Abstract
We propose a new method to derive aortic pressure from peripheral pressure and velocity by using a time domain approach. Peripheral pressure is separated into its forward and backward components, and these components are then shifted with a delay time, which is the ratio of wave speed and distance, and added again to reconstruct aortic pressure. We tested the method on a distributed model of the human systemic arterial tree. From carotid and brachial artery pressure and velocity, aortic systolic and diastolic pressure could be predicted within 0.3 and 0.1 mmHg and 0.4 and 1.0 mmHg, respectively. The central aortic pressure wave shape was also predicted accurately from carotid and brachial pressure and velocity (root mean square error: 1.07 and 1.56 mmHg, respectively). The pressure transfer function depends on the reflection coefficient at the site of peripheral measurement and the delay time. A 50% decrease in arterial compliance had a considerable effect on reconstructed pressure when the control transfer function was used. A 70% decrease in arm resistance did not affect the reconstructed pressure. The transfer function thus depends on wave speed but has little dependence on vasoactive state. We conclude that central aortic pressure and the transfer function can be derived from peripheral pressure and velocity.
 forward and backward waves
 wave propagation
 reflection coefficient
 carotid and brachial arteries
knowledge about the magnitude and shape of the central aortic pressure wave is of importance in several aspects. Central aortic pressure determines the systolic load on the heart through its relationship with wall stress and, to a large extent, determines coronary perfusion in diastole. The augmentation of the aortic pressure wave (12, 13) appears to be age dependent and plays a role in left ventricular and carotid anatomy (10, 14, 17). The diastolic aortic pressure wave is often used in the derivation of total arterial compliance through several methods [for review, see Stergiopulos et al. (15)]. The aortic pressure wave allows for the derivation of aortic flow (18).
The central aortic pressure waveform, however, cannot be obtained noninvasively. Therefore, a number of research groups have recently tried to obtain aortic pressure from (noninvasive) measurement of peripheral pressures, such as carotid artery (4), brachial artery (6, 7), and radial artery pressure (3). Others have determined the relationship between finger pressure and brachial pressure (1, 5). The methods are based on the determination of a pressure transfer function between the peripheral and central aortic locations. With the use of invasive techniques (3, 4, 7) or models (6), the transfer function, averaged over a group of patients, is derived and then used to predict central aortic pressure from peripheral pressure in individual patients.
The transfer function of the vasculature of the arm may be affected by age and disease (2), and the vasculature of the carotids, although only correcting over a short distance, may also not be constant (4). Thus differences between individual patients and the averaged pressure transfer function may exist, and the use of a generalized “average” transfer function may result in errors in the prediction of central aortic pressure (3).
The goal of the present study was to develop a method, based on the separation of peripheral waves into their forward and backward components, to derive the central aortic pressure from noninvasively determined peripheral pressure and flow velocity. In contrast to previous transfer function methods, this new time domain method can be applied on a perpatient basis. On the basis of this wave separation and reconstruction method, we also derived the major parameters that determine the transfer function and thus gave it a physical basis. Finally, we studied how pulse wave velocity and vasodilation affect the reconstructed aortic pressure.
MATERIALS AND METHODS
Principle.
The principle of the wave separation and reconstruction method is based on the breaking up of the peripheral pressure into its forward and backward running components and then shifting these waves in time according to their travel times from and to the central aorta. The forward pressure is delayed with respect to the central aortic pressure and needs to be shifted “backward in time.” The backward pressure is running toward the central aorta and thus needs to be advanced in time. This shift in time is determined by the distance between the peripheral measuring site and the central aorta and by the wave speed. The shifted waves are then added to reconstruct the central pressure.
To obtain the forward and backward pressure (and flow velocity) waves, we made use of the method first proposed by Westerhof et al. (21) and later presented in a simpler time domain form by Murgo et al. (12). We used velocity, and not volume flow, in the present study, because noninvasive determination of crosssectional averaged velocity is easier than measurement of volume flow. The same method, however, applies for volume flow. Peripheral pressure (P_{p}) and flow velocity (V
_{p}) form the basis of the separation into forward and backward pressures as follows (12)
The artery is assumed to be uniform and sufficiently large in diameter so that it can be considered “lossless” (no frictional losses due to blood viscosity and wall viscosity). These assumptions imply that the characteristic impedance is a real number so that separation is allowed in the time domain. We also assume the vessel to be linearly elastic. None of these assumptions is true for the arterial system, and they will be evaluated in discussion.
