Authors: Dževad Belkić
Publish Date: 2014/02/11
Volume: 52, Issue: 5, Pages: 1253-1291
Abstract
Michaelis–Menten secondorder chemical kinetics is used to describe the three main mechanisms for surviving fractions of cells after irradiation These are a direct yield of lethal lesions by single event inactivation metabolic repair of radiation lesions and transformation of sublethal to lethal lesions by further irradiations The mass action law gives a system of timedependent differential equations for molar concentrations of the invoked species that are the DNA substrates as lesions enzyme repair molecules the product substances etc The approximate solutions of these coupled rate equations are reduced to the problem of finding all the roots of the typical transcendental equation axmathrmebx=c with xge 0 being a real variable where ab and c are real constants In the present context the unique solution of this latter equation is given by x=1/bW 0bc/a where W 0 is the principalbranch realvalued Lambert function Employing the concept of Michaelis–Menten enzyme catalysis a new radiobiological formalism is proposed and called the “Integrated Michaelis–Menten” IMM model It has three doserange independent parameters ingrained in a system of the rate equations that are set up and solved by extracting the concentration of lethal lesions whose time development is governed by the said three mechanisms The indefinite integral of the reaction rate is given by the Lambert W 0 function This result is proportional to the sought concentration of lethal lesions Such a finding combined with the assumed Poisson distribution of lesions yields the cell surviving fraction after irradiation Exploiting the known asymptotes of the Lambert W 0 function the novel doseeffect curve is found to exhibit a shoulder at intermediate doses preceded by the exponential cell kill with a nonzero initial slope and followed by the exponential decline with the reciprocal of the D 0 or D 37 dose as the final slope All three dose regions are universally as well as smoothly covered by the Lambert function and hence by the ensuing cell surviving fractions The outlined features of the proposed IMM model stem from a comprehensive mechanistic description of radiationlesion interactions by means of kinetic rate equations They are expected to be of critical importance in new doseplanning systems for high doses per fraction where the conventional linearquadratic radiobiological modeling is demonstrably inapplicable
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