A Monte Carlo study of proton LET calculations: Comparison between the codes PENHAN, FLUKA, and TOPAS

Abstract

Background

Linear energy transfer (LET) is frequently used to characterize radiation quality in proton therapy and is an important quantity for radiobiological modeling. However, LET calculations can, in principle, vary between Monte Carlo (MC) codes, affecting consistency in treatment planning and relative biological effectiveness (RBE) estimation.

Purpose

This study aims to investigate the variability of LET results when using different MC codes.

Methods

An intercomparison of LET results was performed using penhan, topas MC, and CERN fluka, together with recently published experimental data. Track‐averaged (${\overline{L}}_{\mathrm{t}}$) and dose‐averaged (${\overline{L}}_{\mathrm{d}}$) LET were evaluated under different irradiation configurations, including pencil and broad beams, with and without nuclear reactions.

Results

The results obtained indicate that discrepancies in LET calculations are primarily linked to the choice of stopping powers and the handling of nuclear interactions. Calculations considering only primary protons exhibited great agreement with recent experimental LET measurements, validating our simulations.

Conclusions

${\overline{L}}_{\mathrm{t}}$ showed higher robustness and consistency across codes, suggesting it as potentially a more reliable metric for LET‐based treatment planning and RBE modeling. Our findings emphasize the importance of consistent LET calculations, result reporting, and code benchmarking in proton therapy calculations.

1. INTRODUCTION

Proton therapy has established itself as a solid alternative to conventional photon and electron radiotherapies for certain patient profiles and tumor types. Its main advantage lies in its precision when depositing dose, which allows greater preservation of healthy tissues, ¹ , ² resulting in an almost complete elimination of the exit dose and a clear reduction of the proximal one. For this reason, proton therapy is indicated for some specific tumor types, such as head and neck or pediatric cancers. ¹

However, absorbed dose is not the only factor that determines the damage that proton beams will produce in tissue. For the same absorbed dose, protons are generally more effective than photons. The difference in effectiveness is usually measured by the relative biological effectiveness, or RBE, ³ defined as the ratio of absorbed doses of a reference radiation, usually , and the radiation of interest, to produce the same biological response (or endpoint). Today, most proton therapy treatments use the recommended RBE value of 1.1. ⁴ , ⁵ , ⁶ , ⁷ , ⁸ This is being reevaluated due to the observation of unexpected side effects occurring at the distal edge of the beams. ⁹ Characterizing RBE is not easy since it depends, among other factors, on the kind of tissue, its oxygenation level, the endpoint considered and the quality of the radiation, that is, its type and energy.

In this context, radiation quality is usually assessed with the concept of linear energy transfer (LET) for the case of charged particles. This practice is also increasingly questioned due to evidence showing that LET may not reliably predict RBE in complex mixed fields, particularly near the distal edge of proton beams. ¹⁰ , ¹¹ Yet, it remains the preferred clinical metric for characterizing RBE due to its prevalence and lack of a consensus alternative. ¹²

One of the problems of modeling RBE in a more realistic way is, not only the obvious difficulty of obtaining biological data for all relevant tissues, but also the variability introduced when such data are derived from different research groups or methodologies. ¹³ Then, to improve treatment plan optimization without requiring explicit biological input, a possible approach is to optimize the LET distribution, considering that RBE tends to increase with LET in clinical proton beams. ¹⁴ , ¹⁵ Consequently, accurate LET information becomes essential. ¹⁶

Monte Carlo (MC) simulations are widely recognized as the gold standard for particle transport and interactions in matter due to their high physical accuracy. Despite their widespread use in proton LET calculations ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ comparisons between different MC codes have been largely neglected. The aim of this work is to analyze the potential differences that may exist between various MC codes in the determination of this quantity, with the objective of identifying the possible uncertainties due to the physics and proton electronic stopping powers included in each code, as well as to aspects related to the tracking algorithms and the contribution of primary and secondary protons. We also evaluate our calculations with the recent experimental results of Muñoz et al. ²⁰

2. MATERIAL AND METHODS

2.1. LET averages: Definitions

For a charged particle of a given type and energy traversing a material, LET (or restricted linear electronic stopping power, $L_{\mathrm{\Delta }}$) is defined as the quotient between the mean energy lost due to electronic interactions while traveling a distance, restricted to energy transfers less than $\mathrm{\Delta }$, and the distance traversed by the particle. Consequently, unrestricted LET ($L_{\infty }$ or simply $L$) is the corresponding electronic stopping power. ²¹ , ²²

A radiotherapy beam passing through a volume is composed of several particles, of different types and energies. In order to characterize it with a single parameter we need an average value of LET. The two most commonly used averages are the track‐averaged LET, ${\overline{L}}_{\mathrm{t}}$, and the dose‐averaged LET, ${\overline{L}}_{\mathrm{d}}$. ²¹ , ²³

In the International Commission on Radiation Units and Measurements (ICRU) Report, ²¹ the track‐averaged LET, ${\overline{L}}_{\mathrm{t}}(\mathrm{\Delta }V)$, and dose‐averaged LET, ${\overline{L}}_{\mathrm{d}}(\mathrm{\Delta }V)$, in a region of interest, $\mathrm{\Delta }V$, centered in $\vec{r}\equiv (x,y,z)$, are defined as

$$\begin{array}{cc}& {\overline{L}}_{\mathrm{t}}(\mathrm{\Delta }V)={\int }_0^{\infty }L\mathbf{t}(L,\mathrm{\Delta }V)\mathrm{d}L;\hfill \\ & {\overline{L}}_{\mathrm{d}}(\mathrm{\Delta }V)={\int }_0^{\infty }L\mathbf{d}(L,\mathrm{\Delta }V)\mathrm{d}L,\hfill \end{array}$$ (1)

where $\mathbf{t}(L,\mathrm{\Delta }V)$ and $\mathbf{d}(L,\mathrm{\Delta }V)$ are both probability density functions and are normalized to unity. $\mathbf{t}(L,\mathrm{\Delta }V)\mathrm{d}L$ is the probability that a particle in the region of interest has a value of LET between $L$ and $L+\mathrm{d}L$ ²² and $\mathbf{d}(L,\mathrm{\Delta }V)\mathrm{d}L$ is the fraction of the total absorbed dose, $D$, imparted in the region of interest by particles with LET between $L$ and $L+\mathrm{d}L$. Both distributions, $\mathbf{t}(L,\mathrm{\Delta }V)$ and $\mathbf{d}(L,\mathrm{\Delta }V)$, are related as follows: ²¹

$$\mathbf{d}(L,\mathrm{\Delta }V)=\frac{L\mathbf{t}(L,\mathrm{\Delta }V)}{{\overline{L}}_{\mathrm{t}}(\mathrm{\Delta }V)}.$$ (2)

