Effects of Detrending for Analysis of Heart Rate Variability and Applications to the Estimation of Depth of Anesthesia

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1 Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp Effects of Detrending for Analysis of Heart Rate Variability and Applications to the Estimation of Depth of Anesthesia C. S. Yoo and S. H. Yi Department of Computer Aided Science, Inje University, Gimhae (Received 8 October 2003) We investigated how preprocessing used to remove a trend from heart rate variability can have some effects on the linear and nonlinear analysis for estimating the depth of anesthesia. Because heartbeat signals frequently contain either slow trends or very slow frequency oscillations, detrending was necessary as a preprocessing step to prepare for analysis by using methods that assume stationarity. For heartbeat time series obtained from 30 female patients undergoing surgery, linear and nonlinear measures were calculated in raw and detrended heart rate variability, respectively. Linear measures did not show significant differences between raw and detrended heartbeat signals, while nonlinear measures were strongly affected by the detrending. We conclude that nonlinear analysis could be performed without detrending, but removal of trends from time series is still often required in linear analysis. PACS numbers: Hh, Ca, Tp Keywords: Nonlinear analysis, Time series, Heart rate variability I. INTRODUCTION Heart rate variability (HRV) describes variations between consecutive heartbeats. The regulation mechanism of HRV originates from sympathetic and parasympathetic nervous systems. Thus, HRV can be used as a quantitative marker of the autonomic nervous system, and HRV parameters have been used to predict the mortality risk in patients with heart disease, such as lifethreatening arrhythmias and acute coronary events [1, 2]. HRV analysis has become an important tool in cardiology because its measurements are noninvasive, easy to perform, have relatively good reproducibility and provide prognostic information on patients with heart disease [3]. In the time or frequency domain, linear measures have most commonly been used to quantify the fluctuation in heart rate, but there is increasing evidence to suggest that the heart is not a periodic oscillator under normal physiologic conditions [4]. Moreover, commonly used statistics of HRV, which are average heart rate and standard deviation of all normal-to-normal R-R intervals over a specific time period [1], are not able to describe accurately changes in beats of heart rate dynamics. Therefore, nonlinear methods have been developed to quantify the dynamics of heart rate fluctuations [5 11]. Linear and nonlinear methods inherently assume that the signal is at least weakly stationary. However, real HRV is yish@chaos.inje.ac.kr usually non-stationary, so that non-stationarities such as slow linear or more complex trends have to be considered before analysis. To obtain the reliable results of analysis of HRV, it is essential to distinguish trends from the heart rate fluctuations intrinsic in the data. Trends are caused by external effects, and they are usually supposed to have a smooth and monotonic or slowly oscillating behavior. Often, experimental data are affected by nonstationarities. Such trends have to be well discriminated from the intrinsic fluctuations of the system in order to find the correct dynamics of fluctuations, but if trends are present in the data, they may give erroneous results. In order to perform spectral analysis, detrending schemes have been used as a preprocessing step to prepare HRV for analysis by using methods that assume stationarity. Tarvainen et al. reported the effect of detrending on time and spectral analysis of HRV [12]. This study demonstrated that temporal statistics measures were not affected by detrending, but low-frequency trend components increased the power of a very-low-frequency (VLF) component of HRV. In recent years, detrended fluctuation analysis (DFA) was developed to accurately quantify long-range powerlaw correlations in a non-stationary time series [5,13,14]. DFA is a well-established method for determining the scaling behavior of noisy data in the presence of trends, without knowing their origin and shape. A few recent works considered different types of non-stationarities associated with different trends (e.g., polynomial, sinusoidal, and power-law trends) and systematically studied

