Distribution of the number of clonogens surviving fractionated radiotherapy: a long-standing problem revisited

Purpose: A long-standing problem is addressed: what form of the probability distribution for the number of clonogene tumor cells remaining after fract ionated radiotherapy should be used in the analysis aimed at evaluating the efficacy of cancer treatment? Over a period of years, a lack of theoretica l results leading to a closed-form analytic expression for this distributio n, even under very simplistic models of cells kinetics in the course of fra ctionated radiotherapy, was the most critical deterrent to the development of relevant methods of data analysis. Materials and methods: Rigorous mathematical results associated with a mode l of fractionated irradiation of tumors based on the iterated birth and dea th stochastic process are discussed. Results: A formula is presented for the exact distribution of the number of clonogenic tumor cells at the end of treatment. It is shown that, under ce rtain conditions, this distribution can be approximated by a Poisson distri bution. An explicit formula for the parameter of the limiting Poisson distr ibution is given and sample computations aimed at evaluation of the converg ence rate are reported. Another useful limit that retains a dose-response r elationship in the distribution of the number of clonogens has been found. Practical implications of the key theoretical findings are discussed in the context of survival data analysis. Conclusions: This study answers some challenging theoretical questions that have been under discussion over a number of years. The results presented i n this work provide mechanistic motivation for parametric regression models designed to analyze data on the efficacy of radiation therapy.