Nonlinear measures of QT interval series: novel indices of cardiac repolarization lability: MEDqthr and LLEqthr

Psychiatry Research 117 (2003) 177–190

Nonlinear measures of QT interval series: novel indices of cardiac repolarization lability: MEDqthr and LLEqthr Vikram Kumar Yeragania,*, K.A. Radhakrishna Raob a

Department of Psychiatry, Wayne State University School of Medicine, Detroit, MI, USA b Department of ECE, Indian Institute of Science, Bangalore, India

Received 19 December 2001; received in revised form 26 September 2002; accepted 14 November 2002

Abstract In this study, we investigated nonlinear measures of chaos of QT interval time series in 28 normal control subjects, 36 patients with panic disorder and 18 patients with major depression in supine and standing postures. We obtained the minimum embedding dimension (MED) and the largest Lyapunov exponent (LLE) of instantaneous heart rate (HR) and QT interval series. MED quantifies the system’s complexity and LLE predictability. There was a significantly lower MED and a significantly increased LLE of QT interval time series in patients. Most importantly, nonlinear indices of QTyHR time series, MEDqthr (MED of QTyHR) and LLEqthr (LLE of QTyHR), were highly significantly different between controls and both patient groups in either posture. Results remained the same even after adjusting for age. The increased LLE of QT interval time series in patients with anxiety and depression is in line with our previous findings of higher QTvi (QT variability index, a log ratio of QT variability corrected for mean QT squared divided by heart rate variability corrected for mean heart rate squared) in these patients, using linear techniques. Increased LLEqthr (LLE of QTyHR) may be a more sensitive tool to study cardiac repolarization and a valuable addition to the time domain measures such as QTvi. This is especially important in light of the finding that LLEqthr correlated poorly and nonsignificantly with QTvi. These findings suggest an increase in relative cardiac sympathetic activity and a decrease in certain aspects of cardiac vagal function in patients with anxiety as well as depression. The lack of correlation between QTvi and LLEqthr suggests that this nonlinear index is a valuable addition to the linear measures. These findings may also help to explain the higher incidence of cardiovascular mortality in patients with anxiety and depressive disorders. 䊚 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Anxiety; Depression; Nonlinear; Chaos; Spectral; Heart rate; QT variability; Lyapunov exponent; Posture

1. Introduction Panic disorder is an anxiety disorder characterized by several autonomic symptoms such as *Corresponding author. Wayne State University School of Medicine, Flat No. 16, K.C.N. Mansion, Bangalore 560001, India. Tel.: q91-80-2287715; fax: q91-80-3563640. E-mail address: [email protected] (V.K. Yeragani).

palpitations, shortness of breath and tremulousness, in addition to a feeling of intense anxiety. Major depression is characterized by feelings of low selfesteem, negative thoughts, and sleep and appetite disturbances among many other symptoms, including suicidal ideation (Spitzer et al., 1987). Several studies suggest that patients with anxiety and

0165-1781/03/$ - see front matter 䊚 2002 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 1 6 5 - 1 7 8 1 Ž 0 2 . 0 0 3 1 9 - 0

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depression are at a higher risk for significant cardiovascular mortality and sudden death (Coryell et al., 1986; Weissmann et al., 1990; Kawachi et al., 1994; Frasure-Smith et al., 1993; Carney et al., 1997; Musselman et al., 1998). The recent literature has shown the utility of heart rate (HR) variability as a noninvasive tool to study cardiac autonomic function (Malik and Camm, 1990; Malliani et al., 1991; Yeragani, 1995b). Spectral power in the high frequency (HF: 0.15–0.5 Hz) band reflects respiratory sinus arrhythmia (RSA) and, thus, cardiac vagal activity. Low frequency (LF: 0.04–0.15 Hz) power is related to baroreceptor control and is dually mediated by vagal and sympathetic systems. Very low frequency (VLF: 0.0033–0.04 Hz) power appears to be related to thermovascular mechanisms and renin-angiotensin systems (Akselrod et al., 1981; Pomeranz et al., 1985; Lindqvist et al., 1990). Decreased HR variance (HRv) is an important predictor of sudden cardiac death in patients with cardiac disease as well as normal subjects (Kleiger et al., 1987; Bigger et al., 1992; Molgaard et al., 1991). QT interval on the surface electrocardiogram (ECG) reflects time for repolarization. The usual duration of QT interval corrected for HR is approximately 400 ms and is dependent upon HR to some extent. Thus, it is customary to correct QT interval for HR (QTc) in clinical situations. Prolongation of QTc may be more dangerous in the setting of a higher HR. QT interval can be prolonged in several different conditions including congenital long QT syndrome. This condition can be associated with serious cardiac arrhythmias including torsades de pointes. Several studies have shown a relationship between prolonged QTc and life-threatening arrhythmias (Jervell and LangeNelson, 1957; Schwartz and Wolf, 1978). Recent literature also has implicated abnormal repolarization in serious arrhythmias (Binah and Rosen, 1992; Tomaselli et al., 1994). We recently found that patients with panic disorder and depression have significantly increased QT variability compared to normal controls (Yeragani et al., 2000b). An increase in QT variability is reportedly associated with symptomatic patients with cardiomyopathy and also sudden cardiac death (Berger et al., 1997; Atiga et al., 1998,

