The modified arterial reservoir: An update with consideration of asymptotic pressure (P_∞) and zero-flow pressure (P_zf)

Abstract

This article describes the modified arterial reservoir in detail. The modified arterial reservoir makes explicit the wave nature of both reservoir (P_res) and excess pressure (P_xs). The mathematical derivation and methods for estimating P_res in the absence of flow velocity data are described. There is also discussion of zero-flow pressure (P_zf), the pressure at which flow through the circulation ceases; its relationship to asymptotic pressure (P_∞) estimated by the reservoir model; and the physiological interpretation of P_zf . A systematic review and meta-analysis provides evidence that P_zf differs from mean circulatory filling pressure.

Introduction

The concept of an arterial reservoir dates back to Borelli1 and Hales;² it was developed further by Weber³ (from whom the term Windkessel is derived in Salisbury et al.⁴) and by Frank,⁵ who provided a mathematical framework for it. More recently, in the early 2000s, a revised form of the arterial reservoir was proposed by Parker⁶ and the results using this approach were first published by Wang et al.⁷ According to this model, the pressure waveform was envisaged as the sum of a Windkessel pressure and (mainly forward) travelling waves.⁷ While this proposal elicited interest, it also received criticism,^8–11 with criticisms related to the assumption of a uniform Windkessel pressure being particularly pertinent. More recently, the model was revised to address this problem.¹² In the revised model, the Windkessel pressure was replaced by a reservoir pressure, which was made up from waves and was delayed by the time taken for waves to travel from the aortic root to the location of measurement.¹² This modification makes explicit the wave nature of reservoir pressure, and this modified definition has achieved some degree of acceptance.¹³ The aim of this review is to describe the modified arterial reservoir in more detail and to provide more information regarding the asymptotic pressure, P_∞, its relationship to zero-flow pressure (P_zf), the arterial pressure at which flow through the circulation ceases, also termed critical closing pressure,¹⁴ and the physiological interpretation of P_zf.

Wave travel in arteries and its relation to the reservoir

The existence of wave travel in arteries is undisputed.¹³ A number of studies have envisaged the arterial system as a single or a T-tube; however, this approach to arterial hemodynamics is too simplistic, and a more sophisticated model of wave propagation in arteries is necessary.¹⁵ More realistic one-dimensional models show that the branching pattern of the arterial circulation gives rise to myriad reflected waves, which are themselves re-reflected and re-re-reflected before returning to the aortic root.^16,17 Tapering of the arterial system may also make an important contribution to wave reflection patterns,¹⁸ and inclusion of visco-elastic behaviour may also be important in intermediate size vessels.¹⁹ The reservoir pressure can be understood as the pressure due to the cumulative effect of these reflected and re-reflected waves, which decrease in magnitude but increase in number as they travel. Another implication of the branching nature of the arterial tree is that the reflection coefficient at a bifurcation depends on the direction of travel of the wave. At bifurcations in large arteries, the combined admittance of the offspring arteries is similar to that of the parent artery (i.e. most bifurcations are well-matched for forward travelling waves); however, this also means that they are poorly matched for backward travelling waves.^17,20 Thus, large reflections from peripheral reflection sites are dispersed by the re-reflections they undergo, while travelling back to the aortic root. These considerations account for a ‘horizon effect’ where the apparent time of reflection of the initial compression wave, as indicated by a peak in the backward wave intensity, is independent of the site where the measurements are made.¹⁶

A modified definition of reservoir pressure

The modified definition of the reservoir pressure (P_res) assumes that the reservoir is made up of a network of N arteries. It is also assumed that the root artery (the aorta), A₀, is connected to the left ventricle and receives the stroke volume, Q_in, and that there are K terminal vessels – these are assumed to be connected to the microcirculation which is not considered part of the reservoir. A time-varying average pressure in each vessel P(t, n) is defined as the integral of pressure over the length of each arterial segment. A simplified version of this scheme is illustrated in Figure 1.

Figure 1.

A simplified schematic showing a network of branching arteries corresponding to an arterial reservoir (where N = 11 and K = 6). The inlet Q_in to the network is labelled as A₀ and the flow into the reservoir is indicated by the large arrow directed into the system. The termini linking to the microcirculation are the smallest vessels and the outflow Q_out is indicated by arrows going out of the system. The microcirculation (through which the reservoir discharges) and venous system are not shown as they are not considered part of the arterial reservoir.

