## 4 Discussion  
Proton RBE was evaluated for five cell types with \(\left(\alpha/\beta\right)_{R}\) 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 \(\upmu\)m) 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, LET\({}_{D}\) and \(\left(\alpha/\beta\right)_{R}\). However, the application range of empirical models is generally limited by the experimental data that they use. For example, the [PERSON] et al., [PERSON] et al., and [PERSON] et al. models are based on data for a single cell line (V79), while the [PERSON] et al. and [PERSON] et al. models are based on two cell lines covering a small range of \(\left(\alpha/\beta\right)_{R}\) 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 [PERSON] et al. and [PERSON] et al. are based on a more extensive set of experimental data covering a  
Figure 7: Absolute relative percentage difference (Equation (37)) between the RBE models and the conventional value RBE = 1.1 at three LET\({}_{D}\) values corresponding (approximately) to the proximal SOBP region (\(\sim\)1 keV/\(\upmu\)m), the center of the SOBP (\(\sim\)2.5 keV/\(\upmu\)m) and the distal edge of the SOBP (\(\sim\)8 keV/\(\upmu\)m) for five cell types at 2 Gy, 5 Gy and 7 Gy dose fractions. broad range of different cell lines. Biophysical models that incorporate well-studied cellular damage (including DNA damage) and repair mechanisms of action (reviewed in [PERSON] 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 (\(\text{LET}_{\text{D}}\)) vs. dose-averaged lineal energy (\(y_{\text{D}}\)), at spatial scales relevant to cellular damage. Most empirical models use \(\text{LET}_{\text{D}}\) as a radiation quality index whereas biophysical models commonly use \(y_{\text{D}}\) instead. Thus, the calculation of proton \(\text{LET}_{\text{D}}\) and \(y_{\text{D}}\) values over a broad energy range is a useful prerequisite for many proton RBE models. In the present work, \(\text{LET}_{\text{D}}\) and \(y_{\text{D}}\) 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 \(y_{\text{D}}\) and \(\text{LET}_{\text{D}}\) values compared to the analytic model over the proton energy range examined. Specifically, MCDS-based \(\text{LET}_{\text{D}}\) values are constantly \(\sim\)10% lower than the analytic-based \(\text{LET}_{\text{D}}\) values, mostly likely because of the delta-ray effects.