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.