\(P_x\left(x,t\right)=\ \zeta Q_{in}\left(t\right)\)
where \(\zeta\) is is a constant with the dimensions of impedance. With this assumption and the definition of excess pressure the equation for overall mass conservation in the arteries can be written
\(\frac{dP_r}{dt}=k_s\left(P-P_r\right)-k_d\left(P_r-P_{\infty}\right)\)
where \(k_s=\frac{1}{\zeta C}\) is the systolic rate constant. This equation can be solved by quadrature for any measured pressure
\(P_r\left(t\right)=e^{-\left(k_s+k_d\right)t}\int_0^tP\left(t'\right)e^{\left(k_s+k_d\right)t'}dt'+\frac{k_s}{k_s+k_d}\left(1-e^{-\left(k_s+k_d\right)}\right)P_{\infty}\)
It should be stressed that the assumption that excess pressure is proportional to the flow in to the arteries is not necessary for the definition of \(P_r\), only for the calculation of \(P_r\) from \(P\) alone. It should also be emphasised that the constant of proportionality \(\zeta\) is not equal to the aortic input impedance, \(z\), or the characteristic impedance, \(Z_0\), although it has the same dimensions. This point is important because Westerhof et al. \cite{26015448} argue that \(P_r=2P_b\) where \(P_b\) is the backward wave calculated by the methods introduced by Westerhof et al. \cite{4656472} and Laxaminarayan \cite{Laxminarayan_1979}. Their argument (paraphrased) is: by definition \(P_b=\frac{1}{2}\left(P-Z_0Q\right)\); and \(P_x=P-P_r\ =\ Z_0Q\), which can be rearranged \(P_r=P-Z_0Q\); therefore \(P_r=2P_b\). There are two errors in this line of reasoning; 1) \(\zeta\ne\ Z_0\) since \(P\left(x,t\right)\) and \(Q_{in}=Q\left(0,t\right)\) are not measured at the same location, 2) for the special case of pressure measured at the aortic root, \(P\left(0,t\right)=Z_0Q\left(0,t\right)\) is only true if there are no reflections, in which case \(P_b=0\). The value of \(\zeta\) will depend on the distribution of flow from the aortic root to the measurement site. Since this cannot be deduced from the pressure measurement, \(k_s\) which depends on \(\zeta\) is treated as a fitting parameter together with \(k_d\) and \(P_{\infty}\).
The most extreme criticism of the concept of reservoir pressure is found in \cite{Westerhof_2015} who conclude that 'The reservoir pressure concept and the diastolic instantaneous pressure flow ratio are both physically incorrect and should be abandoned'. Readers referring to this paper should also refer to a published response \cite{Davies_2015} and the subsequent correction from the original authors \cite{Westerhof_2015a} acknowledging misinterpretation and misunderstanding of the relevant definitions in their original paper; although they did not withdraw the claims based on these misinterpretations.
The second part of this comment refers to the instantaneous wave-free ratio (iFR) which has been developed to clinically assess the functional effect of a coronary stenosis which, contrary to the claims of the authors, has absolutely nothing to do with the reservoir pressure. Furthermore, as the authors acknowledged in a follow-up letter to the journal \cite{Westerhof_2015a} their analysis of iFR was based on an equation which was incorrectly quoted from the papers describing iFR (incorrectly claiming that iFR was a ratio of pressure and flow instead a ratio of pressures across the stenosis as originally defined). We believe that basing their argument on an incorrect equation invalidates their conclusion about the physical correctness of iFR.
Their arguments concerning the reservoir pressure are also invalid, but for more subtle reasons. The primary fallacy in their argument is the statement that reservoir pressure is not a wave. Our current view of the reservoir pressure is that it is made up of wave fronts, both forward and backward, but only that complement of wave fronts that have been in the system long enough to have visited an extensive part of the arterial system and have been influenced by the global properties of the system. The calculated excess pressure accounts for the 'new' pressure wave fronts generated by the most recent left ventricular contraction or reflections from nearby locations that have not been in the arterial system long enough to have travelled to, and been affected by, an extensive part of the system; these wave fronts underlying the excess pressure are therefore dominated by local conditions.
The deduction that \(P_r=2P_b\) is also wrong in general for reasons already discussed. However, this relationship is true during diastole as can be seen by the following argument: during diastole when inflow is zero the forward and backward pressures must be equal so that the forward and backward flow waves will cancel, thus \(P_b=P_f=\frac{1}{2}P\); also during diastole \(P_r\approx\ P\) and therefore \(P_r\approx2P_b\) during diastole. During systole the \(Pb\) and \(\Pr\) waveforms differ although there are many similarities because they are convergent during diastole. The most obvious difference occurs during very early systole while \(Q_{in}\) is increasing but still less than \(Q_{out}\) which, if we assume outflow occurs through a simple 'Ohmic' resistance, is driven by the relatively high mean arterial pressure. By mass conservation \(P_r\) will continue to fall reaching a minimum at the time when \(Q_{in}=Q_{out}\). This period of falling \(P_r\) during very early systole is clear in all of the \(P_r\) waveforms calculated in this study. This behavior is not generally observed in \(P_b\) waveforms where it is assumed that the backward wave is delayed by the time it takes to travel back to the measurement site from its distal reflection sites. In summary, we believe that the objections to the reservoir pressure expressed in \cite{Westerhof_2015} are based on false assumptions about the nature of \(P_r\), an erroneous confluence of ideas that are not related, and misinterpretation of the assumptions used in the calculation of \(P_r\). For all of these reasons we reject the claim that \(P_r\) is 'physically incorrect'. As an heuristic hypothesis, the decision about whether or not it should be used must rely on experimental testing of its usefulness.
