Department of Anaesthesia, Waikato Hospital, Hamilton, New Zealand*Corresponding author
Accepted for publication: August 28, 2001
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Abstract |
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Methods. We observed 49 patients 36 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: 614
Keywords: statistics, data analysis; lung, respiration; analgesics, opioid; statistics, extreme value theory
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Introduction |
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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.
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Methods |
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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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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 seriesthe 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 MannWhitney 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 rankwhich is the distribution function for a type I distribution. It is given as follows:
yi:n=loge [loge(pi:n)] where pi:n=(i0.5)/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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Results |
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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.
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Discussion |
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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 doseresponse 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.
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Appendix |
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References |
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2 Zikria BA, Spencer JL, Kinney JM et al. Alterations in ventilatory function and breathing patterns following surgical trauma. Ann Surg 1974; 179: 14[ISI][Medline]
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4 Smith RL. Extreme value theory. In: Lederman W, ed. Handbook of Applicable Mathematics: Supplement. Chichester: Wiley, 1990; 43772.
5 Matthews R. Far out forecasting. New Scientist 1996; October: 3740
6 Gumbel EJ. Statistics of Extreme. New York, NY: Columbia University Press, 1958
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