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\title{Factual errors in a recent paper by Westerhof, Segers and Westerhof in Hypertension}


\author{Kim H. Parker}
\affil{Affiliation not available}

  
\author{Alun Hughes}
\affil{Affiliation not available}
  



\date{\today}

\maketitle




\begin{quote}
\begin{quote}
But facts are chiels that winna ding
\\An downa be disputed
\\-- from \textit{A Dream} by Robert Burns (1786)

(But facts are fellows that will not be overturned,
\\And cannot be disputed)
\end{quote}
\end{quote}
\textit{Wave separation, wave intensity, the reservoir-wave concept, and the instantaneous wave-free ratio (2015) N Westerhof, P Segers and BE Westerhof, Hypertension, DOI: 10.1161/HYPERTENSIONAHA.115.05567 } Hereinafter referred to as [WSW].

This paper by three distinguished workers in the field of cardiovascular mechanics, concludes that both the reservoir pressure and instantaneous wave-free ratio are '... both physically incorrect, and should be abandoned'. These are very strong conclusions which, if they were opinions could only be debated. Reading the paper in detail, however, reveals that it contains numerous factual errors in their discussion of these two entities. Since facts are different from opinions, we believe that it is essential that these errors be corrected before they gain credence by repetition.

\begin{quote}
\begin{quote}
False facts are highly injurious to the progress of science, for they often endure long; but false views, if supported by some evidence, do little harm, for every one takes a salutary pleasure in proving their falseness.
\\-- Charles Darwin (1871)
\end{quote}
\end{quote}

Because we are naturally prejudiced about the validity of both the reservoir pressure ($P_{res}$) and instantaneous wave-free ratio (iFR), having been involved in the conception and development of both ideas, we will try to present our arguments as transparently and fairly as possible. As far as possible we will demonstrate the errors by direct quotations from the paper. The whole paper$^1$ is available from the Hypertension web site\href{http://hyper.ahajournals.org/content/early/2015/05/26/HYPERTENSIONAHA.115.05567.full.pdf}{} and should be consulted directly if there are any questions about our treatment of the text.

Approximately two thirds of the paper is taken up with a discussion of wave separation and wave intensity from the point of view of the more usual Fourier-based methods of analysing cardiovascular mechanics, frequently called the impedance method. This part of the paper is, as far as we can see, both insightful and free of major errors. We found some of the discussion about wave intensity analysis thought-provoking and agree with most of their conclusions. We recommend the first two-thirds of this paper to anyone interested in arterial mechanics.

In contrast, the last third of the paper, starting with the final sentence of the section 'Summary of Wave Separation and WIA' is riddled with errors of interpretation and, more importantly, contains a number of mistakes (or in Darwin's terms 'false statements of fact') that need to be corrected. Instead of dealing with these errors chronologically, we will point out the fundamental errors first and then deal with their sequelae.


    
    
    
    
    
    
  
  
  
  
  
  
  
  
  
  
  
  
  

\textbf{The fundamental errors}

The first major error is the assumption that RWC (Reservoir-Wave Concept) and iFR (Instantaneous wave-Free Ratio) are directly related and that conclusions drawn from either of the two ideas can be directly applied to the other. This assumption is not stated overtly but it permeates almost all of their discussion.

In fact, iFR makes no use of the reservoir-wave concept and we are unaware of any publications before WSW that imply that it does. We did look into the idea of applying the reservoir pressure to our coronary measurements and very quickly decided that it added nothing to the wave intensity analysis based on the measured pressure and flow.  It is our belief that the reservoir pressure hypothesis is very likely to be inappropriate in the coronary arteries because of their limited compliance and their proximity to terminal reflection sites and other sources of backward travelling waves. We have used reservoir pressure analysis of pressure measured in various distal locations, e.g. the radial artery in the analysis of the CAFE study and while we have some reservations about their interpretation in relation to current theories about the reservoir pressure, one cannot contest their 
epidemiological predictive power.

Conversely, iFR played no role in the development of the reservoir-wave hypothesis which antedated iFR by more than a decade. After the development of iFR there has never been any attempt to apply that principle to the reservoir pressure Pr.

Examples of this basic error will be discussed at various points in the following discussion which concentrates on first Pr and then iFR.


