Silicon and diamond microdosimeters have been realized with sensitive volumes as low as 1 μm in thickness and a few hundred μm2 area and collection charges approaching 100%. Their small geometry provides also high-spatial resolution and the possibility to measure full therapeutic beam intensities, as the electronic chain is not saturated by the large number of particles required for clinical use. A main limitation of semiconductor microdosimeters is the electronic noise, as the devices work with little or no electronic gain due to the small voltage that can be applied to the small-volume semiconductor. This limits the lowest energy events that can be detected. The fixed size of the crystal also implies that different detectors may need to be used to obtain microdosimetric information for different sensitive volumes while gas-based detectors can achieve this by varying the gas-pressure. The advantages and disadvantages of both detector types are confronted in Table 4.2.Table 4.2
Comparison of tissue equivalent proportional counters (TEPC) and solid-state microdosimeters

TEPC

Solid-state microdosimeter

Advantages

Tissue equivalence
Easy handling and manufacturing
Operation for clinically relevant beams

High-spatial resolution
Compact size
Economic development
Low sensitivity to vibrations
Suitable for clinical environment

Disadvantages

Wall-effects
High electric fields

No tissue equivalence


In the framework of radiation biology, either the linear energy transfer LET or the lineal energy can be used to specify the radiation quality. While the LET is frequently used, at least for broad classifications of different radiation qualities (high vs low LET, i.e., densely vs sparsely ionizing radiation), the use of the lineal energy is less common due to the limited experimental data and the complexity in analyzing the microdosimetric spectra. However, the use of LET to determine radiation quality is affected by some intrinsic limitations such as different particles of different mass and energy having the same LET being still characterized by a different energy distribution of the secondary electrons. In general, the microdosimetric spectra provide information that is not captured in the LET and it may be very beneficial for fundamental radiobiological studies aimed at linking biological response to energy deposition events, as well as for radiation protection and clinical work, e.g., predicting treatment efficacy.


Several LET-based RBE models have been developed over the years. The Microdosimetric Kinetic Model (MKM) is a model based on the dual radiation action theory and specifically developed to link microdosimetric measurements to radiobiological effects [19]. The central hypothesis of the dual radiation action theory is that the number of lethal lesions is, through a linear quadratic relationship, proportional to the specific energy deposited in a microscopic site. The specific energy (z) is defined as the ratio between the energy imparted (ε) and the mass of the microscopic volume (m) [17]:


As both the specific energy (z) and the lineal energy (y) measured by microdosimetry are related to a microscopic volume, the two quantities are linked through the microscopic volume mass (m) and mean chord length (l):

 (4.20)


with l, m, ρ, and rd the mean chord length, the mass, the density, and the radius of the microscopic volume, respectively.


Kase et al. [20] formalized the link between cell survival fraction SF and the microdosimetric measurements:

 (4.21)


with

 (4.22)


where α0 and β0 are the linear quadratic parameters specific for each cell line (usually taken from X-ray measurements), D is the macroscopic dose absorbed by the cell and ρ is the cell density. y0 and rd are fixed parameters accounting for the overkill effect observed at high lineal energy values (usually set at y0 = 150 keV/μm) and for the sensitive critical volume of the specific cell line, respectively. The parameter y* includes the measured microdosimetric spectrum (f (y)) providing therefore a direct link between radiobiological response and physical measurements. The use of the MKM and microdosimetry is the only approach providing a link between physical and biological measurements, considering that LET values cannot be experimentally determined. Supported by the fast development of technologies that will facilitate microdosimetric measurements, there is renewed interest in this approach. However, the precise estimation of the y0 and rd parameters requires further investigation (Box 4.5).


Microdosimetry quantifies individual energy deposition events through the lineal energy

Microdosimetry is performed through tissue equivalent proportional counters (TEPC) or solid state microdosimeters

Microdosimetry is able to directly link radiobiological response to physical measurements


Microdosimetry is performed through tissue equivalent proportional counters (TEPC) or solid state microdosimeters


Microdosimetry is able to directly link radiobiological response to physical measurements


When ionizing radiation (IR) interacts with a biological sample (which is composed of ~70% water in weight and the rest biological molecules), it can either directly hit the biological molecules or the water molecules. In both cases, these interactions can lead to an energy deposition at the interaction point (inelastic scattering) and the production of secondary particles, mostly electrons that can, in turn, also interact with the target. The ionized or excited molecules, particularly those of water, generate radicals (water radiolysis) that also can attack the biological unit molecules or aggregates, leading to subsequent structural damage of these molecules, which could ultimately have consequences on the functioning of the cell and its outcome.


In this section, we will review the current state of knowledge concerning the different stages that lead from the physical interactions between ionizing radiation and biological matter (known as the physical stage) to the formation of damage to biomolecules as schematically depicted in Fig. 4.12. As indicated above, this includes the description of the production of radicals and their chemical interactions with molecules (chemical stage) but also the consideration of the geometrical and chemical structure of the target molecules. In particular, we will look at the effect of these interactions on the DNA contained in the cell nuclei, because it is well established that the DNA is a privileged target with regard to the effects of radiation on cells [21]. This continuous description will reveal differences at each stage level between the various types of radiation that will enable us to categorize them according to their capacity to produce these damages in terms of number and complexity.

A schematic diagram exhibits different stages of physical, physicochemical, chemical, biochemical, and biological stages of physical interaction with target molecules, radical creation, radical diffusion, chemical reactions, biological effects, D S B, aberration, and mutations.

Fig. 4.12
Schematic representation of the different processes leading to the damage produced by irradiation in the cells and their characteristic times


A schematic diagram exhibits different stages of physical, physicochemical, chemical, biochemical, and biological stages of physical interaction with target molecules, radical creation, radical diffusion, chemical reactions, biological effects, D S B, aberration, and mutations.


Different experimental techniques have been developed and used in recent years to measure this damage, as we will see in Sect. 4.3.3. However, in most cases, these techniques do not allow to have access to the total number of strand breaks or double strand breaks as well as to the base damage or to the complexity of the damage cluster. Thus, the Monte Carlo simulation method has become the “gold standard” for the prediction of these damages. This means however that it is necessary to know, with the least possible uncertainty, all the data allowing the description of these stages to feed the codes. In Sect. 4.3.4, we will detail what these data or parameters are, their current uncertainties, and therefore the current simulation capabilities with the different codes.


Ionizing radiation interacts with the exposed target through a cascade of random interactions with the atoms of the medium. The result concerns as much the interactions of the primary radiations as the slowing down of the secondary radiations emerging from them. During these interactions, radiation can be scattered or absorbed by gradually losing energy. If these absorption processes lead to sufficiently high energy transfers (typically >10 eV in radiobiology), they can lead to the ejection of electrons and modify the electronic layer of atoms and molecules and their chemical properties, which gives them the power to induce effects. Indeed, IR can be categorized as directly and indirectly ionizing depending on whether it is composed of charged or uncharged particles, respectively, but in all cases with enough energy to produce ions in matter.


While ionization is considered the most important physical phenomenon to explain radiation-induced effects, excitation, a phenomenon in which electrons are transferred to higher atomic or molecular levels, is also considered among the possible events to be precursors of the radiation-induced effect. It is assumed that the ratio of energy loss by ionization and excitation is stable between radiations of different natures and energies, and, therefore, that the measurement of ionizations alone is sufficient. This approximation is important for the validity of reference dosimetric and microdosimetric measurement techniques using gas detectors. These techniques only “see” the ionizations but apply global physical data such as the average ionization energy (W) that accounts for both phenomena. It is generally recognized that the distinction between ionization and excitation is more blurred in condensed states, which are ultimately the ones targeted in dosimetry using other measurement methods than gas meters, and in radiobiology.


From a mechanistic perspective, if one wants to identify which energy deposition will result in damage to the structure of the target biomolecules, this proportionality between the number of ionizations in a volume (however small it may be) and the deposited energy is therefore not detailed enough. Indeed, in this context, it is necessary to “zoom in” on the scale of the target’s constituents at the nanometric scale to look at all the energy deposits (or energy transfers) produced by the initial radiation, as well as the secondary particles, notably the electrons. This is the study of the so-called track structure of radiation. At this scale, the differences between the spatial distribution of energy deposits defined by the tracks produced by different types of radiation (photons, electrons, energetic ions of different energies, etc.) lead to variations in early damage sufficient to produce a great diversity of later effects at both the cellular and tissue levels.


Thus, for example, in the case of irradiation by high energy ions, we can look at the track they produce as being formed by a “core” and a “penumbra” region. The core is formed by the energy deposits of the projectile itself and is almost straight as elastic scattering does not have an important influence on the ion direction at energies under 10 MeV. The penumbra region is formed by the energy deposits of secondary electrons produced during ionizations with energies of ~1–100 MeV, interacting with many molecules in the target [22, 23].


However, when the primary particle ionizes water molecules, the main component of biological matter, many of the electrons are produced with low energy [24, 25]. Indeed, the energy of the emitted electrons for a given material is mainly determined by the oscillator-strength distribution of its valence electronic structure. The long-range of Coulomb interactions and the cross section that peaks at ~20–30 eV and decreases to very low values at 100 eV leads to the formation of electrons with energies, in general, less than 100 eV [26]. These low energy electrons (more extensively defined as those ≤10 keV) have a small penetration range (<1 μm) and inelastic mean free path (IMFP) (<10 nm) in typical condensed media [27] like water or DNA components. Therefore, most of the direct damage is produced around the track and, more specifically, at the track ends, where they are produced in high quantity.


In fact, the electrons below ~20 eV seem to be particularly effective because, in addition to participating in the production of direct damage by ionizations or excitations of the constituents of the DNA, they can undergo resonant scattering with molecules, generating reactive radicals and molecular species, which can themselves contribute to DNA breaks [28] and oxidative damage. Experiments have indicated that electrons (or photons) with energies as low as ~10 eV can still induce double strand breaks, possibly through a resonance mechanism [29, 30].


