## Abstract

It is generally recommended that an esophageal balloon-catheter possess an adequate frequency response up to 15 Hz, such that parameters of respiratory mechanics may be quantified with precision. In our experience, however, we have observed that some commercially available systems do not display an ideal frequency response (<8-10 Hz). We therefore investigated whether the poor frequency response of a commercially available esophageal catheter may be adequately compensated using two numerical techniques: *1*) an exponential model correction, and *2*) Wiener deconvolution. These two numerical techniques were performed on a commercial balloon-catheter interfaced with 0, 1, and 2 lengths of extension tubing (90 cm each), referred to as configurations L0, L90, and L180, respectively. The frequency response of the balloon-catheter in these configurations was assessed by empirical transfer function analysis, and its “working” range was defined as the frequency beyond which more than 5% amplitude and/or phase distortion was observed. The working frequency range of the uncorrected balloon-catheter extended up to only 10 Hz for L0, and progressively worsened with additional tubing length (L90 = 3 Hz, L180 = 2 Hz). Although both numerical methods of correction adequately enhanced the working frequency range of the balloon-catheter to beyond 25 Hz for all length configurations (L0, L90, and L180), Wiener deconvolution consistently provided more accurate corrections. Our data indicate that Wiener deconvolution provides a superior correction of the balloon-catheter's dynamic response, and is relatively more robust to extensions in catheter tube length compared with the exponential correction method.

- esophageal pressure
- balloon-catheter
- frequency response
- compensation
- Wiener deconvolution

## NEW & NOTEWORTHY

Measurement of esophageal pressure affords calculation of many insightful and clinically important parameters of respiratory mechanics. It is recommended that an esophageal balloon-catheter possess an adequate frequency response up to 15 Hz. In this report we show that when this requirement is not met, it is possible to digitally compensate for the dynamic response of an esophageal balloon-catheter using an exponential model correction, or Wiener deconvolution (whereby superior results are obtained via the latter method).

the measurement of pleural pressure via its surrogate, esophageal pressure (*Pes*) (19, 22, 26), allows for the determination of numerous parameters of respiratory mechanics such as respiratory muscle work, diaphragmatic tension time index, and dynamic lung compliance. Given the importance of obtaining *Pes* in assessing respiratory mechanics, it follows that any device used to measure its value must not distort the signal's amplitude and/or temporal phase. To this end, it is recommended that an esophageal balloon-catheter system should possess a “flat” frequency response up to 15 Hz (22), such that the majority of the spectral content in the respiratory waveform may be observed and faithfully recorded (33). Esophageal balloon-catheter systems cited in the literature have often met this requirement, particularly those fabricated by hand (4, 9, 38). However, it is our experience that some commercially available systems display comparatively poor frequency response characteristics (i.e., <8–10 Hz). A poor frequency response impairs the ability of an esophageal balloon-catheter to faithfully record “true” variations in *Pes* at high respiratory frequencies, such as those engendered by strenuous exercise (i.e., >35 breaths/min). When a poor frequency response is observed, it has been common practice for our laboratory to digitally compensate the esophageal balloon-catheter using an exponential correction method adapted from Arieli and Van Liew (1). However, we have not formally explored whether other numerical methods would provide more accurate corrections. On this point, Bates et al. (2, 3) have consistently reported that Wiener deconvolution (filtering) outperforms the exponential method when compensating for the dynamic response of respiratory mass spectrometers, and bag-in-box systems.

This study investigated whether Wiener deconvolution would indeed prove superior to the exponential method as a numerical technique to correct the poor frequency response of a commercial esophageal balloon-catheter. And given that length is an important determinant of a pressure catheter's dynamic response (17, 18, 39), we examined the relative performance of these two correction methods after the commercial esophageal balloon-catheter had been interfaced with additional lengths of extension tubing.

## METHODS

### Esophageal Balloon-Catheter

The commercial esophageal balloon-catheter (47–9005; Ackrad Laboratories, CooperSurgical, Trumbull, CT) used in this study consisted of an 86-cm-long catheter section (ID ≈1.03 mm) and a 9.5-cm thin-walled polyvinylchloride balloon affixed to the catheter's distal end. The introducing stylet was removed from the balloon-catheter assembly before testing. This commercial system is packaged with a 90-cm extension tube (ID ≈1.9 mm). The esophageal balloon-catheter was also interfaced with +0, +90, and +180 cm of extension tubing. These catheter-length configurations are henceforth referred to as L0, L90, and L180.

