Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis

TL;DR

This paper presents a mathematical analysis of a two-dimensional model for metastatic growth incorporating angiogenesis, focusing on well-posedness and long-term behavior of solutions.

Contribution

It introduces a novel structured equation model for metastasis density that accounts for angiogenesis, and provides rigorous mathematical analysis of its properties.

Findings

Proved well-posedness of the model.

Analyzed asymptotic behavior of solutions.

Investigated properties of the function space $ ext{Wd}( ext{Om})$.

Abstract

Angiogenesis is a key process in the tumoral growth which allows the cancerous tissue to impact on its vasculature in order to improve the nutrient's supply and the metastatic process. In this paper, we introduce a model for the density of metastasis which takes into account for this feature. It is a two dimensional structured equation with a vanishing velocity field and a source term on the boundary. We present here the mathematical analysis of the model, namely the well-posedness of the equation and the asymptotic behavior of the solutions, whose natural regularity led us to investigate some basic properties of the space $\Wd(\Om)=\{V\in L^1;\;\div(GV)\in L^1\}$, where $G$ is the velocity field of the equation.