## Abstract

We have developed a new technique to estimate the clinically relevant aortic pressure waveform from multiple, less invasively measured peripheral artery pressure waveforms. The technique is based on multichannel blind system identification in which two or more measured outputs (peripheral artery pressure waveforms) of a single-input, multi-output system (arterial tree) are mathematically analyzed so as to reconstruct the common unobserved input (aortic pressure waveform) to within an arbitrary scale factor. The technique then invokes Poiseuille's law to calibrate the reconstructed waveform to absolute pressure. Consequently, in contrast to previous related efforts, the technique does not utilize a generalized transfer function or any training data and is therefore entirely patient and time specific. To demonstrate proof of concept, we have evaluated the technique with respect to four swine in which peripheral artery pressure waveforms from the femoral and radial arteries and a reference aortic pressure waveform from the descending thoracic aorta were simultaneously measured during diverse hemodynamic interventions. We report that the technique reliably estimated the entire aortic pressure waveform with an overall root mean squared error (RMSE) of 4.6 mmHg. For comparison, the average overall RMSE between the peripheral artery pressure and reference aortic pressure waveforms was 8.6 mmHg. Thus the technique reduced the RMSE by 47%. As a result, the technique also provided similar improvements in the estimation of systolic pressure, pulse pressure, and the ejection interval. With further successful testing, the technique may ultimately be employed for more precise monitoring and titration of therapy in, for example, critically ill and hypertension patients.

- arterial tree
- blood pressure
- model
- system identification
- transfer function

as the arterial pressure wave traverses from the aorta to the peripheral arteries, its contour becomes significantly distorted due to complex wave reflections in the distributed arterial tree (18). For example, both systolic pressure and pulse pressure usually become amplified with the extent of the amplification dependent on the particular peripheral site and state of the arterial tree (22). Thus it is the systolic and diastolic pressures measured specifically in the aorta that truly reflect cardiac afterload and perfusion (5). Perhaps, as a result, central measurements of systolic pressure and pulse pressure have been shown to be superior in predicting patient outcome compared with corresponding measurements made in more peripheral arteries (20, 26, 27). Moreover, aortic pressure (particularly in the descending thoracic aorta) is less complicated by wave reflections than peripheral artery pressure (4, 17), and the entire waveform usually reveals the cardiac ejection interval through the dicrotic notch (6).

The measurement of the aortic pressure waveform involves introducing a catheter into a peripheral artery and guiding the catheter against the flowing blood to the aorta. However, placement of an aortic catheter is not commonly performed in clinical practice (5) because of the risk of blood clot formation and embolization. On the other hand, related, but distorted, peripheral artery pressure waveforms may be measured less invasively and more safely via placement of a catheter in a distal artery. Indeed, radial and femoral artery catheterizations are routinely performed in clinical practice (14). Moreover, over the past few decades, totally noninvasive methods have been developed and refined to continuously measure peripheral artery pressure based on finger-cuff photoplethysmography (7) and applanation tonometry (12). These noninvasive methods are even available as commercial systems at present (see, for example, the Finometer and Portapres, Finapres Medical Systems, The Netherlands, and the T-Line Blood Pressure Monitoring System, Tensys Medical, San Diego, CA).

Several techniques have therefore been recently developed to mathematically derive the clinically more relevant aortic pressure waveform from less invasively measured peripheral artery pressure waveforms. Most of these techniques have involved applying an average transfer function derived over a group of subjects to measured peripheral artery pressure from another subject to predict the unobserved aortic pressure waveform (5, 6, 11, 22). The principal assumption underlying these “generalized transfer function” techniques is that arterial tree properties are constant over all time and between all individuals. Because of known intersubject and temporal variability of the arterial tree, a few techniques have been more recently proposed toward partial individualization of the transfer function relating peripheral artery pressure to aortic pressure through modeling (9, 10, 21, 24).