In the previous section, we were interested in the interactions between IR and the target molecule (DNA) and how some of these interactions can cause damage in a direct way. However, IR interacts in the same way with the surrounding water medium and induces local electronic instability. The physicochemical stage corresponds to the set of rapid electronic and atomic modifications resulting from the readjustments of the medium in order to return to thermal equilibrium. Thus, water molecules that are in an excited or ionized state can dissociate into new chemical species (radiolysis):


Among these species, the OH° (hydroxyl) radical is particularly interesting in radiobiology, because it can be the origin of DNA damages that are difficult to repair by the cell. This radical is mainly produced from the radiolysis of pure water following different mechanisms (dissociation directly after an ionization or an excitation of the water molecule).


Moreover, under-excitation electrons (with an energy lower than the last excitation shell of the water molecule, 8.22 eV) will undergo elastic scattering and will continue to lose energy by vibrational and rotational interactions until reaching the energy of the medium, the so-called thermalization energy. This thermalization process is in competition with two processes of electron capture, either by a neutral water molecule (“dissociative attachment”) or by an ionized water molecule (“geminal recombination”) and is supposed to be completed within a picosecond after the irradiation.


Beyond the picosecond, the newly created radiolytic species are free to diffuse randomly in the medium and to interact with each other, which is the chemical stage. Initially localized around the energy deposits of the track, they propagate and distribute more homogeneously in the medium as time evolves. The initial distribution of species depends strongly on the LET of the incident particle. In the case of high-energy electron projectiles (low LET), the initial distribution in the form of clusters will be more strongly marked than in the case of ions, where the LET is more important, and thus the energy depositions are more homogenously located all over the track. It is generally accepted that beyond the microsecond, most of the reactions between different clusters are completed and the chemical stage can be considered as finished for a given track.


As indicated, all the simultaneous reactions are thus in competition and the temporal evolution of the chemical species, as shown in the example in Fig. 4.13, can strongly depend on the initial parameters. These reactions are very numerous in a liquid water medium [31] and increase even more in complex biological media. Thus, even in rigorous radiation chemistry experiments studying the kinetics of elementary chemical reactions, it can be difficult to measure the impact of secondary and competing reactions. In this context, simulation becomes a powerful tool to predict the complex dynamics of macroscopic observables, starting from elementary mechanisms [32].

6 schematic circles of liquid water of 0.01 p s, 6, 23, 55, 76, 100 n s of radiolysis 1-kilo electron volt of electrons present at the center, and the scale of 200-nanometer present at the bottom. The electron is a single dot first and spreads as several dots.

Fig. 4.13
Spatial and temporal evolution of the radiolysis products of a 1 keV electron in liquid water computed by Monte Carlo simulation (Geant4-DNA)


6 schematic circles of liquid water of 0.01 p s, 6, 23, 55, 76, 100 n s of radiolysis 1-kilo electron volt of electrons present at the center, and the scale of 200-nanometer present at the bottom. The electron is a single dot first and spreads as several dots.


To do so, one category of numerical simulations consists in dividing the modeling into two phases with different levels of granularity and acceptable simplifying assumptions. In the first one, each radical species is considered individually, and we are interested in the calculation of the reaction rate, the diffusion coefficient, or the branching ratios. This first phase can be simulated using molecular dynamics (like Born–Oppenheimer or Car–Parrinello) and/or quantum mechanical calculations like TD-DFT. However, this approach is unfortunately prohibitive in terms of computation time for a high number of molecules, which limits their application to systems such as a cell. In the second phase, approximations can be made to significantly reduce the computation time. For example, molecules of the same species can be grouped in order to describe their evolution by a unique variable (concentrations) and two types of methods are often applied: either probabilistic (Gillespie algorithms) or based on the solution of differential equations.


A second category of numerical simulations consists of describing the medium as a solvent or continuum and only calculating the diffusion and the chemical reactions of particular interesting species. This method is well adapted when the number of molecules is relatively small and, more particularly, when their distribution is inhomogeneous like in this case. Therefore, most of the track structure codes including the simulation of the chemical stage use this approach (Sect. 3.3.4) and include other simplifications as considering each molecule spherical and diffusing independently of the other molecules. In this frame of a diffusion-reaction model, their diffusion in the medium can be solved with the Green Function of the Diffusion equation (GFDE). The eventual reaction of two particles is considered when the interparticle distance is smaller than their reaction radius. The reactions can be either fully or partially diffusion-controlled and involve neutral or charged particles. This gives four classes of reactions that were introduced by Green et al. [33]. For totally diffusion-controlled reactions (type I), the rate constant is assumed to be infinite, meaning that the particles react whenever they collide. In this case, the GFDE solution can be calculated using the Smoluchowsky boundary conditions in three dimensions [34]. This reaction mechanism is the one most often triggered when radiolytic species diffuse and encounter a reactive site that is either representing other radicals or a DNA constituent (sugar-phosphate backbone or bases with high rate constants). For other reaction types, including those representing the scavenger effects, please refer to the literature [35, 36].


Within this frame (GFDE), different stochastic simulation techniques have been proposed in order to calculate the probability of reactions to happen depending on the position of each molecule at a given time [33, 37, 38] as the step by step method or the IRT for Independent Reaction Time method.


Indirect damages are the consequence of these reactions for the DNA molecule and can represent between 30 and 90% of the total DNA damage depending on the LET of the irradiation. Among them, of importance are the strand breaks produced by the hydroxyl radical capturing the hydrogen of the deoxyribose at the C4 position or the addition of hydroxyl radical to a nitrogenous base, resulting in base alterations. These altered bases are often unstable and can either decompose or react with environmental molecules and radiolytic species. The underlying reactions are therefore multiple and complex [39]. DNA-protein or DNA–DNA bridging can also occur under the effect of radical species produced by radiation [40].


It should be noted that the description of the chemical stage process as explained above becomes much more complex if we take into account a more realistic chemistry of the cellular environment adding factors such as the pH, the oxygen concentration, or the presence of more complex molecules around the DNA, commonly called “scavengers” because of their action on the radical species. In particular, the concentration of oxygen has been shown to have a significant impact on radiation resistance: indeed, carcinogenic cells, which are hypoxic, are 2–3 times more resistant to radiation than healthy, normoxic cells. This “oxygen effect” is also believed to be one of the possible explanations for the protective effect on healthy tissue in the case of FLASH radiotherapy as the depletion of oxygen during irradiation could create a temporary hypoxic environment for both healthy and cancer cells. Nevertheless, this hypothesis is still not completely proven and the mechanism behind this FLASH effect however remains unknown [41].


Radiation-induced damage is multiple and depends on numerous factors such as the type of radiation, the DNA configuration, or the irradiated medium condition. They are the result of the physical, physicochemical, and chemical processes explained in the previous sections and thus generated either by direct or indirect effects. The main DNA damages are strand breaks (simple, double, or clustered), base alterations, protein-DNA, and DNA–DNA bridges. Of these, the radiobiology and simulation communities have historically been most interested in double strand breaks (DSB) or clustered damage including at least one DSB. Indeed, in most repair models this type of DNA damage is called “lethal” or “semi-lethal,” as they are considered to lead to misrepair and cell death [42–44]. In all cases, and even if they can sometimes be correctly repaired by cellular repair mechanisms, it is established that these complex damages can have important consequences on the cellular survival or its functioning. Moreover, DSB can be detected experimentally and compared to the results of predictions from simulations. Several detection techniques exist, which are adapted according to the irradiation configuration, the dose used, or the cell type. Historically, comet assay or pulsed field electrophoresis (PFE) has been used with high-dose irradiation in order to generate DNA fragments that can be separated and measured leading to a given number of DSB detected. Data obtained in this way, for example, in the case of proton irradiations at different energies or gamma rays [40, 45], have been used extensively to validate codes such as PARTRAC [46], KURBUC [47], or, more recently, Geant4-DNA [48].


Other techniques, used at low dose, consist in using immunofluorescent probes to localize the radio-induced DSB within the genome. For example, in the case of H2AX immunofluorescence; the histone closest to a double strand break that contains the H2AX variant of histone H2A (approximately present at 25% of H2A histones and evenly distributed in the DNA) allows the detection of DNA double strand breaks through its phosphorylation. This phosphorylation is visible using specific antibodies, containing a fluorochrome substance, making the double strand breaks appear as luminous points called “foci” or IRIF (ionizing radiation-induced foci) [49].


An important quantity of experimental data has been obtained recently using this technique or with other fluorescent biomarkers such as the 53BP1 protein, which allows to quantify the DSB produced by different types of radiation and to compare them with the simulation results. However, an important bias of this technique is that, in general, one detectable focus does not correspond to a single DSB formed in the DNA [50], and therefore the irradiation conditions and the geometry of the target must be explicitly considered in the simulation for such validations [51, 52] .


As we described earlier in this chapter, particle transport through matter using MC codes is generally handled via a “condensed history” (CH) approach [53], currently used for dosimetry and the majority of microdosimetry applications for very energetic particles. In such a CH approach, many scattering events are grouped into fewer artificial steps, much longer than the mean free path of the particle, using multiple-scattering theories and a continuous energy loss along those steps. However, in order to simulate the physics at the nanoscale and to possibly link it to the biological effects of radiation with track structure properties in the nm regime [54], an event-by-event tracking of the different physical events is necessary to allow for better spatial resolution. Therefore, so-called track structure codes have been developed for applications in micro- and mostly nanodosimetry. In Table 1 taken from [55], we present the list of the main track structure codes that have been developed since the 80 s of the last century. In this table, it is indicated if the code includes the possibility of simulating the chemical stage and the materials available for the simulation of the physical stage.


