The statistical theory of the angiogenesis equations

TL;DR

This paper develops a statistical theory of angiogenesis using stochastic PDEs with noise, deriving invariant measures and comparing them to numerical simulations, thus providing a new probabilistic framework for understanding blood vessel growth.

Contribution

It introduces a novel stochastic PDE model for angiogenesis incorporating noise, and analytically derives the invariant measure, linking it to the deterministic tip density.

Findings

Invariant measure obtained for the stochastic angiogenesis model

Comparison shows good agreement between theory and simulations

Invariant measure relates to Korteweg-de Vries soliton approximation

Abstract

Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological induced by a cancerous tumor. A mean field approximation for a stochastic model of angiogenesis consists of partial differential equation (PDE) for the density of active tip vessels. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite dimensional L\'evy process and is the basis of a statistical theory of angiogenesis. The associated functional equation has been solved and the invariant measure obtained. The results are compared to a direct numerical simulation of the stochastic model of angiogenesis and invariant measure multiplied by an exponentially decaying factor. The results of this theory are compared to direct numerical simulations of the underlying angiogenesis model. The invariant measure and the moments are functions of the Korteweg-de Vries soliton which approximates the deterministic density of active vessel tips.