Abstract
A mathematical model of deoxygenation of blood in the microcirculation is used to estimate the mass transfer resistance in the blood and to examine certain assumptions used in prior work on simulation of the microcirculation: the treatment of blood as a continuum and the use of a single-step reaction kinetics model. The erythrocytes are treated as cylindrical slugs which alternate with plasma gaps such that oxygen transport is by radial diffusion in the cell. The system of equations including reaction kinetics and oxyhemoglobin diffusion is solved numerically. The results are of direct applicability in estimation of oxygen concentration profiles in tissue. The results also indicate that the resistance to oxygen transport in the capillary (relative to that in the surrounding tissue) is much higher than predicted by the continuum approach used by most prior workers. The resistance in the capillary is a significant fraction of the overall resistance. Other results give quantitative estimates of the error incurred from use of a single-step kinetic model.
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References
Aroesty, J. and J.F. Gross. Convection and diffusion in the microcirculation.Microvasc. Res. 2:247, 1970.
Baxley, P.T. Mathematical modeling of oxygen transport in the capillaries of the microcirculation. M.S. thesis in chemical engineering, Rice University, 1981.
Bennett, C.O. and J.E. Myers.Momentum, Heat, and Mass Transfer. New York: McGraw-Hill, 1962, pp. 305–312.
Bugliarello, G. and G.C. Hsau. A mathematical model of the flow in the axial plasmatic gaps of the smaller vessels.Biorheology 7:5, 1970.
Douglas, J.. The application of stability analysis in the numerical solution of quasilinear parabolic differential equations.Trans. Am. Math. Soc. 89:484–518, 1958.
Duda, J.L. and J.S. Vrentas. Steady flow in the region of closed streamlines in a cylindrical cavity.J. Fluid Mech. 45:247–261, 1971.
Gijsbers, G.H. and J.J. Van Ouwerkerk. Boundary layer resistance of steady-state oxygen diffusion facilitated by a four-step chemical reaction with hemoglobin in solution.Pfluegers Arch. 365:231–241, 1976.
Hellums, J.D. The resistance to oxygen transport in the capillaries relative to that in the surrounding tissue.Microvasc. Res. 13:131–136, 1977.
Hellums, J.D. Deformation of blood cells in capillaries: A commentary.Blood Cells. 6:815, 1980.
Krogh, A.The Anatomy and Physiology of Capillaries, 2nd edition. New Haven, Conn.: Yale University Press, 1930, pp. 391–405.
Lipowsky, H.H., S. Kovalcheck, and B.W. Zweifach. The distribution of blood rheological parameters in the microvasculature of cat mesentery,Circ. Res. 43:738–749, 1978.
Lipowsky, H.H. and B.W. Zweifach. Application of the two-slit photometric technique to the measurement of microvascular volumetric flow rates.Microvasc. Res. 15:93–101, 1978.
Madsen, M.K. and R.F. Sincovec. Software for non-linear partial differential equations.ACM Trans. Math. Software. 1:3, 1975.
Madsen, J.K. and R.F. Sincovec. General software for partial differential equations. InNumerical Methods for Differential Analysis, edited by L. Lapidus and W.E. Schiesser, New York: Academic Press, pp. 228–249, 1976.
Moll, W. The influence of hemoglobin diffusion on oxygen uptake and release by red cell.Respir. Physiol. 6:1–15, 1969.
Sheth, B.V. and J.D. Hellums. Transient oxygen transport in hemoglobin layers under conditions of the microcirculation.Ann. Biomed. Eng. 8:183–196, 1980.
Wiedman, M.P. Lengths and diameters of peripheral arterial vessels in the living animal.Circ. Res. 10:686–690, 1962.
Wiedman, M.P. Dimensions of blood vessels from distributing artery to collecting vein.Circ. Res. 12:375–378, 1973.
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This work was supported by the National Institutes of Health under Grant 2R 01 HL18584.
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Baxley, P.T., Hellums, J.D. A simple model for simulation of oxygen transport in the microcirculation. Ann Biomed Eng 11, 401–416 (1983). https://doi.org/10.1007/BF02584216
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DOI: https://doi.org/10.1007/BF02584216