Additionally, we can define the particle fluence in $\vec{r}$, $\mathrm{\Phi }(\vec{r})$, as the number of particles, $\mathrm{\Delta }N$, incident on a small sphere, $\mathcal{S}$, of traversed cross sectional area $\mathrm{\Delta }S$, centered at $\vec{r}$

$$\mathrm{\Phi }(\vec{r})=\frac{\mathrm{\Delta }N}{\mathrm{\Delta }S};\overline{\mathrm{\Phi }}=\frac{1}{\mathrm{\Delta }V}{\int }_{\mathrm{\Delta }V}\mathrm{\Phi }(\vec{r})\mathrm{d}\vec{r},$$ (3)

with $\overline{\mathrm{\Phi }}$ being the corresponding average particle fluence in $\mathrm{\Delta }V$. In a similar way, the charged particle fluence spectrum in energy, $\mathrm{\Phi }(E,\vec{r})$ is defined as

$$\mathrm{\Phi }(E,\vec{r})=\frac{\mathrm{\Delta }n(E)}{\mathrm{\Delta }S};\overline{\mathrm{\Phi }}(E)=\frac{1}{\mathrm{\Delta }V}{\int }_{\mathrm{\Delta }V}\mathrm{\Phi }(E,\vec{r})\mathrm{d}\vec{r},$$ (4)

where $\mathrm{\Delta }n(E)\mathrm{d}E$ is the number of particles with energies between $E$ and $E+\mathrm{d}E$ incident on $\mathcal{S}$ and $\overline{\mathrm{\Phi }}(E)$ is the corresponding average charged particle fluence spectrum in energy in $\mathrm{\Delta }V$. Finally, the particle fluence spectrum in LET, $\mathrm{\Phi }(L,\vec{r})$, is defined as

$$\mathrm{\Phi }(L,\vec{r})=\frac{\mathrm{\Delta }n(L)}{\mathrm{\Delta }S};\overline{\mathrm{\Phi }}(L)=\frac{1}{\mathrm{\Delta }V}{\int }_{\mathrm{\Delta }V}\mathrm{\Phi }(L,\vec{r})\mathrm{d}\vec{r},$$ (5)

where $\mathrm{\Delta }n(L)\mathrm{d}L$ is the number of particles with LET values between $L$ and $L+\mathrm{d}L$ incident on $\mathcal{S}$ and $\overline{\mathrm{\Phi }}(L)$ is the corresponding average particle fluence spectrum in LET in $\mathrm{\Delta }V$. From this definition, and after normalizing $\overline{\mathrm{\Phi }}(L)$, we have (see chapter 5 of Salvat ²⁴ ):

$$\frac{\overline{\mathrm{\Phi }}(L)\mathrm{d}L}{{\displaystyle \int \overline{\mathrm{\Phi }}(L)\mathrm{d}L}}=\mathbf{t}(L,\mathrm{\Delta }V).$$ (6)

Taking into account that

$$\overline{\mathrm{\Phi }}(L)=\overline{\mathrm{\Phi }}(E){\left(\frac{\mathrm{d}L}{\mathrm{d}E}\right)}^{-1},$$ (7)

the previous definitions and the relation given by Equation (2), we can write ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ from Equation (1) as:

$$\begin{array}{cc}& {\overline{L}}_{\mathrm{t}}(\mathrm{\Delta }V)=\frac{{\displaystyle {\int }_0^{\infty }S(E)\overline{\mathrm{\Phi }}(E)\mathrm{d}E}}{{\displaystyle {\int }_0^{\infty }\overline{\mathrm{\Phi }}(E)\mathrm{d}E}};\hfill \\ & {\overline{L}}_{\mathrm{d}}(\mathrm{\Delta }V)=\frac{{\displaystyle {\int }_0^{\infty }{\left(S(E)\right)}^2\overline{\mathrm{\Phi }}(E)\mathrm{d}E}}{{\displaystyle {\int }_0^{\infty }S(E)\overline{\mathrm{\Phi }}(E)\mathrm{d}E}},\hfill \end{array}$$ (8)

where we have written $S(E)$, the electronic stopping power of the charged particle for the energy $E$, instead of $L$. To be precise, in Equation (8) we would have to take into account the contribution of all charged particles to the spectrum. Since we are interested in these quantities in the context of proton therapy, we will follow the recommendations given in the literature ¹² , ²⁵ and calculate ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ in water including only primary and secondary protons.

2.2. LET averages: MC calculation

The strategy to calculate LET averages with MC methods is to use Equation (8) and determine the energy spectra using MC simulations. $\overline{\mathrm{\Phi }}(E)$ is given by Equation (4), and in order to calculate it, we have to determine $\mathrm{\Phi }(E,\vec{r})$ at each point $\vec{r}$ of the region of interest, $\mathrm{\Delta }V$, integrate it in that region, and divide by $\mathrm{\Delta }V$. Assuming that $\mathrm{\Delta }V$ is small, we can approximate the integral as

$$\begin{array}{cc}\hfill \overline{\mathrm{\Phi }}(E)& =\frac{1}{\mathrm{\Delta }V}{\int }_{\mathrm{\Delta }V}\mathrm{\Phi }(E,\vec{r})\mathrm{d}\vec{r}\hfill \\ & =\frac{1}{\mathrm{\Delta }z}{\int }_{\mathrm{\Delta }z}\mathrm{\Phi }(E,z_0)\mathrm{d}z\approx \mathrm{\Phi }(E,z_0).\hfill \end{array}$$ (9)

Here, $\mathrm{\Phi }(E,z_0)$ is the energy fluence spectra at $z_0\in \mathrm{\Delta }V$ in a $\mathrm{\Delta }S$ defined by the accumulation voxel. We chose $z_0$ to be at the beginning of $\mathrm{\Delta }V$. The particular choice of $z_0$ is not important in this study as long as it is chosen consistently for all codes. $\mathrm{\Phi }(E,z_0)$ can be easily obtained by simulation and, once one has this quantity, it is trivial to calculate ${\overline{L}}_{\mathrm{t}}(\mathrm{\Delta }V)\equiv {\overline{L}}_{\mathrm{t}}(z_0)$ and ${\overline{L}}_{\mathrm{d}}(\mathrm{\Delta }V)\equiv {\overline{L}}_{\mathrm{d}}(z_0)$.