2 -562- Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004 their effects on the scaling behavior of long-range correlated signals [15 17]. In this paper, we study how preprocessing used to remove a trend from HRV can have some effects on the linear and nonlinear analysis. The existence of trends in time series generated by physical or biological systems is so common that it is almost unavoidable. The immediate problem facing researchers applying methods of analyzing HRV is that of knowing whether trends in data arise from external conditions, and have little to do with the intrinsic dynamics of the system generating noisy fluctuations. In this case, a possible approach is to first recognize and filter out the trends before we attempt to quantify the dynamics of HRV. Alternatively, trends may arise from the intrinsic dynamics of HRV, rather than being an influence of external conditions. Until now, new methods for analysis of HRV were proposed and applied to HRV time series, but it was not clear if the detrending procedure would lead to a significantly different characterization of the dynamics of HRV. The paper is organized as follows: in Sec. II, we review characteristic measures of linear and nonlinear analysis, and compare the effect of the detrending between raw HRV and detrended HRV. In Sec. III, we show the results of detrending applied to estimating the depth of anesthesia. Finally, we summarize the results in Sec. IV and discuss the role of the detrending procedure in various linear and nonlinear measures. 1. Preprocessing II. METHODS The discrete HRV signal is defined to be the series of time intervals between two consecutive normal R-peaks of an electrocardiogram (ECG). R-peak detection includes one of the most important preprocessing parts of HRV analysis. Several methods of R-peak detection on digitized ECG signals have been reported in the literature, including schemes based on QRS morphology [18] and pattern recognition [19]. In our study, we used the R-peak extraction method using successive differences between adjacent samples of the differentiated ECG. In order to obtain the single R-peak point from the ECG, we performed 40Hz low-pass filtering over the original signal. The R-R interval time series was generated using the automatic scheme to detect R-peak in the ECG, after which careful manual editing was performed by visual inspection of all R-R intervals. Heart-period signals, constituted by successive complex heartbeats (cardiac R-peak event series), are irregularly spaced in time. In order to analyze heart period signals in the frequency domain, the cardiac event series should become a regularly sampled time series. Therefore, it is necessary to rebuild event time series into evenly sampled time series with a certain sampling rate [20]. Conventionally, power spectra can be estimated only from regularly sampled signals, although some techniques have proposed the direct estimation of power spectra of unevenly sampled data [21,22]. The resampling method of HRV time series was carried out by means of widely used techniques based on a cubic polynomial interpolation of inverse interval function values [23]. In the spectral analysis, the assessment of regularly sampled data by means of cubic polynomial interpolation was more successful than the conventional method (Berger s method) for the analysis of HRV signals [23, 24]. 2. Detrending of HRV Trend in a time series is a slow, gradual change in some property of the series over the whole interval under investigation. In traditional time series analysis, a time series was decomposed into trend, seasonal or periodic components, and irregular fluctuations, and the various parts were studied separately. Detrending is the statistical or mathematical operation of removing trend from the time series. Because HRV signals frequently contain either slow trends or very slow frequency oscillations, detrending was necessary as a preprocessing step to prepare HRV for analysis by using methods that assume stationarity. Many alternative methods are available for detrending. Simple linear trend in mean can be removed by subtracting a least-squares-fit straight line. More complicated trends might require different procedures. Detrending was usually based on first- [25,26] or higher-order polynomial models [27]. In this paper, we present detrending procedures based on the smoothness priors approach [12]. The smoothness priors detrending algorithm was recently proposed as an advanced detrending scheme [28], and will be briefly summarized here. We denote the R-R interval time series as S = (R 2 R 1, R 3 R 2,..., R N R N 1 ), (1) where N is the number of R peaks detected. The R-R interval series could be considered to have two components: S = S stat + S trend, (2) where S stat is the nearly stationary R-R series of interest and S trend is the low frequency aperiodic trend component. The trend component can be modelled with a linear observation model such as S trend = HΘ + υ, (3) where H is the observation matrix, Θ are the regression parameters and υ is the observation error. A widely used method for the solution of the estimate Θ is the least squares method. The regularized least squares solution ˆΘ is ˆΘ λ = arg min Θ { HΘ S 2 + λ 2 D d (HΘ) 2 }, (4)

