Numerical solution of partial differential equation describing oxygenation rate of the red blood cell

The non-linear partial differential equation for O2 diffusion was solved numerically in the three-dimensional red cell model by using the alternating-direction implicit method. The oxygenation rate factor of hemoglobin (FS) was assumed to decrease as the O2 saturation (SO2) increases, as given by FS = 2.1 x (1--S)2 (sec-1 . (mmHg)-1). The result obtained was compared with the solutions of the equations derived by Threws and Moll and also with those obtained from the sheet model. The oxygenation rate of the red cell largely depended on the diffusivity across the diffusion barrier around the red cell (eta). When eta = 2.5 x 10(-6) cm . sec-1 . (mmHg)-1 was inserted into the present equation, the numerical solution showed a good correlation with the experimental data. When the sheet model was applied, the eta value obtained from the same experimental data was about twice as great as that obtained in the disc model. One of the characteristic features of the SO2-time curves of the red cell was the decrease in steepness at a high SO2 range, which has been thought to occur due to the decrease in the oxygenation rate of hemoglobin. Therefore, the difference of the actual PO2 in the red cell from the fictitious, so-called "back-pressure" which is evaluated from the O2 dissociation curve through the actual SO2 has been expected to become greater as the SO2 increases. The result obtained from the present equation revealed that the above PO2 difference became as great as 20 mmHg at the maximum point. In the solutions obtained from Thew's and Moll's equations, however, the slope of the SO2-time curve was not significantly reduced at a high SO2 range.