#
Comparative Evaluation of Two Analytical Functions for the Microdosimetry of Ions from ^{1}H to ^{238}U

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{1}H to

^{238}U over a broad energy range (1–1000 MeV/n). The newer AMF was found to offer superior performance, particularly for very heavy ions, producing results that align more closely with published in vitro clonogenic survival experiments. These findings suggest that the updated AMF provides a more reliable tool for microdosimetric calculations and RBE modeling, essential for ion radiation therapy and space radiation protection.

## 1. Introduction

_{2}O

_{3}:C optically stimulated luminescent detectors [31], and BaFBr:Eu optically stimulated luminescent detectors [32]. Furthermore, microdosimetric calculations with the AMF were instrumental for the development and benchmark of biophysical models describing and predicting in vitro clonogenic cell survival [33,34,35,36,37,38,39] and in vivo skin reactions [40].

^{1}H to

^{238}U.

## 2. Methodology

#### 2.1. RBE

_{10%}). This quantity was defined as the ratio (D

_{ref}/D

_{ion}) of the absorbed doses needed to obtain a 10% cell surviving fraction when irradiating the cells with reference photons (D

_{ref}) and the ion beam under study (D

_{ion}).

_{ref}and β

_{ref}) and the radiation under investigation (α and β), the RBE

_{10%}was calculated with Equation (2) [37] in the case of S = 10%.

#### 2.2. Experimental RBE Data

_{ref}and β

_{ref}). The data filtering process follows that of previous studies [17,38,39]; thus, only a summary is given here. At first, we excluded experiments in which the cells were irradiated with ion energies lower than 1 MeV/n. In the case of hydrogen ions (

^{1}H and

^{2}H), we included the results of exposures along monoenergetic and spread out Bragg peaks (SOBP), as the LET-dependence of the RBE was shown not to markedly depend on the irradiation scenario [45,46]. For heavier ions, we selected only results for monoenergetic beams since the LET-dependence of the RBE appears to differ between monoenergetic and SOBP exposures [45]. The V79 cell line was chosen as it is the most used mammalian cell line in the PIDE [2,44]. We selected the results only for the ten ions with most entries, as listed in Table 1.

#### 2.3. Computer Simulations

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U with 400 logarithmically spaced kinetic energy values between 1 and 1000 MeV/n.

^{−3}and 10

^{7}keV/µm (50 bins per decade). The production of secondary particles was deactivated by setting cmin = 10

^{10}MeV in the PHITS input file. As recommended by the International Commission on Radiation Units and Measurements (ICRU), the mean excitation energy of water was set to 78 eV [47].

#### 2.4. RBE Modeling

_{ref}is the linear term of the LQM for the reference photons, β

_{ref}is the quadratic term of the LQM for the reference photons, ${\overline{y}}_{D,ref}^{}$ is the dose-mean lineal energy for the reference photons, ρ is the density (=1 g/cm

^{3}), r

_{d}is the mean radius of the subnuclear domains, and ${R}_{n}$ is the mean radius of the cell nucleus.

_{ref}and α

_{ref}/β

_{ref}used in the MCF MKM calculations match those of the in vitro experiments and are listed in Table 1. A representative, average value of ${\overline{y}}_{D,ref}^{}$ = 4 keV/µm was used in all calculations, as the cell irradiations were carried out with different photon radiation types, the details of which are generally underreported. Furthermore, previous results suggest that the choice of ${\overline{y}}_{D,ref}$ has a relatively minor effect on the calculated RBE, especially for ions heavier than protons [40,46]. The cell-specific values of r

_{d}and ${R}_{n}$ for the V79 cell line were previously determined from measurable cell characteristics and equal to 0.27 µm and 4.0 µm, respectively [38,39].

#### 2.5. Structure of the Study

^{1}H,

^{12}C,

^{56}Fe, and

^{238}U) and energies (1 MeV/n and 100 MeV/n) in the case of spherical water targets with radii of 0.3 µm and 0.5 µm. The spherical targets with a 0.3 µm radius were chosen due to their relevance for the RBE calculations, as they represent the mean calculated dimension of giant chromatin loops [17,38,39]. On the other hand, a spherical volume with a radius equal to 0.5 µm represents the most common target size for experimental measurements with microdosimetric gas detectors [7,8].

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U), the two AMFs, and the two microdosimetric targets (water spheres with radii equal to 0.3 µm and 0.5 µm).

_{10%}of the V79 cell line using the MCF MKM [38,39] in combination with the microdosimetric distributions calculated with the two AMFs [25,41] for the reference microdosimetric targets (water spheres with radii equal to 0.3 µm). Corresponding in vitro values from the PIDE [2,44] for

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U ions were used as validation.

_{10%}of the V79 cell line is affected by the target size used in the microdosimetric calculations (spheres with radii equal to 0.3 µm or 0.5 µm, i.e., reference targets vs. the most used target size).

