the justification of reservoir pressure (2)

One of the most convincing demonstrations of the utility of the reservoir-wave hypothesis comes from work carried out by J-J Wang, formally of the University of Calgary where the work was done, currently at the Fu Jen Catholic University, Taiwan. The reference is below and should be consulted for the full description of the work.

The study involved simultaneous measurements of pressure and flow at four locations along the aorta: the aortic root, the descending aorta just beyond the left subclavian artery, just above the diaphragm and just before the aorto-iliac bifurcation. The reservoir and excess pressures were determined at each measurement site using the measured pressure and flow. The excess pressure was then separated into its forward and backward components and these data formed the basis of the study. Additionally, the pressure was measured sequentially at intervals of 2 cm from the ascending aorta to the femoral artery. These data were ensemble averaged and related to the simultaneously measured data using the peak of the R-wave on the ECG.

The primary goal of the study was to see if the pattern of forward and backward waves measured at different sites was consistent; i.e. are forward waves measured at one site seen at the more distal sites after the a delay time determined by the wave speed and are the backward waves seen at more proximal sites? The answer these questions was yes and a very interesting and convincing pattern of wave travel emerged from the analysis.

The best way to illustrate the results is pictorially with the following figures:

This figure shows the pressure measurements. The 4 sites where pressure and flow were measured are the aortic root (black), descending aorta (red), at the level of the diaphragm (green) and the aorto-iliac bifurcation (magenta). The blue lines represent the pressure measurements at 2 cm intervals. The black dots identify the start of the initial compression wave and the slope of the line connecting these points in the x-t plane indicates the local wave speed. This 'wave track' is important in the subsequent analysis because other waves must follow parallel tracks (ignoring the effects of convection, which are relatively small since the Mach number (ratio of convective velocity to wave speed) is small).

This figure shows the reservoir pressure (thick line) calculated using the measured pressure (thin line) and the measured flow (not shown). The excess pressure is the difference between the two and is used in the rest of the analysis. The colour coding for the 4 measurement sites is retained in all of the figures.

The excess pressure is then separated into its forward (thin lines) and backward (thick lines) components, again using the measured velocity. Salient features of these waveforms are then identified; the feet of waves or, when the magnitudes are too small to make the identification of the foot of the wave, the wave peak.

This figure shows the points identified as the feet of compression waves marked by solid circles. The black line to the left is the line which best fits the feet of the initial compression wave using data at the four principle sites of measurement and the feet of the pressure measurements taken at 2 cm intervals along the arteries (the black circles in the first figure). The second black line indicates the wave track in the backward direction obtained by negating the sign of the slope of the forward wave track. For reasons that will become clear, only the most distal data shows a clear compression wave foot (thick magenta line) and so the backward wave track is positioned to pass through that point. This wave track represents a backward compression wave apparently originating from a reflection site at approximately 70 cm from the aortic root. There are secondary forward compression waves in the 3 most distal measurements and the third black line is determined as the forward wave track that best fits the three wave feet. This is a forward compression wave and its origin seems to be the reflection of the backward compression wave from the aortic valve, which is closed at the time of arrival of the backward compression wave.

At the two middle measurement sites there is evidence of a decompression wave in the backward waveforms (thick lines) which are marked by open circles in the figure. The dotted line is the backward wave track that best fits these two points. This line intersects the initial compression wave at approximately 30 cm which indicates that there is a negative reflection site (i.e. a reflection site with a negative reflection coefficient) at that point in the aorta. There is no evidence that this backward wave is reflected in the same way that the backward compression wave was. As we will see in the next figure, this is probably because the aortic valve is still open at the time of arrival of the backward decompression wave and the reflection from the ventricle is either non-existent or attenuated to the degree that it is not discernable.

