Mechanistic description of survival of irradiated cells: repair kinetics in Padé linear-quadratic or differential Michaelis–Menten model

Abstract

Biophysical models for repair mechanisms for cell surviving fractions \(S_\mathrm{F}(D)\) after exposure to radiation are studied. The principal focus is on a novel theory, the Pade-linear quadratic (PLQ), or equivalently, the differential Michaelis–Menten radiobiological model, which predicts \(S_\mathrm{F}(D)\) as a function of the absorbed dose \(D\) in the form \(S_\mathrm{F}(D)=\exp {\{-(\alpha D+\beta D^2)/(1+\gamma D)\}}\), with a clear biological and clinical meaning of the three parameters \(\alpha , \beta \) and \(\gamma \). It is shown that this functional form in the PLQ model emerges directly from the simultaneous fulfillment of the requirements for the correct asymptotic behaviors of the repair function at low and high doses. Moreover, this automatically secures the purely exponential cell kill modes at both small and large \(D\), as also encountered in the corresponding experimental data for cell surviving fractions. Further, it is demonstrated that the PLQ-based repair function, given by a rectangular hyperbola, coincides with the reaction velocity for enzyme catalysis from the Michaelis–Menten mechanism. This repair velocity is the halved harmonic mean of the low- and high-dose asymptotes of the catalytic repair function. Such circumstances constitute a firm mechanistic basis of the PLQ model, which is shown to exhibit excellent agreement with measurements. Robust applications of the PLQ model are anticipated, especially in hypofractionted radiotherapy, such as stereotactic radiosurgery.