Indeed, in order to model all the physical interactions taking place in the physical stage, these codes need to include cross sections for simulating ionization, electronic excitations below the ionization threshold, and, ideally, vibrational or rotational excitations of the medium, in principle for all the interacting particles but particularly for secondary electrons, for the reasons explained in Sect. 4.3.2. Therefore, track structure codes either rely on pre-parameterized or tabulated sets of total and differential elastic and inelastic cross sections in order to calculate the energy deposition in condensed matter. An important point to consider is that at these low energies, the interaction cross sections depend on the composition of the material but also on its state. That is to say that the cross sections are not the same for a medium in a gaseous or a solid state. This leads to a particular difficulty because it is very difficult (not to say, almost impossible) to obtain experimental cross sections for biological media in their condensed state [24]. Only a few data obtained under very specific conditions exist for liquid water [56, 57] and these data are the basis for the set of models utilized to calculate the cross sections used by most track structure codes.


However, still, some track structure codes use atomic ionization/electronic excitation cross sections [58] obtained in the gas phase even if, in principle, they are not suitable for low energy excitations of valence electrons in water, since such excitations are sensitive to the electronic structure of the target [54, 59–61].


Nevertheless, most of the theoretical models for the calculation of the cross sections used in these codes are based on the first Born approximation that uses the dielectric formalism. Here, the properties of a given material in terms of characterizing the inelastic interactions with charged particles are given in what is called the Energy Loss Function (ELF). This function allows calculating the mean free path and thus the inelastic cross sections. However, this function depends on the energy and momentum of the charged particles. As the existing experimental data have been obtained in the optical limit (i.e., for a zero momentum transfer), it is necessary to extend the calculation of this function for non-zero momentum transfers. Different dispersion algorithms based on the electron gas theory [53] are then used to redistribute the imaginary part of the function between the different ionization and excitation levels while preserving the agreement of their sum with the initial experimental data.


However, differences in results of inelastic scattering obtained with different dispersion algorithms to extrapolate optical data to finite momentum transfer reach about a factor 2 in the range 50–200 eV (and even further at still lower energies) [62] and consequently, these differences impact the obtained results. Recent studies have reported a potentially relevant effect of ionization clustering [63] or DNA damage induction [64].


The description of the dielectric function of water also continues to be studied. Thus, only recently have works been published that address exchange and correlation effects based on the electron gas model or that improve the description of effects beyond the first Born approximation [27, 65]. The objective is to improve previous dispersion algorithms [66], to develop new TS codes [67, 68], and to clarify differences in inelastic scattering between different condensation phases [69]. Besides, other authors still work on measuring or adapting the theoretical model, using, for example, pre-parametrized models [70], to obtain cross-sections for targets other than water to be included in TS codes.


Concerning the elastic scattering models for low-energy electrons, different theoretical approaches are also developed and included in TS codes. Some use screening parameters derived from experiments to enlarge the applicability of the first Born approximation [71] and others use the Dirac partial wave analysis [72, 73].


Overall, the accuracy of the results for water at energies below 100 eV remains questionable, and it would be desirable to have results for the dielectric function, the electron energy loss and the inelastic mean free path from ab initio TD-DFT approaches, i.e., with no free parameters and which, as a consequence, are prone to have predictive power and to be extended to a variety of targets (Table 4.3).Table 4.3
MC track structure codes used in various radiation effects studies in biological cells

Code

Particles

Energy range

Target materials

Chemical stage

CPA100

e−

Thermalization −256 keV e−

Water (l), DNA

Yes

DELTA

e−

≥10 eV–10 keV e−

Water (v)

Yes

EPOTRAN

e−, e+

≥7.4 eV–10 keV

Water (l, v)

No

ETRACK

e−, p, α

≥10 eV–10 keV e−

Water (v)

Yes

ETS

e-

≥10 eV–10 keV

Water (l, v)

Yes

Geant4-DNA

e−, p, H, α, ions

Thermalization −1 MeV e−,
100 eV–100 MeV p, H
1 keV–400 MeV α
0.5–106 MeV/u ions

Water (l), DNA, gold, N2, and C3H8 (in progress)

Yes

IONLYS/IONLYS-IRT

e−, p, ions

0.2 eV–150 keV e−, p
0.1–300 MeV ions

Water (l)

Yes

KAPLAN

e−

≥1–10 keV e−

Water (l, v)

Yes

KITrack

e−, ions

≥10 eV–100 keV

Water (l)

No

KURBUC (KURBUC/LEAHIST/LEPHIST/CHEM-KURBUC)

e−, p, α, C

10 eV–10 MeV (10 keV, liq.) e−,
1 keV–300 MeV, p, 1 keV/u–2 MeV/u α,
1 keV/u–10 MeV/u carbon
≥0.3 MeV/u

Water (l, v)

Yes

LEEPS

e−, e+

0.1–100 keV

All materials

Yes

LEPTS

e−, e+, p

Thermalization −10 keV e−, Thermalization −10 MeV p

Water (v), CH4, C2H4, C4H8O, SF6, C4H4N2

No

Lion track

e−, p, ions

>50 eV e−, 0.5–300 MeV/u p, ions

Water (l)

No

MC4

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

No

MOCA8B

e−

10 eV–100 keV e−

Water (v)

Yes

NASIC

e−

Thermalization −1 MeV e−

Water (l)

Yes

NOTRE DAME

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

Yes

OREC/NOREC

e−

7.4 eV–1 MeV e−

Water (l)

No

PARTRAC

e−, e+, p, H, α, ions

1 eV–10 MeV e−
1 keV–1 GeV p, H, α
1 MeV/u–1 GeV/u ions

Water (l), DNA

Yes

PITS04

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l)

No

PITS99

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (v)

Yes

PTra

e−, p, α

1 eV–10 keV e−,
1–10 MeV α,
300 keV–10 MeV p

Water (l, v), DNA, N2, C3H8

No

RITRACKS/RETRACKS

e−, ions

0.1 eV–100 MeV e−, ions 10−1–104 MeV/u

Water (l, v)

Yes

SHERBROOKE

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

Yes

STBRGEN

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

Yes

TILDA-V

e−, p, H, ions

≥7.4 eV e−, 10 keV/u–100 MeV/u ions

Water (l, v), DNA

No

TRAX

e−, p, ions

1 eV—few MeV e−
10 eV—few hundred MeV/u ions

Water (v)

Yes

RADAMOL (TRIOL/STOCHECO)

e−, ions

≥7.4 eV–2 MeV e−,
≥0.3–200 MeV/u ions

Water (l)

Yes

TRION

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

No

TRACEL/RADYIE/RADIFF

e−, ions

≥10 eV e−,
≥0.3 MeV/u ions

Water (l, v)

Yes

Associated particles, energy ranges, and target media (e.g., whether vapor or/and liquid phase cross sections are used) are indicated. (Taken from [55])


DNA damage is calculated from the energy depositions at nanometric scale in liquid water simulated with track structure codes and overlaid onto DNA models. DNA geometrical description can be as simple as cylindrical models of the DNA [74, 75] or as complex as a full atomistic description of human chromosomal DNA [76]. Nowadays, some of these models are directly included in the physical stage simulation (see Fig. 4.14) [48, 78], in order to facilitate the use of DNA material cross-sections instead of liquid water if they are available in the MC TS code. Besides, some subcellular structures are implemented in some TS codes [79] for the calculation of energy deposited in mitochondria or cellular membranes, for instance.

A schematic representation of the geometrical model of D N A double helix to two linked nucleosomes, chromatin fiber loop built with voxels, and endothelial cell nucleus model.

Fig. 4.14
Example of DNA target geometrical model used in the mechanistic simulation of DNA radiation-induced damage with the Geant4-DNA code [48]. The generation of this geometrical model was done with the DNAFabric software [77] from the nucleotide description to the complete genome of an eukaryotic cell nucleus in the G0/G1 phase


A schematic representation of the geometrical model of D N A double helix to two linked nucleosomes, chromatin fiber loop built with voxels, and endothelial cell nucleus model.


From the resulting energy deposition values or interactions registered in the DNA volumes, direct damages are calculated using different approaches depending on the TS code. For instance, in some cases, an energy threshold value (often of 17.5 eV) in the nucleotide backbone is used to define a direct strand break [30, 80]. Others, as in the case of the PARTRAC code, use a uniform probability linear function from 5 to 37.5 eV [81] in order to calculate the resulting direct strand breaks, taking into account that very small energy depositions from vibrational excitations can also lead to this kind of DNA damage.


After the simulation of the physical stage, the geometrical model of the DNA target (essentially the position of all its constituents) as well as the position of the surrounding ionized or excited liquid water molecules are “translated” in terms of chemical species and injected in the code for the simulation of the chemical stage as described in Sect. 4.3.3. Here also, different codes use different parameters for the definition or the calculation of the indirect strand breaks depending on the DNA geometrical model; the number of included reactions or the duration of the chemical stage simulation [32].


Finally, in order to quantify the results, an important issue is the definition of double strand breaks and, above all, of clustered damage. Indeed, these notions are fundamental if we want to be able to compare the results of the simulation predictions with the experimental data, representing either the fragments produced (PFE, comet assay) or the signaling of a repair process set in motion by the cell (foci). The way of quantifying the damage predicted by the modeling of the physical, physicochemical, and chemical stages must thus be adapted each time to the characteristics of the experimental observable used for the validation. Nevertheless, for a relative comparison of different radiations, other types of classification can be used. Finally, in order to extend the modeling to later stages and include the repair mechanisms, the scoring method must also be adapted to the initial damage definitions of each repair model. Thus, the definition of a double stranded break is relatively well established as two breaks in the sugar-phosphate group on opposite strands separated by less than 10 base pairs (bp). More complex breaks or clustered damages are very author-dependent: DSBs accompanied by altered bases or single breaks at less than 10 bp, two double breaks separated by less than 25 bp, for instance [82], or more complete definitions as the classification proposed by Nikjoo et al. [83].