The elastic properties of the commercial balloon-catheter system (in each configuration) were assessed by constructing its pressure-volume curve using previously described methods (4, 9). In brief, the distal end of the catheter was suspended in a glass jar filled with water such that the balloon was completely submerged (horizontally) to a depth of 10 cm. At this depth, the balloon was considered devoid of air. The distal end of the catheter was then removed from the jar and left to dry, minimizing the surface tension acting on the balloon wall due to any residual water. While in air, the balloon was inflated via a glass syringe in 0.1–0.2 ml increments. Intraballoon pressure was recorded at each volume, and repeated five times for L0, L90, and L180. The pressure-volume relationship of the esophageal balloon-catheter was not different between L0, L90, and L180 configurations. Thus the mean of all pressure-volume relationships was obtained. The volume at which intraballoon pressure first became atmospheric (0 cmH_{2}O) was defined as the zero-pressure volume (*V*_{0}) (4). The working range of the esophageal balloon (*V*_{range}) was delimited by the volumes corresponding to intraballoon pressures of −0.2 to +0.2 cmH_{2}O [see Fig. 2 in (9)]. Within this range of volumes, the balloon does not itself augment or attenuate the pressure that is transmitted across its walls.

#### Dynamic response.

For all configurations, the esophageal balloon was inflated with a volume of air equal to *V*_{0} determined above. The dynamic response of L0, L90, and L180 configurations were assessed by characterizing the relationship between externally applied pressures (input), and those measured by the balloon-catheter system (output). In so doing, the balloon-catheter's dynamic response (transfer function) was obtained empirically. In brief, the distal end of the balloon-catheter was suspended inside a small-volume chamber. Pressure inside the chamber was rapidly increased to a peak within 3 ms. Due to a small leak, chamber pressure then decayed to 0 cmH_{2}O over the subsequent 10 ms. This process of pressurizing/depressurizing the chamber was repeated 60 times at 6 different levels of peak chamber pressures ranging between ∼40 to 140 cmH_{2}O, for L0, L90, and L180 configurations (1,080 trials in total). The apparatus and procedures used to establish the balloon-catheter's dynamic response are outlined in greater detail in appendixes a and b.

### Data Acquisition

Intraballoon and chamber pressures were recorded using two separate differential pressure transducers (response time <1 ms, range ±352 cmH_{2}O, PX138-005D5V; Omega Engineering, Stamford, CT). Pressure transducers were calibrated using a water manometer before each test. The analog voltage output from the pressure transducers were sampled continuously at 4,000 Hz (PowerLab 16SP; ADInstruments, Castle Hill, Australia).

### Correction Methods

The issue of correcting the dynamic response of the commercial balloon-catheter was addressed by examining the performance of two numerical techniques: *1*) an exponential model correction (1, 3), and (*2*) a Fourier-based method known as Wiener deconvolution (2, 3). The exponential model correction assumes that the behavior of the commercial balloon-catheter adheres to second-order, overdamped linear dynamics—the impulse response of which may be fully determined by two time constants, τ_{1} and τ_{2}. Using these time constants, the observed *Pes* signal may then be corrected using Eq. B3. On the other hand, Wiener deconvolution corrects the dynamic response of the balloon-catheter system by convolving the *Pes* signal with a regularized inverse filter in the frequency-domain (see Eq. B4 and Eq. B5). The theory and implementation of the exponential model correction and Wiener deconvolution are described in greater detail in appendix b.

### Performance of Correction Methods

#### Step response analysis.

The 10–90% rise times (*t*_{10–90%}) of the balloon-catheter system were taken from the simulated step responses as described in appendix c. Settling time was quantified as the time required for the balloon-catheter's simulated step response to settle within ±2% of the final value (i.e., 100%). Overshoot was taken as the percentage difference between the peak and final value of the esophageal balloon-catheter's step response curve. The accuracy of each balloon-catheter's simulated step response was assessed by computing the sum of squared differences (SSD) between the reconstructed step and a synthesized true signal.

#### Frequency response analysis.

The working region of a balloon-catheter's frequency response was defined as those range of frequencies delimited by 0 Hz and the minimum of either the amplitude error (*f*_{A5%}) or phase error (*f*_{φ5%}) frequencies; that is, the bandwidth of frequencies in which the amplitude response may be considered flat to within ±5%, and the phase response to be of a constant negative slope (9, 18, 24, 25). Furthermore, the minimum of either *f*_{A5%} or *f*_{φ5%} was used to estimate the maximal working respiratory frequency (*f*R,max) of each balloon-catheter system (9). The *f*R,max represented the highest respiratory frequency that an esophageal balloon-catheter will transmit the *Pes* wave form with ≤5% amplitude and/or phase distortion. The *f*_{A5%}, *f*_{φ5%}, and *f*R,max were computed using methods described previously by Cross et al. (9).

