title: 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.
mesh_terms: D002470:Cell Survival; D003602:Cytotoxicity, Immunologic; D007694:Killer Cells, Natural / Q000276:immunology; D008954:Models, Biological; D009369:Neoplasms / Q000276:immunology* / Q000473:pathology; D011336:Probability; D013269:Stochastic Processes