    
    
    
  
  

\textbf{The basic error about $P_{res}$}

The first sentence of the second paragraph of the section \textbf{Reservoir-Wave Concept} states:
\begin{quote}
\textit{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). (my italics)
\end{quote}

This is simply wrong. Nowhere in our work do we assume that diastole is 'wave-free'. Wave-free is used in the description of iFR. In the derivation of Pr we assume only that the major part of the diastolic pressure waveform can be described by a falling exponential function, which is predicted by the overall mass conservation equation when there is no flow into the aortic root. This is also predicted in the Frank Windkessel model and that model was important in our original conception of the RWC. In fact, in our first paper on the subject (ref 36), we did not define a reservoir pressure, but called it the Windkessel pressure. Further work on the subject, experimental and theoretical, showed that the 2-element Windkessel was not a viable model. It was clear that what we had called the Windkessel pressure was propagating down the aorta and so we introduced the term 'reservoir pressure' in our next paper (Wang et al. Am J Physiol 2005) and were careful to describe how it differed from the Windkessel pressure. All of our subsequent papers on RWC should make it clear that Pr is not the Windkessel pressure.
    
  
  

\textbf{The basic error about iFR}

The first sentence of the Section \textbf{The Instantaneous Wave-Free Ratio} states:

\begin{quote}
\textit{The iFR is the ratio of pressure and flow in the latter 75\% of diastole.} 
\end{quote}

This is blatantly untrue. The iFR is defined as the ratio of the pressure measured downstream of a coronary stenosis to the pressure measured upstream at a particular time during diastole (see our published equation below). We have never asserted that this ratio is a measure of coronary resistance. As shown below, this fallacy is repeated at least five times.


    

\textbf{Detailed commentary on the WSW paper}

Having pointed out the basic factual errors of the WSW paper, we now provide a detailed commentary on the final two sections of the paper, section by section. The indented sections are direct transcription of the different sections of the WSW paper with comment numbers \textbf{[Cn]} inserted to indicated the numbered comments following. The original paper does not have numbered section headings but these seem to be necessary in the latex to HTML compiler that we are using. Apart from the numbering of the sections, the indented text should be an accurate transcription of the paper. References given by superscripts refer to references in WSW; references given by author and date refer to references at the bottom of this document.

\begin{quote}
\section{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 \textbf{[C1]} 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 \textbf{[C2]} 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) volume25,36,39 and the excess pressure accounts for waves and reflections. \textbf{[C3]}

\end{quote}

    
    
    
  
  
  

\textbf{[C1].} This is the basic error about Pr pointed out above. Pr is not based on the assumption that there are no waves during diastole. In our first paper we suggested that the arterial pressure $P(x,t)$ could be separated into a Windkessel pressure $P_{Wk}(t)$ and an excess pressure 
$P_{ex}(x,t)$, where $P = P_{Wk} + P_{ex}$. We did not use the term 'reservoir pressure' in that paper. After publishing that paper, we realised that this definition was not viable, mainly because we observed in our experiments that what we had defined as the Windkessel pressure was propagating down the aorta which is incompatible with the Windkessel pressure as defined by Frank. In our subsequent papers we coined the term 'reservoir pressure' so that it would not be confused with the Windkessel pressure. Obviously our efforts to distinguish the two failed because WSW have confused reservoir pressure with Frank's Windkessel pressure.

