## Abstract

A method for estimation of central arterial pressure based on linear one-dimensional wave propagation theory is presented in this paper. The equations are applied to a distributed model of the arterial tree, truncated by three-element windkessels. To reflect individual differences in the properties of the arterial trees, we pose a minimization problem from which individual parameters are identified. The idea is to take a measured waveform in a peripheral artery and use it as input to the model. The model subsequently predicts the corresponding waveform in another peripheral artery in which a measurement has also been made, and the arterial tree model is then calibrated in such a way that the computed waveform matches its measured counterpart. For the purpose of validation, invasively recorded abdominal aortic, brachial, and femoral pressures in nine healthy subjects are used. The results show that the proposed method estimates the abdominal aortic pressure wave with good accuracy. The root mean square error (RMSE) of the estimated waveforms was 1.61 ± 0.73 mmHg, whereas the errors in systolic and pulse pressure were 2.32 ± 1.74 and 3.73 ± 2.04 mmHg, respectively. These results are compared with another recently proposed method based on a signal processing technique, and it is shown that our method yields a significantly (*P* < 0.01) lower RMSE. With more extensive validation, the method may eventually be used in clinical practice to provide detailed, almost individual, specific information as a valuable basis for decision making.

- arterial tree
- parameter identification
- mechanical model
- transfer function

the possibility to obtain an accurate estimate of the pressure waveform in central arteries such as the abdominal aorta, is of great clinical interest. On its own, the pressure waveform enables the computation of the augmentation index (13), a measure of wave reflection used to assess aortic stiffness, and a potent indicator for cardiovascular diseases (29). Together with simultaneous measurement of vessel radius, the systolic (SP) and diastolic pressures are often used for compliance calculations (see, for instance, Ref. 23 or 31). More recent methods, however, utilize the entire waveform to compute wall stresses and determine material parameters (20, 26, 27). A potential use for these new methods is as diagnostic tools for prediction of aneurysm growth and rupture, which are linked to the material properties of the arterial wall and the applied load, i.e., the pressure wave. The abdominal aorta is a common site for aneurysm formation (34), and the development of methods that can provide the abdominal aortic pressure wave with good accuracy is, therefore, highly motivated.

Unfortunately, it is currently impossible to measure the central blood pressure directly by noninvasive methods. Therefore, measurements on peripheral vessels, most commonly the brachial or radial arteries, are often used as a substitute for the actual waveform. The shape of the pressure waveform may, however, differ considerably between central and peripheral vessels (11, 15), and substituting one for another may yield misleading results (18, 33). To overcome this problem, a number of different methods for estimation of central pressure have been developed. Roughly, the methods can be divided into two groups: *1*) methods based on signal processing or machine learning (see, for example, Refs. 4, 16, and 28), and *2*) methods based on a more direct modeling of the underlying physiology, e.g., transmission line models (6, 21, 36).

Many methods based on signal processing suffer from at least two inherent limitations. First, they are calibrated on a group basis (see, for instance, Ref. 4 or 16) and are thus not subject specific. Second, these models are calibrated to estimate pressure in a specific location in the arterial tree and can, therefore, not be used to give information about the pressure in other parts of the tree. In a recent paper (28), the authors tried to overcome the first limitation by using a signal processing technique called multichannel blind system identification (MBSI). The MBSI method does not require the use of a training set, and it is, therefore, possible to get a patient-specific waveform estimate. The method is, however, limited to estimating the common source of two or more pressure signals, and estimation in an arbitrary location may, therefore, still be difficult.

The aim of the present paper is to demonstrate the feasibility of using the transmission line theory, together with a distributed model of the human arterial tree, to make individual specific estimates of the abdominal aortic pressure waveform. The idea is illustrated in Fig. 1. Using an optimization routine, we take a generic model of the arterial tree and adjust it to fit peripheral pressure measurements. In return, we get a specific model, which can be used to compute the pressure in any part of the arterial tree. Compared with earlier signal processing methods, the most obvious advantage of our method is that it gives a patient-specific estimate, and, compared with the MBSI method, it gives higher accuracy.