3 Effects of Detrending for Analysis of Heart Rate Variability C. S. Yoo and S. H. Yi Temporal Statistical Measures The time domain parameters are the simplest ones calculated directly from the R-R interval time series. The simplest time domain measures are the mean (MEAN) and standard deviation (SDNN) of the R-R intervals. The RMSSD describes the root mean square of the differences of successive normal sinus intervals. There are also other commonly used parameters like NN50, which is the number of consecutive R-R intervals with absolute differences of more than 50 ms. The pnn50 is the percentage value of NN50 with respect to all consecutive R-R intervals. In this study, we calculated NNx and pnnx with the threshold x of 10 ms or 20 ms, to be compared with the conventional 50 ms [29]. Fig. 1. Effect of the detrending procedure on HRV. (a) raw R-R interval series without detrending and fitted trend (dashed line). (b) R-R interval series after detrending performed by the smoothness priors method. where λ is the regularization parameter and D d indicates the discrete approximation of the d-th derivative operator. The solution of Eq. (4) can be written in the form ˆΘ λ = (H T H + λ 2 H T D T d D d H) 1 H T S (5) Ŝ trend = H ˆΘ λ, (6) where ˆΘ λ is the estimated trend which we want to remove. The selection of the observation matrix H can be implemented according to some known properties of the data S. In this paper, we used the trivial choice of identity matrix for the observation matrix; H = I. The regularization part of Eq. (4) can be understood to draw the solution towards the null space of the regularization matrix D d, where the side norm D d (HΘ) becomes smallest. The null space of the second order difference matrix contains all first order curves and thus D 2 is a good choice for estimating the aperiodic trend of the R-R series. Then, the detrended nearly stationary R-R series can be written as Ŝ stat = S H ˆΘ λ = (I (I + λ 2 D T 2 D 2 ) 1 )S. (7) Fig. 1(a) shows the raw R-R interval time series, including trend and fitted trend. Trends like nonperiodic fluctuations alternate with periodic high-frequency(hf) fluctuations. These trends are quite low in frequency and can be filtered out by a detrending algorithm employing subtraction of fitted least square methods. Fig. 1(b) shows the detrended R-R interval time series after detrending performed by the smoothness priors method. The R-R interval of the nonperiodic fluctuations is diminished significantly, compared to non-detrending, as seen in Fig. 1(b). 4. Spectral Measures The R-R interval time series is an irregularly timesampled signal. This is not an issue in the time domain, but in the frequency domain it has to be taken into account. If the spectrum estimate is calculated from this irregularly time-sampled signal, implicitly assuming it to be evenly sampled, additional harmonic components are generated in the spectrum. Therefore, the R-R interval signal is usually to be interpolated before the spectral analysis to recover an evenly sampled signal from the irregularly sampled event series [1]. In the frequency domain, spectral analysis is generally performed with recording of consecutive R-R intervals by either fast Fourier transform (FFT) or autoregressive (AR) algorithm. The former is easily available from most mathematical libraries, but requires a sufficient length of data and an appropriate editing of the data before power spectral density (PSD) computation. AR algorithms are based on a parametric approach, which assumes that each R-R interval time series is an output of a given mathematical model. PSD estimation, however, requires a careful choice of the order of AR model of the signalgeneration mechanism [30]. In the latter approach, the R-R time series is modelled as an AR(p) process S t = p a j S t j + e t, t = p + 1,..., N 1, (8) j=1 where p is the model order, a j are the AR coefficients and e t is the noise term. A modified covariance method is used to solve the AR model. The power spectrum estimate P s is then calculated as P s (ω) = σ p j=1 a je iωj 2, (9) where σ 2 is the variance of the predicted error of the model [31].

