Extreme value theory applied to postoperative breathing patterns

Y. P. Leong, J. W. Sleigh* and J. M. Torrance

Department of Anaesthesia, Waikato Hospital, Hamilton, New Zealand*Corresponding author

Accepted for publication: August 28, 2001


    Abstract
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
Background. There has been little published work on the statistical features of breath times in postoperative patients. We applied extreme value theory (a statistical method) to the variation in the timing of postoperative breathing.

Methods. We observed 49 patients 3–6 h after a variety of surgical procedures, once they had achieved a stable breathing pattern. The breathing patterns could be one of the three types predicted by the extreme value model. ‘Finite’ breathing patterns (n=30) have a finite upper limit of duration for any apnoea. Patients that displayed one of the other two patterns (‘standard’ and ‘extended’) have, potentially, no limit in duration of apnoea.

Results. The type of breathing pattern observed in each patient was not reliably identified by most of the commonly used risk factors (age, type of surgery, opioid type, dose, and route of administration). A finite pattern was observed in 13 of 26 patients receiving epidural (vs 17 of 23 parenteral analgesia: P=0.15), and 15 of 19 receiving morphine (vs 15 of 30 other opioids: P=0.05). The patients with ‘finite’ patterns were also significantly less drowsy (score 1.04 (0.92) vs 1.62 (0.62), P<0.05).

Conclusions. The breathing pattern was not related to mean breath times, suggesting that the prevalence of apnoeas cannot be reliably predicted by measurement of the respiratory rate alone.

Br J Anaesth 2002; 88: 61–4

Keywords: statistics, data analysis; lung, respiration; analgesics, opioid; statistics, extreme value theory


    Introduction
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
It is well known that the duration of the breaths in postoperative patients may vary. Irregular unpredict able processes such as these are usually described statistically by using the probability distribution of a random variable. Surprisingly, there have been relatively few published reports describing the statistical modelling of these variations in postoperative breathing patterns.1 2 A number of factors contribute to postoperative respiratory irregularities including the use of opioids for analgesia.3

Apnoeas can be described as breaths that have a very prolonged breath time (TTOT) and can, therefore, be identified as points that lie in the right-sided ‘tail’ of the TTOT frequency-distribution curve. There is a statistical method called extreme value theory, which describes the different possible ‘tails’ of frequency distributions.4 The extreme value theory is concerned with probability calculations and the statistical inference associated with the extreme values of random processes. It is widely used in the prediction of climatic processes (floods), financial events (insurance claims), and athletic records.5 Tipett laid the theoretical foundations in 1928 when he showed that there could be only three possible types of extreme value limit distributions. The type I, or Gumbel distribution (which we will call the ‘standard’ distribution) has a tail that decays exponentially. It is found in data that follow normal and log normal distributions. In the type II, or Frechet (we term ‘extended’) distribution the tail decays more slowly than the exponential. The type III, or Weibull (we term ‘finite’), distribution has a tail that decays more quickly approaching a finite upper limit. Gumbel wrote a seminal book in 1958 that further defined the theoretical basis for the method.6 For a description in non-technical terms, there is a readable review by Matthews.5

We applied the methodology of extreme value theory to postoperative breathing patterns.


    Methods
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
Clinical data collection
We used extreme value theory7 to analyse data from an observational study of postoperative breathing patterns. Ethics committee approval and written consent had been obtained. Respiratory waveforms were monitored in 49 adult patients (31 male, mean (range) age 63 (range 21–82) yr), for a 3–4 h period starting 3–6 h postoperatively, once their breathing had stabilized, using an inductance plethysmograph (Respitrace, Studely Data Systems, Oxford, UK). All patients received supplemental oxygen and were monitored with continuous pulse-oximetry. Data were sampled at 20 Hz, converted with a 14-bit analogue-to-digital interface, and analysed using purpose written software which determined the TTOT (time between successive breaths). This program marked each breath by detecting expiration–inspiration flow reversal. The signal was calibrated at the start of each session by the patient breathing to empty and fill 800-ml calibration bags. Breaths of less than 50 ml were ignored. We recognized that this could miss shallow breaths, but a threshold was necessary to reduce artifact from movement or cardiac pulsations. A single inductance band at the level of the nipple was used. We did not detect airflow at the mouth, and thus could not reliably differentiate obstructive and central apoeas.

