## Abstract

Central aortic pressure gives better insight into ventriculo-arterial coupling and better prognosis of cardiovascular complications than peripheral pressures. Therefore transfer functions (TF), reconstructing aortic pressure from peripheral pressures, are of great interest. Generalized TFs (GTF) give useful results, especially in larger study populations, but detailed information on aortic pressure might be improved by individualization of the TF. We found earlier that the time delay, representing the travel time of the pressure wave between measurement site and aorta is the main determinant of the TF. Therefore, we hypothesized that the TF might be individualized (ITF) using this time delay. In a group of 50 patients at rest, aged 28–66 yr (43 men), undergoing diagnostic angiography, ascending aortic pressure was 119 ± 20/70 ± 9 mmHg (systolic/diastolic). Brachial pressure, almost simultaneously measured using catheter pullback, was 131 ± 18/67 ± 9 mmHg. We obtained brachial-to-aorta ITFs using time delays optimized for the individual and a GTF using averaged delay. With the use of ITFs, reconstructed aortic pressure was 121 ± 19/69 ± 9 mmHg and the root mean square error (RMSE), as measure of difference in wave shape, was 4.1 ± 2.0 mmHg. With the use of the GTF, reconstructed pressure was 122 ± 19/69 ± 9 mmHg and RMSE 4.4 ± 2.0 mmHg. The augmentation index (AI) of the measured aortic pressure was 26 ± 13%, and with ITF and GTF the AIs were 28 ± 12% and 30 ± 11%, respectively. Details of the wave shape were reproduced slightly better with ITF but not significantly, thus individualization of pressure transfer is not effective in resting patients.

- brachial pressure
- augmentation index
- travel time

central aortic pressure has been shown to give more accurate information about the load on the heart, is of more prognostic value (40), and may give better indication of treatment (17) than brachial pressure or radial pressure (2). The differences in mean and diastolic pressure between peripheral sites and the aorta are small, but the differences in systolic pressure and pressure wave shape may be considerable. Therefore, the use of transfer functions (TFs) to reconstruct central pressure from peripheral pressure, has received a great deal of attention, and has generated lively debate (12, 24). Generalized transfer functions, i.e., averaged over groups of subjects, have been shown to provide mean and diastolic aortic pressure values within limits of accuracy as set by the AAMI (1). However, systolic pressure, being the most variable, is more difficult to derive correctly. Additionally, waveform analysis, such as the calculation of the reflection index (21, 22, 37), augmentation index (16), and estimation of cardiac output (33) rely on detailed information on aortic systolic pressure waveform and magnitude.

Several approaches have been taken to acquire transfer functions, starting as early as 1970 (5, 7, 11, 15, 23, 24). Calibration (4, 8, 10, 30, 35) has become an issue in itself (32).

Usually, a generalized transfer function (GTF) is used; however, more exact results may be expected when a TF is individualized (ITF), i.e., optimized for a particular subject. For instance, an ITF can account for age or sex (13); with extra measurements an ITF might be improved even more (14, 27, 29).

Earlier we showed that the time delay between central and peripheral pressure is the most important parameter determining the pressure transfer function (27, 36). As this time delay can be noninvasively obtained, we set out to investigate if this parameter could be used to individualize the TF and if the use of an individual TF predicts aortic pressure better than a generalized TF. We used a set of invasively, almost simultaneously measured brachial and aortic pressures to test this.

## METHODS

#### Measurements.

Blood pressure recordings were taken from an earlier study by Lasance et al. (18). In short, pressure recordings were made in 74 subjects in whom elective coronary arteriography was performed. A fluid-filled catheter, inserted with the Sones technique, was placed in the ascending aorta and pulled back to the brachial artery. The frequency response of the fluid-filled system was regularly checked during the measurements. The resonance frequency was always higher than 50 Hz, thus the fluid-filled system did not distort the pressure signal. The data were sampled at 100 Hz. Since measurements were not performed simultaneously, aortic and brachial pressure beats were selected such that heart period differed <90 ms and mean pressures differed <9 mmHg, since in these patients a larger mean pressure difference must result from instability of the cardiovascular system. For each patient, a single aortic and brachial pressure beat was used since the catheter was kept at each site for only a few seconds. Recordings of Lasance et al. (18) were made during routine patient care and at the time of acquisition no permission of an Institutional Review Board was needed to use the recordings for scientific purposes.

