## 4 Discussion
Although the approximate microdosimetry model included in the MCDS is \(\sim\)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 \(\text{LET}_{\text{D}}\) than for \(y_{\text{D}}\), 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 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Characteristics** & **Empirical Models** & **Biophysical Models** \\ \hline \multirow{7}{*}{Strengths} & Parameters and functional form & Reflects plausible biophysical \\  & relative to LET directly derived & mechanisms related to cellular \\  & from measurements & damage and repair \\ \cline{2-3}  & & Able to interpolate and \\  & & extrapolate across multiple ion \\  & Often simple algebraic formulas & cell types with fewer purely \\  & that relate LET to cell survival & adjustable parameters. Tends to \\  & & reduce number of ad hoc \\  & & adjustable parameters \\ \cline{2-3}  & & Explicit consideration of cellular \\  & Often specific to one particular & endpoints that facilitates \\  & type of ion, cell line and endpoint & comparisons of RBE estimates \\  & & across cell lines and ion types for \\  & & a wide range of LET values \\ \hline \multirow{7}{*}{Limitations} & Estimates of parameter values are & May involve auxiliary \\  & uncertain and highly specific to & microdosimetric concepts, more \\  & the cell line, endpoint and & specific to the cellular endpoint \\  & experimental design & of interest \\ \cline{2-3}  & May give a false sense of & \\  & confidence when applied to & May require additional \\  & alternate in vitro and & computational complexity \\  & in vivo conditions & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of empirical and biophysical models of cellular RBE. and at proton energies below a few tens of MeV. In these cases, significant deviations from the fixed value \(\mathrm{RBE}=1.1\) are noticed (see Figure 3). More specifically, for cells with high \((\alpha/\beta)_{R}\), the mean \(\mathrm{RBE}\) of the empirical models (\(\mathrm{RBE}_{\mathrm{emp}}\)) varies from \(\sim\)1.0 (\(>\)50 MeV) to \(\sim\)2.0 (\(<\)50 MeV) whereas for cells with lower \((\alpha/\beta)_{R}\) it can reach values up to \(\sim\)2.5. However, the variation among the individual (empirical) \(\mathrm{RBE}\) models may be significant. For example, at high energies, the individual (empirical) models differ from the mean value (\(\mathrm{RBE}_{\mathrm{emp}}\)) by \(\pm\)10%, while at low energies the variation reaches \(\pm\)30%. The results of biophysical models also support a variable \(\mathrm{RBE}\) yielding a similar trend with the empirical models. However, biophysical models predict systematically lower \(\mathrm{RBE}\) values from the empirical models. For example, at high energies (\(>\)50 MeV), biophysical models predict up to 10% lower values from \(\mathrm{RBE}_{\mathrm{emp}}\), whereas at lower energies (\(<\)50 MeV) the differences can reach up to 30%. For cells with high \((\alpha/\beta)_{R}\), the \(\mathrm{RBE}\) of the biophysical models varies (on average) from \(\sim\)1.0 (\(>\)50 MeV) to \(\sim\)1.7 (\(<\)50 MeV), whereas for cells with lower \((\alpha/\beta)_{R}\) it can reach values up to \(\sim\)2.0. As regards the differences among the biophysical \(\mathrm{RBE}\) models (TDRA vs. MKM), they are up to \(\sim\)5% at high energies but reach \(\sim\)30% at the low-energy end.