The principle of the wave separation and reconstruction method is shown schematically in Fig. 1. We consider waves propagating in a single tube with convective acceleration and fluid friction to represent the human common carotid artery under control conditions. The crosssectional area of the tube is 0.93 cm^{2} at the proximal end and 0.43 cm^{2} at the distal end. Length is 21 cm and wave speed is 4.62 m/s for the entire segment. The Womersley parameter (α) is 5.1 at the heart rate (1 Hz). In Fig. 1, the pressure at the entrance is given at top left, and the pressure and velocity at the distal end of the tube are given at top right. With the use of Eqs. 1 and 2 , the forward and backward components were calculated. Characteristic impedance at the distal end can be obtained from the pressurevelocity plot in early systole (9) or from Fourier analysis of pressure and velocity, calculation of input impedance, and averaging of the impedance moduli at high frequencies (12). The result of the separation is described in detail by Westerhof et al. (21) and Murgo et al. (12) and is shown in Fig. 1(bottom right). Subsequently, these waves are shifted in time as shown in Fig. 1 (bottom left). The time shift is the distance over the wave speed, which is the true phase velocity, given that we deal with reflectionless waves. Addition of the shifted forward and backward waves yields the reconstructed pressure wave at the proximal end of the tube. The reconstructed pressure is compared with the true proximal pressure wave in Fig. 1 (top left). The same procedure can be applied to the velocity signal to obtain the velocity at the entrance of the tube.
Transfer function.
The present method can be formulated in terms of a pressure transfer function in the frequency (ω) domain between distal and proximal location. The calculation of the forward and backward waves and their shifts in time were carried out as described in the previous paragraph. The resulting transfer function, T, reads
The transfer function in the frequency domain was calculated by using standard Fourier analysis of the two pressures and subsequently calculating, per harmonic pair, the ratios of the moduli and the differences of the phase angles (1, 3, 7).
Arterial model.
A distributed computer model of the systemic arterial circulation was used for all simulations. The systemic arterial tree is modeled by 55 arterial segments, accounting for all major arteries (16). Each terminal arterial segment is loaded with a peripheral impedance represented by a threeelement windkessel (20). The mathematical model is based on the onedimensional flow equations and accounts for the nonlinearities due to convective effects and elastic properties of the arterial wall. A detailed description of the computer model, the governing equations, and the physiological parameters defining the geometric and elastic properties, as well as the numerical solution scheme, can be found in Ref. 16. The model is an improvement of the model earlier published by Westerhof et al. (19). Pressures and velocities were sampled at 1,000 Hz for further analysis.
The effect of a reduction of vascular compliance in all arteries was studied. Aortic pressure was reconstructed from brachial pressure and velocity by using both the wave separation and reconstruction method and the transfer function obtained under control conditions. The effect of vasodilation of the vascular bed of the arm (reduction of peripheral resistance to 30% of control) on the pressure was also studied. With this approach we tested two things. First, we investigated whether the transfer function determined under control conditions could be applied in the presence of such vascular changes. Second, we tested whether the wave separation and reconstruction method applies under different vascular conditions.
Data analysis.
The reconstructed pressure and aortic pressure were compared in terms of systolic and diastolic values and in terms of wave shape. Differences in systolic and diastolic pressure were given in millimeters of Hg. Wave shapes were compared according to their rootmeansquare errors (RMSE)
RESULTS
Figure 2 shows the peripheral pressure and velocity in the common carotid artery of the distributed model directly, together with the reconstructed central aortic pressure (at the entrance of the left common carotid artery) obtained by the shift in time of the forward and backward waves. Figure 2 also shows the transfer function determined from aortic and carotid pressure as well as the theoretical transfer function according to Eq.4 with the use of Eq.5 as reported by Westerhof (21). In Fig.3 the same data are shown for brachial pressure and velocity and for reconstructed central aortic pressure at the entrance of the left subclavian artery. In the two examples the reconstructed pressure is, in terms of both systolic and diastolic values and overall shape (RMSE), close to the central aortic pressure. When reconstructed from the carotid signals, aortic systolic and diastolic pressure deviated <0.1 mmHg. The RMSE was 1.07 mmHg. When reconstructed from the brachial signals, aortic systolic pressure was underestimated by 0.4 mmHg, whereas diastolic pressure was overestimated by 1 mmHg. The RMSE was 1.56 mmHg. The transfer functions calculated directly from central and peripheral pressures agree well with the theoretical transfer function (Eq.4 ) for the low frequencies (Figs. 2 and 3). For frequencies >3 Hz, the direct transfer function shows more scatter than the theoretically derived transfer functions, and the moduli are somewhat higher and the phases more negative.