4 -564- Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004 Fig. 2. Effect of the detrending procedure on spectral analysis. PSD estimates for (a) raw and (b) detrended R-R interval series by using a 20-th order AR model. Fig. 3. Sample graph of the scaling exponent representing the slope of the line relating to, and fitted line for: (a) Raw R-R interval series. (b) Detrended R-R interval series. The PSD is analyzed by calculating powers and peak frequencies for different frequency bands. The commonly used frequency bands are very low frequency (VLF, Hz), low frequency (LF, Hz), and high frequency (HF, Hz). The most common frequency-domain parameters include the powers of VLF, LF, and HF bands in absolute and relative values, the normalized power of LF and HF bands, and the LF to HF ratio. Also, the peak frequencies are determined for each frequency band. For the AR-model-based spectrum, powers are calculated by integrating the spectrum over each frequency band. The physiological explanation of the VLF component is much less well defined. Thus, VLF assessed from short-term recording is a dubious measure and should be avoided when interpreting the PSD of HRV. According to Ref. 1, the LF component has been proposed as an index of sympathetic modulation, whereas HF, which is a measure of respiratory sinus arrhythmia, is used as an index of vagal activity [32]. The ratio between the absolute power of the LF component and the HF component was as an index of autonomic balance of sympatho-vagal interaction [33,34]. Fig. 2 shows the difference of the PSD calculated from AR modelling between raw and detrended HRV. Each band power showed a significant difference between before and after detrending. Before detrending, the power of the VLF component is clearly greater than the power of the LF or HF component (see Fig. 2 (a)). After detrending, the VLF components are appropriately removed, while the LF and HF components are not significantly altered by the detrending (see Fig. 2 (b)). 5. Nonlinear Measures A. Detrended Fluctuation Analysis The detrended fluctuation analysis (DFA) method was used to quantify the fractal-like scaling properties of the R-R interval data. This technique is a modification of root-mean-square (RMS) analysis of random walks applied to non-stationary data [13]. The RMS fluctuation of an integrated and detrended time series is measured in different observation windows and plotted against the size of the observation window on a log-log scale. The DFA index is based on the unsegmented R-R interval series S(i). Briefly, the DFA computation involves the following steps (for further details, see Ref. 13 or 14). First, the R-R time series (of total length k) is integrated by using Eq. (10): y(k) = k [S(i) S avg ], (10) i=1 where y(k) is the k-th value of the integrated series, S(i) is the i-th interbeat interval, and S avg is the average interbeat interval over the entire series. Next, the integrated time series is divided into windows of equal length n. In each window of length n, a least-squares line is fitted to the R-R interval data (representing the trend in that window). The y-coordinates of the straight line segments are denoted by y n (k). We detrend the integrated time series y(k) by subtracting the local trend y n (k) in each window. The RMS fluctuation of this integrated and detrended series is calculated using Eq. (11): F (n) = 1 N [y(k) y n (k)] N 2. (11) k=1 This computation is repeated over all time series (window size) to obtain the relationship between F (n) and the window size n. Typically, F (n) will increase with window size. The fluctuations in small windows are related to the fluctuations in larger windows in a powerlaw-like fashion. Under such conditions, the fluctuations can be characterized by a scaling exponent (selfsimilarity parameter) α by using Eq. (12). α represents the slope of the line relating log 10 F (n) to log 10 n, as shown in Fig. 3; α = log 10 F (n) log 10 n. (12) In this method, a fractal-like signal results in a scaling exponent value of 1 (α = 1.0). White Gaussian noise

