1 1 1 −1 −1 −1 −1 A″ 2 1 1 1 −1 −1 −1 −1 1 z E″ 1 2 2 cos α 2 cos 2α 0 −2 −2 cos α −2 cos 2α 0 (xz, yz) (Rx, Ry) E″ 2 2 2 cos 2α 2 cos α 0 −2 −2 cos 2α −2 cos α 0 D4h, 4/mmm E 2C4 C2 2C′ 2 2C2 ″ i 2S4 σh 2σv 2σd h = 16 A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2 A2g 1 1 1 −1 −1 1 1 1 −1 −1 Rz B1g 1 −1 1 1 −1 1 −1 1 1 −1 x2 −y2 B2g 1 −1 1 −1 1 1 −1 1 −1 1 xy Eg 2 0 −2 0 0 2 0 −2 0 0 (xz, yz) (Rx, Ry) A1u 1 1 1 1 1 −1 −1 −1 −1 −1 A2u 1 1 1 −1 −1 −1 −1 −1 1 1 z B1u 1 −1 1 1 −1 −1 1 −1 −1 1 B2u 1 −1 1 −1 1 −1 1 −1 1 −1 Eu 2 0 −2 0 0 −2 0 2 0 0 (x, y) DATA SECTION 1027 I E 12C5 12C5 2 20C3 15C2 h = 60 A 1 1 1 1 1 z2 + y2 + z2 T1 3 1 – 2(1 + ) 1 – 2(1 − ) 0 −1 (x, y, z) T2 3 1 – 2(1 − ) 1 – 2(1 + ) 0 −1 (Rx, Ry, Rz) G 4 −1 −1 1 0 G 5 0 0 −1 1 (2z2 −x2 −y2, x2 −y2, xy, yz, zx) Further information: P.W. Atkins, M.S. Child, and C.S.G. Phillips, Tables for group theory. Oxford University Press (1970). 5 5 5 5 Oh (m3m) E 8C3 6C2 6C2 3C2 (= C4 2) i 6S4 8S6 3σh 6σd h = 48 A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 + z2 A2g 1 1 −1 −1 1 1 −1 1 1 −1 Eg 2 −1 0 0 2 2 0 −1 2 0 (2z2 −x2 −y2, x2 −y2) T1g 3 0 −1 1 −1 3 1 0 −1 −1 (Rx, Ry, Rz) T2g 3 0 1 −1 −1 3 −1 0 −1 1 (xy, yz, xy) A1u 1 1 1 1 1 −1 −1 −1 −1 −1 A2u 1 1 −1 −1 1 −1 1 −1 −1 1 Eu 2 −1 0 0 2 −2 0 1 −2 0 T1u 3 0 −1 1 −1 −3 −1 0 1 1 (x, y, z) T2u 3 0 1 −1 −1 −3 1 0 1 −1 Td, ∞3m E 8C3 3C2 6σd 6S4 h = 24 A1 1 1 1 1 1 x2 + y2 + z2 A2 1 1 1 −1 −1 E 2 −1 2 0 0 (3z2 −r2, x2 −y2) T1 3 0 −1 −1 1 (Rx, Ry, Rz) T2 3 0 −1 1 −1 (x, y, z), (xy, xz, yz) The cubic groups The icosahedral group
2581	https://casmusings.wordpress.com/2012/04/25/4x4-grid-and-extensions/	4×4 Grid and Extensions \| CAS Musings CAS Musings Just another WordPress.com site Skip to content Home About Conference Presentations My Publications Resources ← Blokus Recognizing Patterns → 4×4 Grid and Extensions Posted onApril 25, 2012\|1 Comment Ben Vitale’s Fun with Num3ers ‘blog is a prolific source of all sorts of interesting number patterns. He just posted a great problem that would be appropriate for students from elementary school through algebra. Here it is: Any students who understand nothing more two-digit addition could enjoy the magic that comes from getting the same answer every time. Older students who are beginning to understand something about variables can handle the generalized question Ben asks. Depending how one approaches the proof, a student might discover that this problem generalizes even a bit further than Ben suggests in his initial post. SOLUTION ALERT: Don’t read any further if you want to solve this problem on your own. PROOF: Let the number in the upper left of the grid be a. One way to tackle this proof is to write the grid elements with the upper left number in parentheses, values added to that number along a row placed inside the parentheses, and values added to that number down a column placed outside the parentheses. The revised grid looks like this: Following the rules of selecting a number and then crossing out any other entries in that numbers row and column, every sum of four numbers selected this way will contain exactly one element from every row and every column making the overall sum contain an (a) from column 1, an (a + 1) from column 2, an (a + 2) from column 3, and an (a + 3) from column 4. Also, every set of four numbers will have outside the parentheses nothing from row 1, a “+4” from row 2, a “+8” from row 3, and a “+12” from row 4. That means the numbers you add for this sum will be some arrangement of . Because for the given problem, the magic sum for this problem is 34. That solves an arithmetic problem. EXTENSION 1: Now think a bit more mathematically. Notice that all my proof requires is that the upper left number be (a). That means any consecutive integer run starting at any integer a in the upper left corner of a 4×4 grid would produce a constant sum of . Encourage your mathematical explorers to start with or include all types of integers, including zero; include negative numbers if they’re ready for that. Advertisement EXTENSION 2: How many different ways are there to pick numbers from a 4×4 grid in this manner, no matter what value (a) you place in the upper left corner? EXTENSION 3: Pushing just a little further, can you prove why any square grid of any size filled with any consecutive elements of any arithmetic sequence produces a constant sum? Share this: Click to share on Facebook (Opens in new window)Facebook Like Loading... Related Multiplication Puzzle for the Very YoungMay 13, 2012 In "Parenting" FiveThirtyEight Riddler Express SolutionJuly 30, 2018 In "problem-solving" Deep Unit Circle UnderstandingAugust 27, 2018 In "Pedagogy" This entry was posted in Parenting, problem-solving and tagged addition, algebra, arithmetic, elementary, proof. Bookmark the permalink. ← Blokus Recognizing Patterns → One response to “4×4 Grid and Extensions” Pingback: 4X4 Grid \| Fun With Num3ers Leave a comment Cancel reply Δ Author chrisharrow Search this site Search for: ### Follow Blog via Email Enter your email address to follow this blog and receive notifications of new posts by email. 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2582	https://zhuanlan.zhihu.com/p/132357116	数学 \| 小学数学各类型应用题解答方法公式汇总，简单易懂！ - 知乎关注推荐热榜专栏圈子 New付费咨询知学堂直答切换模式登录/注册数学 \| 小学数学各类型应用题解答方法公式汇总，简单易懂！切换模式数学 \| 小学数学各类型应用题解答方法公式汇总，简单易懂！学大教育整数和小数的应用 01 简单应用题（1）简单应用题：只含有一种基本数量关系，或用一步运算解答的应用题，通常叫做简单应用题。（2）解题步骤： a 审题理解题意：了解应用题的内容，知道应用题的条件和问题。读题时，不丢字不添字边读边思考，弄明白题中每句话的意思。也可以复述条件和问题，帮助理解题意。 b选择算法和列式计算：这是解答应用题的中心工作。从题目中告诉什么，要求什么着手，逐步根据所给的条件和问题，联系四则运算的含义，分析数量关系，确定算法，进行解答并标明正确的单位名称。 C检验：就是根据应用题的条件和问题进行检查看所列算式和计算过程是否正确，是否符合题意。如果发现错误，马上改正。 02 复合应用题（1）有两个或两个以上的基本数量关系组成的，用两步或两步以上运算解答的应用题，通常叫做复合应用题。（2）含有三个已知条件的两步计算的应用题。求比两个数的和多（少）几个数的应用题。比较两数差与倍数关系的应用题。（3）含有两个已知条件的两步计算的应用题。已知两数相差多少（或倍数关系）与其中一个数，求两个数的和（或差）。已知两数之和与其中一个数，求两个数相差多少（或倍数关系）。（4）解答连乘连除应用题。（5）解答三步计算的应用题。（6）解答小数计算的应用题：小数计算的加法、减法、乘法和除法的应用题，他们的数量关系、结构、和解题方式都与正式应用题基本相同，只是在已知数或未知数中间含有小数。答案：根据计算的结果，先口答，逐步过渡到笔答。 ( 7 ) 解答加法应用题： a求总数的应用题：已知甲数是多少，乙数是多少，求甲乙两数的和是多少。 b求比一个数多几的数应用题：已知甲数是多少和乙数比甲数多多少，求乙数是多少。 (8 ) 解答减法应用题： a求剩余的应用题：从已知数中去掉一部分，求剩下的部分。 -b求两个数相差的多少的应用题：已知甲乙两数各是多少，求甲数比乙数多多少，或乙数比甲数少多少。 c求比一个数少几的数的应用题：已知甲数是多少，乙数比甲数少多少，求乙数是多少。 (9 ) 解答乘法应用题： a求相同加数和的应用题：已知相同的加数和相同加数的个数，求总数。 b求一个数的几倍是多少的应用题：已知一个数是多少，另一个数是它的几倍，求另一个数是多少。 ( 10) 解答除法应用题： a把一个数平均分成几份，求每一份是多少的应用题：已知一个数和把这个数平均分成几份的，求每一份是多少。 b求一个数里包含几个另一个数的应用题：已知一个数和每份是多少，求可以分成几份。 C 求一个数是另一个数的的几倍的应用题：已知甲数乙数各是多少，求较大数是较小数的几倍。 d已知一个数的几倍是多少，求这个数的应用题。（11）常见的数量关系：总价= 单价×数量路程= 速度×时间工作总量=工作时间×工效总产量=单产量×数量 03 典型应用题具有独特的结构特征的和特定的解题规律的复合应用题，通常叫做典型应用题。（1）平均数问题：平均数是等分除法的发展。解题关键：在于确定总数量和与之相对应的总份数。算术平均数：已知几个不相等的同类量和与之相对应的份数，求平均每份是多少。数量关系式：数量之和÷数量的个数=算术平均数。加权平均数：已知两个以上若干份的平均数，求总平均数是多少。数量关系式（部分平均数×权数）的总和÷（权数的和）=加权平均数。差额平均数：是把各个大于或小于标准数的部分之和被总份数均分，求的是标准数与各数相差之和的平均数。数量关系式：（大数－小数）÷2=小数应得数最大数与各数之差的和÷总份数=最大数应给数最大数与个数之差的和÷总份数=最小数应得数。例：一辆汽车以每小时 100 千米的速度从甲地开往乙地，又以每小时 60 千米的速度从乙地开往甲地。求这辆车的平均速度。分析：求汽车的平均速度同样可以利用公式。此题可以把甲地到乙地的路程设为“ 1 ”，则汽车行驶的总路程为“ 2 ”，从甲地到乙地的速度为 100 ，所用的时间为，汽车从乙地到甲地速度为 60 千米，所用的时间是，汽车共行的时间为 + = , 汽车的平均速度为 2 ÷ =75 （千米）（2）归一问题：已知相互关联的两个量，其中一种量改变，另一种量也随之而改变，其变化的规律是相同的，这种问题称之为归一问题。根据求“单一量”的步骤的多少，归一问题可以分为一次归一问题，两次归一问题。根据球痴单一量之后，解题采用乘法还是除法，归一问题可以分为正归一问题，反归一问题。一次归一问题，用一步运算就能求出“单一量”的归一问题。又称“单归一。” 两次归一问题，用两步运算就能求出“单一量”的归一问题。又称“双归一。” 正归一问题：用等分除法求出“单一量”之后，再用乘法计算结果的归一问题。反归一问题：用等分除法求出“单一量”之后，再用除法计算结果的归一问题。解题关键：从已知的一组对应量中用等分除法求出一份的数量（单一量），然后以它为标准，根据题目的要求算出结果。数量关系式：单一量×份数=总数量（正归一）总数量÷单一量=份数（反归一）例一个织布工人，在七月份织布 4774 米，照这样计算，织布 6930 米，需要多少天？分析：必须先求出平均每天织布多少米，就是单一量。 693 0 ÷（ 477 4 ÷ 31 ） =45 （天）（3）归总问题：是已知单位数量和计量单位数量的个数，以及不同的单位数量（或单位数量的个数），通过求总数量求得单位数量的个数（或单位数量）。特点：两种相关联的量，其中一种量变化，另一种量也跟着变化，不过变化的规律相反，和反比例算法彼此相通。数量关系式：单位数量×单位个数÷另一个单位数量 = 另一个单位数量单位数量×单位个数÷另一个单位数量= 另一个单位数量。例修一条水渠，原计划每天修 800 米， 6 天修完。实际 4 天修完，每天修了多少米？分析：因为要求出每天修的长度，就必须先求出水渠的长度。所以也把这类应用题叫做“归总问题”。不同之处是“归一”先求出单一量，再求总量，归总问题是先求出总量，再求单一量。80 0 × 6 ÷4=1200 （米）（4）和差问题：已知大小两个数的和，以及他们的差，求这两个数各是多少的应用题叫做和差问题。解题关键：是把大小两个数的和转化成两个大数的和（或两个小数的和），然后再求另一个数。解题规律：（和＋差）÷2 = 大数大数－差=小数（和－差）÷2=小数和－小数= 大数例某加工厂甲班和乙班共有工人 94 人，因工作需要临时从乙班调 46 人到甲班工作，这时乙班比甲班人数少 12 人，求原来甲班和乙班各有多少人？分析：从乙班调 46 人到甲班，对于总数没有变化，现在把乙数转化成 2 个乙班，即 9 4 － 12 ，由此得到现在的乙班是（ 9 4 － 12 ）÷ 2=41 （人），乙班在调出 46 人之前应该为 41+46=87 （人），甲班为 9 4 － 87=7 （人）（5）和倍问题：已知两个数的和及它们之间的倍数关系，求两个数各是多少的应用题，叫做和倍问题。解题关键：找准标准数（即1倍数）一般说来，题中说是“谁”的几倍，把谁就确定为标准数。求出倍数和之后，再求出标准的数量是多少。根据另一个数（也可能是几个数）与标准数的倍数关系，再去求另一个数（或几个数）的数量。解题规律：和÷倍数和=标准数标准数×倍数=另一个数例:汽车运输场有大小货车 115 辆，大货车比小货车的 5 倍多 7 辆，运输场有大货车和小汽车各有多少辆？分析：大货车比小货车的 5 倍还多 7 辆，这 7 辆也在总数 115 辆内，为了使总数与（ 5+1 ）倍对应，总车辆数应（ 115-7 ）辆。列式为（115-7）÷（5+1） =18 （辆）， 18 × 5+7=97 （辆）（6）差倍问题：已知两个数的差，及两个数的倍数关系，求两个数各是多少的应用题。解题规律：两个数的差÷（倍数－1 ）= 标准数标准数×倍数=另一个数。例甲乙两根绳子，甲绳长 63 米，乙绳长 29 米，两根绳剪去同样的长度，结果甲所剩的长度是乙绳长的 3 倍，甲乙两绳所剩长度各多少米？各减去多少米？分析：两根绳子剪去相同的一段，长度差没变，甲绳所剩的长度是乙绳的 3 倍，实比乙绳多（ 3-1 ）倍，以乙绳的长度为标准数。列式（ 63-29 ）÷（ 3-1 ） =17 （米）…乙绳剩下的长度， 17 × 3=51 （米）…甲绳剩下的长度， 29-17=12 （米）…剪去的长度。（7）行程问题：关于走路、行车等问题，一般都是计算路程、时间、速度，叫做行程问题。解答这类问题首先要搞清楚速度、时间、路程、方向、杜速度和、速度差等概念，了解他们之间的关系，再根据这类问题的规律解答。解题关键及规律：同时同地相背而行：路程=速度和×时间。同时相向而行：相遇时间=速度和×时间同时同向而行（速度慢的在前，快的在后）：追及时间=路程速度差。同时同地同向而行（速度慢的在后，快的在前）：路程=速度差×时间。例甲在乙的后面 28 千米，两人同时同向而行，甲每小时行 16 千米，乙每小时行 9 千米，甲几小时追上乙？分析：甲每小时比乙多行（ 16-9 ）千米，也就是甲每小时可以追近乙（ 16-9 ）千米，这是速度差。已知甲在乙的后面 28 千米（追击路程）， 28 千米里包含着几个（ 16-9 ）千米，也就是追击所需要的时间。列式 2 8 ÷ （ 16-9 ） =4 （小时）（8）流水问题：一般是研究船在“流水”中航行的问题。它是行程问题中比较特殊的一种类型，它也是一种和差问题。它的特点主要是考虑水速在逆行和顺行中的不同作用。船速：船在静水中航行的速度。水速：水流动的速度。顺水速度：船顺流航行的速度。逆水速度：船逆流航行的速度。顺速=船速＋水速逆速=船速－水速解题关键：因为顺流速度是船速与水速的和，逆流速度是船速与水速的差，所以流水问题当作和差问题解答。解题时要以水流为线索。解题规律：船行速度=（顺水速度+ 逆流速度）÷2 流水速度=（顺流速度逆流速度）÷2 路程=顺流速度× 顺流航行所需时间路程=逆流速度×逆流航行所需时间例一只轮船从甲地开往乙地顺水而行，每小时行 28 千米，到乙地后，又逆水航行，回到甲地。逆水比顺水多行 2 小时，已知水速每小时 4 千米。求甲乙两地相距多少千米？分析：此题必须先知道顺水的速度和顺水所需要的时间，或者逆水速度和逆水的时间。已知顺水速度和水流速度，因此不难算出逆水的速度，但顺水所用的时间，逆水所用的时间不知道，只知道顺水比逆水少用 2 小时，抓住这一点，就可以就能算出顺水从甲地到乙地的所用的时间，这样就能算出甲乙两地的路程。列式为 284 × 2=20 （千米） 2 0 × 2 =40 （千米） 40 ÷（ 4 × 2 ） =5 （小时） 28 × 5=140 （千米）。（9）还原问题：已知某未知数，经过一定的四则运算后所得的结果，求这个未知数的应用题，我们叫做还原问题。解题关键：要弄清每一步变化与未知数的关系。解题规律：从最后结果出发，采用与原题中相反的运算（逆运算）方法，逐步推导出原数。根据原题的运算顺序列出数量关系，然后采用逆运算的方法计算推导出原数。解答还原问题时注意观察运算的顺序。若需要先算加减法，后算乘除法时别忘记写括号。例某小学三年级四个班共有学生 168 人，如果四班调 3 人到三班，三班调 6 人到二班，二班调 6 人到一班，一班调 2 人到四班，则四个班的人数相等，四个班原有学生多少人？分析：当四个班人数相等时，应为 168 ÷ 4 ，以四班为例，它调给三班 3 人，又从一班调入 2 人，所以四班原有的人数减去 3 再加上 2 等于平均数。四班原有人数列式为 168 ÷ 4-2+3=43 （人）一班原有人数列式为 168 ÷ 4-6+2=38 （人）；二班原有人数列式为 168 ÷ 4-6+6=42 （人）三班原有人数列式为 168 ÷ 4-3+6=45 （人）。（10）植树问题：这类应用题是以“植树”为内容。凡是研究总路程、株距、段数、棵树四种数量关系的应用题，叫做植树问题。解题关键：解答植树问题首先要判断地形，分清是否封闭图形，从而确定是沿线段植树还是沿周长植树，然后按基本公式进行计算。解题规律：沿线段植树棵树=段数+1 棵树=总路程÷株距+1 株距=总路程÷（棵树-1）总路程=株距×（棵树-1）沿周长植树棵树=总路程÷株距株距=总路程÷棵树总路程=株距×棵树例沿公路一旁埋电线杆 301 根，每相邻的两根的间距是 50 米。后来全部改装，只埋了201 根。求改装后每相邻两根的间距。分析：本题是沿线段埋电线杆，要把电线杆的根数减掉一。列式为 50 ×（ 301-1 ）÷（ 201-1 ） =75 （米）（11 ）盈亏问题：是在等分除法的基础上发展起来的。他的特点是把一定数量的物品，平均分配给一定数量的人，在两次分配中，一次有余，一次不足（或两次都有余），或两次都不足），已知所余和不足的数量，求物品适量和参加分配人数的问题，叫做盈亏问题。解题关键：盈亏问题的解法要点是先求两次分配中分配者没份所得物品数量的差，再求两次分配中各次共分物品的差（也称总差额），用前一个差去除后一个差，就得到分配者的数，进而再求得物品数。解题规律：总差额÷每人差额=人数总差额的求法可以分为以下四种情况：第一次多余，第二次不足，总差额=多余+ 不足第一次正好，第二次多余或不足，总差额=多余或不足第一次多余，第二次也多余，总差额=大多余-小多余第一次不足，第二次也不足，总差额= 大不足-小不足例参加美术小组的同学，每个人分的相同的支数的色笔，如果小组 10 人，则多 25 支，如果小组有 12 人，色笔多余 5 支。求每人分得几支？共有多少支色铅笔？分析：每个同学分到的色笔相等。这个活动小组有 12 人，比 10 人多 2 人，而色笔多出了（ 25-5 ） =20 支， 2 个人多出 20 支，一个人分得 10 支。列式为（25-5 ）÷（ 12-10 ） =10 （支） 10 × 12+5=125 （支）。（12）年龄问题：将差为一定值的两个数作为题中的一个条件，这种应用题被称为“年龄问题”。解题关键：年龄问题与和差、和倍、差倍问题类似，主要特点是随着时间的变化，年岁不断增长，但大小两个不同年龄的差是不会改变的，因此，年龄问题是一种“差不变”的问题，解题时，要善于利用差不变的特点。例父亲 48 岁，儿子 21 岁。问几年前父亲的年龄是儿子的 4 倍？分析：父子的年龄差为 48-21=27 （岁）。由于几年前父亲年龄是儿子的 4 倍，可知父子年龄的倍数差是（ 4-1 ）倍。这样可以算出几年前父子的年龄，从而可以求出几年前父亲的年龄是儿子的 4 倍。列式为：21（ 48-21 ）÷（ 4-1 ） =12 （年）（13）鸡兔问题：已知“鸡兔”的总头数和总腿数。求“鸡”和“兔”各多少只的一类应用题。通常称为“鸡兔问题”又称鸡兔同笼问题解题关键：解答鸡兔问题一般采用假设法，假设全是一种动物（如全是“鸡”或全是“兔”，然后根据出现的腿数差，可推算出某一种的头数。解题规律：（总腿数－鸡腿数×总头数）÷一只鸡兔腿数的差=兔子只数兔子只数=（总腿数-2×总头数）÷2 如果假设全是兔子，可以有下面的式子：鸡的只数=（4×总头数-总腿数）÷2 兔的头数=总头数-鸡的只数例鸡兔同笼共 50 个头， 170 条腿。问鸡兔各有多少只？兔子只数（ 170-2 × 50 ）÷ 2 =35 （只）鸡的只数 50-35=15 （只）分数和百分数的应用 01 分数加减法应用题分数加减法的应用题与整数加减法的应用题的结构、数量关系和解题方法基本相同，所不同的只是在已知数或未知数中含有分数。 02 分数乘法应用题是指已知一个数，求它的几分之几是多少的应用题。特征：已知单位“1”的量和分率，求与分率所对应的实际数量。解题关键：准确判断单位“1”的量。找准要求问题所对应的分率，然后根据一个数乘分数的意义正确列式。 03 分数除法应用题求一个数是另一个数的几分之几（或百分之几）是多少。特征：已知一个数和另一个数，求一个数是另一个数的几分之几或百分之几。“一个数”是比较量，“另一个数”是标准量。求分率或百分率，也就是求他们的倍数关系。解题关键：从问题入手，搞清把谁看作标准的数也就是把谁看作了“单位一”，谁和单位一的量作比较，谁就作被除数。甲是乙的几分之几（百分之几）:甲是比较量，乙是标准量，用甲除以乙。甲比乙多（或少）几分之几（百分之几）：甲减乙比乙多（或少几分之几）或（百分之几）。关系式（甲数减乙数）/乙数或（甲数减乙数）/甲数。已知一个数的几分之几（或百分之几 ) ,求这个数。特征：已知一个实际数量和它相对应的分率，求单位“1”的量。解题关键：准确判断单位“1”的量把单位“1”的量看成x根据分数乘法的意义列方程，或者根据分数除法的意义列算式，但必须找准和分率相对应的已知实际数量。 04 出勤率发芽率=发芽种子数/试验种子数×100% 小麦的出粉率= 面粉的重量/小麦的重量×100% 产品的合格率=合格的产品数/产品总数×100% 职工的出勤率=实际出勤人数/应出勤人数×100% 05 工程问题是分数应用题的特例，它与整数的工作问题有着密切的联系。它是探讨工作总量、工作效率和工作时间三个数量之间相互关系的一种应用题。解题关键：把工作总量看作单位“1”，工作效率就是工作时间的倒数，然后根据题目的具体情况，灵活运用公式。数量关系式：工作总量=工作效率×工作时间工作效率=工作总量÷工作时间工作时间=工作总量÷工作效率工作总量÷工作效率和=合作时间 06 纳税纳税就是把根据国家各种税法的有关规定，按照一定的比率把集体或个人收入的一部分缴纳给国家。缴纳的税款叫应纳税款。应纳税额与各种收入的（销售额、营业额、应纳税所得额 ……）的比率叫做税率。利息存入银行的钱叫做本金。取款时银行多支付的钱叫做利息。利息与本金的比值叫做利率。利息=本金×利率×时间。来源：网络发布于 2020-04-17 06:30 数学小学数学数学教育赞同添加评论分享喜欢收藏申请转载写下你的评论... 还没有评论，发表第一个评论吧关于作者学大教育回答 1文章 474关注者 2,221 关注发私信推荐阅读数学丨小学数学各类型应用题解答方法公式汇总，简单易懂！ =========================== 学大教育小升初数学应用题解答方法公式汇总，考试大题再也不怕啦！ =========================== 整数和小数的应用 1 简单应用题 1、简单应用题：只含有一种基本数量关系，或用一步运算解答的应用题，通常叫做简单应用题。 2、解题步骤： a 审题理解题意：了解应用题的内容，知道应用题… 加州国际教育小学数学四大类应用题详解，方法技巧都在这里了！ ======================= 1、一般应用题一般应用题没有固定的结构，也没有解题规律可循，完全要依赖分析题目的数量关系找出解题的线索。 ● 要点：从条件入手？从问题入手？从条件入手分析时，要随时注意题目的问题 … 加州国际教育初中数学学习重点 ======== 第一章实数 ★重点★ 实数的有关概念及性质，实数的运算 ☆内容提要☆ 一、重要概念 1．数的分类及概念数系表说明：“分类”的原则：1）相称（不重、不漏） 2）有标准 2．非负数：正实… 胖飞想来知乎工作？请发送邮件到 jobs@zhihu.com 打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码
2583	https://www.quora.com/There-are-3-brothers-each-of-them-have-2-brothers-How-many-are-they-all-together	There are 3 brothers, each of them have 2 brothers. How many are they all together? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Family Relationships and ... Math Word Puzzles Three Brothers Its Riddle Time Brain Teasers Siblings Logic Riddles Sibling Relationships Reasoning Puzzles 5 There are 3 brothers, each of them have 2 brothers. How many are they all together? All related (78) Sort Recommended Assistant Bot · 1y If there are 3 brothers and each of them has 2 brothers, they are all referring to the same set of brothers. Therefore, there are a total of 3 brothers. Upvote · 9 1 Sahil Bharti Car enthusiastic, Engineer and Puzzle Solver. ·7y Let is suppose there are three brothers. “A”, “B” and “C”. Now the tricky part of the puzzle comes into place(each has two brothers) But we can simplify it by this method : A has two brothers B&C. B has two brothers A&C. C has two brothers B&A. Hence in total there are only 3 people and they are brothers of each other. (This is what we have to think while solving this). Thanks Upvote · 99 12 Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder ·Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did. Cancel Your Car Insurance You might not even realize it, but your car insurance company is probably overcharging you. 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See every single one have two brothers… . Its very easy …try to understand the question calmly,u will get it. Upvote · 99 25 9 2 Related questions More answers below I have a brother and my brother has three and each has three, how many are we? I have three sisters and each sister has three. How many sisters are there? A person has 3 sons and they each have 2 sisters, how many siblings are they? A woman has 3 sons and they each have 2 sisters, how many children does she have? I have six sisters, and each of them has three brothers. How many brothers do I have? Ak Akshit Kumar High school from Delhi Public School, Agra ·7y They're still 3 in total. Let them be A,B,C. A has B & C as brothers B has C & A as brothers And, C has A & B as brothers. So they're still three. If my answer helps then Upvote. Upvote · 9 4 Abhishek Choudhary (अभिषेक चौधरी) Sub Divisional Officer at Water Resources Department Govt.of Rajasthan (2022–present) ·7y There are 3 brothers,A B C . So we can say that,A has 2 brothers B and C. B has 2 brothers C and A and C has 2 brothers B and A. =Each of them have 2 brothers. Total no. Of person= 3 (A,B,C) Upvote · 9 1 9 1 Related questions More answers below What comes next in the sequence 2, 4, 3, 6, 5, 8, 7? If 2 + 3 = 10, 6 + 5 = 66, 3 + 4 = 21, 7 + 2 = 63, how much is 9 + 7? How many are we if I have 3 brothers, and each brother has 3 brothers? What is the answer of this sequence 29, 34, 32, 37, 35? If I and my brother each have 3 sisters and 2 other brothers, how many children does our mother have? Pooja Bagri Mathematics Expert at Self-Employment (2019–present) · Author has 2K answers and 18M answer views ·5y Solve. There are 3 brother 1.A 2.B 3.C Each of them have 2 brother A have two brother B and C B have two brother C and A and And C have two brother A and B So that totally 3 brother Upvote · Promoted by Spokeo Spokeo - People Search \| Dating Safety Tool Dating Safety and Cheater Buster Tool ·Apr 16 Is there a way to check if someone has a dating profile? Originally Answered: Is there a way to check if someone has a dating profile? Please be reliable and detailed. · Yes, there is a way. If you're wondering whether someone has a dating profile, it's actually pretty easy to find out. Just type in their name and click here 👉 UNCOVER DATING PROFILE. This tool checks a bunch of dating apps and websites to see if that person has a profile—either now or in the past. You don’t need to be tech-savvy or know anything complicated. It works with just a name, and you can also try using their email or phone number if you have it. 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Upvote · 6.1K 6.1K 99 84 Madhusudhana Rao Sankarapu Director (Business Development) at Kenra Technologies LLP, Secunderabad (2017–present) · Author has 603 answers and 1.2M answer views ·7y Again another simple problem They all together 3 brothers only If we consider these THREE Brothers as a TRIANGLE placing each brother as a corner. Hence, each corner will have two sides which is similar to each one having 2 brothers each So, there are 3 brothers only Upvote · 9 2 Imbeshat Aslam Anonymous ·7y There are three brothers. Let us call them as Jon, Robb and Brandon. Each of them has two brothers, that means Jon has two brothers, Robb and Brandon. Robb has two brothers, Jon and Brandon. Brandon has two brothers, Jon and Robb. So, basically there are only three of them Jon, Robb and Brandon. Upvote · 9 1 Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. Learn More 999 116 Rishabh Yadav Btech from SHUATS, Allahabad (Graduated 2015) ·7y Its simple. All together there are 3 brothers. Suppose Ram,Shyam & Raju are three brothers.. Then you can see that Ram has two brother namely Shyam & Raju. Shyam has two brother namely Ram & Raju. Raju has two brother namely Ram & Shyam. So now its quite clear that Ram ,Shyam & Raju each have two brothers. Upvote · Steven McGlinn Former Selective High School Teacher · Author has 5.1K answers and 5.4M answer views ·7y Originally Answered: There are 3 brothers each brother has 2 brother then how many are thein in total? · Three. So each of the brothers has two brothers. If we call the brothers A B and C. A s brothers are B and C. B has brothers A and C. C has A and B as brothers. So each of the three has two brothers Upvote · Sponsored by CDW Corporation What’s the best way to protect your growing infrastructure? Enable an AI-powered defense with converged networking and security solutions from Fortinet and CDW. Learn More 999 263 Nooh Hussain Mirza Teacher ·7y Obviously, there are only 3 people here. Let these people be called Mr X, Mr Y and Mr Z. Now according to the question above, Each of them has 2 brothers. If we talk about Mr X, his brothers are Mr Y and Mr Z If we talk about Mr Y, his brothers are Mr Zand Mr X If we talk about Mr Z, his brothers are Mr Xand Mr Y. Hope this helps! Upvote · Pranjal Studied Joint Entrance Examination&Indian Institute of Technology, Delhi at Kendriya Vidyalaya, Pragati Vihar · Author has 76 answers and 39.9K answer views ·7y The answer is 3 brothers because:- The first let it be A has two brothers ,let say B and C The second (B) has two brothers,A and C Similarly,The third has two brothers A and B Upvote · 9 1 Rohit Parkar Digital Media Planner and Strategist (2018–present) ·7y The question says there are 3 brothers, say a, b, c are 3 brothers So ‘a’ has 2 brothers ‘b’ and ‘c', 'b' has 2 brothers 'a' and 'c' 'c' has 2 brothers 'a' and 'b' So here each of them have 2 brothers and there are just 3 people here and hence this answers your question. There are only 3 people all together. Upvote · Related questions I have a brother and my brother has three and each has three, how many are we? I have three sisters and each sister has three. How many sisters are there? A person has 3 sons and they each have 2 sisters, how many siblings are they? A woman has 3 sons and they each have 2 sisters, how many children does she have? I have six sisters, and each of them has three brothers. How many brothers do I have? What comes next in the sequence 2, 4, 3, 6, 5, 8, 7? If 2 + 3 = 10, 6 + 5 = 66, 3 + 4 = 21, 7 + 2 = 63, how much is 9 + 7? How many are we if I have 3 brothers, and each brother has 3 brothers? What is the answer of this sequence 29, 34, 32, 37, 35? If I and my brother each have 3 sisters and 2 other brothers, how many children does our mother have? A person has 3 sisters, each of them has 2 brothers and each brother has 2 kids. How many brothers do I have? I have three sisters. Each sister has three brothers. How many brothers do I have? I have seven sisters, and each has two brothers. How many brothers do I have? I have 2 sisters each of them has 2 brothers. How many brothers do I have? If 1 = 5, 2 = 25, 3 = 125, and 4 = 625, then what does 5 equal to? Related questions I have a brother and my brother has three and each has three, how many are we? I have three sisters and each sister has three. How many sisters are there? A person has 3 sons and they each have 2 sisters, how many siblings are they? A woman has 3 sons and they each have 2 sisters, how many children does she have? I have six sisters, and each of them has three brothers. How many brothers do I have? What comes next in the sequence 2, 4, 3, 6, 5, 8, 7? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
2584	https://medicaljournalssweden.se/actadv/article/download/13858/17591	Acta Derm Venereol 2001; 81: 205–223 LETTERS TO THE EDITOR Superior Vena Cava Syndrome: An Important Di Œerential Diagnosis in Patients with Facial Edema Sir ,Facial edema is a frequent symptom in dermatological patients. At rst glance, facial edema is often mistaken for classic angioedema, but may also be related to the administration of angiotensin-converting enzyme inhibitors (1), the Melkersson-Rosenthal syndrome (2), dermatomyositis and viral or bac-terial diseases (e.g. erysipelas). A further but rarer di Œerential diagnosis is Morbus Morbihan, i.e. persistent facial edema which, for example, occurs in patients with acne, rosacea or after erysipelas (3). Furthermore, patients with kidney diseases often exhibit facial swelling, especially of the eyelids (4). In this paper we describe three patients initially examined by dermatologists because of their most visible symptom – acute or chronic facial swelling. In all three patients, the examination of facial edema resulted in detection of a superior vena cava syndrome caused by malignant tumours. Clinically, this diagnosis should be suspected by the presence of associated ndings such as dilatation of the two external jugular veins and increasing symptoms when the patient is in a horizontal position. Rapid detection of this condition is essential, otherwise acute life-threatening complications can be expected. CASE REPORTS Case 1 Within the course of 7 weeks, a 65-year-old man developed swelling and ushing of the face and upper extremities which increased overnight when he lay in a horizontal position. He also complained of headaches and the formation of dilated skin vessels on the upper trunk ( Fig. 1). Computed tomography of the chest some weeks previously had not shown any pathological ndings. A physical examination disclosed swelling of the face and hands and dilatation of the super cial veins of the chest and of both external jugular veins. Repeated contrast-medium-enhanced computed tomography showed compression of the superior vena cava caused by a tumour in the Fig. 1. Dilated super cial veins on the upper trunk in patient 1. upper mediastinum. Histopathological examination revealed a low di Œerentiated aden-ocarcinoma. The patient underwent a stent implantation into the superior vena cava and radiation treatment. Just a few days after stent the tumour was found to be inoperable. At present, the patient is implantation, the clinical symptoms improved dramatically. undergoing chemotherapy. Case 2 Case 3 A 50-year-old man was admitted with edema of the face and neck, ushing and a dilated external jugular vein on the right side, which A 50-year-old woman had noticed swelling of her face, especially of the eyelids, for the rst time several weeks previously. In addition, she had developed within the course of 6 months (Fig. 2). The symptoms increased when the patient was lying down or doing hard physical had developed dyspnea and a cough as well as pain and weakness of the upper arms. An increase in weight and tension of the mammae work. Six months previously, thrombosis of the right internal jugular vein, the subclavian vein and the superior vena cava had been were further symptoms. Physical examination disclosed periorbital edema, swelling of the mammae and dilated super cial veins on the diagnosed. At that time computed tomography of the chest did not reveal any underlying cause. Laboratory tests revealed an activated chest. Auscultation of the lung revealed that the respiratory sound of the right lung was reduced. Computed tomography of the chest was protein C (APC) resistance and Paget-von Schroetter syndrome (thrombosis of the axillary or subclavian veins following muscular performed immediately, disclosing a tumour in the upper mediastinum with narrowing of the right pulmonary artery, the right main bronchus eŒort) was suspected owing to the hard physical exertion which the patient had to perform daily in his job as a welder. A stent was and the superior vena cava. Histopathological examination showed a parvicellular bronchial carcinoma. Computed tomography of the brain implanted in the cranial superior vena cava to improve the clinical symptoms. After stent implantation all clinical ndings showed regres- and the abdomen showed metastasis of the brain, the liver and the suprarenal gland. Following initial radiotherapy of the chest, the sion. Three months later, further computed tomography of the chest showed a tumour growing in the upper mediastinum. The histopathol- tumour shrank and all symptoms disappeared. Furthermore, the patient underwent chemotherapy with carboplatin /etoposide and ogical diagnosis was embryonal rhabdomyosarcoma. Because of com-pression of the aorta and the esophagus as well as severe heart disease, radiotherapy of the whole brain. © 2001 Taylor & Francis. ISSN 0365-834 1 Acta Derm Venereol 81 206 Letters to the Editor retrosternal goiters (7). Rare causes described in the literature have been intrathoracic plasmocytoma (8), Behcet’s syndrome (9), syphilitic aneurysm of the ascending aorta (10) and superior vena cava syndrome in association with infectious diseases such as Klebsiella pneumoniae pneumonia (11). In another case, superior vena cava syndrome was the main symptom of a mediastinal amelanotic melanoma (12). The onset of superior vena cava syndrome demands immedi-ate therapy. Depending on the underlying diagnosis, treatment may include surgical reconstruction of the superior vena cava (13), radiation treatment (14) or percutaneous stenting (15). In conclusion, the three cases presented here clearly demon-strate that, in patients with facial edema, superior vena cava syndrome should always be taken into consideration. REFERENCES Pillans PI, Coulter DM, Blac P. Angioedema and urticaria with angiotensin converting enzyme inhibitors. Eur J Clin Pharmacol 1996; 51: 123–126. 2. Rogers RS. Melkersson-Rosenthal syndrome and orofacial granul-omatosis. Dermatol Clin 1996; 14: 371–379. 3. Hoelzle E, Jansen T, Plewig G. Morbihan disease – chronic persistent erythema and edema of the face. Hautarzt 1995; 46: 796–798. 4. Dyken JR, Pagano JP, Soong VY. Superior vena caval syndrome presenting as periorbital edema. J Am Acad Dermatol 1994; 31: 281–283. 5. Goerdt S, Krengel S, Tenorio S, Tebbe B, Geilen C, Orfanos CE. Vena cava superior syndrome. Hautarzt 1997; 48: 122–126. 6. Hirschmann JV, Raugi GJ. Dermatological features of the super-ior vena cava syndrome. Arch Dermatol 1992; 128: 953–956. 7. Jansen T, Romiti R, Messer G, Stu ¨cker M, Altmeyer P. Superior vena cava syndrome presenting as persistent erythematous oedema of the face. Clin Exp Dermatol 2000; 25: 198–200. Davis SR, King HS, Le-Roux I, Bolding E. Superior vena cava Fig. 2. Flush and swelling of the face in patient 2. Dilatation of the syndrome caused by an intrathoracic plasmocytoma. Cancer 1991; right external jugular vein. A sutured incision from skin biopsy can 68: 1376–1379. be seen on the left cheek. The patient agreed that the photograph 9. Roguin A, Edelstein S, Edoute Y. Superior vena cava syndrome could be published without masking. as a primary manifestation of Behcet’s disease. A case report. Angiology 1997; 48: 365–368. 10. Omos JM, Fernandez-Ayala M, Gutierrez JA, Val JF, Gonzalez- DISCUSSION Marcias J. Superior vena cava syndrome secondary to syphilitic aneurysm of the ascending aorta in a human immunode ciency Superior vena cava syndrome often shows gradual develop- virus-infected patient. Clin Infect Dis 1998; 27: 1331–1332. ment with increasing swelling of the face and upper extremities. 11. Kim JY, Lim CM, Koh Y, Choe KH, Kim WS, Kim WD. A A further and very characteristic clinical feature is extension case of superior vena cava syndrome caused by Klebsiella pneu-moniae. Eur Respire J 1997; 10: 2902–2903. of the external jugular veins and increasing symptoms when Shishido M, Nagao N, Miyamoto K. Mediastinal amelanotic the patient is in a horizontal position. If the syndrome develops melanoma presenting as superior vena cava syndrome. Nihon slowly, dilated super cial veins of the chest can also be seen. Kyobu Sikan Gakkai Zasshi 1997; 35: 240–244. Dyspnea, acute swelling, ushing and cyanosis of the face 13. Magnan PE, Thomas P, Giudicelli R, Fuentes P, Branchereau A. and neck indicate the immediate onset of superior vena cava Surgical reconstruction of the superior vena cava. Cardiovasc syndrome (5). Surg 1994; 2: 598–604. Numerous di Œerent diseases may cause superior vena cava 14. Hochrein J, Bashore TM, O’Laughlin MP, Harrison JK. syndrome. In up to 85 % of cases, a primary lung carcinoma Percutaneous stenting of superior vena cava syndrome: a case report and review of the literature. Am J Med 1998; 104: 78–84. is the underlying cause (10). In our rst case, a low di Œerenti- Rodrigues CI, Njo KH, Karim AB. Hypofractionated radiation ated adenocarcinoma of the lung could be found, and in the therapy in the treatment of superior vena cava syndrome. Lung third case a parvicellular bronchial carcinoma was the under- Cancer 1993; 10: 221–228. lying cause of the superior vena cava syndrome. Only in the second case was a rare tumour such as embryonal rhabdomyo- Accepted February 26, 2001. sarcoma revealed by histopathological examination. In addition, lymphomas, invasive thymomas, metastatic T. Burgdor Œ, K. E. Douwes, T. Bogenrieder, R. M. Szeimies, U. lymph nodes or brosing mediastinitis can lead to compression Hohenleutner, M. Landthaler and W. Stolz of the superior cava vein. Other causes might be, stenosis after Department of Dermatology, University of Regensburg, D-93042 multiple central venous catheterization, peritoneovenous Regensburg, Germany. E-mail: wilhelm.stolz@klinik.uni-regensburg.de shunts or cardiac pacemakers (6), or benign tumours such as
