A closed-form description of tumour control with fractionated radiotherapy and repopulation

Purpose: To derive a closed form expression of tumour control probability (TCP) following the geometric-stochastic approach of Tucker and Taylor. Methods: A model was constructed based upon a Galton-Watson branching process with cell killing represented by a Bernoulli random variable, and repopulation represented by a Yule-Fury process. A closed-form expression of the probability-generating function was derived, which yielded an explicit expression for the mean number of surviving clonogens and the TCP. Results: The mean number of surviving cells, after i clonogens have been treated with n fractions of irradiation, was i(se(lambda Delta t))(n), where s is the surviving fraction, lambda is the rate of cell division, and Delta t is the interfraction time interval. The tumour control probability was {[s-1-(s-1) (se(lambda Delta t))(n)]/[s-1-(e(-lambda Delta t)-1) (se(lambda Delta t))(n)]}(i). Conclusions: Tucker and Taylor provided improvements upon the conventional Poisson model for TCP, mainly through numerical simulation. Here a model based up on their geometric-stochastic approach has been derived in closed form. The resultant equations provide a simpler alternative to numerical simulation allowing the effects of fractionated radiotherapy on a replicating population of tumour cells to be more easily predicted.