5 Effects of Detrending for Analysis of Heart Rate Variability C. S. Yoo and S. H. Yi (totally random signal) results in a α value of 0.5, and a Brownian noise signal with a spectrum of rapidly decreasing power in the higher frequencies results in an exponent value of 1.5 [13,14,35]. In this study, heart period correlations were here defined separately for short-term ( 11 beats, α 1 ) and intermediate-term (>11 beats, α 2 ) fluctuations in the R-R interval data [13]. The effect of the detrending procedure on DFA is shown in Fig. 3. The short-term scaling exponent α 1 is not significantly altered between raw and detrended R-R interval series, while the intermediate-term scaling exponent α 2 is strongly affected by the detrending. The detrending method on DFA algorithm removes the local linear trend in the specific window. The raw R-R interval series contains the global trend, whereas the detrended R-R series does not keep the global trend as the length of window increases. Because long-range correlations are destroyed by removing the global trend of the raw R-R series, the slope α 2 stops increasing on a large time scale. Thus, the intermediate-term scaling exponent α 2 of detrended HRV decreases more than that of raw HRV. B. Approximate Entropy and Sample Entropy Approximate entropy (ApEn) is a measure quantifying the regularity of time series. It measures the logarithmic likelihood as to which runs of patterns that are close to each other will remain close in the next incremental comparisons. A greater likelihood of remaining close (i.e. high regularity) produces smaller ApEn values, and conversely random data produce higher values [36]. ApEn algorithms have been published elsewhere [7,36], but are also briefly discussed here. Let each heart period of a N data set be represented as s i. From s i, vector sequences S i through S (N m+1) are reconstructed from the scalar heart-period time series: S i = (s i, s i+τ,..., s i+(m 1)τ ). (13) These vectors represent m consecutive s values, commencing with the i-th point. By using the usual norm definition that the distance d[s i, S j ] between vectors S i and S j is represented as the maximum difference in their respective scalar components, the following quantity Ci m (r) relevant to the correlation integral is easily estimated for each i N m + 1: Ci m (r) = (number of s j such that d[s i, S j ] r).(14) (N m + 1) The Ci m (r) values measure, within a tolerance r, the regularity of patterns similar to a given pattern of window length m. Next, calculate an average of logarithmic Ci m(r): N (m 1)τ Φ m 1 (r) = ln Ci m (r), (15) (N (m 1)τ) i=1 Fig. 4. Comparison of the return map for raw R-R interval series and detrended R-R interval series. The same signals as in Fig. 1 are used. (a) Raw R-R interval series without detrending. (b) Detrended R-R interval series. and then define the approximate entropy ApEn(m, r, N) by ApEn(m, r, N) = Φ m (r) Φ m+1 (r). (16) Two input variables, m and r, must be fixed in order to calculate ApEn. In this study, m = 2 and r = 20 % of standard deviation of the data sets were chosen as suitable values [36]. A recently proposed sample entropy (SampEn) that properly deals with a bias induced from finite data length and self-matches in calculating ApEn was also used to quantify the complexity of short heart rate time series [37, 39]. A lower value of ApEn or SampEn reflects a higher degree of regularity and predictability. Fig. 4 compares the return maps of the raw and detrended HRV. ApEn and SampEn together measure the logarithmic likelihood that patterns of the return map that are close to each other will remain close in the next incremental comparison. Because the return map of detrended HRV (Fig. 4(b)) has more complex distributions than that of raw HRV (Fig. 4(a)), complexity is increased for all r values. Therefore, ApEn and SampEn were significantly higher in detrended than in raw HRV. These results may be the effect of reduction in intermediate-range correlation(α 2 ), since the long-range correlation of R-R interval time series is reduced, and high-frequency noise becomes more pronounced with respect to low frequency components than before detrending [38]. III. APPLICATION TO ESTIMATING THE DEPTH OF ANESTHESIA 1. Data Acquisition The Department of Anesthesiology at Busan Medical University (Busan, South Korea) approved this study, and all patients granted their written informed consent. The authors studied 30 patients (American Society of

