© 2018 Institute of Physics and Engineering in Medicine

Introduction

Periodic breathing (PB) is a pathological feature observed in preterm neonates, and is characterized by repetitive periods of hyperventilation, followed by periods of complete breathing cessation (i.e. apnea) occurring during sleep (Fenner et al 1973, Glotzbach 1989). The effects of PB include severe arterial oxygen desaturation

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Quantifying ventilatory control stability from spontaneous sigh responses during sleep: a comparison of two approaches

Leonardo Nava-Guerra1,8, Bradley A Edwards2,3, Philip I Terrill4, Scott A Sands5, Raouf S Amin6, James S Kemp7 and Michael C K Khoo1

1 Department of Biomedical Engineering, University of Southern California, Los Angeles, CA, United States of America2 Sleep and Circadian Medicine Laboratory, Department of Physiology Monash University, Melbourne, VIC, Australia3 School of Psychological Sciences and Monash Institute of Cognitive and Clinical Neurosciences, Monash University, Melbourne, VIC,

Australia4 School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD, Australia5 Division of Sleep and Circadian Disorders, Brigham and Women’s Hospital and Harvard Medical School, Boston, MA,

United States of America6 Division of Pulmonary Medicine, Department of Pediatrics, Cincinnati Children Hospital Medical Center, Cincinnati, OH,

United States of America7 Division of Allergy, Immunology and Pulmonary Medicine, Department of Pediatrics, Washington University School of Medicine, St.

Louis, MO, United States of America8 Author to whom any correspondence should be addressed.

E-mail: navaguer@usc.edu and lnavaguerra@chla.usc.edu

Keywords: periodic breathing, mathematical modeling, ventilatory control stability, loop gain

AbstractRationale: Ventilatory control instability is an important factor contributing to the pathogenesis of periodic breathing (PB) and other forms of sleep-related breathing disorders (SRBD). The development of tools for the quantification of such instabilities from non-invasive respiratory measurements during sleep could be useful to clinicians in identifying subjects that are at risk of developing SRBD. Objectives: To present and compare two different mathematical modeling approaches that allow the quantification of ventilatory control stability from the ventilatory responses to spontaneous sighs. Measurements and methods: Breath-by-breath measurements of normalized ventilation were derived from respiratory inductance plethysmography (RIP) traces collected during sleep from a cohort of 19 preterm infants with various degrees of periodic breathing. A hypothesis-based minimal closed-loop model consisting of a gain, time-constant and time delay; and a data-driven autoregressive model with time delay were used to fit the ventilatory responses to the spontaneous sighs. Loop gain, a quantitative measure of ventilatory control stability, was extracted from both models. Results and discussion: Both approaches accurately described the ensuing responses to the sighs. Significant and robust correlations were found in the loop gain estimates extracted with the two models in the frequency range spanning 2–8 cycles min−1, which corresponds to PB cycling oscillations in infants. In addition, the hypothesis-based model showed a decreased within-subject variability of the estimated stability quantifiers, while the data-driven better resembled the experimental data. There are advantages and limitations associated with each of the modeling approaches which are discussed in the paper. Conclusions: The agreement found between the two mathematical models indicates that either methodology can be used indistinctively providing reliable results and their application can expand to sigh data from other clinical cohorts of preterm infants.

PAPER2018

RECEIVED 31 July 2018

REVISED

4 October 2018

ACCEPTED FOR PUBLICATION

11 October 2018

PUBLISHED 13 November 2018

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and increased heart rate, which can have deleterious consequences such as neurocognitive impairment and hypertension, respectively, when experienced chronically (Hodgman et al 1990, Poets and Southall 1991). The negative feedback configuration of the respiratory system enables maintaining ventilatory homeostasis in the presence of respiratory disturbances, which can be either internal or external (Ben-Tal and Tawhai 2013, Molkov et al 2014). PB can therefore be seen as the inability of the respiratory regulation system to attain homeostasis following a perturbation. It has been hypothesized that such departure from homeostasis occurs as a result of an instability in the feedback network of the respiratory system (Longobardo et al 1982, Cherniack and Longobardo 2010). Drawing from origins in engineering control theory, loop gain (LG)—the propensity of the respiratory control system to attenuate or accentuate ventilatory disturbances through feedback mechanisms—has been employed as an index that quantifies the susceptibility to respiratory instability (Khoo 2001). As such, there is clear motivation to develop tools that allow the quantification of loop gain and therefore the propensity of an individual to develop PB from respiratory measurements.

The modeling approaches that have been used to quantify LG can be broadly classified into two categories: (i) hypothesis-based, and (ii) data-driven models. Hypothesis-based, or parametric models are built upon a mecha-nistic understanding of the biophysical processes that constitute the systems of interest. They are typically repre-sented in the form of integro-differential equations, and have origins in the early 1950s with the pioneering work by Grodins (1954). The advantage of this modelling strategy is that calculated parameters are directly interpret-able in a physiological context (Ljung 1998). In contrast, data-driven, or nonparametric models are those whose structures are directly estimated from observable time-series data (Marmarelis 1997). The main advantage of this modeling approach is that it can effectively summarize highly complex systems by making minimal assump-tions about their biophysical properties (Marmarelis 1993). In particular, they are likely to provide more accurate fitting of the experimental data when the assumptions about the underlying physiology are unknown or incor-rect (Marmarelis et al 2013).

