On the mean field approximation of a stochastic model of tumor-induced angiogenesis

TL;DR

This paper rigorously proves the validity of a mean field approximation for a stochastic model of tumor-induced angiogenesis, simplifying complex multiscale interactions while ensuring the non-extinction of vessel tips over finite times.

Contribution

It provides a rigorous proof of the propagation of chaos for a stochastic angiogenesis model, justifying the mean field approximation previously only heuristically used.

Findings

Proved propagation of chaos for the model

Validated mean field approximation for vessel tip dynamics

Established non-extinction of vessel tips over finite times

Abstract

In the field of Life Sciences it it very common to deal with extremely complex systems, from both analytical and computational points of view, due to the unavoidable coupling of different interacting structures. As an example, angiogenesis has revealed to be an highly complex, and extremely interesting biomedical problem, due to the strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. In this paper an original revisited conceptual stochastic model of tumor driven angiogenesis has been proposed, for which it has been shown that it is possible to reduce complexity by taking advantage of the intrinsic multiscale structure of the system; one may keep the stochasticity of the dynamics of the vessel tips at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. While in previous papers only an heuristic justification of this approach had been offered, in this paper a rigorous proof is given of the so called "propagation of chaos", which leads to a mean field approximation of the stochastic relevant measures associated with the vessel dynamics, and consequently of the underlying TAF field. As a side though important result, the non-extinction of the random process of tips has been proven during any finite time interval.