For future application to studying regulation of microvascular oxygen delivery, a model is developed for O${}_2$ transport within an idealized volume of tissue, that is perfused by a continuous distribution of capillaries. Considering oxygen diffusion, convection, and consumption, an O${}_2$-dependent transfer term between the capillaries and tissue is used to extend previous single-compartment approaches to include separate tissue and capillary compartments. The coupled tissue–capillary PDE system is considered for unidirectional capillary flow in $z$, as a simplified model of O${}_2$ transport in skeletal muscle, and steady-state $2D$ solutions are obtained using boundary conditions in $x$ that are consistent with two experimental situations of interest. To validate the continuous capillary model, comparisons are made of an exact nonlinear solution (for no flux at $x=0$) to results of an established discrete capillary model (solved via finite differences) for varying capillary density, O${}_2$ consumption rate, and red blood cell velocity. In addition, comparisons of an approximate linearized solution (for fixed PO${}_2$ at $x=0$) are made to the corresponding discrete capillary solution. Results of the continuous capillary model are presented for varying inlet O${}_2$ saturation, showing the utility of the new model for studying physiological problems. Numerical solution of the new model for problems with time dependence and complex geometry is expected to be substantially more efficient than for the corresponding discrete capillary problems.

为了将来应用于研究微血管氧输送的调节，开发了一个模型用于 O${}_2$在理想化的组织体积内运输，即由连续分布的毛细血管灌注。考虑到氧气的扩散、对流和消耗，O${}_2$毛细血管和组织之间的依赖转移项用于扩展先前的单隔室方法，以包括单独的组织和毛细血管隔室。耦合组织 - 毛细血管 PDE 系统被认为是单向毛细血管流动$z$, 作为 O 的简化模型${}_2$ 骨骼肌中的转运和稳态 $2D$ 使用边界条件获得解决方案 $X$这与两个感兴趣的实验情况一致。为了验证连续毛细管模型，比较了精确的非线性解（对于在$X=0$) 对不同毛细管密度 O 建立的离散毛细管模型（通过有限差分求解）的结果${}_2$消耗率和红细胞速度。此外，近似线性化解的比较（对于固定 PO${}_2$ 在 $X=0$) 对应的离散毛细管溶液。给出了不同入口 O 的连续毛细管模型的结果${}_2$饱和度，显示了新模型在研究生理问题方面的效用。对于具有时间相关性和复杂几何形状的问题，新模型的数值求解预计比相应的离散毛细管问题更有效。