2585	https://www.merriam-webster.com/rhymes/syn/braggadocio	braggadocio Related Words - Merriam-Webster Related Words Rhymes Near Rhymes Advanced View Related Words Descriptive Words Homophones Same Consonant Similar Sound Chatbot Dictionary Games Word of the Day Grammar Word Finder Slang NewNewsletters Wordplay Rhymes Thesaurus Join MWU More Dictionary Grammar Games Word of the Day Word Finder Slang Newsletters New Wordplay Rhymes Thesaurus Join MWU Shop Books Merch Log In Username My Words Recents Account Log Out Est. 1828 Results for 'braggadocio' Rhymes 2082 Near Rhymes 35 Advanced View 122 Related Words 75 Descriptive Words 90 Homophones 0 Same Consonant 0 Similar Sound 0 Related Words Rhymes Near Rhymes Advanced View Related Words Descriptive Words Homophones Same Consonant Similar Sound Show All Results Show [x] Similar - [x] Opposite Related Words for braggadocio 75 Results \| Word \| Syllables \| Categories \| --- \| bluster \| /x \| Noun \| \| bravado \| x/x \| Noun \| \| bombast \| /x \| Noun \| \| swagger \| /x \| Noun \| \| hyperbole \| x/xx \| Noun \| \| belligerence \| x/xx \| Noun \| \| bombastic \| x/x \| Adjective \| \| arrogance \| /xx \| Noun \| \| hubris \| /x \| Noun \| \| posturing \| /xx \| Noun \| \| pomposity \| x/xx \| Noun \| \| rhetoric \| /xx \| Noun \| \| bragging \| /x \| Noun \| \| grandiosity \| xx/xx \| Noun \| \| unabashed \| xx/ \| Adjective \| \| machismo \| x/x \| Noun \| \| unapologetic \| xxxx/x \| Adjective \| \| nonchalance \| /x/ \| Noun \| \| exuberance \| x/xx \| Noun \| \| nonsense \| /x \| Noun \| \| petulance \| /xx \| Noun \| \| affectation \| xx/x \| Noun \| \| puerile \| /x \| Adjective \| \| pretension \| x/x \| Noun \| \| gloating \| /x \| Verb \| 1 2 3 3 More Ideas for braggadocio Go to the Advanced Search page for more ideas ### Can you solve 4 words at once? Play Play ### Can you solve 4 words at once? Play Play Word of the Day obliterate See Definitions and Examples » Get Word of the Day daily email! Filter Done Result type [x] Rare Words [x] Phrases Reset Merriam Webster Learn a new word every day. Delivered to your inbox! Help About Us Advertising Info Contact Us Privacy Policy Terms of Use Facebook Twitter YouTube Instagram © 2025 Merriam-Webster, Incorporated Information from your device can be used to personalize your ad experience. Do not sell or share my personal information. ✕ Do not sell or share my personal information. You have chosen to opt-out of the sale or sharing of your information from this site and any of its affiliates. To opt back in please click the "Customize my ad experience" link. This site collects information through the use of cookies and other tracking tools. Cookies and these tools do not contain any information that personally identifies a user, but personal information that would be stored about you may be linked to the information stored in and obtained from them. This information would be used and shared for Analytics, Ad Serving, Interest Based Advertising, among other purposes. For more information please visit this site's Privacy Policy. CANCEL CONTINUE
2586	https://www.researchgate.net/publication/325915277_IEEE_Std_1459_power_quantities_ratio_approaches_for_simplified_harmonic_emissions_assessment	(PDF) IEEE Std. 1459 power quantities ratio approaches for simplified harmonic emissions assessment Conference Paper PDF Available IEEE Std. 1459 power quantities ratio approaches for simplified harmonic emissions assessment May 2018 DOI:10.1109/ICHQP.2018.8378832 Conference: 2018 18th International Conference on Harmonics and Quality of Power (ICHQP) Authors: Antonio Cataliotti University of Palermo Valentina Cosentino Valentina Cosentino This person is not on ResearchGate, or hasn't claimed this research yet. Dario Di Cara Italian National Research Council G. Tine Italian National Research Council Download full-text PDFRead full-text Download full-text PDF Read full-text Download citation Copy link Link copied Read full-textDownload citation Copy link Link copied Citations (8)References (19)Figures (2) Figures Single-phase test system. … Three-phase test system … Figures - uploaded by G. Tine Author content All figure content in this area was uploaded by G. Tine Content may be subject to copyright. Discover the world's research 25+ million members 160+ million publication pages 2.3+ billion citations Join for free Public Full-text 1 Content uploaded by G. Tine Author content All content in this area was uploaded by G. Tine on Jul 31, 2018 Content may be subject to copyright. 978-1-5386-0517-2/18/$31.00 ©2018 IEEE IEEE Std. 1459 power quantities ratio approaches for simplified harmonic emissions assessment Antonio Cataliotti, Member IEEE, Valentina Cosentino Departme nt of Energy, Info rmation Enginee ring a nd Mathemat ic Mo dels (D EIM), U nive rsità di P alermo, Palermo, Italy email: acat aliotti@i eee.org, va lentina.cosent ino@uni pa.it Dario Di Cara, Member IEEE, Giovanni Tinè, Member IEEE, Nationa l Rese arch Cou ncil (CN R), Institute of Intelli gent Syste m for Automation (ISSIA) Palermo, Italy e-mail: dicara@pa.is sia.cnr.it, tin e@pa.iss ia.cnr.it Abstract — The paper inves tigate s the s uitab ility o f using p owe r ratio parame ters for harmo nic emissio ns assessmen t at the poin t of common coupling (PCC). The study is carried out starting from the IEEE Std. 14 59-2010 app arent power deco mposition, where power factors are de fined fo r evaluating l ine utiliz ation and harmonic pollutio n levels. In addition, the study investigat es the behavior of new p arameters, which are expressed in terms of ratio between IEEE Std. 1459-20 10 power quantities. The study is carried out for both single-phase and three-phase case, also considering the pres ence of cap acitors. Index Terms -- power measurement, harmonic distortion, powe r definitions, power quality, IEEE S td. 1459-2010, harmo nic sources. I. I NTRODUCTION The ass essme nt of harmo nic emis sio ns leve ls is a v ery impo rtant is sue f or mo dern e lect ricity dist ributio n g rids, where dist ortion levels are progr essively in creasing beca use of the presence of non l inear loads (equipped w ith pow er electroni c devices) or even distri buted generati on from renewable e nergy reso urces (equi pped wit h inverter-i nterface for grid co nnection) . T he definitio n of an effe ctive methodology for harmonic emissions assessment has been deeply deb ated i n lite rature and it is a n esse ntial issue for ensuring norm al power quality levels, promoting reg ulation for har monic miti gation, sharing r esponsibilit y between customers and util ities for power systems distur bances -. Some o f them allow to se parate custome rs a nd util itie s contributio n to harmo nic distortio n, allow ing also to investigat e the i mpact of single har monic components. Th e limitati on of such approach es is that they are di fficult to be imple mented in prac tical measu reme nt inst ruments. On t he contr ary, some pr opose d sol ution s ar e abl e to pr ovide less informati on, giving only in dication on th e prevailin g source of dist urban ce or on th e pres ence of a di stur bing l oad. The advant age is that s uch me thod s ca n be eas ily impleme nted o n smart me ters o r othe r meas uring i nstru ments diffu sed o ver the whole netw ork. Cu rrent inte rnational S tandards o n power quality and harmonics - set limits f or networks an d loads harmo nic dis tort ion lev els; they define also measu rement metho ds fo r harmo nic d istortio n ev aluat ion (in terms of TH D or single harmonics amplitudes) and electri c power qua ntities fo r quantify ing flow of e lectrical ene rgy, as in IEEE Std. 1459-201 0 . This last St andard provide s a set of power defi nitions (active, nonactive, apparent) and re lated line ut ilizat ion, h armo nic pollu tio n and loa d unba lance facto rs, which can be use d for revenue purpo ses, determi nation of major harmo nic pollute rs and so on. As regards the line utilizatio n, some po wer facto r definitio ns are introduc ed. I n sinusoidal co nditio ns, powe r factor is an impo rtant index f or pow er quali ty ev aluatio n an d it is a ve ry suit able p arame ter, which is effecti ve for power tran smission efficiency improv eme nt and it is also simple to b e me asure d (almo st al l measuring i nstrument ation fo r power sy stems applicat ions can easily imple ment its me asu rement). Pow er fac tor is well defined in sinuso idal situ ations, while different def initions exist in nonsinusoidal conditi ons, for exam ple as those reported in . As reg ards this, the IEEE Std. 1459 is based on the se para tion of the fu ndame ntal co mpo nents f rom the harmo nic co ntent o f voltag e and cu rrent. T his app roac h allow s to measu re the t raditio nal qu antitie s (active, reac tive an d apparent po wers and e nergies, and related pow er factor), and to intro duce some o ther qua ntitie s for h armonic pollu tio n asses sment. Star ting from th e appr oach of the IEEE Std. 1459 and the common con cept of power fact or correction, in this paper a study is prese nted, aimed at invest igating t he possibility of using po wer facto r concepts o r other pow er ratio parame ters fo r harmo nic emiss ion asse ssme nt. The adv ant age of s uch solu tion, in co mparis on w ith mo re com plex method s and algorith ms is that, even if only quali tatively, it can be easily integ rated in commo n fiel d meas ureme nt ins trume nts (smart me ters, po wer qu ality a nalyzers, and so o n) II. B ACKGROUND A.IEEE Std. 1459-2010 and the apparent pow er resolution IEEE Std. 1 459 ap parent po wer terms a re defined star ting fr om th e separa tion of fun dament al com ponent s of voltage s and curre nts (a t pow er sy stem fre que ncy) f rom the harmonics. The ap parent pow er decompositio n schemes are summa rized i n TABL E I. and TABLE I I. fo r single-phas e and three-phas e systems, respectivel y. In both tables p owers are divi ded in to thr ee basi c group s: appa rent, acti ve, and nonactive; each group includes combined, fun damental and nonfundame ntal pow ers. The last row s report so me comb ined indices fo r line utilizat ion (pow er factors) and harmonic pollut ion ass ess ment (as we ll as f or lo ad unb alanc e amo unt, in th e thr ee-ph as e ca se). In the s ingle p hase case, funda ment al activ e, re activ e and app are nt p ow ers, re p rese nt the app are n t po we r co mpo ne nts in the ide al cas e of a pure ly sinusoid al sy stem; a ll the othe r apparent pow er terms prov ide a basis f or harmo nic assessment. Fundamental power factor (PF 1) is often referre d to as the displac eme nt pow er facto r and it is the mo st popu lar parameter to evaluate funda mental pow er flow co ndition and to adjust rea ctive power flow by mean s of capacitor banks (po we r fa cto r c or rec tio n). I n no ns inu so ida l c ond iti ons, p ow er factor PF can be interp reted as t he ratio be twee n the energ y trans mitted to the lo ad and the max imum e nergy that co uld be transmitt ed (with the same l ine losses), thus it is a line utilizati on factor, wher e the maximum ut ilization is gained when P = S, i.e. PF = 1 (for given valu es of ap paren t power S and rms voltage V, and ev en with ha rmonics). The o verall amou nt of harmo nic po llut ion is ev aluate d wit h the rat io S N/S 1, that is e qual to ze ro in pure ly s inusoid al co nditio ns. Simi lar considerati ons can be made for th e three-phase case, wh ere the fundamen tal power fact or PF 1+allows ev aluat ing the positive-s equence po wer flow conditions, w hile PF,S eN/S e1 and S U1/S 1 +fac tors allo w ev aluating line ut ilizat ion, h armo nic poll ution a nd load un bal ance amount s.In th e absen ce of unba lanc e, S U1/S 1 += 0, and the eff ective appare nt power decompositi on becomes anal ogous to that of th e single-ph ase case. B.Summary of harmonic emi ssion assessment techniq ues The mos t popular indic es fo r evaluating t he harmo nic distortio n leve l at a given mete ring sec tion are the to tal harmonic d istortio n facto rs (THD V and TH D I fo r voltages and curr ents, respect ively. Su ch par ameter s are con sider ed also in IEEE Std. 1459; generally speaking, they can measure the amo unt of the voltage and cu rrent disto rtio n, but they don’t allow assess ing the disturba nce sour ce. As regard s this last aspe ct, sev eral met hods have bee n propo sed in li terature for harmonic source s detection have been presented, b ased on bot h sin gl e-poi nt and mul ti-poin t ap proa ches. Som e of them allo w qua ntifying the emis sion le vel, p rovidi ng basis for shari ng res ponsibili ty be tween cus tomers a nd uti lities f or power system harmonics; how ever they require the use of complex algorithms, making use also of distributed measure ments i nfrast ructure, t hus the are no t very suitab le for diffuse d and simp le prac tical me asuring instru ments. On t he othe r ha nd, sim ple r s olu tio ns have be en p rop os ed, w hic h c an be able to reveal if a given load is produc ing harmo nics or not, supporting the harmonic source detec tion, upstream or downst ream the me tering s ection. Exa mples o f such appr oach es ar e th ose bas ed on ha rmon ic a cti ve power fl ow direction or on a circui tal approach based o n impedance measurements . Active power flow direction method can prov ide misleading informatio n, depending on the ope rating cond itions. Co mpensat ing eff ects amo ng harmonic c ompo nents, p hase ang les be tween ha rmonic voltage a nd curre nt compo nents or me asureme nt uncertainties can af fect the info rmation correctness . On the o ther han d imped ance me thods a re quite c omplex to be implemente d due to the pract ical c hallenge o f the evaluatio n of utility an d custo mer harmo nic impe dance s. Various re search w orks hav e bee n conducted to estab lish methods that can me asure t hese impe dances. Unfortu nately, impedance measure ment is a v ery diffic ult proble m and research pr ogress has been slow, i.e. independent compon ent analy sis metho d –ICA . Also the autho rs have de alt wit h this issu e, fo cusing t he attention o n the analy sis of non-act ive powers ;in very bri ef the prop osed st ra tegy wa s bas ed on the c ompar is on of three dif ferent no nactiv e powe r quantitie s, whic h were derived form the IEEE Std. 1459 app roach and measured a t the same me tering s ectio n. Such me thod w as teste d in several situatio ns (bo th in simul ation a nd expe rimental ly), prov iding satisfac tory re sults for the de tectio n of the prev ailing disturbance source (upstream of dow nstream the meteri ng sectio n) . Howeve r, some diff iculties arose in defining the thres holds f or compa rison, whic h can depe nd from differ ent elem ents, such as th e depen dence of th e po wer quantitie s value s on the influe nce of other load s connec ted to the sam e PCC, th e harmoni c state of th e system or the presence of c apacitors for powe r factor correctio n . III. P ROPOSED APPROACH AND SIMULATION RESULTS In order to investig ate the p ossib ility to ov ercome the afor esai d pr oblem s, a s impl ifi ed ap proa ch coul d be u sed, in acco rdance with that c ommonly used fo r PF correctio n. This would al low the follow ing adv antages: assess ing the glo bal ha rmonic e missio ns, by means of the co mpariso n with a tole rable t hreshold f or harmonics; providi ng a simp lified tool fo r billi ng purpo ses, regulato ry framew orks, incen tives fo r mitigati ng harmonics on power systems; impleme nting the c orrespo nding measu remen t in simple and prac tical me asuring inst ruments (even the existing mete rs, wit h few modif ications). To th is aim, in thi s paper a prel iminar y simulat ion st udy is carried o ut, wit h respe ct to li ne utiliz ation a nd harmo nic pollution factors of IEEE Std. 1459. The study is c arried out for both the single-phase and the t hree-phase case (see tes ts systems of F igure 1. and Figure 2. respe ctively). The beha viour of su ch in di cator s is inves tig ated in di ffer ent operatin g condi tions. Fu rthermo re, the feasib ility is also studied of some o ther simplifie d new indica tors, w hich are always derived f rom the IEEE Std. 1459 appa rent power resolutio n. More i n detail, apart f rom the de fined PF 1,PF and S N/S 1, the be haviour of the follow ing new pow er ratio pa ramete rs is inves tigated: P 1/S,S 1/S, Q 1/N. As rega rds t he f irst p ara met er, P 1/S, it can allo w evaluati ng the total a mount of line utilizat ion, co nsidering not only the fundamental po we r flow conditio n, but als o the p resence of harmo nics (w hich are included in S); in p urely sinusoidal case,P 1/S =PF 1, thu s the indicator behave s as a power fac tor indicator (whose maximum ac hievab le value is equal to PF 1). As regards the second pa ramete r, S 1/S, it can allo w ev aluating the who le harmonic e missio n level, w ith respe ct to both activ e and reactiv e powe r compo nents; i n purely sinuso idal co nditio ns, S 1/S = 1, thus the indicator be haves as a powe r factor indicato r (who se maximum achiev able v alue is e qual to 1). As regards the last pa rameter, Q 1/N, in p revious papers the auth ors showed t hat th e reacti ve powers beh avior depends on t he lo ad natu re (li nea r or no nli ne ar); in a giv en d ist orte d working condition, Q 1 is a minim um refer ence val ue, si nce it is the on ly non active po wer comp onen t in th e sinusoi dal conditio n; N is a maximu m ref erence value since it g roups al l the nonactive co mponents of the apparent power ; thus N Q d 1 . In sinuso idal co ndition the tw o quant ities are e qual. When the lo ad is line ar, the amo unt of current dis tortio n is low and i t is due only to the disto rtio n of the su pply voltage; in this case the diff erence be tween Q 1 and N is small. If the load is non l inear the amo unt of current disto rtion is hig her, Q 1>N. Thus, the co mparis on be tween Q 1 and N, calcul ated in the s ame mete ring sec tion and in the sa me w orking conditio n, can g ive a pie ce o f informa tion on the pres ence of dist urbi ng l oads. A.Single-phase study A preliminar y validation of th e proposed appr oach was carried out o n a simple sing le-phase tes t system, w hich represe nts a simp lifie d situ atio n, in w hich b oth the sup ply and the l oad can be re sponsi ble for the ha rmon ic di stort ion . A scheme of the test sys tem is re porte d in Fig ure 1. The sy stem consis ts o n: a sup ply v oltage E 1 = 230 V at the f undamenta l power supply frequency (50 Hz), with a phase a ngle D 1 = 0°; a line imp ed anc e Z_ Line with R L= 0,1172:and L L= 3,93410-4 H; a res istive-ind uctiv e load Z_ Load with capacitor C for power factor corr ection(In = 5 A, c os M= 0,95). TABLE I. IEEE S TD 1459-2010 A PPARENT POWER RESOLUTION –S INGLE-PH ASE C ASE Quanti ty Co mbined Fundame ntal Nonfund amenta l Apparent p ower resolut ion sche me Appare nt [VA] S = V I S 1= V 1 I 1 2 1 2 S S S n  S H = V H I H Active [W] ¦ n h h h h I V P 1 cos T 1 1 1 1 cos T I V P 1 1 cos P P I V P h h h h h  ¦ z T Nonact ive [VAR] 2 2 P S N  1 1 1 1 sin T I V Q D I ,= V 1 I H D V= V H I 1 2 2 H H H P S D  Line Utilization PF=P/S PF 1=P 1/S 1– Harmoni c pollution –– S N/S 1 V h and I h are the rms values of t he harmon ic components of volt age and cur rent, T h is their displaceme nt and h is the harm onic orde r. TABLE II. IEEE S TD 1459-2010 E FFECTIVE APPARENT POWER RESOLUTION –T HREE-PHASE CASE Quant ity Combined Fundame ntal Nonfund amenta l Effective apparen t power resolut ion sche me Appare nt [VA] S e= 3 V e I e S e1 = 3 V e1 I e1 , S 1+=3 V 1+I 1+ 2 1 2 1 1  S S S e U 2 1 2 e e eN S S S  S eH = 3 V eH I eH, Active [W] ¦¦ c b a n h h h h I V P ,,1 cos T P 1+=3 V 1+I 1+ cos T 1+ 1 ,,2 cos P P I V P c b a n h h h h H  ¦¦ T Nonact ive [VAR] 2 2 P S N e  Q 1+=3 V 1+I 1+sin T 1 + D eI = 3 V e1 I eH De V= 3 V eH I e1 2 2 H eH eH P S D  Line utilization PF=P/S e PF 1+=P 1+/S 1+ – Harmoni c pollution ––S eN /S e1 Load Unbala nce –S U1 /S 1+– V e,V e1,V eH, are th e rms valu es o f effe ctive vo ltag es; I e, I e1, I eH, are the rms values of effe ctive currents (total, fundament al, harmonic) Differe nt harmonics c an be added o n both the su pply voltage and t he lo ad curre nt, by means o f v oltage and cu rrent gene ra to rs (re pre se nte d i n t he f ig ure wi th E h a nd I h).; thus, it is possib le to s imulate the p resence of a source of disto rtion o n the s upply side and/or t he load s ide. Sev era l s imu la tio ns w e re ca rri ed o ut i n di ff ere nt wo rki ng conditio ns, which w ere obtained b y intro ducing vario us harmonics o n voltage and c urrent. Vo ltage and cu rrent were measu red at the load te rminals (as rep resente d by the vo ltage and current m eters of Figure 1. As an example, the firs t simulatio n was carried out by introduc ing a fift h harmo nic on the sup ply v oltage, wit h rms va lue E 5= 0,1 E 1; no harmo nics were inject ed by the load. The mea sured quantiti es are reported in TABLE III. As shown in the table, the values of PF 1, PF, P 1/S, S 1/S and Q 1/N are between 0.94 and 1, i n accordan ce with the lin ear load simulated conditions (S N/S 1 is small, a s the distor tion amount is low). Further simulations were car ri ed out by in tr oducin g the fi fth ha rmoni c on both the supply voltag e and t he load c urre nt, wit h E 5= 0,1 E 1 and phase ang le D 5= 0 and I 5= 0,4 I 1 and phase angle E 5,var iable, from 0° to 360°. Thus, in this case the simulate d load is nonlinear. T he obtained results are sy nthesize d in Figure 3. In all cas es the v alues of PF, P 1/S, Q 1/N and S 1/S are lower than PF 1 and S N/S 1 is hig her tha n the p revious simul ated cas e, because of th e harmonic inject ed by the load and d epending on the phase angle E 5,value. B.Three-phase study Further simu lations wer e carried out on a s imple three- phase test sy ste m whic h is ab le to si mula te dif ferent w orki ng conditio ns, with both sinuso idal or disto rted supply and linear (RL) or non lin ear (N.L.) l oads [1 1]. A linear l oad with capacitor bank for power fact or correcti on (RLC) has been also added a t PCC, i n order to take under co nsideratio n the presence of capacit or banks. A block di agram of th e deve loped syste m, wit h its m ain c haracte ris tics, is sho wn i n Figu re 2. Sim ul atio ns we re c a rrie d o ut fo r dif f ere nt w or kin g conditions. Some of the obt ained result s are summarized in Figure 4. T hey are ref erred to the follow ing load condi tion s, all balanced and with no nsinusoidal sup ply vo ltage (switch 1 open, switch 2 c losed): Test A. linear lo ad, RL (switches: 3 clo sed; 4 and 5 ope n); Test B. linea r lo ad with capac ito rs, RL C (sw itches: 5 clo sed; 3 and 4 open); Test C. non line ar load, N.L. (switc hes: 4 clos ed; 3 and 5 open); Test D. RL a nd RLC loa ds (s witche s: 3, 5 close d; 4 open); Test E. N.L. and RL l oads (switch es: 3, 4 cl osed; 5 op en); Test F. N.L. and RLC l oads (swit ches: 4, 5 closed; 3 open); Test G. Al l lo ads (s witc hes: 3, 4, 5 c lose d). It can b e obse rved that the ob tained v alues f or linea r load are simi lar to th ose obtain ed in th e singl e phase tes t of TABLE III. In fact, the values of PF 1,PF, P 1/S, Q 1/N and S 1/S are high (betw een 0.94 and 1), while S N/S 1 is small, as the load harmonic emiss ion is nil. On the contra ry, values for non line ar load are s imilar to thos e obtained i n the si ngle- phase tes t with load ha rmonic inje ction. I n fact the value s of PF, P 1/S, Q 1/N and S 1/S are lower than PF 1 and S N/S 1 is higher t han that obtai ned fo r the line ar load, b ecaus e of the harm oni cs i nje ct ed b y the no n li nea r lo ad. A s re ga rds the beha viour of th e l oad wi th ca pa citor s ban k (RL C), it can be obse rved that, whe n the RL C load is the o nly one lo ad at PCC or it is co nnected togethe r with the l inear R L load, the v alues of PF, P 1/S and S 1/S are simila r to PF 1 w hile the mo st signific ant diff erence is obtained f or S N/S 1 and Q 1/N becau se the capac itors amp lify the d istortio n from the nonsinusoid al supply vo ltage. On the co ntra ry, when b oth RLC and N.L. loads ar e connect ed to th e PCC, the va lues of PF, P 1/S and S 1/S and Q 1/N are lower th an PF 1 because the capacitors amplify the disto rtion inje cted bo th by the N.L. lo ad and the supply; thus the RL C load behaves quite simil arly to the non linear lo ad, as expec ted. V Z _Load I h_Load E h Z _Line PCC E 1 A C Figure 1.Single-phase t est syst em. TABLE III. S INGLE-PHASE TEST–M EASUREMENTS WITH E 5=0,1 E 1. PF 1 PF S N/S 1 P 1/S Q 1/N S 1/S 0.95 0.95 0.11 0.94 0.96 0.99 Sinusoidal Supply (230 V, 50 Hz) Non Sinusoidal Supply (V 1 = 230 V, 50 Hz, THD = 6,9%) Line impedance (R = 0,1172 :, L = 3.934 mH; R n=2 R,L n=2 L) Linear load (RL) (P = 5716 W, Q L = 1281 var) 1 2 3 PCC Nonlinear load (diode bridge rect. + DC load; P 1 = 5716 W, Q 1 = 1281 var) 4 Linear load (RL) + capacitor bank (C) (P = 5716 W, Q L = 3542 var, Qc = 2261 var) 5 Figur e 2.T hree-pha se t est syst em (a) (b) Figure 3. Simulation results. Sin gle-phase case. (a) IEEE 1459 line utilization and harmonic po llution fac tors; (b) new power r atio paramet ers 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 PF1 0,98 0,98 0,97 0,98 0,98 0,98 0,97 0,98 0,97 0,98 0,98 0,97 PF 0,97 0,96 0,86 0,97 0,96 0,97 0,86 0,86 0,87 0,97 0,89 0,87 Sn/S1 0,09 0,16 0,55 0,09 0,16 0,09 0,55 0,53 0,53 0,11 0,46 0,53 Linear load (RL) Linear load + Cap. bank (RLC) Nonlin ear load (N.L.) RL and RLC values for RL - RL and RLC values for RLC - NL and RL -values for RL - N.L. and RL values for N.L. - NL and RLC -values for RLC - N.L. and RLC values for N.L. - All loads values for RL - All loads values for RLC - All loads values for N.L. - (a) 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 P1/S 0,97 0,96 0,85 0,97 0,96 0,97 0,85 0,86 0,86 0,97 0,89 0,86 Q1/N 0,98 0,82 0,44 0,97 0,82 0,98 0,43 0,38 0,40 0,96 0,43 0,41 S1/S 1,00 0,99 0,88 1,00 0,99 1,00 0,88 0,88 0,88 0,99 0,91 0,88 Linear load (RL) Linear load + Cap. bank (RLC) Nonlin ear load (N.L.) RL and RLC values for RL - RL and RLC values for RLC - NL and RL -values for RL - N.L. and RL values for N.L. - NL and RLC -values for RLC - N.L. and RLC values for N.L. - All loads values for RL - All loads values for RLC - All loads values for N.L. - (b) Figure 4. Simulation results. Three-phase case. (a) IEEE 1459 li ne utilizat ion and harmonic po llution f actors; (b) new power r a tio par amete rs IV. C ONCLUSIONS The pape r has i nvestigate d the s uitabili ty of using power factors and the ot her new pow er ratio parameters for harmonic e missions asses sment at t he poin t of co mmon coupling (PCC). The most sensitive power ratio parameter to harmonic e mission is Q 1/N while PF 1 could b e consi dered as referen ce valu e. Th e study ha s been ca rried ou t start ing from the IEEE Std. 1 459-2010 appare nt powe r decompositio n, by using only quantities derived from the I EEE Std. 1459 apparent powe r resolution. The study has been carried o ut for both si ngle-phase and three-phas e case, also con sidering the presence of ca pacitor bank s. Th e obtain ed r esul ts sh ow that th e prop osed ap proa ch allow obtain ing a qu alitativ e informatio n on the p rese nce of disturbing lo ads connected a t PCC. The emplo yed powe r ratio parameters are very e asy to be measu r ed, thus they could be easily implemented in c ommo n inst rume ntatio n for po wer syste m measu r eme nts. R EFERENCES Spelko, A., Bla zic, B., Papic, I., Poura rab, M., Meyer, J., Xu, X., Djokic, S.Z., “CIGRE/CIRED JWG C4.42: Overview of comm on methods f or assessme nt of h armonic con tribution f r om customer install ation”, (2017) 2017 I EEE Manchest er PowerTe ch, Powertech A. E. Emmanuel, “On the Assessment of Harmonic Pollution” IEEE Trans. On Power De livery, Vol. 10, No 3, January 1995, pp 1693- 1698. A. P. J. Rens, P. H. 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Citations (8) References (19) ... 