A two-compartment model of oxygen transport in skeletal muscle using continuously distributed capillaries

For future application to studying regulation of microvascular oxygen delivery, a model is developed for O₂ transport within an idealized volume of tissue, that is perfused by a continuous distribution of capillaries. Considering oxygen diffusion, convection, and consumption, an O₂-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 O₂ 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, O₂ consumption rate, and red blood cell velocity. In addition, comparisons of an approximate linearized solution (for fixed PO₂ at x=0) are made to the corresponding discrete capillary solution. Results of the continuous capillary model are presented for varying inlet O₂ 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.