An algorithm for calculating reservoir pressure from the arterial pressure waveform

The reservoir-wave hypothesis was introduced in order to resolve a number of difficulties in the interpretation of arterial mechanics. The capacitive nature of the compliant arteries was recognised by Borelli (Borelli 1680), popularised by Hales (Hales 1733) and made quantitative by Frank (Frank 1899). The Windkessel model introduced by Frank is an excellent representation of arterial mechanics during diastole when the arterial flow is isolated from the left ventricle by the closed aortic valve. However, despite decades of work, the Windkessel model could not describe the observed pressure and flow in the arteries during systole. As a result, it fell into disfavour.

The wave nature of arterial mechanics was obvious to the earliest observers and a great deal of attention was given to the arterial pulse. In the second half of the twentieth century, the wave analysis of arterial mechanics became the dominant interpretation of arterial physiology. The measured pressure and flow were Fourier analysed and state of the cardiovascular system was characterised by its impedance, derived by dividing the pressure coefficient by the flow coefficient for each frequency. This method proved to be a very powerful descriptor of the state of the cardiovascular system (Nichols).

The impedance method provided a way of separating the measured pressure and flow waveforms into forward and backward travelling waves. This is very convenient in the arterial system with the ventricle at the proximal end of the arteries and the highly resistive microcirculation at the distal end. This separation into forward and backward waves has also been very successful in describing many of the puzzling features of the mechanics of the arterial system. Features that can be explained include the difference between the flow and pressure waveforms, the increase in pulse pressure in more distal arteries and the change in pressure waveform with age [Refs to be included??]. There are, however, some features of the separated forward and backward waves that are difficult to explain. Most prominent of these difficulties is the large wave that occurs during diastole in which the two pressure waves sum to give the diastolic decrease in pressure that is characteristic of all pressure waveforms and the two flow waves subtract to give the very small flow waveforms that is characteristic of all flow waveforms during diastole.

The reservoir-wave hypothesis resolves this problem by assuming that the measured arterial pressure \(P\) is the sum of a reservoir pressure \(\bar{P}\) reflecting the capacitive nature of the arteries and an excess pressure \(p\) describing their wave nature. Under this hypothesis, the exponential decay of pressure during diastole is described by the reservoir pressure and the excess pressure during diastole is very small, similar to the flow. The reservoir pressure is not exactly the Windkessel pressure described by Frank’s theory, which assumes that the wave speed in the arteries is infinite, because it is seen as the result of waves and can vary from place to place in the arterial system, unlike the Windkessel pressure. There have been several attempts to define the reservoir pressure theoretically but none are entirely satisfactory. It is best seen as an ad hoc hypothesis to be judged by its usefulness in explaining physiological and pathological arterial behaviour. Because of this, it is important to have a well-defined algorithm for calculating the reservoir pressure from the measured arterial pressure.

In the original presentation of the reservoir-wave hypothesis (Wang 2003), the reservoir pressure was calculated from the flow waveform that was simultaneously measured. It was recognised that the difficulty in simultaneously measuring pressure and flow clinically is a serious limitation of the method. Recently, we published a method for estimating the reservoir pressure using only the measured pressure waveform (J. 2007). This method was based on the observation in canine experiments that the excess (or wave) pressure waveform in the ascending aorta, derived by subtracting the flow-derived reservoir pressure from the measured pressure, was very nearly proportional to the measured flow; the constant of proportionality being the characteristic impedance of the aorta. The results obtained by this method, applied in a number of arteries in both canines and humans, were encouraging. We have found, however, that the application of the method described in that paper resulted in the failure of the algorithm in an unacceptable fraction of cases. This problem is also reported in another study comparing the reservoir pressure to the windkessel pressure obtained from a three-element windkessel model (Vermeersch 2009). In this study, they report that the reservoir pressure algorithm failed in 307 of 2019 subjects.

This paper describes a new Matlab algorithm for calculating reservoir pressure from a measured pressure waveform. It is based on the same principles as the previous algorithm but incorporates a number of differences that make it much more robust. The major changes include a method for fitting the pressure during diastole based on exponential moments and a global parameter fitting iteration that minimises the sensitivity of the method to the accurate determination of the end of systole and start of diastole.

As in our previous paper (J. 2007), the calculation of the reservoir pressure is based upon the mass conservation arguments used by Frank in the derivation of the windkessel model

\begin{equation} \frac{dV}{dt}=Q_{in}-Q_{out},\nonumber \\ \end{equation}where \(V\) is the volume of the arterial system, \(Q_{in}\) is the volume flow rate into the arteries from the ventricle and \(Q_{out}\) is the volume flow rate out of the arteries through the microcirculation. We assume that the compliance of the arteries, \(\mathcal{C}=\frac{dV}{d\bar{P}}\), is constant and that flow through the microcirculation can be describe by the linear relationship

\begin{equation} Q_{out}=\frac{\bar{P}-P_{\infty}}{\mathcal{R}},\nonumber \\ \end{equation}where \(P_{\infty}\) is the pressure at which flow through the microcirculation ceases and \(\mathcal{R}\) is the net resistance of the microcirculation. We note that \(P_{\infty}\) is not necessarily the venous pressure but could be related to the interstitial pressure through a waterfall effect. In the algorithm, \(P_{\infty}\), \(\mathcal{R}\) and \(\mathcal{C}\) are treated as model parameters. With these assumptions, the mass conservation equation can be written

\begin{equation} \label{eqwkflow}\frac{d(\bar{P}-P_{\infty})}{dt}+b(\bar{P}-P_{\infty})=\kappa Q_{in}\nonumber \\ \end{equation}where \(b=1/\mathcal{RC}\), \(\kappa=1/\mathcal{C}\) and we have assumed that \(P_{\infty}\) does not vary with time.

This equation can be solved by quadrature for any \(Q_{in}(t)\) using the integrating factor \(e^{\frac{t}{\mathcal{RC}}}\), which leads to

\begin{equation} \bar{P}-P_{\infty}=\kappa e^{-bt}\int_{0}^{t}Q_{in}(t^{\prime})e^{bt^{\prime}}dt^{\prime}+e^{-bt}(\bar{P}_{d}-P_{\infty}),\\ \end{equation}where \(\bar{P}_{d}=\bar{P}(0)\) is the reservoir pressure at \(t=0\), taken as the end diastolic time. This is the classical result which has been used to calculate the reservoir pressure when the volume flow rate into the arteries \(Q_{in}\) is measured.

We note that the solution is particularly simple during diastole when \(Q_{in}=0\). If we define \(P_{N}=\bar{P}(T_{n})\) as the pressure at \(t=T_{n}\), the time of the dicrotic notch which denotes the start of diastole, then the solution during diastole is

\begin{equation} \label{eqwkflowdiastole}\bar{P}-P_{\infty}=e^{-b(t-T_{n})}(P_{N}-P_{\infty}),\qquad T_{n}\leq t\leq T,\\ \end{equation}where \(T\) is the cardiac period.

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