Quantifying Sensitivity of Carbon RBE Models to Reference Parameter Variations

Abstract

Models used to calculate the relative biological effectiveness (RBE) of carbon-ion radiotherapy include the microdosimetric kinetic model (MKM), stochastic MKM (SMKM), repair-misrepair-fixation (RMF) model, and local effect model I (LEM). We compared the sensitivities of these models to variations in input biological and reference parameters. We used Monte Carlo simulations of clinically realistic carbon-ion beams incident on a phantom and scored input parameters for RBE models (kinetic energy, microdosimetric spectra, double-strand break yield, and physical dose). We combined data with cell- and model-specific parameters to calculate the linear (α) and quadratic (β) components of the carbon-ion beam, which were used along with the reference α and β values and dose to calculate RBE. Model sensitivity to parameters was quantified by statistically introducing uncertainty into independent parameters and sampling the resultant RBE. To assess histological differences contributing to variations in the RBE, we also used various reference cell lines. We recalculated the RBE using different reported datasets within individual cell lines to compare inter- and intra-cell line variability. The variability introduced by inherent measurement and estimation uncertainty was typically 26% for the microdosimetric models, 25% for the RMF model, and 30% for the LEM at the $1-\text{σ}$ level. The variability across cell lines, which averaged 27% for the microdosimetric models and 2.5% for the RMF model, was similar to the intra-cell line variability in the RBE as calculated with unique datasets for an individual cell line. While the focus is largely on comparing models, the results of this study indicate that the variation in RBE within each model, based solely on reference parameters, is substantial. Our findings indicate that the selection of input parameters is of comparable importance to the choice of cell line and even the RBE model. This study provides insight into model robustness and emphasizes the need for continued computational and in-vitro RBE research.

INTRODUCTION

Carbon-ion radiotherapy has a range of distinct capabilities, including treating tumors that are traditionally unresectable or resistant to conventional radiotherapy. Although access to carbon-ion radiotherapy is currently limited to a small number of institutions worldwide, the demand for this treatment is predicted to increase by as much as 25% in the upcoming years. In fact, the number of patients eligible for carbon-ion radiotherapy in the United States alone is estimated to reach 115,000 by 2025, and the first center in North America is currently being built at Mayo Clinic Florida (1, 2). The growing interest in carbon-ion radiotherapy stems from its beneficial dose distributions, lower oxygen enhancement ratios, and higher biological effectiveness.

The relative biological effectiveness (RBE) is a metric used to quantify the efficiency of a radiation type in causing the same biological endpoint, or surviving fraction, relative to a reference radiation type, usually photon radiation. In carbon-ion radiotherapy, however, the RBE is both elevated compared to conventional radiotherapy and variable. This variability is a characteristic of many parameters, including the reference cell line, radiation beam quality, dose fractionation, and biological endpoint. Researchers have proposed several biophysical models to describe the mechanisms of the variable RBE in carbon-ion radiotherapy, and many of these models have been implemented in clinical and research-based treatment planning systems.

The two models used to calculate the RBE in clinical treatment planning procedures are the modified microdosimetric kinetic model (MKM) and the local effect model I (LEM). The principal difference between these two models lies in how each derives the cell survival curve. The MKM employs a microdosimetric energy deposition approach to estimate cell survival, whereas the LEM utilizes an approach based on a theoretical track structure model. Other models used in research-based treatment planning systems include the stochastic MKM (SMKM), which is an adaptation of the MKM that accounts for the stochastic nature of specific energy in the nucleus, in addition to the submicron domains, and the repair-misrepair-fixation (RMF) model, which correlates DNA damage and repair mechanisms with microdosimetry to determine cell death.

The RBE estimated with each model varies significantly, both qualitatively and quantitatively, currently presenting one of the largest uncertainties in carbon-ion radiotherapy (11–15). Thus, the management and implementation of RBE models remain substantial challenges in clinical practice. Although most studies have focused on determining which model most accurately predicts the relative biological effectiveness (RBE), few have investigated the uncertainties contributing to the variability in these models’ performance. One such study examined uncertainties in the MKM, highlighting how variations in input parameters can significantly impact the model’s RBE predictions (16). Building on this approach, we aim to quantify the robustness of multiple models, including MKM, SMKM, LEM-1, and RMF, by assessing the variability introduced by input parameters and reference conditions. This expanded analysis not only provides a consistent framework for evaluating the robustness of each model but also enables a direct comparison of uncertainty within and across different models. In this study, we undertook this effort by evaluating the impact of several other factors on estimating the RBE. In addition to quantifying the effect of individual input parameters on RBE and biological dose, we also quantified the effects of using different cell lines and different input parameter datasets within a single cell line. The various approaches employed in this assessment enabled a comprehensive description of the robustness of different models, particularly in terms of clinical implementation, despite the challenges in quantifying the interplay between input parameters.

METHODS

To thoroughly quantify uncertainty within the models and to compare this uncertainty to the magnitude of variability across the models, we quantified the inherent uncertainty in model-specific input parameters, variability from different reference cell lines, and variability within cell lines based on various reported datasets.

Monte Carlo Simulation

To understand the sensitivity of RBE models to various input parameters in a clinically realistic framework, we calculated input parameters for various RBE models from phase space data simulated at the National Center for Oncological Hadrontherapy (CNAO) in Pavia, Italy. The phase space files were calculated downstream of the passive beamline elements and 60 cm upstream of the isocenter. A 14 × 14 × 30 cm³ water tank was simulated with the proximal z-axis face centered at the isocenter. The phase space of the monoenergetic beams consisted of 6 × 6 cm² mono-energy layers with 2-mm spot spacing and simulated three different beams with penetration ranges of 50, 150, and 250 mm in water. For simulating a spread-out Bragg peak (SOBP) of carbon ions, the scoring plane of the phase space consisted of an 8 × 13 cm field with a 2-mm spot spacing, creating an SOBP with a range of 180 mm and a width of 60 mm. Each beamlet of the SOBP was normalized to produce a uniform physical dose throughout the SOBP, providing a comparative basis across models. As the physical dose is used in the linear-quadratic RBE formula, an SOBP optimized to produce a flat biological dose for each model would not allow for reasonable RBE comparisons. The toolkit for particle simulation (TOPAS), which wraps and extends the Geant4 Simulation Toolkit, was used to read each phase space file and score various physical and biological parameters throughout the water phantom (17). The following validated physics lists were used in each simulation: “g4em-standard_opt4” “g4h-phy_QGSP_BIC_HP” “g4decay” “g4ion-binarycascade” “g4h-elastic_HP” “g4stopping”, with the range cutoff for all particles set to 0.05 mm.