The conservation of mass for the arterial network constrains the rate of change in the total volume, V, of the system to be equal to the difference between volume flow rate into the root, Q_in, and the sum of the flow out of the terminal vessels, Q_out

$$\frac{\mathrm{dV}}{\mathrm{dt}}=Q_{\mathrm{in}}-Q_{\mathrm{out}}$$ (1)

We assume that the end of each terminal vessel is coupled to a resistance, R_k, which is assumed to be constant (i.e. independent of pressure). Under these conditions, the flow out of the kth terminal vessel is given by

$$Q_k=\frac{(P\left(n\right)-P_{\mathrm{zf}})}{R_k}$$ (2)

where P_zf (zero-flow pressure/critical closing pressure) is the pressure at which flow through the microcirculation ceases. As will be discussed later, P_zf can be greater than zero (or venous pressure), and for the purpose of this model, it is assumed that it is the same for all termini. We also assume that the compliance of the nth vessel is C_n, where $C_n={\int }_0^{L_n}A(n)/(\rho \nu {(n)}^2)$ and L_n, A_n and v_n are the length, cross-sectional area and wave speed, respectively, of the nth vessel. The mass conservation equation can now be written in terms of the properties of the individual vessels

$$Q_{\mathrm{in}}-\sum _N{C}_n\frac{\mathrm{dP}\left(n\right)}{\mathrm{dt}}-\sum _K\frac{(P_k-P_{\mathrm{zf}})}{R_k}=0$$ (3)

where $\sum _{\mathit{N}}$ is the sum over all of the vessels and $\sum _{\mathit{K}}$ is the sum over the terminal vessels.

Windkessel pressure

Frank’s two-element Windkessel model has similarities and differences from reservoir pressure. The Windkessel pressure P_wk is assumed to be uniform.⁵ With this assumption, the pressure, P_wk, can be taken outside of the summations and the mass conservation equation reduces to

$$Q_{\mathrm{in}}-C\frac{d{P}_{\mathrm{wk}}}{\mathrm{dt}}-\frac{(P_{\mathrm{wk}}-P_{\mathrm{zf}})}{R}=0$$ (4)

where $\underset{\mathit{N}}{C=\sum }{C}_n$ is the total arterial compliance and $(1/R)=\sum _{\mathit{K}}(1/R)$ is the total peripheral resistance, using the usual formula for resistances in parallel.

This is, in essence, the approach originally used by Frank to solve for the Windkessel pressure, although Frank assumed that P_zf = 0.

P_wk takes account of the compliant nature of the large arteries but the assumption of a uniform pressure implies an infinite wave speed, which is physiologically implausible. The modified reservoir pressure does not share this defect.

Reservoir pressure and excess pressure

The modified reservoir pressure is defined as a pressure that is similar in form throughout the extent of the arterial reservoir, but which is delayed by the time it takes for waves to travel from the root to that location; hence, the reservoir pressure in the nth vessel is

$$P_{\mathrm{res}}(t,n)=P_{\mathrm{res}}(t-\tau \left(n\right))$$ (5)

where t is time, and τ(n) is the wave transit time, the time it takes for waves to travel from the root to vessel n. With this assumption, the mass conservation equation takes the form

$$\begin{array}{c}\multicolumn{1}{c}{Q_{\mathrm{in}}-\sum _n{C}_n\frac{d{P}_{\mathrm{res}}(t-\tau \left(n\right))}{\mathrm{dt}}}\\ -\sum _k\frac{(P_{\mathrm{res}}(t-\tau \left(n\right))-P_{\mathrm{zf}})}{R}=0\hfill \end{array}$$ (6)

This equation involves only one pressure, P_res, instead of involving N different pressures, P(n). This equation is a first-order time-delay differential equation with constant coefficients. These have been studied extensively in the context of control theory and there are existence and uniqueness theorems that ensure that a solution of this equation exists, with suitable boundary conditions, and that it is unique. Unfortunately, there is no established way to find the solution for a particular case, and most solutions are found by iterative methods. Solving the equation would require knowledge of all of the individual compliances and resistances of all of the arteries – knowledge that is impossible to obtain in practice, since there are too many vessels.

The excess pressure (P_xs) in the nth vessel is defined as the difference between the measured pressure and the reservoir pressure

$$P_{\mathrm{xs}}(t,n)=P(t,n)-P_{\mathrm{res}}(t-\tau \left(n\right))$$ (7)

with these definitions P_res and P_xs can be calculated as shown below.

