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.

Pulmonary artery wave propagation and reservoir function in conscious man: impact of pulmonary vascular disease, respiration and dynamic stress tests

“…This may be attributed to the increase in P ∞ , which will result in a decrease of the estimated τ (Parragh et al . ). RC TPR calculated as the product of TPR and total arterial compliance gives a subtly different, but related estimate of the diastolic pressure decay.…”

The modified arterial reservoir: An update with consideration of asymptotic pressure (P∞) and zero-flow pressure (Pzf)

“…This is one of several well-established ways to estimate total arterial compliance, 23 although it should be noted that inclusion (or not) of P zf has a substantial effect on estimates of the time constant or arterial compliance using this method. 24 – 26 We use maximum negative rate of pressure change ( max −dP/dt ) as the indicator of end-systole to determine P es , since the timing of this event has been shown to agree very closely (mean error < 0.4 ms) with the time of cessation of aortic flow at the end of systole in invasive studies in dogs 27 and is easy to identify in recorded pressure waveforms.…”

Estimating central blood pressure from aortic flow: development and assessment of algorithms

“…P out ), performance may be improved via simultaneous or iterative estimation, as suggested by Parragh et al (56), though this was beyond the scope of our study.…”