Substrate Transport in Cylindrical Multi-Capillary Beds with Axial Diffusion

Abstract

It is known that in oxygen concentration profiles for capillary beds of skeletal muscles, radial diffusion most likely has considerably more effect on oxygen transport in long and parallel capillary beds than axial diffusion. However, axial diffusion may still play a significant role in oxygen transport in tissue, especially in relatively short pathways. Our model adds to known solutions the component of axial diffusion to multi-capillary beds inside a tissue cylinder, where arbitrary characteristics include random locations and uneven oxygen strengths. Discussion of the solutions for oxygen supply in multicapillary beds near the arterial ends, in the central regions, and near the venous ends in capillaries is introduced in the remainder of the article. Our prime model builds on known solutions involving circular regions by adding a $Z$-axis and by accounting for circular cylindrical tissue around multiple capillaries. To account for relatively small longitudinal diffusivities, we use perturbation methods to solve the associated governing equations.