\textbf{[C2].} The sequence of definitions is wrong. No \textit{a priori} assumptions are made about $P_{exc}$; it is simply defined as the difference between $P$ and $P_{res}$. $P_{res}$ is defined in the following way: During diastole we assume that $P_{res}$ falls exponentially 
$P_{res}(t - T_N) - P_{inf} = (P(T_N) - P_{inf}) e^{-(t - T_N)/\tau}$, where $T_N$ is the time of the start of diastole. $P_{inf}$ is, as WSW state, the asymptotic pressure that would be reached after a long asystole.  $P_{inf}$ is included in the definition because we found that this 2-parameter model achieved a significantly better fit to measured diastolic pressures than the more common 1-parameter equation that fits the time constant $\tau$ to a curve that decays to zero pressure. We note that the existence of $P_{inf}$ (i.e. a positive pressure at zero flow) is supported by considerable experimental evidence from several groups. [supply refs] The form of $P_{res}$ during systole is then determined in one of two ways. If local flow 
measurements $Q(x,t)$ are available, as they are in our experiments on dogs, then $P_{res}$ is obtained by solving the differential equation describing overall mass conservation (assuming that net arterial compliance $C$ is constant so that $dV/dt = C dP/dt$) 
\[
 P_{res}(x,t) - P_{inf} = \frac{e^{-k_d t}}{C} \int\limits_{t'=0}^t Q(x,t') e^{k_dt'} dt' + (P(x,0) - P_{inf}) e^{-k_d t}
\]
where $k_d$ is the diastolic rate constant (the inverse of the more usual diastolic time constant $\tau = 1/(RC)$, where $R$ is the net resistance of the microcirculation) and $t=0$ is taken to be the time of end diastole, when $P(x,0)=P_D(x)$. Using this definition of $P_{res}$ we observed 
experimentally that for measurements in the ascending aorta the morphology of the excess pressure waveform was remarkably similar to the flow waveform. Since local measurements of flow are not usually available we developed an algorithm for determining $P_{res}$ during systole from $P(t)$ alone. The critical assumption in this method is the assumption that $Q(0,t)$ is proportional to $P_{exc}(x,t)$ which is purely an assumption which has not yet been tested experimentally. We recognised that the constant of proportionality, which is a fitting parameter in our algorithm, bears some relationship to the characteristic impedance of the aorta, $Z_c$ , but we do not assume that it is equal to $Z_c$ as is stated here.

\textbf{[C3].} It is generally agreed by the originators of the reservoir-wve hypothesis that $P_{res}$ propagates down the aorta, as observed experimentally. By the very broad definition of 'wave' in the first sentence of WSW, this would mean that $P_{res}$ is a wave, but some disagree with WSW's definition of 'wave' and prefer not to think of $P_{res}$ as a wave. Others see $P_{res}$ as the summation of those waves that are responsible for the exponentially falling pressure during diastole. If one prefer to think of arterial waves as sinusoidal wavetrains, then these will be the long wavelength components that are affected by the whole of the arterial circulation. If one prefers to think of arterial waves as successive wavefronts, then these will be the wavefronts that have been in the arterial system long enough to have visited all of the arterial circulation, in a statistical sense. These long-wavelength/old waves that determine $P_{res}$ during diastole will also be present in systole and their summation determines $P_{res}$ during systole. The rest of the waves determine, by definition, the excess pressure. In terms of sinusoidal wavetrains, these will be the small-wavelength, higher-frequency waves. In terms of wavefronts, these will be the newer waves generated by the most recent ventricular contraction or reflections from local sites of impedance mismatching which have not had sufficient time to travel extensively throughout the arterial system. This is the sense in which we have asserted that $P_{res}$ is determined by the global compliance and resistance and $P_{exc}$ is determined primarily by local conditions.
    
    
    
    
  
  
  
  
  
  
  
  
  
  
  
  

\begin{quote}

\subsection{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 $P_{res} = 2 P^b$. \textbf{[C4].}

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 $2P^b$ and in diastole the sum of $P_{res} = P^f + P^b$. Hametner et al.40
have shown in patients that indeed $P_{res} = 2.01 P^b$ , $r^2=0.94$. \textbf{[C5].}

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 Mynard41,42 where they concluded that. "...foot of $P_{res}$ clearly constituting a propagated disturbance, or by definition, a wave". \textbf{[C6].}
\end{quote}

    
    

