Influence of an Asymptotic Pressure Level on the Windkessel Models of the Arterial System

Abstract

Windkessel models are lumped-parameter models of the arterial system that describe the dynamic relation between blood flow and pressure in the aorta. Despite their simplicity, they are used in many applications including methods for the non-invasive stratification of cardiovascular risk. However, even though they have been studied extensively, there is still disunity regarding the question if pressure should be modelled as resulting solely from the ejection of the heart or if an asymptotic pressure level should be included which is independent from cardiac beating. The aim of this work is to mathematically analyse the influence of such a pressure level P∞ on the model behaviour of the four most widely used Windkessel models (two-, three- and four-element Windkessel, the latter in a series as well as a parallel configuration). Therefore, the model equations are introduced and Fourier analysis is performed to clarify the impact of P∞ on the other parameters. Then, a typical aortic flow wave is used as input to the models and simulation experiments with varying values of P∞ are performed. Theoretical considerations as well as numerical results show that, in all four models, including P∞ mainly affects the diastolic part of the modelled pressure wave and could potentially improve the fitting performance during diastolic decay. However, further research is needed to clarify the physiological interpretation of P∞ as well as its appropriate size.