From Equation (8), and taking into account the corresponding histograms for $\mathrm{\Phi }(E,z_0)$, we can write:

$$\begin{array}{cc}& {\overline{L}}_{\mathrm{t}}(z_0)=\frac{{\sum }_{i=1}^N\mathrm{\Phi }(E_i,z_0)S(E_i)\mathrm{\Delta }{E}_i}{{\sum }_{i=1}^N\mathrm{\Phi }(E_i,z_0)\mathrm{\Delta }{E}_i};\hfill \\ & {\overline{L}}_{\mathrm{d}}(z_0)=\frac{{\sum }_{i=1}^N\mathrm{\Phi }(E_i,z_0)S^2(E_i)\mathrm{\Delta }{E}_i}{{\sum }_{i=1}^N\mathrm{\Phi }(E_i,z_0)S(E_i)\mathrm{\Delta }{E}_i},\hfill \end{array}$$ (10)

where $E_i$ is the central value of the energy bin, $\mathrm{\Delta }{E}_i$, and $N$ is the number of bins. Then, we will focus on depth‐distributions for ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$.

The uncertainties associated with the MC calculation of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ were estimated by first calculating the standard deviation of the mean of the numerator and denominator separately using linear error propagation and then estimating the uncertainty of the quotient with MC, similarly to Chica et al. ²⁶ Our simulations and results are reported following Sechopoulos et al. ²⁷

2.3. Monte Carlo codes

2.3.1. $\mathrm{PENHAN}$

One of the MC codes used in our study of LET is penhan (version 2024), ²⁸ , ²⁹ an extension of the electron, positron, and photon transport code penelope, ²⁴ which also simulates the transport of protons, alphas, and neutrons. ³⁰

It is a class II simulation code, meaning that it simulates hard interactions in a detailed way, from the corresponding differential cross sections, while the combined effect of soft interactions is simulated by using approximate multiple scattering theory. Hard (soft) interactions refer to events in which the energy loss and/or the angular deflection of the transported particle are above (below) certain cutoff values.

Energy cutoffs ($W_{\mathrm{cc}}$, $W_{\mathrm{cr}}$) and absorption energies ($E_{\mathrm{abs}}$) are user‐defined, as are the tracking parameters $C_1$ and $C_2$ (ranging from 0 to 0.2), with $C_1=C_2=0$ giving a fully detailed simulation. The values of the simulation parameters adopted in the present penhan simulations are listed in Table 1. Finally, in order to obtain the spectra, we used the impact detector IMPDET in 1 $\mu \mathrm{m}$ thick surfaces centered in the corresponding value of $z_0$. IMPDET is a command that tallies the energy spectrum entering a volume.

TABLE 1.

Values of the parameters considered in $\mathrm{PENHAN}$ for the tracking of electrons, positrons, photons, protons, alphas and neutrons for the simulations regarding Sections 2.4 and 2.5.

photons	$E_{\mathrm{abs}}$	$10$ keV
electrons	$E_{\mathrm{abs}}$	${10}^6$ keV
and	$C_1,C_2$	0.05
positrons	$W_{\mathrm{cc}}$, $W_{\mathrm{cr}}$	$100$ keV
protons	$E_{\mathrm{abs}}(p)$	$25$ keV
and	$E_{\mathrm{abs}}(\alpha )$	$0.01·E_i$
alphas	$C_1$, $C_2$	0.05
	$W_{\mathrm{cc}}$	$2$ keV
neutrons	$E_{\mathrm{abs}}$	$10$ keV
	FNABS	0.8
	FORCEN	500

$E_i$ denotes the initial energy in the case of the monoenergetic beam, and the minimum energy considered in the case of SOBP simulations. The SOBP was used to simulate the experiment in Muñoz et al. ²⁰

Regarding the proton stopping powers $S(E)$, penhan computes them from the differential cross section between projectile and the atoms of the material using the plane‐wave Born approximation (PWBA) with the Sternheimer–Liljequist generalized oscillator strength model. ³⁰ The Rao–Sahib and Wittry extrapolation ³¹ is used at low energies (in water, from approximately 0.27 MeV). $S(E)$ values calculated in this way are included in the penhan material files.

In contrast, sbethe ³³ is a Fortran program that provides hadronic $S(E)$ values directly by means of a modified Bethe formula with the shell, Lindlhard–Sørensen and Barkas corrections. At low $E$, two approaches are adopted. If the material is included in the International Atomic Energy Agency (IAEA) experimental database, sbethe uses the empirical formula in the ICRU Report ³² fitted to the IAEA experimental database for $E<0.75$ MeV. If there is no experimental data available, sbethe uses an analytical formula. The penhan material files can be modified to use the $S(E)$ from the code sbethe.

We studied how these two different sets of $S(E)$ values affect the results obtained with penhan. To the best of our knowledge, it is the first time that penhan (or its earlier versions) are used for LET calculations.

2.3.2. $\mathrm{FLUKA}$

The CERN code fluka (version 4.3) ³⁴ , ³⁵ was also used in our simulations. It is based on a class I tracking algorithm for charged particles. To prepare the fluka input files, the FLAIR Graphical User Interface (version 3.1‐15.1) ³⁹ was used. We used the DEFAULTS card to activate the PRECISION mode. An energy transport cutoff of 25 keV was set for protons. In order to score the spectra, the USRYIELD card was activated at the specific value of $z_0$. We used the $S(E)$ calculated using the function “GETLET”.

2.3.3. $\mathrm{TOPAS}$

We also performed simulations with topas MC version 3.9, a wrapper of the Geant4.10.07.p03 simulation toolkit. ⁴⁰ , ⁴¹ The topas default physics modules (G4EmStandardPhysics_opt4, g4h‐elastic_HP, g4decay, g4stopping, g4ion‐binarycascade, and g4h‐phy_QGSP_BIC_HP) were used, ⁴² modifying only the proton production cut, which was set to $6.413·{10}^{-5}$ cm, in order to have a similar value to the absorption energy used in penhan, which is 25 keV (see Table 1). Spectra were obtained using the scorer SurfaceTrackCount in 1 $\mu \mathrm{m}$ thick surfaces centered in the corresponding value of $z_0$. We used the $S(E)$ calculated using the Geant4 method “ComputeElectronicDEDX”.

