Two dimensional soliton in tumor induced angiogenesis

TL;DR

This paper models the movement of blood vessel tips in tumor-induced angiogenesis as a two-dimensional diffusive soliton, providing a deterministic approach that matches stochastic simulation results.

Contribution

It introduces a deterministic equation for the mean density of vessel tips and characterizes the soliton shape and trajectory in general geometries.

Findings

The vessel tip density forms a stable diffusive soliton.

The soliton's velocity and trajectory are described by collective coordinate equations.

The deterministic model aligns well with stochastic simulation ensemble averages.

Abstract

Ensemble averages of a stochastic model show that, after a formation stage, the tips of active blood vessels in an angiogenic network form a moving two dimensional stable diffusive soliton, which advances toward sources of growth factor. Here we use methods of multiple scales to find the diffusive soliton as a solution of a deterministic equation for the mean density of active endothelial cells tips. We characterize the diffusive soliton shape in a general geometry, and find that its vector velocity and the trajectory of its center of mass along curvilinear coordinates solve appropriate collective coordinate equations. The vessel tip density predicted by the soliton compares well with that obtained by ensemble averages of simulations of the stochastic model.