Appendix A.1. Simulation of Micrometric Volume Irradiation
RITRACKS [
7] is a MC tool that simulates event-by-event energy deposition of ions of various energy and atomic numbers in liquid water, the main constituent of cells. It thus provides a detailed description of ion tracks at the sub-cellular scale and can thus be used for microdosimetry calculations or the study of DNA damages in the context of space radiation. In the present study, calculations were performed by defining a parallelepiped irradiation volume
encompassing a spherical volume of radius
, as depicted in
Figure 1. The number of ions
crossing
is modeled as a Poisson distribution,
where
represents the average number of tracks traversing
,
is the surface of irradiation of
, and
is the beam fluence obtained from the well-known equation,
where
is the irradiation dose, and the
is obtained using Bethe’s equation with corrections [
26]. For poly-energetic beams, the particle types and energies are obtained differently. The number of tracks of each ion type is calculated by numerically integrating the spectra,
The contribution of each ion type to the dose is calculated as
The total dose is calculated by summing over each ion, i.e., . To simulate a given total dose, , the fluences are multiplied by the ratio . The number of tracks for each Z is obtained by sampling the Poisson distribution using . For each track, the energy is determined by using a rejection method. Essentially, a random energy is generated between and , which are the minimum and maximum energies over which the spectra are defined. A random number U is drawn between 0 and the maximum value of the spectra for Z, . If , the energy value is accepted. The process is repeated until an energy is accepted.
For ions simulated in the present study, ejected δ-electrons have an energy distribution spanning between a few eV and hundreds of keV, with paths in tissue that can extend beyond a few millimeters [
27]. Simulating such large volumes with RITRACKS would result in a prohibitive long calculation time. Thus, to model a realistic geometry of a cell located within a larger tissue structure, and therefore account for δ-electrons generated in neighboring volumes by tracks that may have missed the cell, we applied periodic boundary conditions (PBCs). PBCs are used to approximate large systems using a small, representative volume of space called the unit cell. As illustrated in
Figure 1, when a secondary particle leaves the irradiated volume, it appears on the opposite side with the same velocity vector. Despite the use of PBC, the irradiation volume still must be set sufficiently large compared to the target volume to avoid simulation artifacts such as energetic δ-electrons crossing the boundary multiple times. Such artifacts would have negligible impact on single-track microdosimetric spectra but would influence total dose and resulting chromosome aberration yields. We analyzed the effect of irradiation volume size (not shown) and found that taking a side length for the irradiation area equal to 15 µm was large enough to avoid such artifacts.
Appendix A.3. Chromosome Aberrations
Chromosome aberrations were computed with the RITCARD model [
10,
13,
14], which is briefly described next. RITCARD consists of different parts that are illustrated on
Figure 1: (step 2) energy scoring in nanovoxels; (step 3) a random walk (RW) algorithm that simulates the geometrical distribution of chromosomes during interphase; (step 4) a DNA damage algorithm that assesses the number of double-strand breaks (DSBs); (step 5) a break repair algorithm; and (step 6) a function to categorize and count chromosome aberrations.
First, RITCARD requires the spatial map of energy deposition in the nucleus, as simulated by RITRACKS and explained in
Appendix A.1. Once the tracks have been simulated for an irradiation dose
, nanometric dose was scored in 20 × 20 × 20 nm
3 voxels that mapped the cell nucleus. In this study, the cell nucleus was of spherical shape with a radius of 4 μm to match the size of the target considered for microdosimetry calculations.
The RW algorithm was used to model the 3D position of all 46 chromosomes within the nucleus during interphase, as in [
29,
30]. Each chromosome was roughly modeled by a random coil and simulated by a sequence of monomers of lattice period of 20 nm, corresponding to the size of the dosimetry voxels. Each monomer contained 2 kbp of DNA. The initial position of the chromosome was sampled within a spherical chromosome domain [
31] and the chromosome contained sub-structures representing chromosome loops of 60 monomers each.
The 3D voxel dose map and chromosome RW were then used to compute DSBs, by first locating intersections between chromatin fibers and voxels for which the dose was higher than 0 Gy. The number of DSBs
contained in a monomer was determined by sampling the Poisson distribution in Equation (A1) with,
where
is the dose in the voxel of spatial coordinates
in lattice unit, and
Gy
−1 is an adjustable parameter representative of the intensity of DSB formation. The number of breaks in a monomer was rarely greater than 1, except in high-dose voxels in the core of high-LET tracks. On average, RITCARD yielded ~35 DSB/Gy/cell with little dependence with ion LET [
15], as reported elsewhere [
16]. Each break in a chromatin fiber led to the formation of two chromatin free ends.
