Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

Abstract

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is T' = gamma 0T - k'6T/(K1 + T) where gamma 0, k'6 and K1 are positive constants. The parameter of most import is lambda 0 = gamma 0 - k'6/K1 which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case lambda = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.