Data-driven ion-independent relative biological effectiveness modeling using the beam quality Q

Beam quality Q = Z2/E (Z = ion charge, E = energy), an alternative to the conventionally used linear energy transfer (LET), enables ion-independent modeling of the relative biological effectiveness (RBE) of ions. Therefore, the Q concept, i.e. different ions with similar Q have similar RBE values, could help to transfer clinical RBE knowledge from better-studied ion types (e.g. carbon) to other ions. However, the validity of the Q concept has so far only been demonstrated for low LET values. In this work, the Q concept was explored in a broad LET range, including the so-called overkilling region. The particle irradiation data ensemble (PIDE) was used as experimental in vitro dataset. Data-driven models, i.e. neural network (NN) models with low complexity, were built to predict RBE values for H, He, C and Ne ions at different in vitro endpoints taking different combinations of clinically available candidate inputs: LET, Q and linear-quadratic photon parameter α x/β x. Models were compared in terms of prediction power and ion dependence. The optimal model was compared to published model data using the local effect model (LEM IV). The NN models performed best for the prediction of RBE at reference photon doses between 2 and 4 Gy or RBE near 10% cell survival, using only α x/β x and Q instead of LET as input. The Q model was not significantly ion dependent (p > 0.5) and its prediction power was comparable to that of LEM IV. In conclusion, the validity of the Q concept was demonstrated in a clinically relevant LET range including overkilling. A data-driven Q model was proposed and observed to have an RBE prediction power comparable to a mechanistic model regardless of particle type. The Q concept provides the possibility of reducing RBE uncertainty in treatment planning for protons and ions in the future by transferring clinical RBE knowledge between ions.


Introduction
Compared to conventional photon therapy, ion therapy is characterized by, first, an energy deposition peak at the end of its range, called the Bragg peak, and, second, an increased relative biological effectiveness (RBE). While the RBE of certain ions, e.g. carbon, has been studied in some detail for decades (Raju and Carpenter 1978, Hawkins 1996, Ando and Kase 2009, more research on other particles is desired. For example, a constant RBE of 1.1 is widely applied for proton beam therapy (Heuchel et al 2022, Paganetti et Eulitz et al2019, 2023. In addition to particles that have already been applied clinically, i.e. proton and carbon, new applications using e.g. helium (Mein et al2019), oxygen (Chang et al2014) and multi-ion beams (Ebner et al2021) are emerging. An ion-independent model, would help to enrich the data pool by assembling data of different ions and to transfer knowledge from better-investigated ones. In order to quantify and predict RBE, different RBE models have been proposed. Phenomenological proton RBE models (Tilly et al 2005, Carabe-Fernandez et al 2007, Wedenberg et al 2013, McNamara et al 2015, Mairani et al 2017, McMahon 2021 were built by fitting fixed formulas on the RBE data for protons. For these models, the transferability of their specific formalism needs to be verified against clinical data as the clinical endpoint differs from the biological in vitro endpoint used for modeling. Furthermore, these models are driven by linear energy transfer (LET), which only quantifies the integral energy deposition while ignoring the microdosimetric features of the beam; thus, it can hardly be used for the many different ions that are naturally involved in clinical ion beams. Mechanistic models, e.g. the local effect model (LEM) (Scholz et al 1997, Friedrich et al 2012 and the microdosimetric-kinetic Model (Hawkins 1998), take into account the microdosimetric features and are based on generally believed mechanisms, including that the enhanced RBE of ions is determined by the microscopic dose distribution in the cell nucleus. However, some quantities required for those models, e.g. the cell nucleus size or microscopic dose distribution (nanometer scale) (Kase et al 2008), may be difficult to determine in clinical application.
Recently, a new concept, namely, beam quality Q was proposed for RBE modeling (Lühr et al 2017, Tian et al 2022. The beam quality is defined as Q = Z 2 /E, with Z and E being the ion's charge and kinetic energy per nucleon, respectively. It has been shown that a Q-driven model is able to predict the RBE, regardless of ion type and for individual ions, comparable to another widely used ion-specific model (Tian et al 2022). This opens up the possibility of using RBE data from different ions for model building and thereby improving the precision of RBE predictions. However, the proposed Q-dependent model is a simple linear model and, thus, only works in the region of low to intermediate Q values, i.e. for LET values below the so-called overkilling region (Tian et al 2022). Accordingly, the general validity of the ion-independent Q concept still needs to be shown.
Therefore, the purpose of this work was, first, to demonstrate the validity of the ion-independent Q concept for a broad LET range including larger Q values and the overkill domain and, second, to propose an experimental data-driven, non-linear Q model describing the RBE for different ions while focusing on clinically available input variables.