To score lineal energy and calculate parameters for the microdosimetric RBE models, we employed the microdosimetry extension created by Zhu et al. (18) in TOPAS. A tissue-equivalent proportional counter (TEPC; model LET-1/2, Far West Technology, Goleta, CA) was simulated at 5-mm intervals throughout the z-axis of the beam in the water phantom. The active volume (6.35-mm radius) of the LET-1/2 TEPC was simulated with propane-based tissue-equivalent gas, the density of which was fixed at 0.0787 mg/cm³. This density corresponded to a tissue-equivalent diameter of 1 μm within the detector and a subsequent mean chord length of 0.67 μm, which was selected to represent the scale of the cell nucleus. The simulated 1 μm diameter, which has been used in several studies with Monte Carlo simulations of carbon-ion microdosimetry, enabled robust comparisons among the RBE models (3, 19–22). The results of this simulation were used in all calculations of RBE using the microdosimetric models, as the lineal energy distributions remain largely consistent across domain sizes within the range used in microdosimetric models (23–25).

To score the other input parameters, including the fragment-specific dose, fragment-specific kinetic energy, and double-strand break (DSB) yield, we aligned a scoring cylinder with a radius greater than the width of the phantom along the z-axis to capture all laterally scattered particles. The scoring cylinder was divided into 0.5-mm intervals along the z-axis. The dose was scored by particle atomic number to determine the extent to which primary carbon ions and secondary particles, including nuclear fragments, recoil nuclei, electrons, and photons, contributed to the dose. The kinetic energy was scored in bins with a linear scale ranging from 1 to 400 MeV/u, with a step size of 1 MeV/u per ion. These simulations were validated by comparing and finding good agreement with various published studies that present results of dose contributions and energy spectra of secondary fragments produced in carbon beams simulated with various Monte Carlo platforms (7, 26–28). The DSB yield was calculated using custom scripts adapted from the microdosimetry extension, which scores the DSB yield for RMF applications (18). The DSB yield was validated by comparing it to previous DSB yield data calculated with Geant4.

RBE Calculation

In each model, the RBE was calculated using the linear quadratic formula

$$RBE=\frac{-{\text{α}}_x+\sqrt{{{\text{α}}_x}^2+4{\text{β}}_x\left({\text{α}}_p{D}_p+{\text{β}}_p{D_p}^2\right)}}{2{\text{β}}_x{D}_p},$$ (1)

in which the cell- and radiation-specific reference parameters, ${\text{α}}_x$ and ${\text{β}}_x$, represented the linear and quadratic components of reference cell survival curves for photon radiation. The additional parameters needed to calculate the RBE include the physical dose, along with the ${\text{α}}_p$ and ${\text{β}}_p$ of the mixed particle beam. The primary differences in calculating the RBE using the different models arise from the calculation of ${\text{α}}_p$ and ${\text{β}}_p$, the methods for which are described below.

MKM

We used the MKM formalism described by Kase et al. to both calculate the RBE and identify the measured and estimated cellular parameters that may have uncertainty. In this case, ${\text{β}}_p$ is a fixed value ($={\text{β}}_x$; “$\text{β}$” is used hereafter), whereas ${\text{α}}_p$ is calculated using

$${\text{α}}_p={\text{α}}_0+\text{β}{z_{1D}}^{\ast },$$ (2)

which depends on ${\text{α}}_o$, ${\text{β}}_x$, and the saturation-corrected specific energy $\left({z_{1D}}^{\ast }\right)$. As described by Kase et al., the ${\text{α}}_o$ parameter is used to describe the ${\text{α}}_x$ term in the limit of linear energy transfer (LET) approaching zero. The latter term, ${z_{1D}}^{\ast }$, can be further reduced and calculated as a function of domain density $\left(\text{ρ}\right)$, domain radius $\left(r_d\right)$, and the saturation-corrected dose-mean lineal energy $\left({y_D}^{\ast }\right)$, using

$${z_{1D}}^{\ast }=\frac{{y_D}^{{\ast }^{\ast }}}{\text{ρπ}{r_d}^2}.$$ (3)

Whereas the former parameters are estimated based on physical cellular characteristics, ${y_D}^{\ast }$ is calculated using frequency-weighted lineal energy spectra as well as the saturation correction parameter $\left(y_o\right)$:

$${y_D}^{\ast }=\frac{{y_0}^2\int \left(1-e^{-\frac{y^2}{{y_0}^2}}\right)f\left(y\right)dy}{\int yf\left(y\right)dy}.$$ (4)

The $y_o$ parameter is calculated through exclusively cell-specific parameters, including the radius of the nucleus $\left(R_n\right)$, in addition to the aforementioned $r_d$, $\text{ρ}$, and $\text{β}$ values:

$$y_0=\frac{\text{ρπ}{r}_d{R_n}^2}{\sqrt{\text{β}\left({r_d}^2+{R_n}^2\right)}}.$$ (5)

Consequently, the model-specific parameters for the MKM that were manipulated at various statistical levels included $R_n$, $r_d$, and ${\text{α}}_o$. The reference cells that were used for this analysis were human salivary gland (HSG) cells exposed to 200-kVp X rays, as described by Kase et al. (3). This is a commonly reported cell line and is utilized in at least one commercial treatment planning system for the MKM.

SMKM

The SMKM, a modification of the original MKM, was developed to describe the RBE at high LET and high dose more accurately (5, 6). The calculation methods for the SMKM are similar to those for the MKM; the primary difference is that for the calculation of ${\text{β}}_p$, the SMKM uses both the dose-mean specific energy $\left({\overline{z}}_D\right)$ and the dose-mean specific energy corrected for saturation $\left({z_{1D}}^{\ast }\right)$

$${\text{β}}_p={\text{β}}_o\ast \frac{z_{1D}^{\ast }}{{\overline{z}}_D}.$$ (6)

This can be simplified to

$${\text{β}}_p={\text{β}}_o\ast \frac{{y_D}^{\ast }}{{\overline{y}}_D},$$ (7)

where ${y_D}^{\ast }$ and ${\overline{y}}_D$, the dose-mean lineal energy, are the primary inputs.

The same model-specific parameters were statistically manipulated for both the MKM and the SMKM. In addition, the same HSG reference cell line was used to enable consistent comparisons of RBE between the models.