1459. The IEEE Working Group is in the process of improving the IEEE 1459 standard considering the physical meaning of the measurement theory and reactive power definitions [24,25], but this does not affect the application of the IEEE 1459 standard in line loss calculations and line loss analyses, due to the fact that the line loss itself is caused by the current flowing through the line. The power decomposition for both single-phase and three-phase systems is depicted in Table 1. ... ... In Figure 4, R is the line resistance. When a non-linear load is connected to the system, the three-phase currents IA, IB, IC, and neutral current In produce line loss in the line resistance R. The three-phase voltage, current, and phase angle information at measurement Point 2 is collected, and the original three-phase system is equivalently transformed to decompose the load measurement equivalent apparent power Se [24,25]. The equivalent system has the same line power losses as the actual distribution system, and Figure 4 shows a schematic diagram of the three-phase, four-wire low-voltage distribution system after introducing the equivalence. ... ... In Figure 4, R is the line resistance. When a non-linear load is connected to the system, the three-phase currents I A , I B , I C, and neutral current In produce line loss in the line resistance R. The three-phase voltage, current, and phase angle information at measurement Point 2 is collected, and the original three-phase system is equivalently transformed to decompose the load measurement equivalent apparent power S e [24,25]. ... Measurement and Assessment of Reactive, Unbalanced and Harmonic Line Losses Article Full-text available Apr 2024 Qun Zhou Yulin Dian Xueshan Liu Haibo Liu This study investigates the feasibility of utilizing the line loss power factor to assess the reactive, unbalanced, and harmonic line losses in low-voltage distribution networks and explores the method of calculating decoupled line loss values based on this factor. To achieve this objective, we establish preliminary definitions of single-phase and three-phase reactive, unbalanced, and harmonic line loss power factors, drawing upon the principles of electrical theory outlined in IEEE Standard 1459. These power factors serve as crucial indicators for evaluating the severity of line losses caused by reactive power, unbalance, and harmonic problems. Subsequently, the values of line loss attributed to reactive, unbalanced, and harmonic components are decoupled and quantified using the line loss power factor as a fundamental parameter. The effectiveness and accuracy of the proposed method were verified in Matlab simulation and physical experiments. View Show abstract ... In this framework, in the authors presented a preliminary study focused on the feasibility of using IEEE 1459 indicators and new power ratio parameters for assessing harmonic emission levels at PCC. The study was based on IEEE 1459 approach and the common concept of power factor correction. ... ... Starting from the results of , in this paper an extended study is presented on the measurement of the considered power ratios in both single-phase and three-phase sinusoidal, nonsinusoidal, balanced and unbalanced cases. For the singlephase and three-phase balanced cases main findings of - are summarized and new results are presented and discussed. For the three-phase unbalanced case, further indicators are introduced for both harmonic and unbalance assessment, which are meant to be exploited with the same approach used for the balanced case. ... ... To avoid these limitations, in - the feasibility of some new indicators was initially investigated, with the aim of replacing S N /S 1 indicator of IEEE 1459 with one or more parameters conceptually similar to the power factors, i.e. approaching 1 in ideal conditions. Such parameters were expressed as a function of IEEE 1459 power quantities, in order to keep the advantage of easy implementation in practical measuring instruments (even in existing PQAs and SMs). ... Measurement of simplified single-phase and three-phase parameters for harmonic emission assessment based on IEEE 1459-2010 Article Full-text available Nov 2020 Giovanni Artale Giuseppe Caravello Antonio Cataliotti G. Tine The paper investigates the feasibility of using a simplified approach, based on the measurement of power ratio parameters, for harmonic emissions assessment at the point of common coupling (PCC). The proposed approach comes from the common concept of power factor correction and the definitions of IEEE Std. 1459-2010, where line utilization and harmonic pollution levels are evaluated by means of ratios between power quantities of the apparent power decomposition. In addition to IEEE Std. 1459-2010 indicators, in this paper the behavior is studied of additional parameters, which are conceptually similar to those defined by IEEE Std. 1459-2010. The suitability of such parameters is discussed, for both single-phase and three-phase balanced/unbalanced case, taking into account both their behavior in different scenarios and their effectiveness when the measurement uncertainty is taken into account. The study is supported by some simulation results that have been obtained on a IEEE benchmark power system, which allows reproducing linear and nonlinear load conditions, balanced and unbalanced operating conditions and the presence of capacitors for power factor correction. View Show abstract ... Finally, harmonic active (S eH , P eH and D eH ) and distortion (D eI and D eV ) powers are described as well. Cataliotti et al. demonstrate the value of using IEEE 1459-2010 for quantifying harmonic content . ... Modeling Harmonic Impacts of Electric Vehicle Chargers on Distribution Networks Conference Paper Full-text available Sep 2018 Nicole Woodman Robert B. Bass Mike Donnelly View ... Finally, harmonic active (S eH , P eH and D eH ) and distortion (D eI and D eV ) powers are described as well. Cataliotti et al. demonstrate the value of using IEEE 1459-2010 for quantifying harmonic content . ... Impacts of electric vehicle charging on electric power distribution systems Article Jan 2013 N. Zimmerman Robert B. Bass View Caracterização de Carga para Análise de Emissão de Harmônicas em Sistemas Elétricos Conference Paper Sep 2023 Matheus B. Arcadepani Lícia Takahashi Dos Santos Alexandre Candido Moreira Helmo Kelis Morales Paredes View Measurement Uncertainty of Harmonic Emission Indicators based on IEEE Std. 1459-2010 Conference Paper May 2020 Giovanni Artale Giuseppe Caravello Antonio Cataliotti G. Tine View Harmonic Interaction Effects on Power Quality and Electrical Energy Measurement System Conference Paper Nov 2019 Roberto Perillo Barbosa da Silva Rodolfo Quadros Hamid Reza Shaker Luiz Carlos Pereira da Silva This paper presents an investigation of the harmonic interaction effects on power quality (PQ) and electrical energy measurement and billing system. In this study two nonlinear loads, a programmable AC source and two power meters are used. An oscilloscope also was used to record the voltage and current signal. The results show that the total harmonic current distortion (THDi) level is a function of the harmonic interaction between each harmonic order of each load. The electrical quantities are influenced as well. Therefore, the characteristics of each load will influence the results. Furthermore, the analysis through Conservative Power Theory (CPT) confirms that the low power factor (PF) of the loads is mainly due to the nonlinearity, and not the reactive power. This is an important question because the PF affects the electrical energy measurements and, consequently, the electrical energy billing. Other contribution of this paper is to show the harmonic interaction effects on electrical energy measurement and billing systems quantitatively. View Show abstract Measurement of Electric Power Quantities and Efficiency in Nonsinusoidal Conditions Conference Paper Oct 2018 Francesco Grasso Antonio Luchetta S. Manetti Stefano De Giorgis View CIGRE/CIRED JWG C4.42: Overview of common methods for assessment of harmonic contribution from customer installation Conference Paper Jun 2017 Aljaz Spelko Bostjan Blazic Igor Papic Sasa Z. Djokic View Method for determining utility and consumer harmonic contributions based on complex independent component analysis Article Nov 2015 Farzad Karimzadeh Seyed Hossein Hosseinian Saeid Esmaeili This study presents a new method for determining the harmonic contributions (HCs) of utility and consumer at the point of common coupling (PCC). The proposed method is based on a well-known statistical signal processing technique, the so-called complex independent component analysis (ICA). The complex ICA technique is applied to the harmonic voltage and current measured at the PCC to estimate the parameters of the Northon equivalent circuit model. Then, the HCs of utility and consumer are calculated through quantitative indices defined by the superposition method. The proposed method is robust against the background harmonic disturbances and works well in resonance condition. Moreover, the impact of consumer harmonic impedance is appropriately considered in calculations. The effectiveness of the proposed method is verified through the computer simulation and the real case study. View Show abstract Harmonic sources detection in power systems via nonactive power measurements according to IEEE Std. 1459–2010: Theoretical approach and experimental results Article Sep 2010 Antonio Cataliotti Valentina Cosentino In this paper an enhanced decision-making strategy is presented for the detection of harmonic sources in power systems able to detect also which is the prevailing nature of the disturbance (nonlinearity or unbalance). It makes use of some simple indices, which are evaluated by means of the measurements of some nonactive power quantities, proposed by the authors and derived from the approach of the IEEE Std. 1459-2010. The decision-making rules for the proposed strategy are presented and discussed by means of simulation and experimental tests. The results obtained are presented, showing the effectiveness of the proposed strategy for the detection of the dominant harmonic source upstream or downstream the metering section. View Show abstract On Techniques for the Localization of Multiple Distortion Sources in Three-phase Networks: Time-Domain Verification Article Sep 2001 J. Rens Piet Hermanus Swart Frequency domain network modelling techniques have shown that the localisation of steady-state distortion sources by monitoring the direction of active power in the harmonic components can not be accomplished through continuous single-point measurements in the presence of multiple distortion sources. Because frequency domain models of line commutated AC/DC converters could so easily be susceptible to error, the same analysis in the time domain will enhance the credibility of the findings. A time-domain approach has therefore been adopted by generating the equivalent power system data through repeated ATP (alternative transients programme) simulations and interpreting the resulting trends by means of a Mathcad package. The results obtained confirm the previous findings. View Show abstract Disturbing Load Detection in Three-Wire and Four-Wire Systems Based on Novel Nonactive Powers From IEEE 1459-2000 Article Jun 2010 Antonio Cataliotti Valentina Cosentino This paper presents an enhanced approach for the detection of disturbing loads in distorted and/or unbalanced three-phase power systems, both three-wire and four-wire. It is based on the IEEE Std. 1459-2000 effective apparent power resolution, and it requires only the separation of the fundamental components of voltage and current from the remaining harmonic content. In this paper, the formulation of the novel approach is given, and its effectiveness is discussed by means of several simulation tests, which were carried out on a simple test system. View Show abstract A New Measurement Method for the Detection of Harmonic Sources in Power Systems Based on the Approach of the IEEE Std. 1459–2000 Article Feb 2010 Antonio Cataliotti Valentina Cosentino In this paper, a novel single-point strategy is proposed for the detection of the harmonic sources in power systems. The method is an improvement of a strategy previously proposed by the authors; it is developed from the IEEE Std 1459-2000 approach and it can be entirely implemented in the time domain, thus simplifying the measurement system. The effectiveness of the proposed method is evaluated by introducing simple decision-making rules and by taking the presence of the measurement transducers into account. Several simulations and experimental tests are presented, showing that the enhanced strategy is able to give an useful information on the detection of the dominant harmonic source upstream or downstream the metering section. View Show abstract Harmonic Contributions Evaluation With the Harmonic Current Vector Method Article Feb 2008 T. Pfajfar Bostjan Blazic Igor Papic A method to evaluate harmonic contributions at the point of common coupling is presented in this paper. The proposed approach is based on the harmonic current vector method where reference impedances are introduced. Resistance defined at fundamental frequency with measurements is used as the customer-side reference impedance. The presented method allows calculation of harmonic contributions without knowing the actual customer impedances. It also enables better evaluation of the customer and utility harmonic contributions in resonance conditions. The method is verified through a simulation study and with extensive field measurements. Long-term measurements are proposed for determination of harmonic contributions. View Show abstract A Method for Determining Customer and Utility Harmonic Contributions at the Point of Common Coupling Article May 2000 Wilsun Xu Yilu Liu A new method is proposed in this paper to determine the harmonic contributions of a customer at the point of common coupling. The method can quantify customer and utility responsibilities for limit violations caused by either harmonic source changes or harmonic impedance changes. It can be implemented in current power quality monitors and digital revenue meters. The method makes it possible to develop fair and consistent billing schemes for harmonic distortion control View Show abstract Closure to discussion of "Evaluation of single-point measurements method for harmonic pollution cost allocation" Article Feb 2000 Eric J. Davis Alexander E. Emanuel D.J. Pileggi This paper reports on the reliability of single-point measurements for harmonic pollution cost allocation. Simulated results give the correlation between the cost and four physical electrical quantities that can be measured at the end-user bus: harmonic active power; harmonic apparent power; total harmonic current squared; and nonfundamental apparent power squared. The nonfundamental apparent power squared produced the best correlation, with errors less than 5% View Show abstract On Separating Customer and Supply Side Harmonic Contributions Article May 1996 Krishnaswamy Srinivasan In this paper, we propose an approach to quantifying the distortion caused by a single customer, when there are many customers in the network. We have described a method for isolating the contributions to waveform distortion from a customer side and the supply side. The voltage and current measurements at a single point are sufficient. This approach will be useful in arriving at equitable ways of settling customer complaints, sharing the cost of waveform distortion through rate structures, penalties, etc. The principle is illustrated using six example loads. The method proposed can be implemented in any power quality measurement device, which monitors the voltage as well as the current harmonics simultaneously View Show abstract Show more Recommended publications Discover more Article Full-text available Improvement of Induction Machine Performance using Power Factor Correction June 2011 I. Daut Mohd Irwan Yusoff Muhammad Irwanto [...] C. B. Risnidar View full-text Article Energetical Factors in Power Systems with Nonlinear Loads May 1977 · Electrical Engineering A. E. Emanuel The classical definitions for the energetical determining factors in power systems are reviewed and a new model for the apparent power is suggested. — Based on the volt-ampere characteristic of the nonlinear load, an equivalent circuit, containing linear elements, can be determined. In this way, load-flow and harmonic compensation studies can be simplified.In der Arbeit werden zunchst die ... [Show full abstract] klassischen Definitionen errtert; darauf basierend wird ein neues Modell fr die Scheinleistung vorgeschlagen, das zu einer Ersatzschaltung mit linearen Elementen fhrt. Damit lassen sich z. B. Lastfluberechnungen und Kompensationsstudien vereinfachen. Read more Article The parameter analysis of the large signal mathematical model of Gunn devices operating in the harmo... January 1993 · International Journal of Infrared and Millimeter Waves Ling Chen Yun-Yi Wang This paper analyses in detail the parameters of the large signal mathematical model of a Gunn device operating in the harmonic mode using the frequency-domain harmonic balance method and indicates the variation trend of the output power and the operating frequency of a harmonic oscillator under the influence of the model parameters of the device. The analysis results offer the theoretical basis ... [Show full abstract] for the production of Gunn devices and for the design of harmonic oscillators Read more Article AC Current Harmonics Reduction and Improve Power Factor for Single Phase Controlled Rectifier Using... November 2011 Basil Mohammed Saied Hussein Ibzar Zynal Aasef Saleh The AC electrical distribution system operates with limited voltage and frequency values. Voltages and currents in the electric power grid are not sinusoidal as a result of the increased use of nonlinear loads and loads connected by power electronics switching devices. For obtaining DC current and voltage, controlled and uncontrolled single-phase and three-phase rectifiers are utilised. This ... [Show full abstract] impacts the shape of the supply current, which contains numerous harmonics. Consequently, both the quality and quantity of the obtained power are declined. Consequently, the form of the current waveform has garnered the interest of numerous research groups. The line injection by current harmonics is one of the most important and effective techniques used for such treatment. The purpose of this research is to reduce input current harmonics and improve the power factor of single-phase controlled and uncontrolled rectifiers. Typically, this is achieved by employing an injection technique with reversible polarity and connecting a passive resonance circuit to the rectifier's input and output. Total harmonic distortion (THD) is mitigated by the passive resonance circuit, which generates second harmonic current that circulates between the AC and DC sides of the rectifier. In this research, the ratio of THD has been lowered from 50% to 10% while the power factor is nearly enhanced to unity at different trigger angle (α) values. Read more Discover the world's research Join ResearchGate to find the people and research you need to help your work. Join for free ResearchGate iOS App Get it from the App Store now. 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2587	https://www.nagwa.com/en/explainers/890151902620/	Lesson Explainer: Using Determinants to Calculate Areas Mathematics • First Year of Secondary School Join Nagwa Classes Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher! In this explainer, we will learn how to use determinants to calculate areas of triangles and parallelograms given the coordinates of their vertices. There are a lot of useful properties of matrices we can use to solve problems. We can use the determinant of matrices to help us calculate the area of a polygon given its vertices. To do this, we will start with the formula for the area of a triangle using determinants. Theorem: Area of a Triangle Using Determinants The area of a triangle with vertices (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) is given by areadet=12\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. We take the absolute value of this determinant to ensure the area is nonnegative. There are other methods of finding the area of a triangle. For example, we know that the area of a triangle is given by half the length of the base times the height. However, this formula requires us to know these lengths rather than just the coordinates of the vertices. Let’s see an example of using this formula to evaluate the area of a triangle from the coordinates of its vertices. Example 1: Finding the Area of a Triangle on the Cartesian Coordinate Using Determinants Find the area of the triangle below using determinants. Answer In this question, we could find the area of this triangle in many different ways. For example, we could use geometry. However, we are tasked with calculating the area of a triangle by using determinants. To do this, we will need to use the fact that the area of a triangle with vertices (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) is given by areadet=12\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. So, we need to find the vertices of our triangle; we can do this using our sketch. It is worth pointing out that the order we label the vertices in does not matter, since this would only result in switching the rows of our matrix around, which only changes the sign of the determinant. Therefore, the area of our triangle is given by areadet=12\|\|\|\|0514513−41\|\|\|\|. Expanding over the first column, we get areadetdetdet=12\|\|0×51−41−4×51−41+3×5151\|\|=12\|−4(5×1−(−4)×1)+3(5×1−5×1)\|=12\|−36\|=18, giving us that the area of our triangle is 18 square units. We can check our answer by calculating the area of this triangle using a different method. For example, the area of a triangle is half the length of the base times the height, and we can find both of the values from our sketch. Taking the horizontal side as the base, we get that the length of the base is 4 and the height of the triangle is 9. So, we can use these to calculate the area of the triangle: areabaseheight=12××=12×4×9=18. This confirms our answer that the area of our triangle is 18 square units. We can use the formula for the area of a triangle by using determinants to find the possible coordinates of a vertex of a triangle with a given area, as we will see in our next example. Example 2: Finding Information about the Vertices of a Triangle given Its Area Fill in the blank: If the area of a triangle whose vertices are (ℎ,0), (6,0), and (0,3) is 9 square units, then ℎ=. Answer In this question, we are given the area of a triangle and the coordinates of two of its vertices, and we need to use this to find the coordinates of the third vertex. We could find an expression for the area of our triangle by using half the length of the base times the height. This would then give us an equation we could solve for ℎ. However, let us work out this example by using determinants. We can find the area of the triangle by using the coordinates of its vertices. A triangle with vertices (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) has an area given by the following: areadet=12\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. Substituting in the coordinates of the vertices of this triangle gives us areadet=12\|\|\|\|ℎ01601031\|\|\|\|. This area is equal to 9, and we can evaluate the determinant by expanding over the second column: 9=12\|\|\|\|ℎ01601031\|\|\|\|=12\|\|−0×6101+0×ℎ101−3×ℎ161\|\|=12\|−3(ℎ−6)\|=32\|ℎ−6\|.detdetdetdet Therefore, rearranging this equation gives 6=\|ℎ−6\|. This gives us two options, either 6=ℎ−6 or −6=ℎ−6. We can solve both of these equations to get ℎ=0 or ℎ=12, which is option B. Thus far, we have discussed finding the area of triangles by using determinants. It is possible to extend this idea to polygons with any number of sides. We begin by finding a formula for the area of a parallelogram. There are two different ways we can do this. The first way we can do this is by viewing the parallelogram as two congruent triangles. If we choose any three vertices of the parallelogram, we have a triangle. It does not matter which three vertices we choose, we split he parallelogram into two triangles. The side lengths of each of the triangles is the same, so they are congruent and have the same area. We can then find the area of this triangle using determinants: areaparallelogramareatriangledetdet()=2()=2×12\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|=\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. We can summarize this as follows. Formula: Area of a Parallelogram Using Determinants The area of a parallelogram with any three vertices at (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) is given by areadet=\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. There is a second way we can find the area of a parallelogram by using determinants. Since translating a parallelogram does not alter its area, we can translate any parallelogram to have one of its vertices at the origin. Thus, we only need to determine the area of such a parallelogram. Consider a parallelogram with vertices (0,0), (𝑎,𝑏), (𝑐,𝑑), and (𝑒,𝑓), as shown in the following figure. We can find the area of this parallelogram by splitting it into triangles in two different ways, and both methods will give the same area of the parallelogram. For example, we can split the parallelogram in half along the line segment between (𝑎,𝑏) and (𝑐,𝑑). We can see that the diagonal line splits the parallelogram into two triangles. These two triangles are congruent because they share the same side lengths. Hence, the area of the parallelogram is twice the area of the triangle pictured below. We can find the area of this triangle by using determinants: areatriangledet()=12\|\|\|\|001𝑎𝑏1𝑐𝑑1\|\|\|\|. Expanding over the first row, we get areatriangledetdetdetdet()=12\|\|\|0×𝑏1𝑑1−0×𝑎1𝑐1+1×𝑎𝑏𝑐𝑑\|\|\|=12\|\|\|𝑎𝑏𝑐𝑑\|\|\|. Since the area of the parallelogram is twice this value, we have areaparallelogramdet()=\|\|\|𝑎𝑏𝑐𝑑\|\|\|. We could also have split the parallelogram along the line segment between the origin and (𝑒,𝑓) as shown below. Once again, this splits the triangle into two congruent triangles, and we can calculate the area of one of these triangles as 12\|\|\|\|001𝑒𝑓1𝑐𝑑1\|\|\|\|=12\|\|\|𝑒𝑓𝑐𝑑\|\|\|.detdet The area of the parallelogram is twice this value: areaparallelogramdet()=\|\|\|𝑒𝑓𝑐𝑑\|\|\|. In either case, the area of the parallelogram is the absolute value of the determinant of the 2×2 matrix with the rows as the coordinates of any two of its vertices not at the origin. We summarize this result as follows. Theorem: Area of a Parallelogram If a parallelogram has one vertex at the origin and two of its other vertices at (𝑎,𝑏) and (𝑐,𝑑), then its area is given by areadet=\|\|\|𝑎𝑏𝑐𝑑\|\|\|. Let’s see an example of how we can apply this formula to determine the area of a parallelogram from the coordinates of its vertices. Example 3: Computing the Area of a Parallelogram Using Matrices Use determinants to calculate the area of the parallelogram with vertices (1,1), (−4,5), (−2,8), and (3,4). Answer Let’s start by recalling how we find the area of a parallelogram by using determinants. The area of a parallelogram with any three vertices at (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) is given by areadet=\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. We can choose any three of the given vertices to calculate the area of this parallelogram. For example, if we choose the first three points, then areadet=\|\|\|\|111−451−281\|\|\|\|. Expanding over the first row gives us areadetdetdetsquareunits=\|\|1×5181−1×−41−21+1×−45−28\|\|=\|(5−8)−(−4+2)+(−32+10)\|=23. Therefore, the area of this parallelogram is 23 square units. We could also use the fact that if a parallelogram has one vertex at the origin and any two of its other vertices at (𝑎,𝑏) and (𝑐,𝑑), then its area is given by areadet=\|\|\|𝑎𝑏𝑐𝑑\|\|\|. To use this formula, we need to translate the parallelogram so that one of its vertices is at the origin. Since one of the vertices is the point (1,1), we will do this by translating the parallelogram one unit left and one unit down. This gives us the following coordinates for its vertices: (1−1,1−1)=(0,0),(−4−1,5−1)=(−5,4),(−2−1,8−1)=(−3,7),(3−1,4−1)=(2,3). We can actually use any two of the vertices not at the origin to determine the area of this parallelogram. Hence, areadet=\|\|−54−37\|\|=\|−35−(−12)\|=\|−23\|=23. Therefore, the area of this parallelogram is 23 square units. We were able to find the area of a parallelogram by splitting it into two congruent triangles. Similarly, we can find the area of a triangle by considering it as half of a parallelogram, as we will see in our next example. Example 4: Computing the Area of a Triangle Using Matrices Use determinants to work out the area of the triangle with vertices (2,−2), (4,−2), and (0,2) by viewing the triangle as half of a parallelogram. Answer First, we want to construct our parallelogram by using two of the same triangles given to us in the question. This means there will be three different ways to create this parallelogram, since we can combine the two triangles on any side. We can see this in the following three diagrams. All three of these parallelograms have the same area since they are formed by the same two congruent triangles. However, we do not need the coordinates of the fourth point to find the area of a parallelogram by using determinants. Recall that if a parallelogram has one vertex at the origin and two other vertices at (𝑎,𝑏) and (𝑐,𝑑), then its area is given by areadet=\|\|\|𝑎𝑏𝑐𝑑\|\|\|. We can use this to determine the area of the parallelogram by translating the shape so that one of its vertices lies at the origin. We translate the point (0,2) to the origin by translating each of the vertices down two units; this gives us (0,2−2)=(0,0),(2,−2−2)=(2,−4),(4,−2−2)=(4,−4). We use the coordinates of the latter two points to find the area of the parallelogram: areaparallelogramdet()=\|\|2−44−4\|\|=\|−8+16\|=8. Finally, we remember that the area of our triangle is half of this value, giving us that the area of the triangle with vertices at (2,−2), (4,−2), and (0,2) is 4 square units. If we can calculate the area of a triangle using determinants, then we can calculate the area of any polygon by splitting it into triangles (called triangulation). Let’s see an example where we are tasked with calculating the area of a quadrilateral by using determinants. Example 5: Computing the Area of a Quadrilateral Using Determinants of Matrices Consider the quadrilateral with vertices 𝐴(1,3), 𝐵(4,2), 𝐶(4.5,5), and 𝐷(2,6). By breaking it into two triangles as shown, calculate the area of this quadrilateral using determinants. Answer We want to find the area of this quadrilateral by splitting it up into the triangles as shown. This means we need to calculate the area of these two triangles by using determinants and then add the results together. We have two options for finding the area of a triangle by using determinants: We could treat the triangles as half a parallelogram and use the determinant of a 2×2 matrix to find the area of this parallelogram, or we could use our formula for the area of a triangle by using the determinant of a 3×3 matrix. Since we have a diagram with the vertices given, we will use the formula for finding the areas of the triangles directly. Let’s start with triangle 𝐴. We can see from the diagram that 𝐴(1,3), 𝐵(4,2), and 𝐶(4.5,5). We recall that the area of a triangle with vertices (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) is given by areadet=12\|\|\|\|𝑥𝑦1𝑥𝑦1𝑥𝑦1\|\|\|\|. So, we can find the area of this triangle by using our determinant formula: areadet(𝐴𝐵𝐶)=12\|\|\|\|1314214.551\|\|\|\|. We expand this determinant along the first column to get areadetdetdetsquareunits(𝐴𝐵𝐶)=12\|\|1×2151−4×3151+4.5×3121\|\|=12\|1(2−5)−4(3−5)+4.5(3−2)\|=12\|−3+8+4.5\|=4.75. Similarly, the area of triangle 𝐴𝐶𝐷 is given by areadetdetdetdetsquareunits(𝐴𝐶𝐷)=12\|\|\|\|1314.551261\|\|\|\|=12\|\|1×5161−4.5×3161+2×3151\|\|=12\|1(5−6)−4.5(3−6)+2(3−5)\|=12\|−1+13.5−4\|=4.25. Summing the areas of these two triangles together, we see that the area of the quadrilateral is 9 square units. There is another useful property that these formulae give us. Since det𝑥𝑦1𝑥𝑦1𝑥𝑦1 tells us the signed area of a parallelogram with three vertices at (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦), if this determinant is 0, the triangle with these points as vertices must also have zero area. The area of this triangle can only be zero if the points are not distinct or if the points all lie on the same line (i.e., they are collinear). Theorem: Test for Collinear Points If we have three distinct points (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦), where det𝑥𝑦1𝑥𝑦1𝑥𝑦1=0, then the points are collinear. Let’s see an example of how to apply this. Example 6: Determining If a Set of Points Are Collinear or Not Using Determinants By using determinants, determine which of the following sets of points are collinear. Answer We first recall that three distinct points (𝑥,𝑦), (𝑥,𝑦), and (𝑥,𝑦) are collinear if det𝑥𝑦1𝑥𝑦1𝑥𝑦1=0. We note that each given triplet of points is a set of three distinct points. So, we can calculate the determinant of this matrix for each given triplet of points to determine their collinearity. We compute the determinants of all four matrices by expanding over the first row. Option A would be det−641−8413101=−6(4−10)−4(−8−3)+(−80−12)=−12. Since this is nonzero, the area of the triangle with these points as vertices in also nonzero. Hence, these points are not collinear. Option B would be det−10−41−8−21−511=−10(−2−1)+4(−8+5)+(−8−10)=0. Since this is equal to zero, the area of the triangle with these points as vertices is 0. Hence, these points must be collinear. Option C would be det−3618−71−3−81=−3(−7+8)−6(8+3)+(−64−21)=−154. Since, this is nonzero, the area of the triangle with these points as vertices in also nonzero. Hence, these points are not collinear. Option D would be det−10−61−2110−91=−10(1+9)+6(−2−0)+(18−0)=−94. Since, this is nonzero, the area of the triangle with these points as vertices in also nonzero. Hence, these points are not collinear. Hence, the points 𝐴(−10,−4), 𝐵(−8,−2), and 𝐶(−5,1) are collinear, which is option B. Let us finish by recapping a few of the important concepts of this explainer. Key Points Lesson Menu Join Nagwa Classes Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher! Nagwa is an educational technology startup aiming to help teachers teach and students learn. Company Content Copyright © 2025 Nagwa All Rights Reserved Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy
