The history of reservoir/excess pressure

The concept of the reservoir/excess pressure is still evolving. To understand this evolution it may be interesting to consider the history of the idea.

The germ of the idea sprang from a telephone conversation with John Tyberg of University of Calgary in the early 2000's. We had been collaborating for years on the development and application of wave intensity analysis (see the bibliography in the wave intensity pages). The conversation was about the meaning of the large, self-cancelling forward and backward waves during diastole that were a ubiquitous feature of all of our wave separation analyses. We were having trouble thinking of a physically and physiologically reasonable explanation for them. I remember remarking that if only we could subtract the Windkessel pressure from diastole then the problem would 'go away'. John correctly responded that it would be inconsistent to do the subtraction only during diastole. And the idea was born.

The first publication of the idea was Wang et al. (2003) where we suggested that the measured pressure be divided into a Windkessel pressure that varied only in time and an excess pressure that varied both in time and space

P(x,t) = P_Windkessel(t) + P_excess(x,t)

 

This figure is from that paper. The upper panel shows the aortic pressure (black), the Windkessel pressure calculated from the measure flow (red) and the left ventricular pressure blue. The bottom panel shows the measured flow (black) and the excess pressure (red) scaled so that the peak values were equal. The results of the analysis exceeded our expectations. As expected, wave intensity analysis applied to the excess pressure did not exhibit the large, self-cancelling waves in diastole. Unexpectedly, we also observed, as seen in the bottom panes, that the excess pressure that we calculated was virtually identical in shape to the measured flow. The excess pressure and flow waveforms were so similar that when I saw the first plot of the results I responded saying that we couldn't possibly publish that figure because it was so good that no one would believe it. JJ Wang responded saying that they all looked like that. In the end, we did another experiment introducing a large backward wave in the descending aorta using an intra-aortic balloon pump to show that the analysis would detect the artificially induced wave so that the results were not some artifact of the analysis algorithm.

By the time of our second publication Wang et al. (2005), we had realised that the reservoir pressure was subtly different from the classical Windkessel and so altered our notation from 'Windkessel' to 'reservoir' pressure. We also, at the time, suggested that the excess pressure was responsible for all of the waves in the arteries and changed our notation from 'excess' to 'wave' pressure.

P(x,t) = P_reservoir(t) + P_wave(x,t)

 

In this paper we looked at the reservoir-wave separation applied to measurements taken in the large systemic veins. In some ways the results were even more impressive that the previous results for the aorta. The figure shows simultaneous measurements of pressure and flow in the inferior vena cava (top) and the superior vena cave (bottom) with the ECG at the top as an indication of the timing of the heart beats. In both cases, the measured pressure (black) and reservoir pressure calculated from the measured flow are shown in the first plot and the measured flow (red) and excess pressure (black) are shown in the second plot. First we must note that because blood is flowing into the veins during diastole, the Windkessel pressure in the veins rises rather than falls during diastole. In order to obtain a clear diastole, the heart was temporarily stopped, as indicated in the ECG, during the last second of the measurements, giving the exponentially rising curve at the end of the time shown in the figure. Fitting this rise in pressure gave us the time constant for the veins which enabled the calculation of the reservoir and the excess pressure. As previously, the flow and the excess pressure curves have been scaled to fit their respective maxima and minima. Again, the correspondence between the two waveforms is remarkable. Venous haemodynamics is a very neglected field. This is probably because venous pressure and flow are not periodic and therefore amenable to analysis using the impedance methods that have been developed to analyse arterial flow. The correspondence between flow and excess pressure indicates that wave analysis is valid in the veins as well as the arteries and may provide the means for studying venous haemodynamics. We have not followed up this possibility and encourage anyone reading this to consider doing so.

In retrospect, the change of nomenclature in this paper was both good and bad; good because the reservoir pressure is different in subtle ways from the classical Windkessel pressure and the change in nomenclature removed the direct association of the two, bad because 'wave' pressure implied that the excess pressure was driving all of the wave activity in the arteries (and veins). In fact, all pressure changes in the arterial system must be caused by or cause waves. The reservoir pressure must therefore also be the product of wave motion. For this reason, we have returned to the original nomenclature of 'excess' pressure. This nomenclature was originally inspired by that used by Lighthill in has book on acoustic waves where the pressures due to the sound waves were defined as the excess over the ambient pressure.

For the past few years we have devoted a lot of effort to the understanding and application of the reservoir-wave idea. The experimental and clinical studies have been very encouraging and we have developed an algorithm to calculate the reservoir/excess pressure which is proving to be very reliable and robust. Theoretical efforts to define the reservoir pressure, however, have been much less successful and we are still unsure of the precise meaning of the reservoir pressure.

Jordi Alastruey has recently completed some impressive theoretical and numerical work using a linearised method of characteristics solution of waves on a model arterial system that is very promising. He defines a 'conduit' component of the pressure by solving for the case when peripheral resistances are matched to the peripheral characteristic impedance so that none of the waves generated by the contraction of the ventricle get reflected in the terminal vessels. He then subtracts that result from the results obtained with the physiological boundary conditions to find the 'peripheral' component of the pressure waveform. The conduit and peripheral components share many of the features of the reservoir and excess pressure but more analysis is necessary before the theory is fully satisfactory.

We have also developed a theory based on spatially averaged pressures in the arterial vessels rather than the full wave solution arising from the method of characteristics analysis. This overcomes the combinatorial problems of dealing with the exponentially growing number of waves that are generated in any realistic arterial network because of the reflection and re-reflection of the waves generated by the ventricle. This theory is very promising and has resulted in some very interesting observations about the reservoir pressure defined in the theory. This work and the full definition of the 'new' reservoir pressure will be described in some detail in the theoretical part of these pages.

Briefly, the reservoir pressure is defined as the solution of the mass conservation equation relating the change in volume of the arterial system to the flow into the system from the ventricle and the flow out through the microcirculation, which has the form

P_reservoir(x,t) = P_reservoir(t-τ_n)

where τ_n is the time it takes a wave to travel from the aortic root to vessel n. In other words, it is assumed that the reservoir has the same waveform throughout the arterial system, but that this waveform is delayed by the wave travel time.

This definition of the reservoir pressure eliminates one of the obvious inconsistencies in previous definitions - the assumption that the reservoir was uniform throughout the arterial system. This is obviously not a viable assumption when measurements clearly show that the arterial pressure waveform propagates progressively throughout the arterial system. It also removes a basic contradiction from the Windkessel model: Uniform pressure implies an infinite wave speed, an infinite wave speed can only be achieved if the arteries were rigid, rigid arteries cannot exhibit the Windkessel effect which relies on the compliance of the arteries. This approach to the definition of the reservoir pressure and hence the excess pressure will be the basis for the rest of the web site.