5.1 The reservoir-wave hypothesis

The physiologically meaningless results of the separation of waves into their forward and backward components during diastole resulted in the reservoir-wave hypothesis. The hypothesis is very simple, but has important implications for haemodynamics. The hypothesis is

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

where P_res is the Windkessel pressure defined by Otto Frank (1899) which varies in time but is uniform throughout the arterial system and P_wave is the pressure that drives the waves.

The reservoir pressure is derived from the measured pressure (the theory follows) by fitting the fall-off of pressure during diastole by a exponential decay as predicted by the Windkessel analysis when the aortic valve is closed and there is no flow into the arteries from the heart. The time constant thus determined is related to the net compliance of the arterial system C and the resistance to flow out of the arteries R.

τ = RC

If the flow into the aorta is measured, then the reservoir pressure can be calculated from an integral relationship that also follows from the Windkessel analysis {mathematical details}. If the flow into the aorta is not measured, we have shown that it is still possible to estimate the reservoir pressure from the measured pressure waveform assuming that the contribution from reflected waves is not too large, a good approximation in the proximal arteries that gets progressively worse as you move distally.

Once P_res is determined, P_wave is just the difference between it and the measured pressure. (In our first publications we referred to P_wave as the excess pressure.)

The figure shows the result of separating the measured pressure into Pres and Pwave. The data are the same human ascending aorta data showed previously in the discussion of wave separation. Now the measured pressure (black) is separated into a reservoir pressure (green) and a wave pressure. This wave pressure is then separated using the method previously discussed into its forward (blue) and backward (red) components. Note first of all that the problem of physiologically meaningless, self-cancelling forward and backward waves during diastole has disappeared. The reservoir pressure accounts for virtually all of the measured pressure during diastole and so the wave pressure (like the velocity) is effectively zero during diastole. Note also that the magnitude of the forward wave is similar during early systole, which follows because the capacitance-like reservoir pressure follows the sudden increase in pressure during the initial contraction wave only sluggishly. Finally, note that the backward pressure wave is greatly reduced once the reservoir pressure is accounted for. The reservoir-wave hypothesis is relatively new and we are still working out all of its implications. It is a simple hypothesis that combines two old but, in some ways, contradictory theories. Interestingly, Otto Frank made major, innovative contributions to both the Windkessel and the wave theory of arterial mechanics. He was aware of the contradictions between them and discussed them at some length (without resolving the contradiction, it must be said) in a paper in 1930 [Frank, O. Z. Biol. (1930) 90, 405-409.]

Here are the two examples side by side to show their differences more clearly:

wave separation without reservoir pressure	wave separation with reservoir pressure

The reservoir-wave hypothesis is related conceptually to the three-element Windkessel model first proposed by Westerhof in 1970 and widely used in compartmental electrical-analogue models of the cardiovascular system.