## Abstract

We investigated the quantitative contribution of all local conduit arterial, blood, and distal load properties to the pressure transfer function from brachial artery to aorta. The model was based on anatomical data, Young's modulus, wall viscosity, blood viscosity, and blood density. A three-element windkessel represented the distal arterial tree. Sensitivity analysis was performed in terms of frequency and magnitude of the peak of the transfer function and in terms of systolic, diastolic, and pulse pressure in the aorta. The root mean square error (RMSE) described the accuracy in wave-shape prediction. The percent change of these variables for a 25% alteration of each of the model parameters was calculated. Vessel length and diameter are found to be the most important parameters determining pressure transfer. Systolic and diastolic pressure changed <3% and RMSE <1.8 mmHg for a 25% change in vessel length and diameter. To investigate how arterial tapering influences the pressure transfer, a single uniform lossless tube was modeled. This simplification introduced only small errors in systolic and diastolic pressures (1% and 0%, respectively), and wave shape was less well described (RMSE, ∼2.1 mmHg). Local (arm) vasodilation affects the transfer function little, because it has limited effect on the reflection coefficient. Since vessel length and diameter translate into travel time, this parameter can describe the transfer accurately. We suggest that with a, preferably, noninvasively measured travel time, an accurate individualized description of pressure transfer can be obtained.

- blood pressure
- transfer function
- personalization
- brachial artery
- aorta

aortic pressure appears a better indicator of cardiovascular morbidity and mortality than peripheral, e.g., brachial or radial, pressure (5, 15, 16, 37, 38). Also, ascending aortic pressure defines the systolic load on the heart through ventricular wall stress. Effects of therapy are preferably studied using aortic pressure (19). However, whereas peripheral pressures can be obtained noninvasively, measurement of aortic pressure requires invasive techniques. Transfer functions that calculate aortic pressure from peripheral pressure circumvent this invasive measurement and provide an opportunity to noninvasively obtain this cardiovascular information.

Several approaches have been taken to arrive at transfer functions. Chen et al. (4) and Fetics et al. (7) used a special mathematical transformation to obtain a transfer function from human data, and the averaged transfer function of a group of patients was used as “standard” transfer. Karamanoglu et al. (18) used a segmented model of the arterial tree to derive the transfer function. Gizdulich et al. (9, 10) proposed a method to obtain brachial pressure from finger pressure by fitting a second order filter to averaged data. Stergiopulos et al. (27) showed that splitting the brachial pressure in its backward and forward waves and by shifting these waves with respect to each other over the travel time between aorta and brachial artery, an accurate wave transformation can be obtained. For this method, however, pressure and flow or flow velocity measurements are needed. Here we intend to investigate all determinants of pressure-to-pressure transfer.

Another topic in this field that has received much attention is the calibration and level correction of noninvasively obtained pressures (2, 10, 11, 30, 31). Thus there are several approaches to describe pressure transfer from periphery to aorta.

Although several groups are successful in using generalized transfer functions (8), others have criticized this approach (14). However, newer publications from the same group (12, 13) superseded this paper and suggest that their findings may have been influenced by methodological issues, as was proposed by O'Rourke and Nichols (25) in reaction to the earlier article (14). We therefore conclude that whether or not a generalized transfer function can be used is not resolved.

To acquire more insight into the determinants of pressure transfer, we investigated the quantitative contribution of all conduit arterial properties, blood density, and viscosity, as well as the properties of distal arterial bed to the transfer function and to the reconstructed aortic pressure in terms of systolic and diastolic values and pulse pressure. This information allows us to determine the main contributing factors and to arrive at a description of the transfer function based on the major contributing parameters only.

## METHODS

For evaluation of the effect of arterial parameter changes on the transfer, we use models between brachiocephalic and brachial artery.

### Tube Model Based on Anatomy

We constructed a model, representing the human subclavian, axillary, and brachial artery, based on Womersley's theory for an artery under stiff longitudinal constraint and including viscous fluid damping (39). The wall is taken to be linear and viscoelastic, wall viscosity is modeled with a second order polynomial, and the constants are taken from Westerhof and Noordergraaf (35). The tube system has a length of 42 cm and is tapering, according to actual anatomical data, from a diameter of 8.1 mm proximal to 4.7 mm distal (33). This tube system is divided into seven segments, with length, radius, and wall thickness given in Table 1. Wave speed and damping [wave propagation coefficient γ and characteristic impedance (*Z*_{c})] follow directly from longitudinal impedance based on Womersley's theory and from transverse impedance that is based on the (viscoelastic) wall properties.

