Relative Biological Effectiveness (RBE) of Monoenergetic Protons: Comparison of Empirical and Biophysical Models

Abstract

A constant proton relative biological effectiveness (RBE) of 1.1 for tumor control is currently used in proton therapy treatment planning. However, in vitro, in vivo and clinical experiences indicate that proton RBE varies with kinetic energy and, therefore, tissue depth within proton Bragg peaks. A number of published RBE models capture variations in proton RBE with depth. The published models can be sub-divided into empirical (or phenomenological) and biophysical (or mechanistic-inspired) RBE models. Empirical RBE models usually characterize the beam quality through the dose-averaged linear energy transfer (${\mathrm{LET}}_{\mathrm{D}}$), while most biophysical RBE models relate RBE to the dose-averaged lineal energy ($y_{\mathrm{D}}$). In this work, an analytic microdosimetry model and the Monte Carlo damage simulation code (MCDS) were utilized for the evaluation of the ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ of monoenergetic proton beams in the clinically relevant energy range of 1–250 MeV. The calculated ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values were then used for the estimation of the RBE for five different cell types at three dose levels (2 Gy, 5 Gy and 7 Gy). Comparisons are made between nine empirical RBE models and two biophysical models, namely, the theory of dual radiation action (TDRA) and the microdosimetric kinetic model (MKM). The results show that, at conventional dose fractions (~2 Gy) and for proton energies which correspond to the proximal and central regions of the spread-out Bragg peak (SOBP), RBE varies from 1.0 to 1.2. At lower proton energies related to the distal SOBP, we find significant deviations from a constant $\mathrm{RBE}$ of 1.1, especially for late-responding tissues (low ${(\alpha /\beta )}_R$ of ~1.5–3.5 Gy) where proton RBE may reach 1.3 to 1.5. For hypofractionated dose fractions (5–7 Gy), deviations from a constant $\mathrm{RBE}$ of 1.1 are smaller, but may still be sizeable, yielding RBE values between 1.15 and 1.3. However, large discrepancies among the different models were observed that make the selection of a variable RBE across the SOBP uncertain.

1. Introduction

Treatment planning in proton beam therapy (PBT) often relies on clinical experiences from conventional photon therapy to set the tumor prescription and clinical goals for organ-at-risk (OAR) tolerance dose [1,2]. Despite well-documented differences in proton linear energy transfer (LET) with decreasing kinetic energy, we continue to use a constant, spatially invariant relative biological effectiveness (RBE) when comparing proton radiation therapy (RT) plans to low-LET X-ray treatment plans, which are spatially invariant with depth in-field and out-of-field.

To more fully exploit the potential of PBT, patient-specific plan optimization needs to consider the well-documented differences in proton LET and RBE with depth on- and off-axis. A constant proton RBE of 1.1, which is commonly used in treatment planning [3,4], may be sufficient in many clinical settings, although plans optimized with a constant, spatially invariant RBE are most likely sub-optimal in terms of the therapeutic ratio for PBT, i.e., overestimating the RBE-weighted dose (RWD) on target while underestimating the RWD to OAR of most concern [5,6]. Published studies imply significant deviations from a constant RBE of 1.1 [1,7,8,9,10,11,12]. These studies provide evidence that RBE is not spatially invariant—rather, it varies within the target volume depending on several factors, including target size, treatment delivery technique and other physical and biological parameters [2].

RBE values that exceed 1.1 at the end of a proton track (i.e., low-energy protons distal to the Bragg peak) are not uncommon in PBT, due to the change in the radiation quality of the beam [13,14,15]. Radiation quality (type and energy of radiation) leads to distinct spatial patterns of energy depositions in (sub) cellular dimensions (DNA helix, chromosomes, cell nucleus) that in turn change the relative effectiveness of a dose of one type of radiation relative to another [14,16]. These differences in the biological effectiveness of different types of radiation are often quantified relative to the linear energy transfer (LET) of the relevant charged particle(s). LET is a measure of the ionization density per unit distance traveled by an ion in matter, and tends to increase with decreasing ion kinetic energy. Therefore, ions at the end of their path exhibit a higher LET and are expected to be more effective in causing localized DNA damage and cell death than lower LET radiations [13,17,18,19]. The adoption of a constant, spatially invariant proton RBE of 1.1 runs counter to fundamental physics and a large body of published in vitro, in vivo and clinical evidence [1,9,10,11].

Several proton RBE models have been developed in recent years that attempt to incorporate tissue-specific and variable-with-LET RBE into PBT treatment planning. Most proton RBE models are empirical (or phenomenological) and derived from linear–quadratic (LQ) model fits to measured data for in vitro cell survival [20]. In contrast, biophysical (or mechanistic-inspired) models, like the theory of dual radiation action (TDRA) and the microdosimetric kinetic model (MKM) [18,21], seek a connection between the stochastic energy deposition pattern on microscopic (subcellular) target volumes and reproductive cell survival [18]. Other biophysical approaches, like the Repair–Misrepair Fixation (RMF) model, relate reproductive cell death to the pairwise interaction of DSB from an independently validated Monte Carlo model, thus using microdosimetric concepts indirectly [17,22,23]. The TDRA and MKM models employ the microdosimetric (i.e., stochastic) analogue of LET, lineal energy (y) [24]. Microdosimetric quantities are preferable to non-stochastic metrics, such as LET, because they account for energy-loss straggling and the finite range of δ-rays [15,25,26,27]. Calculations of these quantities are typically carried out by Monte Carlo track structure simulations (MCTSs). However, among other things, such simulations require significant processing power, and they are often time-consuming for direct microdosimetric calculations in PBT planning and patient-specific optimization. Therefore, various (semi) analytical microdosimetry methods have been proposed as alternatives to full-scale MCTSs [27,28,29,30,31].

In this study, an established analytic microdosimetry model is utilized to calculate the dose-averaged lineal energy ($y_{\mathrm{D}}$) and the dose-averaged linear energy transfer (${\mathrm{LET}}_{\mathrm{D}}$) of monoenergetic proton beams with energies from 1 to 250 MeV. The calculated ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values are used for the estimation of RBE for five different cell types based on nine empirical RBE models and two biophysical RBE models (TDRA and MKM). Comparisons between an analytic approach to estimate $y_{\mathrm{D}}$ and ${\mathrm{LET}}_{\mathrm{D}}$ and the fast algorithm used in the MCDS code are presented.

2. Materials and Methods

2.1. Lineal Energy (y)

To describe radiation effects in subcellular structures (like DNA, nucleosomes, or chromosomes), microdosimetric quantities must be used, since they consider the finite range of secondary ejected electrons and the energy-loss straggling effect [32]. A widely used microdosimetric quantity to characterize radiation quality is the lineal energy (y), which is defined as the ratio of the imparted energy in a volume from a single event (ε₁) (i.e. from the primary particle and all its secondary particles) by the mean chord length ($\overline{l}$) of that volume. Due to the stochastic nature of energy deposition, biophysical RBE models typically use the fluence-averaged lineal energy ($y_{\mathrm{F}}$) and the dose-averaged lineal energy ($y_{\mathrm{D}}$) given by

(1)$y_F={{\displaystyle \int }}_0^{+\infty }y f(y) dy,$

(2)$y_{\mathrm{D}}={\displaystyle \frac{{{\displaystyle \int }}_0^{+\infty }{y}^2 f(y)dy}{{{\displaystyle \int }}_0^{+\infty }y f(y) dy}}={\displaystyle \frac{1}{y_F}}{{\displaystyle \int }}_0^{+\infty }{y}^2 f(y)dy,$

where $f\left(y\right)$ is the probability distribution of y.

