# Wave Separation, Wave Intensity, the Reservoir-Wave Concept, and the Instantaneous Wave-Free RatioNovelty and Significance

## Presumptions and Principles

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## Abstract

Wave separation analysis and wave intensity analysis (WIA) use (aortic) pressure and flow to separate them in their forward and backward (reflected) waves. While wave separation analysis uses measured pressure and flow, WIA uses their derivatives. Because differentiation emphasizes rapid changes, WIA suppresses slow (diastolic) fluctuations of the waves and renders diastole a seemingly wave-free period. However, integration of the WIA-obtained forward and backward waves is equal to the wave separation analysis–obtained waves. Both the methods thus give similar results including backward waves spanning systole and diastole. Nevertheless, this seemingly wave-free period in diastole formed the basis of both the reservoir-wave concept and the Instantaneous wave-Free Ratio of (iFR) pressure and flow. The reservoir-wave concept introduces a reservoir pressure, *P*_{res}, (Frank Windkessel) as a wave-less phenomenon. Because this Windkessel model falls short in systole an excess pressure, *P*_{exc}, is introduced, which is assumed to have wave properties. The reservoir-wave concept, however, is internally inconsistent. The presumed wave-less *P*_{res} equals twice the backward pressure wave and travels, arriving later in the distal aorta. Hence, in contrast, *P*_{exc} is minimally affected by wave reflections. Taken together, *P*_{res} seems to behave as a wave, rather than *P*_{exc}. The iFR is also not without flaws, as easily demonstrated when applied to the aorta. The ratio of diastolic aortic pressure and flow implies division by zero giving nonsensical results. In conclusion, presumptions based on WIA have led to misconceptions that violate physical principles, and reservoir-wave concept and iFR should be abandoned.

## Introduction

Waves are signals that vary in time and space. Pressure and flow waves in the arterial system are generated by the cardiac pump and these waves travel over the elastic arteries at a speed termed the pulse wave velocity (PWV, *c*). The waves are reflected at all discontinuities in the arterial tree giving rise to (compound) reflected waves returning to the heart.

Measured pressure and flow waves can be separated into their forward and backward (reflected) components^{1–3}; the amount and timing of the reflected waves may provide information about arterial function. When reflections are absent aortic pressure and flow have the same wave shape, and their relation is given by the aortic characteristic impedance, *Z*_{c}=ρ*c*/*A*, with *c*=PWV, *A*=aortic area, and *ρ*=blood density. When pressure and velocity (flow/*A*) are used, the water hammer formula replaces *Z*_{c} by *ρc*. We will here use flow (rather than velocity).

In this controversy series article, we first discuss and compare 2 approaches to analyze arterial pressure and flow waves, namely wave separation analysis (WSA) and wave intensity analysis (WIA). We then discuss and demonstrate the inconsistencies of 2 WIA-based methods used to interpret arterial function: the reservoir-wave concept (RWC) and the Instantaneous wave-Free Ratio (iFR). The analyses and methods can be applied to any arterial bed, (systemic, pulmonary, and coronary system) but emphasis is here on ascending aortic pressure and flow.

## WSA and WIA

WSA uses the measured pressure, *P*^{m}, and flow, *Q*^{m}, waves to decompose them into their forward and backward (reflected) waves, *P*^{f}, *P*^{b} and *Q*^{f}, *Q*^{b}.^{1–3} Originally, the calculations were carried out in the frequency domain, via Fourier analysis and addition of harmonics.^{2} Because aortic characteristic impedance is independent of frequency,^{4} the calculation is more easily carried out in the time domain, that is, from pressure and flow waveforms directly.^{3,5}

Forward pressure and flow waves are similar in shape, and so are the reflected pressure and flow waves, with proportionality factor *Z*_{c}, *P*^{f}/*Q*^{f}=−*P*^{b}/*Q*^{b}=*Z*_{c}. The *Z*_{c} may be obtained either from high frequency input impedance^{1,3} or from the slope ratio of measured pressure and flow in early ejection.^{6,7} Mean pressure and flow, are related through peripheral resistance, and because they are not waves they are subtracted from *P*^{m} and *Q*^{m} before the calculation.

WIA is also based on measured pressure and flow (or velocity) where the changes in these signals are considered as wavelets used for analysis. This means that time derivatives of pressure and flow are used, *dP*^{m}(*t*)/*dt* and *dQ*^{m}(*t*)/*dt* (Figure 1).

