Mathematical simulation of the transport of oxygen and important metabolites in the human brain
Description
Mathematical models of the transport of oxygen and the simultaneous transport of carbon dioxide, glucose and lactic acid within the microcirculation of the human brain have been developed. These were solved for both steady-state and dynamic cases which included normal and pathological conditions The models are distributed parameter, interactive systems of non-linear partial differential equations. These were reduced to finite difference equations and solved using various numerical techniques. All three models employ the Price-Varga-Warren finite difference approximations for the capillary equations. The steady-state constant tissue metabolism model (SSCM) requires an analytical steady-state radial mass flux from the capillary. The dynamic non-linear tissue metabolism model (DNLM) uses Crank-Nicolson analogs in solving the non-linear, radial diffusion tissue equations. The third model is similar to the DNLM model but also includes axial diffusion in the tissue. Both the Extrapolated Liebmann and the Alternating Direction Implicit methods were used to solve the tissue equations of this model. Steady-state overall mass balances for normal conditions revealed errors of 0.1, 2.0 and 12.0 percent, respectively, for each of the three models The models incorporate in the transport description, the processes associated with the interaction of components through the Bohr and Haldane effects and the non-linear metabolism. Also studied was the oxygen mass transfer coefficient of the red blood cell and its effect on tissue oxygen delivery. Other cases investigated include normal conditions, arterial and venous hypoxia, reduced flow and hematocrit, hypocapnia, reduced glucose and with the dynamic models transient arterial upsets Results indicate that local concentration deficits of oxygen and/or glucose and excesses of lactic acid can exist with the system described by the models and therefore possibly could exist in the tissues of the human brain under similar circumstances. The implications are that the local energy production may be impaired directly or indirectly by the lack of oxygen and/or glucose and the existence of a local acidic environment. It is also proposed that local variations of components can be masked in experimental studies of regions of multiple capillaries, since the total chemical content gives no indication of local deficits and/or excesses