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

A mathematical model of the transport of oxygen from capillaries to skeletal muscle tissue is a diffusion problem in a two-dimensional, bounded domain with Neumann and mixed boundary conditions. We consider N capillaries of small but arbitrary cross-sectional shape and demonstrate, for N > 1, that this is a singular perturbation problem that involves an infinite expansion of logarithmic terms of the small parameter $\varepsilon$, which characterizes the size of the capillary cross sections. For $\varepsilon \ll 1$, we use a hybrid asymptotic-numerical method to calculate the steady-state oxygen partial pressure in the tissue correct to within all logarithmic terms. Our results from this hybrid method illustrate the effect of tissue heterogeneities such as mitochondria, variable permeability of the capillary walls, and the facilitation of oxygen transport by the presence of myoglobin. The results from the hybrid method compare well with full numerical solutions.