A Mathematical Model for the Rate of Oxygenation of Blood in Pulmonary Capillaries using nth-Order One-Step Kinetics of Oxygen Uptake by Haemoglobin

A mathematical model is described for the process of gas exchange in pulmonary capillaries by taking into account the transport mechanisms of molecular diffusion, convection, and the facilitated diffusion due to haemoglobin. The nth-order one-step kinetics of oxygen uptake by haemoglobin has been incorporated. The rate k at which blood becomes oxygenated is determined by setting up an appropriate eigenvalue problem. This method eventually leads to a transcendental equation in k. A multiprecision technique due to Verma and Sharan (1980) is employed to obtain a physically acceptable solution. It is shown that, at equilibrium, the saturation of haemoglobin with oxygen computed from the analysis is fairly close to the data of Severinghaus (1966). It was found that 97.15% of the total haemoglobin combined with oxygen. The blood is oxygenated well before it leaves the pulmonary capillary. The dissolved oxygen takes longer to achieve equilibration whereas the carbon dioxide traverses a comparatively smaller distance in the capillary.