Impact of Microdosimetric Modeling on Computation of Relative Biological Effectiveness for Carbon Ion Radiotherapy

Simple Summary

Modeling energy deposition on a microscopic scale is a useful tool for predicting biological outcomes in carbon ion radiotherapy. This study investigates how different approaches to calculating microdosimetric energy deposition influence predictions of biological effectiveness, which impact delivered patient doses. We compared several established models, along with a novel implementation that enables consistent geometrical comparisons among calculation approaches. Our results show that the choice of microdosimetric model can notably affect the calculated biological effectiveness. These results highlight the need to explicitly consider both the computational method and the assumed target geometry to ensure consistent and biologically meaningful predictions in particle therapy.

Abstract

Microdosimetry plays a critical role in particle therapy by quantifying energy deposition within microscopic domains to assess biological effects. This study evaluates the influence of different microdosimetric functions (MFs) and domain geometries (DGs) on relative biological effectiveness (RBE) predictions in carbon ion radiotherapy. Specifically, we compare the analytical microdosimetric function (AMF), calculated for spherical domains and implemented in PHITS, with the Kiefer–Chatterjee (KC) track structure model, which is conventionally applied to cylindrical geometries. To enable a direct comparison, we also introduce a novel implementation of the KC model for spherical domains. Using both models, specific energy distributions were calculated across a range of domain sizes and geometries. These distributions were input into the modified microdosimetric kinetic model (mMKM) to calculate RBE for the HSG cell line and compared against published in vitro data. The results show that both microdosimetric function and domain geometry significantly affect microdosimetric spectra and the resulting RBE, with deviations exceeding 10% when fixed mMKM parameters are used. Parameter optimization within the mMKM enables alignment across models. Our findings emphasize that microdosimetric function and domain geometry selection must be explicitly accounted for in microdosimetry-based RBE modeling, and that model parameters must be tuned accordingly to ensure consistent and biologically accurate predictions.

1. Introduction

Carbon Ion Radiation Therapy (CIRT) relies on biophysical models to quantify radiation effectiveness, which can depend on both dosimetric and microdosimetric inputs specific to the patient’s treatment plan. Microdosimetry, which assesses the spatial distribution of energy deposition at microscopic levels, is useful for predicting the relative biological effectiveness (RBE) of heavy ion radiation like carbon and is an integral component of some RBE models [1,2,3]. It links microscopic track structure to macroscopic dose distribution and helps to optimize biological dose delivery by maximizing tumor exposure while minimizing damage to healthy tissue [4].

Microdosimetric calculations involve defining a target domain such as a cell nucleus or a subnuclear structure, where individual energy depositions in the domain are typically presented as frequency and dose-weighted distributions of the microdosimetric quantities specific energy or lineal energy [1,2]. These microdosimetric distributions can be experimentally measured, modeled using computationally expensive Monte Carlo simulations [5,6,7], or derived from empirical or semi-empirical approaches. They can then be used as inputs in radiobiological model calculations of RBE and other related quantities [8,9,10]. This study will consider two common approaches to calculating microdosimetric spectra, namely the analytical microdosimetric function (AMF) [11,12], and the Kiefer–Chatterjee (KC) track structure model [13,14]. In this work, the KC model is implemented within a microdosimetric formalism and referred to as a distinct microdosimetric function for convenience. These two approaches are traditionally used in combination with different radiobiological models: the AMF is used in conjunction with the Mayo Clinic Florida microdosimetric kinetic model [15], while the KC model forms the original basis of the modified microdosimetric kinetic model (mMKM) [10,16].

The KC model is a pioneering track structure model that was used for the original biological dose calculations using the mMKM at NIRS/QST in support of CIRT, a foundation upon which the majority of modern clinical data was gathered [16,17]. In the NIRS/QST implementation, the KC model was applied using a cylindrical domain geometry as an approximation of the spherical geometry of a cell nucleus, as described by Kase et al. [16,17]. In contrast, the AMF [11,12], which has been implemented in the Particle and Heavy Ion Transport code System (PHITS) [18], calculates microdosimetric spectra for spherical water domains. The AMF was developed by fitting the results of detailed track structure simulations [19], which represents its most significant distinction from and potential refinement over the KC. The Mayo Clinic Florida microdosimetric kinetic model relies on the AMF, and therefore spherical domains, while the mMKM relies on the KC implemented for cylindrical domains.