2000). Certain drugs such as nortriptyline, a tricyclic antidepressant (TCA), increase QT variability in patients with panic disorder (Yeragani et al., 2000c). Thus, these noninvasive measures may prove valuable to study cardiac autonomic function and cardiac side effects. We have studied QT variability in several of our recent studies and have shown the relevance of these new measures to psychiatric research (Yeragani et al., 2000c,d). Recent reports have repeatedly stressed the importance of the additional value of nonlinear techniques to study HRv in health and disease (Goldberger and West, 1987; Yeragani et al., 1993b; Guzzetti et al., 1996; Lombardi et al., 1996; Voss et al., 1996; Ho et al., 1997; Yeragani et al., 1997; Braun et al., 1998; Storella et al., 1988; Yeragani et al., 1998b; Kagiyama et al., 1999; Silipo et al., 1999; Yeragani et al., 2000a, 2002a,b). Some of these techniques appear to yield information that is different from the more traditional measures of time and frequency (Pincus et al., 1991; Guzzetti et al., 1996; Lombardi et al., 1996; Voss et al., 1996; Ho et al., 1997; Yeragani et al., 1997, 2000a; Braun et al., 1998; Kagiyama et al., 1999; Makikallio et al., 1999). Time series analysis using methods of nonlinear dynamics includes estimation of minimum embedding dimension (MED) and Lyapunov exponents (LE). Complexity is reflected by MED and predictability by LE. Kanters et al. (1997) suggest that though the correlation dimension of R–R intervals is due to linear correlations in the R–R intervals, a small but significant part is due to nonlinear correlations between R–R intervals and thus these nonlinear measures give additional information to the linear techniques. Guzzetti et al. (1996) and Casaleggio et al. (1997) also reported similar findings with cardiovascular signals. Thus heart rate variability cannot be a single chaotic system and consists of intertwined periods with different nonlinear dynamics. The same thing may apply to the time series of beat-to-beat QT intervals as QT interval follows HR to some extent. In our ongoing studies on the pathophysiology of panic attacks, we have found that these patients show a decrease in cardiac vagal function and a relative increase in sympathetic activity (Yeragani et al., 1990, 1992, 1993a, 1994, 1995b, 1998a;

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Yeragani, 2000). Several reports suggest decreased HR variability in patients with depression as well (Carney et al., 1995; Krittiyaphong et al., 1997; Stein et al., 2000; Yeragani, 2000). We have also shown recently that patients with major depression have significantly decreased largest Lyapunov exponent (LLE) of HR time series, which is similar to the findings we observed in patients with panic disorder (Radhakrishna and Yeragani, 2001; Yeragani et al., 2002b). We are in the pursuit of new nonlinear techniques that might better discriminate between normal controls and patients with even subtle cardiovascular abnormalities. The aim of the present study was to evaluate the utility of LLE of beat-to-beat QT interval time series, a measure of chaos and predictability in patients with panic disorder and depression, and normal controls. The second specific aim was to obtain the ratio of MED of QTyHR (MEDqthr) and LLE of QTyHR (LLEqthr) for these groups as we specifically hypothesized that patients with panic disorder and depression would have significantly lower values of MEDqthr and higher values of LLEqthr compared to controls. 2. Methods 2.1. Subjects Thirty-four normal controls (16 males and 18 females), 45 patients with panic disorder (17 males and 28 females) and 18 patients with major depression (5 males and 13 females) participated in this study. Twenty-eight controls (29.8"6.9 years), 36 patients with panic disorder (32.8"6.3 years) and all 18 patients with major depression (37.3"6.6 years) had nonlinear data available for HR as well as QT interval time series in supine and standing postures. We report means and standard deviations throughout the text and tables. These studies were approved by the Institutional Review Boards at the Wayne State University School of Medicine, Detroit, MI and the Wright State University School of Medicine, Dayton, OH. All subjects were healthy, and informed consent was obtained prior to their participation in these studies. The subjects were physically healthy with no history of hypertension, and their routine blood chemistry and

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ECG were within normal limits. These subjects had not taken any medication for at least 2 weeks prior to the studies except for occasional nonopioid analgesics. All patients were diagnosed according to DSM-III-R criteria (Spitzer et al., 1987) and they were symptomatic at the time of recruitment. At the baseline evaluation, the subjects were rated on Spielberger’s State Anxiety Inventory (SAI) (Spielberger et al., 1970). ECG was recorded by a Hewlett-Packard 78173A ECG Monitor in lead II configuration using limb lads. The signal was recorded onto a PC using a 12-bit AyD board, at a sampling rate of 500 Hz in supine and standing postures. Standing data were recorded only after 2 min of equilibration time. We used 256 s of supine as well as standing data for the analyses in this study. We used a peak-detection algorithm to identify the R– R intervals (in milliseconds) from the ECG. 2.2. QT variability All these analyses were conducted on 256-s segments of data sampled at 500 Hz. The QT variability algorithm has been described by Berger et al. in detail and has been used by his and our groups in previous studies (Berger et al., 1997; Atiga et al., 1998, 2000; Yeragani et al., 2000c,d). This was performed on a PC using Solaris Desktop Unix software (Sunsoft, Mountainview, CA). It uses a graphical interface of digitized ECG where the time of the ‘R’ wave is obtained using a peakdetection algorithm. Then the operator provides the program with the beginning and the end of the QT wave template. This algorithm finds the QT interval for each beat using the time–stretch model. If the operator chooses a longer QT template, all the QT intervals will be biased accordingly. This algorithm’s purpose is mainly to study QT variability and not the mean QT. The HR (beats per minute: bpm) time series were sampled at 4 Hz using the technique of Berger et al. (1986). The amplitude spectrum of this HR signal more closely matches that of the input signal to an integral pulse frequency modulation model of the heart’s pacemaker than do the spectra of other ECG-derived HR signals. The HR signal produced by this algorithm is like a step-

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wise continuous instantaneous HR signal convolved with a rectangular (boxcar) window. This signal maintains an amplitude equal to the reciprocal of the current R–R interval, for the duration of that R–R interval. It also works as an antialiasing filter. It behaves as a low-pass filter, passing very little power beyond the Nyquist rate. It preserves all the frequencies up to 1y4th of the sampling rate. As used in all our studies and in this article as well, it does not affect the information up to 1 Hz because we sample the signal at 4 Hz. We used HR time series free of ventricular premature beats and noise. The data were then detrended by using the best-fit line prior to the computation of spectral analyses. 2.3. Length of instantaneous HR series We have performed the MED and LLE estimations on several different lengths of time series (ranging from 1024 points or 256 s of data to 14 400 points or 3600 s of data or 3600 beats (interbeat intervals)) (Yeragani et al., submitted for publication). We were able to show that one could obtain reliable information using 1024 data points as it is difficult to obtain longer stationary segments for analysis from biological systems. We also have shown that using an interpolation technique such as Berger’s algorithm (Berger et al., 1986) did not interfere in any significant way with the results of MED and LLE as it passes all the frequencies up to 1 Hz when the sampling rate is 4 Hz. Usually the frequency content of these signals is limited to 0.5 Hz. Skinner et al. (1993) have reported that the correlation dimension of R– R interval series ranged between 3 and 5 and that data lengths of approximately 1000 heart beats were suitable to distinguish patients at imminent ventricular fibrillation from matched controls who were not at such risk. Though our MEDs are in the range of 11–16, the correlation dimensions of HR series in our previous report (Radhakrishna and Yeragani, 2001) were in the range of 6–7 in normal controls and patients with panic disorder. One other reason to use 1024 time points is also to ensure stationarity of the HR series as this is very important for these calculations.