2.2. Calculations of LET_D

Empirical models use the macroscopic quantity LET to characterize the radiation quality. In practice, LET is commonly calculated through the electronic stopping power ($S_{el}$), which is defined as the mean energy loss of a charged particle per unit path length due to electronic collisions. For a clinical beam which consists of many particles with different energies, the fluence-averaged LET (${\mathrm{LET}}_{\mathrm{F}}$) or the dose-averaged LET (${\mathrm{LET}}_{\mathrm{D}}$) are used instead, and can be calculated as follows:

(3)${\mathrm{LET}}_{\mathrm{F}}={{\displaystyle \int }}_0^{+\infty }{S}_{el}\left(E\right) \phi \left(E\right) dE,$

(4)${\mathrm{LET}}_{\mathrm{D}}={\displaystyle \frac{{{\displaystyle \int }}_0^{+\infty }{S}_{el}^2\left(E\right) \phi \left(E\right)dE}{{{\displaystyle \int }}_0^{+\infty }{S}_{el}(E) \phi \left(E\right) dE}}={\displaystyle \frac{1}{{\mathrm{LET}}_{\mathrm{F}}}}{{\displaystyle \int }}_0^{+\infty }{S}_{el}^2\left(E\right) \phi \left(E\right)dE,$

where $\phi \left(E\right)$ represents the local energy spectrum. For monoenergetic proton beams, the energy spectrum is narrow; therefore, ${\mathrm{LET}}_{\mathrm{F} }$ ≈ ${\mathrm{LET}}_{\mathrm{D}}$ [33]. In clinical applications, there exist various methodological approaches for calculating average LET (i.e., ${\mathrm{LET}}_{\mathrm{F}}$ or ${\mathrm{LET}}_{\mathrm{D}}$) which are based either on experimental data and analytic models or the interpretation and analysis of measurements through Monte Carlo simulations. For assessments of biological radiation damage, it is the dose-averaged LET (${\mathrm{LET}}_{\mathrm{D}}$) that is most relevant. Different methods of interpretation can substantially influence the reported values of average LET. One significant difference among these methods lies in the types of particles that they consider in the calculation of ${\mathrm{LET}}_{\mathrm{F}}$ or ${\mathrm{LET}}_{\mathrm{D}}$. Some studies take into account only the primary particles, while others include the secondary ejected electrons, secondary protons or/and heavier particles produced from nuclear reactions, like hydrogen or helium isotopes [34,35]. Other differences arise from the choice of medium for calculating the average LET. Most calculations pertain to liquid water, and therefore using other tissue-equivalent material may slightly affect the LET spectrum. If other non-unit density materials are examined, the corresponding averaged LET values must be normalized to unit density in order to be suitable for clinical applications. Additionally, if MC codes are employed, discrepancies arise, since the different simulation procedures and physics lists vary among the different MC codes and impact the calculations of LET [34,35].

To eliminate the methodological uncertainties surrounding the calculation of ${\mathrm{LET}}_{\mathrm{D}}$ [34,35], which will further obscure the influence of the choice between ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$, in the present work we determine ${\mathrm{LET}}_{\mathrm{D}}$ from $y_{\mathrm{D}}$ using the approach of Kellerer [33], which was further refined by Xapsos [36].

Microdosimetric Calculation of LET_D

For a first approximation, the average energy deposited inside a microscopic volume from an ion that crosses it (usually referred to as a direct event) depends on the LET of the incident ion and the average chord length of the volume ($\overline{l}$):

(5)$\overline{\epsilon }=\overline{\mathrm{LET}} \overline{l} .$

To account for the fraction of the ion’s energy loss that is deposited outside the volume due to the escape of secondary electrons, the above expression was refined by Xapsos as [28,29,37]:

(6)${\overline{\epsilon }=f}_{ion}\overline{\mathrm{LET}} \overline{l} ,$

where the coefficient ($f_{ion}$) represents the fraction of the ion’s energy loss that is imparted in the volume of interest.

Fluctuations of the deposited energy can be observed due to LET variations or the different path lengths that the particles may follow inside of the volume of interest. However, ions crossing the volume with the same LET and path length may still deposit different energy values due to the energy loss straggling. It follows from Kellerer’s investigation [33] that the relative variance of the imparted energy ($V_{\epsilon }$) is:

(7)$V_{\epsilon }{=V}_{\mathrm{LET}}{+V}_l{+V}_{\mathrm{LET}}{V}_l{+V}_{\delta },$

where $V_{\mathrm{LET}}$ is the relative variance of an ion’s LET, $V_l$ is the relative variance of the ion’s path length and $V_{\delta }$ is the relative variance of the energy loss straggling. The relative variances are additive and the additional term $V_{\mathrm{LET}}{V}_l$ enhances the combined effect of LET and path-length variations [33].

The relative variance of a random variable ($x$) can be expressed as:

(8)$V_x={\displaystyle \frac{{\sigma }_x^2}{{\overline{x}}^2}}={\displaystyle \frac{x_{\mathrm{D}}}{\overline{x}}}-1,$

where ${\sigma }_x$ is the standard deviation of the variable, $\overline{x}$ is the mean value and $x_{\mathrm{D}}$ the weighted average. If $V_{\epsilon }$, $V_{\mathrm{LET}}$ and $V_l$ are replaced in Equation (7) from Equation (8) and the relative variance of the energy loss straggling is taken as: [33],

(9)$V_{\delta }={\displaystyle \frac{{\delta }_2}{\overline{\epsilon }}},$

where ${\delta }_2$ is the dose-averaged deposited energy by the particle in one collision, it follows that

(10)${\displaystyle \frac{{\epsilon }_{\mathrm{D}}}{\overline{\epsilon }}}-1={\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{\overline{\mathrm{LET}}}}-1+{\displaystyle \frac{l_{\mathrm{D}}}{\overline{l}}}-1+\left({\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{\overline{\mathrm{LET}}}}-1\right)\left({\displaystyle \frac{l_{\mathrm{D}}}{\overline{l}}}-1\right)+{\displaystyle \frac{{\delta }_2}{\overline{\epsilon }}}.$

If the dose-averaged lineal energy is taken as $y_{\mathrm{D}}{=\epsilon }_{\mathrm{D}}/\overline{l}$, and by using the mean imparted energy from Equation (6), an expression that relates ${\mathrm{LET}}_{\mathrm{D}}$ with $y_{\mathrm{D}}$ can be obtained:

(11)${\mathrm{LET}}_{\mathrm{D}}={\displaystyle \frac{1}{f_{ion}{ l}_{\mathrm{D}}}}(\overline{l} y_{\mathrm{D},direct}-{\delta }_2).$

Equation (11) is valid under the short track-segment condition, i.e., when the dimensions of the volume of interest is small enough that the ion’s LET can be considered constant along its entire path within the volume [33]. The evaluation of $y_{\mathrm{D},direct}$, ${\delta }_2$ and $f_{ion}$ are described in the next section.

