A mathematical method for determining oxygen distribution in skeletal muscle with multiple capillaries and axial diffusion

Abstract

A mathematical method is presented for analysing oxygen transport in skeletal muscle that includes both the interaction of very large number of capillaries and the effect of axial diffusion in the tissue. The analysis takes into account differences in the blood flow velocity among the capillaries and in the oxygen concentration and haematocrit of the blood entering the capillaries at the arterial end. First, the oxygen concentration is determined within a functional unit consisting of a single capillary surrounded by a region of tissue, in which a flux is prescribed on the outer boundary of the region. The method of matched asymptotic expansions is used to include the effect of axial diffusion in the tissue. The flux that is prescribed on the boundary of the functional unit is a result of the interaction among all the capillaries comprising the vascular bed. These fluxes are found by matching the oxygen concentration along the boundaries of adjacent functional units. This leads to a system of ordinary differential equations for the capillary concentrations coupled with a system of algebraic equations for the fluxes. Results are presented for a two-dimensional array of capillaries when each capillary has a different blood flow velocity. Axial diffusion causes the oxygen concentration to drop more rapidly at the arterial end and remain below the level determined when axial diffusion is neglected. This difference decreases toward the venous end, because the excess oxygen that left the capillaries at the arterial end and diffused axially is available to the tissue there. In some cases a single capillary model, which neglects interaction, incorrectly indicates that portions of the tissue become anoxic, whereas including interaction shows that all of the tissue is in fact supplied with oxygen.