2.4. Intercomparison between codes

This section outlines the approach adopted to intercompare ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ calculations among the different MC codes. The simulation configurations and settings and the statistical methods used to quantify discrepancies are described below.

2.4.1. Configurations considered

In order to compare the results of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ between the different codes, we considered a monoenergetic proton beam of 160 MeV, in the middle of the proton therapy range, impinging on a water phantom of dimensions $50\times 50\times 30.1{\mathrm{cm}}^3$, with two different geometries:

• (a)Pencil beam (PB), using scoring voxels of $50\times 50\times 0.2{\mathrm{cm}}^3$, eliminating in that way any possible difference due to how the codes treat proton scattering.
• (b)Broad beam (BB) of $5\times 5{\mathrm{cm}}^2$ with scoring voxels of $4\times 4\times 0.2{\mathrm{cm}}^3$, reflecting a more realistic therapy setting.

We also considered simulations activating (On) and deactivating (Off) nuclear reactions.

$\mathrm{\Phi }(E,z_0)$ were calculated considering $N=640$, $E_1=12.5$ keV, and $E_N=160.125$ MeV. A bin width of $\mathrm{\Delta }E\approx 0.25$ MeV was chosen following Wilkens and Oelfke. ¹⁷ However only bins with $E_i>0.3$ MeV were included in the calculation of Equation (8) since that is the minimum energy for which there are experimental data on $S(E$) for liquid water (see the IAEA online experimental). ⁴⁴ Data below that minimum can only be interpreted as semi‐quantitative. ²⁴ Hence, in practice, the sums start at $i=3$ in Equation (8).

The comparison between codes was done as follows. First, we considered the configuration 160 PB Off and analyzed the influence of changing proton $S(E)$ in the case of the penhan code, as we explained in section 2.3.1. Once this was done, we compared the results of the three codes for the same configuration. After that, we activated nuclear reactions (configuration 160 PB On) to investigate how their implementation affect the LET values. Finally, we considered the scenario most relevant to the clinic practice in which a broad beam was used (160 BB On). It is important to note that electrons were not tracked since only protons were scored at the spectra and at these energies electrons cannot produce them in water. Only points $z$ in the depth distributions with more than 0.1% of the maximum dose were included. ¹⁸ The number of histories was ${10}^7$ in all simulations.

All uncertainties in this section are given with a coverage factor $k=3$ (or a confidence interval of 99.7%) unless otherwise stated. The term compatible is used to describe two results (either individual data points or complete curves) whose uncertainties overlap.

${\overline{L}}_{\mathrm{t}}$ results will be presented along with their relative differences, ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z)$, defined as,

$${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z)=\frac{{\overline{L}}_{\mathrm{t}}{(z)}^{\alpha }-{\overline{L}}_{\mathrm{t}}{(z)}^{\beta }}{{\overline{L}}_{\mathrm{t}}{(z)}^{\alpha }}·100$$ (11)

where $\alpha$ and $\beta$ are P (penhan), SB (sbethe), T (topas), and F (fluka). Only the points that are not compatible (within statistical uncertainty) are included when plotting ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z)$. We calculated the uncertainty of ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z)$ using a MC procedure, as described in Chica et al. ²⁶ We proceed similarly with ${\overline{L}}_{\mathrm{d}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}(z)$. Analogously to ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$, ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z)$, and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}(z)$ will be written as ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$.

2.4.2. Statistical methods

The gamma evaluation method or $\gamma$‐test ⁴⁵ gives a measure of the agreement between two datasets $g_1({\mathbf{r}}_1)$ and $g_2({\mathbf{r}}_2)$, defined at the points ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. We will define $f_{\gamma ({\mathrm{\Delta }}_r,{\mathrm{\Delta }}_g)}$ as the percentage of points ${\mathbf{r}}_1$ that has not passed the $\gamma$‐test for given spatial, ${\mathrm{\Delta }}_r$, and function, ${\mathrm{\Delta }}_g$, thresholds. The bigger $f_{\gamma ({\mathrm{\Delta }}_r,{\mathrm{\Delta }}_g)}$, the worse the agreement. We are going to consider the local version of the method, in which ${\mathrm{\Delta }}_g$ is expressed as a percentage of the reference value in the specific point, that is, ${\mathrm{\Delta }}_g\propto g_1({\mathbf{r}}_1)$.

The standard $\gamma$‐test does not explicitly incorporate the uncertainties of the functions being compared, it assumes deterministic input values for $g_1({\mathbf{r}}_1)$ and $g_2({\mathbf{r}}_2)$. However, since we aim to ensure that observed discrepancies between simulation codes are statistically meaningful and not simply due to uncertainty, we propose a slight relaxation of the gamma criterion. Specifically, if the evaluation is performed at the same point (${\mathbf{r}}_1={\mathbf{r}}_2$), and the corresponding values $g_1({\mathbf{r}}_1)$ and $g_2({\mathbf{r}}_2)$ are compatible within their uncertainties (i.e., their uncertainty intervals overlap), we will assign $\gamma ({\mathbf{r}}_1)=0$, indicating agreement at that point.

We will present our results for $f_{\gamma (1,3)}$ using ${\mathrm{\Delta }}_z=1$ mm and ${\mathrm{\Delta }}_L=3$%, where ${\mathrm{\Delta }}_L$ denotes both ${\mathrm{\Delta }}_{{\overline{L}}_{\mathrm{t}}}$ and ${\mathrm{\Delta }}_{{\overline{L}}_{\mathrm{d}}}$. It is important to note that the choice of ${\mathrm{\Delta }}_z=1$ mm enforces a point‐by‐point comparison for the $\gamma$‐test. On the other hand, a ${\mathrm{\Delta }}_L=3$% value was selected, following the criterion for absorbed dose in photon ⁴⁷ and proton ²⁵  therapy.

Finally, we remark that the $\gamma$‐test is typically applied to absorbed dose comparisons. To our knowledge, this is the first application of the method to compare ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ distributions.

2.5. Comparison with experiment

Recently, Muñoz et al. ²⁰ have developed an experimental study in which LET measurements for protons and ions have been carried out using fluorescent nuclear track detectors. Our goal in this work was to compare MC simulations done with penhan with these experimental data. According to the reference paper, ²⁰ we implemented a geometry that consists of a nozzle and a $30\times 30\times 20{\mathrm{cm}}^3$ PTW RW3 solid water box with the entrance surface placed at 1.0 m from the nozzle exit. The space between nozzle exits and phantom is filled with air. The nozzle is composed of a 2 mm water slab representing the beam application and monitoring system and a 1.43 mm PMMA slab accounting for the ripple filter.

