![[image]](pics/puloop.jpg)
The wave speed of elastic waves in the arteries is very important for both clinical and theoretical reasons. The theory tells us that the wave will travel with the speed U ± c where the wave speed (with no blood velocity) is
| c = | 1 (ρD)1/2 |
where D is the distensibility of the vessel. The wave speed is therefore a measure of arterial distensibility which is clinically important. Theoretically, the wave speed figures prominently in virtually every aspect of the analysis. The separation of forward and backward waves is particularly sensitive to the value of c that is used.
Wave speed in arteries has traditionally been determined by measuring the time it takes for the pulse wave to travel between two measurement sites. Although the peak of the pressure or velocity is probably the easiest to measure, it has been shown that it is more accurate to measure the time of the foot of the wave since that alters less as the waveform changes as it propagates. For this reason these methods are generally known as foot-to-foot methods.
For the purposes of wave intensity analysis these measurements are not really adequate. They measure some average of the wave speed over the distance between measurement sites rather than the local wave speed at the site of measurement. (It is actually a wave speed weighted average because the wave spends more time in vessels with low wave speed than in vessels with higher wave speeds.) A great deal of effort was expended in the early days of wave intensity analysis in finding a way to measure the local wave speed accurately.
Probably the best method takes advantage of the water hammer equation
dP± = ± ρc dU±
This equation says that if there are only forward wavefronts present then dP is proportional to dU and the constant of proportionality is ρc. The easiest way to exploit this relationship is to plot P vs U for the cardiac cycle - the PU-loop.
A PU-loop (top left) plotted together with P - Pd vs. t (top right) and U vs. t (bottom left). The dotted red line shows the slope during early systole and is equal to ρc.
Since there is little or no wave motion during late diastole, it is almost certain that there is some interval at the start of systole when the only wavefronts are those generated by the contraction of the ventricle; it will take some time for reflections downstream to propagate back to the measurement site. In both the systemic and pulmonary arteries we always observe a linear portion of the PU-loop at the start of systole. This provides a very simple way to determine the local c: measure the slope of the linear portion of the PU-loop and divide it by the density of blood to obtain c.
Although the method is simple, there are several points that should be stressed:
1) The method is very sensitive to relative delays between the pressure and velocity measurements
Because the pressure and velocity are changing rapidly during the initial upstroke of the pulse wave, any relative delay between the two measurements will cause problems in the PU-loop method. This is illustrated in the figure where the same data are plotted with different shifts relative to each other. We see that a shift of only one or two sampling intervals can make the linear portion of the curve become significantly convex or concave and make the determination of the appropriate section to use for the linear fitting very difficult or even impossible.
One solution is the accurate calibration of the measuring instruments in time as well as sensitivity. In addition to electronic delays in the measuring instruments, there is also the problem that simultaneous measurements of P and U at exactly the same location in the artery are impossible for practical reasons; the two sensors frequently interact with each other and to there usually has to be at least a small distance between the two sensors. This will introduce a lag between the two measurements that depends on the wave speed and the velocity of blood in a very complex way.
A practical solution that we frequently use is to assume that there will be a period of only forward waves at the start of systole and shift the P and U data until the most linear line is obtained. This may sound circular, but it has a very firm theoretical foundation. The water hammer equation is based on conservation of mass and momentum and is a very fundamental relationship. The absence of wavefronts in late diastole strongly implies that there will be some period in early systole when the forward wavefronts generated by the initial ventricular contraction are the only waves in the vessel since the backward wavefronts are the result of reflections of these forward wavefronts and for the length of time it takes these wavefronts to travel to a site of reflection and back again they will be the only waves present. Therefore there should be a linear portion of the PU-loop at the start of systole.
2) The time at which the PU-loop first deviates from linear correlates well with the time of arrival of the first reflection
A corollary to the existence of a linear portion of the PU-loop in early systole is that the relationship will cease to be linear when significant backward waves arrive at the measurement site. This has been verified in in vitro tests where the wavefronts can be controlled and measured with great accuracy.
3) The method will fail if there is a reflection site too close to the measurement site
It can be shown theoretically that a reflection site very close to the measurement site will cause a systematic error in the wave speed determined by this method. If there is only one reflection site with a known reflection coefficient, the measurements could be corrected theoretically. However, the correction factor varies from 0 to ∞ depending upon the usually unknown value of R, which makes it a singularly bad technique of correction.
The reason for the systematic error is easy to understand. If the time delay between a forward wavefront and its reflection, the effect of the reflected will be very strongly correlated with the forward wave causing an error in the PU plot whose magnitude depends on the degree of correlation. In practice, the method should not be used if there is a reason to believe that there is a significant reflection site nearby.
The method will fail badly if there are backward wavefronts at the measurement site in early systole
We have tested the method in both systemic and pulmonary arteries using all available methods for comparison of the results and it generally performs very well. The exception to this rule are the coronary arteries where we expect backward wavefronts caused by the compression of the intramyocardial vessels during myocardial contraction. These wavefronts can actually precede the forward wavefronts transmitted from the ventricle into the aorta and the coronary ostia. In these cases the PU-loop fails badly. It proved necessary to find another method for determining c in the coronary arteries. This is presented in the next section. [the sum of squares method]