Given these advantages and disadvantages, one or the other approach may be more appropriate for an appli-cation depending on the specific objectives and data characteristics. As such, it is highly desirable that quantifica-tions from these different approaches are able to be directly compared and interpreted in the literature. However, to our knowledge no study has previously compared their performance in the assessment of respiratory control stability. Therefore, in this work we aim to develop and directly compare two methods to quantify respiratory control stability in preterm infants where spontaneous sighs are present: (i) a hypothesis-based method, and (ii) a data-driven method. Both methods are applied to a cohort of preterm infants born at 24–28 weeks gestational age, and studied at discharge from neonatal intensive care at ≈36 weeks; and demonstrating various degrees of PB. The performance of the two proposed models is objectively compared according to three different criteria: (a) equivalency between the models by using correlation analysis and Bland–Altman plots; (b) reliability in the stability quantifier estimation by examining the intra-subject variability with the coefficient of variation; and (c) goodness of fit by quantifying the normalized mean square error. Some of these results have been recently pub-lished (Edwards et al 2018).

Methods

Subjects and proceduresA total of 19 infants born between 24–28 weeks estimated gestational age were included in this study and were evaluated at 36 weeks post-menstrual age prior to discharge from the neonatal intensive care unit. This cohort of infants is a subsample of a larger subject pool enrolled in the multicenter prematurity and respiratory outcomes project (PROP, Clinical Trials.gov NCT01607216) (Pryhuber et al 2015).

Participants were studied with respiratory inductance plethysmography (RIP, Great Lakes Neurotech, Cleve-land, Ohio, USA) measurements while they were laying in the supine position. All centers utilized the same equipment to monitor respiratory activity and after careful evaluation of the RIP tracings we found no apparent differences in signal characteristics among the different sites. Recordings were typically 60 min long (average sleep duration during this time was 19.3 ± 7.0 min) and the percentage of the time that the infant exhibited PB during sleep, defined as more than three consecutive cycles of breathing and apnea, was quantified. The subset of 19 infants analyzed in this study was chosen on the basis of the amount of PB exhibited while spontaneously breathing during sleep such that we could have a representation of various degrees of PB. This included nine infants with 0% PB; six infants with percent PB ranging between 0% and 10%; and four infants with more than 10% PB.

The PROP protocol was approved by local institutional review boards at each of the clinical research sites and the PROP Observational Study Monitoring Board. Parents of the PROP enrollees provided written informed consent for all study procedures prior to their participation, including tidal breathing analysis.

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Data analysisEach recording was sleep scored via observation as active or quiet sleep (Grigg-Damberger et al 2007). Instances of quiet sleep were selected to identify episodes of spontaneous sighs. Sighs were defined as a doubling of the volume of preceding breaths accompanied by elevated thoracic and abdominal movements and were visually detected from the RIP signals. Following the identification of the sighs, we computationally extracted breath-by-breath measurements of flow and tidal volume for the 30 s preceding and 90 s following the sighs and used them for further analysis. The ratio of contributions to tidal volume by thoracic and abdominal RIP (Vol = Th + k.Ab, i.e. ratio = k) were determined using the quantitative diagnostic calibration technique (Adams et al 1993), which is a method that relies only on spontaneous variability in thoracic and abdominal signals. Derived flow and tidal volume measurements were values calculated from raw uncalibrated RIP data. Because our analyses required only uncalibrated volume signals, there was no need to use an external known volume for calibration in our study (Palmer et al 2004). Subsequently, breath-by-breath measurements of ventilation were derived using the Vivosense software (Vivonetics, CA, USA) and normalized by the mean for each of the analyzed segments. In order to be included in our analysis, segments had to be free of movement artifact within the 120 s segment and a minimum of two segments with sighs were required per infant.

Mathematical modelingThe overall loop model represents the lumped effects of several mechanisms that participate in the ventilatory control process: (a) the gas exchange occurring in the lungs and tissues; (b) the delay in signal transduction through the circulatory system and the processing within the medullary circuitry; (c) the process of chemoreception by the peripheral and central chemoreceptors; and (d) the activation of the respiratory muscles (Ben-Tal and Smith 2010, Feldman et al 2010, Ben-Tal 2012). Both of our proposed models aim to find estimations of the overall loop gain by using the ventilatory responses to the spontaneous sighing episodes. The conceptual basis of these methods in its application to neonatal data is the recognition that sigh events act as external perturbations to the respiratory control system; and thus the subsequent breathing patterns reveal the underlying dynamics of the transient response of the respiratory control system. By studying the transient response to the sighs, we can subsequently quantify the propensity toward ventilatory instability.

