Proton RBE models: commonalities and differences

Uncertainties in the relative biological effectiveness (RBE) of protons remains a major barrier to the biological optimisation of proton therapy. While a constant value of 1.1 is widely used in treatment planning, extensive preclinical in vitro and in vivo data suggests that proton RBE is variable, depending on proton energy, target tissue, and endpoint. A number of phenomenological models have been developed to try and explain this variation, but agreement between these models is often poor. This has been attributed to both the models’ underlying assumptions and the data to which they are fit. In this brief note, we investigate the underlying trends in these models by comparing their predictions as a function of relevant biological and physical parameters, to determine where models are in conceptual agreement or disagreement. By doing this, it can be seen that the primary differences between models arise from how they handle biological parameters (i.e. α and β from the linear–quadratic model for photon exposures). By contrast, when specifically explored for linear energy transfer-dependence, all models show extremely good correlation. These observations suggest that there is a pressing need for more systematic exploration of biological variation in RBE across different cells in well-controlled conditions to help distinguish between these different models and identify the true behaviour.


Introduction
Protons offer significant physical dosimetric advantages over photons for many cancers, with their finite range and Bragg peak enabling dose to be more accurately conformed to tumours. This has the potential to significantly reduce radiotherapy side-effects and has driven a dramatic expansion in the usage of proton therapy in recent years (Mohan and Grosshans 2017).
Alongside these physical benefits, however, proton therapy also has biological differences. A given dose of proton therapy will (typically) lead to greater cell-killing than the same dose of photons. This is quantified in terms of their relative biological effectiveness (RBE), which is the ratio of a reference photon dose to the dose of a radiation of interest (such as protons, alpha particles, etc) which causes the same biological effect. However, unlike the well-understood physical characteristics of proton therapy, these biological effects remain poorly quantified and have not seen significant clinical exploitation (Underwood andPaganetti 2016, Lühr et al 2018).
Proton therapy is typically planned assuming a fixed RBE of 1.1 compared to photons, independent of proton energy or target tissue. This approach is straightforward and largely successful, with no clear clinical evidence supporting significant deviations from a constant RBE of 1.1 for most tissues in current proton therapy practice (Paganetti et al 2019). However, it is likely that the approach is a significant simplification. There is extensive evidence from preclinical in vitro and in vivo studies that the RBE of protons depends on their energy (and thus linear energy transfer (LET)), the biology of the target tissue (Paganetti 2014) as well as a number of other parameters such as dose, dose-rate and tissue oxygenation. As these studies indicate proton RBE can range from less than 1 to more than 2, there is significant scope to better optimise radiotherapy to exploit these effects in tumours, and avoid unintended consequences of high RBEs in normal tissues .
A full understanding of the mechanisms underpinning proton RBE remains elusive. It is broadly accepted that, as proton energies reduce, they deposit energy more densely around their tracks (reflected by an increase in Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. their LET), which leads to more densely distributed and complex DNA damage. Cells have a greater difficulty in repairing this damage, and thus see greater rates of death for a similar delivered dose. However, the exact nature of the types of damage and biological processes leading to these responses are still the subject of some debate, despite extensive mechanistic modelling (McMahon and Prise 2019).
An alternative approach to quantify proton RBE is using phenomenological models, which do not attempt to incorporate all of the underlying biology, but instead identify key trends from experimental or clinical data. Most of these models are designed to modify the linear-quadratic (LQ) dose response model to predict proton a and b parameters for a given irradiation by applying simple scaling functions. These functions typically incorporate parameters such as the proton LET and the cell's a and b parameters for photon irradiation, or their ratio a b / (Rørvik et al 2018, McNamara et al 2020. These models were recently reviewed by Rørvik et al, who investigated eleven different phenomenological RBE models in clinically-relevant scenarios, together with two plan-based models (Rørvik et al 2018). This work highlighted a high degree of variability, with some dosimetric parameters varying by up to 50% when different models were used, particularly in normal tissue exposed to high LETs. Both the datasets used to fit the models (which often include primarily tumour measurements and data from non-human cell lines) and the models' underlying assumptions were identified as sources of this variability. In principle, these models could be re-fit to a common dataset in an attempt to standardise their predictions. However, there are also systematic differences in their underlying assumptions, which would persist even if common datasets were used.
This works seeks to evaluate these systematic differences in predictions, to determine how well (or not) these models' assumptions agree with one another, and explore what aspects of proton RBE are most poorly characterised. This may then enable a better direction of future research in this area, help distinguish which models make clearly different predictions which can be benchmarked against future RBE studies, and potentially better inform future treatment optimisation.