#### Validation.

Leave-one-out cross-validation (LOOCV) was used to assess how well the two correction methods would generalize to a new (future) data set (42). In general, cross-validation approaches require partitioning of data into “training” and “validation” sets. The desired analysis is performed on the training set, and its performance is evaluated on the validation set. For LOOCV, training data are the entire data set save one, and the validation data are simply the data that were left out. Analyses of training and validation data are performed at this step. This process of analysis/evaluation is repeated until all possible combinations of LOOCV data sets are analyzed. In the present study, we treated each pressurization/depressurization event as a discrete sample. Thus the balloon-catheter transfer functions [i.e., *H*(*s*)], double-exponential correction models, and Wiener filters for all catheter configuration were established using the LOOCV training data sets. Correction performance (e.g., *t*_{10–90%}, SSD, *f*R,max, etc.) was evaluated on all available validation data sets produced by the LOOCV procedure. Average values and 95% confidence intervals were then computed for each index of correction performance for all catheter configurations (L0, L90, and L180).

## RESULTS

The *V*_{0} and *V*_{range} of the commercial esophageal balloon-catheter was 1.2 and 1.2 ml, respectively. The volume displacement coefficient of the entire balloon-catheter-transducer system was 0.00032, 0.0031, and 0.0031 ml/cmH_{2}O for the L0, L90, and L180 configurations, respectively. The dynamic response characteristics of the uncorrected and corrected esophageal balloon-catheter systems are reported in Table 1. The *t*_{10–90%}, settling time and SSD of the uncorrected balloon-catheter increased as the catheter tubing was extended (i.e., from L0 to L180). In contrast, the working range of the uncorrected catheter's frequency response progressively decreased as the extension tube length was increased.

The optimal values for τ_{1} and τ_{2} used in the exponential model corrections for L0, L90, and L180 configurations were 4.486 ± 0.631 and 0.214 ± 0.560 ms, 14.025 ± 0.110 and 0.022 ± 0.052 ms, and 24.258 ± 0.524 and 0.006 ± 0.025 ms, respectively. The optimal values for constants *A* and *B* used in the construction of the Wiener filters for L0, L90, and L180 configurations were 0.0004 ± 0.0001 and 0.0312 ± 0.0007, 0.0004 ± 0.0001 and 0.0306 ± 0.0001, and 0.0004 ± 0.0001 and 0.0280 ± 0.004, respectively. Wiener filtering produced the most accurate reconstructions of the simulated pressure step response of all balloon-catheter configurations (Table 1 and Fig. 1). Moreover, Wiener filtering consistently produced the best enhancement of the esophageal balloon-catheter's frequency response for L0, L90, and L180 configurations (Table 1 and Fig. 2).

## DISCUSSION

This report examined whether the dynamic response of a commercial esophageal balloon-catheter can be adequately compensated using the two numerical techniques, exponential model correction and Wiener deconvolution (filtering). Our findings were threefold: *1*) the uncorrected balloon-catheter system exhibited a poor dynamic response that worsened as the catheter length was extended; *2*) exponential correction and Wiener filtering adequately compensated for this poor dynamic response when the catheter system was interfaced with +0, +90, and 180 cm extension tubing (i.e., L0, L90, and L180); and *3*) Wiener filtering provided consistently more accurate corrections of the balloon-catheter's dynamic response.

### Uncorrected Commercial Balloon-Catheter

It was mentioned earlier that an esophageal balloon-catheter system should display a “flat” frequency response up to 15 Hz (22). Here, the term flat refers primarily to the ability of the balloon-catheter to faithfully transmit the *Pes* waveform with minimal amplitude distortion. We extended this definition to incorporate the requirement that the *Pes* waveform must also display a linear phase response over the same frequency broadband (i.e., 0–15 Hz). Thus the frequency interval shared by these two regions was herein defined as the working range of the esophageal balloon-catheter system. To this end, the working range of the uncorrected esophageal balloon-catheter fell below 15 Hz for L0, and progressively worsened as extension tubing was added. The corresponding maximal working respiratory frequencies (i.e., *f*R,max) also declined with increasing tubing length. Thus, according to the recommendation above, the dynamic response of this particular commercial balloon-catheter may be considered inadequate for the recording of dynamic swings in *Pes*. We note, however, that although the working frequency range of the balloon-catheter fell below 15 Hz in the L0 configuration, the corresponding *f*R,max (i.e., 63 breaths/min) remained higher than that typically observed during the tachypnea of strenuous exercise (i.e., >35 breaths/min) (5, 6, 11, 12, 20, 21). It may therefore be reasoned that this balloon-catheter is suitable for recording *Pes* during exercise without correction, provided that no extension tubing is used. The relatively poor dynamic response of the uncorrected balloon-catheter does, however, obviate its use in determining lung and/or chest wall impedance during forced oscillation spirometry, in which applied pressures often vary within the range of 4 to 32 Hz (34).

