The rest of this sentence is not correct, as we realised only after a lot of thought and reflection. Perhaps we could be a bit anecdotal here? 'In the initial data analysis we too thought that variations along the aorta in the parameters used to calculate Pr (kd, ks and Pinf) was evidence that our assumption that Pr is uniform along the aorta was wrong. However, further reflection revealed that this is not the case. In fact, given the obvious variation in Pmeas at the different aortic sites, it is necessary that these parameters be different if Pr is to be constant. Since Pr is calculated as the solution of an ordinary differential equation involving these parameters as coefficients and the measured pressure as the inhomogeneous forcing function, the relationship between the parameters and Pr is dependent on the Pmeas waveform in a highly interactive way.'
In the 1st paragraph they state '...magnitude and shape of Pr are still assumed similar at all locations. Narayan et al. now attempt to underpin this assumption.' (their italics) This has a slightly negative connotation and I think we should emphasise that the purpose of the study was to test the hypothesis that Pr is similar at all locations by calculating Pr from the measurements made at the different sites and testing these results for constancy. This is not an attempt to 'underpin' the assumption but to 'test' it.
In the last paragraph of the 1st page the comment that 'no (statistical) test is reported'. As they know, it is notoriously difficult to compare waveforms statistically and so we opted for comparing scalar measures of Pr using ICC, which we believe is the 'state of the art' statistical test for data measured at a number of sites in a number of different subjects. The data that they calculated from one subject (who incidentally was not selected as a best-case but simply because it was Patient 1) is commensurate with data found in other patients and the statistical results which we report are the result of the data from all of the patients. It may be relevant here that the average SD over the ensemble averages of Pmeas that were used in the analysis was on the order of 4% (I could look up the exact figure, but don't think it would add anything to the argument).
I have dealt with the fallacy of their statement towards the end of this paragraph '...their magnitudes and parameters describing the waveforms still indicate that they are not identical' in my first comment. To repeat; Pr is calculated from the best fit of the solution of an ODE with 3 parameters and the measured pressure as the forcing function. The relationship between Pr and the 3 parameters is thus dependent in a highly non-linear way on Pmeas(t).
2. It seems that they have not understood that the assumption that Px = xi . QAA (QAA is their notation for volume flow rate at the aortic root) is only an assumption and that it remains to be tested experimentallly (as we have discussed). The justification for this assumption is solely from the measurements in dogs by Tyberg. It is possible to find highly ideal cases where this will be true (e.g. arterial bifucations with reflection coefficients so that Q(0,t) propagates without change through the arterial tree. In these idealistic cases it is possible to relate xi to the characteristic impedance but this is not possible in the real cases. Theoretically this is a very important assumption because it allows us to solve for the local Pr(x,t) from measurements of the local Pmeas(x,t). Without the assumption it would be necessary to measure Q(x,t). Unless I was insufficiently alert, we never clamed that xi is a characteristic impedance (although it obviously has the dimensions of impedance). I prefer to view it as an unknown parameter that is determine through fitting (ks = 1/xiC, where C is the local compliance in the same way that kd = 1/RC, where R is described as the net arterial resistance). I think it is confusion between xi and characteristic impedance (Z0 in their notation) and between the local charactistic impedance Z0(x) and the characteristic impedance at the root Z0(0) that leads to almost all of the differences between us in the rest of the letter.
I agree with them that Q(0,t) = xi Px(x,t) is an untested assumption (in humans) that needs to be tested by measurements in humans. The calculations they describe using the data from [6] is a valuable bit of evidence that must be taken into consideration. I do not think there results are conclusive because of differences in the way that Pr is calculated in our work and in [7] which is based on one of our preliminary studies that was published but modified considerably in our more recent calculations of Pr from Pmeas. The untested nature of the basic assumption in the pressure-only calculation of Pr is a serious question that warrants careful experimental testing.
3. It is probably unfortunate that we used old data in Fig.1 instead of 'typical' data from our study. In defense of our choice, Figure 1 is only used as a sketch to define the various parameters that we calculated in our study. We never assert that it is anything more than a sketch.
Their point about the uniformity of Pr in patients with less 'rounded' pressure waveforms is a good one. There is a partial answer to their question in our paper where 'all except one of the participants exhibited a pre-systolic inflection point...' and we still found a statistically uniform Pr. Perhaps we should go back and compare Px(x,t) as an indirect way to test that Q(0,t) is proportional to Px(x,t). Since Q(0,t) is the same for measurements at all of the sites (apart from minor temporal variations between our sequential measurements) the differences in Px(x,t) at different sites would be an indication of the validity of our assumption. If I can find the time, I might look into this question.
My intuition says that, if the uniformity that we found is confirmed by other studies using patients with other 'irregularities' in their pressure waveforms, the Pr analysis will be valuable because Px in these case will emphasise the irregularities and help to disentangle them from the variations in Pr which is primarily determined by the net properties of the arterial tree. This, of course, is speculation until the the whole concept is tested more extensively.
