The process of gas exchange in systemic circulation in a hyperbaric environment: an analytical approach

A mathematical model is proposed to deal with the simultaneous transport of oxygen (O2) and carbon dioxide (CO2) in systemic capillaries and the surrounding tissue in a hyperbaric environment. The transport in the capillary region depends on molecular diffusion (radial as well as axial), the convective effect of the blood, and the saturation of haemoglobin with O2 and CO2. The corresponding equation in the tissue region describes the transport of the species due to radial and axial diffusion in the tissue and consumption of O2 in the metabolic process. The production of CO2 inside the tissue is incorporated through the respiratory quotient. The saturation of blood with O2 and CO2 have been approximated by linear functions to simulate the conditions of the hyperbaric environment. The resulting system of governing equations with the physiologically relevant boundary conditions is solved analytically. The concentration of O2 is shown to decrease from the core of the capillary to the tissue periphery, whereas the concentration of CO2 increases. It is shown that very little of the CO2 is transported radially. The location of the vulnerable point from the point of view of CO2 accumulation is found to be the corner (x = R2, z = L) situated at the periphery of the tissue near the venous end of the capillary. The accumulation of O2 and CO2 in the tissue is discussed in terms of various dimensionless parameters. It is found that the accumulation of CO2 increases whereas that of O2 decreases in the hyperbaric environment. Finally, it is surmised that one of the major causes of discomfort among divers could be excessive accumulation of CO2 in the tissue.