Multiscale methods for modeling tissue perfusion and angiogenesis: a work in progress

d’Angelo, Zunino and Quarteroni have developed a multiscale methodology to handle numerically complex problems that arise naturally in microcirculation and perfusion of biological tissues through networks of vascular beds. The problem is composed by two scales: (i) at the macroscale, the tissue is treated as a homogenized porous media, usually saturated with plasma where relevant chemical species diffuse, advect and react; and (ii) the microscale is a finite and discrete network of small vessels (arterioles and capillaries), from which plasma permeates, and are naturally considered as a network of one dimensional models of fluid flow in circular tubes. Starling’s filtration law, a widely established description of the permeation of plasma from the arteriole into the tissue, couples both scales in a nonlinear fashion, i.e. the amount of fluid delivered from the vessel into the tissue (or vice versa) generally depends locally on the dependent variables of both scales. Currently, the mathematical modeling of angiogenesis is generally treated with two concurrent modeling approaches: (i) on one hand, continuum models of angiogenesis are based on reaction-diffusion-advection equations that describe endothelial cell conservation and quantify their motility accordingly to the distribution of other important fields, such as the concentration of angiogenic factor or presence of fibronectin, and fall short in the discrete identification and description of the vascular network; and (ii) stochastic discrete models based on lattice descriptions of the tissue together with a set of phenomenologically based cellular automata rules of network behavior and evolution have provided interesting qualitative results to describe a multitude of different patterns of angiogenesis. The multiscale methodology developed by d’Angelo and co-workers seems to be a promising strategy to develop quite a fundamental model of angiogenesis, based on evolution of the microscale (i.e. with an evolving network in response to biological stimuli provided by the tissue) or even on the evolution of both scales (with a growing tissue in response to improved conditions provided by the enhancement of the microscale). Several challenges naturally arise with the mathematical description of this complex problem and are currently a work in progress, such as: (i) the ability of the method to handle very complex arterial networks downstream after multiple rounds of capillary sprouting, (ii) anastomosis (when two capillary tips join and form a loop, changing the topology of the microscale domain), and among others (iii) the choice of proper one dimensional models for fluid flow in one-ended tubes and the treatment of anisotropic (directional) plasma sources at the macroscale.