We surmise two reasons for the generally poor dynamic response of the commercial balloon-catheter examined in this study. First, the luminal diameter of this catheter (∼1.0 mm) is narrower than is generally recommended (1.4 mm) for measuring esophageal pressure swings in adults (22). The relatively narrow catheter lumen likely increased the resistance of propagating pressure waves along the catheter tubing. Second, we observed only four pairs of lateral holes (eight total) on the commercial catheter's tubing within the region encompassed by the balloon. These perforations are extremely important because they are the only means of communicating pressure swings from within the balloon to the catheter lumen and, ultimately, the pressure transducer further downstream. When esophageal balloon-catheters are fabricated by hand, catheter tubing within the balloon region is typically perforated >8–10 times bilaterally, and in a spiral-like fashion (i.e., >16 holes). The larger the number of holes, the smaller is the effective flow resistance between the balloon volume and catheter lumen. We conjecture that the poor dynamic response of this particular commercial balloon-catheter system is secondary to the increased resistance and, therefore, greater viscous damping imposed by the relatively narrow catheter lumen and small number of communicating holes.

### Evaluation of Correction Methods

We used the esophageal balloon-catheter manufactured by Ackrad Laboratories to quantify *Pes* dynamics in several previous studies (7, 8, 10, 12). In those experiments, the balloon-catheter's frequency response was digitally compensated with custom-written software using the exponential correction method. The current report serves to vindicate this approach, insofar as we show that exponential model correction improves the accuracy of the balloon-catheter's simulated step response, and extends its working frequency region to beyond 15 Hz for L0, L90, and L180 configurations. The primary advantage of using this correction method is the simplicity of its numerical implementation. Once τ_{1} and τ_{2} are determined, only a small amount of data are required at any given time point to compute the first- and second-order derivatives for inputting into Eq. B3. Furthermore, in our analyses, the optimal time delays for exponential corrections did not exceed 12 samples (i.e., 3–6 ms). This method is therefore well suited for real-time implementation. A clear disadvantage of the exponential correction method is that it assumes a specific morphology of the balloon-catheter's impulse response (see Eq. B2). As such, this method may perform well for balloon catheters that display second-order overdamped systems behavior, but not those evidencing higher-order, underdamped response characteristics (9). The exponential method may be extended to incorporate more complex impulse response behavior in such cases (2). What is preferable, however, is a correction method that can be applied to all esophageal balloon-catheters, not just those that adhere to the system's behavior defined by a single equation or model.

Wiener deconvolution is a model-free method of signal correction because it does not assume any specific morphology of the balloon-catheter's impulse behavior. Consequently, this numerical method may be used to correct the dynamic response of any esophageal balloon-catheter, provided its transfer function [i.e., *H*(*s*)] can be measured. In the present study, Wiener filtering yielded the most accurate reconstructions of the balloon-catheter's step response, and extended its working frequency region to well beyond 50 Hz for all length configurations. These observations are similar to those made by Bates et al. (2, 3), who reported that Wiener filtering consistently outperformed the exponential correction method when digitally compensating the dynamic response of respiratory mass spectrometers and bag-in-box spirometers. Although our data clearly indicate that Wiener filtering afforded the most desirable correction of the commercial balloon-catheter, there are some aspects of its implementation that are worth mentioning.

Wiener filtering requires significantly more computation than the exponential correction method. For example, if the observed signal *y*(*t*) has a length equal to the Fourier size (*N*) of the Wiener filter [i.e., *H*^{−1}(*s*)·*W*(*s*); see appendix b], Wiener deconvolution may be implemented in three steps:

*1*) *y*(*t*) is Fourier transformed to *Y*(*s*);

*2*) *Y*(*s*) is multiplied by *H*^{−1}(*s*)·*W*(*s*) to obtain *X̂*(*s*); and

*3*) *X̂*(*s*) is inverse Fourier transformed to yield *x̂*(*t*), where *x̂*(*t*) and *X̂*(*s*) are the time- and frequency-domain realizations of the estimated “true” signal.

