Survival of radiation-damaged cells via mechanism of repair by pool molecules: the Lambert function as the exact analytical solution of coupled kinetic equations

Repair of radiation damaged cells can be carried out through their interactions with intracellular substances that can supply the needed energy for repair. These substances may be viewed as forming a pool of repair molecules that through chemical reactions with lesions can lead to cell recovery from the initial radiation insult by deposition of dose . Presently, time evolution of mean concentrations of interacting substances is obtained by solving the corresponding rate equations given by a coupled system of second-order non-linear differential equations that are imposed by the mass action law. For cell surviving fractions after irradiation, the most important quantity is the time-dependent concentration of lethal lesions. Our main working hypothesis is that pool substances are capable of repairing the inflicted injury to any cell molecules, including deoxyribonucleic acid which is generally viewed as the most critical target of radiation. The previous solution of these rate equations is only formal as it is expressed by yet another equation of an implicit, transcendental form. In the earlier applications, this formal solution has only been tackled by numerical means that, however, have no connection with any of the myriad of the usual explicit forms of cell surviving fractions. This drawback effectively discouraged researchers from further explorations of the otherwise attractive pool methodology. Such a circumstance is unfortunate in light of a clear and advantageous radiobiological interpretation of the parameters of this theoretical formalism of chemical kinetics.The present study is aimed at rescuing the pool methodology by solving the underlying transcendental equation for lethal lesions uniquely, exactly and explicitly in terms of the principal value Lambert function. This is a single-valued and dose-dependent function, which can be readily and accurately computed either from the available fast numerical algorithms or by employing the existing simple closed expressions with a quotient of elementary, logarithmic functions. Another distinct advantage of this analytical result is the known behaviors of at small and large doses. This permits an easy and immediate identification of the final (or ) dose and the extrapolation number . Such a circumstance offers new possibilities within the presently proposed "Pool Repair Lambert" (PRL) model for analysis of problems encountered in assessing cell survival after exposure to various modalities of radiation, including different schedules (acute, fractionated) for the same radiation quality. Importantly, the PRL model is universally applicable to all doses with a smooth passage from low through intermediate to high doses. As to applications in radiotherapy, this feature is particularly important for treatment schedules with high-doses per fraction as in stereotactic radiosurgery and stereotactic body radiotherapy.