The mean HR (HRm), detrended HRv, mean QT interval (QTm) and detrended QT variance (QTv) were calculated from the instantaneous HR and QT time series of 1024 points (256 s). A normalized QT variability index was calculated as suggested by Berger et al. (1997). w

z

QTvislog10xyŽQTvyQTm2. y ŽHRvyHRm2.|~ This index represents the log ratio between the QT interval and the HR variabilities, each normalized for the corresponding mean. We used HR time series free of ventricular premature beats and noise. All 28 control subjects and 36 patients with panic disorder and 18 patients with depression had both supine and standing NL data (MED and LLE) during spontaneous breathing. 2.4. Methods of nonlinear analysis The methods were described in great detail in our previous reports (Radhakrishna et al., 2000; Radhakrishna and Yeragani, 2001; Yeragani et al., 2002a,b). The reconstruction of HR time series and the calculation of MED and LLE were all computed automatically using a PC with custom designed software according to the following methods. 2.4.1. Estimation of minimum embedding dimension Proper reconstruction of an attractor is guaranteed if the dimension of phase–space is sufficient to unfold the attractor. It is shown that an embedding dimension of m)2dq1 can achieve this, where d is the dimension of the attractor (Takens, 1980). In most cases of observed time series analysis, we neither have knowledge of d nor m. There are many different algorithms used to estimate these quantities (Grassberger and Procaccia, 1983a,b; Theiler, 1987; Broomhead and King, 1986; Mees et al., 1987; Kennel et al., 1992), but many of them have the disadvantage of either being too subjective, requiring large number of data points or being computationally very intensive. The method proposed by Liangyue Cao

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(1997) overcomes these difficulties and is suitable for short-term time series. Additionally, this method gives more reliable estimates of MED even when the dimension is sufficiently large. After time delay embedding, we compute a quantity )

(

)

is1,2,«,Nymt

(1)

where ≤∆≤ is some measurement of Euclidean distance and is given in this article by maximum norm defined as:

≤Xk(m)yXi(m)≤s

max ZxkqjtyxlqjtZ

(2)

0(j(my1

Xi(mq1) is the ith reconstructed vector in mq1 dimension; n(i, m) is an integer in the range 1(n(i, m)(Nydt such that Xn(i, m)(m) is the nearest neighbor of Xi(m) in the m dimensional reconstructed phase–space. The n(i, m) in Eq. (1) is the same as that of the denominator. If Xn(i, m)(m) equals Xi(m), then we take the second nearest neighbor. A quantity E(m) which is the mean value of all a(i, m)’s is computed as E(m)s

chastic signals, and it is given by EU(m)

1 Nymt Zxiqmtyxn(i,m)qmtZ Nymt 8 is1

and its variation from m to mq1 as E2(m)sEU(mq1)yEU(m)

≤Xi(mq1)yXn i,m (mq1)≤ a(i,m)s ≤Xi(m)yXn i,m (m)≤ (

181

1 Nymt a(i,m) Nymt 8 is1

This averaging removes subjectivity involved in fixing threshold and tolerances of the false nearest neighbor method (Kennel et al., 1992). E(m) is dependent only on the dimension m and lag t. To investigate its variation from m to mq1, we define E1(m)sE(mq1)yE(m) It is found that E1(m) stops changing when it is greater than some value m0; if the series comes from an attractor, then m0q1 is the MED, which accommodates the attractor efficiently. Another quantity is determined which is useful in distinguishing deterministic signals from sto-

where n(i, m) has the same meaning defined earlier (Eq. (1)). For random time series E1(m) will never attain a saturation value as m is increased, but because of limited data samples and practical computations, it may be difficult to ascertain whether E1(m) is slowly changing or has stopped changing. In such a situation E2(m) will be very useful; since for random data future values are independent of past values, E2(m) will be equal to 1 for any m, where as for deterministic signals there exists some m’s such that E2(m)/1. We computed both E1(m) and E2(m). We applied this method on time series of some of the standard maps and found their MED tallying with the literature. 2.4.2. Subjectivity of arriving at MED Though this is a cause for some concern when the MEDs are calculated by many people, this can be substantially reduced by training only a few people to do this and in this particular article one of the authors has calculated all the MEDs blindly to the patients’ condition. We chose the point of the beginning of saturation on the graph after plotting the E1 values. 2.4.3. Largest Lyapunov exponent LE is another invariant which could be used to characterize the dynamical system. It quantifies sensitivity of the system to initial conditions. An m-dimensional dynamical system has m LE. The presence of positive LE indicates chaos. It also quantifies the amount of instability or predictability of the system. A fully deterministic system will have a zero LE since it is fully predictable, whereas a random system will have a large positive exponent indicating no predictability. In most applications it is sufficient to compute only LLE instead of all LE. There are many algorithms available to

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estimate LLE and the Lyapunov spectrum (Sano and Sawada, 1985; Wolf et al., 1985; Sato et al., 1987; Zeng et al., 1991). Most of them are unreliable when operated on small data sets. In our present work we used a method proposed by Rosenstein et al. (1993), which is robust against small data sets. After reconstructing the attractor, this algorithm looks for nearest neighbors of each phase point Xi on trajectory. The distance between two neighboring points at instant ns0 is defined by di(0)sminXi≤XjyXi≤ where ≤∆≤ is the Euclidean norm. This algorithm imposes the constraint that nearest neighbors are temporally separated at least by the mean period of the time series. The LLE is then estimated as the mean rate of separation of nearest neighbors, i.e. we can write dj(i)fCjel1(iDt)

(3)

where Cj is the initial separation. Taking logarithm on both sides of Eq. (3), we obtain lnŽdj(i).slnCjql1(iDt)

(4)

Eq. (4) represents a set of approximately parallel lines (for js1, 2«, M), where slope is roughly proportional to the LLE. In practice, the LE is easily and accurately estimated using a leastsquares fit to the ‘average’ line defined by; y(n)s

Fig. 1. Shows the QT interval time series of a normal subject in supine and standing postures.

