Multiscale modeling of vascularized tissues via non-matching immersed methods

TL;DR

This paper introduces a multiscale immersed method for efficiently modeling vascularized tissues, capturing the interaction between elastic matrices and thin vascular structures with reduced computational complexity.

Contribution

It develops variational formulations that incorporate vascular effects as singular forcing terms, simplifying the simulation of complex tissue structures.

Findings

Validated with exact solutions for simple cases

Simulated tissues with random vessel distributions

Characterized effective mechanical properties statistically

Abstract

We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depend on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean pressure over cross section), thus drastically reducing the complexity of the computational model. We perform several numerical tests, ranging from simple cases with known exact solutions to the modeling of materials with random distributions of vessels. In the latter case, we use our immersed method to perform an in silico characterization of the mechanical properties of the effective biphasic material tissue via statistical simulations.