Model 1: Hypothesis-based closed-loop minimal modelThis method is an adaptation of our previously published method for quantifying the dynamic loop gain of the respiratory control system in adult obstructive sleep apnea patients (Terrill et al 2014). Briefly, a standard chemoreflex model of respiratory control (consisting of gain, time-constant, delay parameters) is used to model the observed dynamics in chemical drive (Vchem):

τdVchem

dt= −Vchem − LG0 xVe(t − δHB), (1)

where Ve represents ventilation, δHB is the delay time (principally the circulation time between the lung and chemoreceptors), τ is the characteristic time-constant (due to time course of the buffering of CO2 in the lung and tissues), and LG0 is the steady state loop gain (see figure 1(A)). Subsequent sigh events within an analysis window are modelled as an independent source of respiratory drive with an additional additive parameter (Vsigh) such that total ventilatory drive (Vdrive) is given by

Vdrive = Vchem + Vsigh. (2)

Vsigh is modelled as a constant increase in ventilatory drive, γ, that accompanies a scored sigh. Such that during a sigh Vsigh = γ, otherwise Vsigh = 0.

In order to model the respiratory control system for each analysis window, an interior-point gradient descent optimization algorithm was used to identify the parameters (δHB, τ, LG0, γ) resulting in an estimated Vdrive that best fits the measured ventilation data in the discrete breath domain.

Model 2: Data-driven autoregressive modelOn the basis that the ventilatory control system is dynamic and that it operates in a feedback fashion, we can therefore assume that the value of ventilation (Ve) at time n depends on M previous samples of Ve itself. This recursive behavior describes an autoregressive process and can be mathematically formulated by the following equation:

∆Ve (n) =M−1∑m=0

k (m)∆Ve (n − m − δDD) + ε (n), (3)

where the operator Δ is used to denote variations around the mean value of ventilation; k(m) is the impulse response of the overall loop; M denotes the number of previous values of Ve that would influence the present

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measurement of Ve and for this case it was assumed to be 50 s (with a sampling frequency fs of 2 Hz this corresponds to 100 samples). By selecting such value for M, we are able to completely characterize the fast peripheral chemoreflex dynamics, which have transient times of approximately 16–32 s in the infant population (Revow et al 1989). Conversely, this value of M only allows us to partially capture the more sluggish central chemoreflex response to the sighs, which has been suggested to be ten times slower than the one associated with the peripheral chemoreflex (Bellville et al 1979). Despite the limitation of only partially capturing the central chemoreflex dynamics, a simulation study revealed that our model was able to accurately recover the loop gain values. Further investigation of the selection of the optimal value of M and its impact on loop gain estimation accuracy is warranted in a larger dataset of preterm infants. ε(n) represents changes in Ve that are not modulated through the chemoreflex; and δDD represents the time it takes for a change in alveolar PCO2 to be transmitted through the circulatory system to the chemoreceptor loci to ultimately produce a compensatory response via the respiratory muscles. Based on previous results obtained in adults and adolescents, and after adapting them to take into account the increased breathing frequency of preterm infants, δDD was assumed to lie within a range between 1 and 5 s (Bellville et al 1979, Asyali et al 2002).

Having set these conditions, the modeling task becomes to find estimations of k(m) that would best describe the responses to the sighs. Accordingly, our approach assumes that k(m) can be expanded as a sum of weighted causal basis functions bq(m), which are defined over the dynamic range of the system [0, M] as follows:

k (m) =

Q∑q=1

cqbq (m), (4)

where Q is the total number of basis functions to be utilized to expand the impulse response, bq represents the basis functions of (q − 1)th order, and cq are the expansion coefficients (Marmarelis 2004). It is important to note that our autoregressive model differs from a standard statistical autoregressive model. For the latter, a total of M * fs parameters (100 in our case) would have to be estimated in order to recover k(m) and thus requiring very long recordings to produce reliable model estimations. By contrast, the impulse response expansion assumption greatly reduces the number of parameters to be estimated from M * fs to q (see figure 1(B)), with the latter

Figure 1. (A) Hypothesis-based model structure with the three parameters (Gain, time-constant and delay) that had to be estimated. (B) Data-driven model structure with number of Meixner basis functions q = 4 and order of generalization = 0. Note that for the latter model, the coefficients cq in addition to the delay δDD had to be estimated.

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normally ranging between four and eight, and thus making the estimation of k(m) more accurate even with short recordings (Marmarelis 1993).

We utilized the set of Meixner basis functions (MBF) because of the built-in exponential term that char-acterizes it and makes it suitable for modeling the onset and relaxation characteristics generally exhibited by physiological systems (Den Brinker1995, Dooge and Garvey 2010). The different onset timings can be captured by varying the order of generalization, which is an MBF parameter that controls the time at which the basis func-tions will start to fluctuate.

By combining equations (3) and (4) we obtain

∆Ve (n) =Q∑

q=1

M−1∑m=0

cqbq (m)∆Ve (n − m − δDD) + ε (n), (5)

where the only unknowns are the expansion coefficients cq and which can be estimated by using ordinary least squares. This process was carried out for different combinations of number of MBF, orders of generalization and time delays. The optimal model was selected by optimizing the minimum description length (MDL) criterion, which aims at minimizing the fitting error and, at the same time, the number of MBF. By minimizing the number of MBF, we reduce the degrees of freedom of the model and thus avoid the data overfitting problem (Rissanen 1982, Ljung 1998). Further details of this methodology can be found in our previous publication (Nava-Guerra et al 2016).

