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On the contrary, if the random walk is correlated, the invariance of [MATH] is broken and the local diffusion coefficients will differ for different levels of coarse graining [MATH] |
Fig. shows an illustrative example of such coarse-graining applied to the motion of DC-SIGN protein on a living-cell membrane discussed above. One can see that the first and the last thirds of the cumulative trajectory [MATH] are only weakly affected by the binary smoothing procedure that indicates that the random moti... |
Thus, the simple pair-wise averaging procedure with the subsequent application of the wavelet-based method provides facts of evidence for the regions, where it give the true diffusion coefficient of the spatial- or time-dependent random walk, which need to be considered, or, as a byproduct, indicates region of time-cor... |
Conclusions The principal results of this work can be summarised as follows. Modern developments in single particle tracking not only open new perspectives for the study of molecular motions in complex environments, but also allow for using these motions as a probe for revealing properties of substrates on which the mo... |
# Source: arxiv 1806.02416 # Title: Use of mathematical modeling to study pressure regimes in normal and Fontan blood flow circulations # Sections: all # Downloaded: 2026-03-03T05:17:27.451708+00:00 |
Use of mathematical modeling to study pressure regimes in normal and Fontan blood flow circulations Abstract We develop two mathematical lumped parameter models for blood pressure distribution in the Fontan blood flow circulation: an ODE based spatially homogeneous model and a PDE based spatially inhomogeneous model. W... |
Introduction to Fontan surgical procedure In a normal biventricular heart, the systemic and pulmonary blood circulations are in series and each one is supported by a ventricle. The Fontan surgical procedure is applied to a malformed heart that is characterized by the presence of only one (left or right) functional vent... |
Fontan surgery is an extraordinary story of success in that it has allowed a generation of newborn babies with the most severe forms of congenital heart disease to survive into adulthood (estimated prevalence of approximately [MATH] per [MATH] births) KPM07 . Though life-saving, a univentricular Fontan circulation does... |
Late Fontan failure might progress gradually over years with an absence of overt symptoms. Fontan patients have lived with less than ideal cardiac output their entire lives and might not recognize decline in functional status until deterioration is significantly advanced. In the medical literature, failure of the Fonta... |
Previous mathematical modeling results for Fontan circulation 2.1 Computational fluid dynamics Computational fluid dynamics is a powerful tool that can be used to gain insight into the local blood flow dynamics in the Fontan circulation. These simulations are used to model the detailed 3-D hemodynamics of a particular ... |
Under the assumptions that vessel walls are completely rigid (according to surgical reports, the vessel diameter change per cardiac cycle is around [MATH] in most of the major arteries) and all vessels are symmetric, numerical simulations of blood flow to the lung after a surgical Fontan procedure are described in DDLP... |
According to the article DeG08 , the main quantities of importance in modeling the Fontan procedure are: Vessel diameters and flow rates representative of the range seen in the patient group under study including resting and exercise states |
Vessel sizes and flow rates matched appropriately Compliant vessels, accurate modeling of surgical anastomosis sites, and surgical material used (unless proven unnecessary) |
Unsteady flow Effects of respiration Correctly shaped vessel anatomy Two different types of boundary conditions, time-averaged and pulsatile, were analyzed in WTT 16 The authors derive a patient-specific sensitivity criterion which provides a guideline for determining when time-averaged boundary conditions can be used ... |
Recent advances in imaging methods and patient-specific modeling now reveal increasingly detailed information about blood flow patterns in healthy and diseased Fontan states. Building on these tools, there is now an opportunity to couple blood flow simulation with optimization algorithms to improve the design of surger... |