The distal part of the arm is modeled as a three-element windkessel (24, 34) as shown in Fig. 1*A*. The windkessel parameters are peripheral resistance of the underarm (*R*_{p} = 56.3 10^{3} g·cm^{−4}·s^{−1} or 42.3 mmHg·ml^{−1}·s), arterial compliance representing the combined elasticity of the large vessels of the underarm (C_{w} = 7.26 10^{−6}·cm^{4}·s^{2}·g^{−1} or 9.66 ml/mmHg) (28, 33), and characteristic impedance, which is taken equal to the characteristic impedance of the distal part of the brachial artery (*Z*_{c} = 6.2 10^{3} g·cm^{−4}·s^{−1} or 4.77 mmHg·ml^{−1}·s). Mathematica (version 4.0, Wolfram Research, Champaign, IL) is used for the analysis. The model is linear, allowing Fourier analysis and treatment per frequency.

First the windkessel impedance is calculated as a function of frequency. Characteristic impedance of the last, most distal segment of the tube system is then calculated. The local reflection coefficient Γ is then derived as (1) Pressure and flow transfer from end to entrance of this segment, length *z*, are calculated from the wave propagation coefficient γ and the reflection coefficient Γ. (2a) (2b) The impedance transfer is as follows: (3a) (3b) Thus the input impedance at the entrance of the last segment can now be calculated.

These calculations are repeated for all segments from distal to proximal end. The pressure transfer, P_{entrance}/P_{end}, is found by the multiplication of the segmental transfers. Because the resistance to mean flow in the tube system is negligible with respect to peripheral resistance (<1%), the transfer function for mean pressure (0 Hz) equals 1, i.e., mean pressures at the entrance and end are equal.

With the use of this model, the effects of changes in conduit artery, blood, and load (windkessel) parameters on the transfer function and blood pressure were calculated. The tube parameters were as follows: (segment) length, radius, wall thickness, Young's modulus of the arterial wall, vessel wall viscosity, blood density, blood viscosity, and the windkessel parameters. The changes in the magnitude of the first peak and in the frequency at which the first peak of the transfer function occurs were determined. These two variables were compared with those in the reference condition.

We used a brachial pressure based on the model by Stergiopulos et al. (28). With our standard transfer function we calculated our reference aortic pressure. The reconstructed aortic pressure obtained by parameter changes was compared with this reference aortic pressure. Comparisons were done in terms of output variables systolic, diastolic, and pulse pressure. Also, the root mean square error (4) (where *n* is the number of data points) between reconstructed and reference aortic pressures was calculated to quantify the error in the reconstructed aortic wave shape.

The magnitude and frequency of occurrence of the first peak of the transfer function and systolic, diastolic, and pulse pressures together with RMSE were called output variables. The sensitivity of these variables to the arterial and blood parameters was calculated in terms of percent error: the percent change in a variable for a change in a parameter. All parameters were increased and decreased by 25%, which is an arbitrary choice. For some parameters it may be too large; for others, it may be too small. However, for easy comparison, we have decided for a single percent change for all parameters.

### Different Pressure Wave Shapes

To investigate the effect of wave shape, two dissimilar aortic pressures, i.e., the type A and type C beat from Murgo et al. (23) were used (note: the participants gave informed consent, and the Institutional Review Board approved the study). The corresponding brachial pressure was calculated by using the control transfer function. Subsequently, these brachial pressures were transformed back with a transfer function of which we varied the segment length by 25%, because this parameter has the largest effect. Systolic, diastolic, and pulse pressure and RMSE were calculated with respect to the reference aortic pressure.

### Changes in Combined Parameters

#### Age.

We have modeled increasing age by a 25% increase in Young's modulus of the conduit arteries, a 25% increase in peripheral resistance, and a 25% decrease in windkessel compliance. Opposite changes were also studied.