Ventilatory control stability quantificationOnce the hypothesis-based (Model 1) and the data-driven (Model 2) optimal models were determined for each of the different windows and for all subjects, we utilized the estimated parameters to obtain loop gain representations of such individual models. Loop gain is a frequency-dependent complex variable, and as such it has magnitude and phase components associated to it. Ventilatory stability was quantified by means of the magnitude of loop gain, which represents the dampening (or in some cases amplification) factor of external inputs provided by the chemoreflex at different frequencies. For the case of the hypothesis-based model, the estimated parameters can be used to calculate the magnitude of loop gain across a range of frequencies f by directly substituting them in equation (6):

|LGHB( f )| =∣∣∣∣

LG0

1 + i2πf τ

∣∣∣∣ . (6)

On the other hand, loop gain is calculated from the data-driven model by transforming the impulse response k(m) into the frequency domain by means of the discrete Fourier transform as follows:

|LGDD ( f )| =∣∣∣∣∣

M−1∑m=0

k (m) e−2πiM fm

∣∣∣∣∣ . (7)

Of particular interest to us was the magnitude of loop gain within the frequency band of 2 and 8 cycles per minute (0.033–0.133 Hz) as it represents the range of periodicities exhibited by PB in preterm infants (Rigatto et al 1972, Weintraub et al 2001).

Within-subject variability assessmentIn order to test the repeatability of the estimations of loop gain provided by the two modeling approaches, we assessed the variability that was present among the multiple analyzed segments per subject. Given that the subjects had a relatively low number of analyzed segments (an average of six), the utilization of parametric descriptive statistics such as the variance, standard deviation or the coefficient of variation was not possible. Accordingly, we quantified the within-subject variability by means of the inter-quartile range and the overall range of the measurements.

Quality of fit evaluationAs mentioned above, the data-driven model utilized MDL as the optimization criterion to select the optimal model, which contrasts with the one that was used for the hypothesis-based model. The optimal parameters of the latter model were selected by minimizing the fitting error by means of the mean squared error. This was due to the fact that the number of parameters to be estimated was fixed, and hence the problem of overfitting was not a concern. Despite the different optimization criteria employed by the two approaches, we compared how well the optimal models fitted the data by means of the normalized mean squared error (NMSE). NMSE was defined as the ratio between the variance of the residuals and the variance of the actual output and takes values that range between zero and one. A value close to zero would correspond to a model that accurately fits the data,

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and conversely, a value close to one would correspond to a model whose output prediction fails to explain the variability found in the measurement.

Statistical analysesThe model-extracted parameters were first tested for normality. Comparisons between the two approaches were carried out by means of paired-samples t-tests. In addition, Pearson’s linear correlation analysis and Bland–Altman plots were utilized to assess the degree of agreement between both methodologies by means of the bias and limits of agreement (Altman and Bland 1983, Bland and Altman 1999). It should be noted that in this case, neither of the two proposed models for loop gain estimation has been clinically established as a ‘gold standard’ and therefore, the term bias should be interpreted with caution as we are not considering any of the two models as a reference. All tests used P ⩽ 0.05 to determine statistical significance.

Results

A representative example of a spontaneous sighing episode in one of the study participants is presented in figure 2. The top panel shows the flow trace, which was derived from the RIP bands. As can be observed, the breathing pattern is stable until the first sigh occurs. This event is followed by hypopnea (i.e. reduced airflow) that in some cases results in brief periods of apnea, which is then terminated by a hyperpneic episode (i.e. increased airflow). This unstable pattern of breathing repeats itself until it eventually stabilizes and eupneic breathing resumes. The bottom panel exhibits the breath-by-breath normalized ventilation, which was derived from the flow trace. The predictions provided by the hypothesis-based (black) and data-driven (red) models are also plotted to compare against the actual measurements. As can be noted, both models provide an accurate fit of the measured ventilation. Although, the data-driven model provided a slightly better fit of the data, as quantified by the NMSE of 0.30, as compared to the hypothesis-based model whose NMSE was 0.38.

Figure 3 shows sample estimations of loop gain using the hypothesis-driven model (left panel) and the data-driven model (middle panel) from two different subjects with high (top row) and low (bottom row) agreement between both methodologies. It can be observed that the hypothesis-driven model produced loop gains that have a frequency profile that resemble that of a low-pass filter and that is similar for the two subjects. On the other hand, the data-driven model exhibited different loop gain patterns for the two individuals. For the infant shown on the top row, the data-driven model produced loop gains whose frequency profiles resemble that of a low-pass filter as well. Conversely, for the infant shown on the bottom row, the loop gain frequency profiles show reso-nances at approximately 4 cycles s−1. This resonating feature is characteristic of second (or higher) order systems and cannot be captured by a first order model such as the hypothesis-driven one. In order to evaluate the degree of agreement between both methods, we computed the difference between each pair of estimated loop gains and is shown on the right panel. It can be observed that for the subject shown on the top panel, the average difference between both methods tends to zero. This is due to the fact that both approaches yielded similar loop gains and, therefore there is good agreement between both approaches in this individual. For the subject shown on the bot-tom row, the observed discrepancies in the loop gain estimations by the two methods result in a difference that does not approach zero, but rather has a negative bias. This indicates that for this subject in particular, the agree-ment between both models is low.

Figure 4 summarizes and compares the hypothesis-based loop gain (LGHB) estimations and the data-driven loop gain (LGDD) in all infants. Panel (A) displays the estimated mean (±95% confidence interval) LGHB (shown in red) and LGDD (shown in blue) as a function of the frequency. As can be seen, the estimations of loop gain with the two approaches significantly differ between each other for frequencies below 2 cycles min−1. The observed differences become less marked within the range of frequencies of 2–8 cycles min−1. Ultimately, both loop gains tended to show similar average values for frequencies above 8 cycles min−1. In order to identify potential biases in our estimations, we calculated the difference between LGHB and LGDD (see figure 4(B)). It can be observed that at the very low frequencies, there is a positive bias indicating that the hypothesis-based model provides higher estimations of loop gain. The bias then transitions to a negative value for all frequencies above 2 cycles min−1 approaching zero for those frequencies above 8 cycles min−1. The panels on the right show the results from the correlation analysis between LGHB and LGDD at the different frequencies. Panel (C) presents the Pearson correlation coefficients (±95% confidence interval) and panel (D) presents their associated P-values. It can be observed that despite the aforementioned biases, there is an overall trend toward a positive correlation between LGHB and LGDD for frequencies below 6 cycles min−1. Importantly, these correlations were found to be either sta-tistically significant (P-value < 0.05) or marginally significant (P-value < 0.1) for those frequencies lying below 6 cycles min−1. Two levels of maximum correlation and statistical significance were observed at approximately 2 and 5 cycles min−1.

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Figure 5 shows the scatter plots (left column) between LGHB and LGDD evaluated at 2 cycles min−1 (top) and at 5 cycles min−1 (bottom), which correspond to the frequencies that exhibited the maximum correlation coef-ficients and lowest p-values. It can be observed from panels (A) and (C) that there is a moderately strong positive association between the estimations of loop gain using the two methods at 2 and at 5 cycles min−1 (r = 0.66, p = 0.003 and r = 0.62, p = 0.004, respectively). Despite the strong correlation that exists between LGHB and

Figure 2. Sample recording of a spontaneous sigh that results in PB. The top panel represents the continuous measurement of flow which was derived from RIP measurements. The bottom panel shows the normalized breath-by-breath minute ventilation (blue). Note that the bottom panel includes the predictions provided by the hypothesis-based (black) and the data-driven models (red).

Figure 3. Example estimations of LGHB (left column, blue) and LGDD (middle column, red) from the multiple analyzed windows (distinguishable by the different markers) in two different infants. The right column depicts the difference between the estimated loop gains using the two different approaches (where a value of zero indicates identical estimations). The data on the top row corresponds to a subject in which both methodologies show good agreement, while the bottom row shows the data from a subject in which the degree of agreement between both methods is low. The thick line in each panel corresponds to the median values and the vertical lines correspond to the inter-quartile range as a function of frequency.

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LGDD, the correlation coefficient alone does not provide us with information related to the degree of agreement between both methods. Indeed, the scatter plot shows that some of the paired-data points fall far from the iden-tity line (dashed). To accomplish this task, we generated Bland–Altmann plots, which help to determine both the magnitude and direction of the bias (Hanneman 2010). On the right-hand side, we present the corresponding Bland–Altman plots for the frequencies of 2 and 5 cycles min−1. It can be noted from panel B that there is a posi-tive bias in loop gain of 0.22 when evaluated at 2 cycles min−1, and one sample that lies outside the upper limit of agreement. This indicates that the data-driven model is providing loop gain values that are higher than the hypothesis-based model. Conversely, panel (D) shows that for the values of loop gain at 5 cycles min−1 there is a negative bias of 0.11 with one outlier on the lower limit of agreement, which means that the data-driven model now results in lower values of loop gain as compared to its hypothesis-based counterpart.

A comparison of the within-subject variability in the estimated LGHB and LGDD being evaluated at 2 cycles min−1 (top row) and at 5 cycles min−1 (bottom row) is shown in figure 6. Two different measures of variability are presented, the inter-quartile range (left column) and the overall range (right column). It can be observed that the within-subject variability, quantified by the inter-quartile range, did not show significant dif-ferences between the two methods for loop gains evaluated at 2 (Panel (A)) and at 5 cycles min−1 (Panel (C)). Although, the inter-quartile of loop gain evaluated at the latter frequency showed a marginally significant result (p = 0.053). On the other hand, when we quantified the within-subject variability by means of the range of the estimated loop gain, we found that the hypothesis-driven model showed a statistically significantly lower vari-ability at 2 (Panel (B)) and 5 cycles min−1 (Panel (D)) (p = 0.011 and p = 0.003, respectively). These results sug-gest that, overall, the hypothesis-driven model tends to produce results that are more repeatable as compared to the data-driven model.

Figure 7 shows the results from the statistical comparison of the NMSE between the two modeling approaches that was carried out by means of a paired-samples t-test. It can be observed that the average NMSE produced by the hypothesis-driven model is approximately 0.68, while the average NMSE produced by the data-driven model lies around 0.59. This improvement in the quality of the fit by nearly 10% results in a significant difference between the two methodologies. It is of note that the hypothesis-driven model performs remarkably well despite its fixed structure and the lower number of free parameters.

Discussion

The objective of this study was to present and directly compare two modeling approaches for the quantification of ventilatory control stability in a cohort of preterm infants. Both methods fundamentally rely on spontaneous sighs to perturb the respiratory control system with the hypothesis-based method fitting a physiologically driven model; and the data-driven method fitting Meixner basis functions. Correlation analysis and Bland–Altmann plots showed a good agreement between the two methodologies, especially within the periodic breathing frequency band, with some minor discrepancies that arise from the differences in model capabilities. However, there was a substantial mean bias at frequencies lower than 2 cycles min−1. The hypothesis-based model had a lower within-subject variability suggesting more repeatable within-subject loop gain estimations; while

Figure 4. (A) Mean for LGHB (red) and LGDD (blue) across the nineteen subject. The faded lines represent the median LGHB (magenta) and LGDD (cyan) for each individual calculated across the multiple analyzed segments. (B) Mean difference between LGHB and LGDD. The faded lines represent the median difference between both models in the 19 subjects. (C) Pearson correlation coefficient resulting from the correlation between LGHB and LGDD as a function of frequency. The dashed line denotes the point of zero correlation. The vertical lines in (A)–(C) represent the 95% confidence intervals at the various frequencies. (D) P-value associated with the correlation analysis between LGHB and LGDD. The dashed line represents the cutoff value that was used to determine statistical significance. Note that the y-axis of panel (D) is presented in logarithmic scale.

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the data-driven model provided better fits to the experimental data. These overall findings suggest that both methods provide reliable results and can be used interchangeably in the quantification of respiratory stability from spontaneous sighing recordings.

Quantitative comparison of the two modeling approachesThere is a good agreement between the two methodologies for frequencies ranging between 2 and 6 cycles min–1 as indicated by a correlation coefficient of approximately 0.5 and with p-values that in general lie below the level of significance (see figure 4). Despite these significant correlations, Bland–Altman plot analyses revealed a positive bias at a frequency of 2 cycles min–1, which then transitioned to a negative bias at frequencies greater than 3 cycles min–1. A positive bias indicates that the hypothesis-based model is overestimating the value of loop gain relative to the data-driven model; and on the contrary, a negative bias indicates that the hypothesis-based model underestimates the value of loop gain. The likely explanation for the negative bias that was observed at higher frequencies is the tendency for the data-driven model to detect a resonant frequency within the frequency band spanning from three up to 6 cycles min–1. In contrast, the hypothesis-based model quantifies only first order dynamics, and therefore does not capture these resonant frequencies. It is unclear whether quantifying such resonant frequency dynamics captures physiologically relevant information; or whether this is likely to reflect ‘noise’ in the system. However, previous studies that have utilized other nonparametric analytical tools to capture the dynamics contained within ventilatory signals have also reported the presence of such resonant behavior (Van Den Aardweg and Karemaker 2002, Nemati et al 2011). This evidence, in conjunction with our findings, might indicate that the observed resonating pattern indeed reflects a physiological mechanism;

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however, more research is required to elucidate the exact mechanism(s) responsible for such behavior. Our recordings were short in duration (approximately 90 s), and therefore limited the study of very low frequency oscillation ventilatory patterns. As such, we were not surprised to identify considerable disagreement between the two methods in loop gain estimations at frequencies below 2 cycles min–1. In order to accurately recover the very low frequency dynamics, artifact-free recordings of significantly longer duration would be required.

An overall trend for reduced within-subject variability was observed in the hypothesis-based model. While the reduced variability could be an indicator of more repeatable and accurate results, it could also reflect the fixed structure of the hypothesis-based model producing less variable loop gain estimations. It is also possible that the responses to the sighs, and therefore the loop gains, change with time depending on the sleep stage or time-of-night, as has been suggested by previous research (Landry et al 2018). The greater flexibility of the data-driven model could allow to capture such within-subject changes slightly better. Extended nocturnal recordings would allow sleep state and time-of-night effects to be investigated in neonates and infants.

Both the hypothesis-based and data-driven models describe the ventilatory responses to the sighs with good fidelity. The data-driven model provided a statistically significantly better fit to the data (p = 0.012), which was not surprising given the greater flexibility in the structure of the model as compared to its hypothesis-based counterpart. One can argue that the better fit provided by the data-driven model could be attributed to data over-fitting, therefore we sought to investigate whether the amount of free parameters of the data-driven model was significantly higher. We found that approximately two thirds of the analyzed segments required five or less basis

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Figure 6. Within-subject variability measurements of loop gain at 2 (top row, LG2) and 5 cycles min−1 (bottom row, LG5) for the hypothesis-based and the data-driven models. Variability was quantified by means of the inter-quartile range (left column), the overall range (right column) of the multiple analyzed segments per infant. The lines plotted to the left and right of the samples represent the estimated population mean of the corresponding variability metric per modeling approach along with the upper and lower 95% confidence intervals. Results from the paired sample t-tests comparing the two approaches are shown on the top of each of the panels.