RMF

The RMF model correlates DNA damage and repair mechanisms with microdosimetry to approximate cell death (7–10). For this model, the methods described by Guan et al. are used to calculate the ${\text{α}}_p$ parameter according to

$${\text{α}}_p=\text{θ}{\Sigma }_p+\text{κ}{\overline{z}}_F{{\Sigma }_p}^2$$ (8)

where ${\text{α}}_p$ depends on a combination of cell-specific parameters along with a microdosimetric quantity, ${\overline{z}}_F$, and the DSB yield of carbon ions $\left({\Sigma }_p\right)$ as calculated with Monte Carlo methods (7).

The cell-specific parameters $\text{θ}$ and $\text{κ}$ respectively describe the fraction of DSBs that submit to lethal first-order misrepair and damage fixation and the fraction of initial DSBs that undergo damage interaction in a pairwise fashion (7). These parameters are calculated using

$$\text{θ}=\frac{{\text{α}}_x}{{\Sigma }_x}\left(1-\frac{2{\overline{z}}_{F,x}}{{\left(\text{α}/\text{β}\right)}_x}\right)$$ (9)

And

$$\text{κ}=\frac{2{\text{β}}_x}{{\Sigma }_x^2},$$ (10)

which depend only on other cell-specific parameters, including the DSB yield of photons $\left({\Sigma }_x\right)$ as well as the reference specific energy $\left({\overline{z}}_{F,x}\right)$, ${\text{α}}_x$, and ${\text{β}}_x$.

In the RMF model, the remaining variable needed to calculate ${\text{α}}_p$ is ${\overline{z}}_F$

$${\overline{z}}_F=\frac{0.204\ast {\overline{y}}_F}{d^2},$$ (11)

which relies on the diameter of the nucleus and the frequency-weighted mean lineal energy, ${\overline{y}}_F$, a quantity calculated using Monte Carlo methods. The beta parameter is calculated using ${\Sigma }_p$ and $\text{κ}$:

$${\text{β}}_p=\left(\frac{\text{κ}}{2}\right){{\Sigma }_p}^2$$ (12)

From these calculations, we determined that the model-specific reference parameters to be varied would be the reference specific energy and the reference DSB yield. The same parameters that were calculated for the carbon-ion beam (${\overline{z}}_F$ and ${\Sigma }_p$) were also varied to determine the effect of these parameters on the RBE.

To allow comparisons between the RMF model and the microdosimetric models, we used the HSG cell line to calculate the RBE_RMF, as reported by Furusawa et al. (29). The HSG cell line has been used in multiple studies involving the RMF model. We also used the chordoma cell line to enable comparisons between the RMF model and the LEM (30, 31).

LEM

In the LEM, the RBE was calculated using pre-tabulated tables of ${\text{α}}_i$ and ${\text{β}}_i$ as functions of kinetic energy for each fragment (denoted with a subscript $i$) (30). Each fragment-specific value was weighted according to the particle’s dose contribution, using

$${\overline{\text{α}}}_p=\frac{{\sum }_{i=1}^n{\text{α}}_i\ast D_i}{{\sum }_{i=1}^n{D}_i}$$ (13)

And

$${\overline{\text{β}}}_p={\left(\frac{{\sum }_{i=1}^n\sqrt{{\text{β}}_i}\ast D_i}{{\sum }_{i=1}^n{D}_i}\right)}^2.$$ (14)

For this study, consistent with clinically used LEM tables, the chordoma cell line was used (30). As a result, there were no reference parameters (other than ${\text{α}}_x$ and ${\text{β}}_x$) that could be statistically varied. Instead, the mean kinetic energy of each fragment (as a surrogate for standard input parameters) was shifted by the variance and used to re-interpolate the fragment-specific ${\text{α}}_i$ and ${\text{β}}_i$. This assessment intends to capture a range of uncertainty associated with Monte Carlo calculations of kinetic energy, as uncertainty can vary even within ion-specific cross section data.

Sensitivity Analysis

The sensitivity analysis was conducted to evaluate the robustness of each model to uncertainties in measured and estimated input parameters. This approach quantified the inherent variability within each model, assessed how parameter uncertainties influenced the calculated RBE and biological dose, and identified the parameters and regions most sensitive to these variations. Specifically, random Gaussian noise was introduced to key input parameters, with the noise scaled by a defined percentage variation $\left(\text{σ}\right)$, simulating a range of realistic uncertainties. For a given parameter ${\text{θ}}_{original}$, the perturbed value ${\text{θ}}_{varied}$ was calculated as:

$${\text{θ}}_{varied}={\text{θ}}_{original}+\in ,\text{where}\in ~\mathcal{N}\left(0,{\left(\text{σ}\cdot {\text{θ}}_{original}\right)}^2\right).$$ (15)

Here, $\text{σ}$ represents the relative standard deviation (e.g., $\text{σ}=0.05$ for 5%, $\text{σ}=0.10$ for 10%, etc.), and $ϵ$ is a random Gaussian-distributed noise term with a mean of 0 and a standard deviation equal to $\text{σ}\cdot {\text{θ}}_{original}$. Gaussian noise was chosen because it is widely used in sensitivity and uncertainty analyses due to its simplicity and its alignment with common sources of experimental variability (22, 32).

Each input parameter was independently varied over 1,000 iterations, and the recalculated RBE and biological dose were recorded for each iteration. The variability introduced by parameter uncertainty was quantified using the standard deviation of the resulting RBE values. Convergence testing confirmed that a sample size of 1,000 was sufficient, as the standard deviation stabilized and varied by less than 2% beyond this number of samples. This approach assumed independence among the input parameters, simplifying the analysis while effectively capturing the effects of parameter-specific uncertainties.

We considered four uncertainty levels $\left(\text{σ}=5\%,10\%,20\%,40\%\right)$ to reflect a range of realistic scenarios. For example, 40% $1-\text{σ}$ uncertainty corresponds to variability observed in reference α values from the particle irradiation data ensemble (PIDE) (33, 34), representing general experimental variability. Conversely, 10–20% $1-\text{σ}$ uncertainty reflects more precise conditions, such as controlled experiments with specific cell lines (e.g., HSG cells). Our approach, therefore, provides a comparative basis across all scenarios, enabling interpretation in different experimental contexts. By analyzing the standard deviation of RBE values at various depths, we identified parameters and regions most sensitive to uncertainty. This insight can inform future experimental design by highlighting areas where reducing uncertainty could have the most substantial impact on model reliability. The inclusion of multiple uncertainty levels also ensures the flexibility to translate the results to other datasets. For instance, a study with a 10% variation in $\text{α}$ for HSG cells can directly estimate the corresponding impact on RBE and biological dose. This comparative approach avoids the limitations of applying a single uncertainty level (e.g., 40% $1-\text{σ}$) across all scenarios.

