A model for diffusion and competition in cancer growth and metastasis

A master equation formalism is used to model the growth and metastasis of a tumor as a function of the diffusion and absorption of a nutrient. Healthy and cancerous (C-) cells compete to bind the nutrient, which is allowed to diffuse starting from a prescribed region. Two thresholds are defined for the quantity of nutrient bound by the C-cells. If this quantity falls below the lower threshold, the cell dies, while if it increases above the upper threshold, the cell divides according to a predefined stochastic mechanism. C-cells migrate when they record a low concentration of free nutrient in the local environment. The model is formulated in terms of a coupled system of equations for the cell populations and the free and bound nutrient. This system can be solved by using the Local Interaction Simulation Approach (LISA), a numerical procedure that permits an efficient and detailed solution and is easily adaptable to parallel processing. With suitable parameter variation, the model can describe multiple tumor configurations, ranging from the classical spheroid with a necrotic core favored by mathematicians to very anisotropic shapes with inhomogeneous concentrations of the various populations. This is important because the nature of the anisotropy may be crucial in determining whether and how the cancer metastasizes. The effects of stochasticity and the presence of additional nutrients or inhibitors can be easily incorporated.