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functions to accurately describe the ventilatory dynamics with an overall mode across the population of four functions (data not shown). The latter number does not drastically differ from the three free parameters of the hypothesis-driven model and we can therefore conclude that both models are comparable in terms of complex-ity.

Previous hypothesis-based and data-driven models for ventilatory stability quantificationHypothesis-based models are the most widely used and have proven to be efficacious at replicating PB instances by simulating different scenarios such as the exposure to high altitude; sleep in adults and infants; and cardiovascular as well as neurological lesions (Khoo et al 1982). The ability of this model to simulate the various conditions can be attributed to its high level of physiological detail and complexity. More recently, simplified versions of the aforementioned model were introduced and used to study how the intrinsic delays in signal transduction that exist in the respiratory control system affect its stability (Batzel and Tran 2000). From these parametric models, stability can be directly quantified by simply applying control theory concepts to the analytical expressions that describe them.

On the other hand, data-driven models have also been utilized to study the respiratory regulation system. For instance, an autoregressive moving average model was employed to fit the ventilatory responses from spon-taneous sighs in anesthetized dogs and chemoreflex sensitivity was estimated (Khoo and Marmarelis 1989). Moreover, respiratory control stability was quantified by means of an autoregressive model using Laguerre basis functions from hyperventilatory responses provoked by an acoustically induced arousal from sleep in humans (Asyali et al 2002). Lastly, a trivariate autoregressive model that utilized spontaneous variations of ventilation, and end-tidal CO2 and O2 partial pressures was used for the assessment of the overall stability of the respiratory system in anesthetized newborn lambs (Nemati et al 2011). Given the lack of an analytical equation of these nonparamet-ric models, stability is generally quantified by means of the average magnitude within a certain frequency band of the estimated model frequency responses. As can be seen, there have been various efforts that aim at building mathematical hypothesis-based and data-driven models for the assessment of chemoreflex stability; however, to the best of our knowledge the results provided by both approaches have not been compared in the past. We believe that such comparative analyses are needed in the literature because they help showcase the strengths and weak-nesses of each methodology, which in turn sets the path for the improvement of these techniques.

Advantages and disadvantages of the modeling approachesOne of the main advantages of the hypothesis-based model is that the estimated parameters have a direct physiological interpretation. This characteristic is of great importance because it facilitates the identification of the mechanism/s that is/are responsible for respiratory instabilities. Such instabilities can result from a high chemoreflex sensitivity, an increased time lag in electric signal transduction and/or an elevated circulatory time

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delay. As such, by employing hypothesis-based models, the source of the observed ventilatory pattern can be easily located. On the other hand, the interpretation of the data-driven models could be difficult because the estimated weights do not necessarily have a direct physiological correlate by themselves and thus, the identification of the altered mechanisms is challenging. Despite such limitation, it should be noted that the impulse and frequency responses, which result from combining the estimated weights with their associated basis functions, can be interpreted based on their temporal and frequency patterns. In fact, these responses are equivalent to those obtained with the hypothesis-driven model and can therefore be directly compared between the two models.

One of the limitations of the minimal structure of the hypothesis-based model applied here is its one- compartment structure. This limits the model to only count with one time-constant, which constrains its ability to capture only one out of the fast and slow dynamics that are characteristic of the peripheral and central chem-oreflexes, respectively (Pedersen et al 1999). While a more complex model could be utilized (i.e. a two- rather than a one-compartment model), this invariably requires more parameters; and fitting these parameters may be less reliable in the short and noisy data frequently encountered in clinical studies. By contrast, the data-driven model, by utilizing basis functions with various dynamic behaviors, permits the fit of both the slow and the fast dynamics at the same time. Additionally, the utilization of data-driven models could help reveal new insights about key physiological mechanisms that might not be well captured by the existing hypothesis-based models. For instance, cases where the chemoreflex response is dominated by positive feedback mechanisms such as short-term potentiation or hypoxic ventilatory depression, which are instances where a change in ventilation in one direction is followed by a change in ventilation in the same direction, have not yet been replicated by minimal hypothesis-based respiratory control models.

Given that both methods have advantages and disadvantages, the selection of one over the other depends on the particular application and the characteristics of the experimental data. For example, if the application of the modeling exercise is to design targeted therapy to treat respiratory instabilities, then the utilization of a hypoth-esis-based model would provide the required information of the particular mechanism that is causing such abnormal respiratory pattern. On the contrary, if the measured data exhibits patterns that are not entirely expli-cable by the current understanding of respiratory physiology, then the utilization of the data-driven approach would be more appropriate given the intrinsic flexibility in its structure. Although, the concomitant utilization of both models is advised when possible, as one can take advantage of the strengths of both approaches. On one hand, the hypothesis-based model could help in the interpretation of the data-driven model results. On the other hand, the data-driven model could help in the improvement of the existing hypothesis-based model by incorpo-rating mechanisms that have not yet been described.

