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During avascular tumor growth, the balance between cell proliferation and cell loss determines whether the colony expands or regresses. Mathematical models describing avascular tumor growth distinguish between necrosis and apoptosis as distinct cell loss mechanisms: necrosis occurs when the nutrient level is insufficient to sustain the cell population, whereas apoptosis can occur in a nutrient-rich environment and usually occurs when the cell exceeds its natural lifespan. Experiments suggest that changes in the proliferation rate can trigger changes in apoptotic cell loss and that these changes do not occur instantaneously: they are mediated by growth factors expressed by the tumor cells. In this paper, we consider two ways of modifying the standard model of avascular tumor growth by incorporating into the net proliferation rate a time-delayed factor. In the first case, the delay represents the time taken for cells to undergo mitosis. In the second case, the delay represents the time for changes in the proliferation to stimulate compensatory changes in apoptotic cell loss. Numerical and asymptotic techniques are used to show how a tumor's growth dynamics are affected by including such delay terms. In the first case, the size of the delay does not affect the limiting behavior of the tumor: it simply modifies the details of its evolution. In the second case, the delay can alter the tumor's evolution dramatically. In certain cases, if the delay exceeds a critical value, defined in terms of the system parameters, then the underlying radially symmetric steady state is unstable with respect to time-dependent perturbations. (For smaller delays, this steady state is stable). Using the delay as a measure of the speed with which a tumor adapts to changes in its structure, we infer that, for the second case, a highly responsive tumor (small delay) has a better chance of surviving than does a less-responsive tumor (large delay). We also conclude that the tumor's evolution depends crucially on the manner and speed with which it adapts to changes in its surroundings and composition.

Original publication




Journal article


Math Biosci

Publication Date





83 - 117


Apoptosis, Cell Division, Humans, Mathematics, Models, Theoretical, Neoplasms