2.3. Analytic Calculation of y_D

In the present work, the calculation of $y_{\mathrm{D}}$ relies upon the application of an analytical microdosimetry model proposed by Xapsos and colleagues [28,29], which has been exploited in space applications [31,38,39]. This model aims to estimate the distribution of deposited energy inside microscopic volumes, taking into account both direct and indirect events. The term ‘direct event’ refers to an ion that crosses the target and deposits an amount of energy inside of it, while the term ‘indirect event’ refers to the energy deposition in the target from δ rays that are produced from an ion that misses the target. The imparted energy ($\epsilon$) from a particle that crosses the target, traversing a path length ($l$) inside of it, is described by a probability density function $f\left(\epsilon \right)$. This function can be expressed as a convolution of an energy loss straggling distribution, $\mathrm{p}(\epsilon ,l)$, with a path length distribution, $c\left(l\right)$ [29]:

(12)$f\left(\epsilon \right)=\int \mathrm{p}\left(\epsilon ,l\right) c\left(l\right) dl.$

If the particle’s path is considered a straight line, $c\left(l\right)$ expresses the chord length distribution, which is a geometrical parameter known for some simple geometries [40,41,42]. For spherical volumes of interest, with the diameter d, the chord length distribution is given by:

(13)$c\left(l\right){=2l/d}^2,$

while the distribution of energy loss straggling, in this model, is approximated by a log-normal distribution [29]:

(14)$\mathrm{p}(\epsilon ,l)={\displaystyle \frac{1}{\sqrt{2}{\pi \sigma }_{\epsilon }\epsilon }}exp\left[-{\left({\displaystyle \frac{ln\left(\epsilon \right)-{\mu }_{\epsilon }}{\sqrt{2}{\sigma }_{\epsilon }}}\right)}^2\right],$

where ${\mu }_{\epsilon }$ is the mean value of the distribution and ${\sigma }_{\epsilon }$ is the standard deviation which are related to the mean deposited energy to the target ($\overline{\epsilon }$) and its relative variance ($V_{\epsilon }$) according to the expressions [29]:

(15)${\mu }_{\epsilon }=ln\left(\overline{\epsilon }\right)-0.5{\sigma }_{\epsilon }^2,$

(16)${\sigma }_{\epsilon }=\sqrt{{ln(1+V}_{\epsilon })}.$

In the case of direct events, the mean deposited energy and its relative variance can be expressed through Equations (6) and (7), respectively, as:

(17)${\overline{\epsilon }}_{ion}{=f}_{ion}{\overline{\mathrm{LET}}}_{ion}\overline{l },$

(18)${\mathrm{V}}_{\epsilon ,ion}{=V}_l{+V}_{\delta ,ion},$

considering that, for monoenergetic proton beams, the LET fluctuations are negligible ($V_{\mathrm{LET}} = 0$). The relative variance of an ion’s path length is calculated by Equation (8) as:

(19)$V_l={\displaystyle \frac{l_D}{\overline{l}}}-1,$

where $l_D$ is the weighted average chord length (for spherical targets, 3d/4), $\overline{l}$ is the mean chord length (for spherical targets, 2d/3) [33], and the relative variance of energy loss straggling is given by Equation (9), where ${\delta }_{2,ion}$ is calculated from [29]:

(20)${\delta }_{2,ion}{=A \Delta }^{{\rm B}}.$

The parameter Δ is the cut-off energy which represents the energy of secondary electrons with a range equal to the mean chord length ($\overline{l}$) of the target, and the parameters A and Β are constants that depend on the type of the particle and the material [29]. The cut-off energy of secondary electrons in liquid water spheres with a diameter of 1 μm was found by Kyriakou et al. [43] to be Δ = 5.562 keV, while the parameters A = 0.195 and B = 0.610 were found empirically from Papadopoulos et al. [27] by fitting Equation (20) to data derived from the Geant4-DNA code.

The parameter $f_{ion}$ can be calculated from [37]:

(21)$f_{ion}={\displaystyle \frac{ln\left(\frac{T_{el,max}\left({\Delta +\Delta }_1{+\Delta }_2\right)}{{{\rm I}}^2}\right)}{2ln\left(\frac{T_{el,max}}{I}\right)}},$

where $T_{el,max}{=4 (m}_{el}{/m}_{ion}{) T}_{ion}=2.179 T_{ion}$ is the maximum energy of a secondary electron (with the mass $m_{el}$) that is produced from a collision with a proton (with the mass $m_{ion}$) of energy $T_{ion}$, Ι is the mean excitation energy of the medium, for which the ICRU (International Commission on Radiation Units and Measurements) recommends a value of 0.078 keV for liquid water [44], and the parameters ${\Delta }_1=\Delta (1-{\displaystyle \frac{\Delta }{T_{el,max}}})$ and ${\Delta }_2=I(1-{\displaystyle \frac{\Delta }{T_{el,max}}})$ are correlated with the deposited energy in the target from secondary electrons that are produced inside the volume but escape from it.

For indirect events, the mean deposited energy and its relative variance are given, similar to direct events, by the subsequent equations, respectively:

(22)${\overline{\epsilon }}_{el}={\overline{\mathrm{LET}}}_{el}\overline{l },$

(23)$V_{\epsilon ,el}{=V}_{\mathrm{LET},el}{+V}_{l }{+V}_{\mathrm{LET},el}{V}_l{+V}_{\delta ,el}.$

In this case, we assumed that secondary electrons do not produce higher-order electrons (tertiary, etc.) that are capable of escaping the target, so the corresponding $f_{el}$ term in Equation (22) is absent. Furthermore, unlike the direct events, the produced secondary electrons are not monoenergetic, but have an energy spectrum. The initial spectrum is further changed due to the slowing-down process. Therefore, the relative variance of the electron’s LET in Equation (23) will be non-zero and it can be evaluated through Equation (8) as:

(24)${\mathrm{V}}_{\mathrm{LET},el}={\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D},el}}{{\overline{\mathrm{LET}}}_{el}}}-1.$

For the calculations of the ${\overline{\mathrm{LET}}}_{el}$ and ${\mathrm{LET}}_{\mathrm{D},el}$, we must take into account the full slowing-down spectrum. For an initially monoenergetic electron beam with the energy $T_{el,0}$, the average LET due to the slowing-down process will be [28]:

(25)${\left.{\overline{\mathrm{LET}}}_{el}\right|}_{T_{el,0}}={\displaystyle \frac{{{\displaystyle \int }}_{\Delta }^{T_{el,0}}{\mathrm{LET}}_{el}\left(T_{el}\right)\frac{1}{{\mathrm{LET}}_{el}\left(T_{el}\right)}{dT}_{el} }{{{\displaystyle \int }}_{\Delta }^{T_{el,0}}\frac{1}{{\mathrm{LET}}_{el}\left(T_{el}\right)}{dT}_{el} }}.$

Equation (25) assumes that the slowing-down spectrum is inversely proportional to the electron’s LET [28]. Then, the ${\left.{\overline{\mathrm{LET}}}_{el}\right|}_{T_{el,0}}$ must be weighted over the initial energy spectrum that the primary proton produces, which is inversely proportional to the square of $T_{el,0}$ [28]:

(26)${\overline{\mathrm{LET}}}_{\mathrm{el}}={\displaystyle \frac{{{\displaystyle \int }}_{\Delta }^{T_{el,max}}{\left.{\overline{\mathrm{LET}}}_{el}\left(T_{el,0}\right)\right|}_{T_{el,0}}\frac{1}{T_{el,0}^2}{dT}_{el,0} }{{{\displaystyle \int }}_{\Delta }^{T_{el,max}}\frac{1}{T_{el,0}^2}{dT}_{el,0} }}.$

Due to the wide electron energy spectrum, the average LET (${\overline{\mathrm{LET}}}_{el}$) can significantly differ from the dose-averaged LET (${\mathrm{LET}}_{\mathrm{D},el}$), which can be given by the fitted expression proposed by Xapsos et al. [28]:

(27)${\mathrm{LET}}_{\mathrm{D},el}=0.925 {\mathrm{LET}}_{el}\left(\Delta +0.05\right), \Delta \ge \mathrm{I} .$

The relative variance of a particle’s path length ($V_l$) and the energy deposited from a single collision ($V_{\delta ,el}$) will be the same as those for direct events, with the ${\delta }_2$ parameter for indirect events (${\delta }_{2,el}$) evaluated from the Mott scattering-based formula [45] as:

(28)${\delta }_{2,el}={\displaystyle \frac{0.2105 {\overline{{\rm T}}}_{el}}{ln\left(\frac{{\overline{{\rm T}}}_{el}}{2I}\right)-0.193}}, {\overline{{\rm T}}}_{el}\ge 2 \mathrm{keV}$

where ${\overline{{\rm T}}}_{el}$ is the mean energy of secondary electrons, which is approximated by the following empirical formula given by Xapsos et al. [28]:

(29)${\overline{{\rm T}}}_{el}=1.25 T_{ion}^{0.229}{ \Delta }^{0.778+0.00142 T_{ion}}, \Delta \le {\overline{{\rm T}}}_{el}\le T_{el,max}.$

Equations (13)–(29) can be utilized to determine the probability density function $f\left(\epsilon \right)$ for both direct and indirect events through Equation (12). Then, by inserting $f\left(\epsilon \right)$ into Equation (2), the total $y_{\mathrm{D}}$, defined as the weighted sum of the direct and indirect $y_{\mathrm{D}}$, can be expressed as [28]:

(30)$y_{\mathrm{D}}{=f}_{ion}{\times y}_{\mathrm{D},direct}+\left(1-f_{ion}\right){\times y}_{\mathrm{D},indirect}.$

The above procedure was implemented in an in-house code written in Mathematica v12.2 code for fast calculations of $y_{\mathrm{D}}$.

2.4. LET_D and y_D from MCDS

The MCDS was originally developed to estimate the clustering of DNA lesions to form SSB, DSB and other complex types of DNA damage [46,47]. Version 3.10A of the MCDS introduced the capability of calculating microdosimetric quantities (e.g., lineal energy and specific energy) for ions passing through a water (or a water-equivalent) medium via a deterministic algorithm that accounts for changes in energy loss and finite ion range within micron-scale targets [17,48]. The irradiation geometry used in the MCDS corresponds to protons (or other ions) randomly entering a spherical target surrounded by a vacuum with options to consider a spherical layer of water (e.g., cell cytoplasm) or a water-equivalent layer representing the bottom of a cell culture dish. The protons are assumed to travel in a straight line until their kinetic energy reaches zero. The algorithm accounts for both crossers and stoppers, i.e., protons that spend either part or all of their kinetic energy within the target volume, respectively, as well as for changes in proton stopping power within the target volume. However, MCDS neglects energy-loss straggling, so, for monoenergetic beams, the equality ${\mathrm{LET}}_{\mathrm{F}}$ = ${\mathrm{LET}}_{\mathrm{D}}$ holds. For polyenergetic beams, the MCDS computes the dose-averaged LET averaged over the incident ion fluence. The MCDS algorithm does not account for δ-rays produced by ions within or external to the target. For conditions in which particle equilibrium holds, $y_{\mathrm{F}}={\mathrm{LET}}_{\mathrm{F}}$ and $y_{\mathrm{D}}={9/8 y}_{\mathrm{F}}$ hold. When stoppers become significant (the range is less than the diameter of the target), which the MCDS accounts for, $y_{\mathrm{F}} \le { \mathrm{LET}}_{\mathrm{F}}$ and $y_{\mathrm{D}}\le {9/8 y}_{\mathrm{F}}$.

2.5. Empirical RBE Models

The majority of proton RBE models in PBT are empirical. Most of them are based on the well-established linear–quadratic (LQ) model, which is widely used in clinical practice for the comparison of different fractionation protocols [49]. Using the LQ formalism, the following general expression for the RBE is [50]:

(31)${\mathrm{RBE}}_{\mathrm{LQ}}={\displaystyle \frac{1}{{2D}_P}}\left[\sqrt{{\left({\displaystyle \frac{\alpha }{\beta }}\right)}_R^2+4{\left({\displaystyle \frac{\alpha }{\beta }}\right)}_R{D}_P{\mathrm{RBE}}_{\mathrm{LD}}{+4\mathrm{RBE}}_{\mathrm{HD}}^2{D}_P^2}-{\left({\displaystyle \frac{\alpha }{\beta }}\right)}_R\right],$

where $D$ is the dose per fraction, $\alpha$ and $\beta$ are the linear and quadratic radiosensitivity coefficients of the LQ-model, respectively, ${\mathrm{RBE}}_{\mathrm{LD}}{=\alpha }_P{/\alpha }_R$ is the low-dose RBE (when the dose tends to zero) and ${\mathrm{RBE}}_{\mathrm{HD}}=\sqrt{{\beta }_P{/\beta }_R}$ is the high-dose RBE [51]. The subscripts P and R refer to protons and the reference radiation, respectively (in practice, R is usually γ-rays from Co-60 or Cs-137 sources).

Empirical models only differ in terms of how they evaluate ${\mathrm{RBE}}_{\mathrm{LD}}$ and ${\mathrm{RBE}}_{\mathrm{HD}}$. Rorvik et al. [52] and McNamara et al. [20] have provided comprehensive reviews of available empirical RBE proton models. Table 1 presents a summary of these models. The model of Belli et al. [53] is not included in Table 1, since it uses the dose-weighted energy spectrum of the beam $d\left(\mathrm{E}\right)$ as a radiation quality index, instead of ${\mathrm{LET}}_{\mathrm{D}}$, making the evaluation of RBE more complex and outside of the scope of the present work. Instead, Tian et al.’s [54,55] model was added, although it uses $Z^2/E$ as a beam quality index (instead of LET), where $Z$ is the ion’s charge and $E$ is the ion’s kinetic energy per nucleon. Peeler’s model [56] is also not examined as it assumes a cubic dependence of ${\mathrm{RBE}}_{\mathrm{LD}}$ and ${\mathrm{RBE}}_{\mathrm{HD}}$ on ${\mathrm{LET}}_{\mathrm{D}}$, which is not consistent with the literature. All the remaining models assume an (almost) linear dependence of ${\mathrm{RBE}}_{\mathrm{LD}}$ on ${\mathrm{LET}}_{\mathrm{D}}$ (see Table 1). Regarding ${\mathrm{RBE}}_{\mathrm{HD}}$, some models assume the same ${\mathrm{LET}}_{\mathrm{D}}$ dependence as for ${\mathrm{RBE}}_{\mathrm{LD}}$, while others take it as a constant equal to 1.