2588	https://en.wikipedia.org/wiki/Negation	Published Time: Thu, 11 Sep 2025 13:53:47 GMT Negation - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 Definition 2 NotationToggle Notation subsection 2.1 Precedence 3 PropertiesToggle Properties subsection 3.1 Double negation 3.2 Distributivity 3.3 Linearity 3.4 Self dual 3.5 Negations of quantifiers 4 Rules of inference 5 Programming language and ordinary languageToggle Programming language and ordinary language subsection 5.1 Usage in colloquial language 6 Kripke semantics 7 See also 8 References 9 Further reading 10 External links [x] Toggle the table of contents Negation [x] 51 languages العربية Azərbaycanca Български Català Čeština Dansk Deutsch Eesti Ελληνικά Emiliàn e rumagnòl Español Esperanto Euskara فارسی Français Galego 한국어 Հայերեն हिन्दी Hrvatski Ido Bahasa Indonesia Italiano עברית Қазақша Latina Magyar Македонски Bahasa Melayu Nederlands 日本語 Norsk bokmål Piemontèis Polski Português Русский Shqip Simple English Slovenčina Slovenščina کوردی Српски / srpski Srpskohrvatski / српскохрватски Svenska Tagalog ไทย Тоҷикӣ Türkçe Українська 粵語中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia Logical operation For negation in linguistics, see Affirmation and negation. For other uses, see Negation (disambiguation). Negation\| NOT \| \| \| \| Definition \| ¬x{\displaystyle \lnot {x}} \| \| Truth table \| (01){\displaystyle (01)} \| \| Logic gate \| \| \| Normal forms \| \| Disjunctive \| ¬x{\displaystyle \lnot {x}} \| \| Conjunctive \| ¬x{\displaystyle \lnot {x}} \| \| Zhegalkin polynomial \| 1⊕x{\displaystyle 1\oplus x} \| \| Post's lattices \| \| 0-preserving \| no \| \| 1-preserving \| no \| \| Monotone \| no \| \| Affine \| yes \| \| Self-dual \| yes \| \| v t e \| \| Logical connectives \| \| NOT¬A,−A,A¯,∼A{\displaystyle \neg A,-A,{\overline {A}},\sim A} ANDA∧B,A⋅B,A B,A&B,A&&B{\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B} NANDA∧¯B,A↑B,A∣B,A⋅B¯{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}} ORA∨B,A+B,A∣B,A∥B{\displaystyle A\lor B,A+B,A\mid B,A\parallel B} NORA∨¯B,A↓B,A+B¯{\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}} XNORA⊙B,A∨¯B¯{\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}} └ equivalentA≡B,A⇔B,A⇋B{\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B} XORA∨_ B,A⊕B{\displaystyle A{\underline {\lor }}B,A\oplus B} └ nonequivalent A≢B,A⇎B,A↮B{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B} impliesA⇒B,A⊃B,A→B{\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B} nonimplication(NIMPLY)A⇏B,A⊅B,A↛B{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B} converseA⇐B,A⊂B,A←B{\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B} converse nonimplicationA⇍B,A⊄B,A↚B{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B} \| \| Related concepts \| \| Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic) \| \| Applications \| \| Digital logic Programming languages Mathematical logic Philosophy of logic \| \| Category \| In logic, negation, also called the logical not or logical complement, is an operation that takes a propositionP{\displaystyle P} to another proposition "not P{\displaystyle P}", written ¬P{\displaystyle \neg P}, ∼P{\displaystyle {\mathord {\sim }}P}, P′{\displaystyle P^{\prime }} or P¯{\displaystyle {\overline {P}}}.[citation needed] It is interpreted intuitively as being true when P{\displaystyle P} is false, and false when P{\displaystyle P} is true. For example, if P{\displaystyle P} is "Spot runs", then "not P{\displaystyle P}" is "Spot does not run". An operand of a negation is called a negand or negatum. Negation is a unarylogical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P{\displaystyle P} is the proposition whose proofs are the refutations of P{\displaystyle P}. Definition [edit] Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false, and a value of false when its operand is true. Thus if statement P{\displaystyle P} is true, then ¬P{\displaystyle \neg P} (pronounced "not P") would then be false; and conversely, if ¬P{\displaystyle \neg P} is true, then P{\displaystyle P} would be false. The truth table of ¬P{\displaystyle \neg P} is as follows: P{\displaystyle P}¬P{\displaystyle \neg P} True False False True Negation can be defined in terms of other logical operations. For example, ¬P{\displaystyle \neg P} can be defined as P→⊥{\displaystyle P\rightarrow \bot } (where →{\displaystyle \rightarrow } is logical consequence and ⊥{\displaystyle \bot } is absolute falsehood). Conversely, one can define ⊥{\displaystyle \bot } as Q∧¬Q{\displaystyle Q\land \neg Q} for any proposition Q (where ∧{\displaystyle \land } is logical conjunction). The idea here is that any contradiction is false, and while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR. Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic. Notation [edit] The negation of a proposition p is notated in different ways, in various contexts of discussion and fields of application. The following table documents some of these variants: \| Notation \| Plain text \| Vocalization \| --- \| ¬p{\displaystyle \neg p} \| ¬p , 7p \| Not p \| \| ∼p{\displaystyle {\mathord {\sim }}p} \| ~p \| Not p \| \| −p{\displaystyle -p} \| -p \| Not p \| \| N p{\displaystyle Np} \| \| En p \| \| p′{\displaystyle p'} \| p' \| p prime, p complement \| \| p¯{\displaystyle {\overline {p}}} \| ̅p \| p bar, Bar p \| \| !p{\displaystyle !p} \| !p \| Bang p Not p \| The notation N p{\displaystyle Np} is Polish notation. In set theory, ∖{\displaystyle \setminus } is also used to indicate 'not in the set of': U∖A{\displaystyle U\setminus A} is the set of all members of U that are not members of A. Regardless how it is notated or symbolized, the negation ¬P{\displaystyle \neg P} can be read as "it is not the case that P", "not that P", or usually more simply as "not P". Precedence [edit] See also: Logical connective §Order of precedence As a way of reducing the number of necessary parentheses, one may introduce precedence rules: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, P∨Q∧¬R→S{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S} is short for (P∨(Q∧(¬R)))→S.{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.} Here is a table that shows a commonly used precedence of logical operators. \| Operator \| Precedence \| --- \| \| ¬{\displaystyle \neg } \| 1 \| \| ∧{\displaystyle \land } \| 2 \| \| ∨{\displaystyle \lor } \| 3 \| \| →{\displaystyle \to } \| 4 \| \| ↔{\displaystyle \leftrightarrow } \| 5 \| Properties [edit] Double negation [edit] Within a system of classical logic, double negation, that is, the negation of the negation of a proposition P{\displaystyle P}, is logically equivalent to P{\displaystyle P}. Expressed in symbolic terms, ¬¬P≡P{\displaystyle \neg \neg P\equiv P}. In intuitionistic logic, a proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two. However, in intuitionistic logic, the weaker equivalence ¬¬¬P≡¬P{\displaystyle \neg \neg \neg P\equiv \neg P} does hold. This is because in intuitionistic logic, ¬P{\displaystyle \neg P} is just a shorthand for P→⊥{\displaystyle P\rightarrow \bot }, and we also have P→¬¬P{\displaystyle P\rightarrow \neg \neg P}. Composing that last implication with triple negation ¬¬P→⊥{\displaystyle \neg \neg P\rightarrow \bot } implies that P→⊥{\displaystyle P\rightarrow \bot } . As a result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko's theorem. Distributivity [edit] De Morgan's laws provide a way of distributing negation over disjunction and conjunction: ¬(P∨Q)≡(¬P∧¬Q){\displaystyle \neg (P\lor Q)\equiv (\neg P\land \neg Q)}, and¬(P∧Q)≡(¬P∨¬Q){\displaystyle \neg (P\land Q)\equiv (\neg P\lor \neg Q)}. Linearity [edit] Let ⊕{\displaystyle \oplus } denote the logical xor operation. In Boolean algebra, a linear function is one such that: If there exists a 0,a 1,…,a n∈{0,1}{\displaystyle a_{0},a_{1},\dots ,a_{n}\in {0,1}}, f(b 1,b 2,…,b n)=a 0⊕(a 1∧b 1)⊕⋯⊕(a n∧b n){\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})}, for all b 1,b 2,…,b n∈{0,1}{\displaystyle b_{1},b_{2},\dots ,b_{n}\in {0,1}}. Another way to express this is that each variable always makes a difference in the truth-value of the operation, or it never makes a difference. Negation is a linear logical operator. Self dual [edit] In Boolean algebra, a self dual function is a function such that: f(a 1,…,a n)=¬f(¬a 1,…,¬a n){\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})} for all a 1,…,a n∈{0,1}{\displaystyle a_{1},\dots ,a_{n}\in {0,1}}. Negation is a self dual logical operator. Negations of quantifiers [edit] In first-order logic, there are two quantifiers, one is the universal quantifier ∀{\displaystyle \forall } (means "for all") and the other is the existential quantifier ∃{\displaystyle \exists } (means "there exists"). The negation of one quantifier is the other quantifier (¬∀x P(x)≡∃x¬P(x){\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)} and ¬∃x P(x)≡∀x¬P(x){\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}). For example, with the predicate P as "x is mortal" and the domain of x as the collection of all humans, ∀x P(x){\displaystyle \forall xP(x)} means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is ¬∀x P(x)≡∃x¬P(x){\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}, meaning "there exists a person x in all humans who is not mortal", or "there exists someone who lives forever". Rules of inference [edit] See also: Double negation There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of P{\displaystyle P} to both Q{\displaystyle Q} and ¬Q{\displaystyle \neg Q}, infer ¬P{\displaystyle \neg P}; this rule also being called reductio ad absurdum), negation elimination (from P{\displaystyle P} and ¬P{\displaystyle \neg P} infer Q{\displaystyle Q}; this rule also being called ex falso quodlibet), and double negation elimination (from ¬¬P{\displaystyle \neg \neg P} infer P{\displaystyle P}). One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination. Negation introduction states that if an absurdity can be drawn as conclusion from P{\displaystyle P} then P{\displaystyle P} must not be the case (i.e. P{\displaystyle P} is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign ⊥{\displaystyle \bot }. In this case the rule says that from P{\displaystyle P} and ¬P{\displaystyle \neg P} follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity. Typically the intuitionistic negation ¬P{\displaystyle \neg P} of P{\displaystyle P} is defined as P→⊥{\displaystyle P\rightarrow \bot }. Then negation introduction and elimination are just special cases of implication introduction (conditional proof) and elimination (modus ponens). In this case one must also add as a primitive rule ex falso quodlibet. Programming language and ordinary language [edit] As in mathematics, negation is used in computer science to construct logical statements. if (!(r == t)) { /...statements executed when r does NOT equal t.../ } The exclamation mark "!" signifies logical NOT in B, C, and languages with a C-inspired syntax such as C++, Java, JavaScript, Perl, and PHP. "NOT" is the operator used in ALGOL 60, BASIC, and languages with an ALGOL- or BASIC-inspired syntax such as Pascal, Ada, Eiffel and Seed7. Some languages (C++, Perl, etc.) provide more than one operator for negation. A few languages like PL/I and Ratfor use ¬ for negation. Most modern languages allow the above statement to be shortened from if (!(r == t)) to if (r != t), which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs. In computer science there is also bitwise negation. This takes the value given and switches all the binary 1s to 0s and 0s to 1s. This is often used to create ones' complement (or "~" in C or C++) and two's complement (just simplified to "-" or the negative sign, as this is equivalent to taking the arithmetic negation of the number). To get the absolute (positive equivalent) value of a given integer the following would work as the "-" changes it from negative to positive (it is negative because "x < 0" yields true) unsigned int abs(int x) { if (x < 0) return -x; else return x; } To demonstrate logical negation: unsigned int abs(int x) { if (!(x < 0)) return x; else return -x; } Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e. will have identical results for any input (depending on the compiler used, the actual instructions performed by the computer may differ). In C (and some other languages descended from C), double negation (!!x) is used as an idiom to convert x to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not special in this regard, it is sometimes important to ensure that a canonical value is used, for example for printing or if the number is subsequently used for arithmetic operations. Usage in colloquial language [edit] "!vote" redirects here. For use of !vote in Wikipedia, see Wikipedia:Polling is not a substitute for discussion §Not-votes. The convention of using ! to signify negation occasionally surfaces in colloquial language, as computer-related slang for not. For example, the phrase !clue is used as a synonym for "no-clue" or "clueless". Another example is the expression !vote which means "not a vote". In this context, the exclamation mark is used at Wikipedia to survey opinions while negating "majority rule", in order "to have a consensus-building discussion, where the proper course is determined by the strength of the respective arguments." Kripke semantics [edit] In Kripke semantics where the semantic values of formulae are sets of possible worlds, negation can be taken to mean set-theoretic complementation[citation needed] (see also possible world semantics for more). See also [edit] Affirmation and negation (grammatical polarity) Ampheck Apophasis Binary opposition Bitwise NOT Contraposition Cyclic negation Negation as failure NOT gate Plato's beard Square of opposition References [edit] ^Virtually all Turkish high school math textbooks use p' for negation due to the books handed out by the Ministry of National Education representing it as p'. ^Weisstein, Eric W. "Negation". mathworld.wolfram.com. Retrieved 2 September 2020. ^"Logic and Mathematical Statements - Worked Examples". www.math.toronto.edu. Retrieved 2 September 2020. ^Beall, Jeffrey C. (2010). Logic: the basics (1. publ ed.). London: Routledge. p.57. ISBN978-0-203-85155-5. ^Used as makeshift in early typewriter publications, e.g. Richard E. Ladner (January 1975). "The circuit value problem is log space complete for P". ACM SIGACT News. 7 (101): 18–20. doi:10.1145/990518.990519. ^O'Donnell, John; Hall, Cordelia; Page, Rex (2007), Discrete Mathematics Using a Computer, Springer, p.120, ISBN9781846285981. ^Egan, David. "Double Negation Operator Convert to Boolean in C". Dev Notes. ^Raymond, Eric and Steele, Guy. The New Hacker's Dictionary, p. 18 (MIT Press 1996). ^Munat, Judith. Lexical Creativity, Texts and Context, p. 148 (John Benjamins Publishing, 2007). ^ abHarrison, Stephen. "Wikipedia's War on the Daily Mail", Slate Magazine (July 1, 2021). Further reading [edit] Gabbay, Dov, and Wansing, Heinrich, eds., 1999. What is Negation?, Kluwer. Horn, L., 2001. A Natural History of Negation, University of Chicago Press. G. H. von Wright, 1953–59, "On the Logic of Negation", Commentationes Physico-Mathematicae 22. Wansing, Heinrich, 2001, "Negation", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, Blackwell. Tettamanti, Marco; Manenti, Rosa; Della Rosa, Pasquale A.; Falini, Andrea; Perani, Daniela; Cappa, Stefano F.; Moro, Andrea (2008). "Negation in the brain: Modulating action representation". NeuroImage. 43 (2): 358–367. doi:10.1016/j.neuroimage.2008.08.004. PMID18771737. S2CID17658822. External links [edit] Horn, Laurence R.; Wansing, Heinrich. "Negation". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. "Negation", Encyclopedia of Mathematics, EMS Press, 2001 NOT, on MathWorld Tables of Truth of composite clauses "Table of truth for a NOT clause applied to an END sentence". Archived from the original on 1 March 2000. "NOT clause of an END sentence". Archived from the original on 1 March 2000. "NOT clause of an OR sentence". Archived from the original on 17 January 2000. "NOT clause of an IF...THEN period". Archived from the original on 1 March 2000. \| v t e Common logical connectives \| \| Tautology/True⊤{\displaystyle \top } \| \| \| Alternative denial(NAND gate)∧¯{\displaystyle {\overline {\wedge }}} Converse implication⇐{\displaystyle \Leftarrow } Implication(IMPLY gate)⇒{\displaystyle \Rightarrow } Disjunction(OR gate)∨{\displaystyle \lor } \| \| Negation(NOT gate)¬{\displaystyle \neg } Exclusive or(XOR gate)⊕{\displaystyle \oplus } Biconditional(XNOR gate)⊙{\displaystyle \odot } Statement(Digital buffer) \| \| Joint denial(NOR gate)∨¯{\displaystyle {\overline {\vee }}} Nonimplication(NIMPLY gate)⇏{\displaystyle \nRightarrow } Converse nonimplication⇍{\displaystyle \nLeftarrow } Conjunction(AND gate)∧{\displaystyle \land } \| \| Contradiction/False⊥{\displaystyle \bot } \| \| Philosophy portal \| \| v t e Common logical symbols \| \| ∧or& and∨ or¬or~ not→ implies⊃ implies, superset↔or≡ iff\| nand∀ universal quantification∃ existential quantification⊤ true, tautology⊥ false, contradiction⊢ entails, proves⊨ entails, therefore∴ therefore∵ because \| \| Philosophy portal Mathematics portal \| \| v t e Formal semantics (natural language) \| \| Central concepts \| Compositionality Denotation Entailment Extension Generalized quantifier Intension Logical form Presupposition Proposition Reference Scope Speech act Syntax–semantics interface Truth conditions \| \| Topics \| \| Areas \| Anaphora Ambiguity Binding Conditionals Definiteness Disjunction Evidentiality Focus Indexicality Lexical semantics Modality Negation Propositional attitudes Tense–aspect–mood Quantification Vagueness \| \| Phenomena \| Antecedent-contained deletion Cataphora Coercion Conservativity Counterfactuals Crossover effects Cumulativity De dicto and de re De se Deontic modality Discourse relations Donkey anaphora Epistemic modality Exhaustivity Faultless disagreement Free choice inferences Givenness Homogeneity (linguistics) Hurford disjunction Inalienable possession Intersective modification Logophoricity Mirativity Modal subordination Opaque contexts Performatives Polarity items Privative adjectives Quantificational variability effect Responsive predicate Rising declaratives Scalar implicature Sloppy identity Subsective modification Subtrigging Telicity Temperature paradox Veridicality \| \| \| Formalism \| \| Formal systems \| Alternative semantics Categorial grammar Combinatory categorial grammar Discourse representation theory (DRT) Dynamic semantics Generative grammar Glue semantics Inquisitive semantics Intensional logic Lambda calculus Mereology Montague grammar Segmented discourse representation theory (SDRT) Situation semantics Supervaluationism Type theory TTR \| \| Concepts \| Autonomy of syntax Context set Continuation Conversational scoreboard Downward entailing Existential closure Function application Meaning postulate Monads Plural quantification Possible world Quantifier raising Quantization Question under discussion Semantic parsing Squiggle operator Strawson entailment Strict conditional Type shifter Universal grinder \| \| \| See also \| Cognitive semantics Computational semantics Distributional semantics Formal grammar Inferentialism Logic translation Linguistics wars Philosophy of language Pragmatics Semantics of logic \| \| v t e Mathematical logic \| \| General \| Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory \| \| Theorems(list) andparadoxes \| Gödel's completenessandincompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor'stheorem,paradoxanddiagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox \| \| Logics \| \| Traditional \| Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram \| \| Propositional \| Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ \| \| Predicate \| First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus \| \| \| Set theory \| Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Mapsandcardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary Set theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive \| \| Formal systems(list), languageandsyntax \| Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list Exampleaxiomatic systems(list) of true arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica \| \| Proof theory \| Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence(from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories \| \| Model theory \| Interpretation function of models Model equivalence finite saturated spectrum submodel Non-standard model of non-standard arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity \| \| Computability theory \| Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory \| \| Related \| Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline 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2589	https://ccnmtl.columbia.edu/podcasts/hs/dental/oralradiology2/2008/cysts.pdf	1 Radiographic features of cysts and benign tumors of the jaws Steven R. Singer, DDS srs2@columbia.edu 212.305.5674 Cyst A Cyst is a benign pathologic cavity filled with fluid, lined by epithelium, and surrounded by a connective tissue wall A = connective tissue wall B = epithelium Effects on adjacent structures Adapted from: White and Pharoah: Oral Radiology-principles and interpretation, page 380 Types ! Odontogenic ! Non-Odontogenic ! Pseudocysts Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic Keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Non-Odontogenic cysts ! Nasopalatine cyst ! Nasolabial cyst ! Dermoid cyst ! Cysts formerly known as “developmental cysts” 2 Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal Bone Cyst ! Mucous Retention Cyst ! Stafne Bone Cyst (aka Stafne Bone Defect) Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Radicular cyts ! Results from the stimulation of the epithelial cell rests in the PDL by the inflammatory products from the non-vital tooth ! Most common type of cysts in the jaws Radicular cyts Radicular cyts Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic Keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst 3 Residual Cyst Residual Cyst Residual Cyst Residual Cyst Residual cyst with Squamous Cell Carcinoma Residual cyst with squamous cell carcinoma 4 Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Dentigerous cyst (follicular cyst) ! Develops around the crown of an unerupted permanent or supernumerary tooth ! Second most common type of cyst in the jaws ! Asymptomatic ! Internal aspect is completely lucent except for the crown of the involved tooth ! Either resorbs or displaces the adjacent teeth ! Follicular spaces >5mm (normal 2-3 mm) should be closely followed for potential development of dentigerous cysts. Dentigerous cyst Dentigerous cyst Root Resorption Dentigerous cyst Root Resorption Dentigerous cyst 5 Dentigerous cyst Dentigerous cyst Dentigerous cyst 2 Dentigerous cyst 2 A B C Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst 6 Odontogenic Cysts ! Paradental cysts (Buccal bifurcation cysts) ! Most common in the 6- to 11-year-old age group. ! Usually associated with the mandibular first molar, occasionally the mandibular second molar. ! The associated tooth has an altered eruption pattern with buccal tilting of the crown. ! The associated tooth is vital. ! Deep periodontal pockets on the buccal aspect of the tooth. ! +/- swelling ! +/- pain or tenderness ! +/- infection. David LA, Sandor GKB, Stoneman DW, The buccal bifurcation cyst: Is non-surgical treatment an option? JCDA 64(9) 712-717 1998. Odontogenic Cysts ! Radiographic Features of the Buccal bifurcation cyst ! Fine radiopaque concave line as lower limit, producing a U-shaped radiolucent lesion that appears superimposed over the roots. ! Intact periodontal ligament space and lamina dura. ! Increased prominence of lingual cusps due to tilting. ! Apices tilted toward lingual cortex. ! Intact inferior border of mandible. ! +/- periosteal reaction on buccal surface. ! +/- bony expansion, thinning and associated swelling of the buccal cortex. ! +/- displacement of adjacent unerupted teeth David LA, Sandor GKB, Stoneman DW, The buccal bifurcation cyst: Is non-surgical treatment an option? JCDA 64(9) 712-717 1998. Buccal Bifurcation Cyst These lesions tend to resolve without intervention David LA, Sandor GKB, Stoneman DW, The buccal bifurcation cyst: Is non-surgical treatment an option? JCDA 64(9) 712-717 1998. Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Odontogenic Keratocyst (OKC) An OKC is a non-inflammatory odontogenic cyst that arises from the dental lamina. The epithelium in OKC appears to have innate growth potential similar to some benign tumors. 7 Odontogenic Keratocyst (OKC) ! First reported by Philipsen in 1956 ! Peak occurence in the 2nd and 3rd decades ! Asymptomatic, swelling on occasion ! Pain from secondary infection ! Aspiration may reveal thick yellow cheesy material (keratin) ! High recurrence rate after surgical enucleation OKC OKC I OKC I OKC I OKC II 8 OKC II Basal cell nevus-bifid rib syndrome ! Age range 5-30 years ! Abnormalities including multiple nevoid basal cell carcinomas of the skin, skeletal abnormalities (bifid ribs, agenesis and/or synostosis of ribs, kyphoscoliosis, vertebral fusion, temporopatietal bossing, etc.), CNS abnormalities (calcification of falx cerebri), eye abnormalities, multiple OKCs Multiple OKC’s Multiple OKC’s Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic Keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Lateral periodontal cyst ! Usually unicystic, it may also appear as a cluster of small cysts "botryoid odontogenic cysts ! Arise from the epithelial rests in the periodontium lateral to the root ! 50-75% develop in the mandible from lateral incisor to the premolar region ! In the maxilla, they appear between lateral incisor and canine 9 Lateral Periodontal Cyst Lateral Periodontal Cyst Lateral periodontal cyst ! Botryoid lateral periodontal cyst ! Origin from dental lamina? [From Greek botruoeid s : botrus, bunch of grapes + -oeid s, -oid.] Odontogenic Cysts ! Radicular cyst ! Residual cyst ! Dentigerous cyst ! Paradental cysts (Buccal bifurcation cysts) ! Odontogenic keratocyst (OKC) ! Basal cell nevus-bifid rib-OKC syndrome ! Lateral periodontal cyst ! Calcifying odontogenic cyst Calcifying odontogenic cyst ! Calcifying odontogenic cysts have a wide age distribution that peaks at 10 to 19 years of age, with a mean age of 36 years ! Clinically, the lesion usually appears as a slow-growing, painless swelling of the jaw. Occasionally the patient complains of pain. In some cases the expanding lesion may destroy the cortical plate, and the cystic mass may become palpable as it extends into the soft tissue. ! Aspiration often yields a viscous, granular, yellow fluid. Calcifying odontogenic cyst 10 Calcifying odontogenic cyst Case courtesy of the KAOMFR Calcifying odontogenic cyst Case courtesy of the KAOMFR Calcifying odontogenic cyst Case courtesy of the KAOMFR Calcifying odontogenic cyst Case courtesy of the KAOMFR Calcifying odontogenic cyst Case courtesy of the KAOMFR Non-Odontogenic cysts ! Nasopalatine cyst ! Nasolabial cyst ! Dermoid cyst ! Former “developmental cysts” 11 Nasopalatine Duct Cyst Nasopalatine duct cyst Courtesy of Department of Oral Surgery, Hornouchi Hospital, Saitama, Japan Nasopalatine duct cyst ! aka incisive canal cyst ! If it involves the posterior hard palate, termed median palatal cyst ! Anteriorly, may be called median anterior maxillary cyst, depending on the radiographic features Non-Odontogenic cysts ! Nasopalatine cyst ! Nasolabial cyst ! Dermoid cyst ! Former “developmental cysts” Nasolabial cysts Source of the epithelium may be embryonic nasolacrimal duct, which initially lies on the bone surface. Courtesy of Dr. Sharon Brooks Thyroglossal duct cyst 12 Pathoses formerly known as “Globulomaxillary” Cysts ! Discredited as a developmental cyst ! Most are found, upon re-examination of histopathological and radiographic evidence, to be radicular or lateral periodontal cysts. “Globulomaxillary” Cyst Image courtesy of Asahi University School of Dentistry Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal Bone Cyst ! Mucous Retention Cyst ! Stafne Bone Cyst (aka Stafne Bone Defect) Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal Bone Cyst ! Mucous Retention Cyst ! Stafne Bone Cyst (aka Stafne Bone Defect) Pseudocysts ! Simple bone cyst (Traumatic bone cyst) Simple Bone cyst 13 Simple Bone cyst Simple bone cyst associated with florid cemento-osseous dysplasia Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal Bone Cyst ! Mucous Retention Cyst ! Stafne Bone Cyst (aka Stafne Bone Defect) Aneurysmal Bone Cyst (ABC) ! The aneurysmal bone cyst (ABC) is an expansible osteolytic pseudocystic lesion that most often affects persons during their second decade of life. Albeit virtually any bone of the skeleton may be affected; ABCs are most frequent in the long tubular bones and spine. There are several reports of the occurrence of this pathological entity in the jaws and other craniofacial bones, rtins.htm Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal bone cyst ! Mucous retention cyst ! Stafne bone cyst (aka Stafne bone defect) Mucous retention cyst ! Dome shaped opacity in the floor of the maxillary sinus ! Non-epithelial lined ! Fluid filled ! Usually asymptomatic 14 Mucous retention cyst Pseudocysts ! Simple bone cyst (Traumatic bone cyst) ! Aneurysmal bone cyst ! Mucous retention cyst ! Stafne bone cyst (aka Stafne bone defect) Mandibular salivary gland depression Image courtesy of University of Athens School of Dentistry Break Time! Back to Work! Benign Tumors of the Jaws 15 Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Benign Jaw Tumors ! Non-odontogenic tumors ! Ectodermal (neurilemoma, neuroma) ! Mixed tumors (neurofibroma, neurofibromatosis) ! Mesodermal tumors (osteoma, Gardner’s syndrome, central hemangioma, A-V fistula,osteoblastoma, osteoid osteoma ! Pseudotumors: Central giant cell granuloma Effects on adjacent structures Adapted from: White and Pharoah: Oral Radiology-Principles and Interpretation, page 380 Torus palatinus Palatal and mandibular tori Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed ( ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma 16 Ameloblastoma Ameloblastoma Image courtesy of University of Athens School of Dentistry Ameloblastoma X,Y and Z Axes Image courtesy of Asahi University School of Dentistry X,Y and Z Axes Image courtesy of Asahi University School of Dentistry 17 The next step ! R/O vascular lesions/A-V malformations ! Auscultate for “bruit” ! Palpate for “thrills” ! Aspirate ! Plan for biopsy ! Advanced imaging ! CT/MR Case 1 Advanced Imaging Courtesy Nagaski University Bone Window Soft Tissue Window Case 2 Advanced Imaging Courtesy Nagaski University Bone Window Coronal CT in bone windows T1W MRI T2W MRI Advanced Imaging: Establish your diagnosis Confirm your diagnosis: Ameloblastoma OKC v. Ameloblastoma Case courtesy of the KAOMFR 18 Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma AOT ! Adenomatoid Odontogenic Tumor ! Most common location : maxillary canine and premolar region. 