## Abstract

We fitted a three-segment transmission line model for the radial-carotid/aorta pressure transfer function (TFF) in 31 controls and 30 patients with coronary artery disease using noninvasively measured (tonometry) radial and carotid artery pressures (P_{car}). Except for the distal reflection coefficient (0.85 ± 0.21 in patients vs. 0.71 ± 0.25 in controls; *P* < 0.05), model parameters were not different between patients or controls. Parameters were not related to blood pressure, age, or heart rate. We further assessed a point-to-point averaged TFF (TFF_{avg}) as well as upper (TFF_{max}) and lower (TFF_{min}) enveloping TFF. Pulse pressure (PP) and augmentation index (AIx) were derived on original and reconstructed P_{car} (P_{car,r}). TFF_{avg} yielded closest morphological agreement between P_{car} and P_{car,r} (root mean square = 4.3 ± 2.3 mmHg), and TTF_{avg} best predicted PP (41.5 ± 11.8 vs. 41.1 ± 10.0 mmHg measured) and AIx (−0.02 ± 0.19 vs. 0.01 ± 0.19). PP and AIx, calculated from P_{car} or P_{car,r}, were higher in patients than in controls, irrespectively of the TFF used. We conclude that*1*) averaged TFF yield significant discrepancies between reconstructed and measured pressure waveforms and subsequent derived AIx; and *2*) different TFFs seem to preserve the information in the pressure wave that discriminates between controls and patients.

- blood pressure
- model
- applanation tonometry

knowledge of central aorta pressure is clinically important because it permits, together with aortic flow, one to compute arterial input impedance and hemodynamic parameters characterizing ventricular-arterial interaction (wave reflection indexes, hydraulic power) of diagnostic value in cardiology (1,5, 14, 19). Using tonometry, one can estimate, noninvasively, the central aortic pressure wave by measuring the arterial pulse *1*) at a superficial artery close enough to the heart (subclavian, carotid artery) so that the effects of wave distortion can be neglected (1,5, 6, 14, 16), or*2*) at a peripheral superficial artery (radial or carotid artery) with use of a pressure transfer function (TFF) to compute the pressure wave at the central aorta (4,7-9, 12).

Averaging transfer functions, measured directly in (relatively) large human populations, yielded generalized TFF for the radial-aorta (7, 12) and the carotid-aorta (11) pathway. Such generalized TFF have been integrated into commercially available systems, predicting central aorta pressure and derived indexes, such as the augmentation index (AIx), from the carotid or radial artery pressure wave (4,23). However, there is large scatter in measured transfer functions, and the morphology of reconstructed central pressure, using such a generalized TFF, may differ considerably from the directly measured pressure.

The aim of this work was to provide a method to individualize TFF, based on a transmission line model for the radial-aortic/carotid pathway (9). If model parameters characterizing the pressure transfer function are related to easily measured patient characteristics (age, blood pressure, etc.), then TFF can be predicted on a patient-to-patient basis, enabling a better estimate of the central aorta pressure wave morphology. First, we fitted the model to a published radial-aorta pressure transfer function that was derived from invasively recorded pressures (7). This way, we obtained reference values for all model parameters. Second, we fitted the model to (noninvasively) measured radial-carotid transfer functions in controls and in patients using a selected number of model parameters, and we studied the correlation between these model parameters and easily measured hemodynamic indexes. Because we were unable to assess any relation, we further studied the impact of using upper and lower limit transfer functions, enveloping measured data, on estimated central blood pressure and on derived indexes such as pulse pressure (PP) and the AIx.

## MATERIALS AND METHODS

### Transmission Line Model For Radial-Aortic/Carotid Pathway

The model simulating the aorta-radial pathway is analoguous to the model used by Karamanoglu and Fenely (9) for the finger carotid pathway and consists of a stepwise tapered transmission line with three segments (proximal, middle, and distal) terminated by a lumped parameter model (Fig. 1). For each segment, the transmission line equations describing the propagation of pressure (P) and flow (Q) harmonics waves apply
with *x* as the longitudinal coordinate and*f* and *b* indicating forward and backward waves, respectively. The wave propagation coefficient (γ), accounting for wave propagation and damping, is given as
with *F*
_{10} as the Bessel function as given by Womersley (24) and *c* as the wave propagation velocity given as
with *c*
_{0} as the inviscid Moens-Korteweg wave velocity. Wall viscosity is taken into account by the viscoelastic phase angle θ as a function of frequency (ω) (2,9, 10)
The relation between pressure and flow harmonics is given by the characteristic impedance (*Z*
_{0})
with *A* as the cross-section of the segments and blood density ρ = 1,050 kg/m^{3}. Values for vessel dimensions and θ_{0} are given in Table1. The impedance mismatch between the terminating windkessel model and the distal segment results in wave reflection. Instead of explicitly modeling the terminating lumped parameter model, the distal reflection coefficient (Γ = P_{b}/P_{f} = −Q_{b}/Q_{f}) is modeled as (9)
where Γ_{0} and τ are the modulus and phase of the distal reflection coefficient, respectively.

