ANALOGY BETWEEN HEAT AND MASS TRANSFER LEADS TO NEW OXYGEN TRANSPORT EQUATIONS IN VASCULARIZED BIOLOGICAL TISSUES

摘要

Mathematical modeling of oxygen transport in living tissues has been an important approach for quantitatively understanding the physiological events. In a vascularized biological tissue, blood flow plays an important role in the local transport of oxygen, nutrients, Pharmaceuticals, waste products and heat through the body. However, most of the existing oxygen transport models take few considerations of the anatomical structure. Therefore, disagreements among the theoretical predictions and the experimental measurements are common in those studies. This is because geometrical irregularity of the vascular structure remained to be a major obstacle for the accurate modeling. In fact, it has long been a desirable objective to establish a quantitative and generalized model, which is mathematically tractable in the region of interest and considering the detailed anatomical vascular geometry. In this paper, following the theoretical strategy through analogy between the heat and mass transport, the well-established Pennes equation, Chen-Holmes equation, and Weinbaum-Jiji (W-J) equation, etc. were implemented to develop the basic equations for characterizing the oxygen transport inside a vascularized tissue. These models have collectively included the contributions of the vascular geometry and the blood flow condition. As an illustration, predictions using the new model from W-J equation on several typical oxygen transfer problems were discussed. The theoretical results were applied to interpret some previous experimental observations. Further, uncertainties for the predicted oxygen concentrations of tissues due to approximate parameters and vascular structures were analyzed based on developing a generalized equation. Contributions from each of the thermal parameters such as diffusion coefficient, blood perfusion rate, and oxygen consumption rate of the tissues etc. can all be attributed to a single source term, which would make the model much convenient for practical use. The theoretical route proposed in this paper may provide a feasible way to comprehensively characterize the oxygen transport behaviors in living tissues with complex vasculature. It can also be extended to more wide mass transfer issues such as drug and nutrients delivery etc.