Mathematical Modeling and Simulation of A Multiscale Tumor Induced Angiogenesis Model.

Abstract

Angiogenesis is the formation of new blood vessels from pre-existing vessels. It is one of the main processes that help in the growth and spread of tumors. The processes that lead to angiogenesis are very complex and involve several pathways. If cancerous cells are successful in inducing angiogenesis then the harmless malignant tumor cell becomes vascularized and is equipped with the ability to spread from one part of its host to other distant sites. We construct a multiscale continuum model for tumor angiogenesis in an attempt to understand the role of angiogenesis induced by hypoxia (lack of oxygen of tumor cells). We model this process as a reaction diffusion system of a system of semi-linear parabolic differential equations. For modeling blood structures, we use a discrete model which comprises of systems of ordinary differential equations and stochastic differential equations. We use analytic semi-groups, functional analysis and complex analysis to study the existence of positive global solutions, linear stability and instabilities due to different diffusion rates (Turing Instability). We perform uncertainty quantification of the reaction system which is modeled using ordinary differential equations and uncertainty quantification via polynomial chaos expansion (the expansions depends on random parameters). For computing the numerical solutions of the continuum model, we use B-spline collocation method. Computations for the numerical methods were done using Python software.