The next part consisted of modeling break repair during the first 24 h after irradiation. The repair kinetics model was recently significantly updated [
13,
14]. It assumes that the number of breaks follows a bi-exponential decay as a function of time after irradiation,
,
,
and
are parameters. Such observations were reported by many investigators and suggests that simple breaks are repaired rapidly (
= 1.7 h) while more complicated breaks take longer to repair (
= 23.7 h) [
17,
32,
33,
34,
35]. The time constants were set based on measured experimental constant times of fibroblasts [
17].
and
were not explicitly set, but breaks were categorized into simple and complex based on a voxel energy threshold of 500 eV. Using this threshold value, each free end associated with a given break was also categorized as either simple (voxel energy < 500 eV) or complex (voxel energy > 500 eV).
The repair algorithm proceeded by small time steps (typically 1 s) over a period of 24 h. At each time step, a repair attempt was made for all free ends. Each pair of simple free ends was assumed to repair properly (i.e., one free end recombined with the free end originating from the same DSB) or to remain unrepaired, with a probability of proper repair equal to
. Complex free ends had an additional outcome, i.e., improper repair. For one complex free end, the total probability of proper and improper repair was
, with the
factor accounting for the fact that each complex free end was counted twice in the complex repair algorithm. If the free end was repaired during a time step, then the Euclidian distance,
, between the selected free end and all other complex free ends was calculated. The probability of any two ends repairing was then equal to,
is an empirically calibrated parameter and μm2 is an adjustable parameter. Equation (A8) reflects the fact that breaks further away from each other have a lower probability to recombine together. The algorithm then used the calculated probabilities to sample one free end for the selected break to repair with, thus leading to either proper or improper repair.
At the end of the 24 h period, the last part of RITCARD analyzed all the fragment sequences that were formed and classified them. The classification includes intact chromosomes, properly repaired chromosomes, and several types of chromosomes aberrations (translocation, inversions, deletions, dicentrics, rings and simple or complex exchanges). The criteria were defined by Ponomarev and colleagues [
29,
30] and are based on the work of [
36]. Aberration types are not necessarily exclusive as, for example, a ring can also be a dicentric. In this work, we focused on simple and complex exchanges. Simple exchanges were defined as exchanges that involved two breaks in two chromosomes (dicentrics and translocations). This is illustrated on
Figure 1. Complex exchanges were defined as exchanges that involved more than two breaks, in two or more chromosomes.
As for the microdosimetry calculation, we assessed the effect of direct and indirect contributions on chromosome aberrations. As described in
Appendix A.2, tracks had an identification number to separate them into direct or indirect contributions. The estimation of the direct contribution to chromosome aberration yields was then performed by scoring energy deposition in voxels due to direct tracks only, which is without scoring energy deposition due to indirect tracks. The chromosome aberrations were then estimated, following the same steps as described above. Likewise, the indirect contribution was performed by scoring energy deposition in voxels due to indirect tracks only. Thus, for a given beam, the simulation was performed three times to obtain the total, direct and indirect contributions.
For a given ion beam, we calculated chromosome aberrations for 7 dose points ranging from 0.05 to 1 Gy. While for microdosimetry calculations, the number of histories depended on the beam energy. For chromosome aberration calculations, each dose point consisted of 10,000 histories. At the end of the simulation, for each dose point, we obtained an average number of exchanges and the statistical standard error.
Appendix A.4. Dose–Response Statistical Analysis
The dose response of simple or complex exchange frequency was then fitted by a linear quadratic (LQ) model,
where
is the number of exchanges (simple or complex) for the dose
and contribution
(total, direct, or indirect), for either mono-energetic or poly-energetic beams. For modeling purposes,
,
and
are assumed to be continuous random variables.
A framework to optimally calibrate Equation (A1) according to random data is available in [
37,
38]. In this article, however, we carry out a suboptimal approach. Each dose–response datapoint
was sampled assuming that exchanges have a normal distribution, with a standard deviation equal to the simple or complex exchange statistical standard error. Random
were drawn from this normal distribution and values of
and
that minimize the least squares error were computed. This process was performed 1000 times, thereby leading to the data cloud of
and
pairs shown in the bottom right of
Figure 1. These data cloud were then used to learn a bivariate correlated normal using the maximum likelihood approach. This distribution is given by
where the first argument is the expected value, the second argument is the covariance, and
is the coefficient of correlation. This distribution, along with the simple structure of Equation (A9), make
a normal distribution having the following closed form,
where the expected value of
is
and the variance
is given by
with
and
For any fixed dose point, the 95% PI is given by
where
.
