An Asymptotic Study of the Steady State Model of Oxygen Diffusion in Tissue Regions
AbstractOxygen plays an important role in the metabolism of cells inside the human body. The transfer of oxygen from blood to tissue takes place in capillaries through a diffusion process. The capillary-tissue region is usually represented by the so-called Krogh cylinder model, 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 the steady state case and solve the equation analytically by means of asymptotic expansion for a particular limit of the oxygen consumption rate. Results show that there exists a critical ratio between supply and consumption of oxygen in the tissue region in order to fulfill the cell's oxygen requirements. Above from this critical ratio, we also found a critical distance in the tissue region above which the oxygen concentration vanishes. We compared our asymptotic results with numerical simulations, which turned out to be quite in agreement.
Middleman, S., Transport Phenomena in the Cardiovascular System, John Wiley Sons Inc., pp. 116-140, 1972.
Krogh, A., The Number and Distribution of Capillaries in Muscles with Calculations of The Oxygen Pressure Head Necessary for Supplying Tissue, J. Physiol. (London), 52, pp. 409-415, 1919.
Salathe, E.P. & Xu, Y.H, Non-Linear Phenomena in Oxygen Transport To Tissue, J. of Mathematical Biology, 30(2), pp. 151-160, 1991.
Titkombe, M.S. & Ward, M.J., An Asymptotic Study of Oxygen Transport from Multiple Capillaries to Skeletal Muscle Tissue, SIAM J. Appl. Math., 60(5), pp. 1767-1788,2000.
Salathe, E.P., Mathematical Analysis of Oxygen Concentration in a Two Dimensional Array off Capillaries, J. Math. Biology, 46(4), pp. 287-308, 2003.
MikeliA , M.P., A Diffusion-Consumption Problem for Oxygen in a Living Tissue Perfused by Capillaries, Nonlinear Differ. Equ. Appl. (NoDEA), 13(3), pp. 349-367, 2006.
Devlin, T.M., Textbook of Biochemistry with Clinical Correlations, John Willey & Sons, pp. 126-129, 1982.
Holmes, M.H., Introduction to Perturbation Method, Springer-Verlag, pp. 47-73, 1959.
Mathews, J.H. & Fink, K.D., Numerical Methods Using Matlab, Prentice-Hall, pp. 505-533, 1999.
Logan, J.D., Applied Partial Differential Equation, Springer-Verlag, pp. 135-138, 1998.
Mattheij, R.M.M., Rienstra, S.W., & ten ThijeBoonkkamp, J.H.M., Partial Differential Equations: Modeling, Analysis, Computation, 16, SIAM, 2005.