2:1 female to male ratio. Average age = ~16 yrs ! Tumors contain specks of calcified material ! Low recurrence rate AOT AOT Adenomatoid Odontogenic Tumor Radiographs courtesy of Akitoshi Kawamata DDS, Ph.D Department of Oral Radiology Asahi University, School of Dentistry Adenomatoid Odontogenic Tumor Radiographs courtesy of Department of Oral Radiology Okayama University, School of Dentistry 19 Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma CEOT (Pindborg Tumor) ! Behaves like ameloblastoma ! Predilection for mandible-premolar/molar area ! half of the lesions will have associated impacted or unerupted tooth ! Periphery well defined to diffuse ! Cystic lesion with numerous scattered, radiopaque foci of varying size and density giving it the appearance of “Driven Snow” ! Presence of amyloid and calcified “Liesegang Rings” Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Odontomas Complex Compound Odontoma Compound Odontomas 20 Odontomas Complex Odontoma Complex Odontoma Compound Odontoma Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Ameloblastic fibroma (Soft odontoma) 21 Benign Jaw Tumors ! Hyperplasias ( tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed ( ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Ameloblastic fibro-odontoma Benign Jaw Tumors ! Hyperplasias ( tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed ( ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Odontogenic Myxoma ! If odontogenic myxomas have a gender predilection, they slightly favor females. Although the lesion can occur at any age, more than half arise in individuals between 10 and 30 years. This tumor often is associated with a congenitally missing or unerupted tooth. It grows slowly and may or may not cause pain. It may also invade the maxillary sinus and cause exophthalmos. Recurrence rate is as high as 25%. This high rate may be explained by the lack of encapsulation of the tumor, its poorly defined boundaries, and the extension of nests or pockets of myxoid (jellylike) tumor into the trabeculae Odontogenic Myxoma Odontogenic Myxoma 22 Benign Jaw Tumors ! Hyperplasias (tori, exostosis and enostosis) ! Odontogenic tumors ! Epithelial tumors ! Ameloblastoma ! Adenomatoid Odontogenic tumor (AOT) ! CEOT/ Pindborg’s tumor ! Mixed (ecto-mesodermal) ! Odontoma ! Ameloblastic fibroma ! Ameloblastic fibro-odontoma ! Mesodermal tumors ! Odontogenic myxoma, Benign cementoblastoma ! Central odontogenic fibroma Benign Cementoblastoma ! Benign cementoblastomas are slow-growing, mesenchymal neoplasms, composed principally of cementum. The tumor manifests as a bulbous growth around and attached to the apex of a tooth root. Its histologic characteristics are similar to those of osteoblastomas, and it is composed of cementoblasts that arise from the mesenchyme of the periodontal ligament. PCD PCD PCD A= mandibular incisor periapical B= Intraoral mandibular occlusal view Benign Jaw Tumors ! Non-odontogenic tumors ! Ectodermal (neurilemoma, neuroma) ! Mixed tumors (neurofibroma, neurofibromatosis) ! Mesodermal tumors (osteoma, Gardner’s syndrome, central hemangioma, A-V fistula, osteoblastoma, osteoid osteoma 23 Neurofibroma Benign Jaw Tumors ! Non-odontogenic tumors ! Ectodermal (neurilemoma, neuroma) ! Mixed tumors (neurofibroma,neurofibromatosis) ! Mesodermal tumors (osteoma, Gardner’s syndrome, central hemangioma, A-V fistula, osteoblastoma, osteoid osteoma Central Hemangioma Benign Jaw Tumors ! Non-odontogenic tumors ! Ectodermal (neurilemoma, neuroma) ! Mixed tumors (neurofibroma,neurofibromatosis) ! Mesodermal tumors (osteoma, Gardner’s syndrome, central hemangioma, A-V fistula, osteoblastoma, osteoid osteoma Osteoblastoma Benign Jaw Tumors ! Non-odontogenic tumors ! Ectodermal (neurilemoma, neuroma) ! Mixed tumors (neurofibroma, neurofibromatosis) ! Mesodermal tumors (osteoma, Gardner’s syndrome, central hemangioma, A-V fistula,osteoblastoma, osteoid osteoma 24 Osteoma Gardner’s syndrome: Gardner’s syndrome, inherited as an autosomal dominant disorder, is characterized by intestinal polyposis, multiple osteomas, fibromas of the skin, epidermal and trichilemmal cysts, impacted permanent and supernumerary teeth, and odontomas. Central Giant Cell Granuloma Central Giant Cell Granuloma Central Giant Cell Granuloma Central Giant Cell Granuloma Acknowledgement ! Thanks to Dr. M. Mupparapu, DMD of the Department of Diagnostic Sciences, Division of Oral and Maxillofacial Radiology at UMDNJ-NJDS for the use of his materials
2590	https://amphibiaweb.org/education/AmphibiaWebIllustratedAmphibiansoftheEarth_v3web.pdf	Created and Illustrated by the 2020-2021 AmphibiaWeb URAP Team: Alice Drozd, Arjun Mehta, Ash Reining, Kira Wiesinger, and Ann T. Chang AmphibiaWeb's Illustrated Amphibians of the Earth This introduction to amphibians was written by University of California, Berkeley AmphibiaWeb Undergraduate Research Apprentices for people who love amphibians. Thank you to the many AmphibiaWeb apprentices over the last 21 years for their efforts. Edited by members of the AmphibiaWeb Steering Committee CC BY-NC-SA 2 Dedicated in loving memory of David B. Wake Founding Director of AmphibiaWeb (8 June 1936 - 29 April 2021) Dave Wake was a dedicated amphibian biologist who mentored and educated countless people. With the launch of AmphibiaWeb in 2000, Dave sought to bring the conservation science and basic fact-based biology of all amphibians to a single place where everyone could access the information freely. Until his last day, David remained a tirelessly dedicated scientist and ally of the amphibians of the world. 3 Table of Contents What are Amphibians? Their Characteristics ...................................................................................... 7 Orders of Amphibians.................................................................................... 7 Where are Amphibians? Where are Amphibians? ............................................................................... 9 What are Bioregions? .................................................................................. 10 Conservation of Amphibians Why Save Amphibians? .............................................................................. 14 Why are Amphibian Populations Declining? ............................................ 15 IUCN Red List of Threatened Species ........................................................ 17 CITES .............................................................................................................. 18 Illustrated Amphibians by Biogeographical Realm Northern America Anaxyrus californicus ................................................................................... 21 Rana sevosa ................................................................................................. 22 Siren lacertina............................................................................................... 23 Central America Agalychnis lemur ......................................................................................... 25 Oedipina carablanca ................................................................................. 26 Sachatamia ilex ........................................................................................... 27 Southern America Atelopus barbotini ....................................................................................... 29 Brachycephalus pitanga............................................................................ 30 Caecilia tentaculata ................................................................................... 31 Afrotropics Breviceps macrops ...................................................................................... 33 Conraua goliath .......................................................................................... 34 Schistometopum thomense ....................................................................... 35 Sechellophryne gardineri ........................................................................... 34 4 Southern Eurasia Pelobates varaldii ........................................................................................ 37 Pelophylax saharicus ................................................................................... 38 Salamandra algira ....................................................................................... 39 Western Eurasia Alytes muletensis .......................................................................................... 41 Bombina bombina ...................................................................................... 42 Proteus anguinus .......................................................................................... 43 Central Eurasia Paradactylodon mustersi ........................................................................... 45 Ranodon sibiricus ......................................................................................... 46 Eastern Eurasia Andrias japonicus ........................................................................................ 48 Hyla japonica ............................................................................................... 49 Indomalaya Ichthyophis bannanicus .............................................................................. 51 Melanobatrachus indicus ........................................................................... 52 Nasikabatrachus sahyadrensis ................................................................... 53 Rhacophorus pardalis ................................................................................. 54 Australasia Leiopelma archeyi ....................................................................................... 56 Myobatrachus gouldii ................................................................................. 57 Pseudophryne covacevichae ................................................................... 58 Subarctic America and Eurasia Ambystoma laterale .................................................................................... 60 Rana arvalis .................................................................................................. 61 Rana clamitans ............................................................................................ 62 Rana temporaria ......................................................................................... 63 Credits and Glossary Literature Cited and Further Reading ....................................................... 64 Illustration Inspirations .................................................................................. 65 Words to Know ............................................................................................. 66 5 www.AmphibiaWeb.org Designed by Nicole Duong (2016) Written by Nicole Duong, Gordon Lau, Ann Chang (2015) What are Amphibians? "Amphibian" comes from the Greek words "amphi-" and "bios" meaning "of both or double kinds" and "life" or "living", referring to the general life history trait of amphibians starting life in water as an aquatic larval form then metamorphosing (transforming) into a terrestrial adult. This is also called a bi-phasic life history. Many frog and salamander species have this "double" life but some amphibians stay in water or on land their entire life. Their Characteristics Amphibians are animals that have a backbone and skin that doesn't have hair, feathers, or scales on it. They are ectothermic, meaning they don't control their body temperature - they have the same temperature as their environment. There are over 8,000 different species of amphibians! Orders of Amphibians Amphibians include frogs, toads, salamanders, and newts. But, they also include another type of animal called caecilians (suh-si-lee-uhn). Together they are in the Class Amphibia. Amphibians are separated more into groups that we call Orders. Orders further divided into groups called families that are genetically and morphologically, or physically, similar. The Order Anura (uh-nyur-uh) includes amphibians that have arms and legs, but no tail. They are frogs and toads. They make up 88% of all amphibians. Species from this order are also called anurans, and all anurans can also be called frogs. So, what are toads? Click Me! 7 In frogs, there are 54 families. One of those families is called Bufonidae - they are the "True Toads". In other words, all frogs in the family Bufonidae are toads. As a result, all toads are frogs, but not all frogs are toads. All true toads have special glands, called paratoid glands, that store chemicals for defense. The Order Caudata (cu-da-ta or ca-da-ta) includes amphibians that have arms, legs, and tails. They are salamanders and newts, and they make up 9% of amphibians. All species in Caudata can be called caudates or salamanders. They are also sometimes called Urodeles. So, why do we call some of them newts? There 10 salamander families. All newts are in the family Salamandridae. All newts are salamanders, but not all salamanders are newts. This is confusing because we call all species of Caudata salamanders! All newts also have paratoid glands. The Order Gymnophiona (gym-no-fee-oh-na) includes amphibians that don't have arms or legs and, most of the time, don't have tails. They are caecilians. These worm or snake-like amphibians are not known well because they are secretive and mostly live underground. Only 3% of amphibian species are in this order. New amphibians are still being described by scientists. There are about 125 - 175 new species described every year. But their descriptions are in the same proportions, with more frogs and fewer caecilians. There are 10 families in Gymnophiona. 8 Where are Amphibians? Amphibians can be found in all sorts of habitats, on every continent except Antarctica. There are also no amphibians in Greenland and most south-Pacific islands. The majority of frogs are found in the tropics of South America, but they can be found all over the world. The majority of salamanders are found in the southeast of North America and in Central America. But salamanders are widely distributed around the world, and can be found in the Americas, Africa, and Eurasia. There are no native salamanders in Australia or New Zealand! 180°0'0" 180°0'0" 120°0'0"E 120°0'0"E 60°0'0"E 60°0'0"E 0°0'0" 0°0'0" 60°0'0"W 60°0'0"W 120°0'0"W 120°0'0"W 180°0'0" 180°0'0" 30°0'0"N 30°0'0"N 30°0'0"S 30°0'0"S Global Amphibian Species Richness Number of Species 0 4 - 10 11 - 20 21 - 30 31 - 40 41 - 50 51 - 60 61 - 80 81 - 100 151 - 188 126 - 150 101 - 125 1 - 3 Data Sources: AmphibiaWeb, IUCN range maps for 7,063 (ca. 85% of known, including native and introduced) species. Map Projection: World Robinson EPSG:54030 Cartography: Michelle S. Koo, with Christina Lew, UC Berkeley Acknowledgements: GIS work supported by the Museum of Vertebrate Zoology, MVZ apprentices, and GIS assistant Zoe Yoo (2019-2021). AmphibiaWeb © 2000-2021 The Regents of the University of California \| Creative Commons Attribution Non-Commercial CC-BY-NC 9 Caecilians are only found along the equator of the world and are most concentrated in South America and India. Most of them live underground, but some can be found in water. What are Bioregions? Surveys of many plants and animals consistently show natural boundaries of geography (for example, mountains and rivers), ecology (similar climate and food webs), and species (communities of plants and animals, which interact with each other). The area set by these boundaries are called bioregions. Bioregions can be grouped into broader categories of biogeographical realms. In amphibians, studies across species show repeated patterns supporting the idea of bioregions. Because of this we use the broader biogeographical realms to organize our book of amphibian diversity. We largely follow the realm definitions from OneEarth.org. Here we provide a summary of the realms used in this book. Click Me! 10 Afrotropics - The Afrotropics realm generally covers the African continental plate, but excludes the driest regions of the north and includes the coastal regions of southern Saudi Arabia and Yemen. This realm has many types of habitats including savannas, scrublands, brushlands, grasslands, mangroves, woodlands, and various types of temperate and tropical forests. There are 5 subrealms and 24 bioregions in the Afrotropics. Southern Eurasia - The Southern Eurasian realm includes northern Africa and most of the Greater Arabian Peninsula. This realm is composed of deserts, savannas, desert marshes, salt marshes, woodlands, and temperate forests. There are 2 subrealms and 5 bioregions. Western Eurasia - The Western Eurasian realm includes the western portion of the Eurasian continental plate from the British Isles to the Mediterranean and to western Russia on its eastern edge. Habitats in this realm include grasslands and various types of temperate forests. The realm is composed of 5 subrealms and 13 bioregions. Central Eurasia - The Central Eurasian realm extends from the border of western Russia to the Gulf of Oman at the Greater Arabian Peninsula and from the Caspian Sea in the west to Tien Shan Mountains in China to the east. Ecosystems found here are deserts, grasslands, meadows, woodlands and temperate forests. This realm has 4 subrealms and 9 bioregions. Eastern Eurasia - The Eastern Eurasia realm includes the eastern portion of Eurasia, but excludes the Arctic portions. The habitats in this realm include deserts, meadows, shrublands, and various types of temperate and subtropical forests. It is made up of 7 subrealms and 17 bioregions. 11 Indomalaya - The Indomalaya realm is made up of the Indian continental plate and the Southeast Asia region of the Eurasian plate. This realm has deserts, grasslands, scrublands, mangroves, and various types of tropical forests. There are 3 subrealms and 18 bioregions. Australasia - The Australasia realm is made up of the Australian continental plate. This includes eastern Indonesia to the north and New Zealand to the south. Ecosystems here include deserts, savannas, shrublands, woodlands, and various types of temperate, subtropical, and tropical forests. Australasia has 3 subrealms and 15 bioregions. Northern America - The Northern American realm includes most of the North American continental plate, but excludes the coldest regions in the north and the tropical regions of Central America. This region includes deserts, grasslands, shrubland, riparian areas, and various types of temperate forests. This realm has 6 subrealms and 22 bioregions. Central America - The Central American realm includes the tropical regions of Mexico south to northwestern Colombia, and includes the Caribbean islands. Habitats in this realm include various types of tropical forests. There are 2 subrealms and 6 bioregions in this realm. Southern America - The Southern American realm covers all of the South American plate except for the northeastern portion of Colombia. Habitats here include deserts, savanna, shrubland, grasslands, and various types of temperate and tropical forests. It is made up of 5 subrealms and 23 bioregions. Subarctic America and Eurasia - Because few amphibians can live in extremely cold conditions, we combined the realms of Subarctic America and Subarctic Eurasia. This area covers the most northern 12 parts of the northern hemisphere and is made up of boreal forests, taiga forests, and tundras ecosystems. Subarctic America has 4 subrealms and 9 bioregions while Subarctic Eurasia has 4 subrealms and 8 bioregions. 13 Conservation of Amphibians Why Save Amphibians? Excerpts by Julianne Oshiro While these little creatures may go unnoticed by most of us, amphibians have greatly influenced our society and planet’s ecosystems. Amphibians have helped to advance the field of medicine, are integral to the ecosystems on which we rely, and are central to many cultural stories and beliefs. Much of the fascinating biology of amphibians have applications beyond understanding their evolution and ecology. They have valuable chemicals that they exude from their skin. These chemicals have already been used in medicine to combat drug-resistant bacteria, cardiac problems, and HIV (Song et al. 2010). In another example, amphibians are a valuable model for researchers to study regenerative tissue (Garg et al. 2007). Salamanders, such as the Ambystoma mexicanum, have the ability to regrow limbs, which brings hope that one day doctors will be able to help people regrow body parts (Voss et al. 2009). Amphibians are central to maintaining a healthy and resilient ecosystem. And because of their permeable skin, amphibians are extremely vulnerable to environmental and water quality degradation. Thus, their decline is an important indicator that an entire ecosystem may be in peril. Click Me! 14 Why are Amphibian Populations Declining? Excerpts from various AmphibiaWeb pages CRX = Critically Endangered, Extinct, or Extinct in the wild, EV - Endangered and Vulnerable The biggest threat to amphibians is habitat destruction (Dodd and Smith 2003). Although amphibians are found in a great variety of ecosystems from tropical rainforests to arid deserts (Stebbins and Cohen 1995), people often think of amphibian habitat as being confined to wetlands and other aquatic environments. Surprisingly, a large number of species are entirely terrestrial (e.g., plethodontid salamanders and eleutherodactylid frogs; for a good overview of amphibian natural history, see Stebbins and Cohen 1995). This diversity in habitat requirements between species and even between life-stages of the same species emphasizes that we can not take a simple approach to amphibian conservation. In order to successfully conserve amphibians, we need a clear understanding of their varied life histories and habitat requirements. There are distinct differences between habitat destruction, alteration and fragmentation (see our glossary at the end of the book). 1098 350 231 2231 406 127 760 162 26 245 60 7 1021 266 41 73 5 4 229 35 19 356 66 10 411 20 1 198 33 6 107 440 41 Afrotropical Australian Madagascan Nearctic Neotropical Oceanina Oriental Palearctic Panamanian Saharo-Arabian Sino-Japanese AmphibiaWeb: Wallacean Biome Map of Amphibian Species in Decline. [web application]. 2014. UC Regents, Berkeley, CA. Available: (12 May 2014). Total No. Amphibians EV CRX 15 What is alarming is that there are many cases where the habitat is protected and amphibians are still disappearing. There are many causes for recent amphibian declines, but global climate change, and diseases caused by fungi, called chytridiomycota (often called chytrid), are thought the be the other big threats to amphibians. Chytridiomycosis is a disease caused by two fungal chytrid pathogens Batrachochytrium dendrobatidis (Bd) and Batrachochytrium salamandrivorans (Bsal). Bd is associated with the global loss of hundreds of species of amphibians and represents a spectacular loss of biodiversity, some say the worst in recorded history. Bsal was identified in 2013 and caused many salamander deaths in Europe. Using lessons learned from Bd, many countries were able to prevent Bsal from entering their region. Find out more about Bd at: Find out more about Bsal at: Click Me! Click Me! 16 IUCN Red List of Threatened Species The International Union for the Conservation of Nature (IUCN) system of evaluating the threat of extinction for a species is used broadly. Below we briefly define their categories. Visit org/ to learn more. Data Deficient (DD) - A taxon is Data Deficient when there isn't enough information to make a decision of a species' risk of extinction. Taxa in this category may be well studied, and have well known biology, but lack appropriate data on abundance and/or distribution in the wild. Data Deficient is therefore not a category of threat. Least Concern (LC) - Taxa are considered Least Concern when they are widespread and abundant. Near Threatened (NT) - A taxon is Near Threatened when it is close to qualifying for or is likely to qualify for a threatened category (Critically Endangered, Endangered, or Vulnerable) in the near future. Vulnerable (VU) - A taxon listed as Vulnerable is considered to be facing a high risk of extinction in the wild. Endangered (EN) - A taxon listed as Endangered is considered to be facing an very high risk of extinction in the wild. Critically Endangered (CR) - A taxon listed as Critically Endangered is considered to be facing an extremely high risk of extinction in the wild. Click Me! 17 Extinct in Wild (EW) - A taxon is Extinct in the Wild when it is only known to survive in captivity or as a naturalized population (or populations) well outside the past range. Extinct (E) - A taxon is Extinct when there is reasonable confidence that the last individual of the species has died. This status is given when exhaustive surveys in known and/or expected habitat, at appropriate times (diurnal, seasonal, annual), throughout its historic range have failed to record an individual. CITES The Convention on International Trade in Endangered Species (CITES) of Wild Fauna and Flora protects species in international plant and animal trade. As of November 2019, 201 amphibian species have a CITES status. Find out more at their website, Click Me! 18 Illustrated Amphibians by Biogeographic Realms Our 2020-2021 research apprentices embraced the AmphibiaWeb mission of student outreach and education by researching global amphibian biodiversity. They chose amphibians that they thought were amazing and hope that you will find these species as beautiful and unusual as they did. Enjoy! 19 Northern America Countries: 3 Includes portions of southern Canada, the continental United States of America, and northern Mexico. Approximate Proportion of Amphibians: 6% Anaxyrus californicus Common Name: Arroyo Southwestern Toad IUCN: Endangered Anaxyrus californicus, the Arroyo toad, is a small, stocky toad found in Mexico and California. This toad can be found near “arroyos,” which are dry streams or creeks that fill with water when it rains; this is probably how the Arroyo Toad got its name! Adult toads burrow in the sand during the day and hunt for prey at night. Arroyo toad larvae are hunted by some fish, and adults are often prey to introduced bullfrogs. While these predators are threatening the Arroyo toad, habitat destruction and human collection also make this population vulnerable. The US government keeps the locations of Arroyo toad secret to protect them from collectors. Still, further limitations should be placed on urban development in order to protect the habitats of Arroyo toads. the habitats of Arroyo toads. Click Me! 21 Rana sevosa Common Name: Dusky Gopher Frog IUCN: Critically Endangered Rana sevosa, dusky gopher frogs, are only found in the southernmost regions of Alabama, Louisiana, and Mississippi. These frogs have distinct breeding and non-breeding habitats - shallow, ephemeral lakes v.s. pine forests, respectively - so they must migrate back and forth during the breeding season. Dusky gopher frogs usually live in burrows abandoned by gopher tortoises (Gopherus polyphemus) or small mammals, and only leave these burrows during times of rain. They regularly forage around their burrow entrance, leaving a smooth area outside of the entrance that many individuals use as a resting place. Dusky gopher frogs are mostly threatened by habitat destruction as pine forests are being urbanized. However, captive breeding, artificial fertilization, wetland restoration, and species reintroduction to the forests have all shown promising results for conserving the dusky gopher frog population. Click Me! 22 Siren lacertina Common Name: Greater Siren IUCN: Least Concern Siren lacertina, the greater siren is a large salamander that lives in aquatic habitats around the southeastern United States. This siren lacks hind limbs and keeps its gills throughout life. Greater sirens are opportunistic feeders, meaning they will eat what they can find, but they mostly consume mollusks, like snails and clams. They make a variety of noises when threatened, including yelping, hissing, and croaking. If their warning calls don’t work, they either thrash and swim away quickly or inflict a strong bite. While they are classified as a Least Concern species, they are threatened by the destruction of wetlands. These areas must be protected in order to ensure that the greater siren has enough space to thrive. Click Me! 23 Central America Countries: 32 Southern Mexico to Northern Colombia and the Caribbean islands. Approximate Proportion of Amphibians: 15% Agalychnis lemur Common Name: Lemur Leaf Frog IUCN: Critically Endangered Agalychnis lemur, the lemur leaf frog, is a nocturnal and territorial species. It has faced declines due to the chytrid fungus. However, it could contribute much to human medicinal research because it has different skin-secreted peptides. These peptides include one that can prevent staph infections from Staphylococcus aureus, another that can stimulate insulin release, which could help treat Type 2 diabetes, and another that could be engineered as an anti-cancer agent. Click Me! 25 Oedipina carablanca Common Name: None IUCN: Endangered Oedipina carablanca is a salamander that has been found in rotting logs and the bark of fallen trees in Costa Rica. Some think this species might climb trees. When they feel threatened, they may coil up or flip their bodies around. They are dark brown with white splotches throughout the body. They have an Endangered status due to habitat fragmentation and loss as well as low population abundance. Click Me! 26 Sachatamia ilex Common Names: Ghost Glass Frog, Limon Giant Glass Frog, Holly’s Glassfrog, Rana de Cristal de Holly, Rana de Cristal Fantasma IUCN: Least Concern Sachatamia ilex, the ghost glass frog, is the largest glass frog in South America. Glass frogs are have glass in their name becuase their bellies are usually transparent. The ghost glass frog is striking for the black reticulations in its eyes. This species has dark green bones and they can change the intensity of their green coloration to match whatever they are resting on. They are considered Least Concern because they live in protected areas. They are vulnerable to habitat loss resulting from deforestation because they need vegetative cover above streams. Click Me! 27 Southern America Countries: 14 From Colombia and Venezuela in the north extending south to the southernmost tip of South America. Approximate Proportion of Amphibians: 31% Atelopus