Recently, a standardized format for the simulation output results [84] has been proposed by different researches of this community, in order to preserve a maximum of information on the DNA damage simulated by the different codes and their location in the genome. This standard output amounts to a mapping of the individual damages produced (and the information of their direct or indirect origin) so that it can then be adapted to the scoring required for each use of the code, validation with experimental data, or use as input to repair models (Box 4.6).


MC Track structure developed over the years allow the simulation of energy deposition at nanometric scale

From these results and a DNA geometrical target model, direct DNA damages can be calculated

Chemical reactions between radiation-induced chemical species in the cell nucleus and the DNA target generate the so-called indirect effects that account for up to 70–90% of the total strand breaks

The way of considering damage and its complexity must be adapted to the different experimental methods


MC Track structure developed over the years allow the simulation of energy deposition at nanometric scale


From these results and a DNA geometrical target model, direct DNA damages can be calculated


Chemical reactions between radiation-induced chemical species in the cell nucleus and the DNA target generate the so-called indirect effects that account for up to 70–90% of the total strand breaks


The way of considering damage and its complexity must be adapted to the different experimental methods


Conventional radiobiological studies are using broad (in the range of cm) irradiation fields for irradiating a whole cell population with a homogeneous dose in order to be able to screen an average reaction of this population to radiation. Already in the 1950s, the reaction of single cells to homogeneous irradiation or even to irradiation of subcellular parts became of interest [85].


Furthermore, in the 1990s the question arose whether there is a reaction of non-irradiated cells when they are located close to an irradiate one—the so-called bystander effect. To address these and other related topics, it is necessary to be able to apply a single, subcellular-sized radiation beam (in the range of sub-micron to a few micron) with an accuracy in the range of 1/a few μm. This is the field of microbeam research, where the term microbeam is used for beam sizes at full width at half maximum in the range of ~1 to ~10 μm for photon as well as particle beams. Additionally, the development of micro-beams makes it possible to not only apply single beams but also arrays of beams, which can then be used to directly study the kinetics of DNA repair, the movement of damage sites, the connection to chromatin organization, and their relation to radiation quality and outcome.


When beam sizes get larger (~100 μm–~1 mm), the beam or beam array is then termed minibeam or minibeam array. Here, the beam sizes become large compared to cell size and the difference in the effects switch from single cell differences to differences in cell population. An effect in this size range was described in the 1980s as the so-called dose-volume effect [86].


This effect is exploited in modern radiotherapy approaches such as Microbeam radiation therapy (MRT) using photon beams with a beam size around 100 μm and particle minibeam radiotherapy (MBRT) using submillimeter-sized beams of protons or heavier ions (Table 4.4).Table 4.4
Definition of micro- and minibeam pattern and corresponding beam size and their application

Type

Single beam size (fwhm)

Application

Single microbeam

~1–~10 μm

– Radiosensitivity of subcellular
Structures
– Bystander effect
– Adaptive effect

Array microbeam

~1–~10 μm

– DNA repair kinetics
– Effects of high-LET particles

Single minibeam

~100 μm–~1 mm

– Dose-volume effect

Array minibeam

~100 μm–~1 mm

– Modern therapy approaches


A new wave of interest worldwide in the use of micro-beams in radiation biology in the 1990s has led to the development of a number of tools that eventually evolved into facilities with potential clinical utility [87, 88]. Single cell micro-beams provide a unique opportunity to control precisely the dose to individual cells in vitro and the localization of dose within the cell. This makes it possible to study a number of important radiobiological processes in ways that cannot be achieved by other methods. Figure 4.15 shows such micro-beams as single or array application visualized on fluorescent nuclear track detectors and also via the foci of 53BP1 repair protein in human HeLa cells.

A set of 3 radiobiological processes of single, and an array of 5 cross 5 matrix proton at 1 and 5 micrometers, respectively.

Fig. 4.15
Proton microbeam with a size of 0.8 μm (fwhm) visualized by a fluorescent nuclear track detector. Array of proton micro-beams with a point distance of 5 μm in both directions. 53BP1 accumulation in HeLa cells after microbeam array irradiation with a single carbon ion per point [beam size 0.8 μm (fwhm) and point distance 5 μm]


A set of 3 radiobiological processes of single, and an array of 5 cross 5 matrix proton at 1 and 5 micrometers, respectively.


Specifically, using charged particle micro-beams, it is possible to deliver exactly one particle per cell providing an ideal method for reproducing in vitro situations relevant to environmental exposure to naturally occurring radioactive radon gas, where virtually no cell receives more than one alpha particle traversal in its lifetime [89]. The high-spatial accuracy offered by micro-beams provides also a useful method to investigate subcellular spatial sensitivity such as the radiosensitivity of DNA close to the nuclear membrane [90] or of specific cellular organelles ([91, 92], p. 2019). Finally, single cell micro-beams have played a crucial role in the understanding of the bystander effect elucidating some of the mechanisms responsible for the transmission of the radiation effects from irradiated to non-irradiated cells [93]. Microbeam facilities can be used to selectively irradiate individual cells that can subsequently be revisited to ascertain what changes have occurred to that cell, and to its unirradiated neighbors.


There are four key aspects for the development of a single cell radiobiological microbeam: the radiation source, the radiation collimation or focusing, the radiation detection, and the cell alignment. A schematic view of a single cell microbeam can be found in Fig. 4.16.

A schematic diagram exhibits a single cell microbeam of the ion source, accelerator, energy selection magnet, aperture, beam switch, beam scanning, focusing unit, sample, detector, signal, and control unit.

Fig. 4.16
Schematic view of a single cell microbeam for radiobiological research using ions. The ions are produced in the ion source and accelerated. Energy selection is carried out with a 90° magnet. Into the focus of this magnet, the aperture needs to be placed, which defines the object that is focused by the focusing unit. The biological sample is placed in its focus. Either in front or behind (shown here) the sample, the ion detector counts the ions and gives the signal to the control unit. Here the signal is processed and the beam switch and scanning unit can be regulated


A schematic diagram exhibits a single cell microbeam of the ion source, accelerator, energy selection magnet, aperture, beam switch, beam scanning, focusing unit, sample, detector, signal, and control unit.


As the main aim of single cell micro-beams is to be able to irradiate individual cells with high-spatial accuracy, the majority of micro-beams utilize low-energy radiation sources as penetration is not a requirement and higher radiation energies have stronger focusing or collimating requirements. Linear particle accelerators [94–96] or lab bench X-ray sources have been mainly used [93], although synchrotron sources have also been employed [97]. Energy resolution and stability are key parameters in order to achieve small radiation probes. The collimation or focusing system is a crucial element, as it provides a method for reducing the radiation beam to a micron or sub-micron size beam with which to probe the cells. Collimation systems (such as devices with high length-aperture ratio) are generally easier to implement and the final opening can be placed close to the cells in their wet environment, although it is very difficult to achieve beams smaller than a few microns [98]. The focusing approach offers the possibility to achieve sub-micron spot sizes while keeping cells in their physiological environment ([99], p. 2019; [100], p. 2017; [101]). The next important element is the particle detection, as a key aspect of the cellular micro-beams is being able to count single ions so one can deliver an exact number of particles (or dose) to a single cell. Charged particle micro-beams achieve this through individual particle counting systems placed either after the biological samples (in which case the radiation energy has to be high enough to traverse the samples) or between the collimation/focusing system and the cells (which may degrade the radiation spot size). Detector systems using a combination of plastic scintillators and photomultiplier tubes have been successfully employed achieving basically 100% detection efficiencies [87, 88]. The final element consists of imaging and micropositioning devices required to identify the biological targets of interest and align them with the radiation probe. Speed is essential because many assays of biological radiation effect require several hundreds, or even thousands of cells to be micro-irradiated individually. The performance of the various single cell micro-beams varies according to the methods adopted and particularly the radiation used. However, state-of-the-art systems can achieve targeting accuracies in the range of a μm and detection efficiency approaching 100% [91]. These systems can also irradiate 10,000 s of cells per hour.


One of the first key studies to make use of micro-beams was completed using the RARAF facility in New York. Miller et al. [102] demonstrated that the transformation frequency of a single alpha particle traversal is not statistically different that of no traversals. The finding has strong implications for radiation protection, and it supports the threshold hypothesis for radiation risks. Many radiobiological studies using micro-beams have been aimed at investigating the bystander effect. In particular, experiments with co-cultured glioma and fibroblast cells showed that micronuclei formation can be induced through bystander signaling across genotypes [103]. These studies also provided information about the signaling processes involved in the bystander response suggesting nitric oxide (NO) and reactive oxygen species (ROS) play a critical role [103]. Another important radiobiological contribution from micro-beams comes from adaptive response studies [104]. The adaptive response manifests itself as a reduction in the effect of a high dose of radiation when a small (<0.2 Gy) priming dose is given first, typically a few hours ahead of the high dose. This observation undermines traditional thinking with regard to radiation effects and has been linked to radiation hormesis; the concept that radiation at low doses may actually be beneficial. Also, the investigation of the radiosensitivity of subcellular structures is a key application for ion micro-beams [91, 92, 105]. For example, it could be shown that radiation-induced localized damage with high-LET particles only triggered localized inhibition of rRNA transcription in nucleoli rather than pan-nucleolar reaction, as it was seen in drug treatment or under UV irradiation [91].


Micro-beams cannot only be used in single beam mode but also with an array of micro-beams. Arrays of particle micro-beams are used especially for two applications. First for understanding the kinetics of DNA repair. The major advantage of micro-beams arrays here is that the damage is induced within a known pattern with defined distances at a defined time. With this method, repair kinetics of various proteins such as 53BP1, Rad52, Mdc1 [106], and PARP1 [107] could be measured. Furthermore, it was found that the sites of DSBs induced by micro-beams show a non-directed, sub-diffusion movement within the cell nucleus [108].