Calculation of reservoir pressure

All of the methods of estimating the reservoir pressure are based on the assumption that the wave transit times are small in comparison with the cardiac cycle (τ(n) ≪ 1 cardiac period – assumed to be ∼1 s in human). This is supported by in vivo measurements of the aorto-iliac transit time in humans (i.e. the time taken for the initial compression wave to traverse the whole of the aorta from aortic root to iliac bifurcation) which is <80 ms,²¹ while the time from foot to peak pressure is approximately 2.5-fold longer (∼200 ms²²).

So, using a Taylor expansion for P_res(t −τ(n))

$$P_{\mathrm{res}}(t-\tau \left(n\right))=P_{\mathrm{res}}\left(t\right)+O\left(\tau \left(n\right)\right)$$ (8)

where $O(\tau (n))$ stands for the terms of order, ${\tau }_n$. Substituting into the mass conservation equation, the terms involving P_res can be taken out of the summations and we obtain the ordinary differential equation (ODE)

$$C\frac{d{P}_{\mathrm{res}}}{\mathrm{dt}}=Q_{\mathrm{in}}-\frac{P_{\mathrm{res}}-P_{\mathrm{zf}}}{R}$$ (9)

where, as in the derivation of the Windkessel pressure, $C=\sum _{\mathit{N}}{C}_n$, and $(1/R)=\sum _K(1/R_k)$. This shows that to $O(\tau (n))$, the equation for P_res is identical to the equation for P_wk, but without the need to assume a uniform pressure throughout the reservoir. This also makes clear that the reservoir pressure travels as waves and is the basis of the method of estimation of P_res.

Calculating P_res when pressure and aortic flow are known

If the aortic inflow, Q_in(t), is measured simultaneously with the pressure, P₀(t), then the calculation of P_res(t) is relatively straightforward. The solution of the ODE is easily found by quadrature

$$\begin{array}{c}{P}_{\mathrm{res}}\left(t\right)-P_{\mathrm{zf}}=e^{-\frac{t}{\mathrm{RC}}}\hfill \\ \underset{0}{\overset{t}{\int }}{Q}_{\mathrm{in}}\left(s\right)·e^{\frac{s}{\mathrm{RC}}}·\mathrm{ds}+(P_{\mathrm{res}}\left(0\right)-P_{\mathrm{zf}})·e^{-\frac{t}{\mathrm{RC}}}\hfill \end{array}$$ (10)

where s is time from the start of systole. During diastole, when the valve is closed, Q_in = 0, and the solution becomes

$$P_{\mathrm{res}}\left(t\right)-P_{\mathrm{zf}}=(P_{\mathrm{es}}-P_{\mathrm{zf}})·e^{-\frac{t}{\mathrm{RC}}}$$ (11)

where P_es is the pressure at end-systole. This is a mono-exponentially falling function of time with a time constant, τ = RC. This is one of several well-established ways to estimate total arterial compliance,²³ although it should be noted that inclusion (or not) of P_zf has a substantial effect on estimates of the time constant or arterial compliance using this method.^24–26 We use maximum negative rate of pressure change (max −dP/dt) as the indicator of end-systole to determine P_es, since the timing of this event has been shown to agree very closely (mean error < 0.4 ms) with the time of cessation of aortic flow at the end of systole in invasive studies in dogs²⁷ and is easy to identify in recorded pressure waveforms.

Frequently reservoir pressure calculations are performed using flow velocity rather than volumetric flow rate (Q); under these circumstances, it should be remembered that while estimates of P_res, P_zf, and τ are unaffected, the values of R and C are equal to resistance × area and compliance/area, where the area refers to the cross-sectional area of the aorta.

This method of calculating P_res can be used in experiments, where both pressure and flow rate (or velocity) are measured in the aorta. Clinically, however, it can be difficult to obtain simultaneous measurements of pressure and flow and so another more approximate method has been devised for calculating P_res that requires only the pressure to be measured.