2.2 Lumped parameter models While computational fluid dynamics models can be used to calculate detailed three-dimensional blood flow in the total cavopulmonary connection, the computational costs of this approach prevent it from being used to simulate the entire circulatory system. Because Fontan failure is a systemic ... |
A lumped parameter model of the Fontan circulation was used by TCT 11 to generate boundary conditions for a computational fluid dynamics model used to design a Fontan assist device. In a study by LSK 14 , lumped parameter models of the Fontan circulation and the normal circulation were compared to determine differences... |
The objective of the present study is to develop lumped parameter models of the Fontan circulation with the goal of understanding the systematic changes that occur during Fontan failure. The outline of this paper is as follows. Section 3 describes an ODE model of the Fontan circulation and presents some basic results f... |
Spatially homogeneous ODE model of blood pressure distribution for the Fontan circulation A simple model of the Fontan circulation can be based on an electric circuit approach. This model consists of five compartments: the heart, the arterial system, the capillary system, the venous system, and the pulmonary system (lu... |
We model the capillary and pulmonary systems as linear resistance vessels. That is, we assume that the pressure drop across the vessel is proportional to the flow through the vessel, with a constant of proportionality called the resistance, labeled [MATH] and [MATH] , for the capillary and pulmonary systems, respective... |
The variables in the system are the volumes [MATH] [MATH] and [MATH] of the compliance vessels (the arterial system, the venous system and the heart), and the pressures [MATH] [MATH] [MATH] and [MATH] at different points along the loop; see Fig. . The parameters of the system are the resistances, compliances and basal ... |
We consider the compliance of the heart to be a piecewise constant function, with value [MATH] in systole and value [MATH] in diastole. To ensure appropriate directionality of the forcing, we assume that there are "perfect" valves where the pulmonary vein enters the heart and where the aorta leaves the heart. Anatomica... |
We use conservation of volume in each compartment to set up the dynamic equations. In particular, we must have the rate of change in volume of a compartment equal to the difference of the flow in and flow out. Due to the assumptions on the valves, there are discontinuities in the variables and their derivatives as the ... |
The three flow rates are the arterial, capillary and pulmonary flow rates, defined as [EQUATION] The three compliance volumes are the arterial, venous and heart volumes, defined as |
[EQUATION] with [EQUATION] Consequently, the dynamic equations are, during inflow (diastole), [EQUATION] so that [MATH] and [MATH] and during outflow (systole) |
[EQUATION] so that [MATH] and [MATH] . In addition, total volume must always be conserved, so that [EQUATION] It is possible to simplify these equations by using the conservation law ( 3.6 ). In particular, during inflow (diastole), |
[EQUATION] and during outflow (systole) [EQUATION] In addition to diastole and systole, the cardiac cycle consists of two isovolumetric phases, during which both heart valves are closed, and the heart undergoes a change in pressure in response to a change in its shape, while maintaining a constant blood volume. Isovolu... |
[EQUATION] and during diastole, [EQUATION] Consequently, while [MATH] and [MATH] are continuous functions of time, [MATH] experiences jump discontinuities at the transitions between diastole and systole. |
Simulations can be done by sequentially integrating the systolic and diastolic equations, and repeating. With model parameter values taken from the table, the simulations of the model equations exhibited realistic values. As shown in Fig. , the stroke volume (the amount of blood pumped out of the heart in one heartbeat... |
Pulmonary vascular resistance [MATH] is known to increase in Fontan failure and an increase in this resistance is known to lead to a decrease in cardiac output. This model can be used to demonstrate the impact of pulmonary vascular resistance on cardiac output. Figure shows the change in average cardiac output as a fun... |