Wave intensity, *dI*, is calculated as:

WIA is typically applied (1) to separate the measured waves into forward and backward components, and (2) to interpret the timing and nature of wave reflections.

In a manner that is similar to WSA, in that it uses decomposition of measured waves, WIA uses the wavelets *dP*^{m} and *dQ*^{m} to separate them into forward (+) and backward (−) components. It follows that:^{7,8}

Again *Z*_{c}=*dP*_{+}/*dQ*_{+}, and *Z*_{c}=−*dP*−/*dQ*_{−}. Summation or integration of all forward and backward travelling wavelets (the convolution) then yields forward (*P*^{f}, *Q*^{f}) and backward (*P*^{b}, *Q*^{b}) pressure and flow waves as obtained in WSA.^{9,10}

Wave intensity can be used to assess the nature and direction of the waves at a given location as a function of time. On the basis of the signs of *dI* and *dP*, 4 possible subwaves are distinguished: forward and backward compression or pushing waves (*dP*>0) and forward and backward expansion or pulling waves (*dP*<0).^{11}

### WSA and WIA are More Similar Than Generally Perceived

Both WSA and WIA (Figure 1) need characteristic impedance to derive forward and backward waves. It is generally assumed that, at least in the ascending aorta, reflections are negligible in early ejection or for the higher harmonics of the input impedance. The *Z*_{c} calculated from high frequency input impedance has been called the impedance method.^{12} When calculated from the slope ratio, *dP*^{m}/*dQ*^{m}, in early ejection it has been called the domain method. Both, however, give similar results.^{6,7,13} It is important to note that the validity of estimation of *Z*_{c} is not guaranteed throughout the complete arterial system. When measuring close to bifurcations (eg, in the common carotid or pulmonary artery), the reflection-free period can be too short. Also, arterial properties may not be constant in time, such as in the coronary arteries. In peripheral arteries, *Z*_{c} may be frequency-dependent requiring the original method to derive separation.^{2} Reservations are warranted in the interpretation of signals in these settings.

The WSA uses *P*^{m} and *Q*^{m} to derive *P*^{f} and *P*^{b}, whereas WIA uses integration of *dP*_{+} and dP_{−} to obtain separated waves.^{9,14} Note that integration introduces a constant, but in WIA the integration constant is either neglected, set at an arbitrary value or taken equal to diastolic pressure.^{9,14} When the integration constant is set to zero (what we recommend), both WSA and WIA lead to the same forward and backward pressure and flow waves, and both the calculations are performed and presented in the time domain. Recently, Mynard and Smolich^{15} proposed a method to account for mean values in forward and backward waves, based on the undisturbed pressure concept as defined by Lighthill.^{16} Nonetheless, the physiological meaning of the integration constant remains to be elucidated because mean pressure and flow do not vary in time and thus are not waves (although they may be a consequence of them).

It is important to realize that both WSA and WIA do not characterize the arterial system alone, but actually characterize the heart and its interaction with the arterial load. The initial forward pressure and flow waves are generated by cardiac contraction and relaxation, and their magnitudes depend on both cardiac *and* arterial properties. This is also reflected in the units that are used in WIA, the *dI*=*dPdQ* (Equations 2 and 3), has the units W/s^{2} (when based on volume flow) or W/s^{2}m^{2} (when based on flow velocity). The product of pressure and flow, *P*_{m}(*t*)×*Q*_{m}(*t*), is (instantaneous) cardiac (output) power, in Watt. Both cardiac output power and wave intensity can be used in time-varying and nonlinear systems but they depend on cardiac contraction and load.

### Where WIA and WSA Differ

Although WSA uses the measured pressure and flow waves, WIA uses derivatives of the pressure and flow waves. Differentiation means that fast changes in time are emphasized namely pressure and flow changes at start of ejection and valve closure. In terms of frequencies, high frequencies are emphasized. Aortic *dP*^{m}(*t*)/*dt* and *dQ*^{m}(*t*)/*dt* seem to show no waves in diastole (Figure 1). However, more accurately formulated, the amplitudes of the waves in diastole are, because of differentiation, small with respect to their maximum values. Measured pressure and flow do differ substantially in diastole as seen in Figure 1.