When vascular compliance is decreased to 50% of control, and thus wave speed is increased by 41%, the wave separation and reconstruction method again predicted central pressure accurately, as shown in Fig.4 A. The aortic pressure predicted using the control transfer function is now, however, much less accurate (Fig. 4 A), showing that wave speed or travel time is an important parameter.
When the peripheral resistance of the arm is reduced to 30% of control, the aortic pressure predicted by means of the wave separation and reconstruction method is accurate (Fig.4 B). The aortic pressure was also well predicted using the transfer function under control conditions, showing that the transfer function from brachial artery to aorta is rather insensitive to changes in peripheral resistance.
We have tested the sensitivity of the reconstructed pressure wave to changes in delay time and characteristic impedance. In Fig.5 we show the effects of changes in the delay time on systolic and diastolic pressure and on the wave shape, expressed as RMSE. We see that a 20% increase in the delay time (from 40 to 48 ms) results in an error in systolic pressure of 3.2 mmHg and in diastolic pressure of 0.5 mmHg. The RMSE increased from 1.07 to 1.79 mmHg. Characteristic impedance obtained from the early systolic rise in pressure and velocity or from the averaged values of the input impedance at high harmonics gave values that differed <5%, a finding in line with the results published by Li (9). When the carotid characteristic impedance was increased by 20%, the reconstructed wave had a systolic value of 122.3 mmHg and a diastolic value of 81.9 mmHg compared with 123.8 and 81.9 mmHg, respectively, when the control characteristic impedance was used. The RMSE was increased from 1.07 to 1.59 mmHg with this 20% increase in characteristic impedance. We thus conclude that reconstructed pressure is accurate and that the deviations in the predicted central aortic pressure are more sensitive to errors in delay time than to errors in characteristic impedance.
DISCUSSION
We derived a time domainbased method to reconstruct central aortic pressure from peripheral pressure and velocity. We have tested the method using carotid and brachial pressure and velocity from a distributed model of the human arterial tree. The results are promising (Figs. 24). We were able to show that the reconstructed aortic pressure depends little on the vasoactive state. The delay time turns out to be an important parameter, and changes in arterial compliance, due to atherosclerosis, age, or, indirectly, through mean pressure, may affect the transfer function. The method should be easily applicable to patients because pressure (e.g., tonometry) and velocity (e.g., ultrasound) can be obtained noninvasively. The time delay may be obtained from simultaneous measurement of an electrocardiogram or heart sounds.
The theoretical transfer function given in Eq.4 and the time shift applied to the forward and backward waves are equivalent operations. The first is the frequency domain approach, and the second is the equivalent time domain description.
The pressure transfer function was calculated directly from pressures in peripheral vessels and the aorta, and the theoretical transfer functions (Eq. 4 ) were similar at low frequencies but different at frequencies above the third harmonic. At high frequencies considerable scatter is observed, particularly in the directly determined transfer function. This is due to the small amplitude of the higher pressure harmonics. However, the prediction of aortic pressure using both transfer functions was virtually the same (Figs. 2 and 3). This suggests that accurate knowledge about the transfer function at higher frequencies is not essential for the prediction of systolic and diastolic aortic pressure. This may explain the reported overall good predictions of aortic pressure from an average transfer function (3, 7).
Assumptions and limitations.
The method uses a simplified approach to wave travel with the assumption of a single frictionless, uniform tube with linearly elastic wall properties between peripheral site and aorta. These assumptions make the calculations very simple and straightforward. We tested the method against data obtained from an extensive model of the systemic arterial tree that has tapered arterial segments, with convective acceleration and losses due to fluid friction. The errors using this simplified approach were small (Figs. 13). The method is derived by assuming transmission in a single uniform tube between the peripheral site of measurement and the central aorta. The two examples derived in this study appear to satisfy this assumption. If major branches are present in between the two sites (e.g., radial artery and aorta or femoral artery and aorta), undesirable errors may result.
For small arteries with linear wall properties in which losses play a role, the calculations should be based on separation into forward and backward waves, wave travel, and attenuation calculated per harmonic (21). When we made the calculations per harmonic and then reconstructed the waves from the harmonics, the reconstructed pressures were somewhat better than those shown in Figs. 2 and 3. However, these calculations require sophisticated programming and good steadystate signals to permit Fourier analysis and are only allowed when the system is linear. In the vessels we studied, we found the errors using these simplifying approximations to be so small that the time domain approach seems appropriate.