Material and methods
2.1. PIDE dataset and data selection The particle irradiation data ensemble (PIDE, version 3.2) (Friedrich et al 2021), consisting of a dataset recording the in vitro experimental data of cell survival experiments of 115 publications covering 1118 data points of 21 types of ion irradiation, was used in this work.
The following data selection criteria were applied. Data from experiments with monoenergetic irradiation of ions no heavier than neon (Z < 11) were considered. The minimum kinetic energy thresholds for different ions were chosen such that the ion ranges in water were at least 25 μm (Lühr et al 2012), i.e. in the order of the size of a single cell. In addition, only experiments with positive and finite α x /β x and an asynchronous cell cycle were considered. Here, α x and β x are the α and β parameters of the linear-quadratic (LQ) model of photon irradiation. Irradiation data of a specific ion were only considered if at least five data points were available for that ion. One proton data point with an α x /β x value much higher (∼70 Gy) than those of all others (<30 Gy) was excluded in this work. Consequently, irradiation data of the following ions were selected: proton (48 data points), helium (30), carbon (148) and neon (58) with a minimum energy of 1.03, 2.29, 4.07 and 5.04 A·MeV, respectively. In the following, this selected dataset is called the PIDE dataset for simplicity.
For each PIDE record, an LET value was provided. These LET values were directly taken for this study, i.e. regardless of their definition as, e.g. dose or track averaged LET. The Q values were calculated using the energy E and charge Z values recorded in the PIDE. Some of the experimental publications covered by the PIDE only provide either an E or LET value. The missing values were calculated by the PIDE group based on the reported counterpart values using the software ATIMA (Geissel et al 2002, Friedrich et al 2021. Two types of α and β values are recorded in the PIDE: first, the data originally reported by the experimenters and, second, the data retrospectively obtained by the PIDE group using the LQ model fitting of the underlying radiation response data. In this work, only the originally reported data were used. RBE values at an isoeffective photon dose d x , RBE dx , were calculated for d x ranging from 1 to 30 Gy using the LQ model formalism (cf appendix) and α x , β x , α i and β i values as recorded in the PIDE. The maximum and minimum RBE values given by Dale et al 2009) were also considered and could be regarded approximately as RBE 0 and RBE ∞ at d x approaching 0 and ∞, respectively. Thus, the RBE dx values derived from the experimental α x , β x , α i and β i values as recorded in the PIDE were regarded as the experimentally derived RBE dx ground truth in this work. RBE values defined by the cell survival S, RBE S , were also calculated (cf appendix) and modeled for S = 0.1%, 1.0%, 10.0%, 50.0% and 90.0% for discussion.

Correlation analysis
This work aimed at building a model that takes clinically available variables, i.e. LET, Q and α x /β x or combinations thereof, as input and predicts the resulting RBE values. Spearman's correlation coefficient values, ρ, between different potential input variables and output data were calculated using the Python package Pandas (Reback et al 2022).

RBE modeling
A neural network (NN) model was used to perform data-driven RBE modeling to avoid any presumption on the functional form of the RBE model. Considering the limited amount of available data (284 experimental records), a model with a comparably simple architecture was applied, i.e. a fully connected NN consisting of two hidden layers of the size of 6. As activation function, the ReLU (rectified linear unit) was used. The machine learning package scikit-learn (Pedregosa et al 2011) was used for the machine-learning application in this work.
Three RBE models with different input variables, i.e. combinations of the physical and biological quantities Q, LET and α x /β x , were compared; namely, RBE 2Gy (Q, α x /β x ), RBE 2Gy (LET, α x /β x ) and RBE 2Gy (Q, LET, α x /β x ), respectively, with RBE 2Gy was explicitly chosen as an example in this manuscript.

