Stochastic pharmacodynamics of a heterogeneous tumour-cell population

Standard pharmacodynamic models are ordinary differential equations that lack the features of stochasti-city and heterogeneity. We develop and analyse a stochastic model of an idealised heterogeneous tumour-cellpopulation treated with a drug, where each cell has a different value of an attribute linked to survival.Once the drug reduces a cell’s value below a threshold, the cell is susceptible to death. The eliminationof the last cell in the population is a natural endpoint that is not available in deterministic models. Wefind formulae for the probability density of this extinction time in a collection of tumour cells, each with adifferent regulator value, under the influence of a drug. There is a logarithmic relationship between tumourpopulation size and mean time to extinction. We also analyse the population under repeated drug dosesand subsequent recoveries. Stochastic cell death and division events (and the relevant mechanistic para-meters) determine the ultimate fate of the cell population. We identify the critical division rate separatinglong-term tumour population growth from successful intermittent treatment.