This study aims to evaluate the sensitivity of microdosimetric spectra to calculation method, including the AMF and the KC model implemented as a microdosimetric function, and to variations in domain geometry. A key component is the novel implementation of the KC track structure model for spherical domains, which enables direct comparisons with the AMF under identical geometrical conditions. Furthermore, this study examines the effects of different microdosimetric calculation approaches on RBE predictions for CIRT within the mMKM framework. Particularly, it explores the necessary adjustments to mMKM parameters to accommodate changes in calculation methods while maintaining consistent RBE predictions.

2. Materials and Methods

2.1. Study Overview

The description of RBE calculation for CIRT using mMKM is well described in the literature and is summarized in Figure 1 [4,10,16]. The input required for the microdosimetric function includes both ion and microdosimetric domain parameters. Once selected, these inputs are used to compute the microdosimetric spectra, the methodology of which varies both by microdosimetric function and domain geometry. These microdosimetric spectra for the ion and the cell-specific parameters are then used to derive a deterministic microdosimetric quantity such as saturation-corrected lineal or specific energy ($y_D^{*}$ or $z_D^{*}$, respectively). This quantity, along with the additional reference cell and radiation-specific parameters, is fed into the RBE model. In this example, the RBE model is mMKM, the output of which includes computational radiobiological quantities, such as RBE and RBE-weighted dose.

In this study, we calculated the mMKM-based RBE (RBE_mMKM) using both the older (2006) and newer (2023) versions of the AMF for spherical domains, along with the KC model implemented for both cylindrical and spherical domains. Detailed implementations of each model are provided in subsequent sections. For each calculation method, we computed the microdosimetric spectra and deterministic microdosimetric quantities across a range of carbon energy spaced on a log scale from 1 to 900 MeV/u, corresponding to an LET range of 700 to 8 keV/µm. The ion stopping powers in water were taken from ICRU Reports 73 and 90 [20,21].

For RBE comparisons, a cylindrical domain radius of 0.32 µm was employed, aligning with cell characterizations in prior research [17,22]. To ensure volumetric equivalence between spherical and cylindrical domains, important because specific energy is defined as energy deposited per unit volume for an approximately consistent density, we calculated the spherical radius using

$$r_{sph}=\sqrt[3]{{\displaystyle \frac{3{r_{cyl}}^3}{2}}}.$$

(1)

This was derived under the assumption that the cylinder height equaled twice its radius, consistent with previous work [16,17]. For the selected cylindrical radius of 0.32 µm, this yields an equivalent spherical radius of 0.37 µm.

Additionally, we compared the effect of varying domain size on microdosimetric spectra for each calculation approach. For this comparison, cylindrical domain radii of 0.050, 0.10, 0.50, 1.0, and 5.0 µm were analyzed, along with corresponding spherical radii of 0.057, 0.11, 0.57, 1.1, and 5.7 µm to maintain volume equivalence across domain geometries.

2.2. Microdosimetry

In microdosimetry, specific energy $\left(z\right)$ is a critical parameter that quantifies the energy deposited by a primary particle and all its secondaries in a domain per unit mass of the domain, typically expressed in Gy. Microdosimetric analysis involves frequency-weighted and dose-weighted probability distributions of the specific energy. The frequency-weighted spectra, $f\left(z\right),$ show the frequency of various specific energy events. Dose-weighted spectra, $d\left(z\right),$ focus on the contribution of these events to the total absorbed dose, emphasizing which are most impactful for the tissue. These distributions are crucial for understanding the biological effects of radiation, as they describe the variability of energy deposition within microscopic targets.

