A numerical model for the simultaneous transport of oxygen and carbon dioxide in the systemic capillaries and surrounding tissue

Abstract

A mathematical model is formulated for the simultaneous transport of oxygen (O₂) and carbon dioxide (CO₂) in the systemic capillaries and surrounding tissue. The model takes into account the molecular diffusion, convection and the consumption of O₂ and the production of CO₂ in the tissue due to metabolism. The interaction of O₂ and CO₂ in the blood is incorporated using non-linear functions to represent O₂ and CO₂ dissociation curves depending on both O₂ and CO₂ in the blood. The formulation leads to a coupled system of nonlinear partial differential equations. A finite difference technique of $\mathrm{O}\mathrm{(}{\mathrm{(}\mathrm{\Delta }{\mathit{r}}_1\mathrm{)}}^2\mathrm{,}{\mathrm{(}\mathrm{\Delta }{\mathit{r}}_2\mathrm{)}}^2\mathrm{,}\mathrm{\Delta }\mathrm{z}\mathrm{)}$ along with a line iterative approach is described to solve the governing equations of the model together with the physiologically relevant boundary conditions.

It is shown that the interaction of O₂ and CO₂ in the blood plays an important role in the removal of CO₂ from the tissue whereas its effect is not significant on the delivery of O₂ to tissue. It is found that by considering the interaction of O₂ and CO₂ in the blood, the accumulation of the total CO₂ is reduced by about 44% of the total CO₂ exchanged in the tissue. The sensitivity analysis shows that the PO₂ (PCO₂) in the tissue decreases (increases) with the decrease of blood velocity and increase of metabolic rate.