Table 1

Summary of empirical models examined in the present work (alphabetic order). Adapted from [14].

Model	Cell Lines (# of Data Points)	${\mathbf{RBE}}_{\mathbf{LD}}$	${\mathbf{RBE}}_{\mathbf{HD}}$	Parameter Values
				${\mathit{p}}_0$	${\mathit{p}}_1$	${\mathit{p}}_2$	${\mathit{p}}_3$
Carabe [57]	V79(44)	$p_0{+p}_1{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	$p_2{+p}_3{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	0.843	0.413644	1.09	0.01612
Chen [58]	V79(14)	${\displaystyle \frac{p_0}{{\alpha }_R}}+{\displaystyle \frac{{1-e}^{-p_1{\mathrm{LET}}_{\mathrm{D}}^2}}{{\alpha }_R{p}_2{\mathrm{LET}}_{\mathrm{D}}}}$	1	0.1	0.0013	0.045	-
Jones [59]	Multiple(28)	$1+{\displaystyle \frac{\left({\mathrm{LET}}_{\mathrm{D}}-0.22\right)}{30.28}}\times \left({\displaystyle \frac{{\alpha }_u}{{\alpha }_R}}-1\right)$	$\sqrt{1+{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}-0.22}{30.28}} \left({\displaystyle \frac{{\beta }_u}{{\beta }_R}}-1\right)}$	${\alpha }_u=2.696 \left(1-e^{-3{.92\alpha }_R}\right)$ ${\beta }_u=0.06 \left(1-e^{-{50\beta }_R}\right)$
McNamara [60]	Multiple(285)	$p_0{+p}_1{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	$p_2+p_3\sqrt{{\left({\displaystyle \raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}\right)}_R}{\mathrm{LET}}_{\mathrm{D}}$	0.99064	0.35605	1.1012	−0.0039
Tian [54]	Multiple(137/48)	$p_0{+p}_1{\displaystyle \frac{1}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}{\displaystyle \frac{Z^2}{E}}$	1	1	15.5	-	-
Tilly [61]	Multiple(11)	$p_0+p_1{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	1	1	0.309 or 0.550964	-	-
Rorvik [62]	Multiple(85)	$p_0+p_1{\displaystyle \frac{{\mathrm{LET}}_{\mathrm{D}}}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	1	1	0.645	-	-
Wilkens [63]	V79(19)	${\displaystyle \frac{p_0+p_1{\mathrm{LET}}_{\mathrm{D}}}{{\alpha }_R}}$	1	0.1	0.02	-	-
Wedenberg [64]	Multiple(19)	$p_0{+p}_1{\displaystyle \frac{{\mathrm{LET}}_D}{{\left(\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}_R}}$	1	1	0.434	-	-

2.6. Biophysical RBE Models

2.6.1. Theory of Dual Radiation Action (TDRA)

A microdosimetric expression for RBE may be derived from the theory of dual radiation action (TDRA) [65]. According to the ‘site’ version of TDRA, any lethal lesion that may lead to a biological effect is caused by the combination of two sub-lesions that are produced in the same site. The number of sub-lesions is proportional to the specific energy ($z$) in the site. A site refers to the average (sub) cellular volume in which a combination of sub-lesions (leading to lethal lesions) may occur. According to TDRA’s site version, RBE may be expressed as follows [66]:

(32)${\mathrm{RBE}}_{\mathrm{TDRA}}={\displaystyle \frac{\sqrt{z_{D,R}^2{+4 D}_P{(z}_{D,P}{+D}_P)}-{ z}_{D,R}}{{2D}_P}},$

where $z_D$ is the dose-averaged specific energy which is related to $y_{\mathrm{D}}$ by (assuming spherical site) [67]:

(33)$z_D={\displaystyle \frac{\overline{l}}{m}} y_{\mathrm{D}}={\displaystyle \frac{y_{\mathrm{D}}}{{\pi \rho r}^2}},$

with $m$ being the mass of the site, $\rho$ the density of the medium and $r$ the radius of the site. If a density of 1 g/cm³ is assumed for the subcellular domains, the above equation can be simplified as [67]:

(34)$z_D={\displaystyle \frac{0.204}{{4r}^2}} y_{\mathrm{D}},$

where $y_{\mathrm{D}}$ is expressed in keV/μm and $z_D$ in Gy.

2.6.2. Microdosimetric–Kinetic Model (MKM)

A more advanced biophysical model, which is based on the TDRA, is the microdosimetric–kinetic model (MKM) developed by Hawkins [68,69,70]. Contrary to the TDRA, the MKM takes into account the cellular repair, as well as the mechanisms that are responsible for the conversion of a single sub-lesion or a pair of sub-lesions into a lethal lesion. More specifically, in the MKM it is assumed that the radiation effects can be caused by two different types of lesions, type I lesions and type II lesions, each of which is produced with a rate proportional to the specific energy ($z$) in a subcellular volume of interest, called the ‘domain’. Type I lesions are lethal (and unrepairable) while type II lesions are potentially lethal if left unrepaired. It can be shown that the average number of lethal lesions per cell is a linear–quadratic function of the absorbed dose ($D$), so the MKM uses the LQ formalism Equation (31). The linear and quadratic parameters of the LQ model are expressed, according to the MKM, as ${\alpha =\alpha }_0{+z}_D{ \beta }_0$ and ${\beta =\beta }_0$, where ${\alpha }_0$ and ${\beta }_0$ are the LQ model coefficients in the limit of zero LET. Then, the ${\mathrm{RBE}}_{\mathrm{LD}}$ and ${\mathrm{RBE}}_{\mathrm{HD} }$ of the MKM are given by the following expressions:

(35)${\mathrm{RBE}}_{\mathrm{LD}, \mathrm{MKM}}={\displaystyle \frac{a_P}{a_R}}=1+{\displaystyle \frac{0.204}{{4r}^2}}{\displaystyle \frac{\left(y_{D,P}-y_{D,R}\right)}{{\left(\alpha /\beta \right)}_R}} ,$

(36)${\mathrm{RBE}}_{\mathrm{HD}, \mathrm{MKM}}=\sqrt{{\displaystyle \frac{{\beta }_P}{{\beta }_R}}}=1.$

3. Results

For the evaluation of proton RBE by the above models, some cell-specific parameters must be known. Specifically, the $\alpha$ and $\beta$ radiosensitivity parameters of the LQ model for the reference radiation (${\alpha }_R$ and ${\beta }_R$) for human tumor cell lines were taken from Chapman et al. [71] and are shown in Table 2. However, the radiosensitive parameters for prostate carcinoma were replaced by those proposed by Brenner and Hall [72], which are the generally accepted values. Also, we added the V79 cell line, as it is widely used in experiments and some of the empirical models of Table 1 are based solely on this cell line.