### Are Self-Cancelling Waves in Diastole Impossible or Possible?

In WIA, only small wave intensities are present in diastole, which has therefore—erroneously—been called wave-free period.^{17,18} However, in diastole measured pressure and flow waves are clearly present (Figure 1). In diastole, measured ascending aortic flow is negligible and forward and backward flow waves have to self-cancel, which was assumed to be impossible.^{19,20} However, self-cancelling waves are perfectly in line with wave theory because the compound forward and compound backward flow waves cancel when reflecting against a closed valve resulting in net zero diastolic flow.

### WSA: Time Domain or Frequency Domain?

It has often been stated that WSA is based on sinusoidal signals^{21,22} requiring a linear arterial system and steady state of oscillation and calculations in the frequency domain. This is, however, not necessarily the case, as in both approaches the waves are separated in the time domain as discussed above. Also the derivation of characteristic impedance can either be carried out in the frequency domain, that is, through input impedance^{1,3} (which then requires a linear and time-invariant system^{8}) or in the time/domain ratio (the slope ratio of measured pressure and flow in early ejection) allowing analysis of nonlinear systems as in the coronary arterial tree. The estimate by the slope ratio of pressure and flow in early ejection avoids the need to determine impedance, and is undoubtedly the easier one. The time domain method requires the identification of a linear segment in the pressure-flow loop (which may carry some arbitrariness) and critically depends on a correct time alignment of pressure and flow. These limitations do not play a role in the frequency domain method because in the calculation of *Z*_{c} from high frequency input impedance the phase information is not used (although discussion exists on the frequency range to be used). Nonetheless, if correctly performed, the 2 methods mostly show only small differences, which are acceptable.^{6,13,23}

### Why do Backward Waves Seem to Travel Forward?

The backward and forward pressure waves obtained by WSA at different locations in the aorta have similar foot–foot time delays (Figure 2).^{24,25} In other words, the foot of the backward wave seems to travel toward the periphery, not backward to the heart. This observation was used as argument against WSA.^{22,24,25} This is, however, not a problem related to the analysis and measurement methods because WIA and subsequent integration of *dP*_{+} and *dP*_{−} gives the same results as WSA.^{14,26} As such, the observation is not incorrect, but rather the interpretation of the findings is incorrect.^{27} This wrong interpretation assumes that the forward running wave travels to the distal aorta and reflects there and returns, as if the aorta were a single uniform tube with a single discrete reflection at its end.^{27} This tube model is too simple because the arterial system contains a continuum of reflection sites because of branching and tapering.^{28} At each reflection site, a backward wave is generated with a certain phase delay,^{29} and these local reflections determine the start of the backward wave.^{28} This phenomenon is comparable with an echo in a gallery with many arcades (Figure 2). At each point, the earliest reflection is determined from the nearest reflection point; when walking to the next arcade, the reflection time does not change. At the heart, the compound central backward wave results from these many individual waves.

This is also the reason why information on timing of reflected waves only weakly relates to aortic PWV and stiffness. For instance, the return time of the compound backward wave seems to decrease little with age while PWV may increase by a factor 2.^{30,31}

### Summary of Wave Separation and WIA

Both WSA and WIA separate waves in the time domain with *P*^{f}, *P*^{b}, and integrated *dP*_{+}, *dP*_{−}, being essentially similar.^{9} Only the estimation of *Z*_{c} is often carried out differently. Therefore, the argumentation that the time domain method is easier to grasp is not strong.^{21} The major difference between WSA and WIA is in their interpretation. The WIA, through differentiation, results in small waves in diastole, which has subsequently been assumed to be a wave-free period. This is incorrect because both WSA and WIA show that nondifferentiated (measured) waves are the same, and are present in diastole. The assumption of a diastolic wave-free period forms the basis of 2 recently proposed methods of analysis of arterial function, the RWC and the Instantaneous wave-Free (pressure-flow) Ratio (iFR). Both are discussed below.

## Reservoir-Wave Concept

The arterial tree is a system of elastic, branching vessels, and travelling waves. However, the arterial system can also be characterized by Windkessel (lumped) models where waves are not considered and cannot be interpreted. The (2-element) Windkessel model, popularized by Otto Frank, is a lumped model of the whole arterial tree. The 3-element Windkessel adds aortic characteristic impedance to Frank Windkessel.^{32,33} In principle at any location in the arterial system, the pressure-flow relation can be mimicked with a Windkessel. Strong and proximal reflections, aortic coarctation, low PWV with (short wavelength) as in young subjects, make the Windkessel less accurate.^{33,34}

