Mechanistic Radiobiological Models for Repair of Cellular Radiation Damage

A comprehensive biophysical and chemical analysis of surviving fractions (SFs) of irradiated cells is reported. Cellular repair of radiation-induced damage is the principal focus in the great majority of the presented derivations. Steady-state and dynamic formalisms are developed using difference and differential equations with and without the delay effect. Within chemical kinetics and nondelayed differential rate equations with time as the independent variable and dose as a parameter, repair mechanism is investigated through enzyme catalysis. Here, the Michaelis-Menten concept is used in the Briggs-Haldane formalism and the accompanying quasi-stationary state (QSS) approximation. This leads to the integrated Michaelis-Menten (IMM) radiobiological model for cell survival in terms of the principal-value Lambert W-0 function. The same problem of repair is also addressed by employing the concept of delayed differential equations (DDEs) with radiosensitivity as the independent variable. Such an investigation gives the Lambert delayed dynamics (LDD) model with the cell surviving fraction which is also expressed by way of W-0. With repair being taken into account, only three biologically interpretable parameters are invoked in both the IMM and LDD model. For this reason, such models are computationally attractive. The Lambert function W-0 can be generated almost instantly from the freely available algorithms. Importantly, the IMM and LDD models possess the correct asymptotic behaviors of the exponential inactivations at both small and large radiation doses as also encountered in the corresponding experimental data for cell survival. The IMM and LDD radiobiological models are universally valid at all doses. This makes them particularly suitable for conventional and nonconventional treatment schedules with small and large doses per fraction, respectively. By contrast, the current workhorse of dose planning systems in radiotherapy, the linear-quadratic (LQ) model, is adequate only at low doses, but flagrantly fails at high doses. In comparisons with experimental measurements, the IMM and LDD models significantly outperform the LQ model both regarding SFs and the so-called full-effect plots that can exhibit a special kind of saturation effect.