Pulmonary resistance also has an impact on the cardiac pressure-volume curve. As shown in the left panel of Fig. , the cardiac pressure-volume curve shifts to the left (i.e. decreased cardiac volumes) for the case of high pulmonary resistance. What this means is that the basal volume of blood in the heart has decreased... |
Figure shows the pressure drop as a function of distance from the heart for healthy and failing Fontan patients based on clinically measured pressure catheter data. This figure illustrates that the majority of the pressure drop occurs near the heart in the systemic arteries and that furthest away from the heart, in the... |
Spatially inhomogeneous PDE models of blood pressure distribution In this section, our ODE approach to modelling blood flow in the Fontan circulation is extended to a PDE model. The PDE model has the advantage of allowing for spatial variation of model parameters such as compliance and resistance. This will allow for m... |
To model the circulatory system as a continuous flow network in a resistive compliance vessel, we assume that blood flow is a Stokes flow, i.e. the Reynolds number is sufficiently small to allow us to neglect inertial effects. Consequently, the flux in a cylindrical tube is a Poiseuille flow for which |
[EQUATION] where [MATH] is the local pressure, [MATH] is the cross-sectional area, and [MATH] is the fluid viscosity. Now we assume that a vessel is a linear compliance vessel with [MATH] , where [MATH] is the compliance. This leads to a flux relationship for a single vessel |
[EQUATION] If we have a total number of [MATH] parallel vessels all with cross-section area [MATH] , the flux is [EQUATION] Notice that in the limit [MATH] , this reduces to Ohm’s Law (as it must) |
[EQUATION] where [MATH] is the resistance per unit length. When combined with the conservation law (the total volume of circulating blood is conserved) |
[EQUATION] this yields [EQUATION] which is a nonlinear parabolic partial differential equation for [MATH] with spatially variable coefficients. In general [MATH] [MATH] and [MATH] |
4.1 Boundary conditions for normal circulation For the normal circulation (see Fig. ), there are two hearts, the left and the right hearts, each of which satisfy a volume-compliance relationship of the form |
[EQUATION] The basal volumes and compliances are different during systole and diastole periods, that is [EQUATION] [EQUATION] Note that in equations ( 4.8 ) and ( 4.9 ), the subscripts [MATH] and [MATH] refer to the left and right hearts, the subscripts [MATH] and [MATH] refer to diastole and systole. In this two heart... |
During systole the input valves (mitral and tricuspid) are closed and output valves (pulmonary and aortic) are open, while during diastole the opposite is the case. For the normal circulation, we let [MATH] |
be the systemic circulation, and [MATH] be the pulmonary circulation, and of course the domain [MATH] is periodic. For convenience, we use the following notation, during systole: systemic pressure is [MATH] , pulmonary pressure is [MATH] |
and during diastole: systemic pressure is [MATH] , pulmonary pressure is [MATH] . Systemic and pulmonary pressures are everywhere continuous functions except two points [MATH] and [MATH] where discontinuity jumps correspond to the left and right heart pressure jumps during the switch between systole and diastole phases... |
4.2 Initial-boundary value problem (systolic regime: [MATH] ). Partial differential equation: [EQUATION] Assume that at the initial time [MATH] the pressure is [MATH] . Systemic circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: |
[EQUATION] Pulmonary circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: [EQUATION] These boundary conditions are defined in such a way as to ensure conservation of the total volume |
[EQUATION] during systole. Indeed, from ( 4.5 ), ( 4.8 ), ( 4.9 ) and the dynamical boundary conditions it follows directly that [MATH] |
4.3 Initial-boundary value problem (switch from systole to diastole at: [MATH] ). Initial data for the systemic circulation model [MATH] (and for [MATH] ) at the beginning of the diastole: |
[EQUATION] Initial data for the pulmonary circulation model [MATH] (and for [MATH] ) at the beginning of the diastole: [EQUATION] |
4.4 Initial-boundary value problem (diastolic regime: [MATH] ). Systemic circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: |
[EQUATION] Pulmonary circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: [EQUATION] Again, it follows directly from ( 4.5 ), ( 4.8 ), ( 4.9 ) and the dynamical boundary conditions that during diastole [MATH] |