The RWC assumes that diastole (diastasis) is wave-free and that, therefore, the arterial system can be described by a reservoir (storage volume) and peripheral resistance (Frank Windkessel). This model explains the diastolic pressure decay in diastole. In systole, however, as Frank Windkessel does not describe pressure well,^{16,32,33,35} an excess pressure, *P*_{exc}(*t*)=*Z*_{c}*Q*^{m}(*t*) was introduced,^{16,36} which is the difference between measured pressure and pressure predicted by Frank Windkessel. The basis of the RWC is described as follows: it is assumed that measured aortic pressure is the instantaneous sum of a constant (*P*_{inf}, pressure reached after long asystole), a Windkessel or reservoir pressure (*P*_{res}), and a wave-related pressure (excess pressure, *P*_{exc}).^{37} In essence, there is a similarity between the 3-element Windkessel and the RWC, as discussed by Vermeersch et al.^{38} An important difference, however, is that the Windkessel is not compatible with waves. The reservoir pressure is assumed to be related to (aortic) volume^{25,36,39} and the excess pressure accounts for waves and reflections.

### The Reservoir Pressure is a Wave and Equals Twice the Backward Wave

The RWC assumes that *P*^{m}=*P*_{res}+*P*_{exc} or *P*_{res}=*P*^{m}−*P*_{exc} and experimental data show *P*_{exc}=*Z*_{c}*Q*^{m},^{32,36} thus *P*_{res}=*P*^{m}−*Z*_{c}*Q*^{m}, and WSA (Equation 1) gives *P*^{b}=(*P*^{m}−*Z*_{c}*Q*^{m})/2 thus reservoir pressure is *P*_{res}=2*P*^{b}.

Also in diastole *P*^{b}=*P*^{f} and *P*_{res}=2*P*^{b}=*P*^{b}+*P*^{f} and *Q*_{res}=*Q*^{f}+*Q*^{b}=(*P*^{f}−*P*^{b})/*Z*_{c}=0 thus ascending aortic flow in diastole is zero, as discussed above. Indeed, *P*_{res} is a wave equal to 2*P*^{b} and in diastole the sum of *P*_{res}=*P*^{f}+*P*^{b}. Hametner et al^{40} have shown in patients that indeed *P*_{res}=2.01*P*^{b}, *r*^{2}=0.94.

When the reservoir pressure is derived at different locations, it is found that it arrives later in the distal aorta,^{22} as also explained by *P*_{res}=2*P*^{b}. The wave property of *P*_{res} was also shown by Mynard^{41,42} where they concluded that. “…foot of *P*_{res} clearly constituting a propagated disturbance, or by definition, a wave”.

### Reservoir Pressure and Excess Pressure

Segers et al^{34} showed that there is no unequivocal relation between the intra-aortic volume and reservoir pressure, as assumed in the RWC. Moreover, pressure–volume loops were found that traverse in a counterclockwise way implying the generation of energy, an impossible condition.^{34}

When *P*_{exc}, the assumed wave-related part of pressure, is split into forward and backward waves the reflected wave is negligible.^{11,20,37,43} Davies et al^{20} suggested that the small reflections were acceptable by referring to Womersley.44 However, Womersley wrote about local reflections at bifurcations.44 The limited reflections found in *P*_{exc} are artifactual and mathematically induced because *P*_{exc}=*Z*_{c}*Q*^{m} implies that *P*_{exc} and *Q*^{m} have similar wave shapes,^{36} resulting in negligibly reflected waves (Equation 1).^{20} It is also ignored that *Q*^{m} is shaped by reflections, and the impact of reflections is not analyzed correctly.

It was shown that aging causes changes in the reservoir pressure and it was suggesting that reservoir function rather than wave reflection changes markedly with age.^{45} Indeed, the decrease in arterial compliance (the reservoir) can explain the pressure changes found in aging.^{46} However, this does not disprove that reflections play a role because *P*_{res}=2*P*^{b} implies that *P*^{b} is increased.^{30}

### Summary of the RWC

The wave-reservoir concept hinges on splitting the function of the arterial system in a reservoir behavior (*P*_{res}, no waves) and a wave behavior (*P*_{exc}). However, the reservoir pressure is a wave equal to 2*P*^{b}, its foot arrives later in the distal aorta and is, therefore, a travelling wave. In addition, *P*_{exc} turns out to be almost reflection-less. In other words the concept is inconsistent.

