Homogenisation of nonlinear blood flow in periodic networks: the limit of small haematocrit heterogeneity

In this work we develop a homogenisation methodology to upscale mathematical descriptions of microcirculatory blood flow from the microscale (where individual vessels are resolved) to the macroscopic (or tissue) scale. Due to the assumed two-phase nature of blood and specific features of red blood cells (RBCs), mathematical models for blood flow in the microcirculation are highly nonlinear, coupling the flow and RBC concentrations (haematocrit). In contrast to previous works which accomplished blood-flow homogenisation by assuming that the haematocrit level remains constant, here we allow for spatial heterogeneity in the haematocrit concentration and thus begin with a nonlinear microscale model. We simplify the analysis by considering the limit of small haematocrit heterogeneity which prevails when variations in haematocrit concentration between neighbouring vessels are small. Homogenisation results in a system of coupled, nonlinear partial differential equations describing the flow and haematocrit transport at the macroscale, in which a nonlinear Darcy-type model relates the flow and pressure gradient via a haematocrit-dependent permeability tensor. During the analysis we obtain further that haematocrit transport at the macroscale is governed by a purely advective equation. Applying the theory to particular examples of two- and three-dimensional geometries of periodic networks, we calculate the effective permeability tensor associated with blood flow in these vascular networks. We demonstrate how the statistical distribution of vessel lengths and diameters, together with the average haematocrit level, affect the statistical properties of the macroscopic permeability tensor. These data can be used to simulate blood flow and haematocrit transport at the macroscale.