For the microdosimetric models, the model-specific parameters that were varied included $\text{α}$ in the case of LET approaching zero, the radius of the domain, and the radius of the cell line. The only carbon beam parameter varied in these models was the saturation correction parameter. For the RMF model, the model-specific parameters included the DSB yield of the reference cell line and the specific energy deposited by a given reference radiation (typically X rays, but dependent on reported values). The carbon-specific parameters mirrored the DSB yield and the specific energy of the carbon beam only. For the LEM, ${\text{α}}_p$ and ${\text{β}}_p$ were calculated using only the kinetic energy of carbon-ion beam fragments and thus were the only model-specific parameters that were varied. Here, the kinetic energy of each particle was shifted by the same random number multiplied by the statistical level during a single iteration. In addition, for all four models, the extra parameters needed to calculate the RBE with the linear quadratic formula (${\text{α}}_x$ and ${\text{β}}_x$ for reference photon radiation) were independently varied using the same methodology.

To conduct a deeper dive into a specific application and estimation of uncertainty, we examined the most probable uncertainty in each parameter. The uncertainty levels were based on the realistic uncertainty in input parameters that was attributed to estimation and/or measurement error. To determine this, we conducted a literature search to find published datasets of input parameters for each model (Table 1). For the microdosimetric models, we found nine datasets for four different cell lines (HSG, V79, CHOK-1, and T1) (3, 4, 29, 35–40). From these datasets, the standard deviations for each input parameter (${\text{α}}_0$, $r_d$ and $R_n$) were calculated across all nine data points. At the $1\text{σ}$ level, the standard deviations were 46%, 20%, and 5.8%, respectively, which were on a similar scale to those calculated from three different datasets for a single cell line: 31%, 15%, and 3.8%, respectively. For the RMF model input parameters, we used the same process; first, we identified eight different datasets for seven unique cell lines (HSG, V79, T1, H460, chordoma, head and neck, and prostate) (7–9, 29, 31, 41). The standard deviation across all reported input parameters was calculated and assumed to be the standard parameter measurement or calculation uncertainty. The calculated uncertainty levels for the RMF model-specific parameters were 2.5% and 42% for ${\Sigma }_x$ and $z_{F,x}$, respectively. For all models, this process was repeated across all 18 collected datasets for the reported ${\text{α}}_x$ and ${\text{β}}_x$, whose $1\text{σ}$ values were 49% and 47%, respectively. The only parameter for which the realistic uncertainty was not determined was the uncertainty in kinetic energy for the LEM. As there are no practical means of determining this, the kinetic energy of each fragment was simply varied at the standard thresholds (± 5%, 10%, 20%, 40%).

TABLE 1.

Cell- and Model-specific Reference Parameters for Different Cell Lines

Model	Cell line	Ref. radiation	Carbon radiation	Ref. $\text{α}$	Ref. $\text{β}$	${\text{α}}_0$	$r_d$	$R_n$	Ref.
Microdosimetric (MKM, SMKM)	CHOK1	Not reported	Not reported	0.228	0.02	0.228	0.2	4.1	Elsässer 2007 (40)
	HSG	200 kVp	290 MeV/u	0.19	0.05	0.13	0.42	4.1	Kase 2006 (3)
	HSG	200 kVp	12–135 MeV/u	0.313	0.0615	0.172	0.32	3.9	Furusawa 2000 (29)
	HSG	200 kVp	12–135 MeV/u	0.313	0.0615	0.313	0.34	4.1	Kase 2008 (36)
	T1	200 kVp	12–135 MeV/u	0.0305	0.0585	0.0305	0.35	3.5	Kase 2008 (36)
	T1	Not reported	Not reported	0.137	0.019	0.137	0.288	3.694	Manganaro 2017 (39)
	V79	Not reported	Not reported	0.104	0.045	0.104	0.364	3.748	Manganaro 2017 (39)
	V79	200 kVp	12–135 MeV/u	0.184	0.02	0.184	0.26	4.1	Kase 2008 (36)
	V79	Not reported	290 MeV/u	0.147	0.02	0.147	0.25	3.63	Bopp 2016 (35)
Model	Cell line	Ref. radiation	Ref. $\text{α}$	Ref. $\text{β}$	$\text{θ}$		$\text{κ}$		Ref.
RMF	Chordoma	Co-60	0.1	0.05	0.0121		0.0015		Krämer 2000 (31)
	Head & neck	Co-60	0.25	0.025	0.0302		0.000724		Girinsky 1993 (41) Stuschke 1999 (45)
	HSG	200 kVp	0.313	0.0615	0.0359		0.0016		Furusawa 2000 (29)
	Prostate	Co-60	0.15	0.048	0.0181		0.0014		Carlson 2004 (46)
	T1	200 kVp	0.0305	0.0585	0.0033		0.0016		Furusawa 2000 (29)
	V79	200kVp	0.0722	0.0502	0.0086		0.0015		Carlson 2008 (8)
	V79	200 kVp	0.184	0.02	0.0211		0.000534		Furusawa 2000 (29)
	NSCLC H460	Cs-137	0.29	0.083	0.03487		0.002407		Guan 2018 (7)
Model	Cell line		Ref. radiation		Ref. $\text{α}$		Ref. $\text{β}$		Source
LEM	Chordoma		200 kVp		0.10		0.05		Mairani 2010 (30)

Finally, the impact of dose on the variance introduced by each parameter was assessed. To do this, the sensitivity analysis was repeated at three different dose levels. The Monte Carlo scored dose was normalized to each 2, 4, and 6 Gy at the Bragg peak or throughout the SOBP, depending on the beamline.

Cell Lines

In addition to quantifying the sensitivity of RBE to variations in model-specific parameters, we quantified the variability of each model with changes in reference cell lines. In current clinical practice, only one cell line is used with each model. Institutions that employ the MKM in treatment planning systems use the HSG cell line, whereas institutions that employ the LEM use chordoma cell lines. Therefore, we quantified the effect that different reference cell lines, as well as different reference datasets for those cell lines, have on the RBE and biological dose. The results show the effect of these selections and how they compare to the selection of the RBE model itself.

