An Asymptotic Study of Oxygen Transport from Multiple Capillaries to Skeletal Muscle Tissue

FROM MULTIPLE CAPILLARIES TO SKELETAL MUSCLE TISSUE MICH ELE S. TITCOMBE AND MICHAEL J. WARD1 Abstract A mathematical model of the transport of oxygen from capillaries to skeletal muscle tissue is a di usion problem in a two-dimensional, bounded domain with Neumann and mixed boundary conditions. We consider N capillaries of small but arbitrary crosssectional shape and demonstrate that, for N > 1, this is a singular perturbation problem that involves an in nite expansion of logarithmic terms of the small parameter ", which characterizes the size of the capillary cross-sections. We construct a hybrid asymptoticnumerical method, that uses the method of matched asymptotic expansions and basic numerical analysis, to solve for the steady-state oxygen partial pressure in the tissue. In general, our oxygen transport model incorporates the e ects of tissue heterogeneities such as mitochondria, variable permeability of the capillary walls and the facilitation of oxygen transport by the presence of myoglobin. We demonstrate the asymptotic results with some speci c examples to illustrate these and other e ects, and in certain cases, compare with the exact or numerical solution.