On the calculation of the radiobiological effect of radiolytic oxygen depletion in FLASH radiotherapy

Abstract: Objective: Radiolytic oxygen depletion (ROD) may play a role in the sparing of cells irradiated with ultra-high dose rates. Different methods have been used to quantify the effect of ROD during FLASH irradiation on cell survival, typically involving some kind of averaging of the oxygen effect and the LQ model. In this work, we compare the results obtained with several of these methods and introduce a novel method based on the non-linear differential form of the LQ model. Approach: We present a novel method to account for a varying oxygen concentration on the dose-response based on the non-linear differential form of the LQ model, and we compare the results obtained with this method with those obtained with other methods that linearize the averaging of the oxygen effect during irradiation. Main results: We found differences in the surviving fractions obtained with the method introduced in this work and other methods that introduce different linearizations (averaging) of the non-linear dependence on the oxygen concentration, especially for oxygenations and doses that lead to important changes in the OERs during the delivery of the dose (initial oxygenations $\approx$5--10 mmHg and doses $>30$~Gy). On the other hand, we showed that the method presented by Zhu \emph{et al.} is equivalent to a first-order Euler numerical method of the differential LQ model. Significance: The method introduced in this work and the method of Zhu \emph{et al.} may allow a more precise quantification of the effect of ROD on dose-response, both for tumors and normal tissues. While all the reviewed methods show an oxygen-dependent sparing effect of FLASH radiotherapy driven by ROD and qualitatively similar results, the method introduced in this work and that of Zhu \emph{et al.} may be more suitable to quantitatively analyze new preclinical (and future clinical) data coming from experimental studies.