Using the datasets in Table 1, we calculated the RBE for each cell line using the corresponding reported input parameters. Typically, input parameters for each model were found as a complete set. However, for the microdosimetric models, either the ${\text{α}}_0$ or ${\text{α}}_x$ parameters were not always individually reported. In these cases, we made the assumption that ${\text{α}}_0={\text{α}}_x$, which is consistent with the assumptions made in several other studies (16, 36, 42). Finally, the variability across the RBE and subsequent biological dose were quantified by calculating the mean, standard deviation, and root mean square for the microdosimetric models and the RMF model, an approach consistent with model comparisons in previous studies (12, 16, 43, 44). For the LEM, the variation by cell line could not be calculated, as pre-tabulated alpha and beta tables were not readily available for cell lines other than chordoma. Because of this, a comparison of all models using a single cell line was not possible; therefore, we elected to focus on comparing the clinical implementation of each model (HSG for MKM and chordoma for LEM), as these cell line and model combinations are used clinically for all treatment sites. Retroactive studies of outcome based on RBE models would necessitate this approach as well, due to this implementation.

RESULTS

Sensitivity Analysis

The variability in the RBE as a function of input parameter variance for each of the four models is shown in Table 2. The variance was averaged across the SOBP, as this is typically the area of most clinical importance. The most impactful parameter for the microdosimetric models was the radius of the domain $\left(r_d\right)$; when this parameter varied by ±20% (the variation reported at the $1-\text{σ}$ level in the literature), it impacted the RBE by ±20%. For the RMF model and LEM, the most impactful parameters were ${\text{α}}_x$ and ${\text{β}}_x$, respectively, whose effects on the variability in the RBE were 18% and 23%, respectively, based on the $1-\text{σ}$ variance of these parameters calculated from the literature.

TABLE 2.

The Percent Standard Deviation Across the Calculated RBE Values as a Result of Statistically Varying Input Parameters by ± 5%, 10%, 20%, and 40% at the $1-\text{σ}$ Level^a

Parameter	${\alpha }_x$				${\beta }_x$				${\alpha }_0$				$R_n$				$r_d$				$y^{\ast }$
Variation (%)	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40
$\text{σ}$ MKM (%) $\text{σ}$ SMKM (%)	1.7 1.7	3.4 3.5	6.9 7.1	14 14	0.6 0.6	1.1 1.2	2.5 2.6	7.1 7.4	0.7 0.7	1.3 1.4	2.7 2.8	5.3 5.5	0.7 0.7	1.3 1.4	2.9 3.0	9.1 9.5	3.7 3.7	8.1 7.9	19 20	79 62	2.2 2.3	4.4 4.5	8.8 9.1	18 19
Parameter	${\alpha }_x$				${\beta }_x$				${\Sigma }_x$				$z_{F,x}$				${\Sigma }_p$				$z_{F,p}$
Variation (%)	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40
$\text{σ}$ RMF (%)	2.2	4.2	8.8	18	1.4	2.9	5.8	16	5.4	10	25	114	0.0	0.0	0.1	0.1	5.1	10	20	40	0.0	0.1	0.1	0.2
	${\alpha }_x$				${\beta }_x$				$K{E}_i$
Variation (%)	±5	±10	±20	±40	±5	±10	±20	±40	±5	±10	±20	±40
$\text{σ}$ LEM (%)	1.6	3.4	6.7	14	1.8	3.8	7.9	23	0.8	1.9	3.7	9.9

Note. The results are presented for the SOBP region (>0.5 cm from the proximal or distal edge), where the physical dose is normalized to 2 Gy throughout.

^aThe reference parameters are in white, whereas the carbon-specific parameters are shaded in gray. For each input parameter, the variation threshold that approximately represents $1-\text{σ}$ uncertainty (estimated from the spread in literature-reported values) is shaded in yellow.

The impact of variations in ${\text{α}}_x$, which is common across all RBE models, consistently ranged from 14% to 18%. In contrast, the impact of variations in ${\text{β}}_x$ ranged from 7.1% for the MKM to 23% for the LEM. With the exception of the DSB yield in the RMF model, the uncertainty in the carbon-ion beam parameters typically had a much smaller effect on the RBE than the uncertainty in the reference parameters did.

Finally, to approximate the most realistic uncertainty in RBE models introduced by input parameters, we added the highlighted uncertainty in Table 2 for each parameter in quadrature. For each microdosimetric model, RMF, and LEM, the $1-\text{σ}$ uncertainty in RBE introduced by these parameters was 26%, 25%, and 30%, respectively.

The impact of parameter uncertainty on the RBE typically decreased with LET for ${\text{α}}_x$ and increased with LET for ${\text{β}}_x$ and most RMF and LEM model-specific parameters. Input parameters for microdosimetric models varied in trend, with the impact of parameter uncertainty on RBE decreasing with LET for ${\text{α}}_0$ and increasing with LET for all other parameters ($y^{\ast }$, $R_n$, and $r_D$). The impact of uncertainty propagated to RBE varied greatly in magnitude as a function of LET by parameter, with the RMF model-specific parameters typically showing practically negligible differences with changes in LET and microdosimetric model parameters the highest change with LET. Additionally, the impact of the dose (2, 4, and 6 Gy) on sensitivity was assessed for all input parameters (Fig. 1). For most parameters, except for ${\text{β}}_x$, the effect on the RBE decreased as the dose increased.

FIG. 1.

The variance introduced into RBE as a result of 20% uncertainty in primary input parameters for each of four RBE models, to compare the impact of each input parameter across models. The variance is plotted as a function of physical dose (2, 4, or 6 Gy).

The effect of $1\text{σ}$ variance in ${\text{α}}_x$ and ${\text{β}}_x$ on the RBE, assessed at a Bragg peak dose of 2 Gy, was compared to the difference in the RBE among all four models (Fig. 2). Typically, the variance in the RBE calculated with a single model exceeded the difference in the RBE across all four models. The exception was in the fragmentation tail distal to the peak, where the microdosimetric models both yielded a higher RBE and biological dose than the RMF model and LEM did. The variability in the RBE (illustrated by the error bars in Fig. 2) was often smaller in this tail region than in the areas proximal to peak dose.

FIG. 2.

RBE calculated as a function of depth for a 18/6 cm (range/width) SOBP using four RBE models with clinically relevant cell lines (chordoma and HSG). Error bars correspond to $1-\text{σ}$ standard deviation resulting from $1-\text{σ}$ variance in ${\text{α}}_x$ and ${\text{β}}_x$, emphasizing that inherent parameter uncertainty is often greater than the magnitude of difference between models. Chordoma and HSG cells were used for the LEM and MKM, respectively per their clinical implementation, and both cell lines were used for the RMF model.