#### Transfer function.

A simple mathematical function, describing pressure transfer over a single uniform tube (27, 36) forms the basis of our models. It has two parameters: (1) with Γ the reflection coefficient at the distal site, taken to be real, Δ*t* the time delay between the distal and proximal site (brachial artery and aorta), j the imaginary unit, and ω the angular frequency, i.e., 2π*f* with *f* heart frequency. This function was shown to give a good description of pressure transfer (27, 36). The Fourier transformation of the brachial pressure wave divided by the TF gave the “reconstructed aortic pressure.” After inverse Fourier transformation, the reconstructed aortic pressure as a function of time was calculated and compared with measured aortic pressure.

#### Calculations.

First, systolic, and diastolic pressures were determined of the aortic and brachial pressure curves. The root mean square error (RMSE) was calculated with the measured aortic and measured brachial pressure shifted in time with respect to each other so that minimal errors were obtained. In this way, the remaining errors are a measure of the difference in wave shape and not artificially large due to differences in timing.

For each patient, only a single aortic and brachial pressure beat was used, since the catheter was kept at each site for only a few seconds. Next, for the pair of pressures of each patient, the optimal Γ and Δ*t* of the TF (*eq. 1*) were found by searching for the Γ and Δ*t* that minimize the RMSE between reconstructed and measured aortic pressure. This was with done with a brute force approach. Aortic pressure was reconstructed with TFs with Γ varying between 0.3 and 0.9 with steps of 0.05 and 2πΔ*t* varying between 0.2 s and 0.4 s with steps of 0.05 s (thus Δ*t* varied between 0.0318 s and 0.0637 s with steps of 0.00796 s). Then the Γ and Δ*t* giving the smallest RMSE were selected. Subsequently, four transfer models were defined. In *model 1*, optimal Δ*t* and Γ for each individual were used. In *model 2* optimal Δ*t* was used while Γ was set at the group average. *Model 3* was a GTF with both Δ*t* and Γ set at the group average. Finally, in *model 4*, we set Δ*t* to the average value and Γ to 1, representing total reflection, to assess the importance of the parameter Γ. Systolic, diastolic, and pulse pressure and RMSE of the reconstructed aortic pressure using the four TF models were compared with measured aortic pressure.

Augmentation index (AI; 16) was calculated from the results of all models and compared with the AI of the ascending aortic pressure waves as another indication of the correctness of the reconstructed wave. Calculations were implemented as described by Segers et al. (25).

#### Statistics.

Results of the four TF models were compared with measured aortic pressure using Dunnett's method for multiple comparisons vs. control. Differences between consecutive models were evaluated with paired *t*-tests. Differences were assumed to be significant for *P* < 0.05.

## RESULTS

Ascending aortic and brachial pressure waves within the limits given in the methods were obtained in 50 patients (43 men and 7 women) aged 51 ± 9 yr (range 28–66). The difference in mean pressure between brachial artery and ascending aortic pressure was −0.4 ± 3.4 mmHg. Difference in interbeat interval (IBI) for these beats was −3.4 ± 27 ms. For Δ*t* we found 0.048 ± 0.010 s and for and Γ was 0.6 ± 0.13. Figure 1 shows the TF calculated from these measured pressure pairs and also the TFs according to *eq. 1* for the generalized cases with the averaged Δ*t* and averaged Γ (*model 3*) and for averaged Δ*t* with Γ = 1 (model 4).