4.5 Initial-boundary value problem (switch from diastole to systole regime: [MATH] ). Initial data for the systemic circulation model [MATH] (and for [MATH] ) at the beginning of the systole: |
[EQUATION] Initial data for the pulmonary circulation model [MATH] (and for [MATH] ) at the beginning of the systole: [EQUATION] |
The problem is periodic in time (i.e systolic and diastolic regimes are repeated). 4.6 Boundary conditions for Fontan circulation |
The Fontan blood flow circulation has only one heart, so the model has only a single loop [MATH] The basal volumes and compliances of the heart are also different during systole and diastole periods, that is |
[EQUATION] For convenience we use the following notation, during systole: blood pressure is [MATH] and during diastole blood pressure is [MATH] |
4.7 Initial-boundary value problem (systolic regime: [MATH] ). Assume that at the initial time [MATH] the pressure is [MATH] One heart circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: |
[EQUATION] 4.8 Initial-boundary value problem (switch from systole to diastole at: [MATH] ). Initial data for the systemic circulation model [MATH] (and for [MATH] ) at the beginning of the diastole: |
[EQUATION] 4.9 Initial-boundary value problem (diastolic regime: [MATH] ). Systemic circulation model [MATH] is given by the partial differential equation ( 4.10 ) and conditions: |
[EQUATION] 4.10 Initial-boundary value problem (switch from diastole to systole regime: [MATH] ). Initial data for the systemic circulation model [MATH] (and for [MATH] ) at the beginning of the systole: |
[EQUATION] The problem is periodic in time (i.e systolic and diastolic regimes are repeated). It is a direct computation to verify that with these boundary conditions [MATH] |
Well-posedness analysis of spatially inhomogeneous PDE model 5.1 Existence and uniqueness of the nonnegative solution in normal case. |
In this section we find restrictions on the parameters of the PDE model with dynamical boundary conditions for the case of the normal blood circulation for which a nonnegative solution exists and is unique. First, we show that the pressure [MATH] stays bounded for any time [MATH] , then we obtain uniform in time bounds... |
We start by introducing the following notations [EQUATION] Consider the following equation [EQUATION] with initial and boundary conditions |
[EQUATION] [EQUATION] for all [MATH] (during systole); [EQUATION] where [MATH] is such that [EQUATION] [EQUATION] for all [MATH] (during diastole); |
[EQUATION] where [MATH] is such that [EQUATION] Let [EQUATION] be a solution to problem ( 5.1 )–( 5.6 ) for all [MATH] such that |
[EQUATION] First, we define a weak solution for our problem. Definition 5.1 A non-negative function [MATH] is said to be a periodic solution of the problem ( 5.1 )–( 5.6 ), i. e. [MATH] , if |
[EQUATION] [EQUATION] and [MATH] satisfies equation ( 5.1 ) in the sense that [EQUATION] for any [MATH] and [MATH] Our main result establishes parameter ranges for which non-negative solutions exist, as follows. |
Theorem 1 If [EQUATION] [EQUATION] initial data [MATH] are non-negative, and [EQUATION] [EQUATION] then the problem ( 5.1 )–( 5.6 ) has a unique non-negative solution in the sense of the Definition 5.1 |
5.2 Proof of Theorem Note that the equation ( 5.1 ) becomes degenerate if [MATH] . Hence, we start by constructing a sequence of positive approximations of initial data [MATH] . We can choose for example [MATH] (if [MATH] then [MATH] ). These approximations allow us to apply the theoretical background developed for uni... |
Volume conservation Integrating ( 5.1 ) on [MATH] , due to 5.3 ) and ( 5.2 ), we arrive at [EQUATION] Integrating ( 5.1 ) on [MATH] , due to 5.6 ) and ( 5.4 ), we have |
[EQUATION] By ( 5.11 ) and ( 5.12 ), we obtain [EQUATION] Due to ( 5.5 ), from ( 5.13 ) we get [EQUATION] Moreover, by ( 5.7 ) and ( 5.14 ) we find that |
[EQUATION] whence, due to ( 5.8 ), we have [EQUATION] Consequently, total volume of the left heart and systolic circulation is identical at |
[MATH] and [MATH] Below we show how to prove, using Moser’s method Mos66 , that the blood pressure [MATH] stays bounded on the whole time interval [MATH] . We start by showing that |
[MATH] is bounded in [MATH] then we show that for any [MATH] the solution [MATH] is bounded in [MATH] and after that we take the limit |
[MATH] Boundedness Multiplying ( 5.1 ) by [MATH] and integrating along [MATH] , due to 5.3 ), we have [EQUATION] Integrating ( 5.16 ) in time, we get |