Mynard and his colleagues have also published a number of papers which are generally critical of the reservoir-wave hypothesis. The most recent \cite{Mynard_2014} summarises their criticisms, consequently we will comment in detail on these - similar arguments hold for their other published comments on applicability of reservoir pressure models. First and foremost, their arguments are based on computational studies on highly idealised arterial models. For example, the discussion in their letter is based on calculations in either a model of a single bifurcation with uniform tubes or a model of a single tapering tube. The computational results for the pressure and velocity waveforms are very 'physiological' in form but anyone experienced in this sort of 1- D modelling of arteries will know that this is because of the particular boundary conditions that are used in the model. The run-off arteries distal to the modelled vessels are modelled by a 3-element Windkessel of unspecified compliance and the ventricle is modelled by a 'validated elastance heart model' where the elastance also contributes a kind of compliance to the system. Without these highly compliant boundary conditions it is certain that the computed pressure waveforms would contain very large fluctuations arising from the two reflection sites in the bifurcation model and the single reflection site in the tapered tube model. This is very different from highly complex physiological arterial systems where it is widely accepted that the single or T-tube modes that were commonly used in the early days of impedance analysis are not only wrong but misleading . Calculations in more complex arterial models, such as the 55 artery model introduced by and used extensively by Alastruey and colleagues \cite{Alastruey_2010}, suggest that the myriad of reflected waves generated by the widely distributed impedance mismatches occurring at bifurcations and at terminal reflection behave qualitatively differently from those observed in simpler models.
We would argue that the application of \(P_r\) to calculations based on these highly idealised models is misguided. As discussed previously, the current view of \(P_r\) is that it is the net effect of the forward and backward wave fronts generated by of the widely distributed reflection sites distributed throughout the arterial system and \(P_x\) reflects the effects of local interactions (potentially including reflections) where the waves have not had time to visit suficient parts of the system to acquire the statistically homogeneous properties of waves comprising \(P_r\). By this criterion, we would argue that all of the waves calculated in these simplistic models are local waves. Separating these local waves from the smoothed pressure changes generated by the compliances at the terminals and, probably more import, the elastance of the heart model is not trivial but it certainly cannot be done by simply subtracting \(P_r\) as calculated from the computed pressure waveform. Attempts to do this could easily give rise to the 'erroneous information' that are cited by critics as examples of the failure of the reservoir-wave hypothesis.
It is our view that the calculations upon which Mynard and his colleagues criticise the reservoir-wave hypothesis are rigorous and believable in the idealized systems used. However, they do not adequately model the physiological arterial system and it is unsurprising that application of the hypothesis to these simplistic, highly idealised computational models lead to inconsistencies and erroneous predictions. For this reason we do not believe that they form a rational basis for rejecting the use of reservoir pressure in clinical measurements. We believe that it is essential that the crucial test of the hypothesis must come from physiological experiments rather than computational experiments on highly idealised models.\cite{Tyberg_2014}
Other papers critical of the reservoir-wave hypothesis are predominantly due to Segers and his colleagues. Segers is a coauthor of \cite{Westerhof_2015} that has been discussed above. Their other comments are summarised in an editorial comment \cite{Segers_2012}. Apart from reporting the criticisms published by Mynard and his colleagues discussed in the preceding paragraphs, we find little to differ with in this commentary. They report finding cases where the calculation of reservoir pressure leads to unrealistic results, particularly in young individuals where it was difficult to fit and exponential decay to the diastolic pressure. This is a legitimate comment which corresponds to our occasional experience in calculating \(P_r\) from the measured \(P\). Segers et al. conclude their commentary 'Reservations on the reservoir' by pointing out that the reservoir-wave concept 'remains and intriguing concept' but is an approximation that may lead to oversimplification of the complexities of arterial wave behaviour. We find nothing to fault in that conclusion.
In summary, we reiterate that the reservoir-wave hypothesis is heuristic and that its merit depends on its usefulness. The results of a recent studies which indicate that various parameters based on either reservoir or excess pressure have prognostic value for the risk of cardiovascular disease provide strong motivation for further studies \cite{24821941}\cite{25534707}. We question the validity of computational studies of simplified, idealistic models as tests of the hypothesis since the homogenising complexity of the cardiovascular system may be the reason that the hypothesis 'works'. Finally, like any approximation, it is certainly not universally valid and finding when it is and is not valid is an essential part of the testing procedure.