It would be desirable to be able to estimate the aortic pressure waveform from peripheral artery pressure in an entirely patient- and time-specific manner. One possible way to do so is with the multichannel blind system identification (MBSI) approach of recent interest in signal processing (1, 28). In this approach, two or more outputs of a single-input, multi-output system are analyzed to reconstruct the common input. To our knowledge, the very recent study by McCombie et al. (15) represents the first application of MBSI to the field of hemodynamic monitoring. However, their study specifically aimed to estimate the shape of the aortic flow waveform from peripheral artery pressure measurements (see discussion).

In this study, we introduce a new technique to reconstruct the aortic pressure waveform from multiple peripheral artery pressure waveform measurements, using the MBSI approach without the need for a generalized transfer function. We then demonstrate proof-of-concept of the MBSI technique with respect to four swine in which femoral and radial artery pressure waveforms and a reference aortic pressure waveform from the descending thoracic aorta were simultaneously measured during diverse hemodynamic interventions.

## MATERIALS AND METHODS

### MBSI Technique

Our technique, which was initially presented in abbreviated form (25), applies standard MBSI algorithms from the signal processing literature (1, 28) to two or more peripheral artery pressure waveforms to reconstruct the aortic pressure waveform to within an arbitrary scale factor and then calibrates the reconstructed waveform to absolute pressure based on known physiology. Below, we describe the technique at a conceptual level while stating its underlying assumptions. See the appendix for a full description of the mathematical details of the technique and the discussion for a justification of its assumptions.

Figure 1 illustrates the single-input, multi-output model of the pressure waveforms in the arterial tree on which the technique is based. In this model, the *m* (>1) measured and sampled peripheral artery pressure waveforms [p_{pi}(*t*), 1 ≤ *i* ≤ *m*] are modeled as outputs of *m* unknown systems or channels driven by the common unobserved and likewise sampled aortic pressure waveform [p_{a}(*t*)] input. Each of the discrete-time channels coupling the common input to each of the distinct outputs characterizes the dynamic properties of a different arterial tree path. These channels are assumed to be linear and time invariant (LTI) over each 1-min interval of analysis (see *Methods*). The LTI channels are further assumed to be well approximated by impulse responses [i.e., time-domain version of transfer functions; h_{i}(*t*), 1 ≤ *i* ≤ *m*] that are finite in duration and different from each other. “Different” in this case precisely means that the finite impulse responses (FIRs) are coprime with each other (i.e., the Z transforms of the impulse responses share no common zeros or roots). In this way, all of the commonality in the measured outputs may be attributed to the input, and the differences in the measured outputs (see discussion) may then be deciphered to estimate the FIRs and ultimately reconstruct the common aortic pressure waveform input. Note that it is generally impossible to determine the scale factor of the FIRs and, therefore, the scale factor of the common input, because any scaling of the common input may be offset with a reciprocal scaling of the FIRs. Thus physiological knowledge must be employed to clarify the ambiguity.

More specifically, first, the FIRs are mathematically estimated based on the cross relations between pairs of measured outputs. These cross relations may be derived from the fundamental properties of the convolution operation governing LTI input-output behavior as follows: where *i* ≠ *j* and ⊗ denotes the convolution operation. To obtain intuition about the above cross relations, note that with *m* known peripheral artery pressure waveforms arising from a single unknown aortic pressure waveform input to *m* unknown systems, there is essentially one less equation (*m*) than unknowns (*m* + 1). Thus the cross relations effectively provide the additional equation needed to determine all of the unknowns. In particular, the FIRs are estimated to within an arbitrary scale factor by solving the homogenous system of equations resulting from the cross relations, using the convenient Eigenvector Algorithm (28). The implicit assumption is that the aortic pressure waveform input contains at least as many frequency components as the number of estimated FIR samples (28).

The aortic pressure waveform input is then reconstructed to within an arbitrary scale factor by deconvolving the estimated FIRs from the measured peripheral artery pressure waveforms. A single reconstructed waveform is specifically obtained by employing the multichannel least-squares deconvolution algorithm (1).

