A MODEL FOR DIFFUSION AND COMPETITION IN CANCER GROWTH AND METASTASIS G.P. PESCARMONA*, M. SCALERANDI**, P.P. DELSANTO**, C.A. CONDAT*** *Dipartimento di Genetica, Biologia e Chimica Medica, UniversitA di Torino, Torino, Italy **INFM, Dipartimento di Fisica, Politecnico di Torino, Torino, Italy ***Department of Physics, University of Puerto Rico, Mayaguez, PR 00680 ABSTRACT 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. INTRODUCTION According to the most recent theories, biological events or even life itself cannot be described by linear equations, but only as a temporal sequence of discrete events, with the choice at every bifurcation depending on the local conditions at the time and place. A suitable model for life- related events like cancer growth must therefore * include a dependence on the environmental variables and cell population properties; * introduce stochastic fluctuations of the variables within preordained ranges; * simulate through a large number of cycles the temporal sequence of life events. To develop a model as general as possible one needs a deep knowledge of the rules driving cell life, modelling at first only some very simple processes. As a second step one increases the complexity of the system by introducing evolution into each parameter, and checking the behavior of the model after each evolutionary step. This approach is expected to mimic the evolution of the living world, attempting a correspondence with real life. According to this methodology, a very basic model for cancer growth consists of two populations (cancerous and normal cells) competing for a single essential nutrient, represented in the present case by iron [ 1-3]. In later more realistic treatments, the model will be, of course, generalized to include additional populations and nutrients, with more complex interaction mechanisms. It is, however, interesting to observe that even this elementary model succeeds in predicting the basic features of tumor growth. 217 Mat. Res. Soc. Symp. Proc. Vol. 489 ©1998 Materials Research Society