Stochastic Versus Deterministic Models of Oxygen Transport in the Tissue

Abstract

The difficulties in describing gas transport in the tissue are caused by the close connexion of diffusion with convection. For gas transport in the tissue the following differential equation is valid:

$${\alpha ^*}(p(\vec r,t),{u_{i)}}\frac{{\partial p(\vec r,t)}}{{\partial t}} = - A(\vec r,t) + Q(\vec r,t) + div\{ (D(\vec r){\alpha ^*}(p(\vec r,t),{u_i}) \cdot grad{\kern 1pt} p(\vec r,t)\} - \vec v(\vec r,t) \cdot grad{\kern 1pt} \{ {\alpha ^*}(p(\vec r,t),{u_i}) \cdot p(\vec r,t)\} ,$$

((1))

$p(\vec r,t)$ describes the pressure of the gas depending on the site, $\vec r$, and the time, $t.{\alpha ^*}(p(\vec r,t){u_i})$ is the solubility coefficient and the apparent solubility, respectively, of the gas. The value of ${\alpha ^*}(p(\vec r,t){u_i})$ depends, for oxygen in tissue containing hemoglobin or myoglobin, on the pressure, $p(\vec r,t)$, and on the parameters u. as, for instance, Pco₂, which here are not specified in detail. $A(\vec r,t)$ and $Q(\vec r,t)$, respectively, describe the sinks and sources of the gas. For oxygen $A(\vec r,t)$ means tissue respiration. $D(\vec r)$ is the diffusion coefficient of the gas examined. $\vec v(\vec r,t)$ is the flow velocity of the blood. In its general form we deal with a nonlinear differential equation.