\textbf{[C4].} Our experimental measurements in the ascending aorta of dogs showed that $P_{exc}$ is proportional to $Q^m$ and, although we pointed out that the constant of proportionality would be the characteristic impedance $Z_c$ if we assumed that the arterial system was a single uniform tube, we never assumed that $P_{exc} = Z_c Q^m$ as stated. If we assume that the constant of proportionality is $\lambda$, i.e. $P_{exc} = \lambda Q^m$, we can define a systolic rate constant $k_s = 1/(\lambda C)$. Again assuming that the reservoir pressure satisfies the net mass conservation equation, we derive the differential equation
\[
 \frac{d(P_{res}(t) - P_{inf})}{dt} + (k_s + k_d)(P_{res}(t) - P_{inf}) = k_s(P^m(t) - P_{inf})
\]
This equation can be solved by quadrature using the initial condition $P_{res}(0) = P_d$, the diastolic pressure
\[
 P_{res}(t) - P_{inf} = k_s e^{-(k_s+k_d)t} \int\limits_0^t P^m (t') e^{(k_s + k_d)t'} dt' + e^{-(k_s+k_d)t}P_d 
 + \frac{k_d}{k_s+k_d}(1 - e^{-(k_s+k_d)t})P_{inf} 
\]
The model parameters $k_s$, $k_d$ and $P_{inf}$ are then found by minimising the difference during diastole between the measured pressure and the model pressure given by this expression. The reservoir pressure obtained using this method is very different from the backward pressure wave found by wave separation analysis (WSA).

\textbf{[C5].} Since $P_{res}$ is very close to $P^m$ during diastole, it follows that the relationships given here are true during diastole. However, $P_{res} \ne P^b$ during systole, and so these results are invalid during systole. The correlation obtained in Hametner \textit{et al.} is, in fact, a correlation between peak($P_{res}$) and peak($P^b$) which invariably occurred at different times during systole; a further indication that $P_{res} \ne P^b)$ during systole.

\textbf{[C6].} We agree that $P_{res}$ propagates down the aorta as shown in our experiments in the dog aorta as well as our later studies in the human aorta. In fact, it was this observation that made us aware that our original paper was naive in calling it the Windkessel pressure and provided the motivation for coining the term 'reservoir pressure'.
    
  
  
  
  

\begin{quote}
\section{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 \textbf{[C7]}

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. \textbf{[C8]}

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 \textbf{[C9]}

\end{quote}
    
    
    
    
    

\textbf{[C7].} The recent paper by Segers et al. warrants further study. A concern, particularly in the light of the factual errors in the WSW paper, is whether or not the reservoir pressure they use is the same reservoir pressure we have defined or another error-based reservoir pressure. We find that deriving $P_{res}$ from clinically measured $P$ is not easy and have spent considerable time in producing an algorithm that robustly calculates a physically reasonable $P_{res}$ from clinical pressure waveforms. A basic problem is that we have no way of knowing the 'true' reservoir pressure in patients. Similarly, because we do not yet have a universally agreed theory of the reservoir pressure, this is also true for pressure waveforms generated by numerical models where the parameters are known. All models that we are aware of contain compliances in the vessels, in the boundary conditions at the end of the terminal vessels and occasionally in the input conditions at the left ventricle in the guise of elastance. Much further work on the theoretical basis of $P_{res}$ is needed.

\textbf{[C8].} We agree that the reference to Womersley$^{44}$ in Davies et al.$^{20}$ is not helpful because it does not deal with the case of a bifurcating tree of arteries, except to point out that 'When a number of junctions are cascaded in series, the direct method of calculation becomes clumsy and tedious...'. A better reference would be to Wormersley's comment reported in \textit{Blood Flow in Arteries} (2nd ed.) by MacDonald 
(p.199),'The case for the contrary is probably best summed up by the remark that Womersley made when he was first confronted with the problem -- "If you wanted to design a perfect sound absorber you could hardly do better than a set of tapering and branching tubes with considerable internal damping such as the arterial tree."'.
    
\textbf{[C9].} This comment propagates the error that $P_{res} = 2P^b$ discussed above. Reiterating our belief that the reservoir pressure is dominated by the global properties of the arterial system and excess pressure is dominated by local, we believe that it is interesting that most of changes in the arterial waveform found with aging relate to the reservoir pressure. Since the reservoir pressure is the summation of many waves, both forward and backward, This does not imply that changes in reflections are not important. Since $P_{res}$ is not directly related to $P^b$ during systole, the observations in Davies \textit{et al.}$^{45}$ provide an alternative view of the effects of aging on global and local arterial properties.
    