1 Nln dj(n)M Dt

where n is the time index and di(n) is the distance between phase–space point Xi and its nearest neighbor at nth instant of time. Operator NM denotes average overall values of i. This averaging allows an accurate evaluation of LLE, l even when we have a short and noisy data set. We used MED and LLE of QT interval and HR time series and also the ratios of MED (qtyhr) (MEDqthr) and LLE(qtyhr) LLEqthr.

2.5. Statistical analysis We used BMDP statistical software (Berkeley, CA) for all the analyses. We used two-way ANOVA for repeated measures with patients vs. controls as the grouping factor and supine vs. standing posture as the repeated measure. Age was used as a covariate as the patients with depression were significantly older. Significant effects were followed up by paired t-tests to compare patients and controls for supine and standing postures separately. All tests were two-tailed, and a probability value of 0.05 was accepted as significant. As we performed two post hoc t-tests after each ANOVA, we used a significance level of 0.025 for these t-tests. Pearson’s product-moment correlations were used to examine the relationship between measures of interest. Fig. 1 shows the QT interval time series of a normal subject, Fig. 2 the attractors, Fig. 3 the calculation of MEDs, Fig. 4 the calculation of LLEs, and Fig. 5 the relationship of LLEqthr and QTvi for normal controls.

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Fig. 2. Shows the attractors of the above time series in Fig. 1.

3. Results Table 1 shows the results of supine and standing HRm, QTm and nonlinear measures of QT. The significance values represent pairwise comparisons after the following ANOVAs were performed using age as a covariate. Patients with depression as well as panic disorder were significantly more anxious

than controls (SAI: controls: 26.9"6.7; panic disorder: 41.1"12.5; depression: 52.7"6.4; P0.0001). Patients had significantly higher HRm (Fs5.4; d.f.s2.80; Ps0.006). For QT interval series, MED was significantly lower (Fs22.2; d.f.s2.80; Ps0.00001) while LLE was significantly higher in patients compared to controls (Fs 11.5; d.f.s2.80; Ps0.00001).

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Fig. 3. Shows the calculation of MEDs for the above data sets in Fig. 1. ‘X’ axis represents embedding dimension and ‘Y’ axis, ‘E1’ values.

MEDqthr (MED of QTyHR) was significantly lower (Fs40.3; d.f.s2.80; Ps0.00001) while LLEqthr (LLE of QTyHR) was significantly higher in patients compared to controls (Fs20.2; d.f.s 2.80; Ps0.00001). This is due to the significant inverse relationship between MED and LLE. There were no significant correlations between LLEqthr, anxiety measures and QTvi in supine or standing postures (Fig. 5).

Fig. 4. Shows the calculation of LLEs for the above data sets in Fig. 1. ‘X’ axis represents evolution of steps and ‘Y’ axis, ‘y(n)’.

4. Discussion To our knowledge, this is the first report on nonlinear analyses of QT interval time series. The main findings of this study are that patients with panic disorder and depression have a significantly higher LLE of QT interval time series, which

Fig. 5. Shows the lack of a significant correlation of LLEqthr and QTvi in supine and standing postures for normal controls.

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Table 1 Heart rate and QT interval variability measures of normal controls (ns28), patients with panic disorder (ns36) and patients with depression (ns18) Normal controls

Panic patients

Depressed patients

HRm—Supine Standing

61.8"7.8 71.9"10.9

69.7"9.3 83.4"10.8

69.1"9.2 83.7"14.6

QTm—Supine Standing

444"27 408"27

422"32 388"28

430"36 389"37

QT-MED—Supine Standing

18.9"0.97 19.2"0.86

17.6"1.55*** 17.3"1.20****

18.0"0.80** 18.6"1.34***

QT-LLE—Supine Standing

0.07"0.01 0.07"0.007

0.08"0.01** 0.08"0.01****

0.08"0.008* 0.08"0.01***

QTvi—Supine Standing

y1.94"0.28 y1.47"0.33

y1.65"0.38*** y1.17"0.38**

y1.48"0.39**** y1.17"0.43**

MEDqthr—Supine Standing

1.53"0.24 1.87"0.28

1.19"0.17**** 1.30"0.21****

1.32"0.20** 1.61"0.37**

LLEqthr—Supine Standing

0.58"0.13 0.41"0.08

0.76"0.19*** 0.61"0.16****

0.67"0.15* 0.53"0.14***

HRm is mean HR in bpm. QTm is mean QT interval in ms. QTvislog10 of detrended QT variability corrected for QTm squared divided by detrended HR variability divided by HRm squared. MEDqthr, MED of QTyMED of HR; LLEqthr, LLE of QTyLLE of HR. * P-0.05. ** P-0.01. *** P-0.001. **** P-0.00001.