Finally, the reconstructed waveform is calibrated to absolute pressure by scaling it to have the same mean value as the measured peripheral artery pressure via a single multiplication. This scaling step is well justified, since the paths from the aorta to peripheral arteries offer very little resistance to blood flow due to Poiseuille's law (17).

It should be noted that the reconstructed absolute aortic pressure waveform will be slightly delayed with respect to the actual aortic pressure waveform, because the time delay shared by the FIRs cannot be identified with MBSI. However, this delay, which is usually <0.1 s, is not important for most clinical applications.

### Methods

We evaluated the MBSI technique with respect to previously collected hemodynamic measurements from swine, which are described in detail elsewhere (16). Below, we briefly describe the experimental procedures employed for collecting these hemodynamic data and then present the methods for the data analysis utilized.

#### Hemodynamic data.

Six Yorkshire swine (30–34 kg) were studied under a protocol approved by the MIT Committee on Animal Care. After the induction of general anesthesia and mechanical ventilation, physiological transducers were placed in each animal as follows. A micromanometer-tipped catheter with high-fidelity frequency response was fed retrograde to the descending thoracic aorta via a femoral artery for reference aortic pressure. Fluid-filled catheters were then inserted in the opposite femoral artery for femoral artery pressure and in an artery as distal as possible to the brachial artery for “radial” artery pressure. Finally, an ultrasonic flow probe was placed around the aortic root following a midline sternotomy for cardiac output. In each animal, a subset of the following interventions was then performed over the course of 75 to 150 min to vary arterial pressures as well as other hemodynamic parameters: infusions of volume, phenylephrine, dobutamine, isoproterenol, esmolol, nitroglycerine, and progressive hemorrhage. Several infusion rates were implemented, followed by brief recovery periods. The hemodynamic waveforms were continuously recorded throughout the intervention period at a sampling rate of 250 Hz and 16-bit resolution.

### Data Analysis

We discarded two of the six swine data sets from the study due to excessive damping of the femoral artery pressure waveform in one data set (16) and an improperly calibrated reference aortic pressure waveform in the other data set. We then applied the technique to all 253 one-minute nonoverlapping intervals of the femoral and radial artery pressure waveforms resampled to 50 Hz (because >99% of the reference aortic pressure waveform energy was usually within 25 Hz) in the remaining four swine datasets. We evaluated the resulting aortic pressure waveform estimates with respect to the measured reference waveforms (likewise resampled to 50 Hz) in terms of standard Bland-Altman plots (to comprehensively illustrate the estimation error including bias μ and precision σ) (3) and the root mean squared error ( to succinctly indicate the total estimation error) of the following parameters: total waveform (i.e., sample to sample), beat-to-beat systolic pressure, beat-to-beat pulse pressure, and beat-to-beat ejection interval. For comparison, we likewise evaluated the peripheral artery pressure waveforms with respect to the measured aortic pressure waveforms in terms of the first three parameters. (Note that we did not attempt to determine the ejection intervals from the peripheral artery pressure waveforms, because the dicrotic notch was generally not visible.) Before conducting these evaluations, we advanced the aortic pressure waveform estimates so that they were temporally aligned with the measured aortic pressure waveforms. To make a fair comparison, we likewise time aligned the peripheral artery pressure waveforms.

## RESULTS

Table 1 shows the hemodynamic parameter range for each of the four analyzed swine datasets. Figures 2 and 3 comprehensively illustrate the estimation errors of the MBSI technique and the corresponding errors of the measured peripheral artery pressure waveforms over all four data sets in terms of Bland-Altman plots, whereas Table 2 succinctly provides these results for each data set in terms of RMSE. (Note that the *x*-axis of the Bland-Altman plots is the measured reference parameter rather than the average of the measured and estimated parameter.) These results generally indicate that the technique was able to reliably estimate the aortic pressure waveform over a wide hemodynamic range with a level of accuracy that was far better than no mathematical analysis of the peripheral artery pressure waveforms.

