Computational modeling of solid tumor growth: The avascular stage

In this paper, we present a mathematical model for avascular tumor growth and its numerical study in two and three dimensions. For this purpose, we use a multiscale model using PDEs to describe the evolution of the tumor cell densities. In our model, cell cycle regulation depends mainly on microenvironment. The cancer growth of volume induces cell motion and tumor expansion. According to biology, cells grow against a basal membrane which interacts mechanically with the tumor. We use a level set method to describe this membrane, and we compute its influence on cell movement, thanks to a Stokes equation. The evolution of oxygen, diffusing from blood vessels to cancer cells and used to estimate hypoxia, is given by a stationary diffusion equation solved with a penalty method. The model has been applied to investigate the therapeutic benefit of anti-invasive agents and constitutes now the basis of a numerical platform for tumor growth simulations.