The theoretical basis of reservoir pressure in arteries

The separation of measured arterial pressure into a reservoir pressure and an excess pressure was introduced nearly 20 years ago as an heuristic hypothesis. We demonstrate that a two-time asymptotic analysis of the 1-D conservation equations in each artery coupled with the separation of the smaller arteries into inviscid and resistance arteries, based on their resistance coefficients, results, for the first time, in a formal derivation of the reservoir pressure. The key to the two-time analysis is the existence of a fast time associated with the propagation of waves through the arteries and a slow time associated with the convective velocity of the blood. The ratio between these two time scales is given by the Mach number; the ratio of a characteristic convective velocity to a characteristic wave speed. If the Mach number is small, a formal asymptotic analysis can be carried out which is accurate to the order of the square of the Mach number. The slow-time conservation equations involve a resistance coefficient that models the effect of viscosity on the convective velocity. On the basis of this resistance coefficient, we separate the arteries into the larger inviscid arteries where the coefficient is negligible and the smaller resistance arteries where it it is not negligible. The slow time pressure in the inviscid arteries is shown to be spatially uniform but varying in time. We define this pressure as the reservoir pressure. Dynamic analysis using mass conservation in the inviscid arteries shows that the reservoir pressure accounts for the storage of potential energy by the distension of the elastic inviscid arteries during early systole and its release during late systole and diastole. This analysis thus provides a formal derivation of the reservoir pressure and its physical meaning.