5.2 The calculation of the reservoir pressure

Although the mathematical determination of the reservoir pressure is a little complex {mathematical details}, the principles involved are very simple. The theory is basically the theory developed by Otto Frank in 1899 to quantify the idea of the Windkessel suggested by Giovanni Borelli in the 17th and Stephan Hales in the 18th century.

We consider the whole of the arterial system with volume V and compliance C = dV/dP. The flow into the arteries from the heart is Q_in and the flow out through the capillaries is assumed to be resistive so that Q_out = (P - P_∞)/R, where R is the resistance of the microcirculation and P_∞ is the pressure at which flow through the capillaries ceases (not necessarily the venous pressure). The conservation of mass requires that

dV/dt = Q_in - Q_out

Using the above assumptions this can be written as an equation for the reservoir pressure P

dP/dt + (P - P_∞)/RC = Q_in/C

This equation can be solved explicitly if Q_in(t) is known {mathematical details}. During diastole, when Q_in = 0 the solution is simple an exponential decay

P_res = P_res,0e^-t/τ + P_res,∞

where τ = RC. The compliance of the arteries is the capacitance in the electrical analogy of hydrodynamics and this time constant is just that of a simple RC circuit, an approach that has been used extensively in the study of blood flow.

Calculation of the reservoir pressure when Q_in is known

If Q_in(t) has been measured simultaneously with the pressure, the calculation of the reservoir pressure is simple. Determine τ from the measured pressure during diastole; determine R from the mean blood pressure and the cardiac output (determined by integrating Q_in); determine C from R and τ and finally calculate P_res from the integral solution of the differential equation {mathematical details}.

This approach has been used in laboratory studies where it was possible to measure both pressure and flow into the ascending aorta simultaneously. Unfortunately, Q_in is not easily available in the clinic and so it would be very useful to have a way of determining P_ref without knowledge of Q_in.

Calculation of the reservoir pressure from the measured pressure alone

The method depends upon two assumptions:

    (i) the fall-off of pressure during diastole is uniform throughout the arteries.
    (ii) the wave generated by the flow into the arteries propagates without change into the arterial system.

The validity of the first assumption can be seen from simultaneous pressure measurements taken at different sites throughout the arteries

3-D plot of pressure vs. distance along the aorta and time	contour plot of the same data

We can see from the figure that the initial pressure pulse at the start of systole propagates into aorta from the slope of the contours at the front of the wave. However, in diastole the contours become remarkably vertical which means that the pressure fall-off during diastole is uniform throughout the large arteries. This supports our first assumption.

The second assumption is more subtle and depends upon a number of different observations. The first is that in our measurements of reservoir and wave pressures in the ascending aorta in animal experiments, there was remarkable similarity between the wave pressure waveform and the velocity waveform.

The top graph shows the measured left ventricular pressure (blue), aortic pressure (black) and the calculated reservoir pressure (red) for a single cardiac cycle. The wave pressure is defined as the difference between the measured pressure and the reservoir pressure. The calculated wave pressure (red) is shown in the bottom graph together the measured volume flow rate (black) where the scales have been adjusted so that the peaks of the two waveforms are identical. It is clear that there is a remarkable similarity between the flow and the wave pressure waveforms.

The fit was so good that we worried that it might be an artifact of the way the calculations were done. In order to check this similar experiments were carried out with an intra-aortic balloon pump placed in the descending aorta with its inflation time adjusted so that the backward wave generated by its inflation and deflation arrived at the measurement site in the ascending aorta during mid-systole. The results are shown in the figure. The top graph shows a normal beat when the intra-aortic balloon was not inflated and the bottom graph shows the next beat when the pump was activated. In the normal beat the remarkable similarity between the wave pressure and the flow rate is seen once again. In the disturbed beat there is a large difference between the two waveforms caused by the backward waves generated by the balloon

With these two assumptions it is possible to calculate the reservoir pressure from the measured pressure waveform alone {mathematical details}. Some examples of these calculations are shown in the next section.