### Assessing Reference Model Parameters: Fitting the Model to the Transfer Function of Chen et al.

The modulus and phase angle data for the steady-state radial-aorta transfer function of Chen et al. (7) were digitized, and the complex form of the measured transfer function (TFF_{Chen}) was calculated. We then derived the transfer function for the transmission line model (TFF_{model}), which depends on the eight model parameters: length *L* and wave propagation speed *c*
_{0} for each of the three line segments, and two distal model parameters (Γ_{0} and τ). Reference values for these parameters were estimated by fitting TFF_{model} to TFF_{Chen}, hereby minimizing the (complex) sum of squared differences between TFF_{Chen} and TFF_{model} (Matlab, The Mathworks, Natic, MA). To reduce the number of parameters in the minimization procedure, *L* was set to fixed values between 10 and 30 cm for the proximal and between 15 and 45 cm for the middle and distal segments. For each set of segment lengths, the remaining five parameters were determined using a least square fitting algorithm. The goodness of fit, expressed as the modulus of the root mean square (RMS) value of the difference between TFF_{Chen} and TFF_{model}, was stored for all combinations of segment lengths. The final set of parameters, further considered as a reference, was the one yielding the lowest ‖RMS‖.

After assessing reference model parameters, we studied the effect of changes in characteristic impedance (half and twice the parameter values following from fitting Chen's function) of the proximal segment and of all segments together, changes in vascular tone , i.e, the distal wave reflection coefficient (Γ_{0} = 0 − 1), and the effect of a 20% variation in segment length.

### Fitting Measured Radial-Carotid Pressure Transfer Functions

#### In vivo measurements.

Applanation tonometry was performed successively at the carotid (P_{car}) and radial (P_{rad}) artery with a Millar SPT-301 penlike transducer (5) in 31 controls and 30 patients, in three different centers, and by three different operators. Patients were subjects with known coronary artery disease confirmed by wall motion abnormalities (rest and/or during dobutamine infusion) on a recent echocardiographic examination and/or a positive treadmill stress test. No patients with uncontrolled hypertension were included. The measured sequence of heartbeats was processed off-line. Using the measured electrocardiogram signal, we identified individual heart cycles, and an average pressure wave was calculated from at least five heartbeats. This signal was then calibrated by setting its mean and diastolic value equal to diastolic and mean brachial blood pressure measured with a cuff sphygmomanometer. Fourier analysis was applied on both pressure waves, and the ratio of corresponding harmonics yielded the measured transfer function (TFF_{meas}). A maximum of 10 harmonics were taken into account, and pressure or flow harmonics with a magnitude <1% of the first harmonic were excluded.

#### Fitting the model to the measured radial-carotid data.

To fit the model to the measured radial-carotid pressure transfer function, we used P_{car} as an input into the model, and the computed distal model pressure (P_{d,model}) was fitted to the measured radial artery pressure (P_{rad}) by minimization of Σ(P_{d,model} − P_{rad})^{2}. We first fitted the data by changing four parameters: the characteristic impedance of the three segments (*Z*
_{0,p},*Z*
_{0,m}, *Z*
_{0,d}) and the modulus of the distal reflection coefficient (Γ_{0}). The remaining four parameters were given their reference value as obtained from fitting TFF_{Chen}. Second, we further restricted the fitting to one single parameter: the characteristic impedance of the proximal segment (*Z*
_{0,p}), with the remaining seven parameters fixed at their reference value. For both fitting methods, we studied correlations between fitted model parameters and (sphygmomanometer) brachial blood pressure, subject age, heart rate, health condition (control subject or patient), and whether model parameters were different for controls and patients.

### Average TFF and Maximal and Minimal Enveloping Transfer Functions

We derived an average transfer function (TFF_{avg}) for the whole group of 31 controls and 30 patients. Measured transfer functions were resampled at 0.25-Hz intervals. Averaging of all 61 transfer functions yielded TFF_{avg}. Furthermore, two TFF were selected from all fitted transfer functions, representing an upper and lower enveloping curve. Because the low frequency harmonics are most important, we chose the transfer function with the highest (TFF_{max}) and lowest (TFF_{min}) amplitude at 2 Hz.

