# Arterial Blood Pressure During Diastole (Draft)

Abstract

Three models of arterial pressure during diastole; a single exponential with zero asymptote, a single exponential with a non-zero asymptote, and a double exponential with a zero asymptote; are fitted to measurements of pressure in the ascending aorta. The results indicate that the single exponential with zero asymptote fits the data relatively poorly. Both the single exponent with a non-zero asymptote and the double exponent fit the measured pressure during diastole remarkably well. These two models, however, diverge significantly at longer times commensurate with extended diastole due to missing or ectopic beats. We conclude that the best choice of model can only be ascertained by looking at the arterial pressure during abnormal extended diastoles.

## Introduction

Arterial pressure waveforms are routinely measured clinically and they contain a significant amount of information about the cardiovascular system. Of prime importance, clinically, are the systolic pressure $$P_{s}$$ (the maximum measured pressure), the diastolic pressure $$P_{d}$$, (the minimum pressure) and the pulse pressure, defined as the difference between these two values $$P_{s}-P_{d}$$. Secondly, the complex structure known as the dicrotic notch is identified and associated with the closure of the aortic valve, giving the approximate times of systole and diastole based on the pressure waveform. Thirdly, various features of the blood pressure waveform during systole have been identified and associated with various cardiovascular phenomena. Finally, the decrease in pressure during diastole is generally described as a falling exponential and its time constant $$\tau$$ is commonly measured to characterise the diastolic behaviour of the arterial system.

We believe that a fuller understanding of the mechanics of the cardiovascular system, particularly the pressure waveform, would be beneficial. The clinical use of the various waveform parameters is historically long and rich. One of the most prevalent cardiovascular diseases, hypertension, is defined and diagnosed almost exclusively from measurements of the systolic and diastolic pressure. Other features of the arterial blood pressure waveform depend upon models of the circulation whose bases are more or less established. Some features of the arterial pressure waveform, particularly the dicrotic notch, are very poorly understood mechanically and could provide a fruitful area of future research. The arterial pressure waveform during systole is particularly difficult to analyse, experiementally or theoretically, because it is affected by the complex interactions between the heart and the arteries which communicate through the open aortic valve. During diastole the aortic valve is closed and the arterial pressure should be easier to analyse because the left ventricle and the arterial system are uncoupled, although the history of previous cardiac output must be considered. It is this pressure that we consider in this work.

## Historical observations

The pulsatile nature of arterial blood pressure has been recognised since the start of medicine. The nature of the pulse has been a subject of great interest to both Chinese and Indian physicians as evidenced by very early manuscripts and an oral tradition dating back to pre-history. In the west, Galen wrote a treatise De Pulsibus describing pulse diagnosis (Osborn 2015). From a mechanistic point of view, modern arterial mechanics effectively began with Frank (Frank 1899). Starting from a much earlier analogy made by Hales , who compared the elastic properties of the arteries to the capacitance of air chamber used in fire-engine pumps to smooth the highly pulsatile nature of the flow due to pumping alone, Frank formulated the Windkessel model of arterial mechanics. This model has formed the basis for our understanding of arterial pressure during diastole.

Although it is available in virtually every textbook of cardiovascular physiology, it is so important to our discussion that we present a very brief resume of the theory in modern notation. The whole of the arterial system is modelled as a single elastic compartment with flow entering from the ventricle and leaving through the microcirculation. Assuming that blood is an incompressible fluid, conservation of mass in the arterial compartment requires

$$\frac{dV(t)}{dt}=Q_{in}(t)-Q_{out}(t)\nonumber \\$$

where $$V(t)$$ is the instantaneous volume of the arteries, $$t$$ is time, $$Q_{in}(t)$$ is the volume flow rate from the ventricle into the arteries and $$Q_{out}(t)$$ is the volume flow rate out of the arteries through the systemic microcirculation (subsequently flowing to the right atrium via the venous circulation). Frank assumed that the flow through the microcirculation can be describe by a simple resistance relationship $$Q_{out}=R(P)$$ where $$R$$ is the resistance and $$P$$ is the pressure of the arterial compartment. He also assumed that the volume of the arterial compartment is related to the pressure through a constant compliance $$C=\frac{dV}{dP}$$. With these two assumptions the mass conservation equation can be written as a first order ordinary differential equation

