# Analytical and numerical properties of an extended angiogenesis PDEs model

## TL;DR

This paper introduces a new mathematical model for tumor angiogenesis that includes oxygen dynamics and proves its mathematical properties.

## Contribution

The novel contribution is extending an angiogenesis model with oxygen dynamics and proving its mathematical properties.

## Key findings

- The model establishes existence, uniqueness, and boundedness of solutions.
- A stable numerical scheme with convergence properties is developed.
- Numerical experiments show biologically plausible vascular formation regulated by oxygen.

## Abstract

This paper presents an extended mathematical model for tumor angiogenesis incorporating oxygen dynamics as a main regulator. We enhance a five-component PDE system describing endothelial cells, proteases, inhibitors, extracellular matrix, and oxygen concentration, with a focus on their spatiotemporal interactions. We establish existence, uniqueness, and boundedness of solutions through a mathematical analysis. A numerical scheme using method of lines and fourth-order Runge-Kutta methods is developed, with proven stability constraints and convergence properties. Numerical experiments demonstrate biologically plausible vascular formation with oxygen-mediated regulation.