Figure A1 presents an example of the results of such a procedure. Note that the parameter dependencies between
and
, which lead to a sizable value for
, play a key role.
Figure A1.
Linear quadratic fit for the dose response for mono-energetic O 325 MeV/n beam and simple exchanges. On the top left (a), dots are the results from RITCARD while the solid line represents the least squares fit, and the dashed line the 95% PI. The marginal PDF of the α and β are plotted on the top right (b) and bottom left (c), while the bottom right (d) shows samples of the joint density.
Figure A1.
Linear quadratic fit for the dose response for mono-energetic O 325 MeV/n beam and simple exchanges. On the top left (a), dots are the results from RITCARD while the solid line represents the least squares fit, and the dashed line the 95% PI. The marginal PDF of the α and β are plotted on the top right (b) and bottom left (c), while the bottom right (d) shows samples of the joint density.
Next, we want to assess whether there is a significant difference between dose–response curves for two cases: first for mono-energetic beam and for poly-energetic (Digimouse) beam to investigate the effect of beam transport; and second and for mono-energetic beam to investigate potential synergy due to the interaction of breaks induced by direct and indirect contributions. We thereafter refer to contributions and as those we compare to each other. To perform the comparison, we defined three figures of merit.
For a given dose point
, the probability of
for a contribution
to be contained in the 95 % PI,
of the contribution
is,
where
is the cumulative distribution function of the normal random variable as defined in Equation (A4). Note that
and the greater
, the more similar both contributions are. Considering a dose range of 0–1 Gy, the first two measures of agreement between the contributions
and
are
where the variables in Equation (A19) are analogous to those in Equation (A18). The closer the responses to both contributions, the closer
and
are to 1.
The third figure of merit is based on the Kolmogorov–Smirnov statistic. At a given dose point
, it is defined as the largest absolute difference between the two CDF of
and
. The figure of merit is defined as the integral of this quantity over the dose range,
This metric is illustrated in
Figure A2 below, by the vertical blue line indicating where the highest difference between the two CDFs is reached. Values of
close to 0 indicate that the two dose responses are similar. All three figures of merit are free to take values between 0 and 1.
Figure A2.
Illustration of the PDF, fy and CDF, Fy, obtained at the dose point Dav = 0.75 Gy, for the contributions i = tot and j = dir + ind, the mono-energetic beam O 325 MeV/n and simple exchanges. The vertical blue line represents the maximum of the difference between Fy,tot and Fy,dir+ind.
Figure A2.
Illustration of the PDF, fy and CDF, Fy, obtained at the dose point Dav = 0.75 Gy, for the contributions i = tot and j = dir + ind, the mono-energetic beam O 325 MeV/n and simple exchanges. The vertical blue line represents the maximum of the difference between Fy,tot and Fy,dir+ind.
Figure A3,
Figure A4,
Figure A5 and
Figure A6 show Equation (A8) as a function of the dose and the value of the corresponding figures of merit. For most figures, we observe that
and
are usually close to 1 across the whole dose range, in agreement with the Kolmogorov–Smirnov statistic is close to 0. There are however few cases, such as in
Figure A6, where the dose responses are clearly distinct, and consequently
and
are ~0. We also observe some cases where we start seeing differences between the two contributions for high doses (>0.5 Gy).
Figure A3.
Dose–response comparison between the total vs. direct + indirect, for simple exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are displayed for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A3.
Dose–response comparison between the total vs. direct + indirect, for simple exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are displayed for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A4.
Dose–response comparison between the total vs. direct + indirect, for complex exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are displayed for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A4.
Dose–response comparison between the total vs. direct + indirect, for complex exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are displayed for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A5.
Dose–response comparison between the total contribution obtained for mono-energetic (ME) vs. poly-energetic (PE) beams, for simple exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are display for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A5.
Dose–response comparison between the total contribution obtained for mono-energetic (ME) vs. poly-energetic (PE) beams, for simple exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are display for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A6.
Dose–response comparison between the total contribution obtained for mono-energetic (ME) vs. poly-energetic (PE) beams, for complex exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are display for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.
Figure A6.
Dose–response comparison between the total contribution obtained for mono-energetic (ME) vs. poly-energetic (PE) beams, for complex exchanges (upper figures). The figures of merit, , and , are plotted in black, red, and blue, respectively (lower figures). The values of the integrals over the dose (Equations (A18)–(A20)), , and are indicated on each sub-figure. Results are display for (a) H 1000 MeV, (b) He 250 MeV/n, (c) C 290 MeV/n, (d) O 325 MeV/n, (e) Si 300 MeV/n and (f) Fe 1000 MeV/n.