$$\frac{dP(t)}{dt}+\frac{P(t)}{RC}=\frac{Q_{in}(t)}{C}\nonumber \\$$

Before discussing the solution of this equation, a few comments about the assumptions are warranted. The equation for the resistance of the microcirculation does not consider any effect of the venous pressure on the other side of the microcirculation. This can be accounted for by rewriting the equation using the resistance relationship $$Q_{out}=R(P-P_{V})$$ where $$P_{V}$$ is the venous pressure. A simpler approach, which we will take in the work, is to assume that the arterial pressure $$P$$ is a gauge pressure relative to the venous pressure $$P_{V}$$. With this definition, the equation above still holds for any constant venous pressure. The second assumption that the compliance of the arterial system is a constant is also a strong assumption, certainly if the change in volume of the arterial compartment is large. [ref] Physiologcially, artery diameters (and hence volumes) normally only vary by 5-10% during the cardiac cycle, hence this assumption may be acceptable in these circumstances. However, it should be remembered that this change in volume between $$P_{s}$$ and $$P_{d}$$ is relative to the pulse pressure at the distending pressure (i.e. mean arterial pressure) and the validity of the constant compliance assumption must be re-evaluated if we consider dropping the pressure to zero. Finally, the assumption of constant resistance over a range of pressures and flow is also a strong assumption. In an intact animal a fall in blood pressure will activate neuro-endocrine reflexes and elicit local autoregulatory (myogenic) responses that will modify resistance.

The mass conservation equation can be solved by quadrature for any given flow from the ventricle $$Q_{in}(t)$$.

$$P(t)=\left(P_{0}+\frac{1}{C}\int\limits_{0}^{t}Q_{in}(t^{\prime})e^{t^{\prime}/RC}dt^{\prime}\right)e^{-t/RC}\nonumber \\$$

where $$P_{0}$$ is the pressure at $$t=0$$. There are several features of this solution that should be pointed out. First, if we take $$t=0$$ as the time at the start of diastole when $$Q_{in}=0$$ (i.e. the time of closure of the aortic valve) the theory predicts that the pressure waveform during diastole will be a simple exponentially falling pressure

$$P(t)=P_{0}e^{-t/\tau}\nonumber \\$$

where $$P_{0}$$ is the pressure at $$t=0$$, and we have defined the time constant, $$\tau=RC$$. As we will see, this prediction for the arterial pressure waveform is very close to measured diastolic pressures. The success of the Windkessel model in predicting diastolic pressure means that it has had a major impact on our common understanding of arterial diastole.

Unfortunately, this solution of the Windkessel model is a poor predictor of the arterial pressure waveform during systole. Despite massive efforts by Frank and his followers, no variant of the single compartment model of the arteries has ever been found which is able to predict systolic pressure satisfactorily and it has been reduced to an historical footnote in current cardiovascular textbooks. The legacy of this model, however, has endured in the concept of the diastolic time constant $$\tau$$ which is still used as a clinically important parameter.

We conclude this historical discussion with some comments about the determination of $$\tau$$ from measured arterial pressure waveforms. Until the advent of modern computers, exponential functions were almost universally analysed by measuring the slope of the curve on a semi-log plot. Taking the logarithm of the equation, we obtain the linear relationship

$$log(P)=log(P_{0})-\frac{t}{\tau}\nonumber \\$$

so that the value of $$\tau$$ is the the negative reciprocal of the slope of $$log(P)$$ vs. $$t$$. This is very straightforward and it is highly probably that this method was used in virtually every experimental determination of $$\tau$$ until the 1980s when advances in computers made it possible to perform non-linear fitting relatively easily. It should be observed that this method of analysis depends on the assumption that the pressure (or gauge pressure) goes to zero as time goes to infinity. If this is not true and there is an asymptote at prolonged times, $$P_{\infty}$$, then the semi-log plot is not linear and the method fails. Practically, if an asymptotic pressure was included in the model, it would require making semi-log plots of log(P - P_∞) vs t for different values of $$P_{\infty}$$, determining which curve was the most linear and using its slope to determine $$\tau$$. This is quite laborious by hand and we suspect that it was seldom used.