In practice, the length of *y*(*t*) is often much greater than *N*, in which case the three steps above are repeated on multiple block subsections of *y*(*t*) using either the overlap-add or overlap-save algorithms (27). Fortunately, with present-day hardware, the implementation of Wiener filtering in one dimension imposes little computational burden, particularly for applications in which signal correction may be deferred until postprocessing. On the other hand, if real-time signal correction is desired, one must first wait until at least *N* samples become available in the input stream before deconvolution can be performed. The ideal *N* is the minimum base-2 number of samples in the time-domain that capture the entire impulse response of the balloon-catheter. In this study, a length of 4,096 samples was found to be adequate. Accordingly, the expected latency incurred with real-time implementation of our Wiener filter would be at least 1 s (i.e., latency = *dt*·*N* = 1.024 s). Thus although Wiener filtering offers superior correction performance, one may still prefer to use the exponential correction method if real-time monitoring of *Pes* is desired.

### Practical and Methodological Considerations

It is emphasized here that only a single balloon-catheter was examined in this report. Although our experience is that this particular commercial system displays poor dynamic response characteristics (i.e., working range <8–10 Hz), our observations may not generalize to all esophageal balloon-catheters manufactured by Ackrad Laboratories. Rather, the intent of this report is to demonstrate that if one observes an inadequate dynamic response, the esophageal balloon-catheter may be compensated using either of the two numerical techniques examined herein. We further stress that these correction methods should be used only on the particular balloon-catheter system from which they were constructed. Indeed, because of minor differences in transfer functions between systems [i.e., *H*(*s*)], the exponential model and Wiener filter constructed for one balloon-catheter may not yield adequate correction performance when applied to a second catheter, even when produced by the same manufacturer. It would therefore appear prudent to determine *H*(*s*) on a per catheter basis.

We estimated *H*(*s*) via 60 repeated pressurization/depressurization events across 6 different magnitudes of peak chamber pressure for each length configuration of the balloon-catheter—a total of 1,080 events that took well over 90 min to complete. Such an extended period of data collection was chosen to comprehensively assess the linearity of the balloon-catheter system for the purposes of the present report. Certainly, we do not recommend that this exhaustive process be replicated each time an investigator wishes to characterize and correct the dynamic response of a balloon-catheter. It is instead suggested that the investigator determine the least number of trials necessary to estimate *H*(*s*) with less than ±5% variability over the frequency interval of interest (i.e., 0–15 Hz). It is also worth mentioning that investigators are not bound to determine *H*(*s*) using only the apparatus described in this study. There are many other ways to generate the requisite input-output response data for empirical transfer function analysis, including but not limited to rectangular pulses using a square-wave generator (35), sweeping sine waves via a loudspeaker chamber (13, 16, 28), or pseudorandom binary sequences using fast-response solenoids (31). It is, however, difficult to institute these methods in the laboratory given the considerable amount of hardware and software engineering required to properly construct and operate such devices. Rather, simpler, more primitive methods can be fathomed, such as bursting the finger cot of a surgical glove sealed over the end of a pressurized tube (30), or by rapidly withdrawing a plunger from the barrel of a syringe (23) connected to an enclosed rigid chamber. If one can ensure that the input pressure waveform contains “substantially non-zero” spectral content across the frequency broadband of interest, and that proper preprocessing of data is performed, then any given method/device conceivably may be used to estimate of *H*(*s*). It is therefore the prerogative of the investigator to choose a method of estimating *H*(*s*) that is best suited to the materials and expertise available within the laboratory.

Regarding the correction methods, per se, it is acknowledged that only two numerical techniques were examined in this report (i.e., exponential correction and Wiener filtering). Many other techniques may be used for dynamic response correction, such as recursive inverse filtering, neural networks, and causal FIR Wiener filtering (14, 29, 32, 36, 37, 41). It would therefore be helpful to determine whether better correction performance is obtained using such techniques in future investigations.

### Summary

This study demonstrates that one cannot assume that an esophageal balloon-catheter will possess an adequate dynamic response (i.e., ≥15 Hz) simply because it was purchased from a commercial source. Furthermore, this report provides a clear illustration of the effect that extension tube length bears on the dynamic response of an esophageal balloon-catheter (i.e., longer catheters engender poorer frequency response characteristics). Finally, of the two numerical methods examined herein, Wiener filtering offered a superior correction of the catheter's dynamic response. This correction method was also robust to extensions in catheter length; an important point for situations in which the experimental setup necessitates a long intervening distance between the subject and pressure transducer (e.g., during optoelectronic plethysmography). Notwithstanding the above, the exponential model correction, though less accurate, may be preferable in applications in which (near) real-time display of *Pes* is desired. In summary, we encourage investigators to report on the dynamic response of their balloon-catheter systems in future publications, and to state which method of dynamic compensation they used (if applicable). In so doing, a greater confidence is afforded in parameters derived from the *Pes* waveform, particularly when respiratory frequency is notably high (i.e., voluntary hyperventilation, strenuous exercise, etc.).