[EQUATION] On the other hand, multiplying ( 5.1 ) by [MATH] and integrating along [MATH] , due to 5.6 ), we have [EQUATION] and integrating ( 5.18 ) in time from [MATH] , we have |
[EQUATION] By ( 5.14 ) with [MATH] we find that [EQUATION] As a result, from ( 5.19 ), due to ( 5.20 ) and ( 5.17 ), we have [EQUATION] |
provided [EQUATION] Multiplying ( 5.1 ) by [MATH] with [MATH] and integrating along [MATH] and in time, we have [EQUATION] [EQUATION] |
provided ( 5.22 ). Next by Moser’s method Mos66 , taking into account that [EQUATION] due to the periodicity and [MATH] , from ( 5.24 ) we obtain |
[EQUATION] provided [MATH] and [MATH] Now we obtain the main a priori estimates for the gradient [MATH] in [MATH] and for the time derivative [MATH] in [MATH] . Using these estimates we are able to build a weak solution for the problem at hand. |
A priori estimate Multiplying ( 5.1 ) by [MATH] and integrating along [MATH] , we have [EQUATION] whence, due to ( 5.3 ), [EQUATION] |
for all [MATH] . Let us denote by [EQUATION] Then from ( 5.27 ) and using that [MATH] we find that [EQUATION] where [EQUATION] Indeed, taking into account |
[EQUATION] i. e. [EQUATION] from ( 5.28 ) we get [EQUATION] provided [MATH] . So, [EQUATION] where [MATH] and [MATH] Similar to ( 5.27 ), for all [MATH] we deduce that |
[EQUATION] Let us denote by [EQUATION] Then from ( 5.31 ) we find that [EQUATION] By the volume conservation ( 5.14 ), we arrive at |
[EQUATION] Due to ( 5.29 ) and ( 5.33 ), from ( 5.32 ) we get [EQUATION] provided [MATH] , where [MATH] . So, [EQUATION] where [MATH] and [MATH] By the periodicity [MATH] , we get |
[EQUATION] where [MATH] , provided [EQUATION] As a result, we obtain the main a priori estimate [EQUATION] for all [MATH] Now we show that the solution constructed above is unique. We use a proof by contradiction. |
Uniqueness Let [MATH] and [MATH] be two solutions to the problem ( 5.1 )–( 5.6 ). Let us denote by [MATH] satisfying [EQUATION] [EQUATION] |
Multiplying ( 5.36 ) by [MATH] and integrating along [MATH] , due to 5.3 ), we have [EQUATION] for all [MATH] . Using Cauchy inequality, boundedness of [MATH] and [MATH] [MATH] , due to Grönwall lemma, we get |
[EQUATION] On the other hand, multiplying ( 5.36 ) by [MATH] and integrating along [MATH] , due to 5.6 ), we have [EQUATION] for all [MATH] , where [MATH] . As |
[EQUATION] then [EQUATION] So, integrating ( 5.40 ) in time from [MATH] to [MATH] , using ( 5.41 ), we have [EQUATION] provided [MATH] . Applying Grönwall lemma to ( 5.42 ), we get |
[MATH] for all [MATH] . As a result, we obtain that [MATH] for all [MATH] We proceed by obtaining well-posedness conditions for the second part of the interval, namely [MATH] where, to compare to the first part [MATH] , Neumann amd dynamical flux boundary conditions are switched. Consider the following problem |
[EQUATION] with initial and boundary conditions [EQUATION] [EQUATION] for all [MATH] [EQUATION] where [MATH] such that [EQUATION] |
[EQUATION] for all [MATH] [EQUATION] where [MATH] such that [EQUATION] Let [EQUATION] be a solution to problem ( 5.43 )–( 5.48 ) for all [MATH] such that |
[EQUATION] Introduce the following notations [EQUATION] Definition 5.2 A non-negative function [MATH] is said to be a periodic solution of the problem ( 5.43 )–( 5.48 ), i. e. [MATH] , if |
[EQUATION] [EQUATION] and [MATH] satisfies equation ( 5.43 ) in the sense that [EQUATION] for any [MATH] and [MATH] Theorem 2 Assume that |
[EQUATION] [EQUATION] and initial data [MATH] is non-negative satisfying [EQUATION] [EQUATION] then the problem ( 5.43 )–( 5.48 ) admits a unique non-negative solution in the sense of Definition 5.2 |
The proof of Theorem is similar to the one of Theorem Finally, to get well-posedness for the whole interval [MATH] , the restrictions on the parameter values obtained in Theorem should be combined with the restrictions obtained in Theorem |
5.3 Existence and uniqueness of the nonnegative solution in Fontan case. Let us introduce the following notation [EQUATION] Consider the following equation |
[EQUATION] for all [MATH] [EQUATION] where [MATH] such that [EQUATION] Let [EQUATION] be a solution to problem ( 5.53 )–( 5.58 ) for all [MATH] such that |
[EQUATION] Definition 5.3 A non-negative function [MATH] is said to be a periodic solution of the problem ( 5.53 )–( 5.58 ), i. e. [MATH] , if |
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