More specifically, the overall total waveform RMSE of the estimated aortic pressure was 4.6 mmHg (after a modest time alignment as described above). For comparison, the average overall total waveform RMSE between the measured peripheral artery pressures and the measured aortic pressure was 8.6 mmHg (after a more significant time alignment). Thus the technique was able to effectively reduce the total wave distortion in the measured peripheral artery pressure waveforms by 47%. Furthermore, the overall beat-to-beat systolic pressure RMSE and the overall beat-to-beat pulse pressure RMSE of the estimated aortic pressure were 6.1 and 7.1 mmHg, respectively. These errors represent an average overall improvement of 63 and 50% with respect to the corresponding parameters from the measured peripheral artery pressure waveforms. In addition, the overall beat-to-beat ejection interval RMSE of the estimated aortic pressure was 20 ms. Finally, the errors in the four studied parameters of the estimated aortic pressure were generally uncorrelated with the respective reference values of these parameters.

Figure 4, *A* and *B*, provides two visual examples illustrating the significant differences between the measured peripheral artery pressure waveforms and the corresponding measured aortic pressure waveforms, whereas Fig. 4, *C* and *D*, shows the resulting aortic pressure waveforms estimated from these peripheral artery pressure waveforms along with the reference aortic pressure waveforms. As is evident in these examples at two different mean pressure levels, the estimated and reference aortic pressure waveforms agree very closely, and much of the wave distortion in the measured peripheral artery pressure waveforms has been eliminated.

## DISCUSSION

In summary, we have introduced a new technique to mathematically reconstruct the clinically more relevant aortic pressure waveform from multiple, less invasively measured peripheral artery pressure waveforms distorted by wave reflections. Our technique capitalizes on the powerful MBSI approach of recent interest in signal processing in which the differences in two or more outputs of a single-input, multi-output system are assessed to reconstruct the common input to within an arbitrary scale factor. The technique then calibrates the reconstructed waveform to absolute pressure by using Poiseuille's law. As a result, in contrast to previous, related efforts, our technique neither employs a generalized transfer function nor requires any training data and is therefore entirely patient and time specific. We have demonstrated proof-of-concept of the technique through its experimental evaluation in four swine in which radial and femoral artery pressure waveforms and a reference aortic pressure waveform from the descending thoracic aorta were simultaneously measured over a wide hemodynamic range. Our results specifically show that the technique was able to reliably estimate the entire aortic pressure waveform and thereby significantly improve on the determination of systolic pressure, pulse pressure, and the ejection interval compared with measurement of these clinically significant parameters directly from the peripheral artery pressure waveforms.

### Assumptions of the MBSI Technique

As stated above, our MBSI technique is based on a set of assumptions. We make physiological arguments to justify each of the underlying assumptions below.

#### Assumption 1: the channels relating the common input to each distinct output in Fig. 1 are LTI over each 1-min interval of analysis.

Over such short time intervals, the arterial tree is usually operating in near-steady-state conditions in which the statistical properties of the arterial pressure waveforms vary little over time. Such steady-state conditions clearly justify the time-invariance approximation. Moreover, these conditions also support the linearity approximation as argued by McCombie et al. (15) and references therein.

#### Assumption 2: the LTI channels are characterized with FIRs.

Although not widely appreciated, it is known that arterial pressure waveforms measured from distinct sites only differ significantly in terms of their high-frequency detail while being quite similar at lower frequencies (16, 17). Thus the dynamics of each of the channels in Fig. 1 are fast [e.g., effectively vanishing within ∼0.5 s (29)], thereby supporting the FIR approximation.

#### Assumption 3: the FIRs are coprime with each other.

If the FIRs were not coprime with each other, then the noncoprime or common FIR dynamics would be erroneously attributed to the common input. As discussed above, peripheral artery pressure waveforms from distinct sites in the arterial tree appear different. Thus the dynamics of each channel cannot be the same, and the coprime channel approximation is at least somewhat tenable.

