Well posedness of an integrodifferential kinetic model of Fokker-Planck type for angiogenesis

TL;DR

This paper establishes the existence and uniqueness of solutions for a coupled integrodifferential kinetic and diffusion model describing tumor-induced angiogenesis, incorporating stochastic vessel motion and branching.

Contribution

It develops a mathematical framework proving well-posedness for a novel integrodifferential Fokker-Planck type model of angiogenesis.

Findings

Existence and uniqueness of solutions are proven.

Fundamental solutions for linearized problems are constructed.

Comparison principles and estimates are used to establish results.

Abstract

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker-Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators.