Abstract
Respiratory flow profiles have been of interest as an output of the respiratory controller. In determining average flow profiles, however, previous methods that align individual breaths in the time domain are susceptible to distortions caused by the great variability, both between breaths and within breaths. We aimed to develop a method for determining typical flow profiles that circumvents such distortions. Our method aligns different breaths by phase of respiratory cycle, which is defined as the angle associated with the point on the normalized flowvolume diagram (a phaseplane plot). Over a number of breaths, median values for flow, volume, and elapsed time from the start of the breath at each phase angle are determined. Because these estimates are mutually semiindependent and in general violate the laws of mass balance, an adjustment was performed such that the volume was precisely the time integral of the flow. The method produced typical flow profiles with characteristics that were significantly closer to the mean values obtained from the individual cycles than those obtained by the technique of Benchetrit and coworkers (Benchetrit G, Shea SA, Dinh TP, Bodocco S, Baconnier P, and Guz A, Respir Physiol 75: 199–210, 1989), which reconstructs the typical flow profile from Fourier coefficients.
 timedomain alignment
 respiratory phasebased alignment
 Benchetrit method
the pattern of respiratory airflow during steady breathing has been studied as a manifestation of various respiratory neural, chemical, and behavioral control mechanisms. Several studies have suggested that the airflow pattern may be optimized according to chemical and mechanical criteria (5, 9, 12, 17). Painter and Cunningham (11) indicated that different respiratory stimuli generated different airflow patterns without affecting the inspiratory or expiratory duration or the tidal volume in the normalized breath. On the other hand, halothane, an inhalational anesthetic, did not affect the airflow profile, although it changed both the tidal volume and the respiratory rate in a dosedependent manner (8, 13). There may be some genetic determinism in the function of the respiratory controller by which the airflow pattern is individually determined (2, 14, 15). Patients with chronic obstructive lung diseases have been reported to possess a characteristic inspiratory flow waveform (1, 7).
Awake humans essentially exhibit irregularities in the depth, frequency, and general pattern of breathing. Although the irregularity, or the temporal variability, of breathing itself has been studied as an expression of oscillatory regulation by the respiratory center (3), it makes the process of obtaining a representative, or average, breath difficult and open to various forms of distortion. Consequently, different approaches to this problem have been adopted. The first is, instead of obtaining a complete description of the respiratory cycle, to calculate averages for certain features of the respiratory cycle, such as the ratios of inspiratory and expiratory durations to total respiratory duration, and values for the peak and mean inspiratory flows (3, 13). The second is to partition the respiratory airflow into many equal segments for inspiration and for expiration and then averaging the flows in these segments across breaths (8, 11). The third is to obtain the Fourier coefficients for the fundamental and first three harmonics of the airflow for each breath and then averaging the Fourier coefficients over a number of breaths (2, 14, 15). Of these techniques, only the third developed by Benchetrit et al. (2) produces a continuous value for flow throughout the respiratory cycle.
The first purpose of the present study is to explore the degree of distortion inherent in the method of Benchetrit et al. (2). In this technique, alignment of the breaths at the beginning of inspiration is implicit in the process of averaging across breaths. A particular problem of averaging in the time domain is that, in general, individual breaths do not all reach the same stage of their respiratory cycle at the same time. For example, if each breath is deemed to start at the beginning of inspiration and have a duration of unity, then averaging the flows at time = 0.4 is likely to result in averaging the flows of some breaths that are still in inspiration with other breaths that are already in expiration. We compare the results of this method with those obtained from a comparable method, which we term “reverse Benchetrit,” where the Fourier representation of each breath is the same but the process of averaging across breaths involves aligning the breaths at the inspiratoryexpiratory transition.
One possible way of avoiding this problem is, instead of considering the relationship between flow and time, to consider the relationship between volume and flow. It allows a phase to be defined based on the angle subtended by a flowvolume point with respect to the origin. The second purpose of the present study is to develop a method for averaging respiratory cycles aligned with respect to this phase rather than with respect to time.
METHODS
Data.
Data were obtained from 6 subjects who were at rest breathing air through a mouthpiece. For each subject, 45 min of data were collected. Respiratory flow was obtained by using an unheated pneumotachograph and was corrected for variations in gas composition and temperature as described in the .
Removal of the drift in the volumetime record.
Despite careful corrections, a volume signal such as ours, obtained by integration of a flow signal, is likely to show a slow drift or “integration error.” To remove such drift, a smoothing spline was employed to estimate the drift in the volume signal (6). This drift was then subtracted from the volume signal. After the subtraction of the drift, the volume signal was given zero mean, and the associated flow signal was obtained by differencing the volume signal.
Selection of data for analysis.
All analyses in this study were conducted on the signals after the removal of drift. For each subject, the first 5 min of data were discarded. Data were then split up into 3min epochs, giving 13 such sections of data within each subject for study.
Calculation of the Benchetrit and reverseBenchetrit average flows.
The calculation of the Benchetrit average flow is as described by Benchetrit et al. (2) and is illustrated in Fig.1. Each breath within a 3min period is treated as of unit duration, with time starting at the beginning of inspiration. The fundamental and first three harmonics of the response were estimated for each breath individually by using multiple linear regression. The original Benchetrit method performs the Fourier transform to obtain the amplitudes and phases of the four harmonics. Our procedure is simply the timedomain version of the Benchetrit method and reconstructs exactly the same airflow profile for each breath. The Fourier coefficients were then averaged to produce one set of coefficients describing the average cycle. Because time 0 was defined as starting at the beginning of inspiration, this had the effect of aligning all breaths at the beginning of inspiration. A problem in this method is that the reconstructed flow is bound neither to start inspiration with the value of zero nor to end expiration with the value of zero, because both the Fourier transform and averaging are purely mathematical processes and do not consider physiological consistency. This phase inconsistency was corrected simply by shifting the respiratory phase of the average breath such that the inspiration started with zero flow and the expiration ended with zero flow.
The reverseBenchetrit method is also illustrated in Fig. 1. Again, each breath within a 3min period is treated as of unit duration, but in this case time 0 is treated as occurring at the inspiratoryexpiratory junction. The fundamental and first three harmonics of the response were estimated for each breath individually by using multiple linear regression. The Fourier coefficients were then averaged to produce one set of coefficients describing the average cycle. In this case, because time 0 was defined at the inspiratoryexpiratory junction, this had the effect of aligning all breaths at the inspiratoryexpiratory junction. The respiratory phase inconsistency produced in the average breath was corrected as in the Benchetrit averaging.
Calculation of phasealigned averaged flow.
This calculation is illustrated in Fig.2. First, the volume and flow signals were divided by their respective standard deviations to give an approximately even scaling between the volume and flow axes (Fig.2 B). Next, for each individual breath, values for flow, volume, and elapsed time since the start of the breath were obtained at each integer degree of phase by interpolation between the two closest values on either side of the angle (Fig. 2 C). After this procedure, median values for flow, volume, and elapsed time since the start of the breath could be obtained for each degree of phase (Fig.2 D). Finally, the normalization of the median volume and median flow signals by their original standard deviations was removed (Fig. 2 E).
One problem remains with these estimates for flow, volume, and elapsed time through the respiratory cycle. Because the values are semiindependent statistical estimates, the requirement for the volume to be precisely the time integral of the flow is lost. To reintroduce this physical requirement, inspiratory and expiratory correction factors for flow were calculated such that the time integral of flow for both inspiration and expiration was exactly equal to the median tidal volume. Values for volume through the respiratory cycle can then be obtained, if required, by integration of this corrected flow signal.
Comparison of the three methods in simulated pathological conditions and effects of the number of harmonics in the Benchetrit and reverseBenchetrit methods.