Calculating P_res when only pressure is known

The method for calculating P_res using only pressure measurement is based on an observation made by Wang et al.⁷ in dogs, who reported that the excess pressure, P_xs, was directly proportional to the flow into the aortic root, Q_in. Subsequent studies in humans employing invasive measurements of pressure and flow velocity in the aorta²⁸ and non-invasive measurements of carotid artery pressure and aortic flow^29,30 have made similar observations. On this assumption, we can substitute Q_in = ζP_xs = ζ(P − P_res) into the mass conservation equation, where ζ is a constant of proportionality. If this relationship is viewed as analogous to a three-element Windkessel model,³¹ then ζ will be related to the characteristic admittance, or 1/Z_c (i.e. the inverse of the characteristic impedance).²⁹ Indeed, Westerhof and Westerhof¹⁵ have proposed that if the analogy with the three-element Windkessel model holds, then P_res will be equal to twice the backward pressure (P_b). (This relationship can be shown to be true in diastole when aortic flow (Q) = 0, but the derivation relies on the assumption that Q = Q_in which may be questionable.)

If we define k_s = ζ/C and k_d = 1/RC, then equation (9) can be written as

$$\frac{d{P}_{\mathrm{res}}}{\mathrm{dt}}+k_d(P_{\mathrm{res}}-P_{\mathrm{zf}})=k_s(P\left(t\right)-P_{\mathrm{res}})$$ (12)

This equation is similar in form to the previous ODE, but the right-hand side depends on P(t) rather than Q_in.

This first-order linear differential equation can be solved as

$$\begin{array}{c}{P}_{\mathrm{res}}=e^{-(k_s+k_d)t}\hfill \\ \underset{0}{\overset{t}{\int }}P(t\prime )e^{(k_s+k_d)t\prime }\mathrm{dt}\prime +\frac{k_d}{k_s+k_d}(1-e^{-(k_s+k_d)t})P_{\mathrm{zf}}\hfill \end{array}$$ (13)

This equation can be solved by iterative non-linear regression based on a three-element Windkessel model³² or alternatively the diastolic parameters k_d and P_zf can be estimated by fitting an exponential curve to the pressure during diastole

$$P_{\mathrm{res}}-P_{\infty }=(P_{\mathrm{es}}-P_{\infty })e^{-k_{d}t}$$ (14)

where the offset of the fit (P_∞) is assumed to be equal to P_zf (NB the validity of this assumption is examined below). Then, k_s is estimated by minimising the square error between P and P_res obtained over diastole in equation (12).

This formulation of reservoir and excess pressure makes the difference between Windkessel and reservoir pressures clear as previously noted by Alastruey³³ (Figure 2).

Figure 2.

Circuit diagrams illustrating the comparison between a two-element Windkessel and reservoir pressures conceptualised as a three-element Windkessel model. (a) Two-element Windkessel model; here, P_in is equal to Windkessel pressure (P_wk). (b) Three-element Windkessel model of reservoir pressure (P_res). In the three-element model, P_res still corresponds to the pressure across the capacitance, C, but P_res is smaller than P_in due to the pressure drop across the characteristic impedance, Z_c. Modified from Alastruey.³³

In principle, the approach described here should only be valid if Q_in = ζP_xs, which implies the absence of reflections. As discussed, reflections are always present in the circulation, but this assumption may hold within reasonable limits in the proximal aorta of healthy individuals.^7,34,35 It is less likely to be true in more peripheral locations where prominent wave reflections are observed in early systole.^36,37 The assumption of proportionality between Q_in and P_xs may also not apply when there is pathology giving rise to marked wave reflections in the proximal aorta.^38,39 Despite these provisos, estimates of P_res made at various locations in the aorta (the transverse aortic arch, the diaphragmatic aorta, the aorta at the level of the renal arteries, and the aortic bifurcation)⁴⁰ and in the brachial and radial artery using invasive methods⁴¹ are very similar to estimates in the proximal aorta (within 5%). Estimates of P_res made using non-invasive methods also show acceptable concordance with aortic measures (intra-class correlation coefficient of 0.77), although they are less accurate, probably due to errors in the estimation of systolic and diastolic pressure by cuff methods.⁴² In contrast, as expected, estimates of P_xs differ substantially between the proximal aorta and more peripheral locations,^40–42 with P_xs being larger in more proximal locations. Alastruey³³ proposed an alternative definition of P_xs, where it would be redefined as proportional to flow at any location; however, such a proposal would also result in a redefinition of P_res and so far the value of this approach seems not to have been explored.

Currently, there appear to be no publications where estimates of P_res based on pressure and flow velocity have been compared with estimates derived from pressure alone, although unpublished data from our group indicate excellent agreement between the approaches (correlation coefficient > 0.9; mean difference in peak P_res = 2 ± 1 mmHg, p < 0.002) based on invasive measurements of pressure and flow velocity in the aorta.