Model evaluation
The models of RBE 2Gy (Q, α x /β x ), RBE 2Gy (LET, α x /β x ) and RBE 2Gy (Q, LET, α x /β x ) were trained and tested using the same training and test set, respectively. The test set (20% of the total selected dataset) was randomly chosen in the domain of Q Î (0, 15) (A·MeV) −1 and α x /β x Î (0, 30) Gy. The remaining data of the PIDE dataset fulfilling the selection criteria specified in section 2.1 were used as training set. The prediction power of the trained models was compared by means of the coefficient of determination (called r2 score in the following) and the mean square error (MSE) between the predicted and experimentally derived RBE 2Gy of the test set regardless of particle type. The ion dependence (95% confidence level) of the models was tested by applying an ANOVA (analysis of variance) test on the residuals between the model calculated and the experimentally derived RBE 2Gy of different particles. For the ANOVA test, all eligible PIDE data, i.e. training and test set, were used due to the limited amount of data of individual particles in the test set.

Uncertainties of the model prediction
The uncertainty of the model prediction in the two-dimensional (2D) space spanned by the two parameters Q and α x /β x was evaluated by the following procedure: (1) randomly divide the PIDE dataset into a training set (80%) and a test set (20%); (2) train a model using the training set and save model parameters; (3) repeat (1) and (2) until 100 models based on different training sets are built and saved; (4) determine the uncertainty of the model by calculating the standard deviation (SD) of the 100 RBE 2Gy values calculated by those 100 models at each (grid) point in the 2D space of Q-α x /β x .

Comparison to other RBE models
The prediction of the proposed data-driven model was compared to RBE results of LEM IV for the biological endpoint of 10% survival fraction, i.e., RBE 10 (Elsässer et al 2010) considering radiation data of human salivary gland (HSG) cells reported by (Furusawa et al 2000). For the fairness of the comparison, the NN model was retrained for RBE 10 and the inputs of Q and α x /β x . The data of the HSG cells reported by (Furusawa et al 2000, Elsässer et al 2010 were used as test set while the remaining PIDE dataset was used for training. The experimentally derived RBE 10 values were calculated as before by applying the LQ model on PIDE-recorded α x , β x , α i and β i values (cf appendix). Note that the ion dependence of the model should not be inferred by this analysis as the training and test set were not split randomly. The prediction of LEM IV was interpolated using the data reported by (Elsässer et al 2010).
For the same test data, the prediction of the recently proposed Q-driven linear RBE model (Tian et al 2022) was also considered and compared in terms of RBE 10 . This model is called linear model in the following, as it assumes RBE max to be linear in Q/( The prediction of RBE 10 by the linear model is described in the appendix.

Data distribution
The distribution of data points of the PIDE dataset in the 2D space of Q-α x /β x is shown in figure 1. All data, except for one data point for protons (Baggio et al 2002) and one for neon ions (Furusawa et al 2000), were within the Q interval of (0, 15) (A·MeV) −1 and α x /β x interval of (0, 26) Gy with a lower data density at high-Q values, especially, when α x /β x values were also high.

Variable correlation
Spearman's correlation coefficients ρ between output (RBE dx at different dose level d x ) and clinically available input variables (LET, Q and α x /β x ) are presented in figure 2. The ρ between RBE dx and either Q or LET were comparable, while the ρ between RBE dx and α x /β x was low. The ρ between RBE dx and LET or Q were highest for d x values within the photon reference dose interval 2-4 Gy. Hence, RBE dx within 2-4 Gy was regarded as the most 'predictable' output.
In line with this finding, the prediction of the RBE dx for d x values between 2 and 4 Gy was observed to be better than that for d x in other domains, i.e. (0, 2) Gy and (4, ∞) Gy. As this is a dose range of particular clinical relevance, results presented in this work focus on the prediction of RBE within this dose domain (cf comparison between the prediction of RBE for different d x domains in the appendix).

