Numerical Simulation of Unsteady Blood Flow through Capillary Networks

Abstract

A numerical method is implemented for computing unsteady blood flow through a branching capillary network. The evolution of the discharge hematocrit along each capillary segment is computed by integrating in time a one-dimensional convection equation using a finite-difference method. The convection velocity is determined by the local and instantaneous effective capillary blood viscosity, while the tube to discharge hematocrit ratio is deduced from available correlations. Boundary conditions for the discharge hematocrit at divergent bifurcations arise from the partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q≥1. When q=1, the cells are partitioned in proportion to the flow rate; as q tends to infinity, the cells are channeled into the branch with the highest flow rate. Simulations are performed for a tree-like, perfectly symmetric or randomly perturbed capillary network with m generations. When the tree involves more than a few generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous self-sustained oscillations in the absence of external forcing. A phase diagram in the m–q plane is presented to establish conditions for unsteady flow, and the effect of various geometrical and physical parameters is examined. For a given network tree order, m, oscillations can be induced for a sufficiently high value of q by increasing the apparent intrinsic viscosity, decreasing the ratio of the vessel diameter from one generation to the next, or by decreasing the diameter of the terminal vessels. With other parameters fixed, oscillations are inhibited by increasing m. The results of the continuum model are in excellent agreement with the predictions of a discrete model where the motion of individual cells is followed from inlet to outlet.