A Model Describing the Unsteady Transport of Substrate to Tissue from the Microcirculation

Abstract

A mathematical model of the interrelationships of substrate concentrations in blood and tissue is given. This model describes (a) unsteady capillary transport by convection, (b) the unsteady substrate exchange kinetics within the capillary, (c) diffusion across a finitely permeable capillary wall and (d) unsteady diffusion and consumption of substrate within the tissue space. The mathematical model is a parabolic partial differential equation with its principal boundary condition (i.e., at the capillary wall) coupled to a system of first order wave equations. These wave equations, which describe the capillary conditions, are “stiff” along their characteristics. The model is treated indirectly by considering the solution of a related initial-boundary value problem which has the solution of the pure boundary value problem as its asymptotic (in time t) limit. An unconditionally stable implicit scheme is developed by coupling a locally one dimensional method (for the parabolic equation) with an implicit scheme for solving the transport equations. The numerical convergence is shown to be first order in the mesh size. The methods and results are illustrated with steady and unsteady flows using parameters of physiologic interest.