Comparison of models using different input
The ability of the RBE 2Gy (Q, α x /β x ) model to predict the experimentally derived RBE 2Gy values is shown in figure 3. The same comparison for the two other models, namely RBE 2Gy (LET, α x /β x ) and RBE 2Gy (Q, LET,  α x /β x ), is shown in supplementary figure S1. A comparison between the model calculated RBE 2Gy by both RBE 2Gy (Q, α x /β x ) and RBE 2Gy (LET, α x /β x ) for the entire dataset are shown in figure S3.
The model performance of the RBE 2Gy (LET, α x /β x ), RBE 2Gy (Q, α x /β x ) and RBE 2Gy (Q, LET, α x /β x ) models is compared in table 1 regarding the r2 score, the MSE between the experimentally derived and modeled RBE 2Gy and the result of the ANOVA test. Note that, according to the result of the ANOVA test, the model of RBE 2Gy (LET, α x /β x ) cannot provide ion-independent predictions, i.e. the model cannot make predictions equally for different ions, even for the training set. Thus, measurement of the prediction power, i.e. r2 score and MSE, should be regarded as invalid, although corresponding numbers could still be obtained and compared to the other two models.
Considering the models of RBE 2Gy (Q, LET, α x /β x ) and RBE 2Gy (Q, α x /β x ), their r2 scores and MSE were comparable and both models were not significantly dependent on ion type. The differences (mean ±SD) between the predictions of the two models RBE 2Gy (Q, α x /β x ) and RBE 2Gy (Q, LET, α x /β x ) for the same data point (0.00 ± 0.11) were much smaller than the differences between the model RBE 2Gy (Q, α x /β x ) and the respective experimental data points (−0.03 ± 0.67), as shown in supplementary figure S2. That means adding LET as an additional variable did not substantially change or improve the predicted RBE 2G values. Therefore, adding LET to the model cannot substantially decrease the observed differences between individual experimental data points and predictions.
The performance of RBE S (Q, α x /β x ) defined by cell survival is shown in table A2 in the appendix and is consistent with the performance of RBE dx (Q, α x /β x ). The r2 score was shown to be highest in the domain near 10% survival fraction; while at all survival fraction levels, the models were not significantly ion dependent.   Figure 4 shows the mean and SD values resulting from the 100 trained RBE 2Gy (Q, α x /β x ) models in the 2D space of Q and α x /β x . RBE 2Gy was observed to increase with increasing Q in the low-Q domain (Q < approx. 3 [A MeV −1 ]) and to decrease with increasing Q in high-Q domain. This resembles the well-known overkilling effect. The same SD values are shown in figure 5 as a 2D color map overlaid by the experimental data points from the PIDE dataset. The model uncertainty was observed to be comparably low in regions of high data point density and particularly high in regions where experimental data points were missing.

Comparison with other RBE models
The experimental RBE 10 values for HSG cells (α x /β x = 5.09 Gy) irradiated with helium and carbon ions as reported by (Furusawa et al 2000) were compared to the model predictions given by LEM IV (Elsässer et al 2010) and the present NN model using Q and α x /β x . They are shown as a function of LET in figure 6(a). In figure 6(b), the same experimental data and model predictions given by the NN Q model were compared to the earlier proposed linear Q model (Tian et al 2022) but shown as a function of Q/(α x /β x ).
For the RBE 10 of helium and carbon irradiation, the r2 and MSE between the NN model prediction and experimental RBE data were 0.85 and 0.08, respectively (figure 7). For the LEM IV model interpolation, the r2 and MSE were 0.82 and 0.10, respectively. Accordingly, both models were comparable in terms of r2 and MSE. Systematic bias for different particles was observed for both RBE models: for the LEM IV model interpolation,  . Color map of the RBE 2Gy standard deviation of 100 trained RBE 2Gy (Q, α x /β x ) models with an overlay of the experimental data points, which are color coded by ion type. The standard deviation is particularly high in regions with no experimental data points. the residuals of helium and carbon RBE 10 were −0.24 ± 0.17 and 0.22 ± 0.17, respectively. For the NN model prediction, the residual values of helium and carbon RBE 10 were 0.19 ± 0.28 and −0.15 ± 0.22, respectively.
Both, the NN and the linear Q model were observed to follow a similar trend in the Q/(α x /β x ) interval of (0, 0.4) (A·MeV·Gy) −1 . However, the linear model cannot predict experimentally derived RBE 10 data in the domain of overkilling, i.e. for Q/(α x /β x ) larger than 0.4 (A·MeV·Gy) −1 in this case.

