MATHEMATICAL MODELING OF OXYGEN TRANSPORT IN THE CAPILLARIES OF THE MICROCIRCULATION

Mathematical models describing oxygen transport within capillaries typical of the human microcirculation are developed in order to investigate the basic physical and chemical processes occurring within the erythrocyte and to establish the validity of some of the assumptions used in previous modeling of the capillary region. These models describe oxygen diffusion in the radial direction within 4 and 6 micron capillaries, taking into account simultaneous reaction and diffusion of oxygen, hemoglobin, and oxyhemoglobin. . The mathematical equations describing this system are solved numerically using a software package employing basis spline collocation on finite elements. ~his method reduces the computation times by more than an order of magnitude compared to the more commonly used finite difference method. The quotations are solved in both linear and cylindrical geometry and indicate that the use of a parallel plate model (linear geometry) can provide adequate predications for oxygen concentrations at the capillary wall under normal physiological conditions, provided the plate separation is chosen to be one half the capillary diameter. In the past, the capillary region has often been considered to be a homogeneous hemoglobin solution, a "continuum." The results of this study indicate that the use of a continuum model of the capillary region can cause an underestimate of the resistance to oxygen transport in the capillary by a factor of two, although the predicted capillary wall concentrations may be in error by only four to seven per cent. The effect of erythrocyte rotation, or "tank treading," on the transport of oxygen to the tissue wall is investigated by assuming the erythrocyte contents to be perfectly mixed. The results of this study indicate that the enhancement of oxygen transport due to this mixing can cause up to an eleven per cent increase in the capillary wall concentrations under normal physiological conditions. Comparison of a simple mass action kinetic model to a more accurate variable rate constant kinetic model indicates that the use of a mass action model can cause severe errors in the predicted oxygen concentrations within the capillary. Also, by varying the rate of reaction between oxygen and hemoglobin, the deviations from local chemical equilibrium within the erythrocyte are found to be less than three per cent near the capillary wall and much smaller than this in the interior of the cell. The capillary model is modified to include a more physically realistic thin plasma gap between the erythrocyte and the capillary wall. It is observed that the oxygen concentration drop across this plasma gap rapidly approaches an asymptotic value. This asymptotic value can be used to accurately predict the influence of the plasma gap on oxygen transport to the tissue and indicates that the fractional concentration drop across the plasma gap in the radial direction is between five and ten per cent of the total concentration drop across both the capillary and tissue regions