TFF_{max}, TFF_{min}, and TFF_{avg} were used to reconstruct the carotid artery pressure (P_{car,r}) from measured P_{rad}, with data from all three centers included. We then calculated pulse pressure (PP = systolic − diastolic pressure) and the AIx from P_{car,r} and P_{car}. AIx is calculated as ±(P_{sys} − P_{infl})/PP, with P_{sys} being systolic pressure and P_{infl} as the pressure corresponding to the first inflection point on the carotid pressure wave. AIx is positive when the inflection pressure precedes systolic pressure and indicates an A-type wave (18); it is negative when the inflection point occurs after systolic pressure (C-type wave). We compared PP and AIx derived from P_{car} and P_{car,r} with TFF_{min}, TFF_{avg}, and TFF_{max} using linear regression analysis. We further calculated RMS as
with *N* as the number of sampling points, indicating the accuracy with which carotid pressure morphology is reconstructed. Finally, we also compared average values for PP and AIx in the control and patient group. Groups were considered statistically different if*P* < 0.05 (*t*-test; SigmaStat 2.0, Jandel Scientific).

## RESULTS

### Fitting Chen's Radial-Aorta Pressure Transfer Function

The agreement between the fitted model TFF_{model} and TFF_{Chen} is shown in Fig. 2. Both modulus and phase angle were well predicted by the model. The derived model parameters for the different segments are given in Table1. The characteristic impedance for proximal, middle, and distal segment was 0.44, 1.33, and 4.08 mmHg · ml^{−1}· s^{−1}, respectively. The total length of the model was 66 cm. The distal reflection coefficient was practically constant (Γ_{0} = 0.86), independent of frequency (τ = 0.00).

With changing *Z*
_{0,p} (proximal segment properties) alone, transfer functions with very distinct morphologies were obtained (Fig. 3). A reduction of*Z*
_{0,p} increases the peak of the transfer function and shifts it to lower frequencies. Because of a lower wave velocity, the time delay between the proximal and distal location increases, and the phase angle becomes more negative. These effects are amplified with an overall change of *Z*
_{0}. With*Z*
_{0} divided by 2, the TFF peak is still shifted to lower frequencies, but there is hardly any amplification for frequencies <3 Hz, and the higher frequencies are damped out. Vascular tone changes are modeled as changes of the reflection coefficient. Γ, being real, does not change the TFF phase angle or the location of the peak. Only its magnitude increases with a stronger reflection (vasoconstriction). The overall length of the path was changed by ±20%. A longer path length dampens the peak, shifts it to lower frequencies, and increases the phase angle; the inverse is found for a shorter path.

### Fitting Measured Radial-Carotid Pressure Transfer Functions

Hemodynamic data for the complete population are given in Table2 as well as average values for*Z*
_{0,p}, *Z*
_{0,m},*Z*
_{0,d}, and Γ_{0}, obtained from fitting the model to measured radial-carotid transfer functions. Overall, there was good agreement between measured and fitted radial artery pressure (RMS = 3.3 ± 2.0 mmHg), but the fitting was slightly better for the control than for the patient group (RMS = 2.8 ± 1.4 vs. 3.9 ± 2.5 mmHg; *P* < 0.05).*Z*
_{0,p} and *Z*
_{0,m} were higher in the patient group, whereas *Z*
_{0,d} was higher in the control group, but differences were not significant. Γ_{0} was higher in the patient group (0.85 ± 0.21 vs. 0.71 ± 0.25; *P* < 0.05). There was no correlation between any of the fitted model parameters and subject age, sphygmomanometer blood pressure (systolic, diastolic, or pulse pressure), heart rate, or the subject being classified as a patient or control.

With the model fitted to the measurements using only*Z*
_{0,p}, the fitting is again better in controls (RMS = 4.6 ± 2.2 vs. 5.4 ± 3.3 mmHg).*Z*
_{0,p} is higher in patients than in controls, but the difference is not statistically significant (1.10 ± 0.64 vs. 1.22 ± 0.60 mmHg · ml^{−1} · s^{−1}). Again, there was no correlation between*Z*
_{0,p} and subject age, sphygmomanometer blood pressure (systolic, diastolic, or pulse pressure), heart rate, or the subject being classified as a patient or control.