Furthermore, by focusing low-LET protons to ~1 μm beam size the RBE can be increased. With this information, it was possible to further understand the enhanced RBE of high-LET particles [99, 109–111], which is an effect on several scales. An enhancement of LET is possible when focusing the ions to ~1 μm beam size but this enhancement does not reach the RBE of a single high-LET particle, where most of the damage is caused in the core region of a few 100 nm diameter. The explanation of this is that when ions are focused to ~1 μm sizes, the DSB get closer together and therefore complex damages occur. If the damage is caused on even smaller scales, single strand breaks will get so close together that they cause further DSB, which enhances the biological effect [110] (Box 4.7).


Micro-beams are beams of photon or particle radiation and have a size of ~1–~10 μm

Micro-beams can be applied as a single beam or array of beams

Collimation is easy to implement but beamsize only a few μm

Focusing is more complex but beamsizes <1 μm are possible

Micro-beams can be used to study bystander effect, radiosensitivity of subcellular structures, and the enhanced RBE of high-LET particles


Micro-beams are beams of photon or particle radiation and have a size of ~1–~10 μm


Micro-beams can be applied as a single beam or array of beams


Collimation is easy to implement but beamsize only a few μm


Micro-beams can be used to study bystander effect, radiosensitivity of subcellular structures, and the enhanced RBE of high-LET particles


A minibeam is a narrow radiation beam, whose width is in the range from ~100 μm to approximately 1 mm. The minibeams play a key role in the development of new therapy approaches, which aim to lower the side effects of external radiotherapy by spatially sparing the healthy tissue, especially in front of the tumor. Using photons, this method is called microbeam radiation therapy (MRT) in order to be able to separate from the particle minibeam therapy (MBRT) using protons and ions. MRT uses beam sizes in the order of 100 μm, whereas in pMBRT the beam sizes are a bit larger up to ~1 mm. There are different approaches of how to irradiate the tumor with minibeams; with photons or heavy ions the minibeam pattern with peaks and valleys is typically maintained, while with protons or light ions homogeneous irradiation of the tumor is feasible.


Nevertheless, both methods rely on the same effect, that the smaller the volume which is irradiated, the more dose is tolerated by tissue, the so-called dose-volume effect [86]. This is attributed to undamaged migratory cells surrounding the damaged tissue, which are able to infiltrate and thus reduce tissue necrosis. A further effect that plays a role in the tissue response to submillimeter beams is the microscopic prompt tissue repair effect. For such small irradiation fields, the capillary blood vessels can be repaired within days or even hours. The intact blood vessels are then able to support the repair of surrounding tissue. The detailed underlying radiobiological effects are yet not completely understood and topic of investigation worldwide. Nevertheless, the use of minibeams in radiation therapy is already used in spatially fractionated radiation therapy such as GRID therapy or is on the way to clinical studies (Box 4.8).


Minibeams are beams of photon or particle radiation and have a size of ~100 μm–~1 mm

Minibeams can be applied as a single beam or an array of beams

Collimation is easy to implement but can give secondary radiation and limits flexibility

Focusing is more complex to implement but has no secondary radiation and full flexibility

Minibeams are used to study the dose-volume effect and the microscopic prompt tissue repair

Minibeams are transferred into clinical application in microbeam radiation therapy (MRT) for photons and minibeam radiation therapy (MBRT) for particles


Minibeams are beams of photon or particle radiation and have a size of ~100 μm–~1 mm


Minibeams can be applied as a single beam or an array of beams


Collimation is easy to implement but can give secondary radiation and limits flexibility


Focusing is more complex to implement but has no secondary radiation and full flexibility


Minibeams are used to study the dose-volume effect and the microscopic prompt tissue repair


Minibeams are transferred into clinical application in microbeam radiation therapy (MRT) for photons and minibeam radiation therapy (MBRT) for particles


Suppose an object (say a macromolecule) is irradiated. Assume that the radiation deposits one or more primary ionizations (i.e., ion clusters) within the molecule. Assume that the molecule has a particular function within our cells and that this function is destroyed only if the ion cluster destroys one particular part of the molecule and that the molecule still works equally well if the ion cluster damages any other part. The sensitive area inside the molecule is then called the target (Fig. 4.17) (Box 4.9).

A schematic diagram of an irregularly shaped macromolecule irradiated object of a small area of the target with volume v.

Fig. 4.17
One assumes that the target only consists of a small area of the object being irradiated. The object may be a macromolecule or an organism


A schematic diagram of an irregularly shaped macromolecule irradiated object of a small area of the target with volume v.


Target theory postulate: only energy deposits in the target can destroy the function of the object


For the sake of simplicity, assume that one hit represents one primary ionization. This can in some cases be an oversimplification since a primary ionization can give rise to many ion pairs however the probability is largest for a primary ionization to give rise to only one ion pair [112].


We can now introduce the dose as the number of hits per cm3. In an elegant experiment, Rauth and Simpson [113] found that the energy deposition per primary ionization is about 60 eV on average. Although this is not the exact average energy per hit, it can be used as an approximate value. Since the dose gives the energy deposition per cm3 (it indicates the energy per g or kg, but when we know the density of the irradiated substance we easily convert it to cm3), we can use Rauth’s and Simpson’s measurement to convert the dose to the number of primary ionizations per cm3 and as a first approach use this as an indication of the number of hits per cm3.


This theory relies on certain key assumptions1.
Ionizing radiation deposits the energy into discrete energy packages that we call hits.
 2.
The response of a molecule (or cell) occurs only if a number of n hits is deposited in the target.
 3.
The number of hits deposited in the target in the irradiated material must be Poisson distributed.


Ionizing radiation deposits the energy into discrete energy packages that we call hits.


The response of a molecule (or cell) occurs only if a number of n hits is deposited in the target.


The number of hits deposited in the target in the irradiated material must be Poisson distributed.


Assumption number 3 can generally only be considered satisfied when the dose is high. Note that the average number of hits in a volume equal to the target volume is μ = vD where the dose is given in hits/cm3 and the target volume, v, is given in cm3. If n is the actual number of hits in the target in a particular irradiated object, the probability of this number of hits being seen is Poisson distributed as:

 (4.23)


If the irradiated object is a macromolecule in a cell, and if this macromolecule is inactivated (i.e., loses its biological function) if it receives n hits in the target, then the molecule retains its function if the number of hits in the target is n−1 or less. We can therefore calculate the probability, pf, for the molecule to retain its function. It must be the sum of the probabilities that it will receive one, two, three, etc., up to n−1 hits in the target [112]:

 (4.24)


or

 (4.25)


Here, pf represents the probability that a target molecule will not be inactivated by the dose D. However, this can also be viewed as pf representing the fraction of the irradiated molecules that do not become inactivated by the radiation [112]. If one irradiates N0 molecules and the number that is not inactivated is N, Eq. (4.25) can be rewritten as:

 (4.26)


If the molecule becomes inactivated by only one hit in its target, n = 1 and Eq. (4.26) becomes:

 (4.27)


This is the Single-Hit Single-Target model of radiation survival. While simple, it is a very powerful equation that can provide insights into the characteristics of cellular response to radiation exposure (Fig. 4.18).

A graph of cellular survival L n versus Dose of single-target model. The linear line plotted has a downward slope.

Fig. 4.18
The relationship between the predictions of the single-hit single-target model on cellular survival versus radiation dose [here N/N0 from Eq. (4.27) is replaced by S/S0 or the ratio of cell survival at any dose D to that at 0 Gy]


A graph of cellular survival L n versus Dose of single-target model. The linear line plotted has a downward slope.


One key insight is that the equation allows us to determine the molecular weight of the target. From the previous derivation, we know that if we express dose in the unit hits/cm3, we can determine the target volume. If we also know the density of the irradiated molecules we can, based on the target theory, determine the molecular mass of the target. The dose is normally given in Gy so the calculation must be based on this unit:

 (4.28)


using that 1 J = 6.242 × 1018 eV. From the experiments by Rauth and Simpson, we know that it takes an average 60 eV to give a primary ionization in an organic material. This value is not necessarily the correct amount of energy needed for a hit, but as an example it can be used. Then we can convert [Gy] into [hits per gram]:

 (4.29)


D37 is the dose that gives on average one hit per target, i.e., v ⋅ D37 = 1 . The surviving fraction at this dose is e−vD= e−1 ≈ 0.37 = 37%, which gives rise to the name of the quantity. If we assume that we irradiate the molecules with different doses and find the D37, we have on average one hit per target at this dose (v ⋅ D37 = 1). Suppose the D37 is given in the unit hits/g. We can then calculate the mass of the target in the unit gas:

 (4.30)


In practice however the dose is in Gy and we must use Eq. (4.29) to convert from hits/g to Gy:

 (4.31)


If the density of the target is ρ = MT/v we can then calculate the target volume:

 (4.32)


In Eq. (4.32) the dose is in Gy. The final calculation of the target volume is left to the reader.


Complicated molecules or cellular organisms may well have more targets and it also may take more than one hit per target to inactivate the molecule or cell.


Recall Eq. (4.25), which calculates the probability that a molecule will not be inactivated if it has one target and that this is deactivated by n hits. The probability of one target being deactivated is then:

 (4.33)


where Nii means the number of molecules that were inactivated. If we now assume that the molecule has a number of m targets that all must be inactivated for the molecule to be inactivated, the probability of inactivation becomes:


and the probability that the molecule will not be inactivated is then:


In the most likely case, it only takes one hit per target for the molecule to be inactivated, that is, n = 1. This gives the following special case:


This is the famous multi-target single-hit equation. For many decades, this was the model radiobiologists fitted to their dose-response curves when they tested the effect of ionizing radiation on human cells. Much of the formalism of this equation and parameter values are still in use when dose-response curves are discussed and described. Therefore, it is important that we perform an analysis of this function:
The equation has a shape with an initial shoulder at small doses followed by, a straight line for large doses. This is seen if the equation is expanded by a power series:


The equation has a shape with an initial shoulder at small doses followed by, a straight line for large doses. This is seen if the equation is expanded by a power series:


Focusing on high-dose regions, all terms with (e−vD)2 and higher power can be ignored. We then end up with the following expression, which only is valid for high doses:

 (4.38)


This is a straight line in a semi-logarithmic plot, and the line intersects the ordinal at point m as shown in Fig. 4.19.