Table 1 lists measured ascending aortic pressure and brachial pressure values together with reconstructed pressure values using the four different TF models. Brachial pressure and each of the reconstructed pressures are all tested against measured aortic pressure (Dunnett) and only brachial systolic and pulse pressure are different. Between the models, best results are obtained with *models 1* and *2* (with both Δ*t* and Γ individualized and with only Δ*t* individualized and with Γ at group average). Between *models 2* and *3* slight differences start to arise, which are significant by *t*-test. Between *models 3* and *4*, results again deteriorate somewhat. The AIs of the models are not different from ascending aortic AI except for *model 4* with Γ = 1. Similarly, the differences in the AIs between the models (*t*-test) are significant only for *model 4* vs. *model 3*. Table 2 lists the linear regression coefficients of the model AIs on ascending aortic AI. Finally, in Table 1, each step of generalization involves an increase in RMSE, which is small, but statistically significant (*t*-test) between the models.

In Fig. 2 the Bland-Altman plots of systolic, diastolic, PP, and AI are shown of *model 1* and *model 3*. The improvement by individualization is negligible.

Because RMSE data only give global information of the fits but no details on the wave shape of the reconstructed aortic pressure wave, four examples are given Fig. 3. In these cases GTF gives distinctly less accurate results compared with the individualized TF. In Fig. 3, *top* (*patient a*), reconstructed pressure using the GTF has no obvious secondary rise in pressure, or pressure augmentation, while the pressure from the individualized TF has. In Fig. 3, *second panel* (*patient b*), both reconstructed pressures show an augmentation but the pressure reconstructed with the individualized TF has a closer fit. In Fig. 3, *third panel* (*patient c*), the GTF does not dampen but augments the oscillations found in the brachial artery pressure. In Fig. 4, *bottom panel* (*patient d*), the pressure augmentation is exaggerated by the GTF.

Thus, although the use of an ITF has on average little advantage, in some cases (*patient c* in Fig. 2) individualization (*model 1*), or even an individualized Δt and Γ = 0.6 (*model 2*, not shown), improves the reconstructed aortic pressure wave shape.

## DISCUSSION

We found that the aortic wave shape reconstruction with the individualized TF is not significantly better than with the generalized one. Thus, in this group of resting patients, a generalized TF can be used without loss of accuracy of reconstruction. If an individualized TF is preferred, the one based on measured Δ*t* and Γ = 0.6 (*model 2*) is the simplest to obtain and gives very accurate results.

We used a simple mathematical description of arterial pressure transfer, which requires only two parameters: Δ*t* and Γ, the travel time of the pressure wave from aorta to measurement site and reflection coefficient at the site of pressure measurement. We hoped that by determining these parameters for an individual patient, a more accurate reconstruction of ascending aortic pressure could be obtained. However, all models gave similar results, except the one with Γ set to 1 (*model 4*), which overestimated pulse pressure. With stringent testing minor differences could be determined between consecutive models, which are statistically significant but not clinically relevant (Table 1).

Obviously, for an individualization method to be useful in practice, the measurements to obtain Δ*t* and Γ should be noninvasive and convenient. Γ could be determined by measuring pressure and flow at the peripheral site (27). This is cumbersome and since using the individualized Γ gives the same results as an average Γ (Table 1, *model 2* vs. *1*), this averaged Γ can be used. Of note, the Sones technique of catheter insertion involves ligation and heparization of the distal artery. Nonetheless, the Γ that was found was not extremely high, i.e., close to unity. Figure 1 demonstrates that the first four harmonics of the models are virtually independent of Γ. This supports the findings in Table 1.

We found travel time Δ*t* to be the more important parameter. The Δ*t* determines the location of the maxima of the TF (Fig. 1). This also came forward from a model study (36). It can be seen in Fig. 1 that the first four harmonics are closely approximated by the TF generated by our procedure, which minimizes RMSE. So the first four harmonics are the most important for accurate pressure reconstruction (5).

To obtain Δ*t*, one option is to simultaneously measure brachial and axillary artery pressure by applanation tonometry, as accessible points most relevant to the trajectory, and determining the foot-foot time delay. Echo- or impedance cardiography could also be considered for the required central information. Another possibility is to use the delay between the R-top in the ECG and the upstroke in peripheral pressure, although the pre-ejection period remains an uncertain factor (31). In the present study we did not perform these extra measurements but analyzed a set of aortic and brachial pressures and calculated for each individual which Δ*t* and Γ was best.

