Comments on ’A new approach to blood pressure analysis’
by Ralph W. Barnes

This is a bit of an experiment about how best to communicate when there is a lot of mathematics involved. I am trying out this new site, Authorea, for producing Latex quality formats on web pages and have found it a convenient way to produce drafts which can be shared with and edited by my co-authors. This is my first attempt to write a commentary. I will be interested to hear how you found it.

First of all, I think that your approach has many virtues and have found many parallels with my current approach to arterial mechanics. I also see some problems in the analysis that will need to be addressed before your approach can be accepted. I hope my discussion of the problems does not discourage you from continuing with the work but suggest avenues for improving it.

A global technical problem with all of your work is that you are using Fourier analysis which is only valid for periodic functions. It is possible to define Fourier integral transforms of non-periodic functions and these are widely used in functional analysis but they require stringent conditions on the functions which can be analysed and generally involve infinite resolution of the time and frequency variables. The Fast Fourier Transform (FFT) which I presume that you are using in your analysis falls into the realm of discrete (rather than continuous) mathematics and it is only valid for periodic functions. Thus when you talk about the pressure waveform during a single cardiac cycle, the Fourier transform automatically assumes that this single waveform is one of an infinite number of identical waveforms. Your can, of course, take the FFT of a large number of cardiac cycles but you still have to remember (as many don’t) that the FFT assumes that this block of waveforms is one of an infinite number of identical blocks. This point is frequently overlooked by people doing impedance analysis.

This problem becomes important in your discussion of the pressure waveform during the diastolic period (which is, I believe what you call the ’Arterial Filter (\(\omega\))’. If you are calculating this by taking the FFT of the diastolic part of your measured arterial pressure \(P(t)\) truncated so that \(t=0\) corresponds to the start of diastole \(T_{d}\), then you have to remember that the analysis is assuming that this truncated time series is actually one of an infinite series of identical truncated waveforms. Not knowing exactly how you obtained Arterial Filter (\(\omega\)) function, I cannot say exactly how this will affect your analysis. It is however a very important point to consider.

The thing I like most about your approach is that you recognise that what we measure in the artery is the result of the interaction between the pressure (or flow) generated by the ventricle and the response of the arteries which, if I am interpreting your work correctly, is what the ’Arterial Filter’ represents. In signal analysis circles this would probably be described as a ’transfer function’. The basic theory behind transfer functions is the observation that for a linear system with an input \(X(t)\) and an output \(Y(t)\), the transfer function \(H(\omega)\) can be calculated as

\begin{equation} H(\omega)=\frac{\widetilde{Y}(\omega)}{\widetilde{X}(\omega)}\nonumber \\ \end{equation}

where \(\widetilde{f}(\omega)=\mathfrak{F}\{f(t)\}\) is the FFT of the function \(f(t)\). The first crucial point, following from the above discussion, is that \(f(t)\) must be (or is implicitly assumed to be) a periodic function. The second crucial point is that this relationship is only valid for linear systems. In my work on waves in arteries I find that the interaction between the input from the heart and the response of the arteries is highly non-linear and so I have reservations. For example, a wave travelling forward in the parent artery will be reflected by a bifucation differently than a wave travelling backwards in one of the daughter arteries. This leads to a type of non-linearity which cannot be described by a transfer function. However, it is a common approach to arterial mechanics and I would not condemn it but suggest that it should be used with caution.

My next comments concern your determination of the parameters describing \(P(t)\) during diastole. This is a subject of great interest to me and I have been working on this for some time now. Because of the lack of details in your MS, I have made several several assumptions about your analysis which may or may not be true. First of all, the arterial waveform in Fig.1 is obviously not a measured arterial pressure which would be much higher. I guess that you have subtracted the diastolic pressure \(P_{d}\) and added the ’Offset = 5 mmHg’. This needs clarification.

I see that you have modelled the diastolic pressure waveform with the equation \(P(t)=A_{0}+A_{1}e^{-\alpha t}\). Did you treat \(A_{0}\) as a fitting parameter? If so, I would be very interested in the results you obtain from this model. I have been using exactly the same model but expressed slightly differently

\begin{equation} P(t)=(P_{0}-P_{\infty})e^{-\alpha t}+P_{\infty}\nonumber \\ \end{equation}

In this formulation, \(P_{0}\) is the value of \(P\) at \(t=0\) (taken to be the start of diastole) and \(P_{\infty}\) is the asymptotic pressure that is approached when \(t\rightarrow\infty\). In your terms \(P_{\infty}=A_{0}\) and \(P_{0}=A_{0}+A_{1}\). We found that including the asymptotic pressure into our model as a free parameter gives much better fits to the measured \(P\) than we obtained using the simpler model where \(P\) asymptotes to zero, which is the model most commonly used to determine the time constant \(\tau=1/\alpha\). The problem is that the values of \(\tau\) we obtain with the model with an asymptotic pressure are generally lower than the textbook values of \(\tau\) obtained by previous workers. The main problem, however, is that the value of \(P\infty\) that we find by fitting our model to measured diastolic pressures is much higher than venous pressure (\(P_{\infty}\) is typically in the range 50 -70 mmHg) and we are somewhat at a loss to explain this finding. I have looked at the effect of fixing \(P_{\infty}\) and fitting \(\tau\) and find universally that \(\tau\) decreases as \(P_{\infty}\) increases but that the goodness of the fit increases as well. Your experience in fitting the diastolic pressure would be of great interest to me.

As I have explained, obtaining your Arterial Filter from your exponential fit to the diastolic pressure is not trivial. If I was doing it, I would point out that

\begin{equation} \mathfrak{F}\{u(t)e^{-\alpha t}\}=\frac{1}{\alpha+i\omega}\nonumber \\ \end{equation}

where \(u(t)\) is the Heavyside function which is the unit step function \(u=0\) for \(t<0\) and \(U=1\) for \(t>0\). It is a discontinuous function at \(t=0\) but with sufficient care in the analysis you can deal with this discontinuity. If you plot \(u(t)e^{-\alpha t}\) it is just the exponential decay for \(T>0\) from the value of 1 at \(t=0\) and asymptoting to 0 at \(\infty\). In electrical terms, this is the response of a simple RC circuit to a delta function input where \(\tau=RC\) (the product of the resistance and the capacitance). I would assume that this was the transfer function of the arterial system \(H(\omega)\). Assuming that the arterial-ventricle system was linear, we could then say (as you do) that

\begin{equation} \widetilde{P_{V}}(\omega)=\frac{\widetilde{P}(\omega)}{H(\omega)}\nonumber \\ \end{equation}

As you have done, we can now obtain \({P_{V}(t)=\mathfrak{F}^{-1}\{\widetilde{P_{V}}(\omega})\}\).

As a passing comment, I was struck by the superficial similarity between your Pressure Source waveform shown in Figure 3 and the expected flow waveform. Since you do not measure the flow waveform, we cannot be sure but your Pressure Source waveform shares many of the features of the ascending aorta waveform that has been measured in other studies: a peak that occurs before the peak in the measured pressure waveform and a not insignificant back flow at the end of systole. We unexpectedly found in our work on the reservoir presssure that the excess pressure waveform defined as the measured pressure minus the reservoir pressure was very similar in shape to the flow waveform in the ascending aorta. We have made some physical arguments to explain this similarity but we actually found the similarity experimentally first and then came up with the physical arguments afterwards. It would be interesting to look at your Pressure Source waveform in a similar way.