## Recent studies of diastolic pressure

Closer study of diastolic pressure in the arteries resulted from the relatively recent reservoir-wave hypothesis (Wang 2002) which postulated that it might be useful to consider the arterial pressure as the sum of a Windkessel-like part depending on global compliance and resistance of the arterial system and an excess pressure depending on local conditions. During the course of this work, analysis of experimental measurements of arterial pressure in dogs and in man showed that the diastolic pressure decay was fitted significantly better by using an exponential relationship with a non-zero asymptote

$$P=P_{\infty}+(P_{0}-P_{\infty})e^{-t/\tau}\nonumber \\$$

and this assumption forms part of the reservoir pressure hypothesis.

The physical origin of the asymptote has not been explored in any depth experimentally but is usually identified with the so-called waterfall effect. This comes from the observation of flow through flexible tubes that are exposed to an external pressure (the Starling resistor). These studies, experimental and theoretical, show that the tubes can collapse when the external pressure is greater than the outlet pressure of the tube and that this collapse can limit the flow so that the flow rate depends on the upstream pressure and the external pressure rather than the upstream pressure and the downstream pressure. This has been referred to as the waterfall effect by analogy to the flow in a river; normally the flow depends upon the difference in height height of the inlet and the outlet of the river, but if there is a waterfall in the river the flow depends on the height of the top of the waterfall and is independent of the downstream topology of the rivers course. By this analogy it is assumed that there is an external pressure, which is greater than the venous pressure, acting on the microcirculation which causes a decrease in microvessel diameter that is large enough to prevent the passage of red blood cells and causes $$Q_{out}=0$$ at $$P_{\infty}>P_{V}$$. The nature of this external pressure is not entirely clear, it is unlikely to be simply tissue interstitial pressure and there is probably some contribution from vasomotor tone of arterial smooth muscle [refs to be added].

With current computers it is relatively simple to fit almost any multi-parameter non-linear function to experimental measurements and this has led to a re-evaluation of previous assumptions. Figure 1 shows the result of fitting a clinically measured pressure waveform in the ascending aorta during diastole using a simple exponential function with and without an asymptote. The fitting was done using the curve fitting tool in Matlab using the three models:
1) the simple exponential

$$P_{1}=P_{0}e^{-t/\tau}\nonumber \\$$

2) a single exponential with an asymptote

$$P_{2}=(P_{0}-P_{\infty})e^{-t/\tau}+P_{\infty}\nonumber \\$$

3) a double exponential

$$P_{3}=P_{0}\left(\alpha e^{-t/\tau_{1}}+(1-\alpha)e^{-t/\tau_{2}}\right)\nonumber \\$$

Figure 1. The diastolic pressure measured in the ascending aorta of a patient fitted with 3 different models. (black curve) measured pressure for 3 successive beats. (black dots) the diastolic pressure selected for fitting. (green line) the single exponential model with zero asymptote. (blue line) the single exponential model with an fitted asymptote.. (red curve) double exponential model with zero asymptote.

The fitted parameter for the 3 models are given in Table 1. There are a number of observations to be made from these results. First, it is clear that the single exponential model with zero asymptote provides an inferior fit to the measured data during diastole (as seen most clearly in Figure 2), although the $$R^{2}=0.9679$$ would generally be taken as a very acceptable goodness of fit. The relatively poor performance of this model might be expected because the pressure transducer was zeroed to atmospheric pressure instead of venous pressure (as is standard practice in most catheter labs). Thus, forcing the exponential to asymptote to zero is unreasonable. The time constant for this patient obtained from this fit $$\tau=0.79$$ s is somewhat smaller than the expected value according to most cardiology textbooks but within the range of values expected from patients in the catheter lab.

