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The implications of P∞

The inclusion of the asymptotic pressure P∞ in the theory may have wider implications than we originally thought. Formally it comes into the theory because we assume that the flow out of a terminal vessel into the microcirculation is given by a resistive law of the form

Q = (P - P∞)/R

where P∞ is the pressure at which flow through the microcirculation ceases. It is usually taken to be the venous pressure because this is the pressure at which the pressure difference across the microcirculation is zero. However, there is some evidence that flow in a microcirculatory vessel will go to zero at an arterial pressure greater than the venous pressure; a phenomenon usually ascribed to the 'waterfall< effect. The usual explanation of the waterfall effect is that the interstitial pressure of the tissue surrounding the vessel can cause the vessel to collapse when the upstream pressure approaches the interstitial pressure. This is not a complete description of the process, but there is a smattering of evidence throughout the physiological literature supporting the idea.

Our initial motivation for including P∞ was more empirical; we found that we could not fit our experimental diastolic pressure measurements satisfactorily when we took P∞ = 0. This is illustrated in the following figure.
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This figure shows the the results of fitting the diastolic pressure measured in the human aorta with the function

(P - P∞) = (P0 - P∞) e-t/τ

where t = 0 is taken as the start of diastole (denoted by the red circle in the figure), P0 is the pressure at t = 0 and the time constant τ and the asymptote P∞ are the fitting parameters. The measured pressure (blue) is a single beat extracted from a sequence of cardiac cycles measured in the ascending aorta using a catheter pressure transducer. The fit obtained assuming P∞ = 0 (black) is obviously less good than the fit when P∞ was taken as a free parameter.

Generally, the difference between the two fits is not obvious during the time of diastole; it is only when the fitted curves are plotted for a period much longer than the time of diastole that the differences between the curves becomes visibly noticeable.

The inability of the model without the inclusion of P∞ to fit the diastolic pressure waveform becomes more and more obvious as the time of diastole increases. It is particularly obvious when we analyse the pressure measured during missing or ectopic beats. In experiments with animals it is possible to stop the heart for several beats by the injection of acetylcholine, and the measured pressure in the artificially extended diastole clearly reaches an asymptote that is much higher than the venous pressure of the animal.

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This figure shows the reservoir pressure calculated beat-by-beat during a sequence of measurements ascending aortic pressure in a dog including an ectopic beat; measured pressure (black), reservoir pressure (red) and ECG (blue) are shown. The numbers under each beat give the fitted values of τ and P∞ found by the fitting algorithm. Note that P∞ is consistently much larger than zero, closer to the diastolic pressure than the venous pressure (not measured but probably less than 2 kPa). Also note that P∞ determined for the missing beat is somewhat lower than the value found for the normal beats but is still much greater than venous pressure.

The effect of P∞ on the time constant τ

An interesting side effect of allowing P∞ ≠ 0 is that the value of τ that results from the fit is invariably lower than the τ calculated assuming P∞ = 0. For example, in the fits shown in the first figure above, when P∞ = 0, &tau = 1.2 s whereas when both were taken as free parameters the best fit occurred for P∞ = 6.2 kPa and τ = 0.5. The first value is consistent with most of the values in the literature for the human arterial time constant, almost all of which have been calculated using the assumption that P∞ = 0 simply because the fitting can be done by linear regression on a semi-log plot of pressure versus time. The model including P∞ as a free parameter requires more sophisticated nonlinear fitting to find the parameters that were extremely difficult before computers.

The possibility that the arterial time constant might have to be reevaluated is clinically important. The most common way to determine arterial compliance C is to
1) measure the arterial resistance R as the ratio of mean arterial pressure to cardiac output
2) calculate the time constant τ by fitting an exponential to the pressure during diastole
3) use τ = RC to calculate C from τ and R

If the true value of τ is half of the value estimated assuming P∞ = 0, then the estimate of C will also be half of the value estimated assuming P∞ = 0.

The effect of P∞ on Windkessel models

The classical Windkessel model can be represented by 2-element RC network. Although it is an excellent model of pressure during diastole, it fails to model pressure during systole. The introduction of the 3-element Windkessel model (pictured at the top of the figure) by Westerhof in 1978 significantly improved the modelling of arterial pressure. Z is the characteristic impedance of the aorta, R is the net arterial resistance and C is the net arterial compliance. P is the pressure that is measured at the root of the aorta for a given flow rate from the ventricle.

There have been many attempts to improve the model by adding other elements to the model (see the papers by Stergiopulos et al. and by Vermeersch et al.). In general, the improvement of the goodness of fit is not significant given the increase in the number of model parameters.

The addition of P∞ to the electrical analogue model is equivalent to adding a battery to the circuit (pictured at the bottom of the figure). This, as far as I am aware, has not been tried and it would be interesting to see if this model would result in a better fit to experimentally measure pressure waveforms than the models which have been tried previously.

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References

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