A concave down, decreasing curve graph of L n of S by S 0 versus dose of G y, and the dashed line extrapolation number.

Fig. 4.19
The relationship between the predictions of the multi-hit single-target model on cellular survival S and radiation dose. S0 is the plating efficiency of the unirradiated controls


A concave down, decreasing curve graph of L n of S by S 0 versus dose of G y, and the dashed line extrapolation number.


Note that in Fig. 4.19, the actual dose-response curve has an initial shoulder followed by a straight line at higher doses. Thus, only the straight line at higher doses is described in Eq. (4.38). The dose-response curve itself is described in Eq. (4.36).


Figure 4.19 illustrates a dilemma with regard to the common definition of radiation sensitivity. It is common to say that the target volume is an expression of radiation sensitivity. For a single-hit, single-targeted model, one obtains a value v = 1/D37. For a single-hit, multi-target model like the one shown in Fig. 4.19, we can say that v = 1/D0 expresses radiation sensitivity. D0 is the dose, which reduces the surviving fraction by 63% in the linear part of the curve (Box 4.10).


This may seem a bit odd: If we have two types of molecules, one with one single target and one with m targets, but where the target volumes are the same, such that D37 = D0, as is the case in Fig. 4.19, then the radiation sensitivity is the same in the two cases and is only given by the slope of the dose-response curves at high doses. Nevertheless, one can immediately see that the curve that has a shoulder shows a higher survival value for a particular dose than the one that does not have a shoulder. This is because it is an advantage for a molecule that the radiation must destroy two or more targets rather than just one to inactivate the molecule. Still, many authors have chosen to use the target size as a mathematical expression of the radiation sensitivity.


One term is important to get into at this stage, namely sublethal damages. So far, we have most talked about irradiating molecules and not cells. However, we can talk about cells in the same way that we have discussed molecules in the hit and target theory. The radiation damage then inactivates some function that the cells usually have. Very often, the effect is referred to as cell death or lethality. This term suggests that radiation should produce some form of death. Often, this will give incorrect associations to the chemical or biological responses we measure. However, the terms lethal, sublethal, and potentially lethal damages have been so incorporated that it is completely impossible to avoid their use.


Note that, based on the formalism of the target theory, sublethal damage is damage to the target. A hit outside the target is no damage according to this theory. When damage in the target does not produce any effect, it is because we have a multi-hit system or a multi-target system (Box 4.11).


Sublethal damage refers to damage, or really ion pairs, which is the cell or molecular target, but does not cause any effect in itself


Later in this chapter, we will talk about dose rate effects. These state that there usually is a stronger effect of a dose when given in a short time than when given over a long period of time. The reason for this is, according to the target theory’s formalism, that the first hit is not enough to inactivate, but that it can interact with the next so that the two or more together can inactivate. However, if the cells or molecules are able to repair the first hit before the next, we will not get such interactions. The fact that this effect decreases with decreasing dose rate is therefore a sign that the radiation damage is repaired.


Note also, that the shoulder of the multi-target curve in Fig. 4.19 has nothing to do with repair in the target theory’s formalism. It is just because the cells or molecules can either tolerate one or more hits in their one target or that they have more than one target.


While target models are useful to generate an initial understanding of the relationship between radiation dose, cell survival, and the process of energy deposition, these models have not been generally adopted because of their use of multiple terms, and also because the “targets” which the models predict have never been identified. Several models have been developed based on target theory, among which the linear quadratic model (LQ) has emerged for application clinically and preclinically [114]. The expression for cell survival according to the LQ model is:

 (4.39)where S represents the probability of cell survival when subjected to dose D and the α and β parameters determine the linear and quadratic components of cell damage, respectively. The dose-squared dependence implies that the survival plot on a logarithmic scale has the characteristic appearance of a quadratic curve (Fig. 4.20). A linear relationship dominated by the α-parameter is observed at very low doses, while for higher doses the quadratic relationship governed by the β parameter becomes dominant. This characteristic feature of the survival curve is commonly referred to as the shoulder.

A decreasing line graph of surviving fractions versus dose of G y. From the top, a vertical line is labeled alpha by beta, beta D power 2, and alpha D. The line is split into a solid line of alpha by beta high, and dashed line of alpha by beta low.

Fig. 4.20
Illustration of LQ curves for high and low α/β ratios. For the low α/β, the shoulder of the curve is more pronounced. The α/β-ratio can be found by drawing a line with the initial slope (α) of the curve and finding the dose where the contribution from the linear and the quadratic terms are equal


A decreasing line graph of surviving fractions versus dose of G y. From the top, a vertical line is labeled alpha by beta, beta D power 2, and alpha D. The line is split into a solid line of alpha by beta high, and dashed line of alpha by beta low.


The linear quadratic model needs only two parameters and shows a good fit for experimental observations. As for its biological interpretation, different approaches have been presented, such as those of Kellerer and Rossi, Chadwick and Leenhouts, and Bodgi and Foray.


Kellerer and Rossi sought to analyze the relationship between dose and effect in a way that was invariant to the quality of radiation, as they considered that the biological effect, in addition to its dependence on the deposited energy, also depended on its microscopic distribution [115]. Having observed the simplicity of the relationship between the doses of two different types of radiation with different LET (see Sect. 1.6) that lead to the same effect (relative biological effectiveness—RBE, see Sect 1.6) they proposed a theoretical model that arrives at a linear and quadratic relationship with the dose. The model assumes three possible states for the biological entity: non-damage, pre-damage, and effect. The probability of transition between states (without allowing for reversion) depends on the dose and a careful choice of these values results in different models [115]. According to the model, the biological effect can be achieved by a direct transition from the non-damage state to the biological effect or by two consecutive transitions between non-damage to pre-damage and pre-damage to biological effect. The first case represents the situation of reaching the biological effect with one hit (single-hit event), which is dominant for high-LET radiation, and for the second case, two hits (double-hit event) are required.


Chadwick and Leenhouts started from the hypothesis that cell death resulted from a double strand break in DNA (DSB) and that the probability of these events was related in a linear quadratic manner with dose. The model assumes:
that DNA is a critical molecule that determines the cell’s ability to reproduce and a DSB is considered critical damage;

radiation produces DNA breaks that can be repaired, and the radiobiological effect reflects the degree of repair [116].

that the rate of critical breaks relative to dose (dN/dD) is proportional to the number of critical bonds (N) and that a critical event (DSB) can occur in two ways: either as a single radiation event that results in a DSB or as two events each inducing a single strand break (SSB), which is close enough in time and space interact to form a DSB.


that DNA is a critical molecule that determines the cell’s ability to reproduce and a DSB is considered critical damage;


radiation produces DNA breaks that can be repaired, and the radiobiological effect reflects the degree of repair [116].


that the rate of critical breaks relative to dose (dN/dD) is proportional to the number of critical bonds (N) and that a critical event (DSB) can occur in two ways: either as a single radiation event that results in a DSB or as two events each inducing a single strand break (SSB), which is close enough in time and space interact to form a DSB.


Therefore, the combination of these assumptions leads to an exponential model with a linear term and a quadratic term, similar to the one developed by Kellerer and Rossi, producing the linear quadratic model of cell survival.


There are various ways of interpreting this model. In one, two DSBs can interact and lead to chromosomal aberrations that impair cell division. In particular, asymmetric aberrations, such as the dicentric, the ring, and the anaphase bridge, make cell division impossible. In another interpretation by Hall, the linear and quadratic terms can be interpreted as asymmetric chromosome aberrations produced in one or two radiation events.


In both the interpretations by Chadwick and Leenhouts and Hall, the shoulder of the LQ-curve is a result of sublethal damage, i.e., damage that is not lethal in itself but can interact with other sublethal damage to become lethal. The difference lies in what is regarded as sublethal damage. Hall assumes that one DSB in itself is not lethal such that lethal damage is created only when two DSBs create an asymmetric chromosome aberration. Chadwick and Leenhouts also acknowledge that asymmetric chromosome aberrations are lethal DNA damage, but they adjust for this by multiplying by a factor, which represents a linear relationship between the number of DSB and the number of asymmetric chromosome aberrations. In their interpretation, sublethal damage is a single strand break (SSB), which needs to interact with another SSB close in time and space to form a DSB.


If the dose is fractionated (i.e., split up into several parts separated in time) or the dose rate is decreased, a linear survival curve will emerge. This is a reflection of the sublethal damage being repaired before it can interact with other sublethal damage to become lethal. With the repair time for DSB and SSB in mind (see Sect. 2.4), this supports Chadwick’s and Leenhouts’ interpretation.


In 2016, Bodgi and Foray proposed a new model for radiation-induced cell death whose mathematical derivation results in the linear quadratic model. In this model, DSB recognition mechanisms are mediated by ataxia telangiectasia mutated monomers (ATM) that are induced in the cytoplasm by radiation and diffuse to the nucleus (nucleo-shuttling of IR-induced ATM monomers). Once in the nucleus, these monomers participate in the DSB recognition mechanism that allows its repair [117]. The rates of DSB production by radiation and monomerization are assumed to have a linear relationship to dose. The same model also includes the notion of cell tolerance, taking into account that not all DSB lead to cell death, which in this case is assumed to be due to unrepaired DSB in cells entering mitosis. Among unrepaired DSB, those that are not recognized and therefore not repaired are distinguished from those that are recognized but not repaired within a suitable time window. The number of unrecognized DSBs in the model has a quadratic relationship to dose, whereas the number of recognized but unrepaired DSBs has a linear relationship to dose. Finally, unrepaired DSBs are assumed to follow a Poisson distribution, which leads to cell survival being modeled by a linear quadratic exponential. In addition to presenting a biological mechanism of cell death by radiation, this model provides an explanation for cellular hypersensitivity at low doses, since it assumes that radiation does not produce enough ATM monomers to cross the membrane and enter the nucleus. Therefore, there is no recognition of DSB and they remain unrepaired, which leads to cell death.