## 3. Results and Discussion

#### 3.1. LET

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U ions) with published data from the Errata and Addenda for ICRU Report 73 (C, O, Ne, and Ar ions) [56], the ICRU Report 90 (H, He, and C ions) [47], and calculations with the track structure code KURBUC (H, He, C, O, Ne, and Fe ions) [19].

^{1}H to

^{40}Ar, the LET values computed with PHITS and SRIM are very similar and agree well with the data from ICRU Reports 73 and 90 [47,56]. By contrast, the results of the KURBUC calculations [19] are generally higher than the other datasets, especially for ions with energy >100 MeV/n. The PHITS and SRIM LET values for

^{56}Fe ions agree well with each other and reasonably well with the high energy KURBUC calculations [19]. No ICRU data was available for ions heavier than argon. For the three heaviest ions included in this article (Kr, Xe, and U), the LET values calculated with PHITS are lower than the SRIM results for ions with energies higher than 10 MeV/n, consistent with what was reported in a previous study [57].

#### 3.2. Lineal Energy Distributions

^{1}H and

^{12}C ions calculated with the two AMFs [25,41] for spherical water targets with radii of 0.3 µm and 0.5 µm. Two energies, 1 and 100 MeV/n, were simulated for each ion. In panels A, B, E, and F, the distributions are plotted in the standard semilogarithmic yd(y) vs. y representation where the area under the curve between two energy lineal points is proportional to the absorbed dose in the considered interval [6]. In contrast, the results in panels C, D, G, and H are plotted on a bilogarithmic scale to highlight the differences between the distributions at low lineal energy values (y < 1 keV/µm).

^{56}Fe and

^{238}U ions (Figure 3), the differences in the results of the two AMFs follow the same pattern as those for the carbon ions of Figure 2, namely, (a) the relative dose contribution from low lineal energy events (secondary electrons) is higher in the case of calculations using the older AMF and (b) the peak in the distribution due to ion events spans over higher lineal energy values for the calculations with the newer AMF.

**Figure 2.**Dose distribution of the lineal energy for

^{1}H and

^{12}C ions (energy: 1 MeV/n and 100 MeV/n) calculated with the newer and the older analytical microdosimetric functions in the case of spherical water targets with radii of 0.3 µm and 0.5 µm.

**Figure 3.**Dose distribution of the lineal energy for

^{56}Fe and

^{238}U ions (energy: 1 MeV/n and 100 MeV/n) calculated with the newer and the older analytical microdosimetric functions in the case of spherical water targets with radii of 0.3 µm and 0.5 µm.

#### 3.3. Dose-Mean Lineal Energy

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U ions is plotted in Figure 4 as a function of the ion energy. The results of both AMFs and both spherical target sizes (sphere radius = 0.3 µm and 0.5 µm) were compared with published results [19,60] obtained with the track structure codes GEANT4-DNA and KURBUC for spherical targets of radius 0.5 µm.

^{1}H and

^{4}He ions and both AMFs (Figure 4A,B), the ${\overline{y}}_{D}$ for the spherical targets with a 0.3 µm radius was found to be generally higher than the corresponding calculations with the larger spheres (radius = 0.5 µm). For a chosen AMF, the differences between the results of the two target sizes become smaller at low energies. For a chosen target size, the ${\overline{y}}_{D}$ values calculated using the newer AMF were generally higher than those for the older AMF.

^{4}He ions and spheres with a 0.5 µm radius, the published ${\overline{y}}_{D}$ assessed with the KURBUC [19] are best described by the corresponding results of the newer AMF.

^{12}C and heavier ions, the ${\overline{y}}_{D}$ values of a given AMF appear to be minorly affected by the choice of the target size (sphere radius = 0.3 µm and 0.5 µm). The results of the newer AMF were systematically higher than those of the older AMF and closer to the corresponding KURBUC data [19] and the theoretical LET-based calculations of ${\overline{y}}_{D}$. It should be noted that the ${\overline{y}}_{D}$ values calculated with the newer AMF were lower than the KURBUC results in a similar manner to how our LET calculations, which are in good agreement with the ICRU LET data [47,56], were lower than the LET values estimated with KURBUC (see Figure 1).

**Figure 4.**Dose-mean lineal energy for

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne,

^{40}Ar,

^{56}Fe,

^{84}Kr,

^{132}Xe, and

^{238}U ions calculated with the newer and the older analytical microdosimetric functions in the case of spherical water targets with radii of 0.3 µm and 0.5 µm. The results are compared with published results from the GEANT4-DNA (

^{1}H ions [60]) and KURBUC (

^{1}H,

^{4}He,

^{12}C,

^{16}O,

^{20}Ne, and

^{56}Fe ions [19]) track structure codes, and corresponding theoretically derived LET-based estimates.