To examine how the three methods might work in pathological conditions, we applied the three methods in subjects breathing against inspiratory resistive and elastic loading. We also tested, using the same data, whether increasing the number of harmonics reduced the distortions in the average airflow profile produced by the Benchetrit and reverseBenchetrit methods. The numbers of the harmonics examined were 8 and 16, compared with 4 for the original Benchetrit method.
Statistics.
The performance of the three methods was tested by comparing flow profile variables and respiratory cycle variables calculated from the average flows with the arithmetic means of the variables obtained from the individual breaths in each 3min epoch. The flow profile variables used were the rate of the change of flow at the beginning of inspiration and the rate of the change of flow at inspiratoryexpiratory transition. Respiratory cycle variables included those for inspiration (inspired tidal volume, inspiratory time, time to maximum inspiratory flow, and maximum inspiratory flow) and for expiration (expired tidal volume, expiratory time, time to maximum expiratory flow, and maximum expiratory flow). For the flow profile variables, statistical comparisons between the methods were performed by applying oneway repeatedmeasures ANOVA to each subject separately. For the respiratory cycle variables, on the other hand, estimation errors were defined as values outside the 95% confidence intervals of the means calculated from individual breaths in the 3min data. The numbers of the 3min epochs in which each method produced estimation errors were compared among the three methods by oneway ANOVA.
RESULTS
Figure 3 illustrates the Benchetrit and reverseBenchetrit average flows for one set of 3min data. Neither average represents the features of the individual breaths well. For example, in the case of the Benchetrit average, it is very apparent that the rate of change of flow at the inspiratoryexpiratory transition is much slower than that observed in the individual breaths. Similarly, in the case of the reverseBenchetrit average, the upstroke of flow at the onset of inspiration is much slower than that observed in the individual breaths.
Figure 4 illustrates the results from the phasealigned averaging for one set of 3min data. It can be seen that the phasealigned average represents the shape of the flow during inspiration better than either the Benchetrit or reverseBenchetrit methods. In particular, the rate of change of flow at the inspiratoryexpiratory transition is greater than that with the Benchetrit method, and the rate of change of flow at the beginning of inspiration is greater than that with the reverseBenchetrit method.
The results from the three methods for the flow profile variables are compared in Fig. 5. Although at the onset of inspiration (Fig. 5 A) the rate of change of flow was 20% larger with the Benchetrit method than with the reverseBenchetrit method, both methods produced significantly smaller values than the mean values calculated from individual breaths in five subjects. By contrast, the phasealigned averaging computed better estimates, and significant differences from the mean values from individual breaths were observed only in two subjects. Essentially similar results were obtained in the rate of change of flow at the inspiratoryexpiratory transition (Fig. 5 B). Both the Benchetrit and reverseBenchetrit methods produced significantly smaller estimates than the means from individual breaths in all subjects. On the other hand, phasealigned averaging produced estimates that were not significantly different from the means from individual breaths in all subjects.
Numbers of estimation errors produced for the respiratory cycle variables by the three methods are shown in Tables1 (inspiratory variables) and 2 (expiratory variables). Phasealigned averaging produced significantly better estimates in the inspiratory variables than did the reverseBenchetrit method (P < 0.05, oneway ANOVA). For the expiratory variables, although statistically significant differences among the three methods were not observed, the phasealigned averaging produced the smallest number of estimation errors.
Figure 6, A and B, shows the results from the Benchetrit and phasealigned averagings, respectively, in a subject breathing against inspiratory elastic loading. A quick inspiration followed by a relatively long endinspiratory pause are the characteristic changes produced by the elastic loading (12). It is clearly seen that the descending limb about the inspiratoryexpiratory transition of the Benchetrit average runs across the band of the descending limbs of the individual breaths (Fig. 6 A). On the other hand, the descent of the phasealigned average runs in parallel to the descents of the individual breaths (Fig. 6 B).
