The Mathematics of Oxygen and Substrate Diffusion

A theoretical model for oxygen and substrate transport to tissues within skeletal muscle involving several interacting capillaries has been developed. It involves an elegant technique that discretizes the transport problem in a tissue with large number of capillaries. The technique can be equally applied to other heat transport problems with actual anatomical information. For oxygen, the role of myoglobin and the effect of axial diffusion have been considered. For the myoglobin model, myoglobin facilitates oxygen diffusion into tissue and can prevent hypoxia due to the interaction of the capillaries. The analysis yields a great deal of information about hypoxia, since as shown, a multicapillary model of the type presented here is needed to determine if the tissue is truly hypoxic. Because capillary length is large compared to capillary spacing, axial diffusion is a small perturbation to the solution without axial diffusion. The effect of axial diffusion is found using perturbation methods. For the substrate model, there can be large differences between the substrate concentration in the capillary and that in the tissue, depending on the permeability of the capillary endothelium.