Methods
The 13  . As these RBE models are complex functions depending on multiple parameters, there is no single quantitative method to determine how well they agree. In this work, a Monte Carlo sampling method has been used to evaluate overall trends. Specifically, a large number of virtual parameter sets were generated, containing a photon a and b response parameters for a virtual cell line and a proton LET. For each model, the corresponding RBE was then calculated, and these RBEs were then correlated between models to evaluate how well they agree about overall trends. This approach is described in more detail below.
In this analysis, cell responses are described by the standard LQ radiation response model (McMahon 2019), which defines cell survival as: where S is the cell survival fraction following exposure to a dose D, and a and b are parameters that characterise the dose response. These parameters depend both on the biology of the exposed cell as well as the type of radiation being delivered. Here, photon (x-ray) response parameters are denoted as a , p RBE is defined as the ratio of doses that lead to equivalent biological effects. The particular biological effect used for the comparison can be chosen in a number of ways, including as a particular survival level (e.g. 10%), or relative to a particular photon dose (e.g. 2 Gy). The choice of reference point can impact on the absolute value of RBE, and often reflects priorities in model design, such as a desire to match clinical photon practice of 2 Gy fractions. In this work, we use the Mean Inactivation Dose (MID) (Belli and Simula 1991). This is defined as the area under the dose-response curve, and so avoids the need to select a particular survival or dose level. It can be interpreted as the average dose (or expected value of dose) needed to kill a cell in the population: This definition has the advantage that it presents a summary over the whole survival curve, rather than focusing on a single investigator-chosen point. The RBE for a given exposure can then be defined as: Thus, given the a and b parameters for exposure to photons and protons, a representative RBE can be calculated. This can be done experimentally by fitting a and b parameters to measured dose response curves, or a p and b p can be predicted by RBE models. Each of the models considered in this work can be expressed as a scaling on the photon dose response parameters, through functions that can depend on the cell a b / ratio and the incident proton LET, that is: a Here F and G are model-specific functions defining the relationship between photon and proton response parameters. These functions are tabulated in the appendix for each of the models used in this work. By using these functions, for any set of input parameters (a , x b x and proton LET), proton dose response parameters (a , p b p ) can be calculated, and used to calculate the RBE. Finally, to evaluate overall trends, 5000 virtual experiments were randomly generated, consisting of sets of (a , x b , x LET). a x was normally distributed with a mean of 0.2 Gy −1 and a standard deviation of 0.04 Gy, a b x x / was uniformly distributed between 2 and 10 Gy, and LET was uniformly distributed between 1 and 15 keV μm −1 . These values were chosen to represent the ranges typically seen in clinical and in vitro experimental scenarios.
For each model, a p and b p were calculated for each virtual experiment using the functions described in the appendix. RBEs were then obtained by calculating the MIDs and RBE MID according to equations (2) and (3) above. Correlations were then evaluated between each pair of models across the entire set of virtual experiments, using the Pearson correlation coefficient. As a high Pearson coefficient indicates a linear correlation between models, this would indicate models had similar underlying forms, to within a parameter rescaling. By contrast, a low Pearson coefficient would indicate fundamentally different trends in predictions for at least some conditions. The same set of experimental parameters were used for all model comparisons.
These models include both biological (a b x x / ) and physical parameters, but their effect on model predictions is difficult to separate. Simply evaluating trends at a single a b x x / or LET value may overlook more complex trends which only arise for certain combinations of these values. One method to partially separate these trends while still sampling the full range of parameter combinations is to normalise predicted additional effects in RBEs to those at a reference LET or a b x x / value. For example, by dividing an RBE by one calculated at a reference LET with the same a b , x x / it is possible to evaluate the relative impact of LET on RBE, while controlling for the contribution of a b x x / which should be an approximately constant scaling in both the numerator and denominator. In practice this is not guaranteed to be a perfect control due to higher-order contributions in more complex models, but it provides a useful first step in evaluating how well model trends agree.
Specifically, the normalised RBE effects can be expressed as: RBE norm / are the LET and a b / -normalised RBEs respectively. LET ref was fixed to 10 keV μm −1 , and a b was fixed at 5 Gy, although the exact values did not strongly impact on the overall trends observed. These normalised RBEs were calculated for the same set of virtual experiments as the unnormalised RBEs, and correlated in the same way.