2.2.1. AMF

We computed the microdosimetric spectra using the original AMF developed in 2006 [11] for use in the Japanese Monte Carlo system PHITS [18] and its improved version that was published in 2023 [12]. The AMF represents a fit to detailed track structure data and uses efficient analytical algorithms to model microdosimetric quantities in water by calculating the spatial coordinates of each ionization event and evaluating the probability densities of the number of ionization events within spherical target sites ranging from 3 nm to 1 μm in diameter [12]. The 2023 AMF [12] incorporates amendments based on updated track structure simulations with PHITS [18]. A detailed comparison between the results of the older and newer versions of the AMF can be found elsewhere [23]. This AMF can directly compute the frequency- and dose-weighted microdosimetric spectra for an ion given its given atomic number, atomic mass, kinetic energy, and LET.

2.2.2. KC Track Structure Model

The KC model is an amorphous track structure model that approximates the energy deposited in a microscopic domain. It combines the Kiefer model to represent the low-dose penumbra region and the Chatterjee model to describe the high-dose core region characteristic of charged particle tracks [13,14]. The KC model has historically been implemented using a cylindrical domain geometry to mimic a spherical cellular geometry in important studies supporting CIRT by Kase et al. and Inaniwa et al. [16,17].

In the KC model, the dose deposited at a radial distance $R_i$ from the ion’s path is given by

$$D_i=\left\{\begin{array}{c} {\displaystyle \frac{ 0.000125·{Z^{*}}^2}{{{R_i}^2\beta }^2}} = {\displaystyle \frac{k_p}{{R_i}^2 }}, R_c<R_i\le R_p\\ \begin{array}{c} \\ \left({\displaystyle \frac{1}{\pi {R_c}^2}}\right)·\left({\displaystyle \frac{LET·0.1602}{\rho }}-{2\pi k}_p\ln \left({\displaystyle \frac{R_p}{R_c}}\right)\right), R_i\le R_c\end{array}\end{array}\right. ,$$

(2)

where $R_c$ and $R_p$ describe the core and penumbra radii, respectively. Here, $\rho$ represents the density of the domain (fixed at 1 g/cm³ in this work), $Z^{*}$ is the effective charge of the ion (calculated with the Barkas expression to account for the gradual electrical neutralization of ions at low velocity [24]), and $\beta$ is the ion velocity relative to the speed of light. LET is provided in keV/µm, and all radii are provided in micrometers. The core and penumbra radii are determined by

$$R_c=0.0116\beta$$

(3)

and

$$R_p=0.0616{\left({\displaystyle \frac{E}{A}}\right)}^{1.7},$$

(4)

where $E$ is the kinetic energy of the ion and $A$ the atomic mass.

To compute specific energy for the cylindrical domain, energy deposition was calculated across 500 concentric annuli centered on the particle track, capturing contributions from both the core and penumbra regions. The full calculation details for this implementation can be found in Appendix A.1. For the spherical domain, the domain was treated as a series of thin cylindrical slices with varying radii stacked along the z-axis, with the same radial dose integration method applied to each slice (Appendix A.2).

For both geometries, energy deposition was calculated as a function of the impact parameter, defined as the perpendicular distance from the particle track to the center of the domain (Appendix A.3). These spatially dependent energy depositions were then used to derive deterministic microdosimetric parameters, such as specific energy and lineal energy distributions (Appendix A.4).

2.3. RBE Calculation

The output of each microdosimetric function was used as input to the mMKM, which was proposed by Kase et al. as an adaptation to the original MKM to calculate RBE [10]. To calculate RBE_mMKM, the saturation-corrected dose-mean specific energy, $z_D^{*}$, was first calculated using microdosimetric spectra and cell specific parameters, including the domain radius (${\mathrm{r}}_{\mathrm{D}}$) and nuclear radius (${\mathrm{R}}_{\mathrm{n}}$), for each microdosimetric function (further described in Appendix A.4). This biologically corrected parameter was then combined with the alpha value in the case of LET approaching zero (${\mathsf{\alpha }}_0$) to calculate ${\mathsf{\alpha }}_{\mathrm{m}\mathrm{M}\mathrm{K}\mathrm{M}}$. Therefore, for a fixed set of reference parameters (${\mathsf{\alpha }}_{\mathrm{x}}$ and ${\mathsf{\beta }}_{\mathrm{x}}$), RBE_mMKM can be expressed as a function of microdosimetric function ($MF$), domain geometry ($DG$), and cell-specific parameters: RBE_mMKM($MF$, $DG$, ${\mathrm{r}}_{\mathrm{D}}$, ${\mathrm{R}}_{\mathrm{n}}$, ${\mathsf{\alpha }}_0$).