Second, the single exponential with a fitted asymptote (the blue line) exhibits a much better fit to the measured data with $$R^{2}=0.9992$$. However, the fitted asymptote $$P_{\infty}=$$ is much higher than the expected value of the venous pressure. The time constant $$\tau=0.27$$ is substantially lower than the expected value. This pattern of behaviour has been observed in virtually every study using an exponential model with an asymptote to model pressure during diastole.

Third, the double exponential model with zero asymptote gives a result that is graphically impossible to differentiate from the single exponential with asymptote model (see Figure 2). The fit is statistically marginally better with $$R^{2}=0.9995$$ but this is expected because the model contains one more free parameter. Interestingly, the 2 time constants $$\tau_{1}=0.19$$ s and $$\tau_{2}=2.20$$ s are bigger and smaller than the time constants derived from the other models.

Parameters from fitting the 3 models to the clinically measured diastolic pressure
model parameters $$R^{2}$$
single exponential $$P_{0}=116$$ mmHg, $$\tau=0.79$$ s 0.9679
” with asymptote $$P_{0}=124$$ mmHg, $$P_{\infty}=54$$ mmHg, $$\tau=0.27$$ s 0.9992
double exponential $$P_{0}=124$$ mmHg, $$\alpha=0.39$$, $$\tau_{1}=0.19$$ s, $$\tau_{2}=2.20$$ s 0.9995

Figure 2. The data and fitted models shown in Figure 1 showing only the diastolic period using the same format. The single exponent with asymptote model (blue line) and the double exponential model (red line) are so similar over this period that it is difficult to discern the two lines separately. The single exponential with zero asymptote (green line) deviates significantly from the measured data.

We have also fitted the three models to the diastolic period of the pressure waveform in the descending aorta calculated numerically (Willemet 2015). The disadvantage of this data is that it is simulated but it has the advantage that we know all of the parameters that were used to calculate it. Pertinent to this study, it was assumed in the calculations that the venous pressure was 10 mmHg. The simulated data and the fits of the 3 models are shown in Figure 3. The fit during diastole is uniformly good ($$R^{2}=0.9995$$ or better) and it is impossible to discern any difference graphically in the fitted curves during diastole. However, during the extended diastolic period the fitted models diverge visible, although not as drastically as for the clinically measured data. The fitting parameters for this case are given in the following Table. The known asymptote $$P_{\infty}$$ is fitted very well in the single exponential with asymptote model. In the other two modes the asymptote is forced to the erroneous value of zero. Adding a fitted asymptote to the double exponential model would almost certainly have resulted in a better fit to this model simply because it has more fitting parameters than the single exponential model.

It is noticeable, once again, that the fitted time constants differ between the models. The fitting of exponential time constants to data is classically known to be a hard problem and these results are in line with that conclusion. In the double exponential model we see once again that the two fitted time constants bracket the time constants calculated for the two single exponential models.

Parameters from fitting the 3 models to the measured diastolic pressure
model parameters $$R^{2}$$
single exponential $$P_{0}=104.9$$ mmHg, $$\tau=1.046$$ s 0.9995
” with asymptote $$P_{0}=97.7$$ mmHg, $$P_{\infty}=10.9$$ mmHg, $$\tau=0.895$$ s 0.9996
double exponential $$P_{0}=97.3$$ mmHg, $$\alpha=0.864$$, $$\tau_{1}=1.158$$ s, $$\tau_{2}=0.437$$ s 0.9995

Figure 3. The 3 models fitted to the diastolic period of the pressure waveform calculated by Willemet et al. (Willemet 2015). (black line) calculated pressure, (green line) single exponential with zero asymptote, (blue line) single exponential with free asymptote, (red line) double exponential with zero asymptote. In each case the fit during diastole was excellent ($$R^{2}>0.995$$) but the models diverge significantly for long times. In the calculations the venous pressure was assumed to be 10 mmHg (1.33 kPa).