Variability Among Cell Lines

The RBE values and biological doses calculated with the MKM, SMKM, and RMF models for each reference parameter combination derived from the data in Table 1 are shown in Fig. 3. Four different cell lines were used for the microdosimetric models, while seven cell lines were used for the RMF model. The spread in each RBE and biological dose across all datasets was much greater for the microdosimetric models than for the RMF model. The mean RBE values calculated across the entrance and SOBP regions for each dataset are shown in Table 3, as are the mean RBE values at each region across cell lines, as well as the variance across cell lines at the $1\text{σ}$ level. The entrance region was defined from 1 to 2 cm depth in each of the monoenergetic beams simulated as well as in the SOBP. Compared with that calculated with the RMF model, the RBE calculated with the microdosimetric models was higher on average (2.3 vs. 1.3) and had much greater variability across cell lines (25–28% vs. 2.5%), even though more reference cell lines were used with the RMF model. The RBE values calculated using the LEM with the chordoma cell line, the only available reference cell line, are also shown in Table 3.

FIG. 3.

The RBE (top) and biological dose (bottom) calculated as a function of depth for a 18/6 cm (range/width) SOBP using each the MKM (left), SMKM (middle), and RMF model (right) with four different cell lines (the microdosimetric models) or seven different cell lines (the RMF model). This provides a comparison of the variability in RBE and biological dose due to different input parameter sets and cell lines. The physical dose throughout the SOBP is 2 Gy.

TABLE 3.

Mean RBE Values as a Function of the Model and Cell Line for Different Reference Datasets^a

	Cell Line	RBE: entrance	RBE: SOBP	Ref.
MKM	T1	2.07	2.47	Kase 2008 (36)
		1.72	2.34	Manganaro 2017 (39)
	V79	1.77	2.41	Kase 2008 (36)
		1.79	2.25	Manganaro 2017 (39)
		1.93	2.60	Bopp 2016 (35)
	CHO-K1	2.06	2.84	Elsässer 2007 (40)
	HSG	1.37	1.80	Kase 2006 (3)
		1.49	1.98	Furusawa 2000 (29)
		1.65	2.07	Kase 2008 (36)
SMKM	T1	2.02	2.38	Kase 2008 (36)
		1.70	2.30	Manganaro 2017 (39)
	V79	1.74	2.38	Kase 2008 (36)
		1.75	2.18	Manganaro 2017 (39)
		1.90	2.56	Bopp 2016 (35)
	CHO-K1	2.04	2.81	Elsässer 2007 (40)
	HSG	1.33	1.74	Kase 2006 (3)
		1.45	1.91	Furusawa 2000 (29)
		1.61	2.01	Kase 2008 (36)
RMF	T1	1.19	1.56	Furusawa 2000 (29)
	V79	1.23	1.61	Carlson 2008 (8)
		1.17	1.53	Furusawa 2000 (29)
	HSG	1.16	1.53	Furusawa 2000 (29)
	H460	1.23	1.61	Guan 2018 (7)
	Chordoma	1.25	1.63	Krämer 2000 (31)
	Head and neck	1.23	1.61	Girinsky 1993 (41)
				Stuschke 1999 (45)
	Prostate	1.23	1.61	Carlson 2004 (46)
LEM	Chordoma	1.8	1.9	Mairani 2010 (30)
Mean $\left(\pm 1\text{σ}\right)$ across cell lines
	MKM	2.1 (25.3%)	2.6 (25.1%)
	SMKM	2.0 (28.1%)	2.6 (27.6%)
	RMF	1.2 (2.5%)	1.6 (2.4%)
RMS across cell lines
	MKM	2.1	2.7
	SMKM	2.1	2.6
	RMF	1.4	1.7
Mean $\left(\pm 1\text{σ}\right)$ within cell lines
	MKM (V79/HSG)	1.9 (12.2%)	2.4 (12.7%)
		1.9 (48.9%)	2.4 (56.3%)
	SMKM (V79/HSG)	1.9 (11.5%)	2.4 (13.4%)
		1.9 (49.3%)	2.4 (56.7)
	RMF (V79)	1.2 (3.1%)	1.6 (3.9%)
Root mean square within cell lines
	MKM (V79/HSG)	2.0	2.5
		1.9	2.4
	SMKM (V79/HSG)	2.0	2.4
		1.8	2.4
	RMF (V79)	1.1	1.3

Note. The entrance region includes all points within 1 and 2 cm depth in each of the three monoenergetic beams and the SOBP, while the SOBP includes all points greater than 0.5 cm from the proximal or distal edge. The dose in the entrance region ranges from 0.50 to 0.65 Gy, while the dose in the SOBP is 2 Gy throughout.

^aThe final rows show the mean and variance, followed by the root mean square, both within and across cell lines.

In addition, we compared the LEM and RMF model using the chordoma cell line as a reference, and we compared the MKM and RMF model using the HSG cell line as a reference (Fig. 4). The cell survival curves as a function of dose and reference parameter combinations are shown in Fig. 5. Owing to limited available reference data, we were unable to use the same cell line with all four models to calculate the RBE; however, comparison using these cell lines seemed reasonable, as current clinical practice is to use only the chordoma cell line for LEM and only the HSG cell line for MKM.

FIG. 4.

The RBE calculated as a function of depth for a 18/6cm (range/width) SOBP using four models with clinically relevant reference cell lines. These include chordoma (LEM), HSG (MKM and SMKM), and both chordoma and HSG (RMF) to serve as a comparative basis.

FIG. 5.

The cell surviving fraction as a function of dose calculated with four models using clinically relevant reference cell lines. These include chordoma (LEM), HSG (MKM and SMKM), and both chordoma and HSG (RMF) to serve as a comparative basis. The surviving fraction was calculated at the middle of the SOBP (15 cm).

In clinical practice, a clinical scaling factor is typically used to align RBE estimates, thereby producing similar RBE-weighted doses at the midpoints of targets. In particular, institutions using the LEM scale their RBE by an arbitrary factor to align RBE-weighted doses with MKM-based predictions (47–50). This has also been done in prescriptions of RBE by the MKM model at NIRS, where the biological dose calculated using 200 kVp X rays was scaled to match that calculated using carbon as a reference (51–53). To capture this clinical scaling of RBE, we normalized the RBE and biological dose calculated using each cell line and reference dataset to the same value at the midpoint of the SOBP (Fig. 6). This approach approximates a best-case scenario, as a scaling factor is used for a specific SOBP, rather than for all treatment fields, as is done in clinical practice. It is clear from Fig. 6 that, regardless of normalization, the RBE and biological dose differ notably among models. In fact, the RBE calculated with the microdosimetric models still exhibits up to 20% variation at the distal edge of the SOBP and in the entrance region across various reported parameter sets and cell lines. The RBE calculated with the MKM also showed a variation of up to 20% compared to that calculated with the LEM, despite normalization, indicating that the parameter set and model yielded similar magnitudes of difference. These results highlight the challenge of attempting to apply a clinical scaling factor and show that no choice of cell line or parameter set can overcome the difference in trend between the different models.

