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.

Observations of spontaneous oscillations in simple two-fluid networks

“…A well-studied network that can exhibit complex dynamic behavior is microvascular blood flow where August Krogh first noted the heterogeneity of blood flow in the webbed feet of frogs in 1921 [8]. Simulations, analysis, and experiments with microvascular networks have demonstrated the possibility of spontaneous oscillations in flow rates and hematocrit distribution though direct validation between model and experiment is lacking [9][10][11][12][13][14][15].…”

Hydrodynamic instability of a suspension of spherical particles through a branching network of circular tubes

“…In Sect. 3, we summarize a continuum model developed by Carr and Lacoin [5] and recently extended by the present authors [1]. In Sect.…”

“…Following Pozrikidis [4], Davis and Pozrikidis [1], and other previous authors, we consider a tree-like model network branching from a single entrance point to multiple exit points, as discussed in Sect. 4.…”

“…Recent numerical studies have shown that a suspension of red blood cells moving through a capillary network may undergo an instability leading to spontaneous oscillations in the flow rate, cell concentration, and accompanying pressure drop [1]. Physically, a large number of cells entering a capillary tube causes an increase in the effective viscosity of the particulate medium and a resultant reduction in the flow rate.…”

“…The network is generated as a deterministic fractal tree disturbed by random perturbations. Analogous models have been employed by other authors in studies of blood flow through normal and neoplastic capillary networks [1][2][3][4].…”