#### Assumption 4: the aortic pressure waveform is persistently exciting of high enough order.

This assumption means that the aortic pressure waveform contains at least as many frequency components as the number of estimated FIR samples. As described above, the channel dynamics are of short duration. Thus the number of FIR samples to be estimated may be small enough to buttress the persistence of excitation approximation.

### Potential Sources of Error of the MBSI Technique

Any violation to the four aforementioned assumptions in the present swine study may indeed represent a source of error of our MBSI technique. However, we note that each of the assumptions must have been at least largely valid, given that the discrepancy between the estimated and reference aortic pressure waveforms was relatively small (see Figs. 2 and 3 and Table 2).

Another potential source of error of our MBSI technique is any damping of the peripheral artery pressure waveforms, which were measured with fluid-filled catheters. However, note that only the noncoprime or shared damping dynamics of the two catheter systems would introduce error, whereas the coprime damping dynamics would be attributed to the channels of Fig. 1 and therefore would not affect the estimate of the aortic pressure waveform. Since the employed peripheral artery catheters were not identical (e.g., different in length), at least part of the damping dynamics was likely coprime. In this way, the technique may have been able to at least partly compensate for any damping by the peripheral artery catheters.

### The MBSI Technique and Its Swine Evaluation in the Context of Previous Efforts

The MBSI technique that we have introduced in this report was inspired by the contributions of several previous investigations described in the hemodynamic monitoring literature. In particular, the idea of mathematically deriving aortic pressure from measured peripheral artery pressure stems from the body of generalized transfer function literature (5, 6, 9–11, 21, 22, 24), whereas the idea of employing MBSI to do so in an entirely patient- and time-specific manner is based on the very recent study by McCombie et al. (15).

McCombie et al. (15) specifically proposed a technique using MBSI to reconstruct the shape of the common aortic flow waveform input from multiple peripheral artery pressure waveform outputs and demonstrated its feasibility in a single pilot swine experiment. However, the channels coupling the aortic flow waveform to each peripheral artery pressure waveform include common dynamics, namely, the channel relating the aortic flow waveform to the aortic pressure waveform, and are therefore not coprime. As a result, these investigators had to develop additional signal processing to estimate the common channel dynamics, which resulted in a considerably more complicated algorithm than standard MBSI. Moreover, their framework did not provide an obvious means to determine the scale factor of the reconstructed input. Thus the technique cannot be utilized to monitor changes in cardiac output. In contrast, our MBSI technique aimed to estimate the aortic pressure waveform input in which the coprime channel assumption is more tenable (thereby rendering a relatively straightforward algorithm), and the arbitrary scale factor of the input is conveniently determined by invoking Poiseuille's law.

The previous generalized transfer function studies have, for the most part, aimed to estimate the aortic pressure waveform specifically from the ascending aorta, presumably because this pressure is clearly reflective of cardiac afterload and perfusion. In contrast, we used the aortic pressure waveform measured in the descending thoracic aorta as our reference measurement. Although this reference measurement may be viewed as a potential limitation of our study, we were interested in this particular site because the diastolic pressure intervals have been shown to best resemble pure exponential decays (4). Furthermore, the time constants of these decays have been shown to be extremely strong predictors of relative changes in total peripheral resistance (4). In this way, total peripheral resistance may potentially be monitored from only arterial pressure without the need for a cardiac output measurement. Moreover, although we observed changes in the morphology of the aortic pressure waveform between measurements made from around the aortic arch and the lower descending thoracic aorta in the swine studied (16), the pressure differences at these sites were small in terms of systolic pressure (∼2 mmHg) and pulse pressure (< 0.25 mmHg). These observations are supported by the study of O'Rourke et al. (19) in which the pressure amplitudes at the mean heart rate measured from the ascending aorta, aortic arch, and descending thoracic aorta were shown to be similar in human patients. It is conceivable that small differences in systolic pressure and pulse pressure from the ascending aorta and descending thoracic aorta could make the former a better predictor of patient outcome than the latter. However, we find no previous studies that demonstrate this in the literature. Finally, we note that although we have shown that the MBSI technique was able to reliably estimate the pressure in the descending thoracic aorta with an overall total waveform RMSE of 4.6 mmHg, it is possible that the technique more accurately reconstructs the pressure at another site in the aorta, including ascending aortic pressure.

