The theoretical basis of reservoir pressure in arteries

The separation of measured arterial pressure into a reservoir pressure and an excess pressure was introduced over 20 years ago as a heuristic hypothesis. Since then it has gained some traction through epidemiological studies that show that various measures of reservoir and excess pressure are independent risk factors for cardiovascular disease. We demonstrate that a two-time asymptotic analysis of the 1-D conservation equations in each artery coupled with the separation of the arteries into inviscid and viscid arteries, based on their resistance or drag coefficients, results in a formal derivation of the reservoir pressure. The most important result of this analysis is that flow in the arteries can be considered quasi-steady during every interval on the slow time scale. The slow-time conservation equations involve a drag coefficient that models the effect of viscosity on the flow. We separate the arteries into the larger inviscid arteries where viscosity is negligible and the smaller viscid arteries where it is not. The slow time pressure in the inviscid arteries is shown to be spatially uniform but varying in time. We identify this pressure as the reservoir pressure. Dynamic analysis using mass and energy 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 ascribes its physical meaning.