For our RBE calculations, we adopted model parameters established by Inaniwa et al. in 2010 [16], which provided the best description of in vitro RBE data for human salivary gland (HSG) tumor cells irradiated with carbon ions [25]: RBE_mMKM(KC, CYL, 0.32 µm, 3.9 µm, 0.172 Gy⁻¹). To ensure volumetric equivalence, we employed ${\mathrm{r}}_{\mathrm{D}}$ = 0.37 µm for spherical domains. The RBE_mMKM was calculated for a surviving fraction of 10% (RBE_10%), for a carbon absorbed dose of 2 Gy (RBE_2Gy), and at low-doses (RBE_α), using previously established methods [15,26] and reference values of ${\mathsf{\alpha }}_{\mathrm{x}}$= 0.313 Gy⁻¹ and ${\mathsf{\beta }}_{\mathrm{x}}$= 0.0615 Gy⁻² derived for HSG cells exposed to 200 kVp x-rays. RBE_mMKM predictions were compared with in vitro data measured by Furusawa et al. [25]. Using the same reference parameters fit by Inaniwa et al. for carbon ions [16], we additionally calculated RBE_mMKM using each of the microdosimetric functions for both helium and neon ions and further compared these calculations with in vitro data.

To additionally explore the impact of microdosimetric function on deterministic microdosimetric parameters and predictions of RBE_mMKM for various ions of the same LET, the parameter calculations were performed for protons, helium, and carbon ions at an LET of 9.9 keV/µm. In a further assessment for carbon ions, we then explored whether the RBE_mMKM calculated using different microdosimetric functions could approximate the KC cylinder model, selected as a reference due to its extensive use in clinical and computational studies [16,17,27], by optimizing the ${\mathrm{r}}_{\mathrm{D}}$, ${\mathrm{R}}_{\mathrm{n}}$, and ${\mathsf{\alpha }}_0$ parameters:

$$\underset{{{\mathrm{r}}_{\mathrm{D}},\mathrm{R}}_{\mathrm{n}},{\mathsf{\alpha }}_0}{\min }\Vert \mathrm{R}\mathrm{B}{\mathrm{E}}_{\mathrm{m}\mathrm{M}\mathrm{K}\mathrm{M}}\left(MF,DG,{\mathrm{r}}_{\mathrm{D}},{\mathrm{R}}_{\mathrm{n}},{\mathsf{\alpha }}_0\right)-\mathrm{R}\mathrm{B}{\mathrm{E}}_{\mathrm{m}\mathrm{M}\mathrm{K}\mathrm{M}}\left(KC,CYL,0.32 \mathsf{\mu }\mathrm{m},3.9 \mathsf{\mu }\mathrm{m},0.172 {\mathrm{G}\mathrm{y}}^{-1}\right){\Vert }^2$$

(5)

3. Results

3.1. Impact Parameter and Microdosimetric Spectra

As part of the validation of our KC model implementation, we calculated specific energy as a function of impact parameter for both cylindrical and spherical domains of equivalent volume. These results are presented in Figure A7. The KC model results for the cylindrical domain reproduce those reported by Inaniwa et al. [16], confirming the accuracy of our implementation. Additionally, this work provides the first direct comparison of KC model predictions for spherical versus cylindrical geometries. At small impact parameters, the spherical domain yields higher specific energy, attributable to the longer path length of the particle track through the center of the sphere. This comparison highlights the influence of domain geometry on microdosimetric outcomes within the KC framework.