When the system is nonlinear a transfer function cannot be calculated, because it is not possible to relate the harmonics of pressure and velocity. However, the nonlinear properties of the arterial tree may not play a major role in this respect. We tested the effect of nonlinearity by introducing nonlinear wall properties according to the data presented by Langewouters et al. (8). Details of the nonlinear model can be found in our earlier work (15). The reconstructed pressure in comparison with aortic pressure in the nonlinear model is shown in Fig. 6. We conclude that the introduction of quantitatively realistic nonlinear wall properties does not introduce large errors. This finding implies that the calculation of the transfer function from Fourier analysis of distal and central pressure is an acceptable approach that does not lead to large errors.
Determinants of transfer function.
The major parameters in the calculations are the characteristic impedance and the delay or shift time. The characteristic impedance can be obtained from pressure and velocity in the time domain or via Fourier analysis. The slope of the relationship between pressure and velocity during the early rise in systole (9) often appears to be an acceptable estimate of characteristic impedance. However, we sometimes found this approximation unsatisfactory compared with the actual characteristic impedance that can be derived from local area and wall properties. The use of Fourier analysis of pressure and velocity and calculation of impedance makes it possible to obtain characteristic impedance from the averaged value of input impedance at high frequencies (11). However, when the impedance modulus oscillates strongly at high frequencies, a good estimation of characteristic impedance may be difficult. The use of flow velocity instead of volume flow implies that characteristic impedance equals blood density multiplied by pulse wave velocity. If pulse wave velocity can be determined accurately over a short length in the distal artery, it may present a good alternative to the other estimates of characteristic impedance. We have shown that a 20% change in characteristic impedance hardly affected the reconstructed diastolic pressure but decreased reconstructed systolic pressure by 1.5 mmHg. The quality of the reconstructed pressure was still good. We conclude that very accurate determination of characteristic impedance is not essential.
The delay time depends on the wave speed, which in turn depends on pressure due to the nonlinear elastic wall properties. We found the delay time to be an important factor in the analysis. We used a sampling rate of 1,000 Hz, and an error of a few milliseconds in the time shift resulted in much poorer reconstructed pressures. The delay time can be estimated using an electrocardiogram or heart sounds on one upper limb and the foot of the pressure or flow velocity wave on the other.
In the case of altered arterial compliance (Fig. 4) and use of the corresponding brachial pressure and velocity, the wave separation and reconstruction method yielded an accurate central pressure. This suggests that the wave separation and reconstruction method can be applied under altered vascular conditions. The use of the transfer function determined under control conditions led to a poorly predicted central pressure. This shows that a single averaged transfer function may not be used under all circumstances. This finding may explain the findings of Chen et al. (3) that in some patients, during special maneuvers, reconstructed pressure deviated from measured pressure. Vasodilation did not appear to affect the reconstructed pressure much (Fig. 4). This may suggest that an averaged transfer function can be used during changes in vasomotor tone.
Karamanoglu et al. (6) used a detailed model of the human upper limb to investigate the effects of arterial parameters on the transfer function between radial artery and the aorta. A change in the reflection coefficient had a considerable effect on the transfer function, as one would expect from Eq. 4 . We found a rather small effect on the reconstructed pressure after a strong vasodilation of the arm. These findings are not contradictory when one realizes that changes in peripheral resistance have only a minor effect on the reflection coefficient. Using our model, we found that for a decrease in arm peripheral resistance to 30% of control, the modulus of the first two harmonics of the reflection coefficient at the brachial artery decreased from 0.88 to 0.72 and from 0.78 to 0.59, respectively.
We have tested the method on the basis of a model. This model may be quantitatively somewhat different from the human systemic arterial tree. Now that we know that the proposed method works, the next step should be the application of the method in the human. To the best of our knowledge, simultaneous peripheral and velocity data, together with central aortic pressure measurements necessary to test the method, are not available.
In conclusion, we have developed a method that permits accurate reconstruction of central pressure from peripheral pressure and velocity. Because the peripheral measurements are noninvasive, all determinations can be performed per patient, thereby avoiding the use of a generalized transfer function.
Footnotes

Address for reprint requests: N. Stergiopulos, Biomedical Engineering Laboratory, Swiss Federal Inst. of Technology, PSEEcublens, 1015 AZ Lausanne, Switzerland.
 Copyright © 1998 the American Physiological Society