Blood flow through the microcirculation is characterized by significant variations in the red blood cell velocity and number density. These variations occur for two primary reasons: (1) at diverging bifurcations, where one parental vessel splits into two descendent vessels, red blood cells are partitioned non-uniformly into the descendent vessels, with a nonlinear dependence on the ratio of the blood flow rates in these vessels; and (2) the effective blood viscosity, which affects the blood flow rate in a vessel, depends strongly on the concentration of red blood cells.
Theoretical models of blood flow through capillary networks, which are generally successful at predicting the variations in the red blood cell concentration and provide various insights, are built on several assumptions. The continuum approximation is typically adopted, and blood is treated as a two-phase, homogeneous medium consisting of plasma and suspended cells. The flow rate through a capillary is described by Poiseuille’s law with an effective viscosity. The dependence of the effective viscosity on the capillary diameter, cell concentration, and flow rate is determined from correlations based on in vivo data. At converging bifurcations, where two vessels merge into one, the red blood cell concentration is deduced from a mass balance, while at diverging bifurcations an empirical cell partitioning law is also utilized. In one extreme case, which occurs for wide vessels, the cells are partitioned in proportion to the blood flow rate. In the opposite extreme case, which is relevant to narrow vessels, all cells are channeled into the descendent branch that receives the highest flow rate. Finally, most models of capillary blood flow are based on the assumption of steady state. In vivo and in vitro evidence, however, reveals that periodic oscillations can occur in the red blood cell velocity and concentration and blood flow rate.
In this talk, the equations governing unsteady blood flow through a capillary network of arbitrary structure are discussed, a solution algorithm is outlined, and results are presented for blood flow through tree-like and mesh-like capillary networks. These networks have different topology and are idealizations of the microvasculature in several tissues. Spontaneous, self-sustained oscillations are found to occur due to an instability associated with a supercritical Hopf bifurcation of the steady solution. These oscillations can occur solely due to the nonlinear rheology of blood coupled to the non-uniform cell partitioning at network bifurcations in the absence of active biological regulation and for static boundary conditions. The predictions of the unsteady, continuum model are validated by comparison to a discrete model in which the motion of individual red blood cells is followed from inlet to outlet, and the agreement is excellent even for vessels with a diameter comparable to the red blood cell size. Results of a linear stability analysis are also presented for tree, honeycomb, and tabulated physiological networks, and extensions to suspension flow through a consolidated porous material are discussed.