Methods based on solving a parameter identification problem to obtain pressure waves have been used before (see, for instance, Refs. 6 and 21). While resembling these previous studies, the goal of the present study is to obtain an estimate of abdominal aortic rather than ascending aortic pressure, and the model used to achieve this goal is also more ambitious than those used previously.

## MATERIALS AND METHODS

#### Mechanical model.

In this subsection, a transfer function that allows us to compute the pressure distribution in the arterial tree is derived. This is done by using the equations of the standard linear one-dimensional wave propagation theory originally developed by Womersley (35) and later on used by several others (see, for example, Refs. 2, 7, and 36). Details on the implementation are given in the appendix.

An arterial segment is modeled as a thin-walled, linearly elastic circular tube with radius *a*, taken as the neutral position of the wall, length *l*, wall thickness *h*, elastic modulus of the wall *E*, and Poisson's ratio ν. Assuming that the arterial wall is incompressible (11), the latter is taken to be 0.5. To model the viscoelastic properties of the arterial wall, the Young's modulus is multiplied by a factor *e*^{jΦ} (30), where *j* is the imaginary number, and Φ is the phase angle between pressure and displacement of the wall, approximated by Φ = Φ_{0}(1 − *e*^{−2ω}), where ω is the angular frequency. For the distribution of values of the phase angle throughout the arterial tree, the expression Φ_{0} = Θ·10*h*/2*a* (6) is used, where Θ is taken to be 0.26 radians.

The blood is treated as an incompressible Newtonian fluid with density ρ and viscosity μ, taken as 1.055 g·cm^{-3} and 0.04 dyn·s·cm^{−2}, respectively, and it is assumed that the velocity profile is the same as that of pulsatile flow in a rigid tube (37). A schematic picture of an arterial segment is shown in Fig. 2, where *r* and *x* denote radial and axial directions, respectively.

Starting from the Navier-Stokes and the continuity equation, the following system of equations for pressure and flow in a one-dimensional transmission line element, with properties as given above, can be derived (36, 37): (1) (2) where q is flow rate, p is pressure, and *t* is time. The pressure wave speed *c* and the characteristic admittance *Y*, defined as the ratio of the forward or backward moving flow and pressure, are given by (3) and (4) where (5) and (6) (Admittance is the reciprocal of impedance and is used here for convenience.) The term F_{10} in *Eqs. 3* and *4* is related to the velocity profile and is given by (7) in which α = *a*(ωρ/μ)^{1/2} is the Womersley number, and *J*_{0} and *J*_{1} are Bessel functions of the first kind, zeroth and first order, respectively (37).

Using the Fourier transform, the pressure wave can be approximated by a sum of harmonic waves, and we, therefore, confine ourselves to study the solution to *Eqs. 1* and *2* for a single harmonic wave. Neglecting all reflections higher than the first order, we obtain the following expression for the pressure (36): (8) where p_{0} = p_{0}(ω) is the amplitude of the applied pressure at the entrance of the segment, *j* is the imaginary number, and *R* is the reflection coefficient at the opposite end of the segment. The reflection coefficient is defined as the ratio between backward and forward traveling waves.

In the presence of wave reflection, *Y* in *Eq. 4* no longer represents the true admittance of the segment. Therefore, it is necessary to introduce a new quantity called the effective admittance, defined as the ratio between flow and pressure at the tube entrance. A slight generalization of the result given in Ref. 36 yields the following formula for the effective admittance: (9) where *Y* is the characteristic admittance of the current segment, *Y*_{d} is the effective admittances of any immediate daughter segments, *m* is the number of daughter segments, and θ = ω*l*/*c*. By assuming pressure continuity and mass conservation at junctions, it is possible to express the reflection coefficient at the distal end of the segment in terms of characteristic and effective admittances by a slight generalization of the results given in (36): (10) Now, by writing *Eq. 8* at the junction between two adjacent segments *i* and *i* − 1, denoting the daughter and parent segments, respectively, and equating these by the pressure continuity, we obtain a transfer function (TF), (11) which relates the initial forward waves in both segments through (12) By applying *Eq. 12* iteratively, it is possible to relate pressures in any two segments of the arterial tree.