The relation of P_zf to P_∞ and the physiological interpretation of P_zf

The presence of positive pressure in the arterial circulation following cessation of flow, P_zf, has been recognised for many years.⁴³ In the context of the arterial reservoir, some authors have assumed that P_zf corresponds to venous pressure (or zero as a rough approximation to venous pressure)^29,44 or else that it should represent mean circulatory filling pressure (MCFP).^32,45 MCFP, as defined by Guyton, is ‘the pressure that would be measured at all points in the entire circulatory system if the heart were stopped suddenly and the blood were redistributed instantaneously in such a manner that all pressures were equal’ (quoted in 46).⁴⁶ Previous work in several species^47,48 including some necessarily limited work in man⁴⁹ has shown that P_zf differs from venous pressure and that in most cases^47,48,50,51 (but not all cases⁵²), there is no equalisation of arterial pressure with venous or right atrial pressure, even after prolonged cessation of flow. Whether P_zf corresponds to MCFP has not been formally examined previously as far as we can tell, so in order to address this question, we undertook a systematic review of the literature; some of these data have been published previously in abstract form.⁵³

A literature search was performed using PubMed and was limited to full articles in English using the search terms ‘MCFP’ OR ‘Mean systemic filling pressure’ OR ‘critical closing’ OR ‘zero-flow’ in publications prior to 01/09/2019. Only data relating to measurements of pressure following cessation of systemic flow were included; other exclusions were individual case reports, pregnancy, non-adult animals, not mammalian, post-mortem, or any non-human models of disease. Meta-analysis was performed using a random effects model, since it was anticipated that there would be heterogeneity between studies. Analyses were conducted in Stata 15.1. Data are shown as means (95% confidence intervals (CIs)).

A total of 1255 unique publications were identified after removal of duplicates; 1235 were excluded during screening. The remaining 20 studies^{48–51,54–69} with P_zf data were included in a meta-analysis (Figure 3); these included data from dog, rat, pig and human; eight of these articles also provided data on MCFP from the same studies. Some further details of these studies are shown in Supplementary Table S1. From this analysis, P_zf = 26.5 (23.4, 29.5) mmHg (Figure 4; 20 studies; mean (95% CI); n = 311; I² = 97%; p < 0.001) and MCFP = 10.6 (9.3, 12.0) mmHg (eight studies; n = 178; I² = 96%; p < 0.001). The difference between P_zf and MCFP was 15.1 (12.0, 18.3) mmHg (eight studies; n = 178; I² = 97%; p < 0.001). The comparison between P_zf and MCFP is shown graphically in Figure 5. There was no evidence of small sample bias based on an Egger test for either analysis (p > 0.05 for both). Further analyses provided no convincing evidence that the duration of cessation of flow was related to the estimate of P_zf based on meta-regression (p = 0.1) although the small sample size precluded firm conclusions. Similarly, the extent of heterogeneity within sub-groups (e.g. species, method of calculation) prevented any reliable conclusions on the importance of these factors in the observed heterogeneity in P_zf between studies. Nevertheless, it seems plausible that methodological differences between studies contribute to variability in the estimates of P_zf . There was evidence that MCFP differed between species (test for heterogeneity between sub-groups, p = 0.007) but there was insufficient data to examine whether reported differences between studies contributed to heterogeneity in MCFP (data not shown).

Figure 3.

A preferred reporting items for systematic reviews and meta-analyses (PRISMA) flow diagram for zero-flow pressure (P_zf).

Figure 4.

Forest plot of meta-analysis of zero-flow pressure (P_zf). Data categorised by species.

Figure 5.

Forest plot of meta-analysis of differences between zero-flow pressure (P_zf) and mean circulation filling pressure (MCFP) in studies in which both were measured.

