Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to d erive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a fu nction only of time. This mathematical model does not allow for spatial var iations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordina ry differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare the m to the solution of the ordinary differential equations described above. I t is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so th e ordinary differential equation should not be used if an accurate calculat ion of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the densit y of the capillaries that supply blood to this tissue. As a result, techniq ues that are based on the ordinary differential equation to calculate the m ixed-venous blood partial pressure may be in error. (C) 2001 Academic Press .