#### Body size.

A large and a small person were modeled by changing all conduit artery lengths, diameters, and wall thicknesses by 25%. Windkessel parameters were adjusted accordingly for size. For the *R*_{p}, we used the following reasoning: linear measures like length and radius are proportional to M^{0.33} in which M is body mass. Cardiac output is proportional to M^{0.75} (36) so that resistance is proportional to M^{−0.75}. Thus, a 1.25 increase in length corresponds to a mass increase of 1.25^{3} and resistance decrease (1.25^{3})^{−0.75}, resulting in a factor 0.6. For the windkessel compliance, C_{w}, we assumed proportionality to body mass: when length increases 1.25, body mass and also C increases by 1.25^{3} or a factor 1.95 (36). Similar reasoning was followed to model a smaller person.

#### Increased blood viscosity.

Increased viscosity implies increased Poiseuille resistance, and, therefore, we studied a combined effect of changes in viscosity and in peripheral resistance: both were changed by 25%.

### Reflection Coefficient as a Function of Vasoactive State

The influence of the vasoactive state of the peripheral load on the reflection index was calculated by *Eq. 1* for changes in peripheral resistance by a factor 4.

### Uniform Tube Model

To investigate whether a single uniform lossless tube would be sufficient to describe the transfer function (Fig. 1*B*), the geometrically correct system was replaced with a uniform tube with the same length of 420 mm, a radius of 3.5 mm, and a wall thickness of 0.65 mm and loaded with the reference windkessel, such that the *Z*_{c} matched to the *Z*_{c} of the tube. This tube model was further simplified by neglecting viscous losses of blood and arterial wall (lossless tube).

## RESULTS

### Tube Model Based on Anatomy

The dimensions of the tube model according to anatomy are shown in Fig. 1. The reference transfer function and the reference aortic and brachial pressures are shown in Fig. 2. This brachial artery pressure is used to calculate the aortic pressure when the tube and load parameters are changed. Values of the conduit artery, blood and windkessel parameters in the reference situation, and the reference output variables are listed in Table 2. Parameters were increased and decreased by 25%. The changes of the characteristic points of the transfer function (frequency and maximum value of the first peak) and of the reconstructed aortic pressure (systolic, diastolic, and pulse pressure) are given in the Table 2 as percentages. Because of the smaller value of the pulse pressure with respect to systolic and diastolic pressure, the percentages of variation in the pulse pressure are largest. The difference between the reconstructed and reference aortic pressure in terms of wave shape is expressed as RMSE (in mmHg). For similar magnitudes of the increase and decrease of a parameter, the output variables may change with unequal magnitude. For instance, a 25% increase in *R*_{p} has less effect than a 25% decrease (see discussion).

Variations in diastolic pressure are negligible for almost all changes. Although most physiological parameters in vivo change <25%, some, like peripheral resistance, may change >25%. This was not considered further since this particular parameter has very little influence (Table 2). A change of 25% in blood density is very unlikely. Therefore, aortic pressure is mainly determined by vessel size (diameter and length), and the transfer function is also determined by vessel size and to a lesser extent by the Young's modulus and windkessel load.

### Different Pressure Waveshapes

In Fig. 3, the effects of 25% changes in segment lengths (as in Fig. 2) are shown for different wave shapes [a type A beat and a type C beat from Murgo et al. (23)]. Resulting errors in pressure are given in Table 3. It can be seen that change in length contributes substantially to pulse pressure in the type C beat but little in the type A beat.

### Changes in Combined Parameters

#### Age.

In Fig. 4, the effects of aging on the pressure wave are shown. The differences are very small (see also Table 3).

#### Body size.

In Fig. 5, the effects of body size on the pressure wave are given and quantified in Table 3. The effect is much larger than aging but comparable with changes in segment lengths (see Table 2).

#### Increased blood viscosity.

In Table 3, it can be seen that the combined change of blood viscosity and peripheral resistance has hardly any effect.

### Reflection Coefficient as a Function of Vasoactive State

In Fig. 6, the effect of a factor 4 decrease (3, 17) and a factor 4 increase (21) in *R*_{p} on the modulus of the reflection coefficient Γ is shown. The strong decrease in *R*_{p} results in a 30% reduction of the first harmonic of Γ, and the difference rapidly diminishes for higher harmonics. The increase in peripheral resistance affects Γ even less (Fig. 6).