#### The arterial tree model.

The arterial tree model used in this article is based on data published in Refs. 25 and 32. The original model, consisting of 55 segments representing the larger arteries, was further segmented to a total of 113 segments in the following way: using results from Ref. 17, the 3-dB cutoff frequency, *f*_{0}, of an arterial segment can be approximated by: (13) where (14) and (15) Now, since the Young's modulus, *E*, will be used in the optimization later, the tree is segmented in such a way that the cutoff frequency of each segment is acceptable for all admissible values of *E*. Substituting *Eqs. 14* and *15* in *Eq. 13* and solving for *E* yields the following expression: (16) To decide whether a segment should be further divided or not, the minimum cutoff frequency of any segment was set to 15 Hz (3), and *E* was calculated from *Eq. 16*. If the resulting minimum value of the elastic modulus was equal to or below some acceptable value (see, for instance, Ref. 14), the segment was not divided further. Otherwise the segment was divided in half, and the procedure repeated for the new segments. It was assumed that the vessels were mildly tapered, and the new radii were chosen accordingly. Wall thicknesses were computed using the wall thickness-to-radius ratio of the original segment.

The peripheral beds are modeled using three-element windkessels consisting of a resistor in series with a parallel combination of a resistor and a capacitor. Each windkessel is, therefore, characterized by three parameters, two resistors, *R*_{1} and *R*_{2}, and a capacitor *C*. In the frequency domain, the admittance of the windkessel models (*Y*_{wk}) are given by (17)

#### Parameter identification.

As is evident from *Eq. 11*, the transfer function depends on the geometric and material properties of the arterial tree model. In general, these properties will differ between individuals, and to account for this, the tree model is individualized. This is done by solving a parameter identification problem, with the parameters describing the terminal beds and the elasticity of the larger arteries as minimization variables. The idea is to minimize the function (18) where is a vector consisting of the windkessel parameters, *n* is the number of terminal segments used in the optimization, * E* is a vector consisting of the Young's modulus of the arteries in the systemic tree,

*S*denotes the current sample,

*N*is the number of samples, P̂

_{FA}is computed femoral pressure,

**P**

_{BA}is measured brachial pressure, and

**P**

_{FA}is measured femoral pressure. The optimization problem is formulated as: where

*i*and

*i*− 1 again denote adjacent segments, with segment

*i*distal to segment

*i*− 1. The last constraint on the Young's modulus is based on the assumption that the arteries gradually stiffen peripherally (10). Lower bounds on

*were calculated using*

**E***Eq. 16*, and upper bounds were set to 40·10

^{6}dyn/cm

^{2}(4 MPa) for all segments. Upper and lower bounds on the windkessel parameters, i.e., κ̄ and κ, were taken, as four times and one-fourth, respectively, of the values published in Ref. 14. This interval was large enough to avoid solutions at the boundaries, while still resulting in physiologically reasonable solutions.

To solve the problem, the Matlab (The Mathworks) routine fmincon was employed. The routine fmincon is a gradient-based solver that utilizes a sequential quadratic programming algorithm (12). To counter problems with the solver being attracted to local minima, a number of randomly distributed start points were generated, and the sequential quadratic programming search algorithm ran until convergence for each of the points. The point corresponding to the minimum objective function value was then selected as the final solution. The termination tolerance was set to 10^{−5}.

#### Mean pressure.

The pressure wave can be divided into a steady and an oscillatory part. The oscillatory part is governed by a wave equation derived from *Eqs. 1* and *2*, whereas the steady part, i.e., the mean pressure, is governed by a different set of equations. For simplicity, however, it is assumed that the mean pressure in the brachial artery and the abdominal aorta, given in mmHg, are related through (19) where P̂_{AA} is the computed abdominal aortic pressure. The correction factors of 1.024 and 0.99 mmHg were found by a linear fit (*r* = 0.99, *P* < 10^{−10}, 95% confidence interval: 0.95–1.10 and −5.90–7.88, respectively) to measured data from 14 individuals (see *Material* section below).