Similarly to the original article, our source was modeled as a square field of $9\times 9{\mathrm{cm}}^2$ impinging perpendicularly to the geometry and made of multiple sources with energies and fluence weights determined to have a 5 cm SOBP (spread‐out Bragg peak), at a depth from 10 to 15 cm in water. ⁴⁶ In total, 17 beams were considered, with energies from 117.656 to 148.234 MeV. The specific energy intervals and weights used are given in Table 2.

TABLE 2.

Lower ($E_{\text{low}}$) and upper ($E_{\text{upp}}$) energy bin limits and weights used for the simulation of the SOBP in the experiment of Muñoz et al. ²⁰

$E_{\text{low}}$ (MeV)	$E_{\text{upp}}$ (MeV)	Weight
117.656	119.612	0.0224394276
119.612	121.545	0.0228275702
121.545	123.455	0.0241901500
123.455	125.344	0.0255957611
125.344	127.212	0.0270413548
127.212	129.059	0.0286190803
129.059	130.888	0.0307890202
130.888	132.697	0.0332651583
132.697	134.489	0.0352156672
134.489	136.263	0.0393529968
136.263	138.020	0.0428432151
138.020	139.760	0.0475185300
139.760	141.485	0.0565008225
141.485	143.194	0.0602297743
143.194	144.889	0.0881681370
144.889	146.568	0.0769104567
146.568	148.234	0.3385678520

In Figure 1 we show the depth dose curve obtained by simulation, together with the experimental data from Muñoz et al. ²⁰ The penhan error bars are too small to be visible. We see that our simulation is compatible with the experimental data, validating in this way the geometry and parameters implemented. Once this validation was done, we carried out simulations in order to obtain and ${\overline{L}}_{\mathrm{d}}$ at the same depths as those considered in the measurements. The number of histories was ${10}^7$ in all simulations. Results of uncertainties for ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ in this case will be given with a coverage factor $k=1$.

FIGURE 1.

Normalized depth dose distribution in the RW3 phantom, compared with experimental results from Muñoz et al. ²⁰ .

We also present the relative error of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$, calculated using Equation (11) with respect to the experimental data. In addition, the relative error of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ of the experimental data against itself were obtained to account for its uncertainty. We will use the relative errors to compare all simulation results against the experiment.

For penhan, two different calculations were performed: one considering the contribution of all protons and another considering only primary protons in the spectrum. In penhan, secondary protons are defined as those produced either via hard elastic scattering with a nucleus or through nuclear reactions.

2.6. Statement on AI use in manuscript preparation

During the preparation of this work, the authors used DeepL and ChatGPT (v. 4o) to improve readability and language. The authors subsequently reviewed and edited the content, and they take full responsibility for the final manuscript.

3. RESULTS

3.1. Intercomparison between codes

Table 3 presents our results for $f_{\gamma (1,3)}$. In all cases, the reference distribution in the $\gamma$‐test is the first mentioned. For example, penhan–topas means that the reference distribution is penhan. Values of $f_{\gamma (1,3)}$ for penhan–sbethe are only presented for 160 PB Off. In general, in all the cases, we have equal $f_{\gamma (1,3)}$, except for ${\overline{L}}_{\mathrm{d}}$ in 160 PB On where it grows to 8.60 (an extra point that fails the gamma test with respect to 160 PB Off). The main trend observed is that the $f_{\gamma (1,3)}$ values for ${\overline{L}}_{\mathrm{t}}$ are consistently lower than those for ${\overline{L}}_{\mathrm{d}}$ across all cases, indicating a greater overall agreement for the ${\overline{L}}_{\mathrm{t}}$ data compared to ${\overline{L}}_{\mathrm{d}}$. Furthermore, the inclusion of nuclear reactions increases $f_{\gamma (1,3)}$ for ${\overline{L}}_{\mathrm{d}}$ significantly.

TABLE 3.

Values of $f_{\gamma (1,3)}$ (%) for the situations considered.

${\overline{L}}_{\mathrm{t}}$	Configuration	penhan–sbethe	penhan–topas	penhan–fluka	topas–fluka
	160 PB Off	6.45	0.0	0.0	0.0
	160 PB On		1.08	1.08	0.0
	160 BB On		0.0	0.0	0.0

${\overline{L}}_{\mathrm{d}}$
	160 PB Off	7.53	3.23	1.08	0.0
	160 PB On		10.75	45.16	44.09
	160 BB On		32.26	55.91	37.63

Figures 2, 3, 4, 5, 6 present representative ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ results (upper panels) with their respective ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$ (lower panels) for the 160 PB Off, 160 PB On, and 160 BB On configurations. Only the values of ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$ that correspond to incompatible values of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ are represented. The ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$ are calculated with the first code appearing on the legend as reference. Similarly to $f_{\gamma (1,3)}$, the example penhan–topas means that the reference distribution is penhan. These results were selected as they best help explain the behavior of $f_{\gamma (1,3)}$ (see Section 4).

FIGURE 2.

Values of ${\overline{L}}_{\mathrm{t}}$ obtained with penhan and sbethe for the configuration 160 PB Off. The black vertical lines in the lower panel denote the boundaries of the upper panel. The gray dotted line indicates the position of the Bragg peak. For the lower panel, only the values of ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ that correspond to incompatible values of ${\overline{L}}_{\mathrm{t}}$ are represented. This figure layout is consistently used for all ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ graphs.

FIGURE 3.

Values of ${\overline{L}}_{\mathrm{t}}$ obtained with penhan, topas, and fluka for the configuration 160 PB On.

FIGURE 4.

Values of ${\overline{L}}_{\mathrm{t}}$ obtained with penhan, topas, and fluka for the configuration 160 BB On.

FIGURE 5.

Values of ${\overline{L}}_{\mathrm{d}}$ obtained with penhan, topas, and fluka for the configuration 160 PB Off.

FIGURE 6.

Values of ${\overline{L}}_{\mathrm{d}}$ obtained with penhan, topas, and fluka for the configuration 160 PB On.