The linear quadratic model is arguably the most used tool in radiation biology and physics, as it provides a simple relationship between the dose absorbed and the number of surviving cells (or the probability that a single cell will survive). In its basic format (SF = exp (−α × D – β × D2)), it has been used to analyze and explain both in vivo and in vitro experiments and after some modest simplifying assumptions, it can be related to a number of mechanistic models such as multi-hit and potentially lethal lesion models. However, despite its widespread usage, questions remain about its applicability, particularly at the very low and very high-dose regions where significant discrepancies have been observed between the model predictions and the experimental data. Such questions spring from the complexity of the underlying biology and modern radiotherapy, where the response of cells and tissues can be modulated by both intrinsic genetic factors as well as the cellular environment and the radiation delivery modality. The linear quadratic model has therefore been the subject of extensive investigations and suggestions for modification to better fit the experimental data and therefore to explain a wide range of radiation conditions.


In the low-dose region, high-resolution in vitro measurements demonstrated increased X-ray effectiveness below 0.6 Gy [118]. The measured survival levels were significantly lower than those predicted by extrapolating the high dose points using the linear quadratic models. The phenomenon, named hypersensitivity, was reported with a range of cell lines and radiation qualities and data suggest that the observed response was unlikely to be due to a subpopulation of radiosensitive cells. In order to account for the increased effectiveness per unit dose at doses lower than 1 Gy and in line with the hypothesis that repair mechanisms are only triggered when sufficient damage has been accumulated, modification to the linear quadratic models has been suggested. Joiner and Johns [119] proposed a simple modification in which the alpha parameter decreases with increasing radiation dose, representing an increased induced radio resistance. The modification only concerns the alpha parameter, as the contribution of the beta parameter is negligible at low doses due to its quadratic influence. The modified linear quadratic models for low doses can therefore be expressed as

 (4.40)


where dc is the dose at which 63% of the induction has occurred and g is the amount by which the alpha parameter changes at low doses (Fig. 4.21).

An error line graph of surviving fraction versus dose for fit by induced repair model of errors a subscript s, H R S, d subscript c, I R R. A dashed linear downward line is the extrapolation from the high-dose fit by L Q model.

Fig. 4.21
Low dose hypersensitivity showing a clear downward bend on the survival curve for doses below 1 Gy, followed by an “increased radio resistance” at doses above 2 Gy. The image also shows the key parameters for the linear quadratic modification


An error line graph of surviving fraction versus dose for fit by induced repair model of errors a subscript s, H R S, d subscript c, I R R. A dashed linear downward line is the extrapolation from the high-dose fit by L Q model.


The interest in radiotherapy treatments delivered with a smaller number of high-dose fractions (hypofractionation) and stereotactic radiosurgery (SRS) has also instigated investigation into the validity of the linear quadratic model at high doses. A number of investigations have shown that the linear quadratic model in its basic form is not suitable in the high-dose region where it underestimates the surviving fraction and does not reproduce the straightening of the curve observed experimentally [120, 121]. To cope with this drawback, modifications of the linear quadratic model have also been suggested at high doses [122]. The starting point is an early modification of the linear quadratic expression to account for repair during a protracted radiation exposure:

 (4.41)


with λ as the repair rate parameter, T is the delivery time for the dose D, alpha and beta as previously described for the basic linear quadratic model. This version of the LQ model is able to predict survival curves taking into consideration scenarios where significant repair occurs during the dose delivery and is in accordance with other mechanistic models (i.e., Lethal, Potentially Lethal model). In order to reproduce the behavior of acute high doses however an additional term needs to be added to the G parameter:


The new parameter (δ) is introduced to match the final slope of the survival curve and can be interpreted as a reduction in survival due to interaction between lesions. Using Eq. (4.41), it can be shown that at high acute doses G (λT + δD) = 1/2 δD and therefore the modified LQ model assumes the form: SF = exp (−(α + β/2δ) × D), which has a linear behavior. Therefore, this model is referred to as a linear quadratic linear or LQL model (Fig. 4.22).

A decreasing line graph of surviving fraction versus dose for L Q alpha by beta = 3, and 10 grays, the values from (0, 1) to (0.0002, 15), (0.03, 15), (0.04, 15), and (0.1, 15). The values are estimated.

Fig. 4.22
Difference in the surviving fraction predicted by the LQ and the LQL model for cell lines with different radiosensitivity (alpha/beta ratio)


A decreasing line graph of surviving fraction versus dose for L Q alpha by beta = 3, and 10 grays, the values from (0, 1) to (0.0002, 15), (0.03, 15), (0.04, 15), and (0.1, 15). The values are estimated.


Although both modifications of the linear quadratic model are able to accurately describe experimental data at low and high doses, they introduce new parameters, which need to be experimentally determined.


The reaction of cells and tissues to radiation damage involves the repair of DNA and a complex interplay between repair and cell survival. The ability of a cell to repair the damage it experiences depends on the part of the dose it receives, the part of the cell cycle in which it is irradiated, and the rate at which the dose is delivered. Therefore, we must give attention not just to the molecular mechanisms of damage and repair but also to cell cycle regulation of repair and ultimately their biological consequences.


Here we return to using the terms sublethal, lethal, and potentially lethal damage. By sublethal damage, we simply mean damage which will not be lethal to the cell even if the damage is not repaired. We will later see that it is still of great importance whether or not these damages are given time for repair hence the temporal aspect.


Lethal damage is fixed in such a way that they cannot be repaired. Potentially lethal damage may well be repaired but will be lethal if not repaired in time, where the notion of “in time” relates to cell cycle regulation.


If the cell passes through S phase with DSBs, the formation of dicentric chromosomes or rings may take place, which is potentially lethal to the cell [123]. If the cell thereafter enters mitosis with such asymmetric chromosomal aberrations, it may not be able to give each of the daughter cells a complete set of genes. If such asymmetric chromosomal aberrations are formed, they are therefore usually lethal for proliferating cells. However, if cells are given time to repair DSB before they can develop into asymmetric chromosomal aberrations, i.e., before the cell enters into S phase, damage such as DSB are only potentially lethal.


These concepts were supported by early experiments by Stapleton [124] and Phillips [125], where, respectively, culture of cells under suboptimal conditions for growth, or in the presence of inhibitors of the cell cycle produced an increased level of cell survival Seminal experimental findings which support this view include work in vivo by Shipley [126], where rat adenocarcinoma cells were irradiated in situ with gamma rays or neutrons, after which explants of the tumor were grown in vitro either immediately after, or from 4 to 24 h after irradiation, whereupon the survival of these cells was assessed in terms of their clonogenic capacity. While situated in functioning tissue within the animal, these cells had limited access to nutrients and growth factors, which set a natural limit on cell density thereby limiting cell growth and proliferation. Within tissues, such cells may well be cycling though they could take several days to do so, and as such have time to repair their DNA. When cultured as explants in vitro post-irradiation they have greater access to nutrients and as such proliferate strongly, with surviving cells able to produce colonies. Cells which were cultured immediately after irradiation exhibited lower survival rates than those which remained in situ for a period of time after irradiation. Clearly, cells that could not proliferate in tissue have an increased opportunity to repair their damage owing to them being prevented from progressing within the cell cycle. Further experimental evidence demonstrated that this repair process could continue up to 24 h after irradiation, indicating the complexity of this repair process [127].


The experiments by Shipley et al. also showed that there is no increase in cell survival for the cells explanted up to 24 h after high-LET-neutron irradiation. The implication of this is, that the damage induced by high-LET-neutron radiation must be too complex to allow for successful repair, which would increase the survival. This finding suggests that complex DSB are not reparable, even with non-homologous end joining (NHEJ), which has been reinforced by observations that not all DSB from high-LET irradiation initiate NHEJ-repair [128].


Following the pioneering development of the clonogenic assay by Puck and Marcus [129], experiments by Elkind and Sutton [130] demonstrated that fractionated irradiation could allow cells to repair their sublethal damage (Fig. 4.23).

A multi-axis graph plots the surviving fraction versus doses. The values are plotted for CLONE A P E = 56.3%, 505 radians PLUS 18.1 hours I N C 37 degree Celsius. 2 decreasing lines of different values.

Fig. 4.23
The surviving fraction of V-79 Chinese hamster cells irradiated either with a single dose or with two dose fractions separated by 18.1 h. The first dose fraction of 5.05 Gy was given at time 0 and then the cells were incubated for 18.1 h at 37 °C before the second dose fraction (varied between 2 and 8 Gy) was given. As seen, the incubation time between the two dose fractions has led to a complete reconstitution of the curve shape. The explanation was that through repair of the sublethal damage induced by the first dose fraction, the cells had regained their sublethal damage potential. Unrepaired, these damages would have added to the new sublethal damages and become lethal [130]. (Adapted with permission from Springer Nature: Elkind and Sutton, X-ray damage and recovery in mammalian cells in culture. Nature, 1959)


A multi-axis graph plots the surviving fraction versus doses. The values are plotted for CLONE A P E = 56.3%, 505 radians PLUS 18.1 hours I N C 37 degree Celsius. 2 decreasing lines of different values.


In this work, V79 cells of the Chinese hamster were irradiated with one single dose or with two dose fractions where the time between the fractions was varied. The results shown in Fig. 4.23 are from an experiment where they kept the time between the dose fractions constant at 18 h.


