Tumour control probability: a formulation applicable to any temporal protocol of dose delivery

An analytic expression for the tumour control probability (TCP), valid for any temporal distribution of dose, is discussed. The TCP model, derived using the theory of birth-and-death stochastic processes, generalizes several results previously obtained. The TCP equation is TCP(t) = [ 1- (S(t) e((b-d)t) / (1+bS(t)e((b-d)t) integral 0t dt'/S(t') e((b-d)t'))](n) where S(t) is the survival probability at time t of the n clonogenic tumour cells initially present (at t = 0), and b and d are, respectively, the birth and death rates of these cells. Equivalently, b = 0.693/ T-pot and dib is the cell loss factor of the tumour. In this expression t refers to any time during or after the treatment; typically, one would take for t the end of the treatment period or the expected remaining life span of the patient. This model, which provides a comprehensive framework for predicting TCP, can be used predictively, or-when clinical data are available for one particular treatment modality (e.g. fractionated radiotherapy)-to obtain TCP-equivalent regimens for other modalities (e.g. low dose-rate treatments).