Figure 6, C and D, shows the effects of the number of harmonics on the average airflow profile in the Benchetrit and reverseBenchetrit methods, respectively. In the Benchetrit method (Fig. 6 C), the average airflow reconstructed with 8 and 16 harmonics improved the flow shape at the beginning of inspiration and reduced the wavy shape at the endinspiratory pause, compared with the average reconstructed with 4 harmonics. However, increasing the number of harmonics could not reconstruct the sudden changes to and from the endinspiratory pause. This also holds for the reverseBenchetrit averages (Fig. 6 D). Increasing the number of harmonics improved somewhat the flow shape at the beginning of expiration and reduced the wavy shape at the endinspiratory pause. However, the discrepancy at the beginning of the inspiration between the reverseBenchetrit averages and the phasealigned average remained unchanged by increasing the number of harmonics.
Figure 7, A and B, shows the averaging results in another subject breathing through inspiratory resistive loading. The resistive loading changed the inspiratory airflow shape so that it became more square (5, 9,12, 17). In the Benchetrit method, there is clearly a distortion around the inspiratoryexpiratory phase transition. Figure 7,C and D, presents the effect of increasing the number of harmonics on the Benchetrit and reverseBenchetrit averages, respectively. Findings similar to those in Fig. 6, C andD, are observed.
In general, increasing the number of harmonics in the Benchetrit method reduces somewhat the distortions around the respiratory phase transition, where individual breaths are aligned, but not around the other phase transition. These examples from loaded breathing indicate that the distortions around the nonaligned phase transition are inherent in the Benchetrit method but do not originate from the small number of harmonics employed.
DISCUSSION
To compensate for differences in respiratory cycle duration in irregular breathing data, the method of Benchetrit et al. (2) normalizes the duration of each breath so that it is of unit length. However, such an adjustment does not ensure that, at any given normalized time, all respiratory cycles will be in phase. This can be seen in Figs. 3 and 4, where, despite starting each inspiration at time 0 and ensuring each respiratory cycle is of unit length, the time of transition between inspiration and expiration differs considerably between breaths. In the present study, the impact of these betweenbreath variations in phase has been assessed by comparing the Benchetrit method, which aligns breaths at the start of inspiration, with a method (reverseBenchetrit) that is identical except that it aligns breaths at the inspiratoryexpiratory transition. The two methods give different results for the average flowtime profile, and this finding suggests that the breathtobreath variations in respiratory phase are a significant factor requiring attention when trying to determine some representative breathing pattern.
As an alternative to the Benchetrit technique, this study developed a method that attempts, via the flowvolume diagram, to align each part of each breath in terms of phase before any averaging process is undertaken. This process resulted in an average flow profile that appeared visually more akin to an average breath than an average flow profile obtained with the Benchetrit technique. Indeed, the values of the flow profile variables and respiratory cycle variables associated with the phasealigned average flows were closer to those estimated from the corresponding individual breaths compared with those from the Benchetrit and reverseBenchetrit methods (Fig. 5, Tables 1 and 2). It is common to use these variables to characterize the airflow pattern (4, 8, 1113, 15). Therefore, the systematic deformations produced by the alignment of breaths in terms of time, as in the Benchetrit method, may be critical in leading to misinterpretation.
The Benchetrit method uses the first four harmonics, which were reported to contain >95% of the power in the original waveform of a breathing cycle (2). However, it does not necessarily imply that excellent representation of each breath circumvents the distortions resulting from the phase variations between breaths.