LimitationsThere are some limitations to our study that should be taken into consideration. First, both approaches are based on the assumption that the respiratory control system follows a linear behavior while operating under natural conditions. While such assumption might not always hold true, it greatly simplifies the assessment of ventilatory stability by the utilization of linear control techniques. In addition, we observed that both linear approaches described the experimentally-measured responses to the sighs with good fidelity. For these reasons we believe that the utilization of more complex nonlinear models was not justified. Second, both methodologies rely on the assumption that the measured ventilation is a good estimate of respiratory drive. While this is likely to be true for central apneic episodes, during obstructive events, the respiratory drive differs from the measured ventilation and thus the assumption fails. Even though most of the PB episodes are associated with central respiratory events in the preterm infant population, it could be the case that some infants exhibit obstructive events during PB pattern. Third, the comparative analysis was carried out solely between the two techniques and is lacking a gold standard measurement to determine which approach more closely matches the ‘ground truth’. One potential solution to this limitation could be the utilization of a mathematical model that could simulate diverse responses to the sighs and use the estimation models to recover the underlying dynamics. There are several simulation models that could be used for this purpose, however most of them have been developed to replicate the adult physiology (Cheng et al 2010). There are few respiratory control models that focus on the infant population. Among them, notable is the work by Batzel and Tran (2000), who adapted the models that were initially developed for adults and made the pertinent physical and physiological adjustments to better represent the infant physiology during sleep. Despite the existence of such simulation models, careful consideration is needed in their selection such that it fairly captures the key characteristics of physiological data (i.e. different simulation models could be used to favor one or other of models). Fourth, this paper lacks the comparison of the estimated loop gains with relevant clinical outcomes such as the presence and severity of PB in the short-term; or health outcomes such as healthcare utilization in the medium-to-long term. A recent study performed by our group found that there is a statistically significant predictive power of these model-derived stability quantifiers on the severity of PB through linear regression models (Edwards et al 2018). Such studies hold promise as they are of great value to the clinical community.

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Future directionsThis comparative analysis could also be extended to include other mathematical models that have been previously proposed in the literature for loop gain quantification. In addition, the various models can be used to fit spontaneous sigh data collected during sleep from other population cohorts of different age groups and/or with different respiratory conditions. After carefully evaluating the performance of all the available models across the different data sets, we could then identify which model yields the most reliable results and establish it as the standard clinical method for loop gain quantification for that particular age and respiratory condition. Once a standard method has been agreed upon for each condition, it can then be applied systematically to the analysis of data collected during diagnostic sleep studies and incorporate the extracted loop gain index to the summary sleep study metrics. The incorporation of such index could be very informative to clinicians, as it could help identify those conditions in which the ventilatory control system is altered. In addition, the loop gain index could also be used for therapeutic purposes. For instance, it could aid in distinguishing individuals who would benefit from supplemental oxygen therapy and also to determine the size of the intervention that is required for respiratory stability restoration.

ConclusionWe presented a comparative analysis between two mathematical models that were used to assess ventilatory control stability from responses to spontaneous sighs in a population of infants. The hypothesis-based model showed more repeatable stability measurements, while the data-driven model resembled the experimental measurements more accurately. Despite these differences, there is a good agreement in the model-derived stability measurements between the two methods within the periodic breathing frequency region. They both have advantages and disadvantages and the selection of one over the other would depend on the particular application; however, when used in combination, they complement each other by aiding with interpretability and by revealing additional insights about respiratory physiology.

Acknowledgments

The authors would like to thank the five clinical centers that participated in the prematurity and respiratory outcomes project (PROP) for sharing the dataset that was used for this analysis. Specifically, the Cincinnati Children’s Hospital Medical Center, the University of California San Francisco, the Vanderbilt University Medical Center, the Washington University School of Medicine and the University of Rochester Medical Center.

This work was supported by National Institutes of Health (NIH), National Heart, Lung, and Blood Institute (NHLBI) and National Institute of Child Health and Human Development (NICHD) through U01 HL101800 to Cincinnati Children’s Hospital Medical Center, AH Jobe and CA Chougnet; U01 HL101798 to Univer-sity of California San Francisco, PL Ballard and RL Keller; U01 HL101456 to Vanderbilt University, JL Asch-ner; U01 HL101465 to Washington University, A Hamvas and T Ferkol. Dr Edwards is supported by a Heart Foundation of Australia Future Leader Fellowship (101167). Dr Khoo and Dr Nava-Guerra were supported by NIH Grants P41-EB001978 and R01-HL105210. Dr. Sands was supported by the American Heart Associa-tion (15SDG25890059) and the National Institute of Health (R01HL102321, R01HL090897, R01HL128658, R35HL135818, and P01HL094307). Dr Terrill was supported by the National Health and Medical Research Council (NHMRC) of Australia project grant (1064163).

Disclosure statement

Dr Sands has received personal fees from Cambridge Sound Management, Nox Medical and Merck unrelated to the current work. All remaining authors have indicated no financial conflicts of interest.

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