## GRANTS

T.J. Cross was supported by a joint Mayo Clinic and Griffith University postdoctoral fellowship. This study was supported by National Heart, Lung, and Blood Institute Grant HL-71478.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

T.J.C. and B.D.J. conception and design of research; T.J.C. performed experiments; T.J.C. analyzed data; T.J.C. and K.C.B. interpreted results of experiments; T.J.C. prepared figures; T.J.C. drafted manuscript; T.J.C., K.C.B., and B.D.J. edited and revised manuscript; T.J.C., K.C.B., and B.D.J. approved final version of manuscript.

## ACKNOWLEDGMENTS

We thank Jennifer Isautier for her insightful comments during preparation of the manuscript. We also thank Alex Carlson for his technical advice and expertise in performing the experiments.

## Appendix A: APPARATUS AND PROCEDURES FOR ASSESSING THE DYNAMIC RESPONSE OF AN ESOPHAGEAL BALLOON-CATHETER

A schematic of the apparatus used to evaluate the dynamic response of the balloon-catheter is illustrated in Fig. 3. The distal end of the balloon-catheter was suspended inside a polyvinylchloride (PVC) pipe with a small internal volume (length 20 cm, diameter 2 cm, internal volume 63 cm^{3}). Internal chamber pressure was recorded from a lateral port bored into the side of the PVC pipe at a site approximately half its length. The ends of the pipe were sealed using prefabricated acrylic plugs. These plugs were sequestered from the drying cartridge of a metabolic cart (MedGraphics CPX/D; Medical Graphics, St. Paul, MN). The plastic barb fitting was removed from one plug through which the esophageal balloon-catheter was passed without damage. A small length of electrical tape was wound circumferentially around the balloon-catheter at a site roughly 1 cm proximal to the balloon. (Care was taken to avoid pinching the catheter as the tape was applied.) The taped section provided a loose seal when the balloon-catheter was retracted toward the proximal plug. The distal plug was directly connected to an electromagnetic solenoid valve (orifice diameter 0.076 mm, VSO LF; Parker Hannifin, Cleveland, OH). The pressure upstream of the solenoid valve was a regulated source of medical-grade helium at 75 pounds per square inch. The solenoid was activated via an optically isolated relay (G4ODC5A; Opto22, Temecula, CA) with 5-V transistor-transistor logic using the analog output channel of the PowerLab data acquisition module (PowerLab 16SP; ADInstruments, Castle Hill, Australia). The width of the 5-V pulse controlled the duration that the solenoid was held open.

The 5-V pulse width was initially set to 1.5 ms, which was sufficient to rapidly increase the pressure inside the chamber to roughly 40 cmH_{2}O within <3 ms. However, because the apparatus was not airtight, the chamber pressure decayed to atmospheric pressure over the proceeding 10 ms. This process of rapid pressurization/depressurization was repeated 60 times on a period of 5 s. The externally applied and intraballoon pressure waveforms were then ensemble-averaged across the 60 repeated events. To test the assumption that the commercial balloon-catheter was linear within a physiological range, it was necessary to repeat this process of pressurization/depressurization at varying levels of peak chamber (input) pressure. As such, the width of the 5-V pulse was progressively increased (approximately 1.5–5.0 ms) in a manner that produced six different peak chamber pressures ranging from ∼40 to 140 cmH_{2}O. (Preliminary testing ensured that the esophageal balloon did not collapse when exposed to external pressures up to 150 cmH_{2}O.) Figure 4 (*top*) displays the ensemble averages of both the externally applied and intraballoon pressure waveforms at each level of peak chamber pressure for the balloon-catheter in the L0 configuration. These “input-output” pressure wave form pairs were then used to construct transfer functions [i.e., *H*(*s*)] that were representative of each peak chamber pressure, for each length configuration (see appendix c).

To adequately characterize the frequency response of any device, it is imperative that the input signal contains substantially non-zero spectral content across the frequency broadband of interest (i.e., at least up to 15 Hz in our case). We defined “substantial non-zero content” as the range of frequencies over which spectral power of the applied (input) pressure waveform was greater than the noise variance of the signal in the time domain. Noise variance was computed from a 10-s recording during which no pressure was applied. Using this criterion, we observed substantial non-zero content in the externally applied pressure spectrum up to a minimum of 200 Hz across all levels of peak chamber pressure.