suggests that these time series are more unpredictable compared to normal controls. This is in line with our findings of a significantly higher QTvi in patients with panic disorder and depression (Yeragani et al., 2000b). Previously we have shown that the LLE of HR is lower in patients with panic disorder and depression (Radhakrishna and Yeragani, 2001; Yeragani et al., 2002b), which shows that the HR time series are more chaotic and unpredictable in normal controls. Together these findings suggest that the condition of health is associated with a higher degree of chaos of HR and a lower degree of chaos of QT intervals, which reflects relatively a higher level of homogeneity of cardiac repolarization reflecting in a lower QT variability. Thus an increase in chaos reflects health only in a few physiological functions (Pool, 1989). It is well known that lifethreatening arrhythmias in various cardiac conditions are associated with heterogeneity of cardiac repolarization, which is associated with

increased QT dispersion in 12-lead ECG as well as increased QT variability in a single lead (Day et al., 1990; Atiga et al., 1998). The ratio of LLE of QT over LLE of HR (LLEqthr) is an approach somewhat similar to the calculation to QTvi in time domain and it is not surprising that LLEqthr is significantly higher in the patients. The interesting finding is that there is little correlation between these NL measures and QTvi in supine or standing postures, which suggests that these new measures could be a valuable addition to the existing linear indices of cardiac repolarization lability. Thus, even for short-term series, supine LLE may yield valuable information from the point of nonlinear measures. It should also be noted that measures such as fractal dimension using certain algorithms and approximate entropy correlate highly significantly with the HF power of the power spectrum of HR, thus reflecting RSA, which is related to cardiac cholinergic function (Yeragani et al., 1993b, 1997, 1998b).

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HRm did not affect the outcome of the analyses, and thus the higher HR in patients did not account for the present findings. If the results were due to better physical fitness in the control group, the HF power should have been significantly lower in the patient group, which was not the case. It will be important to evaluate these measures in future studies, especially in patients before and after treatment with different medications. This is important in view of the association between prolongation of QT interval and sudden cardiac death. Several psychotropic agents produce such a prolongation of QT interval. These drugs include TCAs and also some of the antipsychotics such as thioridazine. It may be more dangerous if a drug results in prolongation of QT interval as well as increased QT interval variability. We have previously shown that nortriptyline, a TCA, significantly increases QTvi (Yeragani et al., 2000c). Here, it is important to note that it may be more dangerous to use some of these medications in subjects who already have an increased QTvi before treatment. Our other studies also suggest that patients with schizophrenia have an increased QTvi at baseline compared to normal controls and treatment with some antipsychotics results in a significant increase of QTvi, which may make these patients vulnerable to serious ventricular arrhythmias (Drs Malaspina and Mujica-Parodi, personal communication). LLE describes the rate of exponential divergence of trajectories and the sensitive dependence on the initial condition. The decrease in predictability (increase in LLE) probably reflects increased variability of the QT interval series. Thus, the decrease in LLE of HR may also partly explain the higher incidence of significant cardiovascular mortality in patients with panic disorder. The significant decrease of LLE of HR in patients with multiple sclerosis and other neurological disorders with autonomic dysfunction suggests that this is a useful measure to study cardiac adaptability (Ganz et al., 1993; Faustmann and Ganz, 1994). It should also be noted that epileptics have a decreased LLE of HR (Faustmann and Ganz, 1994) and they apparently have a higher incidence of sudden death (Chugh et al., 2000; Ficker, 2000; Opeskin et al., 2000). In this context the report of Nei et al.

(2000) is important as it suggests that cardiac rhythm and conduction abnormalities are common during seizures and that these abnormalities may contribute to sudden death in epileptics. Some reports also suggest that that with increasing age, there may be a decrease in nonlinear measures of the cardiovascular system, which may be related to a decrease in cardiac vagal function (Kaplan et al., 1991; Lipsitz and Goldberger, 1992). In fact, two recent reports have shown that vagal blockade significantly decreases LLE (Hagerman et al., 1996; Zweiner et al., 1996) thus linking LLE to cardiac vagal activity. If these findings are replicated, LLE may prove to be a very sensitive indicator of cardiac vagal function. However, the focus of this article is the nonlinear measure of QT interval, LLEqthr, which may help us better understand the relative cardiac sympathetic function. It will also be interesting to study how the NL indices change with advancing age. The values of LLE were inversely related to those of MED, and MEDqthr was very significantly lower in patients compared to controls. This suggests that patients have a lower dimension of QT interval time series and this difference was highly significant between normal controls and patients. We found that MED was more sensitive than CD in differentiating normal controls and patients with panic disorder (Radhakrishna and Yeragani, 2001), and overall, there was no strong correlation between these measures in either posture. Due to the labor-intensive nature of the computation of CD, we have limited this investigation to MED and LLE measures. At the same time, the calculation of CD was more subjective and time-consuming (Radhakrishna and Yeragani, 2001). 4.1. Limitations The main limitation is the use of only 256 s of supine and standing data. However, we were able to show consistent differences with these shortterm records of QT interval and HR time series between groups. It is possible that even a series of approximately 1000 points may be affected by nonstationarity of the data. Studies are underway in our laboratory using longer segments of Holter

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ECG of 3600 inter-beat intervals (45 min to 1 h of data before and after using any interpolation techniques). The results of these analyses showed essentially similar differences between controls and patients with panic disorder for LLE of HR. Though there is a certain amount of subjectivity involved in the calculation of MED, we have standardized it to a great extent using a graphical procedure to identify the point of saturation, which correlates highly even when different people perform these analyses. The rest of the steps are all completely automated and do not lead to any bias on the part of the investigator. 5. Conclusions The present study of QTv and HRv time series using nonlinear techniques has shown highly significant differences between normal controls and patients with panic disorder and depression, and these techniques appear to be valuable additions to the already existing time and frequency domain measures. Measures of chaos such as LLE should be investigated further in future studies of cardiac autonomic function in different illnesses and the effect of successful treatment on these measures. These techniques may be useful to study other nonlinear time series as well. References Akselrod, S., Gordon, D., Ubel, F.A., Shannon, D.C., Barger, A.C., Cohen, R.J., 1981. Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222. Atiga, W.L., Calkins, H., Lawrence, J.H., Tomaselli, G.F., Smith, J.M., Berger, R.D., 1998. Beat-to-beat repolarization lability identifies patients at risk for sudden cardiac death. Journal of Cardiovascular Electrophysiology 9, 899–908. Atiga, W.L., Finanpazir, L., McAreavey, D., Calkins, H., Berger, R.D., 2000. Temporal repolarization lability in hypertensive cardiomyopathy caused by beta-chain heavy gene mutations. Circulation 101, 1237–1242. Berger, R.D., Akselrod, S., Gordon, D., Cohen, R.J., 1986. An efficient algorithm for spectral analysis of heart rate variability. IEEE Transactions in Biomedical Engineering 33, 900–904. Berger, R.D., Kasper, E.K., Baughman, K.L., Marban, E., Calkins, H., Tomaselli, G.F., 1997. Beat-to-beat QT interval variability. Novel evidence for repolarization lability in

