This method of determining the wave speed from the measured P and U waveforms is based on the observation that using the 'wrong' wave speed usually results in self-cancelling forward and backward wave intensity. Remember that the measured wave intensity dI is the sum of the positive wave intensity of the forward wavefront dI+ and the negative wave intensity of the backward wavefront dI-. This means that using the wrong value of c in the equations for the separation of the measured waves into their forward and backward components almost always results in the prediction of more wave intensity than there should be.
This is exploited by finding the value of c that minimises the absolute magnitude of the forward and backward wave intensities over the whole cardiac cycle. The analysis is straightforward but algebraically complex {mathematical details}, but as usual the results are relatively simple. The value of c that minimises the sum of the absolute values of the forward and backward wave intensity is
| (ρc)2 = | Σ dP2 Σ dU2 |
where the sums are taken over an integral number of cardiac cycles.
This method of finding c from the measured data is very easy to implement. The differences in pressure and velocity dP and dU are squared and summed over the cardiac cycle. The square root of the ratio of these two sums, divided by ρ is c.
The sum of squares method generally gives results that are very similar to those obtained by the PU-loop method in the systemic arteries that we have studied.
Prof. M. Sugawara has pointed out that this relationship is exact only if the cross correlation between P and U over a cardiac cycle. Since the cross correlation can be easily calculated from the measured data, this should provide an internal check on the method. In principle, it might also provide a means of correcting the data, although this avenue of research has not yet been explored.
[plot of comparison of results for different methods??]
In principle, this method can be used for measurements made in the coronary arteries and is being used to determine the local wave speed in our clinical measurements. Unfortunately, there are no alternative methods for measuring c in the human coronary arteries and so there is a certain amount of faith involved in its use.
Finally, we note that this method is not sensitive to any relative time shifting between the data. The sums are taken over a cardiac cycle and so it doesn't matter when in the cycle we start. However, any time shift between the data will have significant effects on the calculation of dI and on the separation of the waveforms into their forward and backward components, even if the correct value of c is used.