    
  

\begin{quote}
\subsection{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 $2P^b$, 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. \textbf{[C10]}

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. \textbf{[C11]}

\end{quote}
    
    

\textbf{[C10].} This summary repeats many of the errors discussed previously. In summary: $P_{res}$ is a wave. $P_{res} \ne 2 P^b$ during systole. $P_{exc}$ is only 'almost reflection-less' during systole and, in fact, the backward wave intensity based on excess pressure during systole is very similar in magnitude to the backward wave intensity based on the measured pressure.

\textbf{[C11].} The reservoir pressure shares some characteristics with Frank's Windkessel model (e.g. conservation of mass in the whole arterial system) but cannot be considered to be a lumped model in the sense that the Windkessel is, since $P_{res}$ is seen as a propagating waveform. A viable theoretical description of the reservoir pressure can be found in Parker et al. (2012) where we defined
\[
 P(x,t) = P_{res}(t - \tau(x)) + P_{exc}(x,t)
\]
where $\tau(x)$ is a delay time related to the wave propagation time from the aortic root to location $x$. Using a variational method we showed that the reservoir pressure defined in this way gives the minimum hydraulic work that the ventricle must do to produce a given volume flow rate $Q_{LV}(t)$. 



    
    
  
  
  

\begin{quote}
\section{The Instantaneous Wave-Free Ratio}

The iFR is the ratio of pressure and flow in the latter 75\% of diastole. Although originally proposed for the coronary circulation,17,18 we illustrate its principles here for the systemic arterial tree. \textbf{[C12]}

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. \textbf{[C13]}

\end{quote}
    
    
    

\textbf{[C12].} The first sentence is simply wrong. iFR is not the ratio of pressure and flow, it is the ratio of the pressure downstream of a coronary stenosis to the pressure upstream of the stenosis during a period in diastole when it is observed that waves defined by peaks in the wave intensity are minimal. In the paper introducing iFR$^{18}$, it is defined in Eq. (2)
\[
 iFR = \frac{\overline{Pd}_{wave-freeperiod}}{\overline{Pa}_{wave-freeperiod}}
\]
where $Pd$ is the distal pressure and $Pa$ is the proximal pressure. It is proposed as a clinical measure for stenosed coronary arteries and 'illustrating its principles in the systemic arterial tree' makes no sense whatsoever.

\textbf{[C13].} The choice of the term 'wave-free' for the period during diastole when the iFR is calculated may, on reflection, have been a bad one because it has been interpreted by WSW to mean that there are no waves during diastole; however it is less of a mouthfull than 'the period in diastole when the intensity of waves is minimal'! The main problem in assessing the effect of a coronary stenosis on perfusion is that the contraction and relaxation of the myocardium produces a dynamic flow waveform which is radically different from the flow waveform in other arteries. It is therefore challenging to differentiate the resistance to flow due to the stenosis from the dynamically changing resistance (defined by the ratio of instantaneous pressure and flow) of the intramyocardial microcirculation. The most widely used clinical measure of the functional severity of coronary stenoses has been the functional flow reserve (FFR) defined as the ratio of the mean pressure measured downstream of the stenosis to the mean pressure measured upstream when the microcirculation has been near maximally dilated by infusing adenosine. The genesis of iFR was the observation that the instantaneous microvascular resistance was minimal and very reproducible from beat to beat during a period in diastole when wave intensity was minimal which led to the idea that the pressure ratio measured only during this period might contain clinical information similar to that measured during maximal vasodilatation.




    
    
    
  
  
  
  
  
  

\begin{quote}
\subsection{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. \textbf{[C14]}

\end{quote}
    

\textbf{[C14].} This paragraph convolutes all of the basic errors discussed above. 'Wave-free' period is defined in iFR as a period in diastole when the waves measured by wave intensity are minimal. It does not assume that there are no waves in diastole. RWC does assume that $P_{res}$ is determined by the net compliance and resistance of the arterial system but makes no assumption that there are no waves during diastole. Since iFR is not instantaneous 
pressure/flow the problem of division by zero does not arise. All inconsistency perceived by WSW are the product of their factual inaccuracies, and the misconception that there is some link between $P_{res}$ and iFR.
    