Based on these findings, it seems clear that despite considerable heterogeneity P_zf does not equal MCFP. It is noteworthy that this is consistent with the standard practice in many experiments designed to estimate MCFP which either routinely transfer blood from the arterial to the venous circulation to achieve equilibration of pressure^57,70 or else apply a correction factor to take account of the ‘trapped’ volume of blood in the arterial circulation after cessation of flow.⁵⁷ We note that this finding does not necessarily imply that arterial and venous pressures cannot equilibrate after extremely prolonged cessation of flow.⁷¹ The duration of cessation of inflow in the studies identified in the systematic review was between 3 and 30 s (median 12.5 s), which is longer than the time constant of the decline in pressure (typically ∼2 to 3 s) but short enough to at least partially limit the secondary rises in MCFP due to reflex changes in vasomotor tone, and decreased venous compliance that tend to reduce differences between MCFP and P_zf through elevated MCFP.⁷¹ It is also unlikely that substantial oedema or hemodilution would occur over this time span and affect estimates of MCFP and P_zf. Most studies did not measure flow as well as pressure, but those that did reported that cessation of flow occurred between 3 and 20 s after cardiac arrest or switching off the perfusion pump.^56,58,62 This suggests that the duration of cessation of inflow was probably sufficient to obtain a reliable estimate of P_zf.

Values of P_zf that exceed MCFP are consistent with previous suggestions that P_zf represents a pressure due to a Starling resistor effect, sometimes termed (although some argue inappropriately^72,73) a ‘vascular waterfall’.⁷⁴P_zf is also often termed critical closing pressure after Burton,⁷⁵ although it is now generally accepted that vessel closure does not account for P_zf.^76–78P_zf is not a fixed parameter and varies between species,¹⁴ individuals,⁷⁹ physiological (or pathophysiological) conditions,⁷⁹ tissues^58,61,80 or even within tissues.⁸¹ Given the reported between-tissue differences in P_zf, it is likely that systemic P_zf is a weighted average of multiple P_zf. This may relate to differences in vascular resistance between tissues, since there is evidence that vasoconstriction and vasodilation increase and decrease P_zf quite markedly, respectively.^14,54,82 Indeed, in the rabbit ear, vasoconstriction has been reported to increase P_zf to ∼130 mmHg – in excess of mean arterial pressure in the rabbit!¹⁴ Increased tissue interstitial pressure also influences P_zf, as would be expected from a Starling resistor.⁸³ Another factor that influences P_zf is blood rheology; estimates of P_zf are lower following hemodilution,⁴⁸ although since positive P_zf has been observed using physiological saline,^43,84 it seems unlikely blood-related factors such as red cell aggregation,⁸⁵ the complex viscous behaviour of blood at low flow⁸⁶ or leukocyte plugging⁸⁷ fully account for P_zf.

The duration of cessation of flow has been reported to affect the estimated P_zf.⁸⁸ Short duration cessation of flow may result in over-estimates due to the effects of capacitive discharge from downstream vessels as suggested by Spaan⁸⁹ and Magder.⁹⁰ This effect, which assumes that a simple RC compartment model increases ‘apparent P_zf’ by an amount equal to −¢RdP/dt (where ¢ is the downstream microvascular compliance)⁹¹ will introduce a difference between P_∞ and P_zf that depends on the rate of pressure decline in diastole. Estimates of the effect of capacitive discharge have been made in the coronary circulation and show that this effect could easily account for a ∼10 mmHg difference between P_∞ and P_zf.⁹¹ Alternatively, longer periods of flow cessation may lead to changes in vascular properties due to the lack of flow or ischaemia or alterations in rheology or tissue interstitial pressure. Braakman et al.⁸² presented evidence for two P_zf (instantaneous (arteriolar) and steady-state (venous)) in skeletal muscle, with the ‘instantaneous’P_zf being dependent on vasoconstrictor tone, whereas the ‘steady-state’P_zf was not. It seems likely that the different techniques used to estimate P_zf will be differentially influenced by the factors that influence P_zf and that this will contribute to variation in estimates of P_zf.

Despite these considerations, evidence presented here suggests that P_zf is substantially lower than estimates of P_∞ (based on fitting diastolic pressure to equation (14)). P_∞ in the human aorta has been reported to be between 54 and 75 mmHg,^26,40 while the upper limit of the predictive interval of P_zf in humans in our meta-analysis was 51 mmHg. This is consistent with a previous report in humans where P_∞ calculated from normal beats was ∼29 mmHg greater than P_zf calculated during arrest.⁴⁹ Estimates of P_∞ will include uncertainties associated with fitting only short durations of diastole, particularly if the fit includes the perturbation in pressure that accompanies the onset of isovolumic contraction prior to the foot of the next pressure cycle.⁹² However, this seems insufficient to account for such a large difference and it may indicate that one or more assumptions in the present approach to fitting reservoir pressure is not valid.

