A multiscale tumor model

Abstract

We consider a tumor model with two time scales: the time $t$ during which the tumor evolves and the running time $s_i$ for each of the phases of the cell cycle for the cells in the tumor. The model also includes the effect of genes mutations in the sense that populations of cells with different mutations and in different phases of the cell cycle evolve by different rules. The model is formulated as a coupled system of partial differential equations; a transition from one population to another occurs at the ‘restriction points’ located at the ends of the $G_1$ and $S$ phases. The PDEs for the cell populations are hyperbolic equations based on mass conservation laws. The model includes also a diffusion equation for the oxygen concentration and an elliptic equation for the internal pressure caused by proliferation and death of cells. The tumor region is viewed as a domain with a moving boundary, satisfying a continuity equation at the free boundary. Existence and uniqueness are proved for a small time interval, for general initial conditions, and for all time in the case of radially symmetric initial conditions.