### Average TFF and Maximal and Minimal Enveloping Transfer Functions

The calculated point-to-point TFF_{avg} is shown in Fig.4, together with TFF_{max} and TFF_{min}. The error bars indicate means ± 2 SE. TFF_{max} is the transfer function with the highest amplitude at 2 Hz (2.0) and is calculated using 0.22, 1.15, and 5.34 mmHg · ml^{−1} · s^{−1} for*Z*
_{0,p}, *Z*
_{0,m}, and*Z*
_{0,d}, respectively, and 0.95 for Γ_{0}. All other parameters have reference values (Table 1). TFF_{min} is the transfer function with the lowest amplitude at 2 Hz (0.9) and is calculated with 1.12, 2.39, and 4.32 mmHg · ml^{−1} · s^{−1} for*Z*
_{0,p}, *Z*
_{0,m}, and*Z*
_{0,d}, respectively, and 0.51 for Γ_{0}.

We reconstructed carotid pressure from measured radial artery pressure using TFF_{max}, TFF_{min}, and TFF_{avg}, respectively. For the whole data set, the average difference between reconstructed and measured carotid pressure, expressed as the RMS difference, was 7.8 ± 2.7, 6.9 ± 3.4, and 4.3 ± 2.3 mmHg, respectively. The correlations between PP and the AIx, derived from the reconstructed and measured carotid pressure, are given in Fig. 5. Best results are found with TFF_{avg}, yielding an average of 41.5 ± 11.8 vs. 41.1 ± 10.0 mmHg measured. A paired *t*-test indicated that both values were not significantly different (*P* = 0.59). Correlations were less good for AIx. TFF_{avg} gave the best results, with an average value of −0.02 ± 0.19 vs. 0.01 ± 0.19 mmHg measured directly on the carotid artery pressure wave. However, values were not significantly different (*P* = 0.22; paired *t*-test). PP and AIx, calculated from measured or reconstructed carotid artery pressure, were finally grouped for patients and controls (Fig.6). PP and AIx were higher in patients than in controls, irrespective the calculation on measured or reconstructed pressures.

## DISCUSSION

This study is based on a transmission line model for the aorta-radial pathway. The model is similar to the finger-carotid model of Karamanoglu and Fenely (9), consisting of three tube segments with the same values for tube diameters and wall viscosity. We first assessed reference model parameters by fitting the model directly to a recently invasively assessed transfer function (7,8). The model parameters (Table 1) are within physiological ranges. The total length of the transmission line is 66 cm and is close to the average length of the arm (radial to sternum) that we measured in 13 subjects (66 ± 6 cm). Pulse-wave velocity varies between 6.5 and 8.5 m/s and is within reported ranges (9, 17). A good fit to Chen's data is obtained with a real terminal reflection coefficient, in contrast to Karamanoglu and Fenely (9). The magnitude (0.86) is higher than what Karamanoglu and Fenely (9) found at the finger level (0.64) but is close to the value of 0.8 reported for the femoral artery (15).

It was recently shown that a simple single tube with a single distal reflection site can be used to model the transfer function for “single tubelike” aortic branches such as the carotid artery and to some extent the brachial artery (21). However, such a model is too simple for the radial-aorta pathway, consisting of multiple reflection sites, and it was impossible to fit such a model to Chen's data. Using three tubes in series, three reflection sites are generated: two sites due to the impedance mismatch between the segments and a distal reflection site at the model termination. Varying the model parameters leads to logical and expected changes in the morphology of the transfer function as shown in Fig. 3 and as summarized in results.

The model was used to fit the measured radial-carotid transfer function in 61 subjects. The fitting was first limited to*Z*
_{0,p}, *Z*
_{0,m},*Z*
_{0,d}, and Γ_{0}. Average model parameters in the control and patient group are consistent with reported physiopathological changes in arterial mechanical properties. In hypertension, for instance, the stiffness of the larger elastic arteries is increased, but the effects are far less clear for more peripheral, muscular arteries such as the radial or femoral artery (13, 20, 22). Nevertheless, individually, there was no correlation between model parameters and subject characteristics such as age, blood pressure, heart rate, or subject classification as a control or as a patient.