Discussion
In this work, we aimed to test the concept of Q-driven RBE modeling, i.e. irradiation with different ions has similar RBE at similar Q level over a wide LET range, including the domain of so-called overkilling. For this purpose, a data-driven NN model irradiations was built that only takes Q and α x /β x as input to predict RBE (defined by either reference photon dose or cell survival level) for different ions. The prediction power was evaluated (coefficient of determination) to be near 0.8 for the RBE defined by either the clinically relevant dose interval of 2-4 Gy or cell survival level of about 10%. No significant ion dependence was found in the Q-based prediction of RBE in the mentioned intervals, i.e. the Q concept was not rejected. In addition, the RBE 10 prediction was observed to be comparable to LEM IV regarding accuracy and precision. The relevance of a Q model that does not depend on ion type could be the consolidation of clinical RBE research for different ions in the future.
The considered combinations of candidate inputs, i.e. (Q, LET, α x /β x ), (Q, α x /β x ) and (LET, α x /β x ), were compared in terms of the difference between predicted and experimentally derived RBE 2Gy . The model taking (LET, α x /β x ) showed significant ion dependence and worst performance and, thus, was abandoned. Compared to the model of (Q, α x /β x ), predictions based on the model using (Q, LET, α x /β x ) as input were not found to be better despite the additional information of LET. Considering that unnecessary input dimensions may degrade the data efficiency due to potential overfitting (Hastie et al 2009), the input of the final NN model proposed in this work was set to (Q, α x /β x ). From a modeling point of view, the 'underlying assumption' of the models using (Q, α x /β x ) and (LET, Q, α x /β x ) can be regarded as 'RBE can be predicted given Q and α x /β x ' and 'RBE can be predicted given LET, Q and α x /β x '. As particle type can be deduced if both Q and LET are given, the assumption of (LET, Q, α x /β x ) model is equivalent to 'RBE can be predicted given particle type, LET and α x /β x ', which is generally applied by most (ion-specific) LET-driven RBE models. The Q model was shown to have no worse performance compared to this kind of ion-specific LET model.
It is well-known that the application of NN models should be limited to interpolation. This limitation was clearly observed for the Q-driven NN RBE model. It can be seen in figure 5 that the model uncertainty in the Q-α x /β x domain covered by data points is much smaller (σ around 0.5) than the uncertainty in the remaining 'extrapolation' domain. Generally, model extrapolation should be treated cautiously, since the extrapolation of a model cannot be verified by experimental data. The same limitation applies to the model in terms of the dependence on dose and cell survival. Measures of the model prediction power, r2 and MSE, were shown to be better within a certain photon reference dose (2-4 Gy) or cell survival interval (near 10%). We believe this may be related to our inference that RBE values calculated in these domains are generally better supported by the consistency of experimental measurements (cf appendix). The currently resulting limitations do not prevent Note that the splitting of data in training and test sets used here was not done following unbiased modeling rule, instead it was done non-randomly for the purpose of fair comparison with LEM IV. Thus, the systematic error seen here is not suitable for the evaluation of ion dependence of the model. future improvement of the NN model in these domains, since data could eventually be measured in all domains of relevance and fed into the model. Moreover, this can also be seen as an advantage of the NN model since the prediction uncertainty can be used as an indicator of how strongly the model prediction was supported by the available experimental evidence.
The NN model provided RBE predictions that are comparable to those by LEM IV. The NN model relies primarily on experimental data rather than pre-knowledge of, e.g., microdosimetric dose distribution, detailed information on the cells, biological effect at extremely high dose (near the center of the ion track). This may allow the NN model to be more flexible and less demanding regarding the needed input data when trained in a clinical setting. In fact, the model was intentionally developed based on only two parameters, Q and αx/βx, that are clinically accessible to enable clinical application in the future after successful clinical training and testing.
For the modeling of clinical RBE, factors beyond the physical and biological process within the cell should be considered including institute-specific factors, e.g. specific irradiation conditions and medical decisions (Karger and Peschke 2017). In this work, experimental details on, e.g., energy spectrum, secondary particles, institutional differences including biological protocols and the level of their specification vary between the records in the PIDE dataset. The data-driven Q model showed that the in vitro RBE is predictable in the domain of clinically relevant dose level by using only Q and α x /β x as input but without the need for a specific previously known formalism, e.g. the formulas used in most phenomenological models (Tilly et al 2005, Carabe-Fernandez et al 2007, Wedenberg et al 2013, McNamara et al 2015, Mairani et al 2017, McMahon 2021 and model parameters in mechanistic models (Hawkins 1998, Elsässer et al 2010. Going from an in vitro to a clinical endpoint, the use of Q and α x /β x as an input allows for flexible data-driven RBE modeling. A systematic deviation between experimental data and the prediction for different ions was observed when the NN model was applied to predict the data reported by Furusawa et al (2000). This does not conflict with the conclusion of the ion-dependence test, as for this case, the NN model was trained on all data but those from one specific publication (Furusawa et al 2000), and then tested on this particular publication (Furusawa et al 2000).
As training data and test data were divided systematically (one particular publication), a systematical error was not unexpected. This test design serves only as an example for the comparison with the other RBE models but is unsuited to test a systematic bias of the model. In addition, systematic deviations between the same experimental data and the predictions for LEM IV (Elsässer et al 2010) were observed, too.
Future work on Q modeling needs to focus on investigating how Q can be uniquely investigated in a spreadout Bragg peak as well as demonstrating the validity of the Q concept (i.e. the RBE of different particles follows the same trend when characterized by Q) for in vivo and clinical data. Yet, some clinical studies on brain toxicity associated with a variable RBE have emerged for patients treated with both protons (Bahn et al 2020, Eulitz et al 2019, 2023 and carbon ions (Koto et al 2014, Shirai et al 2017 and could be considered as a potential clinical endpoint of clinically related future studies.
Since Q is a relatively simple physical quantity, it can be easily implemented in treatment planning systems and used in place of LET in biological effectiveness guided treatment plan optimization that is emerging for proton therapy .