A general problem with the microdosimetric models is the lack of consensus on the radius of the spherical target. The experimental measurements of $y_{\mathrm{D}}$ are commonly performed by tissue-equivalent proportional counters (TEPCs) with spherical volumes of 1–3 μm in diameter [66,67]. On the other hand, most theoretical calculations of $y_{\mathrm{D}}$ for use in biophysical models refer to much smaller volumes, typically between 0.01 and 1 μm in diameter, which correlate to critical cellular targets (i.e., nucleosome, chromatin, chromosomes, cell nucleus) [43]. A generic value of 1 μm is used in the present study, which is easily accessible by TEPC measurements. In the TDRA and MKM, the $y_{\mathrm{D}}$ of the reference radiation for the volume of interest must also be specified. Here, the γ-rays of Co-60 were used as the reference radiation, with $y_{\mathrm{D},\mathrm{R}}$ = 2.34 keV/μm for a sphere of 1 μm diameter [73].

3.1. Calculations of y_D and LET_D

Figure 1 shows the $y_{\mathrm{D}}$ and ${\mathrm{LET}}_{\mathrm{D}}$ values, calculated by the present analytic model (see Section 2.2 and Section 2.3) and the MCDS code (see Section 2.4), as a function of proton energy in the clinically interesting range of 1–250 MeV. Protons with an initial energy of up to 250 MeV are used in PBT, but as they propagate through tissues they slow down, so energies as low as a few MeV may be of interest. In practice, protons with lower energies can also be found in the target, but they have significant contributions only in areas that receive very low doses [1,63].

3.2. Calculations of RBE

Figure 2 shows the RBE values for all the examined models as a function of proton energy for the five types of cells (having different ${\left(\alpha /\beta \right)}_R$ ratios) listed in Table 2. These results assume a 2 Gy dose fraction which is commonly delivered at the conventional protocols. In addition, the mean empirical RBE (${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$) is also presented for comparison. Note that the models of Wilkens and Chen appearing in Table 1 are only utilized for cell line V79 (with ${\left(\alpha /\beta \right)}_R$ = 3.76 Gy) as their RBE values at other cell types deviate significantly from all the other models. This is probably due to the fact that those models are optimized solely upon data from this specific cell line, so they fail to predict the RBE for other cell types.

In Figure 3, we present the relative percentage difference (at each proton energy T_i) between each RBE model depicted in Figure 2 and the conventional value of RBE = 1.1 calculated from:

(37)${\Delta ({\rm T}}_i)={\displaystyle \frac{{\mathrm{RBE}}_{\mathrm{model}}{({\rm T}}_i{)-\mathrm{RBE}}_{1.1} }{{\mathrm{RBE}}_{1.1}}}\times 100\%,$

where ${{\rm T}}_i$ is the i-th discrete energy value in the 1–250 MeV interval and ${\mathrm{RBE}}_{1.1}$ is the conventional fixed value of RBE = 1.1.

Using the data of Figure 3, in Figure 4 we show the mean percentage difference (MPD) of each RBE model against the value RBE = 1.1 calculated from:

(38)$\mathrm{MPD}={\displaystyle \frac{1}{\mathrm{N}}}{\displaystyle \sum }_{\mathrm{i}}^{\mathrm{N}}\left|{\Delta ({\rm T}}_i)\right|$

where N is the total number of discrete energies (${{\rm T}}_i$) considered in this study. In Figure 4, in addition to the 2 Gy dose fraction, we also show the MPD at 5 Gy and 7 Gy dose fractions which are used in some hypofractionation protocols, especially on prostate, lung and liver carcinomas [74].

3.3. Comparison of MCDS-Based and Analytic-Based RBE

Figure 5 shows the relative percentage differences in the (mean) empirical and biophysical RBE calculated by the ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values from MCDS and the Xapsos analytic model.

3.4. Spread-Out Bragg Peak RBE

Figure 6 shows the proton RBE for three ${\mathrm{LET}}_{\mathrm{D}}$ values, namely, 1 keV/μm, 2.5 keV/μm and 8 keV/μm, which correspond approximately to different regions of a spread-out Bragg peak (SOBP), namely, the proximal SOBP (~1 keV/μm), the center of the SOBP (~2.5 keV/μm), and the distal SOBP (~8 keV/μm) [1]. Note, however, that the exact ${\mathrm{LET}}_{\mathrm{D}}$ values at different SOBP regions may depend on SOBP width and depth, as lower ${\mathrm{LET}}_{\mathrm{D}}$ values are expected in wide modulations and deep targets [1]. For each SOBP region, RBE values are calculated as an average of the empirical models (${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$), the MKM and the TDRA for the examined cell types at 2 Gy, 5 Gy and 7 Gy dose fractions. Figure 7 shows, for each SOBP region, the absolute relative percentage difference between the examined RBE models and the conventional value of RBE = 1.1.

4. Discussion

Proton RBE was evaluated for five cell types with ${(\alpha /\beta )}_R$ ratios ranging from 1.50 Gy to 12.57 Gy and three dose fractions (2 Gy, 5 Gy, 7 Gy) as a function of proton energy from 1 to 250 MeV. In proton therapy, proton beams with initial energies ranging from 50 to 250 MeV are generally employed to cover tumors at different depths. At the distal edge of the SOBP, proton energies are less than 30 MeV, and a 1 MeV cut-off energy (proton CSDA range < 25 μm) is considered sufficiently accurate for clinical applications [1,63]. For the estimation of RBE, nine empirical models and two biophysical models were utilized. Table 3 provides a comparison between empirical and biophysical modes, indicating their strengths and limitations.

Empirical models aim to develop simple mathematical expressions that describe the experimental data without considering the underlying biophysical mechanisms involved. They provide analytic relations for the dependence of RBE on input parameters like dose, ${\mathrm{LET}}_{\mathrm{D}}$ and ${(\alpha /\beta )}_R$. However, the application range of empirical models is generally limited by the experimental data that they use. For example, the Wilkens et al., Chen et al., and Carabe et al. models are based on data for a single cell line (V79), while the Tilly et al. and Jones et al. models are based on two cell lines covering a small range of ${(\alpha /\beta )}_R$ values. As a result, these models may not be representative for the prediction of RBE for cell types with dissimilar characteristics. On the other hand, the models of McNamara et al. and Tian et al. are based on a more extensive set of experimental data covering a broad range of different cell lines. Biophysical models that incorporate well-studied cellular damage (including DNA damage) and repair mechanisms of action (reviewed in Stewart et al., 2018 [18]) are often able to address the need for RBE modeling across cell lines and endpoints in a computationally efficient manner with a smaller number of purely (ad hoc) adjustable parameters. However, there remains a disconnect between RBE modeling of in vitro cellular responses based on empirical and more mechanistic models compared to in vivo and clinical models of RBE. In this paper, we attempt to reduce uncertainty in the empirical and biophysical modeling of cellular RBE by characterizing differences in fundamental dosimetric quantities, i.e., dose-averaged LET (${\mathrm{LET}}_{\mathrm{D}}$) vs. dose-averaged lineal energy ($y_{\mathrm{D}}$), at spatial scales relevant to cellular damage.