6 -566- Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004 Anesthesiologist physical status class I or class II) aged 30 to 58 years, who were scheduled for myomectomy surgery. Patients were excluded if they suffered from severe ischemic heart disease, congestive heart failure, diabetes mellitus or other disorders known to affect autonomic function. None of the patients was taking medications that affect cardiovascular function. Patients were anesthetized according to the routine clinical judgement of the anesthetist. HRV extraction was performed at 10-min intervals for the following seven sessions: (1) pre-surgery (S1): the day before surgery, (2) post-intubation (S2): from the start of intubation to the start of incision, (3) post-incision (S3): after skin incision to 10 minutes later, (4) maintenance anesthesia (S4), (5) awake state (S5): before the end of the anesthesia to 20 minutes later, (6) pre-extubation (S6): from 10 minutes before extubation to the start of extubation, and (7) post-surgery (S7): the day after surgery. detrending procedure of HRV time series did not cause a significant difference in any linear characteristic measure. In Table 2, the effect of detrending on nonlinear analysis of HRV is demonstrated. The short-term scaling exponent of the detrended HRV was not significantly different from the raw HRV, while the intermediate-term scaling exponent and cross-over point showed a significant difference between raw and detrended HRV. The detrending procedure may reduce the important long-range correlation of the HRV signal, which thus decreases in the case of detrended HRV. ApEn and SampEn were significantly different among raw and detrended HRV at r = 15 or 20 % of standard deviation of data. In the case of raw HRV, we also noticed that ApEn and SampEn tended to increase during the anesthesia stages. Both ApEn and SampEn were estimated to be significantly higher in detrended HRV than in raw HRV, as discussed at the end of section II. 2. Statistics Statistical comparison between the measures of HRV calculated from the raw and the measures computed from the detrended HRV was performed by using independent t-tests. All data were expressed as mean ± SD (standard deviation). The values of HRV measures were also compared among the sessions, by using independent t-tests. A critical value of p < 0.05 was used in the statistical evaluation. All statistical analyses were performed by using SPSS version 10.0 (SPSS, Chicago, IL). 3. Results Changes in linear measure at different stages of anesthesia are presented in Table 1. Temporal linear measures did not show significant differences between raw HRV and detrended HRV. SDNN increased during anesthetic induction, compared with pre-surgery (p < 0.05), and still tended to increase from the S2 to the S5 session, but did not change significantly. RMSSD showed no significant change during anesthesia. pnn10, pnn20 and pnn50 decreased during anesthesia, compared with the pre-surgery and post-surgery sessions (p < 0.001), but did not show significant differences among anesthesia sessions. In the frequency domain, as compared with raw HRV, the LF component decreased in detrended HRV, but the difference was not significant. The HF component and the LF/HF ratio also showed similar results. Both LF and HF components showed no significant difference between pre-surgery and anesthesia sessions. There was a slight increase in both LF/HF during anesthesia induction, compared with pre-surgery, but not to a statistically significant extent. These results showed that the IV. DISCUSSION AND CONCLUSION In this paper, we compared linear and nonlinear measures of HRV between raw R-R interval time series and respective preprocessed signals, and applied these measures to estimating the depth of anesthesia. Temporal statistical measures and spectral measures do not show significant differences between raw and detrended HRV, and also reveal no significant change during anesthesia. As shown in Fig. 1, although the effect of detrending is remarkable in rendering the baseline of HRV stable, temporal measures do not show significant differences. Thus, the detrending procedure does not lose any temporal information in the HRV signal. In order to enable distinct division of the spectrum into VLF, LF and HF components, an AR model of spectral analysis is usually used. In the AR spectrum of the raw HRV signal, LF and HF components are almost entirely obscured by the strong VLF component, but, for the detrended HRV spectrum, the LF and HF components are clearly dominant, whereas the powers of LF and HF components were not significantly different between raw and detrended HRV. While conventional linear measures analyze the time series directly, the DFA method removes the local trends in data. For the reliable detection of long-range correlation, it is essential to distinguish trends from the longrange fluctuations intrinsic in the data. In this study, we have shown that the short-term scaling exponent α 1 does not significantly change between raw and detrended R-R interval series, while the intermediate-term scaling exponent α 2 is strongly affected by the detrending. The recent studies with the DFA method have suggested that DFA is more sensitive to slowly varying trends, while quickly oscillating trends disturb the scaling behavior at large temporal scales [40]. After removing global trends by using the detrending scheme of smoothness priors, the