#### Material.

In this study, right brachial, right femoral, and abdominal aortic pressure waves, recorded invasively in nine healthy, nonsmoking subjects were used. Subjects came from three different age groups: 24 ± 3 yr (*n* = 3), 46 ± 5 yr (*n* = 5), and 69 yr (*n* = 1). The coefficients in *Eq. 19* for the mean pressure estimation were obtained using measured data from 14 healthy, nonsmoking subjects in age groups 26 ± 3 yr (*n* = 4), 51 ± 4 yr (*n* = 3), and 60 ± 2 yr (*n* = 7) (femoral artery pressure wave recordings were not available for these subjects). Informed consent was given by each subject before investigation. The study was approved by the Ethics Committee, Lund University, Sweden.

Pressure measurements were done using a 3-F (SPC 330A) or a 4-F (SPC 340) micromanometer tip catheter (Millar Instruments, Houston, TX) or with a fluid-filled catheter system (pressure monitoring kit DTX+ with a R.O.S.E; Viggo Spectramed, Oxnard, CA). The Millar catheter had a higher frequency response (flat range to 10 kHz) than the fluid-filled system (flat range 35 Hz; 3 dB). However, the amplitude was identical when the curves of one cardiac cycle from each pressure system, created by a Blood Systems Calibrator (Bio Tech Model 601A, Old Mill Street, Burlington, VT) were superimposed on each other. The sampling frequency was ∼870 Hz. For more details on the data acquisition, see Refs. 1 and 23.

Cuff measurements where done on both left and right brachial arteries, resulting in similar means, thus indicating the absence of any significant coarctation. Abdominal aortic pressure was recorded between the renals and the bifurcation. Since abdominal and femoral pressures were not measured simultaneously, there were two sets of measurements for each individual: one containing simultaneously measured abdominal pressure (**P**_{AA}) and **P**_{BA}, and one containing simultaneously **P**_{FA} and **P**_{BA} (**P**_{BA/FA}). The time between the two measurements was ∼2 h.

In the process of parameter identification, we used **P**_{FA} and **P**_{BA/FA}, whereas validation against the abdominal aortic pressure was done using **P**_{AA} and **P**_{BA}. The pressure waves used for parameter identification and validation were constructed by averaging of 10 consecutive pulses.

#### Statistics.

Results are presented as means ± SD with values in millimeters of mercury. The different methods are compared using a paired *t*-test. Values of *P* < 0.05 were considered significant.

## RESULTS

Four different methods for estimation of the abdominal aortic pressure were considered. First, the proposed method [referred to as ICAM (individually calibrated arterial tree model)]; second, the MBSI method, implemented according to the method presented in Ref. 28; and, finally, using brachial or femoral artery waveforms as direct substitutes for the abdominal aortic waveform. As inputs to the MBSI method, we used intervals of ∼11 s of **P**_{FA} and **P**_{BA/FA} for each subject. Mean levels were adjusted using *Eq. 19* with **P**_{BA}, and the resulting waveforms were averaged before comparison to the abdominal aortic pressure wave. Mean levels of the brachial and femoral pressures were not adjusted.

#### Pressure estimation.

In order to refine the comparison of the shapes, the overall waveform errors [root mean square error (RMSE)] were computed with the mean pressure removed from all estimated and measured signals. Thus the effect of any constant offset was eliminated. When computing the error in SP, however, the means were not removed, and these figures, therefore, include errors from the estimation of the mean pressure level.

The plots in Fig. 3 show the results from the ICAM method in detail. The RMSE values are closely centered around the mean (1.61 ± 0.73 mmHg), while the SP plot shows more scatter (3.73 ± 2.04 mmHg). No significant correlation was found between the results from the ICAM method and any of the error measures.

