## 4 Discussion
It follows from the results depicted in Figures 2 and 3 that proton \(\mathrm{RBE}\) is variable over the proton energy range studied (1-250 MeV) and may differ significantly from the fixed value of \(\mathrm{RBE}=1.1\), 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 (\(\sim\)30%), precluding a firm conclusion regarding the exact \(\mathrm{RBE}\) variation at these energies. To obtain a single index of the deviation of the various \(\mathrm{RBE}\) models from the conventional value of \(\mathrm{RBE}=1.1\), we calculated the mean percentage difference (MPD) as defined by Equation (38). Figure 4 shows the MPD of the examined \(\mathrm{RBE}\) models for the various cell types at three different dose levels (2 Gy, 5 Gy and 7 Gy). At hypoffractionated dose fractions (5-7 Gy), the deviations of \(\mathrm{RBE}\) models from the conventional value \(\mathrm{RBE}=1.1\) is clearly less pronounced than for the conventional dose fraction (\(\sim\)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 \(\mathrm{RBE}\) at large dose fractions (5 Gy and 7 Gy) compared to the conventional dose fraction (\(\sim\)2 Gy). To examine how the \(\mathrm{RBE}\) values are affected by the calculation methods of \(\mathrm{LET}_{\mathrm{D}}\) and \(y_{\mathrm{D}}\), we compared MCDS-based and analytic-based \(\mathrm{RBE}\) calculations (Figure 5). For the empirical \(\mathrm{RBE}\) models, MCDS-based \(\mathrm{RBE}\) values are lower from the analytic-based values by up to \(\sim\)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 \(\mathrm{RBE}\) (\(\mathrm{RBE}_{\mathrm{emp}}\)) against the TDRA-based \(\mathrm{RBE}\) and the MKM-based \(\mathrm{RBE}\) for three \(\mathrm{LET}_{\mathrm{D}}\) values which correspond (approximately) to different depths of the SOBP, namely, the proximal SOBP (\(\sim\)1 \(\mathrm{keV}/\upmu\)), the center of SOBP (\(\sim\)2.5 \(\mathrm{keV}/\upmu\)), and the distal SOBP (\(\sim\)8 \(\mathrm{keV}/\upmu\)) for three different dose levels (2 Gy, 5 Gy and 7 Gy) [1]. At conventional dose fractions (\(\sim\)2 Gy), the two biophysical models (TDRA, MKM) predict that, at the proximal SOBP, the \(\mathrm{RBE}\) is \(\sim\)1.0 regardless of the cell type, while \(\mathrm{RBE}_{\mathrm{emp}}\) 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 \(\mathrm{RBE}\) is \(\sim\)1.05, while \(\mathrm{RBE}_{\mathrm{emp}}\) values range from 1.05 to 1.2 (again, increasing with decreasing \((\alpha/\beta)_{R}\) ratio). At the distal SOBP, the \(\mathrm{RBE}\) takes the highest values and its dependence on the cell type is more pronounced, e.g., the \(\mathrm{RBE}\) of the biophysical models varies from 1.1 to 1.3 while the \(\mathrm{RBE}_{\mathrm{emp}}\) varies from 1.2 to 1.5 (in both cases, increasing with decreasing \((\alpha/\beta)_{R}\) ratio). At hypoffractionated dose fractions (5 Gy and 7 Gy), the \(\mathrm{RBE}\) values at the proximal and central regions of the SOBP do not differ from the values observed for the conventional dose fraction (\(\sim\)2 Gy). However, at the distal SOBP, the \(\mathrm{RBE}\) for the 5 Gy and 7 Gy dose fractions is lower compared to the \(\mathrm{RBE}\) for a 2 Gy dose fraction, e.g., the biophysical models predict an \(\mathrm{RBE}\) of 1.05 to 1.15 (vs. 1.1-1.3), while the \(\mathrm{RBE}_{\mathrm{emp}}\) ranges from 1.15 to 1.3 (vs. 1.2-1.5).