Perturbation solutions of a mathematical model for determining the roles of Endothelial, pericyte and macrophage cells in the capillary

Abstract

The purpose of this work is to obtain the regular perturbation solutions of a mathematical model for capillary formation in tumor angiogenesis. The model we study here was originally presented in [Levine HA, Sleeman BD and Nilsen-Hamilton M., 2000]. The regular perturbation method is a well-known and highly effective method to obtain the solutions of coupled non-linear differential equations. In fact, a few terms of the perturbation series obtained by this method are good enough to see the structure of the solutions of the model. These solutions govern the movement of the certain cells, namely endothelial, macrophage and pericyte cells, in the capillary which are necessary for the initiation of tumor angiogenesis. Our MATLAB-generated figures show that our numerical simulations are in good agreement with the biological facts about the tumor angiogenesis. Even though computing the terms of the regular perturbation series are kind of tedious, more stable and accurate solutions of the model can be obtained by adding new terms to the series.