187

ischemic and nonischemic dilated cardiomyopathy. Circulation 96, 1557–1565. Bigger, J.T., Fleiss, J.L., Steinman, R., Rolnitzky, L.M., Kleiger, R.E., Rottman, J.N., 1992. Frequency domain measures of heart period variability and mortality after myocardial infarction. Circulation 85, 164–171. Binah, O., Rosen, M., 1992. Mechanisms of ventricular arrhythmias. Circulation 85, I25–I31. Braun, C., Kowallik, P., Freking, A., Hadeler, D., Kniffki, K.D., Meesmann, M., 1998. Demonstration of nonlinear components in heart rate variability of healthy persons. American Journal of Physiology 275, H1577–H1584. Broomhead, D.S., King, G.P., 1986. Extracting qualitative dynamics from experimental data. Physica D 20, 217–236. Carney, R.M., Sounders, R.D., Freedland, K.E., Stein, P., Rich, M.W., Jaffe, A.S., 1995. Association of depression with reduced heart rate variability in coronary artery disease. American Journal of Cardiology 76, 562–564. Carney, R.M., Freedland, K.E., Sheline, Y.I., Weiss, E.S., 1997. Depression and coronary heart disease: a review for cardiologists. Clinical Cardiology 20, 196–200. Casaleggio, A., Cerutti, S., Signorini, M.G., 1997. Study of the Lyapunov exponents in heart rate variability signals. Methods of Information in Medicine 36, 274–277. Chugh, S.S., Kelly, K.L., Titus, J.L., 2000. Sudden cardiac death with apparently normal heart. Circulation 102, 649–654. Coryell, W., Noyes Jr., R., House, J.D., 1986. Mortality among outpatients with anxiety disorders. American Journal of Psychiatry 143, 508–510. Day, C.P., McComb, J.M., Campbell, R.W.F., 1990. QT dispersion: an indication of arrhythmia risk in patients with long QT intervals. British Heart Journal 63, 342–344. Faustmann, P.M., Ganz, R.E., 1994. Central cardioautonomic disorganization in interictal states of epilepsy detected by phase space analysis. International Journal of Neuroscience 78, 43–47. Ficker, D.M., 2000. Sudden unexplained death and injury in epilepsy. Epilepsia 41, S7–S12. Frasure-Smith, N., Lesperance, F., Talajic, M., 1993. Depression following myocardial infarction. Impact on six-month survival. Journal of the American Medical Association 270, 1819–1825. Ganz, R.E., Weibels, G., Stacker, K.H., Faustmann, P.M., Zimmermann, C.W., 1993. The Lyapunov exponents of heart rate dynamics as a sensitive marker of central autonomic organization: an exemplary study of early multiple sclerosis. International Journal of Neuroscience 71, 29–36. Goldberger, A.L., West, B.J., 1987. Fractals in physiology and medicine. Yale Journal of Biology and Medicine 60, 421–435. Grassberger, P., Procaccia, I., 1983a. Measuring the strangeness of strange attractors. Physica D 9, 189–208. Grassberger, P., Procaccia, I., 1983b. Characterization of strange attractors. Physical Review Letters 50, 346–349. Guzzetti, S., Signorini, M.G., Cogliati, C., Mezzetti, S., Porta, A., Cerutti, S., Malliani, A., 1996. Non-linear dynamics and

188

V.K. Yeragani, K.A. Radhakrishna Rao / Psychiatry Research 117 (2003) 177–190

chaotic indices in heart rate variability of normal subjects and heart-transplanted patients. Cardiovascular Research 31, 441–446. Hagerman, I., Berglund, M., Lorin, M., Nowak, J., Sylven, C., 1996. Chaos deterministic regulation of heart rate variability in time and frequency domains: effects of autonomic blockade and exercise. Cardiovascular Research 31, 410–418. Ho, K.K., Moody, G.B., Peng, C.K., Mietus, J.E., Larson, M.G., Levy, D., Goldberger, A.L., 1997. Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics. Circulation 96, 842–848. Jervell, A., Lange-Nelson, F., 1957. Congenital deaf mutism, functional heart disease with prolongation of the QT interval, and sudden death. American Heart Journal 54, 59–68. Kagiyama, S., Tsukashima, A., Abe, I., Fujishima, S., Ohmori, S, Onaka, U., Ohya, Y., Fujii, K., Tsushihashi, T., Fujishima, M., 1999. Chaos and spectral analyses of heart rate variability during head-up tilting in essential hypertension. Journal of the Autonomic Nervous System 76, 153–158. Kanters, J.K., Hojgaard, M.V., Agner, E., Holstein-Rathlou, N.H., 1997. Influence of forced respiration on nonlinear dynamics in heart rate variability. American Journal of Physiology 272, R1149–R1154. Kaplan, D.T., Furman, M.I., Pincus, S.M., Ryan, S.M., Lipsitz, L.A., Goldberger, A.L., 1991. Aging and the complexity of cardiovascular dynamics. Biophysical Journal 59, 945–948. Kawachi, I., Sparrow, D., Vokonas, P.S., Weiss, S.T., 1994. Symptoms of anxiety and risk of coronary heart disease: the normative aging study. Circulation 90, 2225–2229. Kennel, M.B., Brown, R., Abarbanel, H.D.I., 1992. Determining minimum embedding dimension using a geometrical construction. Physics Review Part A 45, 3403–3411. Kleiger, R.E., Miller, J.P., Bigger, J.T., Moss, A.J., 1987. Decreased heart rate variability and its association with increased mortality after acute myocardial infarction. American Journal of Cardiology 59, 256–262. Krittiyaphong, R., Cascio, W.E., Light, K.C., Sheffield, D., Golden, R.N., Finkel, J.B., Glekas, G., Koch, G.G., Sheps, D.S., 1997. Heart rate variability in patients with coronary artery disease: differences in patients with higher and lower depression scores. Psychosomatic Medicine 59, 231–235. Liangyue, Cao, 1997. Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50. Lindqvist, A., Jalonen, J., Parviainen, P., Antilla, K., Laitinen, L.A., 1990. Effect of posture on thermally stimulated cardiovascular oscillations. Cardiovascular Research 24, 373–380. Lipsitz, L.A., Goldberger, A.L., 1992. Loss of complexity and aging. Journal of the American Medical Association 267, 1806–1809. Lombardi, F., Sandrone, G., Mortara, A., Torzillo, D., Rovere, M.T.L., Signorini, M.G., Cerutti, S., Malliani, A., 1996. Linear and nonlinear dynamics of heart rate variability after acute myocardial infarction with normal and reduced left