There are 2 ways to study the arterial system, one is based on waves (reality) and the other on lumped (Windkessel) models. The wave-reservoir approach, mixing both, is a hybrid view creating confusion.

## The Instantaneous Wave-Free Ratio

The iFR is the ratio of pressures distal and proximal to a (coronary) stenosis, both averaged over the latter part of diastole, the presumed wave-free period. Although originally proposed for the coronary circulation,^{17,18} we illustrate its principles here for the systemic arterial tree.

The iFR again finds its origin in WIA with its assumed wave-free period. The extra assumption is that in this period, when pressure equals diastolic pressure and flow is negligible, their instantaneous ratio thus *P*^{m}(*t*)/*Q*^{m}(*t*) and averaged diastolic *P*^{m} and *F*^{m} are measures of (minimal, vasodilated) resistance.

### The iFR and RWC Compared

The RWC assumes that in the wave-free period, the arterial pressure is determined by resistance and compliance (Windkessel). Contrary to this, the iFR proposes that division of reservoir pressure by flow in diastole gives (vasodilated) resistance only. However, as diastolic flow is negligible the instantaneous pressure/flow ratio implies division by zero, thus physical nonsense. The calculation must be carried out using mean values of pressure and flow in a steady state. This here exemplified inconsistency also pertains to the coronary circulation where errors are mitigated by the fact that flow in diastole is not negligible.

### Application of iFR Is, up till now, Limited to the Coronary System

The iFR is assumed to give a measure of minimal (vasodilated) coronary resistance. If true, it could make estimation (fractional) flow reserve (FFR) possible without the need for drugs to obtain maximal dilation.

Association studies have indeed shown iFR associates with FFR.^{17,18} Yet, it has also been criticized.^{47–49} Thus, although the physical basis for iFR is incorrect, associations between iFR and FFR seem to plead for its practical use.

### Summary of the Instantaneous Wave-Free Ratio

The instantaneous wave-free ratio (of pressure and flow in diastole) is not a correct measure of microvascular resistance, dilated or not dilated, because its calculation violates basic physical principles (Ohm law).

### Perspectives

WSA and WIA are strongly related and their difference is mainly in their derivations. However, WIA suggests, incorrectly, that there is a wave-free period (diastole), and this assumption has led to the RWC and the instantaneous wave-free pressure/flow ratio, iFR. Both the concepts, based on flawed interpretations of arterial hemodynamics, are increasingly used in practice, which is worrisome. The internal inconsistencies in both the concepts are easily demonstrated, the most important ones being:

The reservoir pressure assumed to be without wave properties is a traveling wave, equal to twice the backward pressure wave.

The instantaneous ratio of pressure and flow assumed in the wave-free period is assumed a measure of (dilated) resistance. However, division of diastolic pressure by (zero) diastolic flow gives nonsensical results.

The reservoir pressure concept and the (diastolic) instantaneous pressure flow ratio, are both physically incorrect, and should be abandoned.

## Acknowledgment

We thank Mark I.M. Noble for critically reading the article.

## Disclosures

B.E. Westerhof is an employee of Edwards Lifesciences BMEYE. The other authors report no conflicts.

- Received March 29, 2015.
- Revision received April 11, 2015.
- Accepted April 28, 2015.

- © 2015 American Heart Association, Inc.

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# Novelty and Significance

### What Is New?

Wave intensity analysis wrongly suggests that there is a wave-free period (diastole) in the cardiac cycle.

### What Is Relevant?

Methods based on this assumed wave-free period, the reservoir-wave concept and the Instantaneous wave-Free Ratio of pressure and flow are, therefore, physically incorrect.

### Summary

The reservoir-wave concept and the Instantaneous wave-Free Ratio should not be used.

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- Wave Separation, Wave Intensity, the Reservoir-Wave Concept, and the Instantaneous Wave-Free RatioNovelty and SignificanceNico Westerhof, Patrick Segers and Berend E. WesterhofHypertension. 2015;66:93-98, originally published May 26, 2015https://doi.org/10.1161/HYPERTENSIONAHA.115.05567
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- Wave Separation, Wave Intensity, the Reservoir-Wave Concept, and the Instantaneous Wave-Free RatioNovelty and SignificanceNico Westerhof, Patrick Segers and Berend E. WesterhofHypertension. 2015;66:93-98, originally published May 26, 2015https://doi.org/10.1161/HYPERTENSIONAHA.115.05567Permalink: