A mathematical model of oxygen transport in intact muscle with imposed surface oscillations

A one-dimensional (1D) reaction–diffusion equation is presented to model oxygen delivery by the microcirculation and oxygen diffusion and consumption in intact muscle. This model is motivated by in vivo experiments in which oscillatory boundary conditions are used to study the mechanisms of local blood flow regulation in response to changes in the tissue oxygen environment. An exact periodic solution is presented for the 1D ‘in vivo’ model and shown to agree with experimental data for the case where the blood flow regulation system is not activated. Approximate low- and high-frequency solutions are presented, and the latter is shown to agree with the pure diffusion solution in the absence of sources or sinks. For the low frequencies considered experimentally, the 1D in vivo model shows that as depth increases: (i) the mean of tissue O2 oscillations changes exponentially, (ii) the amplitude of oscillations decreases very rapidly, and (iii) the phase of oscillations remains nearly the same as that of the imposed surface oscillations. The 1D in vivo model also shows that the dependence on depth of the mean, amplitude, and phase of tissue O2 oscillations is nearly the same for all stimulation periods >30 s, implying that experimentally varying the forcing period in this range will not change the spatial distribution of the O2 stimulation.