Stochastic models of tumor growth and the probability of elimination by cytotoxic cells |
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Authors: | Stephen J. Merrill |
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Affiliation: | (1) Department of Mathematics, Statistics and Computer Science, Marquette University, 53233 Milwaukee, WI, USA |
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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 where γ0, k′6 and k 1 are positive constants. The parameter of most import is 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 λ = 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. |
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Keywords: | Cytotoxic cells stochastic models elimination probability tumor growth |
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