The TTOT data were divided into 10-min epochs. The longest TTOT (TTOT max) from each epoch (see Fig. 010F1) was used to produce a distribution of maxima, which was then subject to statistical analysis. This series of maxima can be thought of as analogous to the maximum annual height of floodwaters. The choice of 10 min for each epoch was arbitrary. It allowed a reasonable balance between having too few points to derive accurate estimates of the parameters of the extreme value distribution, and distorting the parameters by including too many data points that lay in the centre and not in the tail of the breath-time distribution. As we are interested in the long ‘outlier’ breaths, collecting more data of short breath-times does not improve our estimation of the probability of occurrence of apnoeas. We repeated the analysis using 1-min epochs without substantially different results.



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Fig 1 A typical time-series of breath times (TTOT) for one patient. (The TTOT max breaths for each 10-min period are circled.)

 
Each patient was constantly observed throughout the study period by the investigator. Recordings were taken every 15 min for pain score, oxygen saturation, and level of sedation (using a simple linear scale:7 sedation: 0=‘active’, 1=‘quiet but eyes open’, 2=‘rouse to voice’, 3=‘rouse to glabellar tap’, 4=‘unrousable’).

Statistical modelling
A time series is said to be ‘stationary’ if the probability laws governing the process that is producing the series do not change during the duration of the series—the process is in statistical equilibrium. After manually checking the raw Respitrace signal for artifacts around any unusually long TTOT, the stationarity of the series of TTOT was checked by comparing the sample means and variances for the first and last half of each series. No non-stationarity was detected. Differences between the three groups were compared using either the chi-squared test for categorical data or the t-test for continuous data. The Mann–Whitney U test was used if the data distribution was skewed. A P<0.05 was considered significant. All data were analysed using the Matlab suite of mathematical functions (Matlab 5.3, The Mathworks Inc., Natick, MA, USA).

The distribution of TTOT maxima was studied by using the Gumbel probability plot (see Appendix for Matlab Code). This plot is conceptually similar to the well-known normal probability plot. Each value from the series of (n) TTOT maxima is ranked (i), and then plotted against its ‘reduced value’ (yi:n). This reduced value is the double negative loge expression of the datum rank—which is the distribution function for a type I distribution. It is given as follows:

yi:n=–loge [–loge(pi:n)]     where pi:n=(i–0.5)/n.

If the plot was well-fitted by a straight line (see Fig. 010F2A), the extreme value distribution was termed ‘standard’. If it curved upwards it was termed ‘extended’, and if it curved down it was termed ‘finite’. The TTOT extreme probability distributions were thus classified according to whether the tail of the reduced-value plot deviated significantly and systematically outside the 95% confidence intervals (see Fig. 010F2) of a linear regression (TTOT max=a+bxyi:n); as fitted using least-squares to the linear Gumbel plot. Because the direction of the ‘tail’ of the plot was the object of interest the regression was done excluding the last four data points. In many cases the curves were obvious, and involved many more points than the last four (see Fig. 010F2). From the form of the distribution and knowledge of the parameter estimates, it is possible to infer the probability of apnoeas.



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Fig 2 Gumbel plots. (A) ‘Standard’ (Gumbel distribution), (B) ‘extended’ (Frechet distribution), (C) ‘finite’ (Weibull distribution). The dashed lines indicate 95% confidence limits for a straight line (least-squares fit to all except the last 4 data points).

 
As shown in Figure 010F2C, the ‘finite’ distribution asymptotically approaches a finite upper limit. As a second analytic step, we therefore fitted a wash-in exponential function to the reduced-value data to predict the longest apnoea (using an iterative non-linear fitting procedure (Matlab: nonlinfit.m)).


    Results
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
We analysed more than 2000 breaths per patient. An example of a typical TTOT series is shown in Figure 010F1. A summary of patient data is shown in Table 1.


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Table 1 Summary of operation type, opioid type and route of administration—grouped according to type of extreme value breathing pattern. {dagger}P=0.05, chi-squared test. *P=0.15, chi-squared test, ‘finite’ vs ‘standard’ and ‘extended’. The morphine equivalent dose was calculated assuming fentanyl 200 µg = pethidine 100 mg = morphine 10 mg. No alteration for epidural vs parenteral potency was made
 
When the TTOT maxima were plotted, all three types of Gumbel plot were demonstrated in the study patients. Thirty patients (61.2%) fitted a ‘finite’ curve, which has a finite upper endpoint. This suggests that—depending on the value of the upper limit—we can have some confidence that an apnoea greater than a certain limit will never occur. Twenty-one patients (42%) had predicted worst apnoeas of less than 60 s.

