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The use of a generalized transfer function: different processing, different results!

Segers, Patricka; Mahieu, Driesb; Kips, Jana,b; Van Bortel, Luc Mb

doi: 10.1097/HJH.0b013e3282ef5c5f
Editorial commentaries

aCardiovascular Mechanics and Biofluid Dynamics, IBiTech, Ghent University, Belgium

bDepartment of Pharmacology, Ghent University Hospital, Belgium

Correspondence to Patrick Segers, Cardiovascular Mechanics and Biofluid Dynamics, IBiTech, Ghent University, De Pintelaan 185, B-9000 Gent, Belgium Tel: +32 9 2403466; fax: +32 9 2404159; e-mail:

It has become common knowledge that, on its path from the central aorta towards the periphery, the amplitude of the pressure pulse is amplified [1,2]. As a result, systolic blood pressure is substantially higher in peripheral vessels, such as the brachial and radial arteries, than in the central aorta. Although there is mixed evidence available indicating that a knowledge of central blood pressure improves diagnosis and is more predictive than peripheral blood pressure in terms of outcome [3–5], it is certainly defendable to assess these central blood pressures from a physiological point of view. After all, it is the central pressure that the heart is facing, and this forms the true afterload. The question is how to assess these central blood pressures in a non-invasive way. One option is to use applanation tonometry at a vessel as close to the heart as possible, where the carotid artery is an obvious choice [6]. Another option is to assess peripheral blood pressure waveforms at the radial artery, where applanation tonometry is somewhat easier to perform, and to use a so-called transfer function (TF) to generate ‘synthesized’ central pressure waveforms from the radial ones [7,8].

In this issue of the Journal, Hope et al.[9] present a study on the generalizability of the transfer function. Although their work focuses on the minimum number of subjects that should be taken into account when constructing the generalized transfer function, they reconfirm their previous findings showing that a generalized transfer function generally performs rather poorly, especially when it comes to an assessment of the augmentation index (AIx) from reconstructed curves [10–12]. Although we have a preference for direct carotid applanation tonometry to assess central waveforms in our own studies [13], the results provided by Hope et al.[9] only partially reflect our own experience with generalized transfer functions. Although we did not find excellent correlations between AIx derived from directly measured (aorta or carotid) and synthesized waveforms (r = 0.64 [14] to 0.75 [13]), the values were still markedly higher than those reported by Hope et al.[9]. With the exception of one study where a correlation coefficient of 0.65 was found [10], most often there are no significant correlations [9,11,12]. The reason for these discrepant findings could be the methodology adopted to assess the generalized transfer function, of which a glance is offered by Hope et al.[9]. To illustrate our point, we first have to provide some background on the physics of the transfer function.

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The transfer function: what it expresses

A transfer function mathematically expresses the relationship between the blood pressure waveform in the aorta and in the radial artery, and it is most commonly displayed in the frequency domain [7,8]. This means that, instead of looking at the pressure waveform as a whole, one considers the waveform as a sum of a steady part (mean blood pressure) and a pulsatile part that consists of sine waves with increasing frequency (Fourier decomposition) [1]. The transfer function thus expresses the relationship between sine waves for a given frequency (also called harmonics) at these two locations. Two parameters suffice to describe the relationship: the ratio of amplitude of the sine waves (the modulus of the TF), which expresses amplification or damping, and the difference in phase (the phase of the TF), which expresses the time delay between them.

Assume, for example, a distance of 1 m between the aortic and radial measuring location, and a pulse wave velocity of 10 m/s along that trajectory. As such, it will take the pressure waves 100 ms to travel that distance. What this means in terms of phase difference is easily calculated if one takes into account that the wavelength λ of a sine wave equals the ratio of PWV and its frequency. In Fig. 1, using an arbitrary example, we have displayed what this implies for the first, fifth and tenth harmonic in case of a heart rate of 70 beats/min. In the absence of wave reflection and assuming that pulse wave velocity is the same for all harmonics, the phase difference (Δϕ) is simply a linear function of frequency (f):

Fig. 1

Fig. 1

where Δx is the distance between the two locations. For a radial/aorta TF, the phase difference is negative; for aorta/radial TF, the phase difference is positive.