Figure 2 displays the frequency- and dose-weighted distributions of specific energy calculated using each microdosimetric approach at four different LET points (9.86, 44.3, 195 and 688 keV/µm). The low limit of the horizontal axis was set at 0.01 Gy, which corresponds approximately to the energy from a single ionization (10.9 eV) within the domain [12]. Spectra calculated using the various microdosimetric functions, and different domain geometries within those functions, clearly converge at high LET, where the range of delta rays approaches and falls below that of the domain radii. The difference in the AMF and the KC models is emphasized in the frequency-weighted distributions, particularly at low LETs, where the impact of the lower-energy delta rays is much higher.

To qualitatively assess the accuracy of the microdosimetric functions, each of them was compared with spectra measured using a tissue-equivalent proportional counter (TEPC) by Tsuda et al. [28], as shown in Figure 3. The sensitive volume of the TEPC was spherical, with a radius of 0.36 µm. Subsequently, spectra using each microdosimetric function in Figure 3 were calculated for domain radii of 0.36 and 0.31 µm for spherical and cylindrical geometries, respectively, to achieve equivalent volumes.

3.2. Deterministic Microdosimetric Parameters

We next compared ${\overline{z}}_F$ and ${\overline{z}}_D$ calculated by each microdosimetric function as a function of LET. We observed a larger spread in ${\overline{z}}_F$ compared to ${\overline{z}}_D$ (Figure A8) due to the significant impact of the low-energy delta ray on the frequency-weighted spectra. The ${\overline{z}}_F$ calculated using the AMFs [11] were notably lower than others due to the substantial delta ray peak that is present in each spectrum in Figure 3. Despite the disagreement in ${\overline{z}}_F$, the ${\overline{z}}_D$ computed with the AMF aligned closely with that of the KC model.

The effect of varying domain size on microdosimetric spectra and deterministic microdosimetric parameters was investigated for domain radii ranging from 0.05 to 5.0 µm. The ${\overline{z}}_F$ values calculated using each microdosimetric function were notably lower for smaller domains than larger domains, which is due to the increased impact of delta ray penumbra seen in spectra calculated for smaller sites. Consequently, due to the weighting of specific energy events, the impact of site size is higher for ${\overline{z}}_F$ than ${\overline{z}}_D$.

The ${\overline{z}}_D$ values calculated using each microdosimetric function and domain geometry combination are shown in Figure 4 for six different domain volumes, with the corresponding radii for each cylinder and spherical domains shown in the legend. This figure additionally shows ${\overline{z}}_D$ for each microdosimetric function normalized to that of the KC cylinder for the analogous domain size. In this figure, the convergence of the AMF with the KC model can be seen above an LET threshold of about 100 keV/µm. Each AMF shows the same general trend in relation to the KC cylinder, though the 2023 model converges approximately to the predictions of the KC cylinder, while predictions using the 2006 model converge to ~80% those of the KC cylinder.

3.3. Biologic Parameters and RBE

In order to validate our calculations using the KC model, we first reproduced the data published in Inaniwa et al. that implemented the KC model for a cylindrical domain and served as the standard lookup table linking microdosimetry and kinetic energy for ions from hydrogen to carbon [16,17]. These results are shown in Figure A9, overlaid upon digitized results from Inaniwa et al. [16]. Small deviations from the Inaniwa et al. data can be seen for some ions at low energies, below 5 MeV/u, which can be attributed to the difference in LET values used as input to the KC model.

We then determined the sensitivity of this parameter to microdosimetric function and domain geometry by calculating $z_D^{*}$ using each microdosimetric approach for carbon ions, as shown as a function of LET in Figure 5. Across all four methods of microdosimetric function and domain geometry calculation, the average $z_D^{*}$ value is shown in

Table 1. Mean, spread (max-min), and standard deviation of the $z_D^{*}$ and RBE_10% calculated for carbon ions using each microdosimetric approach.

