Mathematical simulation of gas transport and acid/base regulation by blood flowing in microvessels

Abstract

A simple model has been developed for simulation of oxygen (O2) transport by red blood cells (RBCs) to and from blood flowing in vessels with diameters of 20 microns and larger where a substantial fraction of the microcirculatory O2 transport occurs. This model is derived from a more complete model which has been validated experimentally. Detailed calculations of the oxygen concentration distribution reveal that the dominant resistance to O2 transport is distributed in the plasma and that relatively little resistance is present within or in the immediate vicinity of the RBCs. Based on these findings, the complete model is simplified from four simultaneous nonlinear partial differential equations (PDEs) to one PDE by: (1) assuming chemical equilibrium within the RBCs, (2) neglecting intracellular and extracellular boundary layer resistances, and (3) incorporating transport in the RBC-free plasma region adjacent to the vessel wall into the boundary condition. The simplified model is much easier to apply mathematically to new situations. A comparison between the two models shows that they give similar predictions which agree well with experimental measurements. In order to investigate the coupling between the oxygen and carbon dioxide (CO2) transport, an extended model is developed to incorporate CO2, as well as the various blood buffer systems that are closely connected to the transport of these gases. The blood is treated as two continuous coexisting phases: a RBC phase and a plasma phase. The microvessel is divided into two regions: the central, RBC-rich and the outer, cell-free region. The radial distribution of RBCs, and transport of various species due to bulk convection are taken into account. Chemical and transport processes which are included in the model are (1) interactions of hemoglobin with O2 and CO2, (2) the Bohr and Haldane effects, (3) CO2 hydration/dehydration reactions, (4) buffering actions of hemoglobin and plasma proteins, and (5) anion exchange across the RBC membrane. Predictions of the discrete model of simultaneous O2/CO2 transport by flowing blood are shown to be in excellent agreement with prior workers' experimental results from large artificial membrane tubes.