A feasible cause of the distortions is the relatively small number of harmonics in the Benchetrit method. When individual breaths exhibit flow profiles deviating far from sinusoidal shapes, four sinusoids might still be insufficient to describe the detailed contour, and this may, in part, cause the distortions. To examine this possibility, we applied the Benchetrit and reverseBenchetrit approaches with 4, 8, and 16 harmonics in the subjects breathing against respiratory loading (Figs. 6 and 7). Increasing the number of harmonics reduced the distortions around the phase transitions where individual breaths were aligned in both methods. However, the distortions around the other phase transitions were not reduced by increasing the number of harmonics. These results indicate that the systematic distortions produced by the Benchetrit method are created inherently by the timedomain alignment of individual breaths and not by the small number of harmonics.
Airflow shape can be considered the respiratory motor output produced by an alternating discharging pattern of respiratory phaserelated neurons in the respiratory center (16). However, it is important to realize that our definition of respiratory phase in the phasealigned technique remains an arbitrary one. Other choices of phase are possible, such as the fraction of inspiration or expiration that has been completed. In the volumeflow diagram, the origin was chosen as the point of zero flow and mean lung volume. Whereas there is some logic to choosing the origin to lie on the axis of zero flow, mean lung volume has much less meaning and will vary under different conditions.
Once values for flow, volume, and elapsed time from the start of inspiration have been estimated, there remains a problem that, in general, these statistical estimates now no longer obey the physical law that volume is the time integral of flow. Ideally, what is required is some algorithm that simultaneously minimizes (in relation to some metric) the deviations in flow, volume, and elapsed time from the start of inspiration subject to the constraint of the physical law. However, it proved much easier to devise an algorithm that estimated flow, volume, and elapsed time semiindependently and then later adjusted the results to obey the physical law. This approach can be justified by the fact that the adjustments to the flow were small (Table3).
In summary, the technique of Benchetrit et al. (2) used to determine an average respiratory flow profile implicitly aligns breaths just at the start of inspiration. In the presence of wide variability, both between the breaths and within the breaths, this method produces significant distortions. As an alternative, we have developed a technique in which different breaths are aligned throughout the respiratory cycle in terms of the phase on a normalized flowvolume diagram. This technique significantly reduces distortions in the average flow profile.
Footnotes

Address for reprint requests and other correspondence: P. A. Robbins, Univ. Laboratory of Physiology, Univ. of Oxford, Parks Road, Oxford OX1 3PT, UK (Email:peter.robbins{at}physiol.ox.ac.uk).

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 Copyright © 2001 the American Physiological Society
Appendix
To obtain an estimate of respiratory flow at btps(an estimate of the swept volume of the chest), we employed the following procedure.
Before each experiment, a turbine volume measuring device was calibrated using a mechanical pump with a tidal volume of 1 liter. Respiratory flow was obtained using an unheated pneumotachograph connected in series with the turbine volume measuring device, which determined the inspiratory and expiratory tidal volumes of each breath separately (7). Respiratorygas compositions at the mouth were also analyzed continuously using a mass spectrometer, and the sampling delays of the signals were adjusted to the flow signal. Both flow and respired gas composition were recorded every 20 ms on a computer for the following offline analysis.
1) The pneumotachograph signal was corrected for changes in viscosity using the gas compositions recorded by the mass spectrometer (air was treated as having unit viscosity). 2) The respiratory flow for each half cycle of respiration was calibrated with the values obtained by the turbine volume measuring device using the fact that the integral of the flow should equal the calibrated volume as measured by the turbine volume measuring device (this approach dealt with any variation in pneumotachograph resistance over the course of the experiment, such as induced by condensation within the pneumotachograph, and thus avoided the need to heat the pneumotachograph). 3) Mean expired temperature was estimated from the fact that the net nitrogen balance over the course of the experiment should effectively be zero. This was achieved by calculating the net nitrogen balance from the gas compositions and respiratory flow over the entire course of the experiment using the measured inspired temperature and a guess for the expired temperature. This initial guess for the mean expired temperature was then adjusted until the nitrogen balance was zero. 4) Finally, using the measured temperature of the inspired gas and estimated temperature of the expiratory gas, the flows could be corrected to btps and then integrated to obtain the swept volume of the chest.