Conclusion
In this work, data-driven non-linear RBE modeling based on Q was proposed, analyzed and compared to experimental in vitro data as well as to a clinically applied RBE model. Using Q and α x /β x as input, the RBE at a clinically relevant dose range (2-4 Gy) can be predicted without explicit knowledge of ion type. This suggests the possibility of an empirical, ion-independent clinical RBE model that supports the transfer of RBE knowledge from better-to less well-studied ions, ultimately advancing clinical RBE research.

Data availability statement
No new data were created or analyzed in this study. Data will be available from 31 January 2023.
The RBE dx is the ratio of the dose of the reference photon d x and the corresponding ion dose d i that result in the same biological effectiveness which is described by formula (A1).
The RBE dx can be calculated by: where S x is the survival fraction of corresponding photon irradiation, α x , β x , α i and β i are recorded in the PIDE. The RBE 10 is the ratio of the dose of reference photon dose d x and the dose of the ion d i when both result in 10% survival fraction. The RBE 10 can be calculated by: x,10 x 2 x x x While the experimentally derived RBE 10 was calculated using the α x , β x , α i and β i recorded in the PIDE, the RBE 10 predicted by the linear model was calculated using the α x and β x recorded in the PIDE and the α i and β i predicted by the model: where k Q = 15.5 A·MeV·Gy (Tian et al 2022).
The performance of the NN model predicting RBE dx at d x levels of 0, 1, 2, 4 and 10 Gy as well as RBE S at cell survival S of 0.1%, 1.0%, 10.0%, 50.0% and 90.0% was compared using the same training (80%) and test (20%) data sets.
As the magnitudes of the experimentally derived RBE at different d x or survival level are different (cf tables A1 and A2, respectively), the MSE of the relative error, instead of the error values discussed in the manuscript, were used for comparison. Other evaluation metrics, i.e. r2 score and ANOVA tests are compared as well. The results are shown in tables A1 and A2. Considering the r2 score and MSE (relative) tradeoff, the performance of the model was considered to be better for d x between 2-4 Gy and cell survival around 10%, this is consistent to the result of correlation analysis (cf figure 2).
Note that the experimentally derived RBE is calculated using PIDE-recorded α and β values, which were obtained by fitting the measured cell survival data points using the LQ model. However, the obtained values for α and β also depend on the applied fitting conditions. Refitting the same experimentally measured data points, the PIDE group obtained and recorded also different sets of α and β values. The resulting effect on experimentally derived RBE values was measured by comparing the two RBE values calculated either based on an α & β set fitted by the original experimenters or by the PIDE group. For a quantitative comparison, the r2 score between the two experimentally derived RBE values was applied at different levels of dose and cell survival. The result is shown in figure A1. Note that no RBE modeling is involved in this analysis. Table A1. The performance of the models of RBE dx (Q, α x /β x ) at given photon reference dose, d x , level of 0, 1, 2, 4 and 10 Gy. The RBE max , i.e. α i /α x , is approximately regarded as RBE 0Gy . The performance is measured by the coefficient of determination (r2 score), mean square error (MSE) of the relative difference, p value of the ANOVA test.  Figure A1. The coefficient of determination (r2) of the RBE calculated using the PIDE and original alpha-beta sets for the same experiments. The RBE shown on the left and right are defined by the endpoints reference photon dose d x and given cell survival, respectively.