ventricular ejection fraction. American Journal of Cardiology 77, 1283–1288. Makikallio, T.H., Koistinen, J., Jordaens, L., Tulppo, M.P., Wood, N., Golasarsky, B., Peng, C.K., Goldberger, A.L., Huikuri, H.V., 1999. Heart rate dynamics before spontaneous onset of ventricular fibrillation in patients with healed myocardial infracts. American Journal of Cardiology 83, 880–884. Malik, M., Camm, A.J., 1990. Heart rate variability. Clinical Cardiology 13, 570–576. Malliani, A., Pagani, M., Lombardi, F., Cerutti, S., 1991. Cardiovascular neural regulation explored in the frequency domain. Circulation 84, 482–492. Mees, A.I., Rapp, P.E., Jennings, L.S., 1987. Singular value decomposition and embedding dimension. Physical Review A 37, 340–346. Molgaard, H., Sorensen, K.E., Bjerregard, P., 1991. Attenuated 24-hour heart rate variability in apparently healthy subjects, subsequently suffering sudden cardiac death. Clinical Autonomic Research 1, 233–237. Musselman, D.L., Evans, D.L., Nemeroff, C.B., 1998. The relationship of depression to cardiovascular disease. Archives of General Psychiatry 55, 580–592. Nei, M., Ho, R.T., Sperling, M.R., 2000. EKG abnormalities during partial seizures in refractory epilepsy. Epilepsia 41, 542–548. Opeskin, K., Harvey, A.S., Cordner, S.M., Berkovic, S.F., 2000. Sudden unexplained death in epilepsy in Victoria. Journal of Clinical Neuroscience 7, 34–37. Pincus, S.M., Gladstone, I.M., Ehrenkranz, R.A., 1991. A regulatory statistic for medical data analysis. Journal of Clinical Monitoring 7, 335–345. Pomeranz, B., Macaulay, R.J.B., Caudill, M.A., Kutz, I., Adam, D., Gordon, D., Kilborn, K.M., Barger, A.C., Shannon, D.C., Cohen, R.J., Benson, H., 1985. Assessment of autonomic function in humans by heart rate spectral analysis. American Journal of Physiology 248, H151–H153. Pool, R., 1989. Is it healthy to be chaotic? Science 243, 604–607. Radhakrishna, R.K.A., Narayana Dutt, D., Yeragani, V.K., 2000. Nonlinear measures of heart rate time series. Influence of posture and controlled breathing. Journal of the Autonomic Nervous System 83, 148–158. Radhakrishna, R.K.A., Yeragani, V.K., 2001. Decreased measures of chaos and increased nonlinearity in patients with panic disorder. Autonomic Neuroscience 88, 99–108. Rosenstein, M., Collins, J.J., De Luca, C.J., 1993. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117–134. Sano, M., Sawada, Y., 1985. Measurement of the Lyapunov spectrum from a chaotic time series. Physical Review Letters 55, 1082–1085. Sato, S., Sano, M., Sawada, Y., 1987. Practical methods of measuring the generalized dimension and largest Lyapunov exponents in high-dimensional chaotic systems. Progress of Theoretical Physics 77, 1–5.

V.K. Yeragani, K.A. Radhakrishna Rao / Psychiatry Research 117 (2003) 177–190 Schwartz, P.J., Wolf, S., 1978. QT interval prolongation as predictor of sudden death in patients with myocardial infarction. Circulation 57, 1074–1077. Silipo, R., Deco, G., Vergassola, R., Gremigni, C., 1999. A characterization of HRV’s nonlinear hidden dynamics by means of Markov models. IEEE Transactions on Biomedical Engineering 46, 978–985. Skinner, J.E., Pratt, C.M., Vybiral, T., 1993. A reduction in the correlation dimension of heart beat intervals precedes imminent ventricular fibrillation in human subjects. American Heart Journal 125, 731–743. Spielberger, C.D., Gorusch, R.L., Luschene, R.E., 1970. StateTrait Anxiety Inventory. Consulting Psychologists Press, Palo Alto, CA. Spitzer, R.L., Williams, J.B.W., Gibbon, M., 1987. Structured Clinical Interview for DSM III-R (SCID). New York State Psychiatric Institute, Biometrics Research, New York. Stein, P.K., Carney, R.M., Freedland, K.E., Skala, J.A., Jaffe, A.S., Kleiger, R.E., Rottman, J.N., 2000. Severe depression is associated with markedly reduced heart rate variability in patients with stable coronary heart disease. Journal of Psychosomatic Research 48, 493–500. Storella, R.J., Wood, H.W., Mills, K.M., Kanters, J.K., Hojgaards, M.V., Holstein-Rathlou, N.H., 1998. Approximate entropy and point correlation dimension of heart rate variability in healthy subjects. Integrative Physiological and Behavioural Science 33, 315–320. Takens, F., 1980. Detecting strange attractors in turbulence. In: Rand, D., Young, L. (Eds.), Dynamical Systems and Turbulence. Lecture Notes on Mathematics, vol. 898. Springer, Berlin, pp. 366–381. Theiler, J., 1987. Efficient algorithm for estimating the correlation dimension from a set of discrete points. Physical Review A 36, 4456–4462. Tomaselli, G.F., Buckelmann, D.J., Calkins, H.G., 1994. Sudden cardiac death in heart failure: the role of abnormal repolarization. Circulation 90, 2534–2539. Voss, A., Kurths, J., Kleiner, H.J., Witt, A., Wessel, N., Separin, P., Osterzeil, K.J., Schurath, R., Dietz, R., 1996. The application of methods of nonlinear dynamics for the improved and predictive recognition of patients threatened by sudden cardiac death. Cardiovascular Research 31, 419–433. Weissmann, M.W., Markowitz, J.S., Ouelette, R., Greenwald, S., Kahn, J.P., 1990. Panic disorder and cardiovasculary cerebrovascular problems. American Journal of Psychiatry 147, 1504–1507. Wolf, A., Swift, J., Swinney, H., Vastano, J., 1985. Determining Lyapunov exponents from a time series. Physica D 16, 285–317. Yeragani, V.K., 1995b. Heart rate and blood pressure variability: implications for psychiatric research. Neuropsychobiology 32, 182–191. Yeragani, V.K., 2000. Major depression and long-term heart period variability. Depression and Anxiety 12, 51–52. Yeragani, V.K., Balon, R., Pohl, R., Ramesh, C., Glitz, D., Weinberg, P., Merlos, B., 1990. Decreased R–R variance in

