Oxygen transport in the human cerebral cortex has been simulated previously by considering a single straight capillary and a concentric tissue cylinder as a representa tive sample (the Krogh tissue cylinder). The same geo metric model has been assumed for this study because of its probable accuracy and to allow comparison with previous calculations. Because of the nonlinear equilibrium relationship be tween dissolved and hemoglobin-bound oxygen (the oxy gen dissociation curve), a differential mass balance yields a nonlinear partial differential equation describing oxy gen transport in the capillary. In the tissue a constant consumption rate is assumed and a linear partial differ ential equation results. The tissue equation and the capillary equation are solved simultaneously, to account for capillary-tissue interaction, thus giving axial and radial oxygen partial pressure profiles for the total system. The Monte Carlo method has been employed to solve the resulting set of equations on a digital computer. This technique involves a numerical scheme wherein a process is simulated directly by a random walk phenomenon (Markov process). A particular advantage of this method is that solutions may be obtained for one point in space independently of all others. Therefore, an analysis of the transient response of the "lethal corner" is possible with out solving for the entire mesh. Computation time with the Monte Carlo method was considerably less than that with standard deterministic numerical calculation tech niques. Two models were considered. In the first model the oxygen dissociation curve was linearized, thus giving a linear partial differential equation describing oxygen transport in the capillary. The second model considered the nonlinear nature of the oxygen dissociation curve, and a nonlinear partial differential equation was ob tained. However, because of difficulties with the Monte Carlo method, it was necessary to solve the equations in a quasi-nonlinear manner. For both the linear and the quasi-nonlinear models, solutions were obtained for the steady state with normal conditions and for dynamic cases. Perturbations of ar terial-oxygen partial pressure and blood flow rate were forced, and the behavior of the system was determined. The pure diffusional model checked with previous de terministic calculations demonstrating very rapid re sponse with time constants in the order of 2 seconds. However, the shapes of the intracapillary and tissue oxy gen tension profiles differed somewhat and no allow ances were made for autoregulatory phenomena such as varying flow rates.
Numerical Schemes and Monte Carlo Method for Black and Scholes Partial Differential Equation: A Comparative Note