barbotini Common Name: None IUCN: No status, likely Vulnerable Atelopus barbotini is a harlequin frog that lives in the primary forests of French Guiana. They have spots in the shape of red commas, rings, and curved lines on their backs. They give a prelude call before beginning a series of calls. As of 2021, this species has no IUCN status because they recognize it as a subspecies of Atelopus spumarius. The Atelopus genus is very sensitive to the chytrid fungus, Bd, and many species in the genus have gone extinct. Click Me! 29 Brachycephalus pitanga Common Name: Red Pumpkin Toadlet IUCN: No Status Brachycephalus pitanga, the red pumpkin toadlet, lives in leaf litter in the Serra do Mar of southeastern Brazil. Like other pumpkin toadlets, they are missing some fingers and toes. They have fluorescent bones on their dorsum and head. Researchers don't know why this is, but it may have to do with mating - we don't know what frogs can see! Lastly, their skin is toxic to protect them from predators. These toadlets do not go through metamorphosis, they hatch as little versions of adults. As of 2021, this species has no IUCN status, but the population could be threatened by habitat loss, climate change, and the chytrid fungus, Bd. the population could be threatened by habitat loss, climate change, and the chytrid fungus, Bd Click Me! 30 Caecilia tentaculata Common Name: None IUCN: Least Concern Caecilia tentaculata is a bluish gray cylindrical caecilian that can be found in loose soil in lowland forests of the Amazon basin. The short-eared dog (Atelocynus microtis) may prey on this species. Caecilia tentaculata eyes are dark spots under the skin. While their population numbers are unknown, they are considered Least Concern because they have a wide range and many parts of that range are in protected areas. There is still a lot of confusion if this is one or many species! Click Me! 31 Countries: 53 All African countries excluding the desert countries in the north, but also including the coastal region of southern Saudi Arabia and Yemen. Approximate Proportion of Amphibians: 14% Afrotropics Breviceps macrops Common Name: Desert Rain Frog IUCN: Vulnerable Breviceps macrops, the desert rain frog, has a balloon-shaped body and huge protruding eyes, which help it to find food in the dark. These frogs are nocturnal and buries themselves under sand dunes to sleep during the day. Their feet are webbed to help them walk on the sand. When they are feeling threatened they let out a scream-like call. Breviceps macrops is native to a strip of coastal land in Namibia that is rich in diamonds and copper, and so it is threatened by mining and habitat loss. Click Me! 33 Conraua goliath and Sechellophryne gardineri Common Names: Goliath Frog and Gardiner's Seychelles Frog IUCN: Both Endangered Conraua goliath, the Goliath frog, is the largest frog in the world, growing over a foot in length. They live in large rivers and rapids in the tropical forests of Cameroon. Due to their size, the Goliath frog is hunted for food. They are also threatened by deforestation and other habitat loss. Sechellophryne gardineri, Gardiner's Seychelles frog, is one of the smallest frogs alive with a maximum length of just over a centimeter. These tiny frogs makes their advertisement calls with a high-pitched squeak, similar to a cricket. They are threatened by climate change and loss of habitat due to frequent wildfires. Click Me! Click Me! 34 Schistometopum thomense Common Name: Cobra bobo IUCN: Least Concern Schistometopum thomense is one of the rare caecelians endemic to an island. This amphibian is found on São Tomé Island, and is commonly called Cobra bobo by people living on the island. They build huge networks of tunnels and usually live underground. However, unlike most caecelians, this species is easy to study because of their high activity levels above ground. In 2021, researchers (O'Connell et al. 2021) found out that Schistometopum thomense is actually two species, one in the north and one in the south. In the middle, where those species meet they mate to make hybrids! To find out more about this species, read our species account using the QR-code on this page. Click Me! 35 Southern Eurasia Countries: 16 Includes the deserts of northern Africa and the peninsular Middle East, excluding the coastal region of southern Saudi Arabia and Yemen. Approximate Proportion of Amphibians: 1% Pelobates varaldii Common Name: Moroccan Spadefoot IUCN: Endangered Pelobates varaldii is a small frog that is covered in irregular brown spots and red warts in some regions. These frogs can be found in northwestern Morocco and live in sandy areas to protect themselves from the sun. They are active in the night and hunt for food in the safety of the night. Pelobates varaldii is threatened by loss and drainage of habitat, as well as pollutants and pesticides. They are also the prey of the Eastern mosquitofish, Gambusia holbrooki. Click Me! 37 Pelophylax saharicus Common Name: Sahara Frog IUCN: Least Concern Pelophylax saharicus can be found across Northern Africa, including Tunisia, Algeria, and Morocco. These frogs are an extremely versatile amphibian, and can tolerate vastly different climates, from the alpine forests, to the Sahara desert. Pelophylax saharicus lives in and around water, and as such, they feeds on aquatic prey like fish eggs, frog eggs, and more. Their population is threatened by climate change, especially droughts. Click Me! 38 Salamandra algira Common Names: Algerian Salamander, North African Fire Salamander, Arous Chta IUCN: Vulnerable Salamandra algira refers to five subspecies that make up the fire salamanders of North Africa. These salamanders are covered in all kinds of yellow and red spots, and in all different patterns. These fire salamander lives in forests, caves, and rivers. In the summer, Salamandra algira stops its activity and rests until the fall when the rain comes. Unfortunately, this salamander's population is threatened by deforestation, habitat loss, and water pollution. Click Me! 39 Western Eurasia Countries: 51 Extends from the British Isles south to the Mediterranean and east to western Russia. Approximate Proportion of Amphibians: 3% Alytes muletensis Common Name: Mallorcan Midwife Toad IUCN: Endangered Alytes muletensis, the Mallorcan midwife toad, is a small frog that lives on the island Mallorca off the coast of Spain. Males carry eggs around their ankles until the tadpoles hatch to protect them. They are Endangered due to urbanization, habitat drainage for water usage, introduced predators including the natricine water snake (Natrix maura), and the competitor Perez's frog (Rana perezi). Click Me! 41 Bombina bombina Common Name: Fire-Bellied Toad IUCN: Least Concern Bombina bombina, the fire-bellied toad, lives across central and eastern Europe. Their belly can be red or orange with blue-black spots and white points. When there is a predator nearby, they turn onto their belly and cover their eyes with its palms to show their warning colors. They have venomous skin secretions. They are a Least Concern species, but wetland destruction and pollution are threats to this species. Click Me! 42 Proteus anguinus Common Name: Olm IUCN: Vulnerable Proteus anguinus, the olm, is a slender salamander that lives in water systems underneath karst formations in southern Europe. The maximum lifespan of an olm is likely over a century, making them the longest living amphibian species. These blind salamanders use various sensory receptors to navigate dark environments. The olm is the only European vertebrate adapted to living in caves. It is Vulnerable due to tourism, pollution, habitat alteration, and overcollection. Click Me! 43 Central Eurasia Countries: 8 Extends from the Russian boarder to the Gulf of Oman and from the Caspian Sea in the west to Tien Shan Mountains in the east. Approximate Proportion of Amphibians: 3% Paradactylodon mustersi Common Names: Afghanistan/ Paghman Mountain Salamander IUCN: Critically Endangered Paradactylodon mustersi, the Afghanistan/ Paghman mountain salamander, is found in the Paghman Mountains of Paghman County, Afghanistan. These salamanders live in cool highland streams fed by glaciers, where the adult species stay near the fast-running water. They are Critically Endangered because their ecological niche is physically disturbed by humans. Additionally, constant irrigation and changing water temperature also limits this species. These salamanders are completely aquatic, meaning that it spends little to no time on land. Click Me! 45 Ranodon sibiricus Common Names: Semirechensk Salamander, Central Asian Salamander IUCN: Endangered Ranodon sibircus, also known as the central Asian salamander, is found in a small range in the mountainous ranges of southern Kazakhstan and northwestern China. Despite only being found in a small region, this salamander has a very diverse range of habitats. They can be found in alpine, subalpine, forest-meadow and forest-meadow-steppe belts. Ranodon sibiricus is Endangered because of habitat loss to deforestation and land conversion to urban and agricultural development. Click Me! 46 Eastern Eurasia Countries: 5 Extends from Mongolia to the Korean peninsula, thru Japan and northern and central China. Approximate Proportion of Amphibians: 5% Andrias japonicus, or the Japanese giant salamander, is one of the largest species of amphibian. This species is endemic to Japan, but has sister species in China. As their common name suggests, this species is huge. They can range from 30 to 150 cm (11.8 - 58 inches) in length. The heaviest recorded individual was a whopping 26.3 kg (58 lbs). Japanese giant salamanders are classified as Near Threatened because of the severe fragmentation of their habitat from urbanization projects like flood and erosion control, agriculture, hydraulic power generation, and road construction. Andrias japonicus Common Name: Japanese Giant Salamander IUCN: Near Threatened the severe fragmentation of their habitat from urbanization projects like flood and erosion control, agriculture, hydraulic power generation, and road construction. Click Me! 48 Hyla japonica Common Name: Japanese Tree Frog IUCN: Least Concern Hyla japonica, or the Japanese Tree Frog is found in many parts of Eastern Europe and Asia, but is known for their populations in Japan. The species lives in a variety of habitats including mixed and deciduous broad-leafed forests, bushlands, forest steppes, meadows, and swamps. In forestless areas, the tree frog primarily inhabits river valleys with shrubs. Male Hyla japnonica are sometimes found to be infected with Batrachochytrium dendrobatidis (Bd) causing their mating calls to differ. These differing mating calls may sometimes attract female frogs more effectively, therefore transmitting the infection faster. This species has some declining populations in the northern edge of their range, but no anthropogenic problems have been studied enough to know whether it has impacted their range or not. Click Me! 49 Indomalaya Countries: 22 Southern (below the equator) Asian countries from India to most of Indonesia and the Philippines. Approximate Proportion of Amphibians: 14% Ichthyophis bannanicus Common Name: Banna Caecilian IUCN: Least Concern Ichthyophis bannanicus, also referred to as the Banna Caecilian, is mostly found in southern China, but their distribution may extend into Vietnam as well. Locally, this caecilian can be found in loose soil around streams, or in the land adjacent to rice-fields. This species is listed as Least Concern on IUCN; however, China's national rating of this species is Endangered. Populations of this species in China are at high risk of decline because of human activities including cultivation of the land, destruction of forest, and pollution, but more studies are need to create an effective conservation plan. Click Me! 51 Melanobatrachus indicus Common Name: Black Microhylid Frog IUCN: Endangered Melanobatrachus indicus, or the black microhylid frog, is endemic to the Western Ghats of south-western India. This species was thought to be super rare, but was found by researchers again in 1997. The black microhylid frog is considered Endangered because their range is fragemented from loss of habitat due to urbanization. This frog is known for its beautiful display of blue dots and a red underbelly. When the black microhylid frog feels threatened, they retract their limbs and arches their back to "contract" themselves! Click Me! 52 Nasikabatrachus sahyadrensis Common Names: Purple Frog, Pig-nosed Frog IUCN: Endangered Nasikabatrachus sahyadrensis is endemic to the Western Ghat Mountain range of Southern India. This species is known for their distinctive skull and unusual appearence. Locals didn't realize they were frogs until researchers identified them! This species is Endangered because of crop farming and dam projects taking place in the Western Ghats. They are known for burrowing beneath the ground and can be spotted from far away by their "bloated" and purplish appearence. Click Me! 53 Rhacophorus pardalis Common Name: Harlequin Treefrog IUCN: Least Concern Rhacophorus pardalis, or the Harlequin treefrog is found in Indonesia, Malaysia, and the Philippines. The Harlequin treefrog is most well known for their webbed hands and feet, which help them glide in the canopies of the forests. This species is commonly found in the dense forests of the Indomalayan contries, but recent surveys suggests that this species' range is being impacted by deforestation and logging. Click Me! 54 Australasia Countries: 10 Includes parts of Indonesia, the New Guinea island, Australia, Tasmina, and New Zealand. Approximate Proportion of Amphibians: 9% Leiopelma archeyi Common Name: Archey's Frog IUCN: Critically Endangered Leiopelma archeyi, or Archey’s frog, can be found in the grassy, moist forests of New Zealand. Archey’s frog is small and usually brown or green, which helps them to blend in with their leafy surroundings. Despite having no true voicebox, Archey’s frog can make chirps and squeaks using resonance frequencies. Archey’s frog also has muscles for wagging a tail despite the fact that they have no tail. Introduced rats are one of the main predators of Archey’s frogs, making the species Critically Endangered. New Zealand’s Department of Conservation has dropped thousands of rat traps in the local forests to curb the rat population and help the Archey’s frog population flourish. Click Me! 56 Myobatrachus gouldii Common Name: Turtle Frog IUCN: Least Concern Myobatrachus gouldii, also called the turtle frog, is found in the dry, sandy regions of Western Australia. This species is called the turtle frog because of their short turtle-like limbs and the way that they dig forward through sand to build underground burrows, where they spend most of their lives. Turtle frogs are never tadpoles; instead, they fully develop in the egg and hatch as small adults. There are very few threats currently facing the turtle frog, but habitat alteration and a changing climate can decrease the size of their habitat. Click Me! 57 Pseudophryne covacevichae Common Name: Magnificent Brood Frog IUCN: Endangered Pseudophryne covacevichae, the magnificent brood frog, is endemic to Australia and can be found only in small areas near Ravenshoe in the state of Queensland. They are usually found in the understory of eucalyptus forests and among leaf litter in grasslands. The magnificent brood frog is nocturnal, meaning it’s most active at night, but they are also active on overcast days. This species is threatened by massive habitat loss to cattle grazing and logging. While some of these frogs live on protected land, like national parks, most do not. Further protections should be enforced to protect this Endangered species. Click Me! 58 Subarctic America and Eurasia Countries: 9 Includes the arctic regions of the North American and Eurasian continental plates. Approximate Proportion of Amphibians: 0.5% Ambystoma laterale Common Name: Blue-spotted Salamander IUCN: Least Concern Ambystoma laterale, the blue-spotted salamander, calls deciduous forests their home and can be found around the Great Lakes or along the Atlantic coast between Quebec and New Jersey. This species hides underground during the day, and will aggressively defend their burrows when needed. They also have aggressive anti-predator mechanisms; when threatened, they will lash their tail around and curl up their bodies. Approximately one-quarter of Ambystoma laterale individuals may be infected with Trypanosoma protozoans in any given year. Some Trypanosoma protozoans are carried by Tsetse fleas and can cause “sleeping sickness” in humans. Habitat destruction is the biggest threat facing the blue-spotted salamander; there are no current conservation efforts underway, but limiting the destruction of wetlands and forests is critical to ensuring their survival. Click Me! 60 Rana arvalis Common Name: Moor Frog IUCN: Least Concern Rana arvalis is a frog that lives in the moors of northern Eurasia. During the year, they have brown coloration with wide stripes on their legs, but during mating season (March through June), the males turn bright blue to attract the females. Although the population of Rana arvalis is relatively stable, industrial pollution and radiation from the Ural Mountains sometimes causes mutations in offspring, including developmental and morphological differences. Click Me! 61 Rana clamitans Common Name: Green Frog IUCN: Least Concern Rana clamitans, the green frog or the bronze frog, is found throughout most of North America. They prefer to live in wetlands, and they are never more than one meter away from a water source, unless it’s raining. When it isn’t breeding season, green frogs are solitary and defend their individual territory. Green frogs aren’t picky -- they’ll eat anything from insects to crustaceans to other frogs! Their unique call sounds like the low snapping of rubber bands. Green frogs may be classified as a Least Concern species, but they are threatened by vehicular traffic, habitat destruction, and game hunters. To ensure this species’ survival, further restrictions should be placed on hunting them and the shoreline development of their habitats. Click Me! 62 Rana temporaria Common Name: Common Frog IUCN: Least Concern Rana temporaria is a very common frog throughout northwestern Europe, and lives in ponds, other damp areas, or in long grass. They hibernate in the winter, and wake up in the spring just in time for the mating season. This frog is quite resilient, and can withstand much change to their environment. In certain areas, habitat loss or drying ponds have caused small populations to migrate. Alarmingly, Batrachochytrium salamandrivoran (Bsal), a pathogenic fungus affecting amphibians, has been found in laboratory populations of Rana temporaria, meaning this species could be a carrier of the pathogen that is lethal to salamanders. Click Me! 63 Literature Cited and Further Reading Conservation of Amphibians: CITES (2019). “New CITES Appendices as amended at CoP18 and new suites of Resolutions and Decisions enter into force." Geneva, 26 November 2019 < new_CITES_Appendices_as_amended_at_CoP18_and_new_suites_of_Resolutions_Decisions_ enter_into_force_26112019> Downloaded 3 May 2021. Dodd, C. K., and L. L. Smith. 2003. Habitat destruction and alteration: historical trends and future prospects for amphibians. Pages 94-112 in R. D. Semlitsch, editor. Amphibian Conservation. Smithsonian Institution, Washington. Garg, Abhishek D, et al. “Toad Skin-Secretions: Potent Source of Pharmacologically and Therapeutically Significant Compounds.” The Internet Journal of Pharmacology, vol. 5, no. 2, 2007, doi:10.5580/18b6. Song, Fengyu, et al. “Amphibians as Research Models for Regenerative Medicine.” Organogenesis, vol. 6, no. 3, 2010, pp. 141–150., doi:10.4161/org.6.3.12039. Stebbins, R. C., and N. W. Cohen. 1995. A natural history of amphibians. Princeton University Press:i-xvi, 1-316. Voss, S. R., et al. “Ambystoma mexicanum, the Axolotl: A Versatile Amphibian Model for Regeneration, Development, and Evolution Studies.” Cold Spring Harbor Protocols, vol. 2009, no. 8, 2009, doi:10.1101/pdb.emo128. Proteus anguinus, Olm: Jewett, K. "Saving Slovenia's 'Human Fish'." Biogeographic downloaded on 3 May 2021 from Schistometopum thomense, São Tomé Caecilian: O'Connell, K., Prates, I., Scheinberg, L., Mulder, K., Bell, R. (2021). " Speciation and secondary contact in a fossorial island endemic, the São Tomé caecilian." Molecular Ecology 00:1–13. California Academy of Sciences (2021). " Study indicates São Tomé island has two species of caecilians found nowhere else on Earth." Eureka Alert releases/2021-05/caos-sis050721.php?fbclid=IwAR28sjgpB-GuDMAr4Hd3BaVKSxunXq-slu_-9v4RYwnphN86BotlBhzjLD0 Downloaded on 12 May 2021. 64 Illustration Inspirations Species Illustrator Based on Photos by: Pg Agalychnis lemur Kira Wiesinger Rob Schell, permission granted in honor of David Wake 25 Alytes muletensis Kira Wiesinger David Daversa 41 Ambystoma laterale Ash Reining Dr. John P. Clare 60 Anaxyrus californicus Ash Reining William Flaxington 21 Andrias japonicus Arjun Mehta Danté B Fenolio 48 Atelopus barbotini Kira Wiesinger Henk Wallays 29 Bombina bombina Kira Wiesinger Boris I. Timofeev 42 Brachycephalus pitanga Kira Wiesinger Alberto López-Torres 30 Breviceps macrops Alice Drozd Robert C. Drewes 33 Caecilia tentaculata Kira Wiesinger The Kentucky University Herpetology Digital Archive 31 Conraua goliath Alice Drozd Marvin Schäfer and Frogs & Friends e.V. 34 Hyla japonica Arjun Mehta B. Thiesmeier and PENSOFT Publishers 49 Ichthyophis bannanicus Arjun Mehta Nikolai Orlov 51 Leiopelma archeyi Ash Reining Simon J. Tonge 56 Melanobatrachus indicus Arjun Mehta Sandeep Das 52 Myobatrachus gouldii Ash Reining Stephen Zozaya - wiki commons 57 Nasikabatrachus sahyadrensis Arjun Mehta Biju Das 53 Oedipina carablanca Kira Wiesinger Brian Kubicki 26 Paradactylodon mustersi Arjun Mehta Theodore Papenfuss 45 Pelobates varaldii Alice Drozd Philip de Pous 37 Pelophylax saharicus Alice Drozd Iulian Gherghel at TrekNature 38 Proteus anguinus Kira Wiesinger Dr. Joachim Nerz 43 Pseudophryne covacevichae Ash Reining Jean-Marc Hero 58 Rana arvalis Alice Drozd Maciej Bonk 61 Rana clamitans Ash Reining James H. Harding 62 Rana sevosa Ash Reining Jeromi Hefner 22 Rana temporaria Alice Drozd Frank Teigler 63 Ranodon sibiricus Arjun Mehta Henk Wallays 46 Rhacophorus pardalis Arjun Mehta Dr. Peter Janzen 54 Sachatamia ilex Kira Wiesinger Twan Leenders 27 Salamandra algira Alice Drozd Henk Wallays 39 Schistometopum thomense Alice Drozd Gonzalo R. Mucientes Sandoval 35 Sechellophryne gardineri Alice Drozd 34 Siren lacertina Ash Reining Todd Pierson 23 World Map none Thomas Kitchin - Wiki Commons 10 65 Words to Know Taxonomy Anura - frogs and toads, see page 7. Caudata - salamanders and newts, see page 7. Gymnophiona - caecilians, see page 8. Taxon, Taxa - A group of similar living things. This could be at the species level or different species that share an evolutionary history. Taxon is singular, Taxa is plural. Taxonomy - grouping living things together to reflect a shared evolutionary history. Scientists use hierarchical taxonomy, meaning we group things, then split each group further until we get to individual species. Those levels are (from biggest group to smallest): Kingdom, Phylum, Class, Order, Family, Genus, and Species. Conservation Batrachochytrium dendrobatidis (Bd) - A species of chytrid fungus that has caused global amphibian declines and extinctions. Also see page 16. Batrachochytrium salamandrivorans (Bsal) - A species of chytrid fungus, identified in 2013, that caused large die-offs of salamanders in Europe. Also see page 16 Chytridiomycota - the group of fungi that make up chytrid fungi. Chytridiomycosis - disease or diseases caused by chytrid fungi. Climate change - the change in our average environmental conditions (for example temperature and rain or snowfall) over a long period of time. Deforestation - clearing large areas of trees. Endemic - local and restricted to a specific place. Habitat - environment or natural home to a living being. Alteration - to change an environment such as adding pollution, exotic species or to overharvest species in that environment. Destruction - to change an environment so much that the species that once lived there are no longer able to. Drainage - to remove the water in the area. 66 Fragmentation - to destroy portions of an environment so that it splits the environment into two or more pieces. Loss - the result of habitat destruction. Pollution - the addition of a harmful substance to an environment. Industrial - the addition of harmful substances to an environment from the large-scale creation of products to sell. Natural History Advertisement calls - sounds created to attract another individual and define their territory. Bi-phasic - two forms or phases of life. Diurnal - active during the day. Larva, Larvae - the immature, or young, form of an animal that looks noticeably different from the adult. Larva is singular, Larvae is plural. Nocturnal - active at night. Mating season - the time of year when individuals gather to make babies. Tadpole - the immature, or young, form of a frog that looks noticeably different from the adult. Developmental Metamorphosis - The process of changing from one form to another during development. In amphibians it is usually associated with a switch from an aquatic larval stage to a terrestrial adult phase. Mutation - change in DNA sequence. This happens to individuals and can happen during reproduction. Some of these changes result in no change to the plant or animal. Other changes can cause big differences that help are harm the individual. Developmental - change in DNA sequence that causes changes in the growth or natural process of the body. Morphological - change in the DNA sequence that causes changes in how a living organism looks. For more terms check out AmphibiaWeb's Glossary at: 67
2591	https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_session27.pdf	“mcs” — 2015/5/18 — 1:43 — page 572 — #580 14.8 The Pigeonhole Principle Here is an old puzzle: A drawer in a dark room contains red socks, green socks, and blue socks. How many socks must you withdraw to be sure that you have a matching pair? For example, picking out three socks is not enough; you might end up with one red, one green, and one blue. The solution relies on the Pigeonhole Principle If there are more pigeons than holes they occupy, then at least two pigeons must be in the same hole. “mcs” — 2015/5/18 — 1:43 — page 573 — #581 14.8. The Pigeonhole Principle 573 B g C 1st sock red 2nd sock green 3rd sock blue 4th sock Figure 14.3 One possible mapping of four socks to three colors. What pigeons have to do with selecting footwear under poor lighting conditions may not be immediately obvious, but if we let socks be pigeons and the colors be three pigeonholes, then as soon as you pick four socks, there are bound to be two in the same hole, that is, with the same color. So four socks are enough to ensure a matched pair. For example, one possible mapping of four socks to three colors is shown in Figure 14.3. A rigorous statement of the Principle goes this way: Rule 14.8.1 (Pigeonhole Principle). If jAj > jBj, then for every total function f W A ! B, there exist two different elements of A that are mapped by f to the same element of B. Stating the Principle this way may be less intuitive, but it should now sound familiar: it is simply the contrapositive of the Mapping Rules injective case (4.6). Here, the pigeons form set A, the pigeonholes are the set B, and f describes which hole each pigeon occupies. Mathematicians have come up with many ingenious applications for the pigeon-hole principle. If there were a cookbook procedure for generating such arguments, we’d give it to you. Unfortunately, there isn’t one. One helpful tip, though: when you try to solve a problem with the pigeonhole principle, the key is to clearly iden-tify three things: 1. The set A (the pigeons). 2. The set B (the pigeonholes). 3. The function f (the rule for assigning pigeons to pigeonholes). “mcs” — 2015/5/18 — 1:43 — page 574 — #582 574 Chapter 14 Cardinality Rules 14.8.1 Hairs on Heads There are a number of generalizations of the pigeonhole principle. For example: Rule 14.8.2 (Generalized Pigeonhole Principle). If jAj > k ! jBj, then every total function f W A ! B maps at least kC1 different elements of A to the same element of B. For example, if you pick two people at random, surely they are extremely un-likely to have exactly the same number of hairs on their heads. However, in the remarkable city of Boston, Massachusetts, there is a group of three people who have exactly the same number of hairs! Of course, there are many completely bald people in Boston, and they all have zero hairs. But we’re talking about non-bald people; say a person is non-bald if they have at least ten thousand hairs on their head. Boston has about 500,000 non-bald people, and the number of hairs on a person’s head is at most 200,000. Let A be the set of non-bald people in Boston, let B D f10; 000; 10; 001; : : : ; 200; 000g, and let f map a person to the number of hairs on his or her head. Since jAj > 2jBj, the Generalized Pigeonhole Principle implies that at least three people have exactly the same number of hairs. We don’t know who they are, but we know they exist! 14.8.2 Subsets with the Same Sum For your reading pleasure, we have displayed ninety 25-digit numbers in Fig-ure 14.4. Are there two different subsets of these 25-digit numbers that have the same sum? For example, maybe the sum of the last ten numbers in the ﬁrst column is equal to the sum of the ﬁrst eleven numbers in the second column? Finding two subsets with the same sum may seem like a silly puzzle, but solving these sorts of problems turns out to be useful in diverse applications such as ﬁnding good ways to ﬁt packages into shipping containers and decoding secret messages. It turns out that it is hard to ﬁnd different subsets with the same sum, which is why this problem arises in cryptography. But it is easy to prove that two such subsets exist. That’s where the Pigeonhole Principle comes in. Let A be the collection of all subsets of the 90 numbers in the list. Now the sum of any subset of numbers is at most 90 ! 1025, since there are only 90 numbers and every 25-digit number is less than 1025. So let B be the set of integers f0; 1; : : : ; 90 25 ! 