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

TL;DR

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 is presented and it is shown that the method presented is equivalent to a first-order Euler numerical method of the differential LQ model.

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.

Figures (4)

• Figure 1: Surviving fractions versus dose at different initial oxygenation levels (1, 5, 10 and 20 mmHg) for CONV-RT (solid line) and FLASH-RT computed with different methods to account for the effect of oxygen depetion: Method 1, averaging of the surviving fraction; Method 2, averaging of $\alpha$ and $\beta$; Method 3, averaging of OERs; Method 4, Zhu's iterative method; and Method 5, the differential LQ method.
• Figure 2: Differences between surviving fractions for FLASH-RT obtained with the different methods used in this work to account for the effect of oxygen depletion, using Method 5 (differential LQ method) as reference. Results are presented for different doses and initial oxygenations.
• Figure 3: Illustration of the problem with method Method 1 for $p(0)$=10 mmHg and $D$=40, 50 Gy. The oxygenation reaches very low values towards the end of the irradiation, which causes a significant spike of $\mathit{SF}(p(t),D)$. When these values are averaged linearly, this spike at the end of the irradiation dominates the average, effectively masking the damage occurring during the early stage of the irradiation.
• Figure 4: Surviving fractions computed with the different methods versus dose for two spatially heterogeneous oxygenation distributions corresponding to poorly and moderately well oxygenated tissue. We also include the surviving fractions for conventional irradiation (i.e., no ROD effect) and the oxygenation histograms before irradiation. Method 1, averaging of the surviving fraction; Method 2, averaging of $\alpha$ and $\beta$; and Method 5, the differential LQ method.