Published ahead of print on May 11, 2006, doi:10.1164/rccm.200601-058OC
© 2006 American Thoracic Society doi: 10.1164/rccm.200601-058OC
Temperature Curve Complexity Predicts Survival in Critically Ill PatientsServicio de Medicina Interna, and Unidad de Cuidados Intensivos, Hospital de Mostoles, Mostoles, Madrid; and Servicio de Microbiologia, Hospital Nuestra Señora de la Candelaria, Tenerife, Spain Correspondence and requests for reprints should be addressed to Manuel Varela, M.D., Ph.D., Hospital de Mostoles, Medicina Interna, Rio Jucar, Mostoles, Madrid 28935, Spain. E-mail: mvarela.hmtl{at}salud.madrid.org
Rationale: Temperature curve complexity is inversely related to clinical status in critically ill patients. Objective: To study if temperature curve complexity analysis predicts clinical outcome and how this test compares to other well-established conventional measures. Methods: Temperature was continuously recorded in 50 patients with multiple organ failure. Time-series complexity was analyzed using hourly approximate entropy (ApEn) and detrended fluctuation analysis (DFA) values. Sequential Organ Failure Assessment (SOFA) score was obtained every other day, and correlation between complexity and SOFA values was evaluated. Differences in complexity between nonsurviving and surviving patients were likewise analyzed. Logistic regression models were calculated to predict outcome, and receiver operating characteristic (ROC) curves were plotted to compare the predictive power of complexity values versus SOFA. Measurements and Results: There was good correlation between complexity results and clinical scores for each patient. Nonsurvivors exhibited lower complexity values than survivors (minimum ApEn = 0.230 vs. 0.378; maximum DFA = 1.636 vs. 1.507; mean ApEn = 0.459 vs. 0.596; mean DFA = 1.376 vs. 1.288; p < 0.001 for all comparisons). In the logistic regression model, a change of 0.1 in the minimum complexity resulted in severe increases in the odds ratio of dying (7.6-fold for ApEn, 5.4-fold for DFA). In terms of predicting outcome, there were no significant differences in the areas under the ROC curves for complexity values versus SOFA scores. Conclusions: Low levels of complexity in the temperature curve are indicators of poor prognosis in patients with multiple organ failure. The predictive ability of temperature curve complexity is similar to that of the SOFA score.
Key Words: multiple organ failure • nonlinear dynamics • Severity of Illness Index Measuring body temperature is one of the oldest clinical tools available. Nevertheless, after hundreds of years of experience with this tool, its diagnostic and prognostic value remains limited. Paradoxically, even though body temperature is a quantitative variable, we only know how to use it dichotomously. If a patient is febrile, we can assume that he or she is ill, but we can say little more about the etiology or prognosis. On the other hand, if a patient has no fever, we can draw no conclusions whatsoever. Complex biological systems are ordinarily characterized by a highly elaborate, apparently random output. Senescence or injury to a system often reduces input or constrains processing, thereby simplifying a system's output (1–11). Such simplification takes the form of lower (pseudo)randomness and the appearance of regular rhythms and patterns that have often been used as clinical diagnostic tools (Cheynes-Stokes breathing, extrapyramidal tremor, convulsive fits, cyclothymia, etc.). Consequently, quantification of the complexity of the output of a biological system may provide significant information about its functioning. Certain methods derived from nonlinear dynamics and chaos theory are useful in analyzing time-series complexity and may reveal information that is "hidden" in the output of complex biological systems (12–17). Specifically, complexity analysis of temperature series may make it possible to overcome the fever/nonfever dichotomy and advance to a truly quantitative analysis of temperature in both febrile and afebrile patients. This should offer an extremely useful view of one of the central phenomena affecting the body's homeostasis and may provide vital information about the physical condition of a critically ill patient. We have recently observed temperature curve complexity to be inversely related to clinical condition in patients with multiple organ failure (18). The measurements provided continuous real-time information that was relevant in the case of both febrile and afebrile patients. The present article attempts to establish the degree to which temperature curve complexity was capable of predicting patient survival or death and how it compares, as a test, with the conventional Sequential Organ Failure Assessment (SOFA) scoring system.