4. My blood pressure was unexpectantly constant while reading the letter until I got to this section. Given my respect for the body of work of the authors of the letter, I cannot understand why this difference persists (although it may be due to my inadequate understanding of impedance analysis and our different understanding of the characteristic impedance. I consider the characteristic impedance to be dP(t)/dQ(t) for a single wave front (which I am careful to call z in my papers to differentiate from the characteristic impedance Z0(w) calculated from P(w) and Q(w) obtained by Fourier analysis from P(t) and Q(t). In my definition z is a real local parameter that can be a function of time which depends on the density of blood and the area and distensibility of the local artery. In the impedance analysis definition it is a function of frequency (w) and is affected by the properties of the entire arterial network including the terminal resistance. Perhaps it would be best if I did not use the term characteristic impedance for z, but this would diminish its meaningfulness because all of us agree that z = Z0 when there are no reflections in the arterial system.
The only way I can think to clarify the differences alluded to in 4 is to express their equations more carefully using (x,t) to indicate position x and time t. Their equation for Pb (including the comment after the equation) becomes Pb(x,t) = (Pmeas(x,t) - Z0(x) Qmeas(x,t))/2. I agree with this equation. The next equation says that if Qmeas.Z0 = Px then Pb = (Pmeas - Px)/2 = Pr/2. Making the (x,t) explicit in this equation gives Qmeas(x,t).Z0(x,t) = Px(x,t) then Pb(x,t)=(Pmeas(x,t) - Px(x,t))/2 = Pr(x,t)/2. This, however, is not what we assume. We assume Q(0,t).xi(x,t) = Px(x,t) where xi(x,t) is determined by fitting the solution to the ordinary differential equation for Pr as a function of zi (through ks), the other fitting parameters kd and Pinf and Pmeas(x,t) as a forcting function. There is no way that we can substitute our assumption about Q0(0,t) into the equation for Pb(x,t) during systole.
During diastole, however, Q0(0,t) = 0 so that the forcing function in the ODE is zero and it follows from the assumption that Pmeas = Pf + Pb and Qmeas = Pf - Pb (which is one of the keystones of wave separation into forward and backward components in both impedance and wave intensity analysis) that during diastole when Qmeas(0,t) Pf = Pb. Since we also require that Pr = Pmeas during diastole, it follows that Pb = Pr/2 during diastole (but not systole). I do not understand why they have a problem with this argument.
Note that their point (ii) in the middle of the paper is right if we did, in fact, say that they say Pmeas(0,t) = Z0.Qmeas(0,t) where Z0 is the characteristic impedance at x. If we did say that we erroneously assumed that Z0 = Z0(0,t) instead of Z0(x,t), probably because of confusion about the meaning of the subscript 0 in Z0? There remains the problem that we do not assert that xi is a characteristic impedance and any assertion that it is a characteristic impedance is wrong.
If you will allow me one outburst of steam in an effort to maintain my blood pressure, the last sentence about our scientific rigour is astounding from the authors who admitted in a corrigenda to their previous letter about Pr and iFR (quite rightly) that their discussion of iFR was based on an incorrect quotation of the equation defining iFR in our paper but (quite wrongly) did not withdraw the conclusions that they made based on this error?
Forgive me if my impatience with this discussion has shown through. I have reluctantly resigned myself to the fact that these erroneous arguments against Pr are going to haunt all of us for the rest of our lives ('the authors have not addressed the problems raised in ...'.). I am very aware that there are basic assumptions in Pr that have not been verified by careful experiments. I think we made significant advances with our pull back paper by testing the hypothesis that Pr is uniform along the aorta. I am very keen to test the assumption that Px(x,t) is proportional to Q(0,t) in humans as it was in the dog experiments, as we have discussed in our talks about where we can go next experimentally. My intuition says that Pr will be found to be a useful concept but I am enough of a scientist to be cautious about its universal applicability, mainly because I have not been able to find a satisfactory derivation of Pr from first principles (conservation of mass and momentum).
The most important immediate problem is to answer the points raised in the WSW letter appropriately. I am pessimistic about erasing all objections but am encouraged by recent comments relayed to me from people like Alun who attended the last Artery meeting that reservoir pressure is generally seen as a much less controversial hypothesis than it has been in previous meetings. The parallels to the slow adoption of wave intensity analysis seem strong and I hope that we can make Pr as well accepted as wave analysis is now.
My personal position is that I should do what I can to make the discussion as rational as possible but there is no way to convert the unconvertable. I have, however, resolved that in the unlikely event that I have another 'good' idea, I am going to publish it anonymously?
I hope these comments are helpful and look forward to your thoughts on the response.
If I don't get a chance, Merry Christmas and Happy New Year. I look forward to collaborating with you in the future.