Figure 2 shows the ${\overline{L}}_{\mathrm{t}}$ distributions for the 160 PB Off configuration obtained with penhan and sbethe. Both results are compatible up to $z=8$ cm, beyond which penhan gives higher ${\overline{L}}_{\mathrm{t}}$ values, with a maximum difference of approximately 6%. These discrepancies increase with depth, reaching the maximum before exhibiting a descent over the last three points. Figure 3 presents the ${\overline{L}}_{\mathrm{t}}$ for 160 PB On from penhan, topas, and fluka. Groups of incompatible data are observed for topas–fluka at the entrance, middle, and distal regions of the phantom, and between penhan–topas and penhan–fluka toward the end. Only the last point of the distribution shows a ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ exceeding 3% for both penhan–topas and penhan–fluka comparisons. Finally, Figure 4 shows the ${\overline{L}}_{\mathrm{t}}$ for the 160 BB On configuration. Incompatibilities between penhan and topas appear at the distal end of the phantom, while those involving fluka (penhan–fluka and topas–fluka) occur at both the proximal and distal regions.

Figure 5 presents the ${\overline{L}}_{\mathrm{d}}$ results for the 160 PB Off configuration obtained with penhan, topas, and fluka. Incompatibilities between penhan–topas and penhan–fluka are observed throughout most of the volume, starting from $z=4$ cm. Additional discrepancies between topas–fluka appear near the distal end, with two isolated incompatible points at the proximal region. Moreover, Figure 6 shows the ${\overline{L}}_{\mathrm{d}}$ distributions for the 160 PB On configuration. Large ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$ values, exceeding 10% for the penhan–fluka comparison, occur at the entrance of the phantom for both penhan–fluka and topas–fluka. Smaller incompatibilities are also observed at the proximal and distal regions across all code combinations.

Finally, Figures 7, 8, 9 show the spectra as a function of the proton energy for penhan, topas, and fluka at selected $z$ points that help explain specific ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ values. We normalized them with respect to their integral (i.e., the denominator in the expression of ${\overline{L}}_{\mathrm{t}}$ in Equation (10)). Because these spectra are linked to specific configurations and require context to interpret, we analyze them in Section 4.

FIGURE 7.

Proton spectra obtained with penhan, topas, and fluka for the configuration 160 PB On at $z=18.34$ cm. Uncertainty has a coverage factor of $k=1$.

FIGURE 8.

Proton spectra obtained with penhan, topas, and fluka for the configuration 160 BB On at $z=18.34$ cm. Uncertainty has a coverage factor of $k=1$.

FIGURE 9.

Proton spectra obtained with penhan, topas, and fluka for the configuration 160 PB On at $z=1.99$ cm. Uncertainty has a coverage factor of $k=1$.

3.2. Comparison with experiment

In Figure 10, the ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ results (upper panel), along with their relative errors (lower panel), from the experiment in Muñoz et al. ²⁰ are presented. Experimental data (gray bands), along with the topas simulation results from the original paper (orange symbols), are shown together with the two results obtained using penhan (blue/red symbols for the calculation with all/primary protons).

FIGURE 10.

(Top) Results obtained in this work with penhan, compared to the experiment and topas simulations of Muñoz et al. ²⁰ Triangles/dotted lines (circles/dashed lines) represent ${\overline{L}}_{\mathrm{t}}$ (${\overline{L}}_{\mathrm{d}}$) data. (Bottom) Relative error of ${\overline{L}}_{\mathrm{t}}$ (left panel) and ${\overline{L}}_{\mathrm{d}}$ (right panel). Gray bands mark the uncertainty of the experimental values.

The upper panel of Figure 10 shows that simulated ${\overline{L}}_{\mathrm{d}}$ values (circles) show greater variability than the simulated ${\overline{L}}_{\mathrm{t}}$ values (triangles). Moreover, topas yields the highest ${\overline{L}}_{\mathrm{d}}$ values across all depths, whereas penhan considering only primary protons gives the lowest.

For ${\overline{L}}_{\mathrm{t}}$, as shown in the lower panel of Figure 10, the relative errors of all simulations are compatible with the experimental data, except at $z=7.5$ cm, where none of the simulations show compatibility. For ${\overline{L}}_{\mathrm{d}}$, the behavior varies depending on the simulation. The simulation from the original paper has relative errors that are compatible with the experimental error only at $z=12.7$ and 14.1 cm. penhan simulation including all protons shows compatibility with the experimental error at $z=9.6,12.7$, and 14.1 cm. Additionally, at all other points, its relative error is smaller than that of the original simulation. The penhan simulation considering only primary protons is particularly interesting. Its relative error is compatible with the experimental error at all points except $z=7.5$ and 14.5 cm. At these two points, however, its error remains smaller than that of the other two simulations.

At $z=14.5$ cm, the difference between the two penhan simulations is minimal, measuring only 0.01 MeV/mm. As we can see in the lower panel of Figure 10, the ${\overline{L}}_{\mathrm{d}}$ results from both penhan simulations show larger differences between them at lower $z$. This is because nuclear reactions are more important at higher energies (which occur at lower $z$), producing secondary protons that contribute only in the simulation that includes all protons.

4. DISCUSSION

First, a comment about our methodology is due. In order to obtain the spectra, we selected an energy bin equal to 0.25 MeV, following a seminal work on the determination of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$, ¹⁷ and serving as a well‐established reference. But, one drawback of our method is the uncertainty that the chosen energy bin introduces in the determination of the spectra. This uncertainty could be important for protons below 1 MeV, where LET varies significantly, potentially affecting the determination of LET averages around the Bragg peak, especially for ${\overline{L}}_{\mathrm{d}}$. Given this, it is useful to consider how changes in the bin width could influence the spectra. To evaluate this impact, the bin width must be increased or decreased. A larger bin width may be required when the resulting spectra appear noisy or not smooth enough; however, this behavior is not observed in our results (see Figures 7, 8, 9). Reducing the bin width would produce a more detailed spectrum, which is important at the low proton energies where $S(E)$ grows rapidly. In our case, only bins with energy greater than 0.3 MeV were included in our calculations (as explained in Section 2.4), and the steepest increase in $S(E)$ for protons in water occurs below this threshold. Consequently, using a smaller bin width would not affect our results.

4.1. Intercomparison between codes

In Table 3, we see a different behavior for ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$. Each quantity will be analyzed in detail in the following sections and a comparison between them will be presented at the end.

