Analytical Models of Oxygen Delivery to the Microcirculation
Advective-diffusive transport of passive or reactive scalars in confined environments (e.g., tubes and channels) is often accompanied by diffusive losses/gains through the confining walls. Furthermore, transport through bifurcating networks drives a myriad of biological and geophysical phenomena. Our analysis is motivated by oxygen transport in arteriolar blood vessels and oxygenation of the surrounding tissue, but is equally applicable to a large number of other biological and engineered systems. We develop transport solutions for the arteriolar vascular tree assuming that the arteriolar tree bifurcates in fractal patterns. In particular we focus on the effects of changing hematocrit (Hct) on the rate of circulatory oxygen delivery (DO2) to the micro-circulation. Hct affects blood’s oxygen carrying capacity (CaO2) and blood viscosity, these two properties oppose each other in relation to DO2. We studied the effects of transfusing 0.5 - 3.0 units of packed red blood cells (pRBC, 300 ml/unit, 65% Hct) to anemic patients. Our results show that maximal DO2 occurs in the anemic range where Hct < 39%.
A portion of this work is applicable to non-biological systems that are relevant to diffusion-reaction solutions for composite materials. Common examples include thermal insulation problems and hydrolysis. We consider both linear and non-linear reaction rates in combination with Fickian diffusion. More specifically, this application may better quantify hydrolysis rates in high explosive components of stockpile weapons. Conventional approaches to analyzing such systems often included employing numerical programs. While useful, these solutions are computationally expensive and cannot easily provide physical insight into simpler geometries. The analytical solutions documented herein can provide physical insight and provide a verification metric, or, trial solutions for more complex schemes. In certain instances, they may even provide upper or lower bounds to effectively certify weapon components. Solution-bounding can thereby sidestep the need to compute expensive numerical solutions to full weapon system geometries. Finally, we emphasize that our transport solutions are generally capable of handling a variety of boundary conditions and do not require re-derivation. For this reason, these solutions are easily integrated to create models for far more complex systems. All of these models are one dimensional and are used as approximations of two dimensional models with minimal error.