A Novel Mechanistic Cell Survival Model to Replace Linear-Quadratic Model for Relative Biological Effectiveness (RBE) in Proton Therapy

Abstract

<h3>Purpose/Objective(s)</h3> Relative biological effectiveness (RBE) is important in proton therapy. Although a large amount of in vitro cell experimental data exists for various linear energy transfers (LETs) and various cell lines, it is difficult to derive a reliable RBE model from these data using the classic linear-quadratic (LQ) model, mainly because its parameters, α_LQ and β_LQ, have no clear mechanistic correlation with LET. In this study, we hypothesize that a mechanistic model with parameters directly relate to LET can be developed to replace the LQ model for cell survival description, and it would be better than the LQ model for RBE model derivation. The 2 parameters in the model are α_M, which represents the ability to generate cell damage and may linearly increase with LET, and β_M, which represents the ability to repair cell damage and may linearly decrease with LET (because higher LET generates more clustered DNA damages, which are difficult to repair). <h3>Materials/Methods</h3> We used the number of unit damages to score the severity of radiation-generated damage in a cell. Assuming that a dose of D generated an average number of α_M*D unit damages, the probability of cells having n (n = 0, 1, 2, ...m) unit damages was thus determined using the Poisson Equation, P(n) = [(α_MD)ⁿ/n!]*exp(-α_MD). We further assumed that a cell with n unit damages has a probability of β_Mⁿ to be repaired, and cells having 4 or more unit damages would not be repaired. Thus, the survival fraction of cells after a dose of D was derived as SF(D) = [1+α_Mβ_MD+(α_Mβ_MD)²/2+(α_Mβ_MD)³/6]*exp(-α_MD). A total of 47 experimental cell survival datasets for 3 cell lines in the literature were used to test this mechanistic model. Correlations of α_M and β_M with LET were determined in terms of RBE_α and RBE_β, which is the ratio of α_M (or β_M) in proton and photon. A RBE model was consequently derived. <h3>Results</h3> The mechanistic model fitted well with all experimental data, similar as the LQ model. Interestingly, for two cell lines with pristine proton beam data, both α_M and β_M were perfect linear functions of LET, with R² = 0.98∼1.0. On the other hand, the linearity was greatly degraded for the corresponding spread-out Bragg Peak (SOBP) data, with same data points deviating at same directions for 2 cell lines, suggesting that the LET values calculated in SOBP were not as accurate as pristine beam. For the 3^rd cell line with majority data points of SOBP beams, R² = 0.89∼0.91. The correlation of α_M and β_M with LET were expressed as RBE_α = 1.1+k_α*LET and RBE_β = 1.0-k_β*LET, where k_α = 0.018, 0.009 and 0.005, k_β = 0.031, 0.028, and 0.027 for 3 cell lines, respectively. A RBE model depending on LET, dose and cell type (represented by α_M and β_M in photon) was finally derived. <h3>Conclusion</h3> A mechanistic cell survival model was developed and it appeared to have advantage over the LQ model in deriving a reliable RBE model.