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The phase of the transfer function: necessity of ‘unwrapping’ the phase

There is a potential danger in the treatment of these phase angles, especially for the higher harmonics. In mathematical terms, a sine wave with a phase delay of −210° is exactly the same as a sine wave that has a phase that leads by +150°, and a phase delay of −420° is exactly the same as a phase delay of −60°. When calculating the TF, any software will, by default, return phase angle values between −180° and +180° (between –3.14 and +3.14 radians), which will lead to phase jumps of 360° in the transfer function. In the case of our example, the phase difference will be negative up to the fourth harmonic (where it reaches −168°), but will jump to the positive value of +150° for the fifth harmonic. To obtain the monotonically decreasing transfer function phase, one must correct for these phase jumps by ‘unwrapping’ the phase angle. In Fig. 2, we have calculated an averaged radial/carotid transfer function from a dataset of 36 carotid and radial pressure waveforms (measured with applanation tonometry) with (‘unwrapped’) and without (‘wrapped’) correction for these phase jumps.

Fig. 2

Fig. 2

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Why is this ‘unwrapping’ important?

The unwrapping becomes particularly important when calculating a generalized TF via averaging of many individual TFs. In that case, wrapped phase angles will show up at the higher frequencies, with random phase jumps and a phase angle that will oscillate around the zero axis. Averaging these TF will result in a phase angle that first decreases monotonically, but then jumps to positive values for the higher harmonics with no clear pattern. Obviously, the use of such an averaged transfer function will have a major impact on calculated waveforms not so much with respect to the low frequency information, which will be fairly reproduced (systolic blood pressure), but certainly with respect to details of the waveform and derived indices (augmentation index). In Fig. 2c and d, we have applied the wrapped and unwrapped transfer functions to two distinct radial artery pressure waveforms to illustrate the effect on the resulting reconstructed pressure wave morphology. Surely, the effect is important, with the unwrapped TF yielding reconstructed curves that much more closely resemble the measured curve.

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How does this relate to the work of Hope and colleagues?

First, it should be stressed that the approach outlined above differs from the approach followed by Hope et al.[9]. In their study [9], they align the aortic and radial waveform in time (using the peak of the first derivative at both curves as a reference point), which is equivalent to shifting the curves in Fig. 1b by 0.1 s. In this idealized example, it can be seen that, after realignment, the phase difference between the harmonics disappears, and the theoretical phase angle of the TF becomes zero, as also stated by Hope et al.[9]. In reality, however, because of wave reflection and nonlinearities, the phase difference will not totally disappear, as also made clear by Hope et al.[9]. Nevertheless, the approach of re-alignment in time is perfectly defendable, and totally valid.

The question is whether other technical aspects of the data processing might explain the discrepancy between these different experiences in the performance of generalized TF. Clearly, in our opinion, the answer is to be sought in data manipulations affecting the high frequency contents of the TF spectra. This issue was previously also raised by O'Rourke et al. [15], who questioned the frequency responsiveness of the measuring equipment. Because the data were measured with high-fidelity catheters, this cannot be the issue. We think, however, that three factors may play a role:

  1. Hope et al.[9] directly derive the aorta/radial TF in the radial-to-aortic direction. This makes a difference on the amplitude of the TF, as 1/average(∑Xi) is not the same as average(∑1/Xi), where Xi is the modulus of the radial/aorta TF of an individual at a given frequency. In Fig. 3, we demonstrate the difference between an inverted averaged radial/carotid TF, and a directly assessed averaged carotid/radial TF. There is a difference in the modulus of the TF, but not in the phase, and the resulting waveforms (Fig. 3c and d) are reasonably similar.
  2. Hope et al.[9] also appear to average the TFs per harmonic. The amplification characteristic of the aortic–radial pathway is, however, frequency dependent, and it would be better to subsample the transfer functions and average them for a given frequency, rather than for a given harmonic. If heart rate varies in the population, this must have an effect on the calculated results.
  3. The bottom panel of Fig. 3 in Hope et al.[9] contains phase angles beyond the range [−3.14, 3.14] rad. Given that these curves were already realigned in time, it is hard to explain where these large negative phase differences arise from, especially given the fact that the figure displays the aortic/radial TF. It appears that these data points should be corrected for by a phase jump of +360°, which would also substantially narrow the error band in the phase angles.
Fig. 3

Fig. 3

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What can we conclude from this?

It is not our intention to undermine the validity of the most valuable work of Hope et al.[9]. We share the concern of this group with respect to the generalizability of transfer functions in large artery research, especially when it comes to the assessment of indices depending on details in the morphology of reconstructed pressure waves. Nevertheless, the overall performance of the transfer function might not be as poor as reported in some of their papers, possibly because of a different treatment of the high frequency information in the transfer function. From our analysis, it follows that the correct processing of the phase angles for the higher harmonics is crucial for the calculation of synthesized pressure waveforms.

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