Patients The study was performed on 50 successive patients with multiple organ failure admitted to the intensive care unit (ICU) at the Hospital de Móstoles from January 2002 to May 2005 (excluding June 2003 and January 2004 due to technical difficulties). Multiple organ failure was defined as two or more impaired organs (central nervous system, coagulation, respiratory, cardiovascular, liver, and renal functions), regardless of the primary diagnosis or cause of admission. The patients comprised 25 women (mean age, 62.4 yr; range, 18–84 yr) and 25 men (mean age, 60.8 yr; range, 36–83). Informed consent was obtained from the patient or a family member whenever possible. In three cases, the patients died before informed consent could be obtained. However, because consent had not been refused (it had proved impossible to contact a family member) and because temperature measurement is a routine form of clinical monitoring that can be regarded as being more of an observational register than a form of intervention, it was decided to include these three cases. The study was approved by the hospital's ethics and research review board.
Temperature Measurement The readings yielded successive series of 30 consecutive hours (180 readings) each, with a 1-h offset and 29 h of overlap between every two adjacent series. The series were used to calculate the approximate entropy (ApEn) and a detrended fluctuation analysis (DFA) value for each successive time period. Accordingly, the method produced one ApEn and DFA value per hour from the day after admission, with each value encompassing the preceding 30 h, until the patient's discharge or death. To avoid the influence of pre- or perimortem conditions, the last four temperature readings preceding discharge or death were not included. Stationarity was verified by means of the Reverse Arrangements Test (19). Patients were occasionally disconnected (for radiologic examinations, surgical procedures, or by accident). Furthermore, accidental disconnection was assumed to have occurred when there was a difference of more than 3°C between any two consecutive readings or when the temperature reading was below 30°C. Where the disconnection spanned one or two readings, the value was calculated by interpolation based on the preceding and the following readings. Where the disconnection lasted more than three readings, the series was halted and restarted from the beginning (meaning that no complexity value was available until 30 h later). Globally, complexity values were obtained for an average of 338 h/patient, representing 84% (range, 40–100%) of the total possible number of hours for each patient (not counting the first 30 h).
SOFA SOFA analyzes the degree of physiologic impairment of various organs or systems (respiratory system, hemodynamic system, coagulation, renal function, liver function, and level of consciousness), with scoring running from 0 (no organ dysfunction) to 24 (maximum organ dysfunction).
Complexity Analysis Two complexity measures that are well established in clinical practice were used in the present study: ApEn and DFA. A more detailed description of each of these methods can be found in References 25–31, but briefly, each uses a different procedure to analyze the degree of internal structure in a time series. The Appendix explains how these complexity measures are calculated.
ApEn(m,r,N)
DFA The minimum level of complexity (i.e., the minimum ApEn value and the maximum DFA value) attained was calculated for each patient, along with the mean value of each measure over each patient's entire stay in the ICU (mean ApEn and mean DFA). In addition, the maximum and mean SOFA score of each patient was also recorded. Reproducibility of the procedure was examined by means of Bland-Altman plots (32), comparing two distinct time series obtained by two separate temperature sensors, each attached to one of the hypochondria of a patient suffering from multiple organ failure over 94 d. The data collection and processing protocol was similar to that for the other patients. A total of 11,906 temperature readings were recorded, from which 1,985 complexity measurements were extracted. The mean difference among temperature readings was 0.04°C (SD, 0.735°C). The 95% confidence interval (CI) was ± 1.47°C. In the case of ApEn, the mean difference between the two temperature sensors was 0.002 (SD, 0.079), with a 95% CI of ± 0.158. In the case of the DFA, the mean difference was 0.018 (SD, 0.112), with a 95% CI of ± 0.228.