7 Effects of Detrending for Analysis of Heart Rate Variability C. S. Yoo and S. H. Yi Table 1. Changes in linear (temporal or spectral) measures of HRV at different stages of surgery. Values are presented as means ± standard deviation. RAW: raw HRV signal, pnn10, pnn20, pnn50: percentage of absolute differences in successive NN intervals exceeds 10ms, 20ms and 50ms, respectively, DT: detrended HRV signal, SDNN: standard deviation of all normal intervals, RMSSD: root mean square of successive differences, HF: high frequency power, LF: low frequency power; sessions are divided into S1: pre-surgery, S2: post-intubation, S3: post-incision, S4: maintenance, S5: awake, S6: pre-extubation, S7: post-surgery. p < 0.05, p < raw HRV vs. detrended HRV. SDNN RMSSD pnn10 pnn20 pnn50 LF HF LF/HF S1 RAW ± ± ± ± ± ± ± ± 0.66 DT ± ± ± ± ± ± ± ± 0.58 S2 RAW ± ± ± ± ± ± ± ± 3.59 DT ± ± ± ± ± ± ± ± 2.86 S3 RAW ± ± ± ± ± ± ± ± 5.03 DT ± ± ± ± ± ± ± ± 2.76 S4 RAW ± ± ± ± ± ± ± ± 1.92 DT ± ± ± ± ± ± ± ± 1.71 S5 RAW ± ± ± ± ± ± ± ± 2.72 DT ± ± ± ± ± ± ± ± 2.00 S6 RAW ± ± ± ± ± ± ± ± 2.81 DT ± ± ± ± ± ± ± ± 2.09 S7 RAW ± ± ± ± ± ± ± ± 1.21 DT ± ± ± ± ± ± ± ± 1.10 Table 2. Changes in nonlinear measures of HRV at different stages of surgery. Values are presented as means ± standard deviation. RAW: raw HRV signal, DT: detrended HRV signal, α 1: short-term correlation exponent, α 2: intermediate-term correlation exponent, α 2/α 1: cross-over, ApEn: approximate entropy, SampEn: sample entropy; sessions are divided into S1: pre-surgery, S2: post-intubation, S3: post-incision, S4: maintenance, S5: awake, S6: pre-extubation, S7: post-surgery. p < 0.05, p < raw HRV vs. detrended HRV. α 1 α 2 α 2/α 1 ApEn15 SampEn15 ApEn20 SampEn20 S1 RAW 0.87 ± ± ± ± ± ± ± 0.27 DT 0.86 ± ± ± ± ± ± ± 0.26 S2 RAW 1.09 ± ± ± ± ± ± ± 0.40 DT 1.03 ± ± ± ± ± ± ± 0.28 S3 RAW 1.19 ± ± ± ± ± ± ± 0.38 DT 1.16 ± ± ± ± ± ± ± 0.38 S4 RAW 1.09 ± ± ± ± ± ± ± 0.55 DT 1.06 ± ± ± ± ± ± ± 0.36 S5 RAW 1.02 ± ± ± ± ± ± ± 0.44 DT 0.98 ± ± ± ± ± ± ± 0.37 S6 RAW 0.95 ± ± ± ± ± ± ± 0.26 DT 0.89 ± ± ± ± ± ± ± 0.35 S7 RAW 0.87 ± ± ± ± ± ± ± 0.31 DT 0.86 ± ± ± ± ± ± ± 0.26 HRV time series was dominated by higher-order trends rather than linear ones. If only linear short-range trends were removed in calculating DFA, one might obtain erroneous results. In order to precisely estimate DFA indices, other studies have suggested a different approach, which can be applied for higher-order DFAs [16]. Further studies extending to higher-order trends will be scheduled in the near future. ApEn and SampEn are nonlinear statistics that are proportional to the degree of regularity of an attractor reconstructed from a signal. These measures are highly dependent on the geometrical structure of the return map. As mentioned in the previous section, the return map in detrended HRV has more complex distributions than that of raw HRV (see Fig. 4). Previously, other studies suggested that a detrending procedure was necessary to

8 -568- Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004 remove slow baseline shift in R-R interval time series, so that ApEn would reflect the true regularity in the signal [41]. In our result applied to estimating the depth of anesthesia, ApEn and SampEn were significantly higher in detrended HRV than in raw HRV. In addition, ApEn and SampEn of raw HRV showed a trend toward an increase during anesthesia sessions, while those of detrended HRV did not. These results gave a warning that detrending should be carefully performed in nonlinear analysis. We conclude that it would be better to perform nonlinear analysis without detrending, but removal of trends from time series is still often required in linear analysis. The results suggest that preprocessing used to remove higher-order trends must be considered in order to study HRV in the field of nonlinear analysis. ACKNOWLEDGMENTS This work was in part supported by the 2002 Inje University research grant. 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