Most empirical models use ${\mathrm{LET}}_{\mathrm{D}}$ as a radiation quality index whereas biophysical models commonly use $y_{\mathrm{D}}$ instead. Thus, the calculation of proton ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values over a broad energy range is a useful prerequisite for many proton RBE models. In the present work, ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values were calculated by an established analytic (microdosimetry) model [28,29,33] and the MCDS code [46,47]. As can be seen from Figure 1, the MCDS predicts lower $y_{\mathrm{D}}$ and ${\mathrm{LET}}_{\mathrm{D}}$ values compared to the analytic model over the proton energy range examined. Specifically, MCDS-based ${\mathrm{LET}}_{\mathrm{D}}$ values are constantly ~10% lower than the analytic-based ${\mathrm{LET}}_{\mathrm{D}}$ values, mostly likely because of the delta-ray effects. Although the approximate microdosimetry model included in the MCDS is ~10% too low for low-energy protons (<1 MeV), it underestimates the target dose by as much as 400% for protons with kinetic energies of 250 MeV. The MCDS offers a much better approximation for ${\mathrm{LET}}_{\mathrm{D}}$ than for $y_{\mathrm{D}}$, which is understandable given the underlying approximations used in the code. The analytic microdosimetry model used in the reported studies explicitly considers indirect delta-ray contributions (see Equation (30)) and is more accurate than the algorithm used in the MCDS [27].

Figure 2 shows a comparison between the proton RBE models examined in this work. All models follow the same trend and predict a variable RBE that increases with decreasing proton energy. The variation in RBE is more evident for cells with a lower ${(\alpha /\beta )}_R$ ratio and at proton energies below a few tens of MeV. In these cases, significant deviations from the fixed value RBE = 1.1 are noticed (see Figure 3). More specifically, for cells with high ${(\alpha /\beta )}_R$, the mean RBE of the empirical models (${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$) varies from ~1.0 (>50 MeV) to ~2.0 (<50 MeV) whereas for cells with lower ${(\alpha /\beta )}_R$ it can reach values up to ~2.5. However, the variation among the individual (empirical) RBE models may be significant. For example, at high energies, the individual (empirical) models differ from the mean value (${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$) by ±10%, while at low energies the variation reaches ±30%. The results of biophysical models also support a variable RBE yielding a similar trend with the empirical models. However, biophysical models predict systematically lower RBE values from the empirical models. For example, at high energies (>50 MeV), biophysical models predict up to 10% lower values from ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$, whereas at lower energies (<50 MeV) the differences can reach up to 30%. For cells with high ${(\alpha /\beta )}_R$, the RBE of the biophysical models varies (on average) from ~1.0 (>50 MeV) to ~1.7 (<50 MeV), whereas for cells with lower ${(\alpha /\beta )}_R$ it can reach values up to ~2.0. As regards the differences among the biophysical RBE models (TDRA vs. MKM), they are up to ~5% at high energies but reach ~30% at the low-energy end. It follows from the results depicted in Figure 2 and Figure 3 that proton RBE is variable over the proton energy range studied (1–250 MeV) and may differ significantly from the fixed value of RBE = 1.1, especially at proton energies below a few tens of MeV. However, the variation among the different models at low proton energies (<50 MeV) is sizeable (~30%), precluding a firm conclusion regarding the exact RBE variation at these energies.

To obtain a single index of the deviation of the various RBE models from the conventional value of RBE = 1.1, we calculated the mean percentage difference (MPD) as defined by Equation (38). Figure 4 shows the MPD of the examined RBE models for the various cell types at three different dose levels (2 Gy, 5 Gy and 7 Gy). At hypofractionated dose fractions (5–7 Gy), the deviations of RBE models from the conventional value RBE = 1.1 is clearly less pronounced than for the conventional dose fraction (~2 Gy), e.g., the MPD at the 5 Gy and 7 Gy dose fractions is up to 26% and 32%, respectively, while at the 2 Gy dose fraction the MPD is up to 54%. Also, the ${(\alpha /\beta )}_R$ ratio seems to have a lower influence on RBE at large dose fractions (5 Gy and 7 Gy) compared to the conventional dose fraction (~2 Gy).

To examine how the RBE values are affected by the calculation methods of ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$, we compared MCDS-based and analytic-based RBE calculations (Figure 5). For the empirical RBE models, MCDS-based RBE values are lower from the analytic-based values by up to ~4%, while for the biophysical models, MCDS-based values are lower by up to 10%. This trend is consistent with the lower ${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$ values of MCDS compared to the analytic model.

In Figure 6, we compare the average empirical RBE (${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$) against the TDRA-based RBE and the MKM-based RBE for three ${\mathrm{LET}}_{\mathrm{D}}$ values which correspond (approximately) to different depths of the SOBP, namely, the proximal SOBP (~1 keV/μm), the center of SOBP (~2.5 keV/μm), and the distal SOBP (~8 keV/μm) for three different dose levels (2 Gy, 5 Gy and 7 Gy) [1]. At conventional dose fractions (~2 Gy), the two biophysical models (TDRA, MKM) predict that, at the proximal SOBP, the RBE is ~1.0 regardless of the cell type, while ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$ varies from 1.0 to 1.1 (increasing with decreasing ${(\alpha /\beta )}_R$ ratio). At the center of the SOBP, both the MKM and TDRA predict that the RBE is ~1.05, while ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$ values range from 1.05 to 1.2 (again, increasing with decreasing ${(\alpha /\beta )}_R$ ratio). At the distal SOBP, the RBE takes the highest values and its dependence on the cell type is more pronounced, e.g., the RBE of the biophysical models varies from 1.1 to 1.3 while the ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$ varies from 1.2 to 1.5 (in both cases, increasing with decreasing ${(\alpha /\beta )}_R$ ratio). At hypofractionated dose fractions (5 Gy and 7 Gy), the RBE values at the proximal and central regions of the SOBP do not differ from the values observed for the conventional dose fraction (~2 Gy). However, at the distal SOBP, the RBE for the 5 Gy and 7 Gy dose fractions is lower compared to the RBE for a 2 Gy dose fraction, e.g., the biophysical models predict an RBE of 1.05 to 1.15 (vs. 1.1–1.3), while the ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$ ranges from 1.15 to 1.3 (vs. 1.2–1.5).

The data of Figure 6 are summarized in Figure 7 where the deviations of ${\overline{\mathrm{RBE}}}_{\mathrm{emp}}$, RBE_TDRA and RBE_MKM from the fixed value of RBE = 1.1 in each region of the SOBP for each dose fraction and for each cell type are presented. Although the deviations from RBE = 1.1 in the proximal and central regions of the SOBP are relatively small (up to ~10%), they can be significant (up to ~20–40%) at the distal SOBP.