Statistical Analysis The statistical analyses were performed using the SPSS version 11.0.1 statistical package (SPSS, Inc., Chicago, IL). The receiver operating characteristic (ROC) curves were compared using the Medcalc program (Medcalc Software, Mariakerke, Belgium).
Table 1 summarizes the main patient characteristics. The sex distributions for surviving and nonsurviving patients were similar. The patients who did not survive tended to be older than the survivors (mean age of nonsurvivors, 65.4 [(95% CI), 60.9–70.0] vs. 58.4 yr [95% CI, 52.3–64.5]), although the difference was not statistically significant (p = 0.075).
Also, the patients who did not survive tended to have longer stays (median length, 17.0 d [interquartile range (IQR), 36.7] as opposed to 13.0 [IQR, 10.7]), although again the difference was not statistically significant (p = 0.081). Similarly, temperature was slightly lower in the patients who did not survive than in the patients who did (mean temperature, 35.16°C [95% CI, 34.86–35.47] vs. 35.69°C [95% CI, 35.46–35.92], p = 0.005 ANOVA), with greater spread (average SD of temperature, 1.05 in the patients who did not survive vs. 0.84 in the patients who did; p = 0.009). As in previous studies (4, 18), patient age was inversely correlated with the minimum complexity values obtained (r = –0.311, p = 0.03, for minimum ApEn; r = 0.369, p = 0.008, for maximum DFA).
Correlation between Complexity Measures in Individual Patients and in the Whole Sample When the whole sample of patients was analyzed as a set, there was a good correlation between the minimum ApEn and the maximum DFA reached by each patient (r = –0.84, p < 0.0001; Figure 1A). Similary, there was good correlation between the mean ApEn and the mean DFA of each patient (r = –0.70, p < 0.0001; Figure 1B).
Correlation between Complexity Measures and Patient's Clinical Status There was a statistically significant inverse correlation between the minimum ApEn value and the maximum SOFA score for each patient (r = –0.643, p < 0.001). In addition, the maximum DFA value and SOFA score for each patient were directly correlated (r = 0.657, p < 0.001). This correlation remained significant after correcting for the effect of age (r = –0.671 for ApEn, p < 0.001; r = 0.700 for DFA, p < 0.001). Furthermore, the mean ApEn value was inversely correlated with the mean SOFA score (r = –0.693, p < 0.001) and the mean DFA value was directly correlated with the mean SOFA score (r = 0.691, p < 0.001). These correlations also remained statistically significant after correcting for age (r = –0.708 for mean ApEn, p < 0.001; r = 0.693 for maximum DFA, p < 0.001).
Differences in Complexity Values between Surviving and Nonsurviving Patients
As expected, significant differences were found between the nonsurvivors and the survivors for both the maximum and mean SOFA scores (Table 1; maximum SOFA score, 15.7 [95% CI, 14.2–17.3] vs. 10.6 [95% CI, 9.1–12.1]; mean SOFA score, 11.2 [95% CI, 9.3 – 13.0] vs. 5.8 [95% CI, 4.9–6.7]; p < 0.001 for all comparisons). Analyzing by quartiles, a gradual increase in mortality was observed with decreasing complexity (Figure 3). For instance, in the case of the maximum DFA value, mortality was 92% in the first (least complex) quartile, 54% in the second quartile, 31% in the third, and 0% in the fourth quartile. Similar percentage mortalities were found for the other variables as well (minimum ApEn: 83, 69, 23, and 0%; mean ApEn: 92, 61, 23, and 0%; mean DFA: 75, 61, 23, and 17%; Figure 3). By way of comparison, the per-quartile mortality distribution for the maximum SOFA scores was 82, 64, 31, and 8%, and for the mean SOFA scores, 89, 69, 23, and 0%.