LET (keV/µm)	${\mathit{z}}_{\mathit{D}}^{\mathbf{*}}$			RBE10%
	Average (Gy)	Spread (Gy)	St. Dev. (%)	Average	Spread	St. Dev. (%)
688	7.14	0.62	4.08%	1.40	1.86%	0.05
508	9.66	0.63	3.49%	1.63	1.99%	0.06
330	14.2	1.30	4.11%	2.07	2.78%	0.13
195	20.9	4.48	9.37%	2.75	7.27%	0.46
92.6	20.6	5.20	10.5%	2.72	8.21%	0.53
44.3	11.4	2.76	10.6%	1.79	6.38%	0.26
19.5	5.07	1.16	10.6%	1.22	3.71%	0.10
9.86	2.50	0.96	18.1%	1.02	3.37%	0.07
8.16	1.93	0.61	13.6%	0.97	2.02%	0.05

for each LET point, along with the subsequent spread and standard deviation. Notably, the standard deviation increased steadily from 4.1% to 14% with increasing LET, indicating that sensitivity to the selection of microdosimetric method is of higher consequence at lower beam energies. Within the KC model alone, calculated for different domain geometries, the spread in $z_D^{*}$ reached 14% at mid-range LET (~150 keV/µm).

To describe the biological effect, the RBE_mMKM was calculated using the reference parameters previously determined by fitting to in vitro data using the KC model for carbon ions [16]. To compare the sensitivity of RBE to microdosimetric function, RBE_mMKM was calculated using these same parameters but varying the microdosimetric function used to calculate the mMKM input, $z_D^{\mathrm{*}}$. To enable a robust comparison, RBE_mMKM was calculated each for 10% surviving fraction, for a physical dose of 2 Gy, and as the ratio of ${\alpha }_{mMKM}$ to ${\alpha }_x$ to approximate low-dose RBE, as shown in Figure 6. The RBE_mMKM using the other microdosimetric functions as input parameters typically followed the same trends as a function of the LET (i.e., reaching a maximum between 100 and 150 keV/µm for carbon and neon ions). The magnitude of RBE_mMKM showed a strong dependence on the microdosimetric function, with the most pronounced variations observed for carbon and neon ions at LET values below 300 keV/µm, and for helium ions across all LET ranges.

In vitro RBE data reported by Furusawa et al. [25] is overlaid with the model calculations for RBE_10% and RBE_α. A clear agreement is observed between the Kase implementation of the KC model and the in vitro data, as the biological parameters used to calculate RBE_mMKM were those specifically fit by Inaniwa et al. using this microdosimetric function. Overall, the sensitivity of the RBE_mMKM increased until an LET of around 100 keV/µm, and then decreased it sharply. At 300 keV/µm and above, the RBE_mMKM calculated with each model generally aligned well. Between RBE calculation methods, RBE_α were notably more sensitive to microdosimetric function than either the RBE_10% or the RBE_2Gy. This lowered sensitivity in RBE_10% and the RBE_2Gy is due to the increasing influence of the β parameter at higher doses, which is constant in the mMKM model and thus impacts all conditions in a similar way.

RBE_mMKM calculated using the KC model implemented for cylindrical domain geometries has long served as a reference microdosimetric input to calculations of RBE_mMKM, and it is therefore of great interest to determine whether each model examined in this study could approximate the results of the KC cylinder through the use of different mMKM parameters. To verify this, we fit the ${\alpha }_0$, $R_n$, and $r_D$ parameters for each model to those that would reproduce the RBE_mMKM calculated using the KC cylinder model, as shown in Figure 7. While the models implemented for spherical geometry were originally calculated using an $r_D$ of 0.37 µm for volume equivalence with the KC cylinder, this value was typically reduced to the same $r_D$ as KC cylinder (0.32 µm), or even below (0.30 µm), in order to achieve reasonably similar RBE values. It is important to note that the parameters shown in Figure 7 do not necessarily represent the best fit to the KC cylinder data but instead demonstrate that reproducing the KC cylinder results using other models might be feasible.

