Modeling Avascular Tumor Growth: Approach with an Adaptive Grid Numerical Technique

The mathematical modeling of complex biological systems leads to system of coupled nonlinear partial differential equations (PDEs). In this paper, we present a short review on the interaction of the urokinase plasminogen activator (uPA) system with a model for cancer cell in the avascular phase, faced using the moving mesh PDE/(MMPDE) numerical technique. The dynamical evolution of the system as a function of the diffusion properties of cancer cells has been considered, as well as the effect of hypoxia to the cancer evolution, introducing a model equation for the nutrient oxygen. The model parameters have been taken from the data existing in the literature, in particular to gauge the oxygen supply, data determined from in vivo experiments on human tumors have been used. The numerical results obtained simulating a one-dimensional portion of the biological tissue are consistent with the data existing in the literature. Our high-resolution computations show that cancer proliferation begins through highly irregular spatio-temporal pattern, which depends on cancer motility characteristics. In presence of hypoxia, the cancer proliferation patterns are still characterized by an inhomogeneous pattern, but other effects are present which depend on the model parameters, triggered by the oxygen.