A comparison between the four methods in terms of RMSE is illustrated in Fig. 4 and Table 1, which also contains a comparison of SP and pulse pressure (PP) errors. The difference between the ICAM and the other methods is marked, and, as can be seen from Table 2, the RMSE of the ICAM method was significantly lower than for the MBSI method (*P* < 0.01) or the brachial artery pressure (*P* < 0.01). Compared with the femoral artery pressure, the difference in RMSE was not significant, but errors (overestimations) in PP and SP were larger for the femoral artery waveform (*P* < 0.01). Differences between the ICAM method and the substitution of brachial artery pressure in terms of SP and PP were not significant.

Two examples of estimated abdominal aortic waveforms from the ICAM method are shown in Fig. 5. A comparison of the estimated and the measured waveforms in the *top* plot yielded an RMSE of 0.69 mmHg. As can be seen, this figure corresponds to a very good fit to the measured signal. A similar analysis of the *bottom* plot gave an RMSE of 1.53 mmHg.

#### Model parameters.

The *top* plot of Fig. 6 shows the total arterial compliance (TAC) for the nine tree models. The volume compliance of each segment was computed using *Eq. 15*, and TAC was then found as the sum of the volume compliances of all segments, together with the compliances of the windkessel terminals, i.e., *C* in *Eq. 17*. The values of TAC are, on average, slightly lower than those found in, for example, Ref. 9. As can be seen in the figure, there is a significant correlation with subject age.

Ascending aortic impedance of the tree models for the nine subjects is shown in the *bottom* plot of Fig. 6. The values are at the upper end compared with those found in Refs. 2 and 11. There is no clear correlation with subject characteristics.

Values of the Young's modulus for three representative arteries, TAC, and total peripheral resistance (TPR) are given in Table 3. The Young's moduli lie in the middle range of the values presented in Ref. 10. TPR was computed by adding the resistance of each segment, approximated using the formula 8μ*l*/(π*a*^{4}) (36) and the 0-Hz impedance of the windkessel terminals. For comparison, a study of TPR can be found in Ref. 5, and it is noted that all TPR values in Table 3, except the highest, fit within the range of values presented therein. No significant correlation between subject age and Young's modulus or TPR was found.

## DISCUSSION

We have presented a subject-specific method for estimation of central aortic pressure from peripheral measurements, based on a direct modeling of the underlying physiology. The method has been tested for estimation of abdominal aortic pressure in nine healthy subjects. Compared with substituting for brachial artery pressure or using the MBSI method, the proposed method offers a significant improvement in terms of RMSE. Although the difference in RMSE between the ICAM method and the femoral artery waveform as a substitute was less significant, the former yielded a better estimation of both SP and PP. Given these results, the ICAM method seems preferable to the substituted femoral artery pressure.

For estimation of SP and PP, the ICAM method is comparable to the MBSI method and the substitution for brachial pressure (it was shown in Ref. 31, that, in elderly, the SP is reasonably predicted by cuff measurement on the brachial artery). However, the improvement of the RMSE compared with these method is very important, because, as applications become more advanced (see, e.g., Refs. 20, 26, 27), it will be possible to extract more information using the entire waveform as input to the analysis, rather than trying to describe it using just a few parameters, such as SP or PP.

It should be noted that the proposed method is not dependent on any particular technique for obtaining the pressure wave, and, although only invasively measured signals were available for this study, the generalization to noninvasive measurements is straightforward.

#### Theory.

In the present paper, the arterial tree is treated as a linear system. While the convective acceleration may be of minor importance (19), neglecting the nonlinearity of the arterial wall may be more questionable. Using a nonlinear model, however, will require a lot more in terms of computational effort, and it is, therefore, not obvious that one should, for this application, abandon the linear theory. Furthermore, in Ref. 22, for example, it was found that the effect of viscoelasticity was as important as the difference between linear and nonlinear models, and peripherally, the representation of terminal impedance had greater impact on the pulse propagation than any other factor.