## Appendix B: NUMERICAL TECHNIQUES FOR CORRECTING THE DYNAMIC RESPONSE OF AN ESOPHAGEAL BALLOON-CATHETER

The pressure that is measured by an esophageal balloon-catheter is a convolution of the “true” input signal, and the catheter's impulse response. Furthermore, any measurement of pressure is likely to be corrupted by additive noise. In this sense, the true pressure signal is related to the observed signal by: (B1)

where ∗ denotes convolution; *h* is the impulse response of the balloon-catheter; and *y*(*t*), *x*(*t*), and *n*(*t*) are the observed, true, and noisy time series, respectively. Thus if the balloon-catheter's impulse response function, *h*(*t*), and the signal noise, *n*(*t*), are known, then the “true” signal, *x*(*t*), may be recovered via deconvolution. The challenge, therefore, is to accurately quantify *h*(*t*) of the balloon-catheter system under test.

#### Exponential Model

Through our experience in working with the commercial balloon catheters manufactured by Ackrad Laboratories, we have observed that these devices exhibit second-order, underdamped linear systems behavior. The impulse response of such a system can be modeled as (2, 3): (B2)

For a real physiological signal, an estimate of the true signal [*x̂*(*t*)] may be obtained via the second-order differential equation (1, 3, 15, 22):
(B3)

where *ẏ*(*t*) and *y¨*(*t*) are the first- and second-order time derivatives of the observed signal, *y*(*t*). It can be seen from Eq. B3 that only two parameters are required a priori to invoke this correction method; namely, τ_{1} and τ_{2}.

We estimated τ_{1} and τ_{2} empirically via least-squares minimization of the difference between the true pressure waveform, *x*(*t*), and the estimated signal, *x̂*(*t*). The true waveform, *x*(*t*), was created by ensemble averaging the chamber pressure response over the 60 repeat trials at a given level of peak chamber pressure and length configuration (see appendix a). Similarly, the observed signal, *y*(*t*), was taken as the ensemble-average of the intraballoon pressure response. Because ensemble averaging significantly reduced the measurement noise, the first and second time derivatives of the observed signal [i.e., *ẏ*(*t*) and *ÿ*(*t*)] were obtained using central finite differences with an interval Δ*t* of 5 ms. Least-squares minimization was performed over a range of time delays between *x*(*t*) and *x̂*(*t*) to accommodate any temporal shifts introduced by the differentiation of the signal. The time delay corresponding to the least-squared error was chosen as the optimal value.

#### Wiener Filtering

The method of Wiener deconvolution (filtering) is perhaps best understood by first transforming Eq. B1 into the frequency domain, such that: (B4)

where *Y*(s), *X*(s), *H*(s), and *N*(s) are the discrete Fourier transforms of the respective time series outlined in Eq. B1. Note that convolution in the time domain corresponds to multiplication in the frequency domain. To recover *X*(s) from *Y*(s), we may write:
(B5)

The true time series, *x*(*t*), is then recovered via inverse Fourier transformation of *X*(s). Unfortunately, the magnitude of *H*^{−1}(s) is inexorably large for all frequencies outside the passband of the original impulse response function. Accordingly, without regularization, Eq. B5 will produce a very noisy estimate of *X*(s). To improve the estimation of *X*(s), *H*^{−1}(s) is regularized by an auxiliary filter function, such that:
(B6)

where *X̂*(*s*) is the estimate of *X*(s) and *W*(s) denotes the auxiliary filter function that regularizes *H*^{−1}(s). The identity of *W*(s) is optimal when the mean-squared error between *x*(*t*) and its estimate, *x̂*(*t*), is minimized. Such minimization is achieved when the noise-to-signal ratio (NSR) of the system is known a priori, whereby:
(B7)

It follows from Eq. B7 that the magnitude of *W*(s) will approach 1 as NSR(s) tends to zero. In which case, *X̂*(*s*) in Eq. B6 is equivalent to that obtained with a simple inverse filter, *H*^{−1}(s). On the other hand, *W*(s) will regularize *H*^{−1}(s) wherever the signal spectrum is dominated by measurement noise (i.e., input frequencies at which NSR is high). It is, however, difficult to quantify NSR without precise knowledge of the measurement noise process. Instead, previous investigators have replaced NSR(s) in Eq. B7 with a surrogate frequency-dependent function (2, 3). This function typically increases with *s*, taking the general form of: Φ(*s*) = *Ae*^{Bs}, whereby the constants *A* and *B* dictate the amplitude and curvature of an exponential function. Thus in light of the above, the identity of *W*(s) is herein defined by:
(B8)

