4.2 Wave separation

If we assume that wavefronts are additive so that dP = dP₊ + dP_- and dU = dU₊ + dU_- then these equations and the water hammer equations can be solved for the forward and backward waves in terms of the measured waves.

dP₊ = (dP + ρc dU)/2

dP_- = (dP - ρc dU)/2

where the corresponding changes in the velocity can be found from the water hammer relationships:

dU₊ = (dP + ρc dU)/2ρc

dU_- = - (dP - ρc dU)/2ρc

Having found the pressure and velocity changes due to the forward and backward wavefronts, the macroscopic wave can be found simply by summing these differences.

P_± = ΣdP_±

U_± = ΣdU_±

To relate these cumulative changes to measured absolute values, it is necessary to specify the starting value. This is equivalent to the 'integration' constant in calculus. For pressure this can be taken as the diastolic pressure if direct comparisons with measured pressure are to be made. Since the velocity in the arteries is usually zero during late diastole, the integration constant can usually be taken as zero, although this is not true in vessels such as the carotid arteries and the umbilical arteries during pregnancy which usually have a positive velocity throughout the cardiac cycle. {mathematical details}

These relationships are very important results from wave intensity analysis since they allow us to separate the forward and backward waves from the measured waveforms. The results of such a separation applied to data measured in the human ascending aorta are shown in the figure.

The top figure shows the pressure with the measured pressure in black, the forward pressure in blue and the backward pressure in red. The bottom figure shows the velocities with the same colour code. Note that, as required by the water hammer equations, the forward pressure and velocity are exactly alike with the scale factor ρc. Similarly the backward pressure and velocity are identical with the scale factor -ρc.

For both the pressure and the velocity, the forward and backward waveforms sum to give the measured waveform. Early in systole the backward waves are effectively zero indicating that there are no reflections in the ascending aorta at this time and the only waves are due to the contracting ventricle. After about 60 ms the backward wave starts to become prominent, but never larger than the forward wave. The ease with which wave intensity analysis can be interpreted in terms of time intervals is obvious here.

Turning our attention to the diastolic part of the cardiac cycle we see a problem with the wave separation. The falling pressure during diastole together with the small, nearly constant velocity gives rise to relatively large, self-cancelling waves during diastole. This is a necessary result of the analysis but it is difficult to think of a physiologically reasonable explanation for these self-cancelling waves during diastole when the aorta is cut off from the left ventricle by the closure of the aortic valve. This problem led us to the reservoir-wave hypothesis which we will discuss in the following pages.