Stochastic pharmacodynamics of a heterogeneous tumour-cell population

Standard pharmacodynamic models are ordinary differential equations without the features of stochasticity and heterogeneity. We develop and analyse a stochastic model of a heterogeneous tumour-cell population 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 elimination of the last cell in the population is a natural endpoint that is not available in deterministic models. We find formulae for the probability density of this extinction time in a collection of tumour cells, each with a different regulator value, under the influence of a drug. There is a logarithmic relationship between tumour population size and mean time to extinction. We also analyse the population under repeated drug doses and subsequent recoveries. Stochastic cell death and division events (and the relevant mechanistic parameters) determine the ultimate fate of the cell population. We identify the critical division rate separating long-term tumour population growth from successful multiple-dose treatment.