The mean complexity values (ApEn and DFA) and SOFA scores in the first and final 24 h were likewise compared in surviving and nonsurviving patients (Figure 4). There were small differences between the complexity values and the SOFA scores between surviving and nonsurviving patients on admission (initial ApEn value: 0.508 vs. 0.426, p = 0.016; initial DFA value: 1.323 vs. 1.379, p = 0.134; initial SOFA score: 10.0 vs. 13.4, p = 0.003). In contrast, the differences during the final 24 h were much more pronounced (final ApEn value: 0.688 vs. 0.441, p < 0.001; final DFA value: 1.240 vs. 1.400, p < 0.001; final SOFA score: 3.0 vs. 12.7, p < 0.001).
Logistic Regression Equations The mean and minimum complexity values were used to construct logistic regression models to predict the outcome (Table 2). All the cases displayed good fits according to the Hosmer-Lemeshow procedure (p > 0.2 in all cases).
A decrease of 0.1 in the minimum ApEn value increased 7.64 times the odds ratio of dying (95% CI, 2.38–24.51). Likewise, an increase of 0.1 in the DFA value increased the odds ratio of dying by a factor of 5.38 (95% CI, 2.05–14.13). Similar results were obtained when the mean complexity values were used (9.43-fold for the mean ApEn value and 4.03-fold for the mean DFA value).
Comparison of the ROC Curves The areas under the ROC curve were 0.875 (95% CI, 0.751–0.951) for the minimum ApEn, 0.841 (95% CI, 0.769–0.960) for the maximum DFA, 0.828 (95% CI, 0.695–0.920) for the maximum SOFA score, 0.890 (95% CI, 0.769–0.960) for the mean ApEn, 0.789 (0.650–0.892) for the mean DFA, and 0.903 (0.786–0.968) for the mean SOFA score (Figure 5).
No statistically significant differences were observable between the different areas under the curves.
These findings confirm that, in critically ill patients, the complexity of the temperature curve is inversely related to the patient's clinical status. Thus, as has been shown for other complex biological systems (5–11), in the case of the body's thermoregulatory system, a poor clinical condition is associated with an impoverished output, which can be measured consistently using simple methods of complexity analysis. The good correlation between two independent measures of complexity (ApEn and DFA) suggests that this finding is not an artifact and probably reflects a real physiologic phenomenon. It could be argued that our measurements are not representative of the core temperature, particularly in patients with hemodynamic instability. However, our purpose is not to measure the "real temperature" (whatever that may mean) but to analyze the thermoregulatory system by measuring the output of one of its pathways. The variables used were the mean value for the entire series and the lowest complexity value (minimum ApEn value and maximum DFA value). The mean value may well be a more representative variable for each patient, but it may be less practical to use in clinical situations: it varies constantly, depending on the length of stay, and for purposes of comparison the final values are only available after the outcome has already taken place. By contrast, the minimum ApEn value tends to be much more constant, and used as a cut-off point, it enables predictions to be made based on current values before the outcome. Alternatively, using the complexity value as a variable in a logistic regression model instead of as a cut-off point allows the odds ratio of a patient's survival to be calculated over the entire course of admission. The predictive power of complexity analysis to forecast the outcome is similar to that of SOFA score. A larger sample may, of course, reveal significant differences between these two measures, but most probably the difference will not be large. However, complexity measurement offers certain advantages over conventional scoring. In the first place, it is harmless, does not require invasive analytic tests, and is very low cost. Furthermore, it provides data continuously in real time and does not depend on the results of supplementary tests that can only be performed sporadically. If corroborated, we believe these methods may be useful in assessing the prognosis of patients suffering from multiple organ failure and consequently may be helpful in making certain clinical decisions (intensifying or curtailing therapeutic efforts or monitoring, etc.). In addition, these methods may hold considerable theoretical interest and may ultimately also prove useful in other clinical contexts (e.g., in the differential diagnoses of febrile syndromes of different etiologies, or in the follow-up of certain neurodegenerative disorders). In short, they are a means of achieving a truly quantitative approach to temperature, thereby overcoming the classical febrile/afebrile dichotomy, and may also become a tool providing relevant information in afebrile patients, for whom thermometry is of no use.