To further illustrate the difficulty in fitting multi-exponential models to pressure measured during diastole, we have calculated the best fit parameters to the same data taken in the ascending aorta for a sequence of multi-exponentials with and without asymptotes. The models tested are 1) a single exponential without an asymptote, 2) a single exponential with an asymptote, 3) two exponentials without an asymptote, 4) two exponentials with an asymptote, 5) three exponentials with an asymptote and 6) four exponentials with an asymptote. The results of the fitting are shown in Table 2 which gives the fitted parameters and in Figure 4 which plots the data and the fitted models. The models have the form

$$P=\sum\limits_{n}a_{n}e^{-t/\tau_{n}}+a_{0}\nonumber \\$$

and the asymptotic pressure $$P_{\infty}$$ equals the sum of the $$a$$’s.

Parameters from fitting different exponential models to the measured diastolic pressure. The coefficients $$a_{n}$$ are in mmHg. The time constants $$\tau_{n}$$ are in s.
model $$a_{1}$$ $$\tau_{1}$$ $$a_{2}$$ $$\tau_{2}$$ $$a_{3}$$ $$\tau_{3}$$ $$a_{4}$$ $$\tau_{4}$$ $$a_{0}$$
1 111 0.790 - - - - - - -
2 67.7 0.262 - - - - - - 51.3
3 54.7 0.240 64.6 3.43 - - - - -
4 49.4 0.207 32.1 0.965 - - - - 37.6
5 48.2 0.229 0.71 0.328 20.7 0.421 - - 49.5
6 8.5 0.191 30.6 0.191 14.8 0.396 22.8 0.741 42.6

The simplest model, a single exponential with no asymptote deviates from the measured data visibly during diastole although the fitting statistic $$R^{2}=0.9655$$. the fit of all of the other models during diastole was extremely good with $$R^{2}>0.9996$$ in all cases. The difference between the fits during diastole are not visible in the Figure. The difference, howevver, become apparent when the models are extrapolated for longer times. The fitted time constants are very different for the different models. In the model with 4 exponential terms two of the time constants are very close to each other which may be an accurate result or may be the result of the algorithm getting stuck in a local minimum.

Figure 4. The result of fitting the same diastolic pressure data with different multi-exponential models. The models are: 1) a single exponential without an asymptote, 2) a single exponential with an asymptote, 3) two exponentials without an asymptote, 4) two exponentials with and asymptote, 5) three exponentials with an asymptote and 6) four exponentials with and asymptote. The values of the fitted parameters are given in Table 2. The models have been extended; for approximately two cardiac cycles which shows the divergence of the model predictions in the case of an extended diastole.

Figure 3. Fitting the 3 diastolic pressure models to a simulated arterial pressure waveform. In the calculations the venous pressure was assumed to be zero. During diastole all three models fitted the data very well. They diverge during the extended diastolic period, but not as much as the fits to clinically measured aortic pressure shown in Figure 1.

## Discussion and conclusions

Extrapolation of exponential functions by fitting models is fraught with difficulty. The results shown in Figure 1 illustrate the difficulties inherent in fitting a exponential model to a segment of data. The single exponential model with zero asymptote gives a fairly good fit to the pressure decay during diastole; very good if we only consider the $$R^{2}$$ value of the fit, less good from visual inspection of the measured data and best fit curve. The simple compartmental model that gives rise to the prediction of exponential decay of the arterial pressure during diastole indicates that we should be plotting the gauge arterial pressure relative to the venous pressure and so it is not surprising that the simple model is deficient in its fit during diastole.