Empirical models seems to be in good agreement with the experimental findings in the literature which, for the conventional dose fraction (~2 Gy), report an RBE~1.1 in the entrance region of the SOBP, an RBE~1.15 in the central region of the SOBP, and an RBE~1.35 in the distal SOBP (with the RBE decreasing with increasing dose fraction, in all cases) as mean values over several cell lines [1]. Similar values were found by Frese et al. [75], who studied proton RBE in a clinical scenario for two cell lines with ${(\alpha /\beta )}_R=2 \mathrm{and} 10 \mathrm{Gy}$ utilizing the RMF model. Specifically, they found RBE values of 1.03 in the entrance region and 1.27–1.30 at the distal edge of the SOBP with an average RBE value of 1.10 in the whole SOBP. Giovanni et al. [76] also examined proton RBE with some of the studied empirical models (i.e., Carabe’s model and Wedenberg’s model) and the Local Effect Model (LEM) in clinical scenarios of two brain tumor cases with ${(\alpha /\beta )}_R=2 \mathrm{Gy}$ and ${(\alpha /\beta )}_R=10 \mathrm{Gy}$ and found RBE values between 1.07 and 1.18 at the entrance, 1.21 and 1.29 in the middle, and 1.6 and 1.9 at the distal edge of the Planning Target Volume (PTV). The examined biophysical models overall predict slightly lower RBE values than the empirical models.

Current studies on the biological effectiveness of protons focus on conventional treatment protocols including dose rate. However, there is growing interest in FLASH radiotherapy in which ultra-high dose rates (UHDRs) (greater than 40 Gy/s as opposed to conventional dose rates of <0.1 Gy/s) are used to spare healthy tissues, as normal tissues appear to recover better with UHDRs compared to tumors [77,78]. Although the current methodology is applicable to conventional dose rates, their application to FLASH radiation therapy (RT) needs further consideration, given the limited RBE models available for FLASH RT [79]. RBE for FLASH experiments is beyond the scope of the present work.

5. Conclusions

The proton RBE is studied for a clinically relevant energy range of 1–250 MeV at three different dose levels (2 Gy, 5 Gy, 7 Gy) and for five cell types (of different ${(\alpha /\beta )}_R$ ratios) using nine empirical proton models and two established biophysical models (TDRA and MKM). Empirical models express the RBE as a function of the macroscopic quantity ${\mathrm{LET}}_{\mathrm{D}}$ while biophysical models commonly use the microdosimetric quantity $y_{\mathrm{D}}$. For the evaluation of those parameters (${\mathrm{LET}}_{\mathrm{D}}$ and $y_{\mathrm{D}}$), we compare an established analytic microdosimetry model with the MCDS code. The results showed that, for conventional dose fractions (~2 Gy) and at proton energies that correspond to the proximal and central regions of the SOBP, a constant value of RBE = 1.1 is reasonable at the ±10% accuracy level. For low-energy protons distal to the Bragg peak, published RBE models predict significant deviations of up to ~20–40% from RBE = 1.1, especially for cells with low ${(\alpha /\beta )}_R$. For hypofractionated dose fractions (5–7 Gy), deviations from the fixed value are less pronounced but still sizeable (up to ~5–20%). However, the comparable discrepancies observed among the different models make the selection of a variable RBE across the SOBP uncertain.

Author Contributions

Conceptualization, D.D., A.P., I.K., R.D.S., P.K. and D.E.; methodology, D.D., A.P. and D.E.; software, D.D. and A.P.; validation, D.D. and A.P.; formal analysis, D.D., A.P., I.K. and D.E.; investigation, D.D., A.P., I.K., R.D.S., P.K. and D.E.; resources, D.D., A.P., I.K. and D.E.; data curation, D.D. and A.P.; writing—original draft preparation, D.D. and A.P.; writing—review and editing, D.D., A.P., I.K., R.D.S., P.K. and D.E.; supervision, A.P., I.K. and D.E.; project administration, I.K. and D.E. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Calculated [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] as a function of proton energy for a liquid water sphere of 1 μm diameter.

Figure 2. Proton RBE values calculated by the different empirical (Table 1) and biophysical (TDRA, MKM) models examined as a function of proton energy for five cell types at a 2 Gy dose fraction. The conventional value of RBE = 1.1 is also shown for comparison.

Figure 2. Proton RBE values calculated by the different empirical (Table 1) and biophysical (TDRA, MKM) models examined as a function of proton energy for five cell types at a 2 Gy dose fraction. The conventional value of RBE = 1.1 is also shown for comparison.

Figure 3. Relative percentage difference (Equation (37)) between the RBE models of Figure 2 and the conventional value of RBE = 1.1 for five cell types at a 2 Gy dose fraction.

Figure 4. Mean percentage difference (Equation (38)) between the RBE models examined in the present work and the conventional fixed value RBE =1.1 for five cell types at 2 Gy, 5 Gy and 7 Gy dose fractions.

Figure 5. Relative percentage difference (Equation (37)) between MCDS-based and analytic-based calculations of proton RBE for five cell types at a 2 Gy dose fraction.

Figure 6. Calculated proton RBE for three [Forumla omitted. See PDF.] values corresponding (approximately) to the proximal SOBP region (~1 keV/μm), the center of the SOBP (~2.5 keV/μm) and the distal edge of the SOBP (~8 keV/μm) for five cell types at 2 Gy, 5 Gy and 7 Gy dose fractions.

Figure 7. Absolute relative percentage difference (Equation (37)) between the RBE models and the conventional value RBE = 1.1 at three [Forumla omitted. See PDF.] values corresponding (approximately) to the proximal SOBP region (~1 keV/μm), the center of the SOBP (~2.5 keV/μm) and the distal edge of the SOBP (~8 keV/μm) for five cell types at 2 Gy, 5 Gy and 7 Gy dose fractions.

Table 2

Representative radiosensitivity parameters for the various cell lines used in the present work. Adapted from [71].

Tumor Pathology	${\mathit{a}}_{\mathit{R}} \left(\mathit{G}{\mathit{y}}^{-1}\right)$	${\mathit{\beta }}_{\mathit{R}} \left(\mathit{G}{\mathit{y}}^{-2}\right)$	${\left(\mathit{\alpha }/\mathit{\beta }\right)}_{\mathit{R}} \left(\mathit{G}\mathit{y}\right)$
Groups A and Blymphoma, myeloma, neuroblastoma, medulloblastoma and lung (SCLC *)	0.73	0.0581	12.57
Groups C and Dbreast, bladder, cervix, pancreas, colon and lung (SCC **)	0.36	0.0581	6.20
Group Emelanoma, osteosarcoma, glioblastoma and renal carcinoma	0.26	0.0581	4.48
V79 [63]	0.112	0.0298	3.76
Prostate carcinoma [72]	0.036	0.024	1.50

* SCLC: small-cell lung cancer. ** SCC: squamous cell carcinoma.

Table 3

Comparison of empirical and biophysical models of cellular RBE.

Characteristics	Empirical Models	Biophysical Models
Strengths	Parameters and functional form relative to LET directly derived from measurements	Reflects plausible biophysical mechanisms related to cellular damage and repair
	Often simple algebraic formulas that relate LET to cell survival	Able to interpolate and extrapolate across multiple ion cell types with fewer purely adjustable parameters. Tends to reduce number of ad hoc adjustable parameters
	Often specific to one particular type of ion, cell line and endpoint	Explicit consideration of cellular endpoints that facilitates comparisons of RBE estimates across cell lines and ion types for a wide range of LET values
Limitations	Estimates of parameter values are uncertain and highly specific to the cell line, endpoint and experimental design	May involve auxiliary microdosimetric concepts, more specific to the cellular endpoint of interest
	May give a false sense of confidence when applied to alternate in vitro and in vivo conditions	May require additional computational complexity

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