In addition to exploring the effect of microdosimetric function on RBE_mMKM for helium and neon ions, we explored the impact of microdosimetric spectra for various ions of the same LET. Here, each function was used to calculate $z_D^{*}$ and RBE_10% for protons and helium and carbon ions, each at an LET of 9.9 keV/µm (

Table A2. Saturation-corrected specific energy and RBE_10% calculated using the mMKM for each ¹H, ⁴He, and ¹²C ion at an LET of 9.9 keV/µm.

Ion	Energy (MeV/u)	Parameter	AMF (2023)	AMF (2006)	KC Sphere	KC Cylinder
1H	3.7	$z_D^{*}$ (Gy) (calculated with Inaniwa 2010 [16] parameters)	4.68	4.21	3.79	4.26
4He	21.6		3.91	2.85	2.92	3.30
12C	500		3.11	2.19	2.18	2.44
1H	3.7	$z_D^{*}$ (Gy) (calculated with fitted parameters)	6.23	5.28	4.88	4.26
4He	21.6		5.19	3.5	3.77	3.30
12C	500		4.18	2.71	2.79	2.44
1H	3.7	RBE10% (calculated with Inaniwa 2010 [16] parameters)	1.19	1.15	1.12	1.16
4He	21.6		1.13	1.04	1.05	1.08
12C	500		1.06	1.00	0.99	1.01
1H	3.7	RBE10%(calculated with fitted parameters)	1.15	1.18	1.13	1.16
4He	21.6		1.06	0.995	1.04	1.08
12C	500		0.987	0.979	0.971	1.01

). Similarly to previous assessments of RBE, HSG cell lines were used as a reference for RBE_mMKM calculations. Predictions of each $z_D^{*}$ and RBE_10% decreased with atomic number for both the AMF and the KC models. The $z_D^{*}$ and RBE_10% parameters for each proton and helium and carbon ions were additionally recalculated using the fitted parameters determined previously, which were calculated to fit predictions of RBE_10% using each microdosimetric function as an input to RBE_mMKM to predictions using the KC cylinder (Table A2). The inconsistency in predictions of RBE_mMKM across models for protons and helium ions is evident in this table, resulting from the mMKM parameters being fit to KC cylinder for carbon ions only. This suggests that a robust fitting to multiple ions would be appropriate for a mixed radiation field.

4. Discussion

This study compared multiple methods for calculating microdosimetric spectra, focusing on their treatment of delta-ray contributions beyond the core ion track. Notably, the AMF predicts higher delta-ray contributions at low LET values compared to the KC model. At high LETs, the delta ray contributions in the AMF become practically negligible in the dose-weighted spectra, and the shape of the AMF distribution resembles that of the KC function. Additionally, the AMF produced spectra with higher maximum energy deposition than all other models due to its incorporation of stochastic energy deposition, so that the maximum energy deposition is not simply determined by the product of the average energy deposition (LET) and chord length.

Comparisons between the calculated microdosimetric spectra and the measurements obtained using a TEPC (Figure 4) found that the 2023 AMF demonstrated the closest agreement with the experimental data, outperforming the KC model. This improved alignment is attributed in part to the AMF’s more accurate representation of secondary electron contributions. These findings are consistent with a previous study, which benchmarked the 2023 AMF against track structure simulations performed using TOPAS-nBio, a Monte Carlo framework based on Geant4-DNA, and reported excellent agreement [5]. Together, the comparisons with both measured TEPC data and detailed Monte Carlo simulations suggest that the 2023 AMF is a more accurate technique to compute microdosimetric spectra. Moreover, by implementing the KC model for spherical domains in this work, we enabled a direct and geometrically consistent comparison between AMF and KC predictions for the first time.