4.1.1. Analysis of ${\overline{L}}_{\mathrm{t}}$

In Figure 2, we saw the ${\overline{L}}_{\mathrm{t}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}$ results for the 160 PB Off configuration obtained with penhan using its default $S(E)$ and the corresponding results from sbethe. It is important to note that the very first point is incompatible. Since we are working with a monoenergetic beam, the energy spectrum at that point is approximately a Dirac delta, ${\mathrm{\Phi }}_{E_i}(z=0)\sim \delta (E_i-160\text{MeV})$. Consequently, the uncertainty in the spectra is very low, and the observed differences can be attributed to $S(E)$.

Table 4 presents the values of $S(160\text{MeV})$, ${\overline{L}}_t(z=0)$, and ${\overline{L}}_d(z=0)$ for both penhan and sbethe and their ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{t}}}(z=0)$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}(z=0)$. It should be noted that $S(E)$ is considered without uncertainty in all the codes, then, the only source of uncertainty in the calculation of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ comes from the uncertainty in the spectra. If we substitute ${\mathrm{\Phi }}_{E_i}(z=0)\approx \delta (E_i-160\text{MeV})$ into Equation (8), we obtain ${\overline{L}}_{\mathrm{t}}\approx {\overline{L}}_{\mathrm{d}}\approx S(160\text{MeV})$. Therefore, the differences of $S(E)$ seen in Table 4 are reflected in the results shown in Figure 2. As seen on the last row of Table 4, these differences are small (below 0.20%).

TABLE 4.

Values of ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ (in MeV/mm) and ${\mathcal{E}}_{\alpha ,\beta }^M(z)$ (in %) for $z=0$ in the case of the configuration 160 PB Off, compared to the corresponding value of the stopping power. The symbol $M$ in ${\mathcal{E}}_{\mathrm{P},\text{SB}}^M(z)$ denotes the magnitude on the column, that is, $S$, ${\overline{L}}_{\mathrm{t}}$, or ${\overline{L}}_{\mathrm{d}}$. Uncertainties are given between parentheses, that is, 0.51951(22) means $0.51951\pm 0.00022$.

	$S(160\text{MeV})$	${\overline{L}}_t(z=0)$	${\overline{L}}_d(z=0)$
penhan	0.519411	0.519422(9)	0.51951(22)
sbethe	0.518571	0.518581(7)	0.51862(6)
${\mathcal{E}}_{\mathrm{P},\text{SB}}^M(z)$	0.161722	0.1618(22)	0.17(4)

Table 3 shows that for the 160 PB Off configuration, $f_{\gamma (1,3)}$ is higher when penhan results are compared to the same code only modifying the $S(E)$ value (i.e., to sbethe) than when they are compared to the other codes (i.e., to fluka or topas). This is consistent with Figure 11, which highlights that the most significant differences in $S(E)$ values occur between penhan and sbethe. Although Figure 11 also shows non‐negligible differences in $S(E)$ values between penhan and fluka or topas, these differences are small enough to result in $f_{\gamma (1,3)}$ $=0$ for the comparisons between penhan and fluka/topas.

FIGURE 11.

Values of liquid water mass $S(E)$ for penhan, sbethe, topas, fluka, and the IAEA experimental database at low energies.

Thus, when computing ${\overline{L}}_{\mathrm{t}}$ without nuclear reactions, variations due to physics models and tracking algorithms are negligible, whereas large discrepancies in $S(E)$ at low energies can induce relatively significant differences. In any case, these differences do not impact $f_{\gamma (1,3)}$ notably.

We now discuss the differences in ${\overline{L}}_{\mathrm{t}}$ between codes when introducing nuclear reactions. In Table 3, we see that $f_{\gamma (1,3)}$ for 160 PB On between penhan and the other two codes increases to approximately 1%. This corresponds to only one point exceeding differences greater than 3%. As seen in Figure 3, this discrepancy occurs at the very last point of the ${\overline{L}}_{\mathrm{t}}$ distribution.

To analyze this case, we plot the proton spectra obtained with penhan, topas, and fluka at that point in Figure 7. We see that the energies for which the spectrum reaches its maximum values are below 10 MeV and in this region penhan presents the highest values, whereas topas and fluka are lower and closely aligned. This discrepancy arises from the different implementations of neutron transport in these codes. After secondary protons, neutrons are the most generated particle by a 160 MeV proton beam in water (see Paganetti et al. ³ and references therein). Secondary protons beyond the Bragg peak are produced through (n,p)‐reactions, which are implemented in topas and fluka but not in penhan, where the process is considered as purely absorptive. The inclusion of these high‐energy, low‐LET secondary protons lowers the resulting ${\overline{L}}_{\mathrm{t}}$.

In Table 3, we observe that for the 160 BB On configuration, $f_{\gamma (1,3)}$ decreases to zero for all combinations. The corresponding graph, Figure 4, shows that the previously incompatible point in the 160 PB On configuration is now compatible. We note that the ${\overline{L}}_{\mathrm{t}}$ values from both fluka and topas have increased. This is due to a reduction in the high‐energy proton spectrum seen earlier. As shown in Figure 8, fluka and topas no longer show the high‐energy proton tail because of the change in beam‐voxel configuration. In this geometry, the high‐energy tail is suppressed because many of these protons, or the neutrons generating them, likely escape the narrower scoring region compared to the PB configuration. Overall, including nuclear reactions does not significantly impact $f_{\gamma (1,3)}$ for ${\overline{L}}_{\mathrm{t}}$.

In summary, we conclude that ${\overline{L}}_{\mathrm{t}}$ is robust against changes in both the $S(E)$ and MC code even when nuclear reactions are included, as $f_{\gamma (1,3)}$ is lower than $7\%$ in all the cases.

4.1.2. Analysis of ${\overline{L}}_{\mathrm{d}}$

Focusing on the $f_{\gamma (1,3)}$ values for ${\overline{L}}_{\mathrm{d}}$ in Table 3 for the 160 PB Off configuration, we observe a similar behavior to ${\overline{L}}_{\mathrm{t}}$. The $f_{\gamma (1,3)}$ values for penhan–sbethe are higher than those for penhan–topas and penhan–fluka. The conclusion remains the same: when nuclear reactions are neglected, the choice of $S(E)$ is more decisive than the choice of the MC code. We must highlight that the $f_{\gamma (1,3)}$ values are higher for ${\overline{L}}_{\mathrm{d}}$ than for ${\overline{L}}_{\mathrm{t}}$ because the first gives more importance to larger values of $S(E)$. In Figure 5, we plot the values of ${\overline{L}}_{\mathrm{d}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$, for this configuration. It is clear that differences between codes are more important for ${\overline{L}}_{\mathrm{d}}$, and higher between penhan and any of the other codes.