It is not valid to compare the results of the MBSI technique reported presently with those of previous studies employing generalized transfer functions due to variations in methods for evaluation (e.g., reference measurements as described above) and evaluation data sets (both subject classes and experimental conditions). To obtain an initial fair comparison, we applied the generalized transfer function approach to the four swine data sets studied. We specifically implemented the autoregressive exogenous input-based generalized transfer function described by Fetics et al. (6), which was shown to be the most accurate among three different generalized transfer functions. We created the generalized transfer function by training on an 8-min contiguous interval (similar to that used in Ref. 6) of arterial pressure waveforms. We then applied the generalized transfer function to the remaining 245 min of data and compared the estimated and measured aortic pressure waveforms exactly as we did for our MBSI technique (e.g., at the same sampling frequency and after time alignment and calibration). To attenuate any bias, we repeated the above steps by training on each contiguous 8-min interval in the four swine datasets and testing on the remaining 245 min of data. We then averaged the results to obtain an average overall total waveform RMSE of 5.4 mmHg. This error represents a 17% increase in error with respect to our MBSI technique. We note that the strength of our MBSI technique is that it does not require any training data. It is therefore conceivable that further improvements can be expected when the technique is applied to the diverse combination of patients and pathophysiological conditions seen in clinical practice, which would invariably include scenarios that would be “foreign” to the generalized transfer function. On the other hand, we acknowledge that the cost of this improvement in accuracy is the requirement of more than one peripheral artery pressure waveform for analysis. However, several convenient methods are currently available for measuring peripheral artery pressure waveforms (see above), and new systems are continually in development. For example, it may be possible one day to chronically monitor peripheral artery pressure waveforms with wearable ring sensors (2).

### Future Directions

The present study opens up the possibility of several different avenues of future investigation. Since 1-min intervals of analysis were selected simply to have a sufficient number of pressure samples per estimated FIR parameter, it would be worthwhile to determine the length of the analysis interval that optimizes the accuracy of the MBSI technique. Moreover, as alluded to above, it would certainly be important to establish the precise point in the aorta that best reflects the waveform reconstructed by the technique. In addition, it would be interesting, from a scientific point of view, to establish the optimal sites and number of the peripheral artery pressure measurements (e.g., the arterial sites that result in the most coprime channels and the smallest number of measurements that do not significantly compromise estimation accuracy). Finally, future evaluations of the technique in humans and with respect to noninvasive peripheral artery pressure waveforms are certainly warranted. Ideally, these subsequent investigations would likewise use the analyzed data sets to conduct further comparisons of the technique with the generalized transfer function approach.

### Potential Applications of the MBSI Technique

Our MBSI technique mathematically derives the clinically more relevant aortic pressure waveform from multiple, less invasively measured, but distorted, peripheral artery pressure waveforms without using any training data. The technique may easily be implemented in near real time (with a 1-min delay) using a standard home personal computer. With further development and successful testing, the technique may ultimately be utilized for more precise monitoring and titration of therapy (5) in, for example, hypertension and coronary artery disease patients instrumented with noninvasive arterial pressure transducers (e.g., calibrated with standard sphygmomanometry). In addition, the technique may possibly be used in critically ill patients with invasive catheters installed, although direct applicability would currently be limited because only one peripheral arterial catheter is commonly employed. Advancements in arterial pressure monitoring technology hold further promise for the application of the technique in the context of chronic ambulatory and home monitoring.