FIG. 6.

The RBE (top) and biological dose (bottom) calculated as a function of depth for a 18/6cm (range/width) SOBP using the LEM and compared to the MKM (left), SMKM (middle), and RMF model (right) for multiple cell lines. The physical dose throughout the SOBP is 2 Gy. The RBE and biological dose are both normalized to the same value the midpoint of the SOBP to simulate a best-case scenario in the case of a clinical scaling factor between RBE model predictions.

Conversely, changes in the trend of the RBE and biological dose by cell line were negligible for the RMF model. The results of normalizing the cell line data indicated that the choice of reference cell line impacts the magnitude of the RBE estimated by the RMF model but not the trend of the RBE with depth and LET. As a result, a single clinical scaling factor could be applied to remove uncertainties in cell lines and input parameters in the RMF model. However, even with a scaling factor, the RMF model presents a different trend than the LEM.

Variability within Cell Lines

In addition to assessing the variability of the RBE across cell lines, we assessed the variability of the RBE within a single cell line using different reference data for the MKM, SMKM, and RMF models. The most reported cell lines were HSG and V79 cells for the microdosimetric models (three unique datasets each) and V79 cells for the RMF model (two parameter sets). For the MKM, the spread in the RBE calculated with both cell lines is shown in Fig. 7; the subsequent survival curve as a function of dose within the same two cell lines is shown in Fig. 8. This is a meaningful comparison, as the formula for the survival fraction relies on ${\text{α}}_p$, ${\text{β}}_p$, and the physical dose as direct inputs, rather than calculations with each of these parameters in addition to ${\text{α}}_x$ and ${\text{β}}_x$ as direct inputs. The variation in the surviving fraction at a single dose level was very high within cell lines, particularly across V79 datasets.

FIG. 7.

The spread in the RBE calculated with MKM using different published parameters for the same cell lines. The RBE was calculated as a function of depth for a 6-cm SOBP (range, 18 cm), normalized to 2 Gy in the middle of the SOBP.

FIG. 8.

The spread in the cell survival fraction of the RBE calculated with the MKM using different published parameters for the same cell lines. The surviving fraction at the middle of the SOBP (15 cm) was calculated.

The mean and variance in the RBE at the entrance and SOBP locations within the cell lines are reported in Table 3. For both the microdosimetric models and the RMF model, the variability within cell lines was similar to the variability across cell lines. In the MKM and SMKM, the variability within V79 cells (12%) was half that across all three cell lines. The variability within HSG cells (52%) was roughly four times that within V79 cells. The variability within cell lines for the RMF model was much lower than that for the microdosimetric models but was higher than the variability across all cell lines (2.5% vs. 3.4% on average).

DISCUSSION

This study investigated the variability introduced into the RBE and subsequent biological dose by uncertainties in both input parameters of reference and carbon-ion beams, along with that introduced by differences in cell lines and reference radiation. The first portion of the study was intended to investigate the inherent uncertainty in each model-specific input parameter, such as that introduced by physical measurements, estimations, or parameter fitting, to quantify the sensitivity of the model. The second portion of the study, investigating changes both across and within reference cell lines and radiation, was intended to supplement the quantification of intra-model uncertainty and determine how it compares to inter-model uncertainty. Although this study did not validate the accuracy of individual models at a cellular level, it did compare each in terms of their variability and robustness to inherent uncertainties.

To first verify our implementation of the various RBE models, we compared the biological doses calculated using the MKM and LEM in the present study with those calculated by Dahle et al. (16). Because biological dose can be optimized, it was important to compare at least two locations (the entrance and SOBP), and the ratio between the two, to be able to make a reasonable comparison between the two studies. When directly compared, the studies’ ratios of the mid-SOBP biological dose to the entrance biological dose agreed to within 0.5% on average for both models and several cell lines, confirming the accuracy of the model implementation that served as the basis for the uncertainty assessment in this study.

Sensitivity of Input Parameters

The variability in the RBE that was introduced by inherent uncertainty in the reference input parameters was typically 26% for the microdosimetric models, 25% for the RMF model, and 30% for the LEM. In each model, this variability depended on many factors, including dose, depth, and LET of reference radiation and carbon radiation (in the case of microdosimetric models, where certain parameters are fit using carbon irradiated data). The effects that the variability of individual parameters for the MKM had on the RBE in the present study are similar to those reported by Dahle et al., who quantified the effects that the uncertainty in the reference parameters for the MKM had on the biological dose (16). Though the Dahle et al. study used different methodology than the current work, each found the MKM parameters to have the same relative effect, where $r_d$ was most impactful and $R_n$ was least impactful. Although the studies’ different methods make a direct comparison difficult due to differences in dose levels, for example, their results are in reasonable qualitative agreement.

The realistic parameter uncertainty levels used for ${\text{α}}_{\text{x}}$ and ${\text{β}}_{\text{x}}$ (determined to be 39% and 40%, respectively, at $1\text{σ}$ standard deviation) in the present study are similar to the variance across all ${\text{α}}_{\text{x}}$ and ${\text{β}}_{\text{x}}$ values (determined to be 35% and 36%, respectively, at $1\text{σ}$ standard deviation) that were reported in the PIDE, which contains more than 615 different datasets for 68 different cell lines (33, 34). The magnitudes of the standard deviations calculated both across and within cell lines in the PIDE are similar to those calculated across the limited reference datasets used in the present study, agreeing to within 5% for both ${\text{α}}_{\text{x}}$ and ${\text{β}}_{\text{x}}$. The $1\text{σ}$ uncertainty levels calculated for each model-specific parameter were also compared to previously reported uncertainties. In a 2006 study, Kase et al. reported an uncertainty in $r_d$ of 9.5% (which we assumed was at the $1\text{σ}$ level). This was about 50% lower than the variance calculated across various reported values of $r_d$ in the present study. Similarly, in the same study, Kase et al. reported an uncertainty in ${\text{α}}_0$ of 23%, which was about half that calculated here. When we used the uncertainty levels reported by Kase et al., rather than those we calculated, the mean effect that $r_d$ and ${\text{α}}_0$ had on the RBE dropped by 11% and 2.5%, respectively (3). The inconsistencies in these values can be attributed to the different methods used to calculate them, as the values reported by Kase et al. were based on fitting uncertainty. This reported uncertainty should vary by study.