Python code implementing the models used in this work is available as supplementary material (available online at stacks.iop.org/PMB/66/04NT02/mmedia) to this paper, and online at https://github.com/ sjmcmahon/RBEModels. Figure 1 presents scatter plots of the correlations between un-normalised RBEs for individual pairs of models by row and column, along with a map of the corresponding correlation coefficients. Strong positive correlation is denoted here in dark blue, and weak or negative correlation in red. Ranges for these sub-plots have been set based on the minimum and maximum RBE values generated by the random sampling. As expected, minimum RBE values are highly similar across all models (mean of  1.02 0.03), while the maximum RBE varied much more (ranging from 1.54 in Frese to 3.36 in Chen, mean of  2.25 0.6). However, as this analysis focuses on correlation, these changes in absolute magnitude do not impact significantly on the subsequent discussion.

Results
While all models are positively correlated (Pearson r>0.4), there are two groups of models which correlate well within group (r>0.9), but poorly between groups (r<0.7). The first group, presented in the upper left of the plot, are the Carabe, Mairani, McNamara, Rørvik and Wedenberg models. All of these models have a linear dependence of a p on LET, which is inversely proportional to the a b x x / ratio. This means that cells with lower a b x x / ratios, in general, see larger RBEs. While these models differ in other assumptions (e.g. the RBE of protons at very low LETs, b p dependence on LET, the presence of other higher-order LET terms), this dependence of a p is the dominant contribution to RBE, and thus these models all correlate very well across variations in both LET and RBE.
The second group of models-Chen, Frese, Jones, and Wilkens-differ in that while a p scaling is still the dominant term in their model, they do not incorporate the inverse dependence on a b , x x / meaning that, in general, cells with lower a b x x / ratios typically see smaller RBEs. This group has a greater diversity in assumptions describing the link between LET and a , p giving slightly less strong intra-group correlations. Figure 1. Correlation of trends in different RBE models. RBEs for Mean Inactivation Doses (MIDs) have been calculated using a panel of models for randomly generated physical and biological parameters and plotted against one another to evaluate correlation. The lower triangle presents scatter plots of modelled RBEs between pairs of models (x-axis models arranged by column, y-axis models arranged by row, models identified on diagonal), while the upper triangle presents Pearson correlation coefficients. While all models are positively correlated, there are two broad groups which correlate well within the group (upper left, lower right), but relatively poorly between groups (lower left).
Three models sit somewhat between these groups. The Tilly model incorporates a weaker inverse dependence of a p on a b x x / ratio, rather than purely inversely proportional. The Unkelbach model was originally developed only as a dose-scaling approach, and so predicts RBEs which are wholly independent of a b x x / ratio. Finally, the Peeler model has a very strong LET 3 scaling for both a p and b , p which depend differently on the a b x x / ratio. This gives a more complex relationship, depending on which term is dominant for a given cell line.
Taken together, these observations suggest that while there is broad agreement in overall trends, there remains significant difference in the details of the structure underlying the RBE predictions of these different models. This can be further explored by considering how the models correlate when the dependence on different parameters is normalised out. Figure 2 shows a correlation plot when responses are normalised to a reference LET exposure within their target cell. This normalisation effectively focuses on the LET-dependence within tissues, and excludes intertissue variability. There is excellent agreement across all models, with no correlation coefficient less than 0.9. Indeed, excepting two models which include very high-order LET terms (RorvikW, which includes terms up to LET 4 , and Peeler which scales as LET 3 ), correlation coefficients are all greater than 0.99. This suggests that, despite the range of model assumptions and datasets, there is little disagreement about the LET-dependence of RBE within a given tissue, to within a linear scaling factor for clinically-relevant LETs. Figure 3 shows a correlation plot when responses are normalised to focus on model dependence on a b , x x / excluding the effect of LET. In contrast to figure 2, many models are no longer well-correlated, with the two / dependence is normalised out to focus on LET dependence, plotted as in figure 1. Despite significant heterogeneity in underlying model assumptions and fitting, if only the LET-dependence within tissues is considered model correlation dramatically improves, with significant deviations from linearity seen only in models with higher-order LET terms. typically changing by no more than a few percent. One outlier in this case is the McNamara model, which can for high LETs and low a b / values have a scaling for b which is less than 1, which can lead to significantly poorer correlation at very high doses. However, even at such high doses the features of the different model groups remain clear.

Discussion
RBE remains an area of significant contention in proton therapy, with numerous models providing significantly different predictions for clinically relevant conditions, making confident RBE-optimisation a distant goal at present. As a result, it is useful to evaluate where models agree and where they disagree, to help direct future investigations to fill knowledge gaps and enable robust model development.