These results aligned with the target theory of the time, whereby the combined effects of several sublethal damage events in DNA may result in lethal damage, such that damage created by hits which are not lethal by themselves may be repaired, but cells will not have time to do such repair if a large dose is given acutely, i.e., at a high dose rate. With a large enough acute dose, the degree of sublethal damage for each cell is so high that it combines to form lethal damage.


It is well worth reflecting on both the differences and the similarities regarding sublethal damages between the traditional multi-target/single-hit model and the newer LQ model. On the one hand, in the multi-target/single-hit model, one does not make any assumption regarding the nature of the molecular damage induced. Still, it introduces the concept of sublethal damage and shows that such damage inevitably leads to an initial shoulder on the survival curve. Thus, Elkind’s and Sutton’s data show that if cells are given time for repair, DNA damage can be repaired.


On the other hand, in the LQ model, one assumes two specific types of molecular damages as being sublethal, namely single strand breaks in DNA (SSB) (which are all sublethal separately) and the repairable double strand breaks in DNA (DSB). In reality, no distinction is made between SSB and DSB concerning repair of sublethal damage observed by dose fractionation in the LQ model. According to the LQ model, the dose-response curve has a downward bending, because two sublethal SSB may give rise to a DSB. The DSB may develop into lethal damage and therefore is potentially lethal but probably may also in some cases be sublethal.


The question then arises as to the timeframe required for the repair of sublethal damage events. While Elkind and Sutton did go some way towards measuring the value of this variable (suggesting that it was as much as 12 h), it was not until experiments by Terasima and Tolmach and further experiments by Elkind that refined this estimate and gave an explanation for its value.


As indicated by Fig. 4.26, the repair curves are different if cells are incubated at room temperature (24 °C) between dose fractions compared to at 37 °C, where this difference has to do with differences in cell cycle progression. The cell cycle is halted almost completely at room temperature but continues almost uninhibited between dose fractions at 37 °C. Thus, the explanation is that the first dose fraction primarily kills more V79-cells in mitosis, G1, and early S than in late S phase where they are resistant (be mindful that V79 cells have a cell cycle duration of approximately 10 h and almost no G1 phase).


At 24 °C, surviving cells stay in the stage of the cell cycle where they are resistant between the dose fractions and therefore their radiosensitivity is constant with time. At 37 °C, they continue through the cell cycle after the first fraction and at some time later will have reached a cell cycle stage where they have maximum radiosensitivity, whereupon the second dose fraction is given. Consequently, survival as a function of the time between fractions will decrease with increasing time.


The customary notion for cell cycle progression between dose fractions is redistribution (also denoted as “reassortment”). So, this notion is used to state that cells, which have not been lethally damaged by a preceding dose fraction, will move to a different cell cycle phase before the next dose fraction (Fig. 4.24).

A multi-line graph plots the surviving fraction versus the hours between 2 doses. The values are plotted for different incubation temperatures of 3, 24, and 37 degrees Celsius for the dose and cells. 5 lines, 3 begin at (0, 0.007), 1 follows a decreasing trend, 1 follows an increasing trend, and 1 fluctuates, and 2 lines begin at (0, 0.15) and follow a decreasing trend.

Fig. 4.24
Chinese hamster V79-cells were irradiated with two dose fractions separated by different time spans (lower abscissa) and with different temperatures in the incubator between dose fractions; respectively 3, 24, and 37 °C. In particular, the curves representing 37 and 24 °C are of interest since the first one represents cells that cycle between the dose fractions while the other one represents cells, which do not cycle between the dose fractions. (Adapted from [131] with permission, © 2022 Radiation Research Society [131])


A multi-line graph plots the surviving fraction versus the hours between 2 doses. The values are plotted for different incubation temperatures of 3, 24, and 37 degrees Celsius for the dose and cells. 5 lines, 3 begin at (0, 0.007), 1 follows a decreasing trend, 1 follows an increasing trend, and 1 fluctuates, and 2 lines begin at (0, 0.15) and follow a decreasing trend.


From Fig. 4.26, one can see that the surviving fraction increases considerably with about 8–10 h repair time at the temperature of 37 °C. This increase is not due to repair. It has to do with the fact that some V79 cells reach completion of cell division before the next dose fraction is given (notice that these cells have a median cell cycle duration of just about 10 h). The consequence of this is that some colony-forming units consist of two daughter cells instead of just one at the time when the second dose fraction is given.


A simple calculation illustrates the importance of this phenomenon. If the probability to kill a cell is p, the probability for this cell to survive is S = 1−p. However, the probability to kill both cells in a doublet or all four cells in a quartet is p2 and p4, respectively. The probability for a doublet or a quartet to form a colony is therefore S = 1−p2 and S = 1−p4, respectively.


If we suppose that the number of V79 cells has doubled during a 12 h period at 37 °C following the first dose fraction in Fig. 4.26, we can understand the increased surviving fraction between 10 and 12 h. It is indicated that the surviving fraction after two dose fractions 2 h apart with full repair is 0.035. The surviving fraction after the first dose fraction alone is about 0.23. This means that normalized survival after the second dose fraction alone, assuming full repair of the sublethal damage induced by the first fraction, is 0.035/0.23 = 0.152. The probability that this dose fraction alone would kill a cell is therefore 1–0.152 = 0.848. However, if the cell reaches cell division between the two dose fractions, the probability for the doublet to become unable to form a colony is 0.8482 = 0.719. The probability for survival therefore increases from 0.152 for the single cell up to 1–0.719 = 0.28 for the doublet. The surviving fraction after both doses and full repair then should be 0.28 × 0.23 = 0.065. Thus, the surviving fraction has increased from 0.035 to 0.065 because of a doubling of the cell number per colony-forming unit. From Fig. 4.26 one can see that this corresponds well with the survival observed by Elkind et al. with 12 h repair time between the dose fractions, quite in agreement with the cell cycle kinetics for V79 cells (cell cycle duration ~10 h).


In radiotherapy, the notion used for cell proliferation between dose fractions is repopulation. By repopulation, we mean that cells which survive a preceding dose fraction get enough time before the next dose fraction to complete the cell cycle and divide. This is an important concept in connection with fractionated radiotherapy. Although cell cycle times in tissues are usually much longer than for V79 cells in culture, 24 h between the dose fractions is sufficient for at least some proliferation of both cancer cells and normal cells between the fractions.


From Fig. 4.26, one can furthermore see that the survival increases almost by the same factor over the first 2 h repair time irrespective of the temperature being 24 or 37 °C. As seen, the surviving fraction increases from the single-dose level of 0.005 and up to 0.02 at 24 °C and 0.03 at 37 °C at 2 h repair time. Thus, the data indicate that the repair itself is less influenced than the cell cycle progression by the temperature (in fact some DNA repair persists at temperatures as low as 3 °C) (Fig. 4.25).

A waveform exhibits surviving fraction versus time between split doses in an hour for repair of S L D, the dotted region is if cells are non-cycling, reassortment or redistribution if cells are cycling, proliferation.

Fig. 4.25
The increased cell survival with increasing time between two dose fractions (up to 2 h) is due to increased time for repair of the sublethal damages induced by the first dose fraction. After about 2 h, all sublethal damage has been repaired. Most surviving cells after the first dose fraction would however be in late S or mid G1, the phases where cells are most radio resistant. If cells are offered optimal growth conditions between the dose fractions (37 °C), these surviving cells will continue cell cycle progression and may after 6 h reach a phase where they are more radiosensitive. If the second dose fraction is given at that instant, the survival will be reduced. Therefore, the curve bends downwards between 4 and 6 h, before an upwards turn between 6 and 8 h, when the cells have proceeded to a phase of higher resistance. After a long time, which depends on cell doubling time (typically >12 h), cell division results in an increased multiplicity of the colony-forming units and we see an increased survival that is caused by repopulation. Curve extracted and generalized from Fig. 4.24


A waveform exhibits surviving fraction versus time between split doses in an hour for repair of S L D, the dotted region is if cells are non-cycling, reassortment or redistribution if cells are cycling, proliferation.


In Fig. 4.25, three concepts are listed that are all related to fractionated radiotherapy: repair (meaning in this connection repair of sublethal damage) redistribution (or reassortment), and repopulation (or proliferation). These concepts cover three of the phenomena usually referred to as the 6 Rs of radiotherapy (see Sect. 5.5). All these mechanisms are interesting from a radiotherapeutic point of view because they can all be manipulated by variations in the fractionation  regime chosen.


Elkind and Sutton demonstrated that cells irradiated with several dose fractions separated by enough time for full repair would repeat the repair of sublethal damage over and over again [132]. Steele also demonstrated that an increase in cell survival is observed when a given dose is delivered at a low dose rate [133]. Some consistent features in the cell survival curves for cells irradiated with fractionated or low-dose radiation were also observed:
While the initial slopes of each cell survival curve differed, the final slopes were consistent. Using the LQ model to describe the cell survival curve would indicate that for a given cell line under fractionated or low dose irradiation, the α-values would change, while the β-values would remain consistent.

The differences between the response of cells of different types (with varying radiosensitivity) were more pronounced with low dose rate irradiation than with acute irradiation. This is a first indication of a more general principle, which is of utmost importance for radiotherapy. The sparing effect of fractionated or low dose rate radiation is most pronounced for cells having a dose-response curve with a broad shoulder (or a shallow initial slope).


While the initial slopes of each cell survival curve differed, the final slopes were consistent. Using the LQ model to describe the cell survival curve would indicate that for a given cell line under fractionated or low dose irradiation, the α-values would change, while the β-values would remain consistent.


The differences between the response of cells of different types (with varying radiosensitivity) were more pronounced with low dose rate irradiation than with acute irradiation. This is a first indication of a more general principle, which is of utmost importance for radiotherapy. The sparing effect of fractionated or low dose rate radiation is most pronounced for cells having a dose-response curve with a broad shoulder (or a shallow initial slope).