In the present study, we considered each parameter to be independent, but this is not necessarily the case in practical situations. For instance, variations in $r_d$, $R_n$, and ${\text{β}}_x$ all affect the calculation of $y^{\ast }$. When we introduced variations in $y^{\ast }$, we assumed the uncertainty to be only measurement uncertainty, rather than a combination of measurement uncertainty and the uncertainty in $r_d$ and $R_n$. However, this still seemed to be a reasonable approach, as $y^{\ast }$ is measured and then calculated with fixed, predetermined values of $r_d$, $R_n$, and ${\text{β}}_x$. Using a TEPC, Hartzell et al. found the uncertainty in $y^{\ast }$ to be 4.8 ± 1.5% (22). Propagating this uncertainty to estimates of the RBE_MKM yielded consistent results (2–3% uncertainty at the $1-\text{σ}$ level) across both studies. It is further worth noting that the parameter dependence of ${\text{α}}_{\text{p}}$ and ${\text{β}}_{\text{p}}$ as predicted by each model was not always linear. Various parameters, such as ${\Sigma }_p$ for ${\text{β}}_{\text{p}}$ calculated by MKM, followed a power law while still other relationships were exponential, which contributed to the variability introduced in RBE by each parameter.

Variability Across and within Cell Lines

The variability in the RBE averaged 25% and 28% across four different cell lines in the MKM and SMKM, respectively, but averaged just 2.5% across seven different cell lines in the RMF model. This difference in variability may have been due to the nature of the input parameters and how they were obtained for each model. Although all three models rely on ${\text{α}}_x$ and ${\text{β}}_x$, which are fit from cell survival data, the microdosimetric models also rely on two additional cell- and model-specific input parameters ($r_d$ and ${\text{α}}_0$) obtained from fitting data. These additional input parameters likely introduce greater uncertainty than those obtained through direct calculation by Monte Carlo, such as those used in the RMF model. One interpretation of the lowered variability in RBE by RMF across cell lines is positive and indicative of model robustness, as a singular cell line is used clinically for all models at present, but this could be a limitation of the model as well. In any case, the variation in input parameters within a single cell line are indicative of inconsistencies in experimental cell line studies, emphasizing the importance of an uncertainty analysis in a clinical framework.

For a given model, the variability within a cell line (33% on average in the microdosimetric models) was typically at least as high as, and often greater than, the variability between cell lines (27% on average in the microdosimetric models). This result reflects the variance in reported reference ${\text{α}}_x$ and ${\text{β}}_x$ values, for which the standard deviation within cell lines was similar to the magnitude of variability across cell lines. This indicates that the selection of input parameters may be as important as the choice of the cell line or even the choice of the model used to calculate the RBE.

This result also emphasizes inconsistencies between cell-based studies in the PIDE database. We were unable to directly assess the effect of reference radiation in comparisons of inter- and intra-cell line variability; however, data reported by Stewart et al. suggest that, compared with using Co-60 as reference radiation, using 60- to 250-kVp X rays as reference radiation yields 1.1 to 1.25 times as many DSBs (54). Implementing this correction factor to estimates of RBE calculated with the RMF model using Co-60 as the reference radiation increased the spread in RBE from 2.5% to 10.0% across all cell lines. Bronk et al. used clinical 6-MV photons as the reference radiation for their experiments with carbon-ion radiation in cells (55). Their RBE data for lung cancer H460 cells differed from those in the PIDE database, but the general variation trend of RBE were in line with each other. This further consolidates our observation that RBE variation depends on the selection of the cell line and reference radiation.

It is common in practice to apply a scaling factor to RBE models to minimize the differences among their results, to compare model predictions, or to compare clinical outcomes from one institution to another (47–49, 51–53). We simulated this factor to determine its effect on RBE. As expected, its application can dramatically reduce the variability in the estimated RBE, as input parameters and cell lines vary, particularly for the RMF model. Nevertheless, variability remained between cell line and input parameters, as it did between models, when a scaling factor was applied. This finding is important for treatment planning, as it quantifies the differences in the RBE and biological dose that exist beyond scaling. Furthermore, this finding indicates that the use of a consistent clinical scaling factor is, at the very least, specific to select reference datasets and should be subject to careful examination. In particular, this finding emphasizes that a single value used across all beam energies, depths, and dose levels is not sufficient for scaling between and across all models. As a final consideration, this variability must be recognized to be the result of a best-case scenario, as all conditions were normalized to the midpoint of a single SOBP. Any scaling factor applied uniformly to all fields would show increased variability.

The primary limitations of this study stem from the specific physical dose applied and the choice of reference cell line used for RBE calculations. In clinical settings, the physical dose for each beamlet is usually normalized to yield a biologically optimized dose across the SOBP. However, for this study, we optimized the dose to achieve a physically flat SOBP. This approach was intentionally chosen to facilitate a consistent comparison across all models, as the physical dose is a component of the linear quadratic calculations for RBE and thus influences RBE calculations directly. Utilizing an SOBP with a uniform physical dose allowed for a more effective comparison of model variability within and across different models using the same data set. The choice of reference cell line was also made with careful consideration. Although using the same cell line for all models might be beneficial for comparison purposes, we opted for the cell line currently used in clinical implementations of the models. Given that clinical treatment planning systems use a single cell line for each of MKM and LEM, regardless of the treatment site, this strategy provided a clinically pertinent basis for comparing RBE models. Furthermore, the requisite reference data to compute RBE for both MKM and LEM, whether for chordoma or HSG cell lines, is lacking. Specifically, the LEM tables detailing α and $\text{β}$ values as functions of kinetic energy are only available for chordoma cells, not for HSG cells. Likewise, the necessary cell parameters for MKM calculations are not documented in the literature for chordoma cells.

CONCLUSION

This study investigated the impact of uncertainty and variability in input parameters and cell lines on the RBE, as calculated using four models commonly implemented in clinical and research-based treatment planning systems. The variability in the RBE calculated using each model was similar to the variability in the RBE calculated across different models. The microdosimetric models, which have unique input parameters that are typically estimated or fitted, were less robust to intrinsic variability and thus less consistent than the other models. This should be considered when comparing outcomes among models. Currently, the clinical implementation of RBE models typically relies on a single cell line for each model (HSG cells for the MKM and chordoma cells for the LEM), regardless of the treatment site. The present study showed that the variability of the RBE as a function of the selected cell line exceeded 25%, which raises concerns about the consistency and robustness of RBE predictions. Furthermore, research and clinical discussions regarding the RBE typically focus on the choice of model and the subsequent effect on treatment. The findings of the present study indicate that the selection of reference parameters, including the cell line and reference radiation, at a minimum, may be as important as the selection of the model and should play a role in informing clinical practice.