10 g, and let f map each subset of numbers (in A) to its sum (in B). We proved that an n-element set has 2n different subsets in Section 14.2. There-fore: jAj D 290 & 1:237 ⇥1027 “mcs” — 2015/5/18 — 1:43 — page 575 — #583 14.8. The Pigeonhole Principle 575 0020480135385502964448038 3171004832173501394113017 5763257331083479647409398 8247331000042995311646021 0489445991866915676240992 3208234421597368647019265 5800949123548989122628663 8496243997123475922766310 1082662032430379651370981 3437254656355157864869113 6042900801199280218026001 8518399140676002660747477 1178480894769706178994993 3574883393058653923711365 6116171789137737896701405 8543691283470191452333763 1253127351683239693851327 3644909946040480189969149 6144868973001582369723512 8675309258374137092461352 1301505129234077811069011 3790044132737084094417246 6247314593851169234746152 8694321112363996867296665 1311567111143866433882194 3870332127437971355322815 6814428944266874963488274 8772321203608477245851154 1470029452721203587686214 4080505804577801451363100 6870852945543886849147881 8791422161722582546341091 1578271047286257499433886 4167283461025702348124920 6914955508120950093732397 9062628024592126283973285 1638243921852176243192354 4235996831123777788211249 6949632451365987152423541 9137845566925526349897794 1763580219131985963102365 4670939445749439042111220 7128211143613619828415650 9153762966803189291934419 1826227795601842231029694 4815379351865384279613427 7173920083651862307925394 9270880194077636406984249 1843971862675102037201420 4837052948212922604442190 7215654874211755676220587 9324301480722103490379204 2396951193722134526177237 5106389423855018550671530 7256932847164391040233050 9436090832146695147140581 2781394568268599801096354 5142368192004769218069910 7332822657075235431620317 9475308159734538249013238 2796605196713610405408019 5181234096130144084041856 7426441829541573444964139 9492376623917486974923202 2931016394761975263190347 5198267398125617994391348 7632198126531809327186321 9511972558779880288252979 2933458058294405155197296 5317592940316231219758372 7712154432211912882310511 9602413424619187112552264 3075514410490975920315348 5384358126771794128356947 7858918664240262356610010 9631217114906129219461111 8149436716871371161932035 3157693105325111284321993 3111474985252793452860017 5439211712248901995423441 7898156786763212963178679 9908189853102753335981319 3145621587936120118438701 5610379826092838192760458 8147591017037573337848616 9913237476341764299813987 3148901255628881103198549 5632317555465228677676044 5692168374637019617423712 8176063831682536571306791 Figure 14.4 Ninety 25-digit numbers. Can you ﬁnd two different subsets of these numbers that have the same sum? “mcs” — 2015/5/18 — 1:43 — page 576 — #584 576 Chapter 14 Cardinality Rules On the other hand: jBj D 90 ! 1025 C 1 0:901 ⇥1027: Both quantities are enormous, but jAj is a bit greater than jBj. This means that f maps at least two elements of A to the same element of B. In other words, by the Pigeonhole Principle, two different subsets must have the same sum! Notice that this proof gives no indication which two sets of numbers have the same sum. This frustrating variety of argument is called a nonconstructive proof. The $100 prize for two same-sum subsets To see if it was possible to actually ﬁnd two different subsets of the ninety 25-digit numbers with the same sum, we offered a $100 prize to the ﬁrst student who did it. We didn’t expect to have to pay off this bet, but we underestimated the ingenuity and initiative of the students. One computer science major wrote a program that cleverly searched only among a reasonably small set of “plausible” sets, sorted them by their sums, and actually found a couple with the same sum. He won the prize. A few days later, a math major ﬁgured out how to reformulate the sum problem as a “lattice basis reduction” problem; then he found a software package implementing an efﬁcient basis reduction procedure, and using it, he very quickly found lots of pairs of subsets with the same sum. He didn’t win the prize, but he got a standing ovation from the class—staff included. The $500 Prize for Sets with Distinct Subset Sums How can we construct a set of n positive integers such that all its subsets have distinct sums? One way is to use powers of two: f1; 2; 4; 8; 16g This approach is so natural that one suspects all other such sets must involve larger numbers. (For example, we could safely replace 16 by 17, but not by 15.) Remarkably, there are examples involving smaller numbers. Here is one: f6; 9; 11; 12; 13g One of the top mathematicians of the Twentieth Century, Paul Erdos, ˝ conjectured in 1931 that there are no such sets involving signiﬁcantly smaller numbers. More precisely, he conjectured that the largest number in such a set must be greater than c2n for some constant c > 0. He offered $500 to anyone who could prove or disprove his conjecture, but the problem remains unsolved. “mcs” — 2015/5/18 — 1:43 — page 577 — #585 14.8. The Pigeonhole Principle 577 14.8.3 A Magic Trick A Magician sends an Assistant into the audience with a deck of 52 cards while the Magician looks away. Five audience members each select one card from the deck. The Assistant then gathers up the ﬁve cards and holds up four of them so the Magician can see them. The Magician concentrates for a short time and then correctly names the secret, ﬁfth card! Since we don’t really believe the Magician can read minds, we know the Assis-tant has somehow communicated the secret card to the Magician. Real Magicians and Assistants are not to be trusted, so we expect that the Assistant would secretly signal the Magician with coded phrases or body language, but for this trick they don’t have to cheat. In fact, the Magician and Assistant could be kept out of sight of each other while some audience member holds up the 4 cards designated by the Assistant for the Magician to see. Of course, without cheating, there is still an obvious way the Assistant can com-municate to the Magician: he can choose any of the 4ä D 24 permutations of the 4 cards as the order in which to hold up the cards. However, this alone won’t quite work: there are 48 cards remaining in the deck, so the Assistant doesn’t have enough choices of orders to indicate exactly what the secret card is (though he could narrow it down to two cards). 14.8.4 The Secret The method the Assistant can use to communicate the ﬁfth card exactly is a nice application of what we know about counting and matching. The Assistant has a second legitimate way to communicate: he can choose which of the ﬁve cards to keep hidden. Of course, it’s not clear how the Magician could determine which of these ﬁve possibilities the Assistant selected by looking at the four visible cards, but there is a way, as we’ll now explain. The problem facing the Magician and Assistant is actually a bipartite matching problem. Each vertex on the left will correspond to the information available to the Assistant, namely, a set of 5 cards. So the set X of left hand vertices will have 52 5 elements. Each vertex on the right will correspond to the information available to the Ma-% & gician, namely, a sequence of 4 distinct cards. So the set Y of right hand vertices will have 52 ! 51 ! 50 ! 49 elements. When the audience selects a set of 5 cards, then the Assistant must reveal a sequence of 4 cards from that hand. This constraint is represented by having an edge between a set of 5 cards on the left and a sequence of 4 cards on the right precisely when every card in the sequence is also in the set. This speciﬁes the bipartite graph. Some edges are shown in the diagram in “mcs” — 2015/5/18 — 1:43 — page 578 — #586 578 Chapter 14 Cardinality Rules zEall sequences of 5 y distinct cards sets of Eall 6 cards g9<L<R<3~h g9<L<R<3~<7~h gL<9<R<3~h gL<9<7~<Rh <R g9<L <:}<7~h Figure 14.5 The bipartite graph where the nodes on the left correspond to sets of 5 cards and the nodes on the right correspond to sequences of 4 cards. There is an edge between a set and a sequence whenever all the cards in the sequence are contained in the set. Figure 14.5. For example, f8~; K; Q; 2}; 6}g (14.2) is an element of X on the left. If the audience selects this set of 5 cards, then there are many different 4-card sequences on the right in set Y that the Assis-tant could choose to reveal, including .8~; K; Q; 2}/, .K; 8~; Q; 2}/, and .K; 8~; 6}; Q/. What the Magician and his Assistant need to perform the trick is a matching for the X vertices. If they agree in advance on some matching, then when the audience selects a set of 5 cards, the Assistant reveals the matching sequence of 4 cards. The Magician uses the matching to ﬁnd the audience’s chosen set of 5 cards, and so he can name the one not already revealed. For example, suppose the Assistant and Magician agree on a matching containing the two bold edges in Figure 14.5. If the audience selects the set f8~; K; Q; 9\|; 6}g; (14.3) then the Assistant reveals the corresponding sequence .K; 8~; 6}; Q/: (14.4) “mcs” — 2015/5/18 — 1:43 — page 579 — #587 14.8. The Pigeonhole Principle 579 Using the matching, the Magician sees that the hand (14.3) is matched to the se-quence (14.4), so he can name the one card in the corresponding set not already revealed, namely, the 9\|. Notice that the fact that the sets are matched, that is, that different sets are paired with distinct sequences, is essential. For example, if the audience picked the previous hand (14.2), it would be possible for the Assistant to reveal the same sequence (14.4), but he better not do that; if he did, then the Magician would have no way to tell if the remaining card was the 9\| or the 2}. So how can we be sure the needed matching can be found? The answer is that each vertex on the left has degree 5!4ä D 120, since there are ﬁve ways to select the card kept secret and there are 4ä permutations of the remaining 4 cards. In addition, each vertex on the right has degree 48, since there are 48 possibilities for the ﬁfth card. So this graph is degree-constrained according to Deﬁnition 11.5.5, and so has a matching by Theorem 11.5.6. In fact, this reasoning shows that the Magician could still pull off the trick if 120 cards were left instead of 48, that is, the trick would work with a deck as large as 124 different cards—without any magic! 14.8.5 The Real Secret But wait a minute! It’s all very well in principle to have the Magician and his Assistant % & agree on a matching, but how are they supposed to remember a matching with 52 5 D 2; 598; 960 edges? For the trick to work in practice, there has to be a way to match hands and card sequences mentally and on the ﬂy. We’ll describe one approach. As a running example, suppose that the audience selects: 10~ 9} 3~ Q J }: ✏The Assistant picks out two cards of the same suit. In the example, the assistant might choose the 3~ and 10~. This is always possible because of the Pigeonhole Principle—there are ﬁve cards and 4 suits so two cards must be in the same suit. ✏The Assistant locates the ranks of these two cards on the cycle shown in Fig-ure 14.6. For any two distinct ranks on this cycle, one is always between 1 and 6 hops clockwise from the other. For example, the 3~ is 6 hops clock-wise from the 10~. ✏The more counterclockwise of these two cards is revealed ﬁrst, and the other becomes the secret card. Thus, in our example, the 10~ would be revealed, and the 3~ would be the secret card. Therefore: “mcs” — 2015/5/18 — 1:43 — page 580 — #588 580 Chapter 14 Cardinality Rules B L 3 R 4 K 5 21 6 : 7 9 8 Figure 14.6 The 13 card ranks arranged in cyclic order. – The suit of the secret card is the same as the suit of the ﬁrst card re-vealed. – The rank of the secret card is between 1 and 6 hops clockwise from the rank of the ﬁrst card revealed. ✏All that remains is to communicate a number between 1 and 6. The Magician and Assistant agree beforehand on an ordering of all the cards in the deck from smallest to largest such as: A\| A} A~ A 2\| 2} 2~ 2 : : : K~ K The order in which the last three cards are revealed communicates the num-ber according to the following scheme: . small; medium; large / = 1 . small; large; medium / = 2 . medium; small; large / = 3 . medium; large; small / = 4 . large; small; medium / = 5 . large; medium; small / = 6 In the example, the Assistant wants to send 6 and so reveals the remaining three cards in large, medium, small order. Here is the complete sequence that the Magician sees: 10~ Q J } 9} “mcs” — 2015/5/18 — 1:43 — page 581 — #589 14.9. Inclusion-Exclusion 581 ✏The Magician starts with the ﬁrst card, 10~, and hops 6 ranks clockwise to reach 3~, which is the secret card! So that’s how the trick can work with a standard deck of 52 cards. On the other hand, Hall’s Theorem implies that the Magician and Assistant can in principle per-form the trick with a deck of up to 124 cards. It turns out that there is a method which they could actually learn to use with a reasonable amount of practice for a 124-card deck, but we won’t explain it here. 14.8.6 The Same Trick with Four Cards? Suppose that the audience selects only four cards and the Assistant reveals a se-quence of three to the Magician. Can the Magician determine the fourth card? Let X be all the sets of four cards that the audience might select, and let Y be all the sequences of three cards that the Assistant might reveal. Now, on one hand, we have jXj D 52 4 ! D 270; 725 by the Subset Rule. On the other hand, we have jY j D 52 ! 51 ! 50 D 132; 600 by the Generalized Product Rule. Thus, by the Pigeonhole Principle, the Assistant must reveal the same sequence of three cards for at least ⇠270; 725 3 132; 600 ⇡ D different four-card hands. This is bad news for the Magician: if he sees that se-quence of three, then there are at least three possibilities for the fourth card which he cannot distinguish. So there is no legitimate way for the Assistant to communi-cate exactly what the fourth card is! MIT OpenCourseWare 6.042J / 18.062J Mathematics for Computer Science Spring 2015 For information about citing these materials or our Terms of Use, visit:
2592	https://www.doubtnut.com/qna/642605233	The most probable speed of 8 g of H2 is 200ms−1. Average kinetic energy (neglect rotational and vibrational energy ) of H2 gas is : 480 J 240 J 120 J none of these More from this Exercise The correct Answer is:b To find the average kinetic energy of hydrogen gas (H₂) given the most probable speed, we can follow these steps: 1. Identify Given Data: - Mass of H₂ = 8 g - Most probable speed (u_mp) = 200 m/s 2. Convert Mass to Kilograms: - Since we need the mass in kilograms for the calculations, convert 8 g to kg: Mass in kg=8 g1000=0.008 kg 3. Use the Formula for Most Probable Speed: - The formula for the most probable speed (u_mp) is given by: ump=√2RTM - Where: - R = universal gas constant = 8.314 J/(mol·K) - T = temperature in Kelvin - M = molar mass in kg/mol (for H₂, M = 0.002 kg/mol) 4. Rearranging the Formula to Find RT: - Squaring both sides: u2mp=2RTM - Rearranging to find RT: RT=M⋅u2mp2 5. Substituting Values: - Substitute M = 0.002 kg/mol and u_mp = 200 m/s: RT=0.002⋅(200)22 - Calculate (200)2=40000: RT=0.002⋅400002=802=40 J/mol 6. Calculate Number of Moles (n): - Using the formula: n=massmolar mass=8 g2 g/mol=4 mol 7. Calculate Average Kinetic Energy (KE_avg): - The average kinetic energy of the gas is given by: KEavg=32nRT - Substitute n = 4 mol and RT = 40 J/mol: KEavg=32⋅4⋅40 - Calculate: KEavg=3⋅4⋅402=4802=240 J Final Answer: The average kinetic energy of H₂ gas is 240 J. --- To find the average kinetic energy of hydrogen gas (H₂) given the most probable speed, we can follow these steps: Identify Given Data: Mass of H₂ = 8 g Most probable speed (u_mp) = 200 m/s Convert Mass to Kilograms: Since we need the mass in kilograms for the calculations, convert 8 g to kg: Mass in kg=8 g1000=0.008 kg Use the Formula for Most Probable Speed: The formula for the most probable speed (u_mp) is given by: ump=√2RTM Where: R = universal gas constant = 8.314 J/(mol·K) T = temperature in Kelvin M = molar mass in kg/mol (for H₂, M = 0.002 kg/mol) Rearranging the Formula to Find RT: Squaring both sides: u2mp=2RTM Rearranging to find RT: RT=M⋅u2mp2 Substituting Values: Substitute M = 0.002 kg/mol and u_mp = 200 m/s: RT=0.002⋅(200)22 Calculate (200)2=40000: RT=0.002⋅400002=802=40 J/mol Calculate Number of Moles (n): Using the formula: n=massmolar mass=8 g2 g/mol=4 mol Calculate Average Kinetic Energy (KE_avg): The average kinetic energy of the gas is given by: KEavg=32nRT Substitute n = 4 mol and RT = 40 J/mol: KEavg=32⋅4⋅40 Calculate: KEavg=3⋅4⋅402=4802=240 J Final Answer: The average kinetic energy of H₂ gas is 240 J. Topper's Solved these Questions Explore 30 Videos Explore 4 Videos Explore 14 Videos Explore 1 Video Similar Questions The root mean square speed of 8 g of He is 300 ms−1. Total kinetic energy of He gas is : What is average kinetic energy of 1 mole of SO2 at 300 K? Knowledge Check The kinetic energy of 1 g molecule of a gas, at normal temperature and pressure, is Two gases O2 and H2 are at the same temperature . If Eo is the average kinetic energy of a molecule of oxygen sample , and EH is the average kinetic energy of a molecule of hydrogen sample , then A particle of mass m has half the kinetic energy of another particle of mass m2. If the speed of the heavier particle is increased by 2ms−1 its new kinetic energy becomes equal to t he original kinetic energy of the lighter particle. The ratio of the orighinal speeds of the lighter and heavier particle is If the speed of a vehicle is increased by 1ms−1, its kinetic energy is doubled , then original speed of the vehicle is A gaseous mixture consists of 16 g of O2 and 16 g of He at temperature T. Neglecting the vibrational modes, total internal energy of the system is Which of the following comparisons of the average kinetic energy and the average molecular speeds of H2andN2 gases at 300 K is CORRECT? The total kinetic energy of a mixture of 4 g of H2and4g of He at 300 K is A rod of mass m spins with an angular speed ω=√gl, Find its a. kinetic energy of rotation. b. kinetic energy of translation c. total kinetic energy. NARENDRA AWASTHI ENGLISH-GASEOUS STATE-Subjective problems The most probable speed of 8 g of H(2) is 200 ms^(-1). Average kinetic... A bubble of gas released at the bottom of a lake increases to four t... A gaseous mixture containing equal mole sof H(2),O(2) and He is subjec... 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2593	https://dsp.stackexchange.com/questions/54606/is-initial-value-theorem-sufficient-to-prove-causality-of-a-system	z transform - Is "initial value theorem" sufficient to prove causality of a system? - Signal Processing Stack Exchange Join Signal Processing By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Is "initial value theorem" sufficient to prove causality of a system? Ask Question Asked 6 years, 8 months ago Modified6 years, 8 months ago Viewed 1k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Textbook states Initial value theorem as follows: if x[n]x[n] is equal to zero for n<0 n<0, the initial value, xx, may be found from X(z)X(z) as follows: x=lim z→∞X(z)x=lim z→∞X(z) Ok...no problem there... then, I was working problems 4.16 in Schaum's outline for DSP. X(z)=z(1−0.5 z−1)(1−0.6 z−2)\|z\|<0.5 X(z)=z(1−0.5 z−1)(1−0.6 z−2)\|z\|<0.5 The textbook gives function X(z)X(z) and says applying "initial value theorem" to X(z)X(z), we see that the limit doesn't exist therefore the system is non-casual. Is applying the "initial value theorem" to a system sufficient to determine if a system is non-casual? or are they inferring this information in some other way? Then, the book goes on to say, we can modify the system by adding delay to make "initial value theorem" converge to a value. function is modified as follows: X(z)=1(1−0.5 z−1)(1−0.6 z−2)\|z\|<0.5 X(z)=1(1−0.5 z−1)(1−0.6 z−2)\|z\|<0.5 Applying initial value theorem to this problem, we see that it converges to a finite value therefore the system is causal. Again is the "initial value theorem" sufficient to prove a system is casual? I guess i'm just confused because the only rule that the textbook gives that links casual systems to the initial value theorem is as a pre-requisite of the system being right-sided before applying the initial value theorem. I was wondering what's the missing rule if there is one? I can kind of see that if the limit isn't met, that might imply the system is not casual, on the other hand, why can't a causal system have an infinite initial value, mathematically speaking... or is that something that just would never happen in the real world therefore it's always non-casual. and what about the other way around? z-transform Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Jan 5, 2019 at 18:06 Matt L. 94.3k 10 10 gold badges 85 85 silver badges 190 190 bronze badges asked Jan 4, 2019 at 18:41 Bill MooreBill Moore 247 2 2 silver badges 10 10 bronze badges 3 1 Your textbook definition is incomplete. It should read "If x[n]=0 x[n]=0 for n<0 n<0, then lim z→∞X[z]lim z→∞X[z]exists and is equal to xx, the initial value of the sequence." Note that X=lim z→∞X[z]X=lim z→∞X[z] as you write is nonsensical: some people might fudge notation a little bit and write X(∞)X(∞) as short-hand or the more formal lim z→∞X[z]lim z→∞X[z] but even such benighted folks will not use XX as short-hand for the limit. Check your book! It probably says x=lim z→∞X[z]x=lim z→∞X[z] and not X=lim z→∞X[z].X=lim z→∞X[z]. Upper and lower case matter....Dilip Sarwate –Dilip Sarwate 2019-01-04 21:22:09 +00:00 Commented Jan 4, 2019 at 21:22 sorry, typo with the uppercase character.Bill Moore –Bill Moore 2019-01-05 00:14:55 +00:00 Commented Jan 5, 2019 at 0:14 1 So how about editing your question to clean it up instead of just acknowledging it in a comment which many people won't even bother reading?Dilip Sarwate –Dilip Sarwate 2019-01-05 03:23:46 +00:00 Commented Jan 5, 2019 at 3:23 Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. One important thing that hasn't been mentioned is the case when the limit does exist. If the limit exists, you cannot conclude that X(z)X(z) is necessarily the transform of a causal sequence. Take as an example the function X(z)=1(1−a z−1)(1−b z−1)(1)(1)X(z)=1(1−a z−1)(1−b z−1) Clearly, lim z→∞X(z)=1(2)(2)lim z→∞X(z)=1 yet, X(z)X(z) as given by (1)(1) is not necessarily the transform of a causal sequence. There exist three sequences with different ROCs with X(z)X(z) as their transform: one of them is causal, one is left-sided, and one is double-sided. In sum, if the limit lim z→∞X(z)lim z→∞X(z) does not exist, we can conclude that there is no causal sequence corresponding to the transform X(z)X(z). If the limit exists, we know that there's at least one causal sequence corresponding to X(z)X(z). However, there are usually also other, non-causal sequences with the same transform. In your example, the fact that the ROC is given by \|z\|<0.5\|z\|<0.5 clearly indicates that the corresponding sequence is left-sided, so there's no need to apply the initial value theorem to check causality. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Jan 5, 2019 at 20:12 answered Jan 5, 2019 at 19:20 Matt L.Matt L. 94.3k 10 10 gold badges 85 85 silver badges 190 190 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Basing on the definition of the z-transform, the initial value theorem can be written and proved as follows: lim z→∞X(z)=lim z→∞∑k=−∞∞f[k]z−k lim z→∞X(z)=lim z→∞∑k=−∞∞f[k]z−k In the case that f[k]=0∀k<0 f[k]=0∀k<0 (causal), obviously this limit will converge to ff (assuming the signal doesn't contain infinite values). If there exists any negative k k for which f[k]≠0 f[k]≠0, it doesn't converge. Now concerning the values of the signal (disclaimer: I'm more of an engineer than a mathematician), in the discrete domain, I've never seen any case where a signal took infinite values, not even theoretically (unlike in the continuous domain, where the Dirac delta takes care of those cases). Having a signal taking infinite values in discrete time would be very problematic, as everything is defined as sums (instead of integrals). To conclude, from my point of view, assuming f[k]∈C f[k]∈C, lim z→∞X(z)∈C⇔f[.]is causal lim z→∞X(z)∈C⇔f[.]is causal meaning that if the limit X(z)X(z) is finite (may as well be complex), then f[.]f[.] is causal. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Jan 4, 2019 at 20:37 Daniel. RDaniel. R 66 5 5 bronze badges 1 thanks! The part I was missing was "non-convergence if negative k present with value", that makes sense now.Bill Moore –Bill Moore 2019-01-05 00:22:29 +00:00 Commented Jan 5, 2019 at 0:22 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Begin with the power series expansion of the Z-transform of a sequence x[n]x[n] : X(z)=∑n=−∞∞x[n]z−n=...+x[−1]z 2+x[−1]z+x+xz−1+xz−2+...X(z)=∑n=−∞∞x[n]z−n=...+x[−1]z 2+x[−1]z+x+xz−1+xz−2+... Now if the sequence x[n]x[n] is causal then those samples of x[n]x[n] for n<0 n<0 are by definition zero and the Z-transform becomes: X(z)=∑n=0∞x[n]z−n=x+xz−1+xz−2+...X(z)=∑n=0∞x[n]z−n=x+xz−1+xz−2+... Now it's very easy to see that when z z is sent to infinity, X(z)=xX(z)=x. If the signal is known to have all finite samples, then this limit will be finite for a causal signal. But if thes sequence is not causal, then there exist at least one nonzero sample x[n]x[n] for n<0 n<0 and the Z-transform includes positive power of z z and, therefore, now if you set the limit of z z to infinity the result will be infinite (or does not converge) which indicates that there is a nonzero sample to the left of n=0 n=0 hence the sequence is nancausal. Yes it's sufficient to conclude (provided all samples are finite) Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications answered Jan 4, 2019 at 20:42 Fat32Fat32 28.8k 3 3 gold badges 25 25 silver badges 51 51 bronze badges Add a comment\| Your Answer Thanks for contributing an answer to Signal Processing Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. 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2594	https://blackpawn.com/texts/pointinpoly/	Point in triangle test Point in triangle test Same Side Technique A common way to check if a point is in a triangle is to find the vectors connecting the point to each of the triangle's three vertices and sum the angles between those vectors. If the sum of the angles is 2pi then the point is inside the triangle, otherwise it is not. It works, but it is very slow. This text explains a faster and much easier method. First off, forgive the nasty coloring. I'm really sorry about it. Honest. Okay, A B C forms a triangle and all the points inside it are yellow. Lines AB, BC, and CA each split space in half and one of those halves is entirely outside the triangle. This is what we'll take advantage of. For a point to be inside the traingle A B C it must be below AB and left of BC and right of AC. If any one of these tests fails we can return early. But, how do we tell if a point is on the correct side of a line? I'm glad you asked. If you take the cross product of [B-A] and [p-A], you'll get a vector pointing out of the screen. On the other hand, if you take the cross product of [B-A] and [p'-A] you'll get a vector pointing into the screen. Ah ha! In fact if you cross [B-A] with the vector from A to any point above the line AB, the resulting vector points out of the screen while using any point below AB yields a vector pointing into the screen. So all we need to do to distinguish which side of a line a point lies on is take a cross product. The only question remaining is: how do we know what direction the cross product should point in? Because the triangle can be oriented in any way in 3d-space, there isn't some set value we can compare with. Instead what we need is a reference point - a point that we know is on a certain side of the line. For our triangle, this is just the third point C. So, any point p where [B-A] cross [p-A] does not point in the same direction as [B-A] cross [C-A] isn't inside the triangle. If the cross products do point in the same direction, then we need to test p with the other lines as well. If the point was on the same side of AB as C and is also on the same side of BC as A and on the same side of CA as B, then it is in the triangle. Implementing this is a breeze. We'll make a function that tells us if two points are on the same side of a line and have the actual point-in-triangle function call this for each edge. function SameSide(p1,p2, a,b) cp1 = CrossProduct(b-a, p1-a) cp2 = CrossProduct(b-a, p2-a) if DotProduct(cp1, cp2) >= 0 then return true else return false function PointInTriangle(p, a,b,c) if SameSide(p,a, b,c) and SameSide(p,b, a,c) and SameSide(p,c, a,b) then return true else return false It's simple, effective and has no square roots, arc cosines, or strange projection axis determination nastiness. Barycentric Technique The advantage of the method above is that it's very simple to understand so that once you read it you should be able to remember it forever and code it up at any time without having to refer back to anything. It's just - hey the point has to be on the same side of each line as the triangle point that's not in the line. Cake. Well, there's another method that is also as easy conceptually but executes faster. The downside is there's a little more math involved, but once you see it worked out it should be no problem. So remember that the three points of the triangle define a plane in space. Pick one of the points and we can consider all other locations on the plane as relative to that point. Let's go with A -- it'll be our origin on the plane. Now what we need are basis vectors so we can give coordinate values to all the locations on the plane. We'll pick the two edges of the triangle that touch A, (C - A) and (B - A). Now we can get to any point on the plane just by starting at A and walking some distance along (C - A) and then from there walking some more in the direction (B - A). With that in mind we can now describe any point on the plane as P = A + u (C - A) + v (B - A) Notice now that if u or v < 0 then we've walked in the wrong direction and must be outside the triangle. Also if u or v > 1 then we've walked too far in a direction and are outside the triangle. Finally if u + v > 1 then we've crossed the edge BC again leaving the triangle. Given u and v we can easily calculate the point P with the above equation, but how can we go in the reverse direction and calculate u and v from a given point P? Time for some math! P = A + u (C - A) + v (B - A) // Original equation (P - A) = u (C - A) + v (B - A) // Subtract A from both sides v2 = u v0 + v v1 // Substitute v0, v1, v2 for less writing // We have two unknowns (u and v) so we need two equations to solve // for them. Dot both sides by v0 to get one and dot both sides by // v1 to get a second. (v2) . v0 = (u v0 + v v1) . v0 (v2) . v1 = (u v0 + v v1) . v1 // Distribute v0 and v1 v2 . v0 = u (v0 . v0) + v (v1 . v0) v2 . v1 = u (v0 . v1) + v (v1 . v1) // Now we have two equations and two unknowns and can solve one // equation for one variable and substitute into the other. Or // if you're lazy like me, fire up Mathematica and save yourself // some handwriting. Solve[v2.v0 == {u(v0.v0) + v(v1.v0), v2.v1 == u(v0.v1) + v(v1.v1)}, {u, v}] u = ((v1.v1)(v2.v0)-(v1.v0)(v2.v1)) / ((v0.v0)(v1.v1) - (v0.v1)(v1.v0)) v = ((v0.v0)(v2.v1)-(v0.v1)(v2.v0)) / ((v0.v0)(v1.v1) - (v0.v1)(v1.v0)) Here's an implementation in Flash that you can play with. :) // Compute vectors v0 = C - A v1 = B - A v2 = P - A // Compute dot products dot00 = dot(v0, v0) dot01 = dot(v0, v1) dot02 = dot(v0, v2) dot11 = dot(v1, v1) dot12 = dot(v1, v2) // Compute barycentric coordinates invDenom = 1 / (dot00 dot11 - dot01 dot01) u = (dot11 dot02 - dot01 dot12) invDenom v = (dot00 dot12 - dot01 dot02) invDenom // Check if point is in triangle return (u >= 0) && (v >= 0) && (u + v < 1) The algorithm outlined here follows one of the techniques described in Realtime Collision Detection. You can also find more information about Barycentric Coordinates at Wikipedia and MathWorld.