The assumption of a mono-exponential decline in diastolic pressure has been examined experimentally by a few authors. From seven patients in whom aortic pressure was measured invasively, Liu et al.²⁴ looked at whether estimates of the slope of the semi-log regression of pressure and time gave consistent estimates of, and whether regression of dP/dt versus P was linear during diastole; they found that results using either method were inconsistent with a mono-exponential decline. Kottenberg-Assenmacher et al.⁴⁹ reported that the goodness of fit (by χ²) of the time-dependent decline in invasive aortic pressure following circulatory arrest in humans was slightly better using a two-exponential model or using a model including a pressure-dependent coefficient, although the effect on the estimates of these more complex models on the estimates of P_∞ was small (<2 mmHg). Schipke et al.⁵¹ reported that the correlation coefficients (presumably to the linearized semi-log transformation of the mono-exponential function) were 0.92 ± 0.05, and that data were less well fitted by linear and quadratic functions, but other functions seem not to have been examined. Brunner et al.⁴⁸ reported that after stopped flow the relationship between the natural logarithm of the declining pressure with time was linear in seven out of 13 dogs (consistent with a mono-exponential decline) but in the remaining six dogs, the data were not consistent with a mono-exponential decline. Sylvester et al.⁵⁰ reported that the decline in pressure following stopped flow was ‘well-described’ by a mono-exponential function with standard deviations on average <2 mmHg but provided no other quantification of fits.

On the basis of theoretical considerations regarding small artery compliance,^89,90 the pressure dependence of large artery compliance^24,93 and resistance,^89,94 it seems unlikely that the assumption of a mono-exponential decline should be valid.^89,90 Still, in view of the well-recognised difficulty in fitting multiple exponentials to complex data,^95,96 we believe that the fitting of multi-exponential functions to diastolic pressure is unlikely to yield much advantage, although it may be worth exploring.

Conclusion

The arterial Windkessel model is undoubtedly a simple and widely used conceptual model of the circulation. In its modified form, the arterial reservoir can be viewed as analogous to the Windkessel but comprising multiple reflected (and re-reflected) waves that arise due to the different forward and backward impedance properties of a branching network,^20,97,98 which give rise to a ‘horizon effect’.¹⁶ The waves that make up the reservoir are indiscernible by wave intensity analysis as their individual magnitudes are very small,³⁷ but together they make up a large store of energy. This energy, which is effectively trapped within the large elastic (conduit) arteries due to reflection and re-reflection, provides the motive force for tissue perfusion during diastole. This wave entrapment⁴⁵ equates to the volume storage of a classic Windkessel and accounts for the apparent similarities between these models. The waves that make up the reservoir persist across several cardiac cycles and account for most of the energy present in any particular cycle at quasi-steady state.³⁵ We have previously proposed that mean arterial pressure should be viewed as largely a product of these waves rather than the equilibrium state of the circulation as envisaged by Fourier-based impedance analysis.⁹⁹

Following cessation of ejection, pressure declines in a quasi-exponential manner towards a value, P_zf, which is the pressure at which outflow through the microcirculation ceases. A review of the experimental evidence suggests not only that P_zf exceeds venous pressure or MCFP but also that estimates of the offset (P_inf) derived from fitting a mono-exponential function to the decline in pressure during diastole of a normal cardiac cycle are substantially greater than P_zf, possibly as a consequence of the mono-exponential assumption. Further work is required to establish on how best to estimate P_zf from recordings of normal cardiac cycles.

Several features of the modified arterial reservoir model (and its underlying wave nature) contrast with interpretations that would be made if the circulation were viewed as analogous to a single tube, and the utility of the single tube model is questionable in our view and that of others.¹⁰⁰ While tangential to the content of this review, it is worth noting that the presence of multiple re-reflections also casts doubt on the utility of the ratio of forward to backward pressure, P_b/P_f (often termed reflection magnitude) as a measure of reflection, since a substantial part of forward pressure will arise from re-reflection of initially reflected (backward travelling) waves. This issue has also been alluded previously,^45,100–102 but its implications for pulse wave analysis seem not to have been fully apprehended.

In summary, the modified arterial reservoir represents a useful, albeit reduced, model of the circulation. The ability of parameters derived from this model to predict future cardiovascular events independent of conventional cardiovascular risk factors^103–111 suggests this model has clinical utility. No model of the circulation is perfect, however, and its limitations need to be recognised.