The wave tracks determined by the previous analysis are all included in this x-t plot. The coloured horizontal lines indicate the position of the measurements. The closed circles indicate the time of the feet of compression waves, both forward and backward. The open circles indicate the time of the feet of decompression waves. The small black circles indicate the feet of the initial compression wave measured at 2 cm intervals from the aortic root to the femoral artery. The black line through them indicates the wave track of the initial compression wave and this wave track is used as the template for all other waves, appropriately inverted for backward waves (i.e. the same slope but with the sign inverted). The other black lines are the wave tracks determined by fitting this template to the discernable wave feet for compression waves and the dotted line is the wave track fitted to the decompression waves. The intersection of the first two lines indicate that there is a positive reflection site approximately 70 cm from the aortic root. Similarly, the intersection of the wave track of the initial compression wave and the wave track of the decompression wave (dashed line) indicate that there is a negative reflection site at approximately 30 cm . The spatial position of the distal reflection site is only approximate because we have no direct measurements of the wave speed in vessels beyond the last measurement site. We observe, however, that the slope of the initial wave track is nearly constant up to the aorto-iliac bifurcation and then seems to increase in the iliac and femoral artery. The faint red lines indicate the estimated location of the distal reflection site if we use the proximal wave speed to estimate distance rather than the local wave speed. It is meant as an indication of the uncertainty in relating temporal events to spatial locations without detailed knowledge of the local wave speeds all along the wave track. Note also that the backward decompression wave wave arrives while the aortic valve is open and there is not discernable reflection. The backward compression wave, however, arrives after it is closed and a reflected, forward compression wave is generated.

All of the previous analysis considered only the waves generated by the initial compression wave. There is also a forward decompression wave generated at the end of systole and it is interesting to follow these waves. Instead of feet, we now look at the maxima of the various excess pressure waveforms as an indication of the start of the decompression waves. The maxima of the forward (thin lines) and backward (thick lines) waveforms are indicated by open circles in this figure. The dashed black lines indicate the wave tracks fitted to these points. We see that the pattern of the decompression waves is very similar to that of the compression wave. A positive reflection site is indicated at 70 cm. The reflected wave propagates back to the aortic root, arriving well after the valve has closed and generating a re-reflected forward decompression wave.

This figure shows that the minima of the backward waveforms, which indicate the start of compression waves, also suggest that there is a negative reflection site at 30 cm. That is, the forward decompression wave is negatively reflected, resulting in a forward compression wave (solid line), at the reflection site already identified from the analysis of the compression waves.

This figure shows the wave tracks of the decompression wave at the end of systole and the reflected and re-reflected waves that it generates. It is in the same format as the x-t plot for the initial compression at the start of systole. The similarity between these two plots, with solid and dashed lines transposed, shows that the pattern of waves that is detected by this analysis is internally consistent.

I believe that these results provide strong evidence for the reasonableness and the utility of the reservoir-wave hypothesis. Similar efforts using the separated measured pressure gave meaningless results, primarily because the self-canceling forward and backward waves during diastole masked the feet of the individual waves. It was only after the reservoir pressure was subtracted and the excess pressure was separated into its forward and backward components that these points aligned in a self-consistent way.

The existence of a strong negative reflection site at ~30 cm (near to the renal arteries) was not anticipated in the experiment. In fact, the examples shown here are from a series of experiments designed to explore the wave mechanics in this region because of the unexpected results of earlier measurements restricted to the proximal aorta. The fact that reflections arising at this point occurred for both the initial compression wave at the start of systole and the decompression wave at the end of systole, supports evidence of its existence.

There are many implications of this work and anyone who is interested should have a look at the paper. It provides a very good illustration of how useful the concept of reservoir and excess pressure can be when analysing the very complex patterns of wave travel in the arteries.

References:

J-J Wang, NG Shrive, KH Parker, AD Hughes and JV Tyberg (2011) Wave Propagation and Reflection in the Canine Aorta: Analysis using a Reservoir-Wave Approach. Can. J. Cardiol. 27 .