A whole family of TFF curves can be obtained only by changing the characteristic impedance of the proximal segment (Fig. 3). It is further expected that changes in arterial mechanical properties are most pronounced for the larger elastic vessels close to the heart (13, 20, 22). Therefore, we studied whether the fitting could be restricted to*Z*
_{0,p} and whether this improved the correlation between the fitted model parameter and the patient data. Fittings were less good than with the four parameters but still acceptable (overall mean RMS of 5.0 ± 2.8 mmHg). The resulting*Z*
_{0,p} were comparable to the values that were found when fitting four parameters (Table 2), with the same tendency for an increased *Z*
_{0,p} in the patient group. However, again, there was no correlation between*Z*
_{0,p} and subject characteristics.

*Z*
_{0,p} was not different between patients and controls (Table 2). A possible explanation is the limited number of subjects included in the study. Also, blood pressure is only slightly higher in the patient group, and the difference in arterial compliance, most prominent in the ascending aorta and aortic arch, may not be high enough to yield significant differences in *Z*
_{0,p}.

The absence of a (simple) relation between model parameters and patient data inhibited the computation of individualized transfer functions. This may be due to the simplicity of the model (and the fitting procedure), with the model parameters not reflecting physiological properties. We also applied a model with reference parameter values derived for the radial-aorta transfer function on measured radial-carotid transfer functions. Still our approach is defendable. The radial-aorta and radial-carotid pathway should at least share the properties of the middle and distal segment; the differences exist in the small part of the ascending aorta and the carotid artery. We assumed the same length for these segments, but their mechanical properties are effectively different: average*Z*
_{0,p} for the radial-carotid data (4 parameter fit) is 1.08 mmHg · ml^{−1} · s^{−1}and is higher than *Z*
_{0,p} that was found for the radial-aorta fit (0.44 mmHg · ml^{−1} · s^{−1}). This agrees with the smaller caliber of the carotid artery. One might think of introducing arm length as a known variable into the model. This will change the values of estimated*Z*
_{0,p}, but it is unlikely that the introduction of a parameter (radial-carotid path length), which cannot be measured with great accuracy either, would change our observations. One should also consider the fact that not only the total path length but also the relative length of the three segments determines TFF.

TFF_{avg} yielded the best approximation of PP and AIx, whereas TFF_{max} or TFF_{min} gave (very) different values for PP and AIx than the values derived from the directly measured pressure wave. However, it is remarkable that when averaged over control subjects and patients, PP and AIx were different for both groups, irrespective the used transfer function. Thus, when (generalized) pressure transfer functions are used in clinical studies, comparing control subjects with patient groups or comparing different interventions, they may potentially yield reconstructed central (aorta) pressures and derived indexes, discriminating between groups or interventions. Nevertheless, individually, there are considerable differences for PP and AIx derived from the measured or reconstructed carotid artery pressure, as indicated by the (modified) Bland-Altman plot (3) in Fig. 7 (data shown for TFF_{avg} only). Furthermore, we feel that derived parameters, such as the AIx, require reconstructed waves in which enough high-frequency information is present to fully capture the details of the curve. Using other algorithms, based on autoregression models (8), may give better results than the frequency-domain approach we used.

This study shows that arterial pressure waves, reconstructed using (generalized) transfer functions, carry analog information as the carotid blood pressure. This, however, does not imply that these reconstructed pressures are a better approximation of central aortic pressure than carotid blood pressure. However, in clinical settings, it is easier to monitor central pressure from an easily accessible peripheral site such as the radial artery, which is more appropriate for applanation tonometry than the carotid artery. Applanation tonometry at the carotid artery is more tedious, more time consuming, and requires skilled people and averaging of the curves. Furthermore, it is better to avoid carotid applanation tonometry in people with carotid plaque. In our study, we had to exclude about 10% of patients because no reliable carotid tonometric signals could be recorded.

We conclude that using averaged, generalized TFF leads to significant discrepancies between reconstructed and measured pressure waveforms and subsequent derived indexes. Nevertheless, very distinct transfer functions (TFF_{max}, TFF_{avg}, and TFF_{min}) seem to preserve the information in the pressure wave that is needed for the computation of indexes as PP and the AIx, discriminating patients from healthy subjects.

## Acknowledgments

This research is funded by a specialization grant of the Flemish Institute for the Promotion of the Scientific-Technological Research in Industry (IWT 943065–IWT 960250) and a concerted action program of the University of Gent, supported by the Flemish government (GOA-95003). S. Carlier was the recipient of a grant from the North Atlantic Treaty Organization and from the Belgian American Educational Foundation.

## Footnotes

Address for reprint requests and other correspondence: P. Segers, Hydraulics Laboratory, Institute Biomedical Technology, Univ. of Gent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium (E-mail:patrick.segers{at}navier.rug.ac.be).

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