    
  
  
  

\begin{quote}
\subsection{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. \textbf{[C15]}

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. \textbf{[C16]}

\end{quote}
    
  

\textbf{[C15].} iFR has been proposed as a clinical measurement in stenosed coronary arteries and there has never been any suggestion that it be applied anywhere other than the coronary system. It should be clear by now that iFR is not a measure of minimal (vasodilated) coronary resistance, but repetition of the fallacy requires repetition of the correction.

\textbf{[C16].} As already pointed out, iFR is exactly the same ratio as FFR, but calculated during the 'wave-free' period in diastole rather than after administration of adenosine. In the first paper on iFR, the correlation between iFR and FFR measured in a small cohort of patients was measured and found to be very high. In subsequent papers and in the ongoing clinical trials it is the clinical efficacy of iFR that is compared to the clinical efficacy of FFR, not the quantities themselves that are compared. We believe that this is a important distinction.
    
  
  

\begin{quote}
\subsection{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). \textbf{[C17]}

\end{quote}
    
\textbf{[C17].} To re-repeat, iFR is not the ratio of pressure and flow in diastole, it is the pressure ratio downstream and upstream of a coronary stenosis at a particular period in diastole when minimal wave activity as indicated by peaks in wave intensity is observed. It is not a measure of microvascular resistance. Taking the ratio of two pressures does not violate any physical laws.
    

\begin{quote}
\section{Perspectives}

WSA and WIA are strongly related and their difference is mainly in their derivations. \textbf{[C18]} 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. \textbf{[C19]} The internal inconsistencies in both the concepts are easily demonstrated, the most important ones being:

1. The reservoir pressure assumed to be without wave properties is a traveling wave, equal to twice the backward pressure wave. \textbf{[C20]}

2. 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. \textbf{[C21]}

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\textbf{[C18].} We agree with this conclusion. 

\textbf{[C19].} We disagree with all of these conclusions: WIA does indicate that waves indicated by peaks in the wave intensity are less prominent during systole than diastole in non-coronary systemic arteries. WIA also indicates that there are large waves in the coronary arteries during diastole but there is a period during diastole, called the wave-free period, when wave intensity is minimal. The similarity between the reservoir pressure and the measured pressure during diastole means that waves that correspond to the excess pressure are small. This does not mean that we assume that there are no waves during diastole; in fact, we see the reservoir pressure as a summation wave that propagates along the aorta. Use of Pr is problematic in the coronary arteries and it played no role whatsoever in the development of the iFR. We believe that the flaws in the WSW paper are flaws in their interpretation of the basis of both the RWC and the iFR. The only 'worrisome' thing is how such able researchers could make so many errors in reporting and interpreting our work.

\textbf{[C20].} Reiterating again: the reservoir pressure does propagate down the aorta and whether or not it is a 'wave' depends on ones definition of a wave. It is approximately equal to twice the backward pressure during diastole but is very different from the backward pressure during systole.

\textbf{[C21].} This error has been repeated at least five times but repetition of a fallacy does not make it true. iFR is a ratio of pressures, not a ratio of pressure and flow, and it does not give nonsensical results. Our preliminary clinical studies suggest that it is at least as effective as FFR as an indication for stenting coronary stenoses and there can be no question that it is easier to measure iFR than FFR, simply because it avoids the administration of Adenosine. If anything needs to be abandoned, it is the factually incorrect representation of reservoir pressure and iFR presented in WSW.
    
    
    
  
  
  

\textbf{References}

N Westerhof, P Segers and BE Westerhof (2015) Wave separation, wave intensity, the reservoir-wave concept, and the instantaneous wave-free ratio. \textit{Hypertension} \textbf{66}. DOI: 10.1161/HYPERTENSIONAHA.115.05567.

R Burns (1786) A Dream in \textit{Robert Burns. Poems and Songs} Volume VI. The Harvard Classics. New York: P.F. Collier & Son, 1909--14; Bartleby.com, 2001. www.bartleby.com/6/.

C Darwin (1871) The Descent of Man, and Selection in Relation to Sex. John Murray, London: 1st ed. (1871) Vol 2; p. 385.
  
KH Parker, J Alastruey and G-B Stan (2012) Arterial reservoir-excess pressure and ventricular work, Med. & Biol, Eng. & Comp. \textbf{50}, 419-424, ISSN: 0140-0118.

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