This analysis shows that, despite a range of differences in the underlying assumptions, fitting approaches, and reference datasets, there is broad agreement about how RBE depends on LET, with RBE predictions in clinically-relevant situations being extremely well correlated (figure 2). While there is some quantitative disagreement about the exact magnitude of this dependence, this could be resolved by a simple model rescaling or re-fitting to a common dataset.
Interestingly, this means that if an analysis only requires relative comparisons within tissues with the same α/β ratio-such as ranking plans by RBE-weighted dose to a particular organ at risk-then most models will produce an identical ranking. This may enable relative comparisons of the impact of RBE on plan quality to be made without a large impact of inter-model uncertainty.
By contrast, there is significant disagreement on the impact of α/β ratio on RBE. The major dividing factor between the two RBE model groups identified here is the inclusion of α/β in the dependence of a , p or equivalently RBE . max One group of models explicitly include a a b 1 ( ) / / term, which leads to RBE increasing with decreasing α/β, while the other does not include an α/β dependence, which tends to lead to a decrease in RBE with decreasing α/β. This means that when RBE comparisons are made between organs-such as comparing tumour dose to a normal tissue dose-much more disagreement is seen between RBE models, and this cannot be resolved by a simple re-scaling. Even re-fitting models to a single dataset will not be able to resolve these issues, as the fundamentally different assumptions in the underlying model structure cannot be reconciled.
The reason for this relatively greater uncertainty in the α/β dependence compared to the LET dependence is partially reflected in the nature of the parameters. LET is a physical parameter, which is usually defined fairly consistently and can be calculated with reasonable precision using well-established computational codes. By contrast, photon α/β values are determined by fitting to sparse biological data, and frequently carry very large uncertainties.
This leads to significant inter-experimental variation in work carried out by different groups. It is well established that there are systematic uncertainties in the dose delivered in radiation biology experiments, which would impact on estimates of both α/β and RBE in non-trivial ways (Seed et al 2016, Draeger et al 2020. As a result, models which seek to determine α/β dependencies by pooling data from a large number of publications suffer from substantial additional uncertainty, which is difficult to quantify and can significantly obscure real biological trends. Decisions made in many experimental designs have also contributed to the relatively greater scale of uncertainty about the biological dependence of RBE. Version 3.2 of the Particle Irradiation Data Ensemble includes 34 papers presenting RBE studies with protons up to 2015 (Friedrich et al 2013). Of these, a majority report RBE measurements at three or more LETs, with a quarter reporting more than five LETs, providing numerous opportunities to test the dependence of RBE on LET within internally consistent datasets.
By contrast, a sizeable majority of papers report RBE measured only in one cell line (22/34, 65%), and only five present three or more cell lines. Additionally, the study with the greatest diversity in cell lines (eight), presents data for DNA repair mutants selected based on their expected impact on RBE, and may not be representative of the natural range of biological factors cause variations in α/β (Gerelchuluun et al 2015). As a result, and in sharp contrast with LET, the majority of studies do not allow for robust testing of even the simplest linear dependence of RBE on α/β within a single dataset. This means biological model building is dramatically more sensitive to inter-experimental variations, and thus subject to significantly more disagreement.
Taken together, these observations help to highlight where experiments may provide the most valuable contributions to improving our understanding of proton RBEs. Most notably, it suggests that studies that span a range of cell lines and genetic backgrounds would be particularly informative, even if it comes at the cost of investigating fewer LETs in a given study. In addition, further developments of our mechanistic understandings of the drivers of proton RBE may be able to help identify which scaling models are reasonable on a biological basis, to constrain the range of possible models and direct further experimental study. However, it is important to note that even within mechanistic models, there is significant variation in predicted dependence of RBE on physical and biological parameters, so further work is also needed in this area (Stewart et al 2018).
Finally, it is important to note that all of the models considered in this work have been based on in vitro observations of radiation sensitivity, quantified through measurements of cell killing. It remains an open question how well such models correspond to clinical endpoints of interest, as in vivo radiation sensitivity is known to potentially differ significantly from in vitro measurements, and it remains unclear if cell killing is the primary driver of a number of forms of normal tissue toxicity. Final validation against clinical data will be an essential step in the validation of any RBE models.

Conclusions
There is broad agreement about the LET dependence of RBE across a range of phenomenological models, but significant disagreement about the role of α/β ratio. This suggests that clinical studies may not be sensitive to model choice when looking at intra-tissue comparisons, but that model differences will play a much more important role when comparing between different tissues. The lack of studies investigating the dependence of RBE on α/β ratio within a single experiment means it is difficult to identify the most biologically accurate model, highlighting the need for greater understanding in the underlying biology of proton RBE.