The transfer function, *H*(*s*), of each balloon-catheter system was determined empirically via spectral division of *Y*(*s*) by *X*(*s*). Here, *Y*(*s*) and *X*(*s*) are the Fourier transforms of the ensemble-averaged intraballoon and chamber pressure responses obtained during the dynamic testing procedure. Data at frequencies at which |*H*(*s*)| <10e^{−6} were set to 0 to avoid numerical instabilities during inversion of *H*(*s*) in Eq. B6 and Eq. B7. The Fourier size of *Y*(*s*), *X*(*s*), and thus *H*(*s*) were equal to the number of samples of each trial (*n* = 5 s·4,000 samples/s = 20,000 samples). The length of *n* was reduced by *1*) inverse Fourier transforming *H*(*s*) to yield the impulse response *h*(*t*); *2*) truncating *h*(*t*) to 4,096 samples; *3*) renormalizing the integrated area of the impulse response to 1; followed by *4*) Fourier transformation of *h*(*t*) back to *H*(*s*).

It was reported earlier that the externally applied pressure waveforms contained a substantial amount of spectral energy up to a minimum of 200 Hz across all levels of peak chamber pressure. Therefore, we sought to regularize *H*(*s*)^{−1} such that the integrated spectral power above 200 Hz in the cross-spectrum of *H*(*s*)^{−1}·*W*(*s*) was no greater than twice that observed in *H*(s) over the same frequency interval.

This regularization was achieved via least-squares optimization of the constants *A* and *B* of the Φ(*s*) in Eq. B8. This criterion for determining Φ(*s*), and thus *W*(*s*), provided a good trade-off between amplifying high-frequency noise, and enhancing the step response (see below).

## Appendix C: OBTAINING THE FREQUENCY RESPONSE AND SIMULATING THE STEP RESPONSE

The transfer function [*H*(*s*)] of each balloon-catheter system was computed via spectral division of the output by the input signal spectrums [i.e., *H(s) = Y*(*s*)/*X*(*s*)]. Obtaining *H*(*s*) in this manner is referred to as nonparametric or empirical transfer function analysis. The amplitude- and phase-frequency responses are then obtained by taking the absolute and complex arguments of *H*(*s*), respectively. An important assumption of empirical transfer function analysis, and of the exponential and Wiener correction methods, is that the balloon-catheter adheres to linear systems behavior. That is, *H*(*s*) must not itself be affected by the magnitude/amplitude of the input signal used to excite the system (40). We tested this assumption by computing *H*(*s*) at six different levels of peak chamber pressure during the rapid pressurization/depressurization events described earlier. Figure 4 (*bottom*) displays the mean ±95% confidence intervals (CI_{95%}) of the amplitude- and phase-frequency responses from these six conditions. It can be seen from this figure that the CI_{95%} width across the frequency broadband of interest (i.e., 0–15 Hz) is negligible for both amplitude and phase data. Moreover, the CI_{95%} width of the amplitude-frequency response is no greater than 0.035 (3.5%) even at its widest point (30–35 Hz). It is emphasized that only data for the L0 configuration is presented in Figure 4 because it was only under this condition that the balloon-catheter displayed a measurable (albeit very minor) degree of variability in amplitude and phase data across the six different levels of peak chamber pressure. Taken together, we believe these data provide convincing evidence that the commercial balloon-catheter examined herein was linear within a reasonable range of input pressures (40–140 cmH_{2}O).

The exponential correction and Wiener filtering were performed on the intraballoon pressure waveforms collected during the dynamic testing described above. Ensemble averages of these corrected waveforms were obtained from the 60 repeat trials. The Fourier transform of these corrected signals yielded *X̂*(*s*). The transfer function, *H*(*s*), of each balloon-catheter system after correction was obtained via spectral division of *X̂*(*s*) by *X*(*s*), where the latter is the Fourier transform of the ensemble-averaged chamber pressure response (i.e., the system input). The empirical transfer function, *H*(*s*), characterizes the frequency response of the balloon-catheter. Having thus determined *H*(*s*) for each method, we were then able to use Eq. B4 to simulate the corrected signal spectrum for any arbitrary input spectrum. Here, we used the Fourier transform of a square-wave as the input spectrum, *X*(*s*). The measurement noise spectrum, *N*(*s*), was estimated as the Fourier transform of intraballoon pressure data acquired during a 10-s quiet period. The corrected spectra for each balloon-catheter were then inverse Fourier transformed to simulate the systems' time-domain response to a unit increment in pressure. These time-domain data were used to analyze features of the balloon-catheter's step response.

- Copyright © 2016 the American Physiological Society