Domain size also plays a crucial role in interpreting measured spectra, particularly when comparing detectors with fixed domain sizes, such as silicon-on-insulator [9,29,30,31] and diamond detectors [32,33], to those with variable domain sizes, like gas-filled TEPCs [34,35,36,37]. Our findings suggest that as target size increases, differences between microdosimetric parameters calculated using the 2023 AMF and the KC cylinder models become less pronounced, converging when the volume-equivalent targets are large enough to capture most of the secondary electron energy. This has important implications for translating RBE estimates across different detector types and for clinical applications where target size and geometry must be accurately reflected. Microdosimetric quantities can be assessed using lineal energy, described by the quotient of the energy deposited to the domain and the mean chord length of the sensitive volume, rather than specific energy, that relates energy deposited to the mass of the sensitive volume. This is a common approach in practical microdosimetry and can help mitigate some of the differences arising from domain size. An example of this is shown in Figure A10, which depicts the ${\overline{z}}_D$ and the dose-weighted mean lineal energy, ${\overline{y}}_D$, calculated using the KC model implemented for a cylindrical domain for six different domain radii.

Beyond the inherent differences among microdosimetric functions, the geometry of the domain employed within these functions also significantly influences the resulting microdosimetric spectra. To isolate this effect, we compared different domain geometries with equivalent volumes. Spherical domains tend to exhibit broader primary peaks due to variations in radiation track lengths, whereas cylindrical domains, with more uniform track lengths, display narrower spectral features. The impact of geometry on $z_D^{\mathrm{*}}$ exceeded 16% between spherical and cylindrical targets, while RBE predictions using the mMKM showed more than a 14% variation.

Noteworthy is the sensitivity of RBE_mMKM (MF, DG, $r_D, { R}_n,{ \alpha }_0$) to microdosimetric function and domain geometry. As seen in Figure 6, RBE_mMKM varied up to 10% across calculation methods, especially at a lower LET (<400 keV/µm). Interestingly, mMKM parameters, such as $r_D$, $R_n$, and ${\alpha }_0$, could be adjusted so that the RBE_mMKM calculated with alternative microdosimetric functions and domain geometries could reasonably reproduce $\mathrm{R}\mathrm{B}{\mathrm{E}}_{\mathrm{m}\mathrm{M}\mathrm{K}\mathrm{M}}\left(KC, CYL, 0.32 \mathsf{\mu }\mathrm{m}, 3.9 \mathsf{\mu }\mathrm{m}, 0.172 {\mathrm{G}\mathrm{y}}^{-1}\right)$. This is important because years of computational and clinical RBE data rely on the mMKM using the KC model implemented for cylindrical domains and allowing $r_D$, $R_n$, and ${\alpha }_0$ to be free fitting parameters [16,17,27,38,39,40]. While this approach is common for the mMKM, it is not applicable for every MKM-based model. Alternative models, such as the Mayo Clinic Florida microdosimetric kinetic model, determine $r_D$, $R_n$, and ${\alpha }_0$ a priori [15].

To specifically assess how different microdosimetric models and domain geometries influence predicted RBE, we maintained constant values for all mMKM input parameters (including nuclear radius, α₀, and the reference α and β) throughout this study. We employed a parameter set derived from X-ray data for an HSG cell line, which is commonly utilized in clinical treatment planning systems and has underpinned extensive prior research and clinical outcome analyses. Although this parameter set may not capture the variability across different cell types, leading to uncertainties in absolute RBE values, our analysis concentrates on the relative differences in RBE predictions attributable exclusively to variations in microdosimetric models and domain geometries. Previous studies have investigated the robustness of the mMKM to variations in input parameters [41,42], and Hartzell et al. further explored the influence of cell line variability [42]. These findings reinforce the importance of understanding how modeling assumptions affect RBE predictions. Our results demonstrate that substituting one microdosimetric model or domain geometry for another, even with fixed input parameters, can significantly alter the predicted biological dose. This highlights the need for careful validation when implementing or modifying RBE models in clinical contexts.

5. Conclusions

This study demonstrated the interplay between an RBE model, the approach used for the computation of the microdosimetric spectra, and the geometry of the sensitive volume. The latter two factors were found to affect the RBE in a comparable way to the variation due to RBE model selection itself, which has been a topic of considerable focus in CIRT [43,44,45,46,47]. Therefore, changes in the computation of the microdosimetric spectra necessitate compensatory tuning of mMKM parameters to maintain consistent RBE calculations.