Epitaxial Growth of a Thin Layer of Putative Cancer Cells under Limited Resources and in the Presence of an Alkylating Agent.

ABSTRACT

The dynamics of the Eden cluster in a 32x32 lattice is implemented using a stochastic model. A single-type of cells solid tumor is assumed. Duplication is probabilistic, and occurs when there is room in the immediate surroundings of a cell, otherwise the cell is inhibited by contact. The growth is epitaxial, the shape of the cluster is disk-like; the ratio between the numbers of perimeter cells; and bulk cells decreases as the cluster grows. Percolation is flagged by an inflection in the rate of growth. We assume that the inflection point actually flags a shortage of nutrients, thereafter the rate of growth decreases to zero. Cancer cells in culture, when deprived of nutrients, actually exhibit a similar behavior. Under the logistic hypothesis, the lattice contains nutrients to sustain the growth up to 1024 cells. The model is expanded to include a drug that pollutes the environment. The drug is an alkylating agent that hinders duplication, eventually causing the death of the cell. The logistic equation accounts for drug consumption. The probability of duplication with the drug decreases as the drug is consumed, eventually leading to relapse. Relapses and survival times are investigated as a function of the dose injected.