## APPENDIX

We outline below all of the mathematical steps of the MBSI technique for the simplest case in which two peripheral artery pressure waveforms are analyzed. See Refs. 1 and 28 for a more general mathematical treatment of the employed MBSI algorithms as well as Refs. 13 and 23 for related background material.

First, the FIRs in Fig. 1 are mathematically estimated to within an arbitrary scale factor based on the following cross relation between the two measured outputs: (A1) The convolution sum has been explicitly written (rather than using shorthand notation as in the initial cross relations equation in the text), and the term e(*t*) has been included to account for any measurement noise and/or modeling error. The variables *L* and *N* in *Eq. A1* respectively represent the number of samples of each FIR (channel order) and the number of measured peripheral artery pressure waveform samples. This equation can be expressed in matrix form by stacking each individual equation corresponding to each time *t*, one on top of the other, as follows: (A2) where are [(*N* − *L* + 1) × *L*] Hankel matrices comprising the respective measured output samples, are [*L* × 1] vectors specifying the samples or parameters of the two respective FIRs, and is an [(*N* − *L* + 1) × 1] vector of the noise samples. For a fixed channel order *L*, the vector **h** in *Eq. A2* is estimated to a certain nontrivial constraint by minimizing the energy in the vector **e**. This optimization problem is specifically solved in closed form by selecting the eigenvector associated with the minimum eigenvalue of the matrix **P**^{T}**P** as a unit-energy estimate of the vector **h** (e.g., see “svd” function in the widely employed MATLAB software package, The MathWorks, http://www.mathworks.com). The channel order *L* is determined by *1*) forming a **P** matrix of dimension [(*N* − *L*_{max} + 1) × 2*L*_{max}], where *L*_{max} = 15 is assumed to encompass the true channel order, *2*) computing the eigenvalues of the matrix **P**^{T}**P**, and *3*) establishing the optimal value of *L* as one-half the number of eigenvalues (rounded up when odd) that are at least 5% of the maximum eigenvalue.

Second, the common aortic pressure waveform input in Fig. 1 is determined to within an arbitrary scale factor from the two determined FIRs (i.e., the estimated vector **h**) and the two measured outputs through multichannel least squares deconvolution. That is, the two measured outputs may be expressed in terms of their common input via the convolution sum as follows: (A3) where n_{i}(*t*) accounts for any noise. This equation also may be expressed in matrix form by stacking each individual equation for each *t* and *i*, one on top of the other, as follows: (A4) where are [*N* × 1] vectors of the respective measured output samples, are the [*N* × (*N* + *L* − 1)] Toeplitz matrices including the estimated samples of the respective FIRs, is a [(*N* + *L* − 1) × 1] vector of unmeasured common input samples, and are [*N* × 1] vectors of the respective noise samples. The vector **p**_{a} in *Eq. A4* is then estimated to within an arbitrary scale factor by minimizing the energy in the vector **n**. This optimization problem is specifically solved in closed form using the following linear least-squares solution: (A5) where the inverse is computed efficiently as described in Ref. 8 (e.g., see “pinv” function in MATLAB).

Third, the reconstructed waveform (i.e., the determined **p**_{a} vector) is calibrated to absolute pressure by scaling it to have the same mean value as that of the measured peripheral artery pressure as follows: (A6) where p(*t*) is the final absolute (scaled) estimated aortic pressure waveform. Note that the above calibration does not explicitly correct for any offset error in the reconstructed waveform. However, both offset and gain are effectively corrected with *Eq. A6*, because the reconstructed waveform is only in error by a scale factor (rather than a scale factor and offset value).

Finally, if the average systolic pressure of the reconstructed aortic pressure waveform is greater than that of the measured peripheral artery pressure, then the solution is considered to be invalid and the above steps are repeated but with the channel order reduced by one. We note that this technique always resulted in a valid estimate of the aortic pressure waveform for every interval of analysis in the present swine study.

## GRANTS

This work was supported by the National Institute of Biomedical Imaging and Bioengineering Grant EB-004444 and an award from the American Heart Association.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2007 by the American Physiological Society