Complexity Analysis ApEn. ApEn (approximate entropy) is a measure of time-series complexity. Given a time series, three parameters are needed to measure ApEn, namely, m, r, and n, m being the length of the vectors of the curve to be compared (usually, m = 2; i.e., the vectors to be compared are composed of two successive points); r being the range defining two measurements as similar (a value of between 0.15 and 0.20 SDs normally being used); and N being the total number of measurements considered (there being a consensus that a value greater than 10m is needed).
By way of illustration, let us suppose we have a time series of N temperature measurements (Figure 6). Let us take the vector consisting of the first pair of successive data points, p1 and p2, from the total series of N points. We then seek all the vectors consisting of points pi and pi+1 in the series that fulfill the conditions:
ApEn is the average logarithm of the conditional probability for each pair of points, after the sign has been changed to ensure a positive value. Thus, ApEn provides a (inverse) measure of the extent to which knowledge of any two successive points allows the next to be predicted. The measure will be higher the greater the irregularity of the series of data points, reaching a maximum value for an entirely random series. In the case of our series, m = 2 (i.e., the vectors to be compared consisted of each pair of successive readings); r = 0.2 SDs for the time series being analyzed; and N = 180 (i.e., 180 readings were analyzed, one reading every 10 min for 30 h).
DFA.
To be able to perform DFA, it is first necessary to integrate the time series
Next, the integrated curve is divided into time segments of size n (Figures 8A–8C). A regression line is calculated for each segment, and the difference between the integrated curve and the different regression lines is computed
This operation is repeated for different time frames (i.e., for different values of n). The smaller the time scale (n), the better the fit of the regression lines to the integrated curve and the lower the value of F(n). Conversely, the value of F(n) tends to increase exponentially as the time frame (n) increases.
Finally, the relation between F(n) and the size of n is analyzed. A plot is drawn with log[F(n)] on the y axis and log(n) on the x axis (Figure 8D). A good fit to a regression line indicates the existence of scaling (self-similarity), and a fractal structure can be assumed. The slope of the regression line ( The program used to calculate both ApEn and DFA was written in Python (http://www.python.org) and is available from the corresponding author on request.
Supported by grant PI 030591 of the Fondo de Investigación Sanitaria, Ministerio de Sanidad y Consumo, Spain. Originally Published in Press as DOI: 10.1164/rccm.200601-058OC on May 11, 2006 Conflict of Interest Statement: None of the authors has a financial relationship with a commercial entity that has an interest in the subject of this manuscript.
* Actually, there is a certain confusion with the terms "complexity" and "random." Strictly speaking, the more random a time series is, the less structure it has, and so it could be considered less complex. Nevertheless, in real-life situations, complex systems produce a highly unpredictable, "pseudorandom," output. When these systems become damaged, the output is simplified, and becomes more predictable and thus less (pseudo)random. A complex power-law 1/f Received in original form January 15, 2006; accepted in final form May 11, 2006
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ABSTRACT
METHODS
2 distribution was used to analyze the sex distribution. Correlations between quantitative variables were analyzed by linear regression. Analysis of variance (ANOVA) and Student's t test were used to evaluate the differences between survivors and nonsurvivors for variables that were normally distributed (age, ApEn, DFA, SOFA score) and the Mann-Whitney U test was used for variables that did not follow a normal distribution (duration of stay). A linear regression equation was developed to correct for the effect of age on complexity, taking age as the independent variable and complexity as the dependent variable. The residuals of the regression equation were then used as the age-adjusted complexity values. A log-likelihood ratio system was used to construct the logistic regression equations. The significance level was p < 0.05. 
) is the scaling exponent and is an indicator of the degree of complexity of the curve. On the whole, a curve is more complex (less predictable) the closer its value of 
-type behavior has a DFA = 1. When it becomes more random, DFA decreases to a minimum of 0.5 for an entirely random series ("white noise"). On the other hand, DFA = 1.5 for a random walk ("brown noise," the integration of a random series), which reflects only trivial complexity. So, in a sense DFA can reflect a loss of complexity both when it increases (from a 1/f