The windkessel model, used to represent the peripheral beds, provides a good representation of the overall behavior of the terminal impedance. It is, however, unable to capture wave propagation effects, resulting in, e.g., local minima in the magnitude plot. A more detailed model of the peripheral beds was proposed in Ref. 14. Therein, the peripheral beds are modeled by asymmetric binary trees, characterized primarily by a minimum truncation radius and a length-to-radius relationship. While this kind of model is attractive, it was found to be somewhat expensive in terms of computational effort compared with the windkessel model, and it was, therefore, not used with the parameter identification routine proposed in this article. Currently, the mean computational time to reach convergence from each start point is ∼11 min on a Pentium 4, 2.4-GHz CPU and 1-GB RAM. If desirable, this number could probably be reduced further by, for instance, a thorough study of suitable convergence criteria.

A distinct feature of transmission line models is the need to divide the arteries into shorter segments with uniform characteristics. Increasing the number of segments reduces the amount of spurious reflection and errors in the computed impedance, at the cost of increased computational times. Using the method outlined in *The arterial tree model*, the tree model was divided into 113, 226, 510, and 1,214 segments, respectively, and abdominal aortic pressure was computed from brachial artery pressure in 24 subjects. The mean RMSE of the computed waveforms were, for respective tree, 3.07, 3.09, 3.09, and 3.09 mmHg. Based on this, it was concluded that the model used in this paper is sufficient for the purpose herein.

#### Parameter identification.

The choice of minimization variables is, in part, based on a study (7) that showed that peripheral wave reflection is a major determinant of the transfer function between the ascending aorta and the radial artery. Therefore, it is reasonable that variables governing the peripheral reflection should be included in the problem. Furthermore, the time of return of the reflected wave at a particular location greatly affects the appearance of the waveform. The return time will depend on the wave speed and, therefore, on both arterial geometry and elastic properties. The geometry of the larger arteries, and possibly arterioles, can be fixated by noninvasive imaging techniques such as, e.g., computed tomography or MRI, and it is, therefore, desirable to have a method that is not dependent on manipulation of the geometry. Instead, to account for differences in the pressure wave velocity between individuals, the elastic modulus of the larger arteries are chosen as variables in the problem.

Given the relatively large number of variables involved in the parameter identification problem, an interesting question is how much information can be extracted from two pulse cycles. The total number of minimization variables was over 100, and concerns about the uniqueness of the solutions may, therefore, be raised. However, considering the results in Table 3, identifiability does not seem to be an issue here. This was also corroborated by checking the eigenvalues of the Hessian for some of the subjects and noting that they were all greater than zero.

The application of constraints can help counter problems with local minima and be used to ensure physiologically plausible solutions. In the present study, with the focus being on pressure estimation, the constraints were relatively loose. Nevertheless, as can be seen from Table 3, the resulting solutions fell within what might be considered reasonable from a physiological point of view, taking into account also the degree of approximations in the theory, and the fact that a generic geometry is used. The implication of this is taken to be that the method may eventually be used, not only for pressure estimation, but also to provide additional, patient-specific information about the arterial system.

The assessment of the model parameters in the results section showed that, while there was no significant correlation between Young's modulus or TPR and subject age, the TAC was clearly correlated to the latter. However, given the relatively few subjects and the age composition of the validation group, the conditions for such analysis are not optimal.

Regarding the parameter values shown in Table 3, it was noted that the values of TAC seemed slightly low. This could be due to the values of the Young's modulus, but might also, since the volume compliance of the arteries is proportional to *a*^{3} and, therefore, sensitive to small changes in radius, be ascribed to the generic geometry, which was not subject to optimization. For *subject 3* in Table 3, it could also be seen that the TPR was slightly high. While issues regarding the values of TAC and TPR could, for instance, be addressed by studying the generic geometry and lower the upper bounds on the resistances of the windkessel models, a more direct way would be to impose constraints directly on TAC and TPR. Although not ensuring that the correct values, i.e., the values that would result from a measurement, are found, such constraints would make sure that the parameters always conform within the range of published data. Clearly, these and the above issues are important, and it is noted that finding appropriate constraints is certainly an area of interest for future investigation.