#### 3.4. RBE

_{10%}of the V79 cell line calculated with the MCF MKM in conjunction with microdosimetric distributions simulated in the reference microdosimetric targets (spheres with a radius of 0.3 µm) using the older and the newer AMFs. The results are shown in Figure 5 as a function of the LET for the ten ions included in this study. Corresponding in vitro data from the PIDE [2,44] are also plotted in Figure 5.

^{1}H and

^{4}He, the RBE

_{10%}values calculated using the newer AMF were systematically higher than those obtained with the older AMF (Figure 5A,B). For ions from

^{12}C to

^{132}Xe (Figure 5C–I), the newer AMF initially yielded lower RBE

_{10%}values compared to the older AMF. Taking the RBE

_{10%}computed with the newer AMF as a reference, an increase in the ion LET led to a relative increase in the results of the older AMF. At LET > 1000 keV/µm, the RBE

_{10%}values obtained with the newer AMF were systematically lower than those of the older AMF.

_{10%}values and the corresponding in vitro data suggests that the newer AMF appears to better reproduce the experimental results, especially at high LET. In order to better visualize the agreement between the modelled and the corresponding n experimental data for a chosen ion, Figure 6 shows the average relative deviation ($\overline{R}=\frac{1}{n}\sum _{i=1}^{n}\frac{\left|{RBE}_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{o}}-{RBE}_{\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{o}}\right|}{{RBE}_{\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{o}}}$) for the ten different ions included in this study. The average relative deviation between in silico and in vitro RBE data was lower for the newer AMF across all tested ions.

^{84}Kr,

^{132}Xe, and

^{238}U), the calculations for a 0.5 µm target radius were closer to the experimental results. However, most of those results were extracted from a relatively old publication [63] whose LET calculations were reported to differ from the PHITS-based LET results [57], thus introducing additional uncertainties which are difficult to quantify.

**Figure 6.**The average relative deviation between the RBE

_{10%}calculated with the MCF MKM and the corresponding in vitro results for the ten ions included in this study. The microdosimetric distributions needed for the RBE calculations were simulated using the older and the newer analytical microdosimetric functions implemented in PHITS.

## 4. Conclusions

^{1}H to

^{238}U) for water spherical targets with radii of 0.3 µm (the reference scale for RBE calculations) and 0.5 µm (the most common target size used in the literature). The two AMFs were compared with regard to the shape of the simulated probability distributions of the lineal energy, their mean values, and the RBE

_{10%}computed in combination with the MCF MKM for the most used mammalian cell line (Chinese hamster lung fibroblasts, V79 cell line). Published data from in vitro RBE experiments and track structure simulations were used to benchmark our results.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Overview of the in vitro data for the V79 cell line: ions, number of survival curves for each ion, mean value of the α for the reference photon exposures ($\overline{{\alpha}_{ref}}$), and the mean value of the ratio of α and β for the reference photon exposures $\overline{\left(\frac{{\alpha}_{ref}}{{\beta}_{ref}}\right)}$.

Ion | Number of Ion- Irradiated Survival Curves | $\overline{{\mathit{\alpha}}_{\mathit{r}\mathit{e}\mathit{f}}}$ [Gy^{−1}] | $\overline{\left(\frac{{\mathit{\alpha}}_{\mathit{r}\mathit{e}\mathit{f}}}{{\mathit{\beta}}_{\mathit{r}\mathit{e}\mathit{f}}}\right)}$ [Gy] |
---|---|---|---|

^{1–2}H | 42 | 0.113 | 4.58 |

^{3–4}He | 31 | 0.176 | 9.70 |

^{12}C | 54 | 0.179 | 10.0 |

^{16}O | 5 | 0.250 | 17.2 |

^{20}Ne | 20 | 0.211 | 12.4 |

^{40}Ar | 16 | 0.237 | 17.0 |

^{56}Fe | 13 | 0.199 | 11.4 |

^{84}Kr | 5 | 0.240 | 17.2 |

^{132}Xe | 14 | 0.244 | 17.2 |

^{238}U | 19 | 0.233 | 16.5 |

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**MDPI and ACS Style**

Parisi, A.; Furutani, K.M.; Sato, T.; Beltran, C.J.
Comparative Evaluation of Two Analytical Functions for the Microdosimetry of Ions from ^{1}H to ^{238}U. *Quantum Beam Sci.* **2024**, *8*, 18.
https://doi.org/10.3390/qubs8030018

**AMA Style**

Parisi A, Furutani KM, Sato T, Beltran CJ.
Comparative Evaluation of Two Analytical Functions for the Microdosimetry of Ions from ^{1}H to ^{238}U. *Quantum Beam Science*. 2024; 8(3):18.
https://doi.org/10.3390/qubs8030018

**Chicago/Turabian Style**

Parisi, Alessio, Keith M. Furutani, Tatsuhiko Sato, and Chris J. Beltran.
2024. "Comparative Evaluation of Two Analytical Functions for the Microdosimetry of Ions from ^{1}H to ^{238}U" *Quantum Beam Science* 8, no. 3: 18.
https://doi.org/10.3390/qubs8030018