2595	https://stackoverflow.com/questions/48752035/calculating-the-area-between-a-curve-and-a-straight-line-without-finding-the-fun	r - Calculating the area between a curve and a straight line without finding the function - Stack Overflow Join Stack Overflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google Sign up with GitHub OR Email Password Sign up Already have an account? Log in Skip to main content Stack Overflow 1. About 2. Products 3. 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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Calculating the area between a curve and a straight line without finding the function Ask Question Asked 7 years, 7 months ago Modified7 years, 7 months ago Viewed 479 times Part of R Language Collective This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I have two points in a 2D space (A and B) and a curve that starts in A and ends in B. I don't have the function of that curve, but an array of n points on that curve. I wish to calculate the area that is locked between the imaginary line AB, and the curve. Any help on how to do that in R would be greatly appreciated. R Language Collective r Share Share a link to this question Copy linkCC BY-SA 3.0 Improve this question Follow Follow this question to receive notifications asked Feb 12, 2018 at 17:19 G.N.G.N. 139 1 1 silver badge 8 8 bronze badges 4 2 It would be great if you can share some example dataset so people can study your question.www –www 2018-02-12 17:26:29 +00:00 Commented Feb 12, 2018 at 17:26 Here is the link to an example dataset: link The first and last points are A and B.G.N. –G.N. 2018-02-12 17:32:48 +00:00 Commented Feb 12, 2018 at 17:32 approximate the function using approxfun, and then use integrate to find the area bouncyball –bouncyball 2018-02-12 17:51:16 +00:00 Commented Feb 12, 2018 at 17:51 When asking for help, you should include a simple reproducible example with sample input and desired output that can be used to test and verify possible solutions.MrFlick –MrFlick 2018-02-12 19:36:43 +00:00 Commented Feb 12, 2018 at 19:36 Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. A solution using the sf package. Assuming that dat is a data frame. If you have a matrix, please begin with the second step. The idea is to create a polygon and then calculate area. ```r library(sf) Convert to matrix dat_m <- as.matrix(dat) Repeat the first row dat_m <- rbind(dat_m, dat_m[1, ]) Convert to polygon dat_pl <- st_polygon(list(dat_m)) Calculate the area st_area(dat_pl) 4874 ``` By the way, here is how the polygon looks like. r plot(dat_pl) DATA r dat <- read.table(text = "-1416.7 214.7 -1418.4 216.8 -1420.3 219.2 -1422.2 221.8 -1424.2 224.5 -1426.3 227.5 -1428.4 230.6 -1430.3 233.9 -1432.2 237.3 -1434 241 -1435.6 244.8 -1437 248.7 -1438.4 252.8 -1439.5 257.1 -1440.5 261.4 -1441.2 265.8 -1441.8 270.4 -1442.2 274.9 -1442.4 279.5 -1442.5 284.1 -1442.4 288.8 -1442.1 293.5 -1441.8 298.3 -1441.4 303.2 -1441 308.2 -1440.6 313.3 -1440.4 318.3 -1440.1 323.3 -1439.9 328.2 -1439.7 333.1 -1439.4 338 -1439.1 342.8 -1438.6 347.6 -1438.1 352.4 -1437.4 357.1 -1436.7 361.8 -1435.8 366.4 -1435 371 -1434 375.5 -1433 379.9 -1432 384.3 -1430.9 388.7 -1429.9 392.9 -1428.8 397.2 -1427.8 401.3 -1426.8 405.4 -1425.9 409.4 -1425.2 413.2 -1424.7 416.8 -1424.3 420.2 -1424 423.4 -1423.8 426.4 -1423.7 429.3 -1423.7 432 -1423.8 434.6 -1423.9 437.1 -1424 439.5 -1424.1 441.8 -1424.2 444 -1424.2 446.2 -1424 448.3 -1423.8 450.3 -1423.4 452.3 -1423 454.3 -1422.4 456.1 -1421.9 457.9 -1421.4 459.6 -1420.9 461.2 -1420.5 462.8 -1420.1 464.3 -1419.8 465.6 -1419.6 466.9 -1419.3 468.1 -1419 469.2 -1418.6 470.2 -1418.2 471.2 -1417.7 472.1 -1417.2 473 -1416.6 473.8 -1415.9 474.5 -1415.2 475.2 -1414.2 475.8 -1413.2 476.4 -1412.2 476.9 -1411.3 477.3") Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 12, 2018 at 17:56 wwwwww 39.3k 12 12 gold badges 52 52 silver badges 93 93 bronze badges Comments Add a comment This answer is useful 2 Save this answer. Show activity on this post. Here is one approach using the pracma package: ```r library(pracma) trapz( c(data[1,2], tail(data[,2])), c(data[1,1],tail(data[,1])) ) - trapz(data[,2], data[,1]) 4276.685 ``` The trapz function finds area under a set of points. The first trapz finds the area under the AB line, and subtracts by the second trapz, the area under the set of points. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 12, 2018 at 17:51 thcthc 9,715 1 1 gold badge 29 29 silver badges 44 44 bronze badges Comments Add a comment This answer is useful 2 Save this answer. Show activity on this post. You can do this without any packages. Here is some sample data. r set.seed(2018) x = sort(runif(150, 0,2)) y = sin(x) + sin(10x)/20 We can get the area from the first point (x[ 1]) to the last (x). But the area between the curves is just the area under the upper curve minus the area under the line. You can use approxfun with your data to get a good approximation to the upper function. Then just do the integrals. ```r Find the line m = (y - y) / (x - x) b = y - m x Get the functions and compute the integrals. F1 = approxfun(x,y) F2 = function(x) { mx+b } integrate(F1, x, x)$value - integrate(F2, x, x)$value 0.4539384 ``` Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 12, 2018 at 18:03 G5WG5W 37.8k 10 10 gold badges 56 56 silver badges 85 85 bronze badges Comments Add a comment This answer is useful 2 Save this answer. Show activity on this post. Here is an approach with sp: ```r library(sp) Polygon(r)@area 4874 ``` data: r dput(r) structure(list(V1 = c(-1416.7, -1418.4, -1420.3, -1422.2, -1424.2, -1426.3, -1428.4, -1430.3, -1432.2, -1434, -1435.6, -1437, -1438.4, -1439.5, -1440.5, -1441.2, -1441.8, -1442.2, -1442.4, -1442.5, -1442.4, -1442.1, -1441.8, -1441.4, -1441, -1440.6, -1440.4, -1440.1, -1439.9, -1439.7, -1439.4, -1439.1, -1438.6, -1438.1, -1437.4, -1436.7, -1435.8, -1435, -1434, -1433, -1432, -1430.9, -1429.9, -1428.8, -1427.8, -1426.8, -1425.9, -1425.2, -1424.7, -1424.3, -1424, -1423.8, -1423.7, -1423.7, -1423.8, -1423.9, -1424, -1424.1, -1424.2, -1424.2, -1424, -1423.8, -1423.4, -1423, -1422.4, -1421.9, -1421.4, -1420.9, -1420.5, -1420.1, -1419.8, -1419.6, -1419.3, -1419, -1418.6, -1418.2, -1417.7, -1417.2, -1416.6, -1415.9, -1415.2, -1414.2, -1413.2, -1412.2, -1411.3 ), V2 = c(214.7, 216.8, 219.2, 221.8, 224.5, 227.5, 230.6, 233.9, 237.3, 241, 244.8, 248.7, 252.8, 257.1, 261.4, 265.8, 270.4, 274.9, 279.5, 284.1, 288.8, 293.5, 298.3, 303.2, 308.2, 313.3, 318.3, 323.3, 328.2, 333.1, 338, 342.8, 347.6, 352.4, 357.1, 361.8, 366.4, 371, 375.5, 379.9, 384.3, 388.7, 392.9, 397.2, 401.3, 405.4, 409.4, 413.2, 416.8, 420.2, 423.4, 426.4, 429.3, 432, 434.6, 437.1, 439.5, 441.8, 444, 446.2, 448.3, 450.3, 452.3, 454.3, 456.1, 457.9, 459.6, 461.2, 462.8, 464.3, 465.6, 466.9, 468.1, 469.2, 470.2, 471.2, 472.1, 473, 473.8, 474.5, 475.2, 475.8, 476.4, 476.9, 477.3)), .Names = c("V1", "V2"), class = "data.frame", row.names = c(NA, -85L)) Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Feb 12, 2018 at 18:04 missusemissuse 19.8k 3 3 gold badges 29 29 silver badges 53 53 bronze badges 1 Comment Add a comment missuse missuseOver a year ago @www the Polygon function creates a polygon from a two column numeric matrix of coordinates (given in dput). The slot area of this object contains the area. 2018-02-12T18:16:01.73Z+00:00 0 Reply Copy link Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. To learn more, see our tips on writing great answers. Draft saved Draft discarded Sign up or log in Sign up using Google Sign up using Email and Password Submit Post as a guest Name Email Required, but never shown Post Your Answer Discard By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions r See similar questions with these tags. 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2596	https://brightchamps.com/en-us/math/measurement/grams-to-kilograms	Our Programs Learn More About us Login Table Of Contents What is Gram? What is Kilogram? Grams to Kilograms Formula How to Convert Grams to Kilograms? Grams to Kilograms Conversion Chart Common Mistakes and How to Avoid Them in Grams to Kilograms Conversion Grams to Kilograms Conversion Examples FAQs on Grams to Kilograms Important Glossaries for Grams to Kilograms Explore More measurement Summarize this article: 301 Learners Last updated on August 5, 2025 Grams To Kilograms Conversion We can measure weight using units like grams, kilograms, pounds, ounces, or tons. Different units are used to measure different quantities. A gram is a small unit of measurement that we use for lighter objects, like a paperclip or a teaspoon of sugar. A kilogram (kg) is a larger unit of measurement, and we use it to measure heavier objects, like a bag of flour or a person's weight. Sometimes we need to change grams to kilograms to make it easier to understand weights. In this topic, we will learn how to convert grams to kilograms. Grams To Kilograms ConversionforUSStudents What is Gram? A gram is a unit of weight that is part of the metric system, which is widely used around the world for measuring small masses. The metric system is based on powers of 10, which simplifies unit conversions. 1 gram is equal to one-thousandth of a kilogram, meaning there are 1,000 grams in 1 kilogram (1,000 g = 1 kg). The symbol for grams is g, and it is commonly seen on food packages and scales. What is Kilogram? A unit of measurement used to measure weight is called a kilogram. One kilogram is equal to 1,000 grams. Kilograms are commonly used to measure heavier objects or larger masses. The symbol used to measure kilograms is kg. Grams to Kilograms Formula To convert grams to kilograms, we use the following formula: 1 kilogram = 1,000 grams Kilograms = So, to convert from grams to kilograms, you divide the number of grams by 1,000. Conversely, to convert from kilograms to grams, you multiply the number of kilograms by 1,000. How to Convert Grams to Kilograms? Converting grams (g) to kilograms (kg) is simple using a standard conversion factor. Since 1 kilogram is equal to 1,000 grams, we can convert grams to kilograms by dividing the number of grams by 1,000. Steps to convert g to kg: Write down the weight in grams. Divide the value by 1,000 to get the weight in kilograms. Formula: Kilograms = Grams to Kilograms Conversion Chart When we measure weights, sometimes we use grams (g) and sometimes we use kilograms (kg). We use simple conversions to understand how much something in grams is in kilograms. Below is a chart that shows us the gram-to-kilogram conversions. Common Mistakes and How to Avoid Them in Grams to Kilograms Conversion When converting grams to kilograms, people often make mistakes. Here are some common mistakes to get a better understanding of the concepts of conversions. Mistake 1 Using the wrong conversion factor People get confused and use incorrect values for conversion, such as 100 instead of 1,000. Mistake 2 Rounding too soon Individuals might round numbers too early while calculating, which can lead to incorrect results. For example, instead of writing 1,650 ÷ 1,000 = 1.65, they might round it to 1.7, which is not accurate. Mistake 3 Misplacing decimal points People get confused and place the decimal point incorrectly, which leads to improper values. For example, instead of writing the value for 2,000 g as 2 kg, they write it as 20 kg. Mistake 4 Incorrect use of formula People get confused and use the incorrect formula. For example, instead of dividing, they multiply. To convert any value from g to kg we divide g by 1,000, not multiply. Mistake 5 Misunderstanding the relationship between kilograms and grams People often get confused about kilograms and grams. They might think 0.5 kg equals 5 grams, but it actually equals 500 grams. Remember, 1 kilogram is equal to 1,000 grams, so half a kilogram (0.5 kg) is 500 grams. Hey! Grams to Kilograms Conversion Examples Problem 1 Convert 3,278 g to kg Okay, lets begin 3,278 g = 3.278 kg Explanation We know the conversion factor: 1 g = 0.001 kg Now, multiply 3,278 by the conversion factor: 3,278 × 0.001 = 3.278 kg. Well explained 👍 Problem 2 Convert 46 g to kg. Okay, lets begin Solution: Converting 46 g to kg gives us 0.046 kg. Explanation Use the conversion factor: 1 g = 0.001 kg 46 × 0.001 = 0.046 kg Well explained 👍 Problem 3 An apple weighs 1,200 g. What is the weight in kg? Okay, lets begin The weight of the apple in kilograms is 1.2 kg. Explanation Convert 1,200 g to kg: 1,200 g = 1,200 × 0.001 = 1.2 kg Weight = 1.2 kg. Well explained 👍 Problem 4 The book weighs 180 g. What is its weight in kg? Okay, lets begin The weight in kilograms is 0.18 kg. Explanation Convert 180 g to kg: 180 g = 180 × 0.001 = 0.18 kg Weight = 0.18 kg. Well explained 👍 Problem 5 Converting 990 g to kg Okay, lets begin 990 g = 0.99 kg Explanation Step 1: Use the conversion factor. 1 g = 0.001 kg Step 2: Multiply 990 by 0.001. 990 × 0.001 = 0.99 kg Well explained 👍 FAQs on Grams to Kilograms 1.How many kilograms is 1 gram? 1 gram is approximately equal to 0.001 kilograms. 2.What is 850 g in kg? 850 g is approximately 0.85 kilograms. 3.Is 1,500 g a heavy weight? 1,500 g is about 1.5 kilograms, which is not very heavy. 4.How do I convert 100 g to kg? 100 g = 100 × 0.001 = 0.1 kilograms. Important Glossaries for Grams to Kilograms Conversion: The process of changing one unit of measurement into another. For example, converting grams to kilograms. Decimal: Numbers in which the whole number and fractional part are separated by a point (dot). For example, 2.5, here 2 and 5 are separated by a dot. Weight: The measure of how heavy something is, often measured in kilograms or grams. Metric System: A system of measurement based on powers of 10, commonly used worldwide. Unit: A standard quantity used in measurement, such as grams or kilograms. Explore More measurement Previous to Grams To Kilograms Conversion Kw to Hp Conversion\|Grams to mL Conversion\|Ml to L Conversion\|Pounds to Kilograms Conversion\|Binary to Decimal Conversion\|Mg to G Conversion\|Imperial to Metric Conversion\|Mg to Mcg Conversion\|Kilometers to Miles Conversion\|Pound to Kg Conversion\|Feet to Inches Conversion\|Decimal to Binary Conversion\|Psi to Bar Conversion\|Gram to Cup Conversion\|Liter to Gallon Conversion\|Kg to Pound Conversion\|Ounces to Ml Conversion\|Kilos to Pounds Conversion\|Pounds to Grams Conversion\|Binary to Octal Conversion Next to Grams To Kilograms Conversion KPA to PSI Conversion\|Miles to Kilometers Conversion\|Binary to Hexadecimal Conversion\|Octal to Binary Conversion\|Ounces to Pounds Conversion\|Liters to Gallons Conversion\|Feet to Miles Conversion\|Kilo to Pound Conversion\|Centimeter to Meter Conversion\|Kilogram to Pound Conversion\|Cm to Cm Conversion\|Feet to Cm Conversion\|Gallon to Liter Conversion\|Km to mph Conversion\|Ml to Kg Conversion\|Millimeters to Inches Conversion\|MB to KB Conversion\|Amps to Watts Conversion\|Feet To Centimeters Conversion\|Meters to Miles Conversion Seyed Ali Fathima S About the Author Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs. Fun Fact : She has songs for each table which helps her to remember the tables Brightchamps Follow Us Email us atcare@brightchamps.com Math Topics Numbers Multiplication Tables Algebra Geometry Calculus Measurement Data Commercial Math Math Formulas Math Questions Math Worksheets Explore by Country United States United Kingdom Saudi Arabia United Arab Emirates Our Programs CodeCHAMPS FinCHAMPS LingoCHAMPS RoboCHAMPS Math Questions Sitemap \| © Copyright 2025 BrightCHAMPS INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. 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2597	https://www.verywellhealth.com/epidermis-anatomy-1069188	Anatomy of the Epidermis The outermost layer of your skin The epidermis is the uppermost layer of your skin. It is responsible for creating skin tone and protecting against toxins and infection. Within the epidermis, there are four major layers of cells called keratinocytes that provide structural support for the skin. In addition to these four layers, you have another layer specific to your soles and palms, called the stratum lucidum. This article describes the layers of cells in the epidermis, including their structure and function. How Many Layers of Skin Are There? There are three main layers. The epidermis sits above the dermis, the middle layer that contains connective tissue, hair follicles, and sweat glands that regulate the integrity and temperature of your skin. The deeper hypodermis layer, also called subcutaneous tissue, is made up of fat and even more connective tissue. Stratum Basale The bottom layer of the epidermis is called the stratum basale. This layer contains one row of column-shaped keratinocytes called basal cells. Basal cells are constantly dividing and pushing already-formed cells towards the skin's surface. As basal cells move into the upper layer, they will also flatten, die, and be shed to make room for newer cells. Melanocytes, the cells that produce melanin—the pigment which provides your skin its color—are also found in this layer. Stratum Spinosum 2007 Heather Brannon, MD licensed to About.com, Inc. The spinosum layer lies just over the stratum basale and is only about five to 10 cells thick. In this layer, also known as the prickle cell or squamous cell layer, cells move in and change from column-shaped to multi-sided. Cells in this layer are responsible for making keratin. This is the fibrous protein that gives skin, hair, and nails their hardness and water-resistant properties. Stratum Granulosum The cells in the stratum granulosum, or granular layer, have lost their center (nuclei). This allows them to contain a high proportion of keratin to form the rigid cell layer of skin. They appear as flattened cells containing dark clumps of cytoplasmic material, which are the parts of the cell minus the nucleus. There is a lot of activity in this layer. Keratin proteins and lipids work together to create many of the cells responsible for the skin's protective barrier. Stratum Lucidum The stratum lucidum layer is only present in the thicker skin of the palms and soles. Its main function is to reduce friction between the stratum corneum and stratum granulosum. The name itself comes from the Latin for "clear layer," which describes the transparency of the cells themselves. Stratum Corneum The cells in the stratum corneum layer are known as corneocytes (or horny cells). These cells have flattened out and are considered dead. Composed mainly of keratin proteins, corneocytes provide structural strength to the stratum corneum but also allow for the absorption of water. They serve as an effective barrier to any chemicals that might harm the living cells just beneath them. The structure of the stratum corneum may look simple, but it plays a key role in maintaining the structural integrity and hydration of the skin. It ensures the continued production of new skin cells. It also provides vital protection against viruses, bacteria, parasites, and any other form of pathogen or toxin. Summary The epidermis is composed of layers of skin cells called keratinocytes. Your skin has four layers of skin cells in the epidermis and an additional fifth layer in areas of thick skin. The four layers of cells, beginning at the bottom, are the stratum basale, stratum spinosum, stratum granulosum, and stratum corneum. In your palms and soles, there's an additional layer called stratum lucidum underneath the stratum corneum. In the bottom layer, keratinocytes divide and push up formed cells toward the upper layer. The cells that reach the surface flatten and die. This provides a barrier to keep out pathogens and protect new skin cells underneath. National Cancer Institute. Layers of the skin. The Nemours Foundation/Kidshealth.org. Skin, hair, and nails. Rogerson C, Bergamaschi D, O'Shaughnessy R. Uncovering mechanisms of nuclear degradation in keratinocytes: A paradigm for nuclear degradation in other tissues. Nucleus. 2018;9(1):56-64. doi:10.1080/19491034.2017.1412027 Évora AS, Adams MJ, Johnson SA, Zhang Z. Corneocytes: relationship between structural and biomechanical properties. Skin Pharmacol Physiol. 2021;34(3):146-161. doi: 10.1159/000513054 Swaney MH, Kalan LR. Living in your skin: microbes, molecules, and mechanisms. Richardson AR, ed. Infect Immun. 2021;89(4):e00695-20. 10.1128/IAI.00695-20 By Heather L. Brannon, MD Heather L. Brannon, MD, is a family practice physician in Mauldin, South Carolina. She has been in practice for over 20 years. Thank you, {{form.email}}, for signing up. There was an error. Please try again.
2598	https://www.youtube.com/watch?v=Lf3KrvnLCug	How to Determine if Lines are Parallel (Using Slope) Wrath of Math 290000 subscribers 50 likes Description 6417 views Posted: 8 Jun 2017 What are parallel lines? If lines don't intersect, if they are not perpendicular lines, then they must be parallel lines. In this video we define parallel lines, giving a parallel lines definition, on the way to determining if two lines are parallel. The given a line, the slope of a parallel line to that line would be equal. That is, parallel lines have the same slop. If you need to know how to find a slope, I go over that here as well, using our slope formula. To determine if two lines are parallel you must determine if they have equal slopes. To do this, you can either pick the information straight out if you have an equation of the line in slope intercept form or point slope form. Or, if you have two points on either line you can figure out its slope using the slope formula. If they have the same slope then they are parallel. I hope you find this video helpful, and be sure to ask any questions down in the comments! Check out my music channel! Check out my blog! Transcript: hello everyone welcome to rocket math I'm your host Shani and in today's video we asked a question how do you know if two lines are parallel of course parallel lines look something like this but you can't just assume things from pictures and say you don't have a picture how do you know if two lines are parallel there's three main ways to do it that all stem from the same basic idea if two lines are parallel like this let's say this is line L and this is line J we say that two lines are parallel like this L is parallel to J with two straight lines probably should use the uppercase J it doesn't really matter so the three main ways to do it all depend on finding out if the two lines we're looking at have the same slope because if two lines are parallel then they have the same slope they're going up or down at the same exact rate and that's why they never intersect so the first two easy ways to figure out if two lines have the same slope is if you have an equation for them in slope intercept form which is y equals MX plus B where you're given the slope is M and the y intercept is B or if you have the equation of the lines in excuse me in point-slope form which is y minus y1 equals M times X minus x1 and this is y minus the y-coordinate of some point on the line equals the slope multiplied by X minus the x coordinate of a point on the line so if you have the equations for the two lines you're looking at in either of these forms then you can just pick out the slope right there it's M in both equations so if you have one line that has the equation y equals 2x plus 3 and another line let's say it's something like Y minus 1 equals 2 multiplied by X minus 4 then here you know the lines are parallel because your M term is equal if they weren't equal you would know that this line is not parallel to this line so those are the two easy ways if you have equations for the line and slope intercept form or point-slope form it's super easy to just look at it and pick it out the slightly harder way but still not all that difficult way is you've got two points on each line so let's say we have one line that has point 1 3 on it and it also has the point 2 4 if we know one line has these two points and another line has points let's say 0 5 & 3 8 then to figure out if these two lines are parallel all we have to do is plug these points into the slope formula so we can figure out the slope just by calculation for each line so for this line we use the slope formula so we subtract our y-coordinates 4 minus 3 and divide by the difference in our x coordinates 2 minus 1 and this is equal to 1 over 1 4 minus 3 is 1 2 minus 1 is 1 so we have a slope of 1 that's equal to M and then if we go over here and do the same thing we've got 8 minus 5 divided by 3 minus 0 and that is equal to 8 minus 5 is 3 3 minus 0 is 3 and that is equal to 1 so here we have that their slopes are equal because they both have a slope of positive 1 and therefore this line the line that is defined by these two points is parallel to the line defined by these two points so I hope this video helped you understand how you can determine if two lines are parallel using two or three different strategies depending on how you look at it they're all pretty pretty easy I think so let me know if you need anything clarified or have any questions in the comments I'd be happy to help you out and talk you through this if you're having any confusion I'll see you next time thank you very much for watching be sure to subscribe for the swanky's math videos on the internet and check out my music channel youtube.com slash shawnee music linked in the end cards and the description thanks for watching all the way that's a light when I saw the faces it [Music]
2599	https://www.teachstarter.com/us/lesson-plan/fraction-decimal-percentage-equivalence-us/	Teach Starter, part of Tes Teach Starter, part of Tes Search everything in all resources Blog Unlock your teaching potential. Podcast Where aspiration meets inspiration. Webinars Expand what's possible for every student. Search everything in all resources Teaching Resources Curriculum-aligned resources to engage and inspire your class. Units & Lesson Plans Take your class on an educational adventure over multiple lessons. Widgets Use simple apps that help you do all kinds of useful things. Curriculums Browse by curriculum code or learning area. Teaching Resource Collections Explore resources by theme, topic, strategies, or events. Teaching Resource Packs Bundles that you can download with one click. Studio Customize and create your own teaching resources and display materials. Free Resource Library Unit Plan Operations with Decimals and Percentages Unit Plan Lessons Fraction, Decimal, and Percentage Equivalence A 60-minute lesson in which students will make connections between equivalent fractions, decimals, and percentages. Login to view the lesson plan. Curriculum CCSS.MATH.CONTENT.6.RP.A.3.C Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. Author Teach Starter Publishing We create premium quality, downloadable teaching resources for primary/elementary school teachers that make classrooms buzz! 4 included resources: Fraction, Decimal, and Percentage Bingo Introducing Percentages PowerPoint Fraction, Decimal and Percentage Match-Up Game I Have, Who Has? Game – Fraction, Decimal and Percentage Equivalence About Us More Ways To Connect Keep In Touch

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