#### Estimation of mean pressure.

It is a quite common assumption that the mean pressure remains constant throughout the arterial tree. However, given *Eq. 19*, it seems more likely that, in general, the mean pressure is slightly higher in central vessels compared with peripheral vessels. Some support for this can also be found in studies described in Refs. 8 and 15 (although the latter concerned nonhealthy subjects). Since, in the linear theory, the computations of the steady and oscillatory pressure are decoupled, the mean pressure estimation only affected the error in SP and not the RMSE. The nonlinear elasticity of real arteries, however, makes it clear that a good estimate of the absolute level is important, and, for the method to be truly subject specific, a statistically derived method is not completely satisfying.

#### Experimental data.

The fact that femoral and abdominal aortic pressures were not recorded simultaneously is a source of error. As pointed out in the previous section, this affects the results from the substitution of the femoral artery pressure, the MBSI method and the ICAM method, respectively. Using the ICAM method, a significant (*r* = 0.69, *P* < 0.05) positive correlation was found for the discrepancy between the two brachial artery signals, i.e., **P**_{BA} and **P**_{BA/FA}, and the RMSE of the estimated abdominal aortic pressure, and it is, therefore, reasonable to believe that using data recorded from the ideal setting, i.e., all three locations measured simultaneously, will yield better results.

Since the MBSI method estimated the abdominal aortic pressure directly from **P**_{BA/FA} and **P**_{FA}, the results were probably influenced by the experimental setup to a higher degree than the ICAM method, and it is likely that the MBSI method would perform better if all measurements had been done simultaneously. Furthermore, we only had access to intervals of ∼11 s, whereas, in Ref. 28, the authors used 1-min intervals of pressure data. Given the character of the MBSI method, it is likely that longer intervals would result in better performance.

Regarding the need for two or more measurements, this may be considered a drawback of both the ICAM and MBSI methods, but it is hard to imagine a subject-specific method that does not rely on measurements of either one parameter in multiple locations, or perhaps multiple parameters in a single location. However, the use of additional measurements might be regarded as an advantage, not only because it provides increased accuracy of the predicted waveform, but also because they might be used to provide additional constraints on the solution. An example of this could be measurements of pulse wave velocity, which could be used to provide more accurate bounds on the Young's modulus.

#### Comparison with the MBSI method.

Briefly, the MBSI method views the arterial tree as a system with a single input that drives a number of linear, time-invariant (LTI) systems, each characterized by its impulse response. The outputs of these LTI systems are the peripheral pressure waveforms. Starting with standard properties of the linear convolution operator, it is possible to derive equations that can be used to estimate the impulse responses of the LTI systems using only the measured peripheral waveforms. Once the impulse responses are estimated, the common input signal, i.e., the central aortic pressure, can be estimated up to an arbitrary scale factor.

The MBSI method has at least two things in common with our method: it is a linear method, and it requires two or more measurements. However, while the ICAM method returns an arterial tree model, thus potentially providing very much information about the system, the MBSI method functions more like a black box, which returns only the impulse responses. Furthermore, as mentioned in the introduction, the MBSI method can only be used to estimate pressure at the location of the common input to the measured peripheral sites. This means that, compared with the ICAM method, the number of locations at which pressure can be estimated are somewhat limited. Thus an important question is, of course, how to interpret the estimated signal, i.e., what is the common input when measuring brachial and femoral pressure? The most reasonable answer is probably the pressure in the aortic arch, since it is the common artery. Therefore, even if the experimental setup were ideal, one problem with the MBSI method might still be, e.g., a slight underestimation of the SP in the abdominal aorta.

#### Future work.

One of the advantages of our method is the possibility to integrate measurements of several quantities in multiple locations in a very natural way. In the present study, measurements of pressure in two locations have been used, see Fig. 7, *left*, but, if available, it is possible to include additional measurements of, e.g., pressure in more locations, geometry, or flow, as illustrated in Fig. 7, *right*. Once these measurements have been made, it would also be interesting to try to determine which measurements are actually needed to achieve, e.g., a given accuracy at a certain maximum cost.

The subjects used in this study were all presumably healthy, and a natural next step is, therefore, to validate the method on nonhealthy subjects. Using a detailed model such as the one presented herein probably means that pathological conditions, e.g., a stenosis, must be modeled with a similar level of detail, thus presenting challenges, but not principal difficulties, for future work on nonhealthy subjects.

