An Asymptotic Study for the Steady Model of Oxygen Diffusion in Tissue Region

Oxygen plays an important role in cell metabolism inside a human's body. The transfer of oxygen from blood to tissues takes place in capillaries through the diffusion process. A capillary-tissue region is usually represented by so called the Krogh Cylinder arrangement in which the distribution of the oxygen concentration in a tissue region leads to a diffusion equation with oxygen consumption rates following the Michaelis-Menten kinetics. In this paper, we restrict ourselves to consider the steady case and solve the equation analytically, by means of the asymptotic expansions, for a particular limit of the oxygen consumption rate. Results show that there exists a critical ratio between supply and consumption of oxygen in tissue region in order to fulfill cell’s oxygen requirements. Above this critical ratio, we also find a critical distance in the tissue region above which the oxygen concentration vanishes. We also compare our asymptotic results with numerical simulations which are both quite in agreement.