Sugimachi et al. (29) also took the work of Stergiopulos et al. (27) as a basis for further research on the TF between radial artery and aorta. They used the distal Windkessel load of their model to determine flow and use wave separation and the “shift theorem” (27) with individualized time delay to reconstruct central pressure. They also concluded this delay is the key parameter to individualize the TF and that Windkessel parameters, and thus reflection coefficient, are of limited importance. Similarly, they were not able to improve reconstruction with individualization.

The AIs of the reconstructed pressures were not significantly different from the AI of measured ascending aortic pressure, except in *model 4*. The coefficients of determination in Table 2 show a decrease with generalization, from *model 1* to *model 4*, consistent with the findings in Table 1. While generalization tends to have some effect on AI, the results of methods using time-integrated values of the central pressure, such as pulse contour cardiac output methods (34) and cardiac oxygen supply/demand indexes (9, 38), are probably even less influenced by the TF.

The finding that a GTF can be used to reconstruct central pressures corresponds with findings in the literature (5), although the use of GTFs has been questioned (13, 19). In the study of Chen et al. (5) the prediction of aortic pressure from radial applanation tonometry showed good results even during changes in left ventricular filling. These authors concluded that the transfer function of arm vessels appears rather unaffected by body morphology, age, sex, and blood pressure. During the Valsalva maneuver in some patients, with similar decrease in arterial pressure, substantial variability in the transfer function was occasionally observed by these authors and the GTF predicted aortic pressure inaccurately. Thus, in certain cases, individualization could be effective (see our Fig. 3), although on average the improvement is small (Fig. 2). From the present findings it emerges that retrieval of detailed information on pressure wave shape, as described by the RMSE, may be somewhat improved by using an ITF while features as systolic and diastolic pressure levels are well represented using a GTF (Table 1). A similar suggestion comes forward from Table 2, showing tighter regressions of the individualized model AIs.

It has been reported that the peripheral pressure (3) or photoplethysmographic wave shape (6, 20) changes little during local infusion of vasoactive drugs, confirming the conclusion that the TF is quite insensitive to changes in Γ. On the other hand, systemic infusion of vasoactive substances has much greater effect on difference between central and peripheral wave shape (3, 6, 20). Therefore, although individualization of TF may have only a minor effect in the present study, the method may prove useful in protocols involving systemic infusion of vasoactive drugs (3, 39).

In exercise it was found that a generalized transfer function can give acceptable results on average at lower exercise load (26) and after calibration with a measured central pressure; however, it was also shown that the interindividual variability can be substantial, especially at higher heart rates (28). With increasing heart rate, harmonics are shifted to higher frequencies, so that the first few harmonics, in which most information is contained, are more influenced by the peak of the TF. In these cases, individualization may help in a more accurate prediction of aortic pressure. Time delay might be obtained from a time difference between ECG and noninvasively measured finger arterial pressure (28), since these methods are reliable during exercise. Whether this method would give delays of sufficient accuracy remains to be determined.

It is a limitation of our study that we included only patients who presented with cardiac problems, most of whom were diagnosed with CAD. This limitation is difficult to circumvent, since central catheterization is rarely performed in healthy subjects. In the present study, ITFs showed only slightly better results than the GTF. Whether the improvement would be greater in a healthy population we cannot predict.

TFs are most useful for estimating central pressure from noninvasively measured peripheral pressures. For this exploratory study we used invasively obtained data. We determined the optimal time delay from the actual measurements of ascending aortic and brachial pressure. In a practical application, the time delay should be determined from other measurements as the central pressure will not be available. Given the limited improvements, if any, by using this time delay, it seems hardly useful to individualize the TF. We conclude that in the resting patient it is sufficient to use a generalized transfer function.

## DISCLOSURES

B. E. Westerhof owns shares of and works for BMEYE. No conflict of interest. I. Guelen owns shares of and works for BMEYE. No conflict of interest.

## Footnotes

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- Copyright © 2008 the American Physiological Society