Since the method is based on a generic model, it is, of course, interesting to try to find the most suitable generic model. It may be that the generic model should be scaled with age or body size for example. Finally, the small parameter study shown in the results section could be extended to see how additional constraints or variations of the geometry affect the resulting parameter values.

#### Conclusion.

It has been shown that it is possible to estimate abdominal aortic pressure by optimizing a linear transmission line model of the systemic arteries to fit measured pressure data. The overall error of the estimated waveforms, RMSE, was smaller than for any of the other methods tested. To sum up, we have presented a physiologically based method that performs well, is easy to implement, allows for a natural integration of multiple measurements, and can be used to compute pressure and other parameters in any location in the arterial tree.

## APPENDIX

This appendix contains a detailed description of the objective function. The parameters **h**, **a**, **l**, ρ, ν, μ, and Φ_{0} are given initially, while **E**, **R**_{1}, **R**_{2}, and **C** are provided by the minimization algorithm at each iteration. In addition to the arterial tree parameters, the amplitude of the applied pressure at the entrance of the brachial artery, see *Eq. 8*, is found by computing the discrete Fourier transform of the input signal as where *ω*_{k} *=* 2π*k*/*N*, *N* is the number of samples, and the term *A* = *e*^{−jωkx/c}+ *R*(ω_{k})*e*^{−jωk(2l−x)/c}(for *k* = 0, we take *A* = 1), in which *c* and *R*(ω_{k}) are defined below, corrects for the fact that we measure the entire signal, not only the amplitude of the applied pressure.

Now, given all necessary parameters, the objective function updates the arterial tree model and computes the objective function value as follows.

In *step 1*, *steps a*–*d* are repeated for each frequency ω_{k} = 2π*k*/*N*, *k* = 1 … *N* − 1.

For *step a*, the first step is to determine wave speed and characteristic admittance for each segment using *Eqs. 3* and *4*: where *c*_{0}, *Y*_{0}, and F_{10} are given by *Eqs. 5*–*7*, respectively, and Φ = Φ_{0} (1 − *e*^{− 2ωk}).

For *step b*, next, the effective admittance of each segment is computed using *Eq. 9*: where *Y*_{d} is the effective admittances of any daughter segments, *m* is the number of daughter segments, and θ = ω_{k}*l*/*c*. In the case where a daughter segment is a terminal bed model, *Y*_{d} is replaced by *Y*_{wk} given by *Eq. 17*:

As is evident from *Eq. 9*, the computation of the effective admittances must start at the most distal segments and proceed proximally.

For *step c*, once the effective admittances are available, the reflection coefficients at the distal ends of all segments can be obtained from *Eq. 10*:

For *step d*, the transfer function between two arbitrary segments is computed as the product of all intermediate transfer functions, i.e., where *S* corresponds to the output segment, and the intermediate transfer function are given by

Should the segments not be numbered from 1 to *S*, *i* and *i* − 1 are readily replaced by ξ(*i*) and ξ(*i* − 1), respectively, where ξ is a function that maps the numbers 1 to *S* onto the actual segment numbers.

In *step 2*, having determined all necessary quantities, the estimated femoral artery pressure, P̂_{FA}, can now be determined as where, for each ω_{k},

Since the transfer function is valid only for the pulsatile part, we take TF(0) = 0, and similar for the measured femoral artery signal, **P**_{FA}, to be used in *step 4* below.

In *step 3*, the time delay between the estimated and measured femoral artery signals is computed using cross-correlation, and the signals are aligned in time accordingly.

In *step 4*, the objective function value is obtained from *Eq. 18*:

For a given tree model, *steps 1*–*3*, using *Eq. 19* for the mean pressure gives the desired output.

## Acknowledgments

The authors acknowledge Toste Länne at the Department of Medicine and Health Sciences at Linköping University for providing the experimental data.

## Footnotes

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- Copyright © 2008 by the American Physiological Society