### Uniform Tube Model

The lossless uniform tube is shown in Fig. 1*B*. The comparison between the transfer function of the model according to anatomy and of the uniform lossless tube is shown in Fig. 7. Peak magnitude of the transfer function is 2.25, at a frequency of 4.0 Hz, whereas with the anatomically correct tube, these values are 2.0 and 4.0 Hz. Aortic pressure was reconstructed by applying the transfer function on the basis of a uniform tube to the brachial pressure and comparing this with the reference aortic pressure. Relative differences are 1%, 0%, and 3% for systolic, diastolic, and pulse pressure, respectively. RMSE is 2.1 mmHg.

## DISCUSSION

This theoretical analysis shows that prediction of aortic pressure from brachial pressure is mainly dependent on vessel size (e.g., length and diameter) and less dependent on other parameters (e.g., peripheral resistance). The uniform lossless tube as simplification of the anatomy is also acceptable, and this outcome stresses that the tapering is only a minor factor as well. The segment lengths used have been shown to be sufficiently short for the physiological frequency range; shorter lengths or more detailed tapering is not required (32).

Our analysis implies that the transfer function from brachial artery to aorta can be simply based on a lossless uniform tube. The tube parameters, segment length, radius, wall thickness, Young's modulus and wall viscosity, and the blood parameters density and viscosity, all contribute to the travel time, an important overall property. Thus the travel time can be viewed as the parameter combining these parameters. Using a uniform lossless tube and accounting for the travel time allow for analytical formulation of the transfer function and underpin the time shift concept (27). We calculated that for a 25% increase and 25% decrease in travel time, the percent changes for systolic pressure (140 mmHg) were 3% and 2%, diastolic pressure (67 mmHg) 1% and 0%, and pulse pressure (73 mmHg) −7% and −5%. If the errors resulting from the introduction of a single uniform tube with known travel time are regarded as acceptable, this model can make it possible to individualize a transfer function from travel time information only, which can be determined noninvasively.

The peripheral bed, i.e., the windkessel parameters, are of limited influence on the pressure transfer (Table 2). We found that changes in peripheral resistance, which can vary over a wide range, had only a small effect on systolic, diastolic, and pulse pressure (Table 2). This is in agreement with earlier findings (1) that local administration of a vasoconstrictor (phenylephrine) and vasodilator (sodium nitroprusside) induced no measurable changes in differences between brachial and finger artery pressure. Similarly, it was found that the wave shape of pressure or diameter as determined by photoplethysmography is not influenced by local infusion of vasoactive drugs (6, 22). In contrast, Karamanoglu et al. (18) found that the distal reflection coefficient had a major influence on mainly systolic pressure. Yet, whereas we calculated our results on basic parameters, Karamanoglu et al. simplified their model by using a reflection coefficient at the distal site and varied this coefficient over a large range of values. However, it turns out that even large variations in peripheral resistance have a limited effect on the modulus of the reflection coefficient (see Fig. 6). It is unlikely that such ranges of change in resistance would occur in resting conditions. By assuming a great influence of *R*_{p} on the reflection coefficient, Karamanoglu et al. overrate the effect of the vasoactive state on the transfer function.

From the current analysis it may be inferred that, once the transfer function is determined, local vasodilation and vasoconstriction are not of great influence. It is important, however, to distinguish two very different aspects of vasodilatation: local and whole-system vasodilatation. Local vasodilatation will change peripheral resistance of the local load but will hardly affect the conduit vessels, the reflection coefficient, and thus the transfer function (1). With whole-system vasodilatation, the transfer function will change (1) due to the decreasing mean pressure with consequences for wave travel (smaller conduit artery diameter and higher vessel compliance).

Segers et al. (26) investigated the possibility to individualize a transfer function based on three segments, and they found that model parameters were not related to heart rate, blood pressure, or age. Optimal reflection coefficients and characteristic impedances of the segments of the model were determined. For convenience, segment lengths were kept constant. Similar results could be obtained by adjusting segment length, but Segers et al. considered it unlikely that this would have a major influence. However, from our study it follows that path length and also travel time are important parameters. This is corroborated by the finding that segment lengths contribute substantially to the pulse pressure in the type C beat (Table 3) but has little effect on pressure values for the type A beat. This implies that the importance of correction for travel time is wave-shape dependent.