Since venous pressure is rarely measured in the catheter catheter lab, it is tempting to use a model that includes an asymptotic pressure as a fitting parameter. This has been done in our second model which has been widely used in studies of the reservoir pressure. The results seen in Figure 1 are typical of the results of using this model to fit the measured diastolic pressure in a large number of studies. The fit of the model during the diastolic period is excellent but analysis of the fitted asymptotic pressure ($$P_{\infty}=54$$ mmHg in this case) raises questions because it is unrealistically higher than physiological venous pressures and higher than any usual estimates of interstitial tissue pressures. From the single compartment model of the arterial system, $$P_{\infty}$$ is the pressure at which the volume flow rate out of the arteries through the microcirculation $$Q_{out}=0$$. It is difficult to conjure up a mechanism that would cause outflow to cease at such high pressures within the arteries.

There is a reasonable explanation for the unrealistic values of $$P_{\infty}$$ that are found from fitting: the physically unrealistic assumption that the arterial compliance is constant over large changes in arterial volume that are expected as the arterial pressure drops below the diastolic pressure. This is certainly a possibility and it would be interesting to incorporate more realistic mechanical models of the arterial compartment. One problem with this approach is that pressure dependent compliance makes the mass conservation differential equation non-linear and analytical solutions are difficult or non-existant. This approach, however, is reasonable and deserves further study.

Another possibility is suggested by recent work on arterial mechanics which suggests that the interaction between input of blood from the heart and the branching arterial system can be described by a number of modes each with its own time constant. This behaviour is common in complex dynamical systems where it is generally argued that the approach to steady state will be dictated by the mode with the largest time constant; the modes with smaller time constants will decay to zero more quickly leaving only the slowest mode. Using a common model of the 55 largest arteries (Matthys 2007), preliminary calculations suggest that there can be as many modes as their are arteries in the arterial network and that the time constants of these modes span a large range from a small fraction of the cardiac period to several cardiac periods. Predicting the asymptotic behaviour of the pressure waveform using this model approach requires extensive knowledge of the anatomy and the mechanical properties of every artery in the system. This clearly is impossible in a clinical setting.

It would be fruitless to try to fit diastolic behaviour with a model that included 55 different time constants. In many cases the number of discrete measurements of pressure during diastole would comparable to the number of free parameters in the model. In order to get some idea of the behaviour of multi-exponential models we have explored the double exponential model described above. In this model there are 2 free time constants $$\tau_{1}$$ and $$\tau_{2}$$ and the free parameter $$0<\alpha<1$$ which describes the influence of these two exponential functions. As in the other models $$P_{0}$$ is a fitting parameter describing the pressure at the start of diastole ($$t=0$$). For the sake of simplicity, it is assumed that the pressure asymptotes to zero.

The results of fitting this double exponential model to pressure during diastole is interesting. The fit during diastole is excellent and it is difficult to discern any difference between this model and the single exponential with an asymptote model. The difference for large times, however, is dramatic. The long time behaviour, effectively the prediction of what would happen if diastole was extended, continues to fall exponentially at a rate dictated by the largest time constant while the smaller time constant modifies the pressure fall during the diastolic period giving a ’local’ time constant that is commensurate with the time constants derived from single exponential modelling.

Results obtained for fitting the 3 models to a simulated arterial pressure waveform show similar but less extreme behaviour as the fits to clinically measured pressure data. The advantage of fitting the simulated data is that all of the parameters that were used in the numerical computation are know. We know, for instance, that the venous pressure was assumed to be 10 mmHg whereas it was not measured in the clinical measurements.

The conclusion from this simple study is that the the monotonic, ’exponential-looking’ decrease in arterial pressure commonly observed during diastole does not mean that it is necessarily a single exponential function with a single time constant. The observed pressure can be modelled equally well with a double exponential with two time constants neither of which corresponds to the single exponential time constant. Both models fit the measured data during diastole extremely well. The extrapolated behaviour for longer times, corresponding to the behaviour that would be expected when the start of systole is delayed, is drastically different.

We conclude that the excellent fit of the double exponential model during diastole does not mean that this model is correct any more than the excellent fit of the single exponential model with an asymptote means that that model is correct. Or results indicate simply that the best choice of model for diastolic pressure cannot be determined solely from analysis of pressure measurements during diastole.

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