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# Variations of the relative biological effectiveness with fractionation in proton therapy: analysis of prostate cancer response

Mar 2023

Purpose: To present a methodology to analyze the variation of the relativebiological effectiveness (RBE) with fractionation from clinical data of tumorcontrol probability (TCP) and to apply it to study the response of prostatecancer to proton therapy. Methods: We have analyzed the dependence of the RBE on the dose per fractionby using the LQ model and the Poisson TCP formalism. Experimental TCPs forprostate cancer (low risk, LR, and intermediate risk, IR) treated with photonand proton therapy for conventional fractionation (2 GyE$\times$37 fractions),moderate hypofractionation (3 GyE$\times$20 fractions) and hypofractionation(7.25 GyE$\times$5 fractions) were obtained from the literature and analyzed. Results: The dependence of the RBE on the dose per fraction presents threetheoretical regions, with RBE monotonically decreasing, increasing or constantwith the dose per fraction, depending on the change of ($\alpha$, $\beta$)values between photon and proton irradiation (the equilibrium point being at($\alpha_p$/$\beta_p$)=($\alpha_X$/$\beta_X$)($\alpha_X$/$\alpha_p$)). Ananalysis of the clinical data shows RBE values that decline with increasingdose per fraction: for LR RBE=1.124, 1.119, and 1.102 for 1.82 Gy, 2.73 Gy and6.59 Gy per fraction (proton doses), respectively; for IR RBE=1.119, and 1.102for 1.82 Gy, and 6.59 Gy per fraction, respectively. Conclusions: Our analysis shows a monotonically decreasing RBE withincreasing dose per fraction for prostate cancer, which is expected from the LQformalism and the changes in ($\alpha$, $\beta$) between photon and protonirradiation. However, due to the limited number of schedules and patients perschedule, the observed decrease of RBE with increasing dose is not significantfor LR prostate cancer. For IR prostate cancer, the RBE obtained at 7.25 GyEseems to be significantly lower than that obtained at 2 GyE.

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