It is interesting to note that the rather large changes in frequency and magnitude of the peak in the transfer function that occur with changing radius have little effect on the reconstruction of pressure. From Table 2 it may be observed that the peak moves to a higher frequency and a greater magnitude with an increasing radius and to a lower frequency and a smaller magnitude with a decreasing radius. The result of this combined change is that the first three harmonics remain at their positions with a changing radius, thus leaving the most important part of the transfer function for pressure reconstruction intact.

Most transfer functions described in the literature were obtained by averaging measured transfers in groups of patients (4, 7) or based on filters fitted to averaged transfers (9, 10). Our analysis shows that transfer functions can be obtained on the basis of actual vascular parameters. A single uniform lossless tube with a distal reflection coefficient is a good approximation. This implies that the basic mathematical description of the transfer from distal to proximal pressure is as follows: (5) as given by Stergiopulos et al. (27). This formula has two nondimensional parameters, and, by determining these parameters, the transfer function can be individualized. In Fig. 6, we show that vasodilation and vasoconstriction affect the reflection coefficient little, so a constant reflection coefficient suffices. Thus the remaining parameter is the travel time of the pressure wave between measurement and derived location (Δ*t*) as the main determinant of the transfer function. This travel time can, for instance, be estimated from the ECG and local pressure or diameter information.

The main objective of this study was to investigate the influence of arterial, blood, and load parameters on the transfer of pressure. We did not evaluate the influence of heart rate or other cardiac factors. Heart rate itself is included in the analysis because the calculations are carried out in the frequency domain.

We investigated the importance of physiological anatomical parameters on a model describing the arteries between the brachiocephalic artery and the brachial artery. This was done so that potential effects of major bifurcations would not obscure our results. It is interesting to note, however, that the transfer function from brachiocephalic artery to brachial artery gives results qualitatively similar to findings in the literature for transfer functions from aorta to brachial, from aorta to radial, and even from carotid to radial artery (8). Apparently, all these transfer functions are mainly determined by the part between the branching from the aorta to the brachial artery. Using the data of Lasance et al. (20), we calculated transfer functions from ascending aorta, aortic arch, and brachiocephalic artery to brachial artery (Fig. 8). The results are very analogous for the most important harmonics between 1 and 4 Hz (4). Particularly, the transfer functions from aortic arch and brachiocephalic artery to brachial artery are strikingly alike. The short and wide aortic segments and the short and narrow segments of the underarm and carotid artery contribute mostly to higher frequencies. In summary, the choice to analyze a model of the arteries between the brachiocephalic artery and the brachial artery is not a major limitation of the study.

Increasing and decreasing a parameter by the same percentage may result in unequal changes in the outcome variables (see, for example, Fig. 7 and Table 2). This is the result of the nonlinearity of the relations. For instance, if an inverse relation is assumed, a 25% increase in a parameter results in a 20% decrease in the variable (1/1.25 = 0.8), whereas a decrease of 25% results in an increase of 13%.

Our findings suggest that for epidemiological studies, with subjects at rest and similar mean pressure, a generalized transfer function can be used. This transfer function may be improved by accounting for length through body height. Would one care to improve the results even more, travel time should be measured and not length, since travel time includes other effects as well, such as arterial wall elasticity and vessel diameter. For example, when exercise or pharmacological interventions change blood pressure (29), the transfer function can be improved by accounting for the changes in large vessel properties by using travel time.

In conclusion, this study shows that a generalized transfer function may be used in most instances. However, an individualized transfer function will improve the calculation of central pressure, when the travel time, preferably measured noninvasively, is accounted for.

## Acknowledgments

We cordially thank Jan Paul Barends of the Laboratory for Physiology, Institute for Cardiovascular Research-Vrije University, Vrije University Medical Center, Amsterdam, The Netherlands, for his inspiring help with the mathematics in this study.

## Footnotes

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- Copyright © 2007 by the American Physiological Society