Although not reaching statistical significance, a greater proportion of the patients who had epidural opioids (13 of 26 vs 6 of 23: P=0.15) had ‘standard’ or ‘extended’ plots. This result is confounded by the fact that these patients had more serious surgery. Conversely, more patients receiving i.v. morphine had ‘finite’ patterns (P=0.05), perhaps because i.v. morphine was given to patients after body-surface surgery.

There was no statistically significant relationship between the extreme value pattern and the mean and standard deviation of the TTOT (Table 1). This suggests that often the shape of the tail of the TTOT frequency distribution is relatively independent of the location and spread of the body of the TTOT distribution. Patients in the group that displayed the ‘finite’ pattern were significantly less drowsy (mean (SD) drowsiness score 1.04 (0.92) vs 1.62 (0.62); P=0.03), suggesting that respiratory irregularity is related to the level of consciousness.


    Discussion
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
In this preliminary study we have shown that extreme value theory can be applied to postoperative ventilatory patterns, and found all three different patterns. The ‘finite’ pattern has a definite ceiling value—the patient is statistically unlikely to have an apnoea that exceeds this value. The other two patterns have no defined maximum value.

Operation type, opioid dose, type, and route of administration did not strongly predict which pattern the patient would display. Less sedated patients were more likely to have a ‘finite’ pattern. The observational study design, and small numbers of patients, do not allow various confounding factors to be separated, and further work is required. It may be that a combination of mean TTOT measurements (or ventilatory frequency), and statistical extreme value pattern, could allow apnoea prediction. With a more robust method of detecting respiration, the statistical methods could be automated and applied to large series of patients. In most cases the Gumbel plot curves are quite obvious, so shorter data series could be used with reasonable accuracy.

We made our observations once breathing patterns were stable. It would be necessary to determine whether the type of pattern that a patient displays reflects an individual response to opioids; or whether this is a dose–response relationship that changes with time. Perhaps the pattern could change following a dose of opioid, and the ‘finite’ pattern may be lost if the opioid dose is changed.

We could not differentiate obstructive from central apnoeas. Most of the apnoeas were probably central. Each patient was continuously observed by an investigator for the whole study period, and clinically detectable airway obstruction was uncommon. The distribution of TTOT did not show multiples of the underlying ventilatory frequency. We may have missed some output from the respiratory centre during ‘apnoea’, because small breaths did not reach the 50-ml threshold required to qualify as a true breath. True monitoring of the respiratory centre output would require a monitor of muscle or nerve activity. We chose 50 ml as the threshold as a compromise between missing true breaths (false negatives) and including cardiac pulsations (false positives). When we used 30 ml as the cut-off, the mean respiratory rate approximated 1 Hz, which probably reflected cardiac motion. If we used thresholds greater than 50 ml there was minimal change in the calculated mean breathing rate as a function of threshold tidal volume. In practical terms, a tidal volume of 50 ml in a spontaneously breathing patient is unlikely to be functionally effective in gas exchange.

We conclude that extreme value methods can be used to study long breath-times in postoperative patients. Further work is needed to establish whether this method might become a useful routine postoperative monitoring tool to identify patients at risk of respiratory depression.


    Appendix
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
Matlab Code to produce a Gumbel plot is shown in Figure 010F3.



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Fig 3 Matlab function to produce a Gumbel plot.

 


    References
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Appendix
 References
 
1 Rosenberg J, Rasmussen GI, Wojdemann KR et al. Ventilatory pattern and associated episodic hypoxaemia in the late postoperative period in the general surgical ward. Anaesthesia 1999; 54: 323–8[ISI][Medline]

2 Zikria BA, Spencer JL, Kinney JM et al. Alterations in ventilatory function and breathing patterns following surgical trauma. Ann Surg 1974; 179: 1–4[ISI][Medline]

3 Frater RA, Moores MA, Parry P et al. Analgesia-induced respiratory depression: comparison of meptazinol and morphine in the postoperative period. Br J Anaesth 1989; 63: 260–5[Abstract]

4 Smith RL. Extreme value theory. In: Lederman W, ed. Handbook of Applicable Mathematics: Supplement. Chichester: Wiley, 1990; 437–72.

5 Matthews R. Far out forecasting. New Scientist 1996; October: 37–40

6 Gumbel EJ. Statistics of Extreme. New York, NY: Columbia University Press, 1958

7 Sleigh JW. Postoperative respiratory arrhythmias: incidence and measurement. Acta Anaesthesiol Scand 1999; 43: 708–14[ISI][Medline]





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