Once nuclear reactions are introduced, $f_{\gamma (1,3)}$ increases significantly for all cases. Figure 6 shows ${\overline{L}}_{\mathrm{d}}$ and ${\mathcal{E}}_{\alpha ,\beta }^{{\overline{L}}_{\mathrm{d}}}$, indicating substantial data incompatibility, especially at the entrance of the phantom, where nuclear reactions are more significant. Furthermore, we see that fluka systematically produces lower results for ${\overline{L}}_{\mathrm{d}}$ compared to penhan and topas in this region. Figure 9 illustrates the proton spectrum at $z=1.99$ cm, a point where the differences are notable. It shows that fluka provides a lower contribution of low‐energy proton, which is crucial for ${\overline{L}}_{\mathrm{d}}$ as it corresponds to high values of $S(E)$. A possible reason could be that proton‐induced nuclear reactions on (the most abundant isotope in water for which non‐elastic nuclear reactions can take place) are producing more secondary protons in penhan and topas than in fluka. penhan uses the experimental database from ICRU Report 63, while topas and fluka implement two different theoretical approaches, both based on the intranuclear cascade framework. One explanation may be that fluka models give a lower production of low‐energy protons in nuclear reactions.

When considering a broad beam, $f_{\gamma (1,3)}$ increases in all cases except for the topas–fluka combination. The behavior of ${\overline{L}}_{\mathrm{d}}$ is similar to that shown in Figure 6, but the differences are more pronounced, as the $f_{\gamma (1,3)}$ values indicate.

In summary, we conclude that when calculating ${\overline{L}}_{\mathrm{d}}$, $S(E)$ plays a more decisive role than the choice of the MC code in the absence of nuclear reactions, as in the ${\overline{L}}_{\mathrm{t}}$ case. The inclusion of nuclear reactions introduces significant differences between codes, reaching $f_{\gamma (1,3)}$ of up to 55.91%. These differences are primarily due to the small‐energy proton spectrum, which plays a key role in ${\overline{L}}_{\mathrm{d}}$ calculations.

4.2. Differences in ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ behavior

In Table  3, we see that, when nuclear reactions are introduced, an increase in $f_{\gamma (1,3)}$ is observed in both ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ (albeit with the comments seen above). For ${\overline{L}}_{\mathrm{t}}$, this increase is due to the region beyond the Bragg peak (see Figure 3), where high‐energy protons are produced in all codes except penhan. In contrast, for ${\overline{L}}_{\mathrm{d}}$, discrepancies also arise at the entrance of the phantom, due to the presence of low‐energy protons, which correspond to high LET values. We attribute these differences to the variations in the implementation of nuclear reactions among codes.

The stability and robustness of ${\overline{L}}_{\mathrm{t}}$ are remarkable, with $f_{\gamma (1,3)}$ values always remaining below 7%. This permits us to conclude that ${\overline{L}}_{\mathrm{t}}$ is inherently less sensitive to variations than ${\overline{L}}_{\mathrm{d}}$ as their definition suggest, serving as a more reliable indicator of radiation quality. If proton RBE models are to be developed, their physical foundation must be as sound as possible to ensure that ${\overline{L}}_{\mathrm{t}}$ calculations based on different methodologies lead to consistent or comparable RBE values.

Concerning ${\overline{L}}_{\mathrm{d}}$, we observe $f_{\gamma (1,3)}$ of more than 40% in some case and differences of up to $\approx 15$% between the codes (see Figure 6) when nuclear reactions are activated. In the region where the differences between codes are higher, ${\overline{L}}_{\mathrm{d}}$ values range between 0.5 and 1 MeV/mm, significantly lower than $\approx 15$ MeV/mm observed near the Bragg peak. Non‐constant RBE models were initially proposed to account for the enhanced biological damage beyond the Bragg peak. ⁴⁸ Since the ${\overline{L}}_{\mathrm{d}}$ differences occur at depths before the Bragg peak, they should not pose an important issue when modeling RBE in the distal region. Furthermore, most widespread proton RBE models produce maximum RBE values (defined when the absorbed dose in a single fraction tends to zero, as occurs beyond the Bragg peak) ⁴⁹ that are linear in ${\overline{L}}_{\mathrm{d}}$. ⁵⁰ , ⁵¹ Hence variations of the extent seen in Figure 6 in the Bragg peak region (around 3%) will propagate to RBE and may need to be considered along with other sources of uncertainty in RBE modeling.

4.3. Comparison with experiment

In Figure 10, we showed that the penhan result with only primary protons produced the lowest relative error with respect to the experiment for ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ of all simulations (at most data points). The better agreement of this simulation compared to the experiment could be due to limitations in the experimental setup. As discussed in the original work, ²⁰ protons produced through nuclear reactions or hard elastic scattering may not be adequately detected due to their minor contribution to the proton spectrum.

Even if future experiments were able to measure this region, discrepancies between the simulations including all particles and the experimental data (particularly in ${\overline{L}}_{\mathrm{d}}$) are expected due to the sensitivity of this low‐energy proton component.

5. CONCLUSIONS

In this work, we have studied the two most commonly used LET averages, and ${\overline{L}}_{\mathrm{d}}$, using the MC codes penhan, topas MC, and CERN fluka. We have observed a better agreement between the results of ${\overline{L}}_{\mathrm{t}}$ for the different configurations considered, which supports its use as an indicator of radiation quality. Furthermore, it has been found that, even if ${\overline{L}}_{\mathrm{d}}$ presents a higher degree of disagreement between codes, as the $f_{\gamma (1,3)}$ values indicate, it should not be an issue when modeling RBE, since most of the discrepancies are minor and occur at the beginning of the phantom, where RBE is generally lower.

Moreover, when only primary protons are included in the spectra, the experimental data on ${\overline{L}}_{\mathrm{t}}$ and ${\overline{L}}_{\mathrm{d}}$ in Muñoz et al. ²⁰ are more accurately reproduced, highlighting the robustness and reliability of the simulations developed. Further improvements are needed in the experimental detection of low‐energy protons in order to validate the simulations considering the full proton spectra.

CONFLICT OF INTEREST STATEMENT

The authors have no relevant conflicts of interest to disclose.