Analytical expression of the β $\beta$ coefficient of cell survival curves predicted by the NanOx model in the low-energy range

Abstract

Background

In cancer research, clonogenic assays are often performed as a means to determine the response of a given cell line to radiation exposure. The resulting cell survival fractions as a function of absorbed dose are usually fitted to a linear-quadratic (LQ) expression involving two coefficients, $\alpha$ and $\beta$, describing the cell's radiosensitivity. However, $\beta$ is particularly hard to compute with accuracy. On the other hand, biophysical models are developed for predicting the enhanced biological efficiency of heavy ions compared to photons. These models provide a more mechanistic description of the biological effects induced by ionizing radiation, while allowing the estimation of the $\alpha$ and $\beta$ coefficients.

Purpose

In this work, we propose an analytical expression for the fast computation of the $\beta$ coefficient for ions with energies ranging from $\sim$1 to $\sim$25 MeV/n.

Methods

The analytical expression for $\beta$ was derived starting from the formalism of the NanOx biophysical model and introducing a set of approximations. The latter consider that the irradiation is carried out under track-segment conditions (as is the case in hadrontherapy) and with doses inducing a low number of impacts (i.e., of the order of some Gy). Moreover, it is assumed that the radiation tracks are narrow with respect to the sensitive volume (the cell nucleus) and that the fluctuations between radiation tracks remain small enough to work with the average values of the specific energy and the number of lethal events. Calculations of $\beta$ were performed for three cell lines (HSG, CHO-K1, and V79) irradiated with hydrogen, helium and carbon ions. The values of $\beta$ predicted with the analytical expression were compared with the results of a more time-consuming approach involving the computation of cell survival fractions with NanOx followed by a LQ fit. Our results were also compared with available experimental data and with the predictions of other biophysical models.

Results

We obtained an analytical expression for $\beta$ as a function of $\alpha$, the linear energy transfer (LET), the $\beta$ coefficient for reference radiation (photons) and the ratio of the chemical yields of hydroxyl radicals (${\mathrm{OH}}^{•}$) for the ion of interest and photons. We found that NanOx predicts a decrease in $\beta$ with increasing LET, showing a similar trend to some other models described in the literature. Moreover, the values of $\beta$ calculated with the analytical expression were in close agreement with the results obtained by applying a LQ fit to the cell survival fractions predicted with NanOx for the energy range in which the approximations underpinning the analytical approach hold.

Conclusions

We derived in this work an analytical expression for the fast calculation of the $\beta$ coefficient for ion irradiations. Although the analytical expression resulted from the NanOx model, it can be also applied to any model used to compute $\alpha$. Overall, the approach presented in this work may provide a reasonable description of the behavior of $\beta$ with LET despite the large spread observed in experimental data.