  We use a stochastic birth-death model for a population of cells to estimate
the normal tissue complication probability (NTCP) under a particular
radiotherapy protocol. We specifically allow for interaction between cells, via
a nonlinear logistic growth model. To capture some of the effects of intrinsic
noise in the population we develop several approximations of NTCP, using
Kramers-Moyal expansion techniques. These approaches provide an approximation
to the first and second moments of a general first-passage time problem in the
limit of large, but finite populations. We use this method to study NTCP in a
simple model of normal cells and in a model of normal and damaged cells. We
also study a combined model of normal tissue cells and tumour cells. Based on
existing methods to calculate tumour control probabilities, and our procedure
to approximate NTCP, we estimate the probability of complication free tumour
control.
