An optimization based 3D-1D coupling strategy for tissue perfusion and chemical transport during tumor-induced angiogenesis

TL;DR

This paper introduces a novel PDE-constrained optimization method for simulating fluid and chemical exchanges in tumor-induced angiogenesis, effectively modeling capillary growth without remeshing.

Contribution

It presents the first application of PDE-constrained optimization to angiogenesis, coupling 3D tissue and 1D capillary models with a mesh-free approach.

Findings

Mesh-free 3D-1D coupling effectively simulates angiogenesis.

Hybrid discrete-continuous strategy captures capillary growth.

Method avoids remeshing during network evolution.

Abstract

A new mathematical model and numerical approach are proposed for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. Thanks to proper modeling assumptions the capillaries are reduced to their centerline: a well posed mathematical model is hence worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). Also the application of a PDE-constrained optimization formulation is here proposed for the first time for angiogenesis simulations. Under this approach no mesh conformity is required, thus making the method particularly suitable for this kind of application, since no remeshing is required as the capillary network grows. In order to handle both the evolution of the quantities of interest and the changes in the geometry, a discrete-hybrid strategy is adopted, combining a continuous modeling of the tissue and of the chemicals with a discrete tip-tracking model to account for the vascular network growth. The tip-tracking strategy, together with some proper rules for branching and anastomosis, is able to provide a realistic representation of the capillary network.