189

panic disorder patients. Acta Psychiatrica Scandinavica 81, 554–559. Yeragani, V.K., Berger, R., Pohl, R., Srinivasan, K., Balon, R., Ramesh, C., Weinberg, P., Berchou, R., 1992. Effects of yohimbine on heart rate variability in panic disorder patients and normal controls: a study of power spectral analysis of heart rate. Journal of Cardiovascular Pharmacology 20, 609–618. Yeragani, V.K., Pohl, R., Berger, R., Balon, R., Ramesh, C., Glitz, D., Srinivasan, K., Weinberg, P., 1993a. Decreased heart rate variability in panic disorder patients: a study of power spectral analysis of heart rate. Psychiatry Research 46, 89–103. Yeragani, V.K., Srinivasan, K., Vempati, S., Pohl, R., Balon, R., 1993b. Fractal dimension of heart rate time series: an effective measure of autonomic function. Journal of Applied Physiology 75, 2429–2438. Yeragani, V.K., Pohl, R., Srinivasan, K., Balon, R., Ramesh, C., Berchou, R., 1995b. Effects of isoproterenol on heart rate variability in patients with panic disorder. Psychiatry Research 56, 289–293. Yeragani, V.K., Nadella, R., Hinze, B., Yeragani, S., Jampala, V.C., 2000a. Nonlinear measures of heart period variability: decreased measures of symbolic dynamics in patients with panic disorder. Depression and Anxiety 12, 67–77. Yeragani, V.K., Pohl, R., Jampala, V.C., Balon, R., Ramesh, C., Srinivasan, K., 2000b. Increased QT variability in patients with panic disorder and depression. Psychiatry Research 93, 225–235. Yeragani, V.K., Pohl, R., Jampala, V.C., Balon, R., Ramesh, C., Srinivasan, K., 2000c. Effects of nortriptyline and paroxetine on QT variability in patients with panic disorder. Depression and Anxiety 11, 126–130. Yeragani, V.K., Pohl, R., Jampala, V.C., Balon, R., Kay, J., Igel, G., 2000d. Effect of posture and isoproterenol on beatto-beat heart rate and QT variability. Neuropsychobiology 41, 113–123. Yeragani, V.K., Pohl, R., Balon, R., Berchou, R., Srinivasan, K., Glitz, D., 2002a. Effect of intravenous sodium lactate and isoproterenol infusions in normal controls and patients with panic disorder on nonlinear indices of heart rate variability. Psychiatry Research 101, 81–92. Yeragani, V.K., Radhakrishna Rao, K.A., Smitha, M.R., Pohl, R., Balon, R., Srinivasan, K., 2002b. Diminished chaos of heart rate time series in patients with major depression. Biological Psychiatry 51, 733–744. Yeragani, V.K., Radhakrishna Rao, K.A., Ramakrishnan, K.R., Srinivasan, S.H. Relationship of measures of LLE of unfiltered time series of heart rate to those in different frequency bands: a possible measure of relative vagal and sympathetic activity. Submitted for publication. Yeragani, V.K., Sobolewski, E., Kay, J., Jampala, V.C., Igel, G., 1997. Effect of age on long-term heart rate variability. Cardiovascular Research 35, 35–42. Yeragani, V.K., Sobolewski, E., Igel, G., Johnson, C., Jampala, V.C., Kay, J., Hillman, N., Yeragani, S., Vempati, S., 1998a. Decreased heart period variability in patients with panic

190

V.K. Yeragani, K.A. Radhakrishna Rao / Psychiatry Research 117 (2003) 177–190

disorder: a study of Holter ECG records. Psychiatry Research 78, 89–99. Yeragani, V.K., Sobolewski, E., Jampala, V.C., Kay, J., Yeragani, S., Igel, G., 1998b. Fractal dimension and approximate entropy of heart period and heart rate: awake vs sleep differences and methodological issues. Clinical Science 95, 295–301. Yeragani, V.K., Srinivasan, K., Balon, R., Ramesh, C., Berchou, R., 1994. Lactate sensitivity and cardiac cholinergic

function in panic disorder. American Journal of Psychiatry 151, 1226–1228. Zeng, X., Eykholt, R., Pielke, R.A., 1991. Estimating the Lyapunov-exponent spectrum from short time series of low precision. Physical Review Letters 66, 3229–3232. Zweiner, U., Hoyer, D., Bauer, R., Luthke, B., Walter, W., Schmidt, K., Hallmeyer, S., Kratzsch, B., Eiselt, M., 1996. Deterministic chaotic and periodic properties of heart rate and arterial fluctuations and their mediation in piglets. Cardiovascular Research 31, 455–465.