Nonlinear Dynamical Metrics Distinguish Between Baseline and 160- and 320-mg Doses of Sotalol

Timothy Callahan, PhD1, William P. Stuppy, MD2

1Biomedical Systems, St. Louis, MO; 2Private Practice, Los Angeles, CA

Correspondence: Timothy Callahan, PhD,
Chief Scientific Officer,
Biomedical Systems,
77 Progress Parkway,
St. Louis, MO 63043.
Tel: 818-470-2066




Nonlinear dynamics have been used recently as a metric for heart rate variability from long-term (Holter) electrocardiograms. These metrics, including Lyapunov exponents and components of Erlang distributions, have been used to characterize disease states such as congestive heart failure and atrial fibrillation. Few studies have been done using current medical therapies that can alter the cardiac safety profile of an individual patient. In this study, we used Holter data from normal, healthy volunteers at baseline and on 160 mg and 320 mg of sotalol, and calculated the maximum Lyapunov exponent and the K and L components of the Erlang distribution as well as the time-domain heart rate variability (HRV) metrics root mean square of the successive differences and standard deviation of all normal RR intervals. The data show that chaotic processes are reduced from baseline on both 160 mg and 320 mg of sotalol. Averaging 5-minute values over the entire recording period, the mean ± standard error of the maximum Lyapunov exponent was –0.0020±0.0162 at baseline, –0.0564 ± 0.0122 on 160 mg of sotalol, and –0.0726 ± 0.0163 on 320 mg of sotalol (P = 0.0042). Around the maximum concentration, the average maximum Lyapunov exponent was 0.0259 ± 0.0191 at baseline, –0.0706 ± 0.0141 at 160 mg, and –0.0825 ± 0.0218 at 320 mg. The decrease in chaotic processes is suggestive of a lessening of cardiac health. Additionally, the K component of the Erlang distribution is increased from 218.9 ± 13.12 at baseline, to 268.3 ± 18.16 on 160 mg of sotalol and 329.6 ± 42.65 on 320 mg of sotalol (P = 0.0057). Changes in the L component of the Erlang distribution did not reach statistical significance. The increases in Erlang components indicate more random processes. Time-domain HRV measured showed slight decreases on dose but stayed above or within normal HRV values. We conclude that sotalol, a pharmaceutical compound known to be proarrhythmic, decreases healthy cardiac dynamics as shown by nonlinear dynamic modeling. Nonlinear dynamical modeling adds valuable information to the analysis of whether a pharmaceutical compound is proarrhythmic.

Keywords: non-linear dynamics, heart rate variability, Lyapunov exponent, sotalol, Holter monitoring


Heart rate variability (HRV) is the science of analyzing components of the autonomic nervous system from the variability of the RR interval of the long-term electrocardiogram (ECG). In general, measurements of the time between each heart beat are taken and analyzed using time- or frequency-domain mathematics, or nonlinear dynamical modeling. HRV values have been assessed using standard statistical methods (time domain) and advanced methods based on calculus functions (frequency domain). These measures have been calculated using a single series of data, specifically the RR intervals from Holter recordings to determine the relative contributions of the sympathetic and parasympathetic nervous systems. Although standard HRV analysis has shown some promise in certain disease conditions, the science of HRV has been controversial because some studies have not been repeatable by different researchers. However, attempts have been made to establish normal values to give clinicians added insight into a patient’s condition.1,2

Nonlinear methods have been applied to HRV.3,4 These methods include, but are not limited to, the Lyapunov exponents and the Erlang distributions. Nonlinear mathematic modeling shows promise, as it can describe the dynamics of time-series data. The Lyapunov exponent characterizes the rate of separation or divergence of trajectories. A positive Lyapunov exponent indicates that a system is ergodic and exhibits chaos. A larger maximum Lyapunov exponent (MLE) is indicative of a healthier state of a system. Erlang distributions are described with two components—K and L. The K component describes the shape of the Erlang distributions. The L component (also known as lambda) is the rate of change of the Erlang distributions. Random (stochastic) systems exhibit larger Erlang components (both K and L).

Sotalol is a class III antiarrhythmic with beta-blocking and inward potassium rectifying current IKr blocking capabilities5 used to treat atrial fibrillation as well as ventricular arrhythmias such as ventricular tachycardia. Even though sotalol is used to treat arrhythmias, it is well understood that it has proarrhythmic potential. Prolongation of the QT interval and torsades de pointes (TdP) has been associated with sotalol dosing.6,7 The package insert suggests that patients be monitored in a facility with ECG monitoring and cardiac resuscitation capabilities for 3 days when dosing is initiated due to the ability of sotalol to cause cardiac arrhythmias.8

The purpose of this study was to analyze long-term ECG data using the MLE and the K and L components of the Erlang distribution, compared with the standard statistical HRV indices of root mean square of the successive differences (rMSSD) and the standard deviation of all normal RR intervals (SDNN), and determine if these measures could potentially discriminate among baseline and low and high doses of sotalol, a potentially torsadogenic compound, and thus possibly find a new metric for determining whether a pharmaceutical compound has proarrhythmic potential.

Materials and Methods


The data used have been characterized previously.9 In brief, sotalol was administered to 39 normal healthy volunteers (28 males and 11 females) in an open-label, nonrandomized, fixed-treatment sequence on 3 successive days. After a baseline day (day –1), subjects were administered 160 mg of sotalol (day 1) and 320 mg of sotalol (day 2). Doses were administered in a fasting condition.

Data Acquisition

The study protocol included the collection of 12-lead ECGs at pre-dose and at 15 time points after dosing, and 12-lead digital Holter monitoring (Mortara H12+, Mortara Instrument Inc., Milwaukee, WI) for 22.5 hours after dosing. Both 12-lead ECGs and Holter recordings occurred during the baseline day and at each dose. The Holter data was captured at 180 samples per second, 16-bit, 2.5-µV resolution. These data were used for this analysis.

Plasma sotalol concentrations were collected and averaged from 1 hour to 5 hours post-dose (the general time of Cmax).

Data Analysis

The Holter data was provided in an International Society for Holter and Noninvasive Electrocardiology (ISHNE)10 format, and scanned using a Century Holter System (version 2.2.1, Biomedical Systems, St. Louis, MO). Holter recordings were scanned by a trained reader using the superimposition mode, in which the current ECG complexes were superimposed on the previous ECG complex. Annotations were placed on the fiducial points of the ECG complex (P wave, Q wave, R wave, J point, peak of the T wave, and at the end of the T wave). If the annotations moved from the fiducial points, the scan was stopped and the annotations replaced in the correct position.

Ectopic atrial beats, premature ventricular contractions, and atrial and ventricular arrhythmias were removed prior to post-Holter analysis. Only the normal RR intervals were exported for calculation of the MLE and Erlang distributions. Missing data were interpolated using a simple spline method. Standard, time-domain indices of rMSSD and SDNN were calculated from the series of normal RR intervals for each Holter recording.

For each 5-minute segment after the beginning of the recording, the MLE and Erlang distributions were calculated until the end of the recording period. The 5-minute values were exported to a database for analysis.

The general formula for the Lyapunov exponent and the Erlang distribution can be found in Jonkheere et al.11

The 24-hour average was calculated from the 5-minute values for the Lyapunov exponent and the K and L Erlang components for each subject.

Because the systems show stochasticity and ergodicity, the percentage of times the MLEs were > 0 were also compared between doses.

Statistical Analysis

Statistical analysis was conducted on the 5-minute maximum Lyapunov exponents and the Erlang values (K and L) representing 3 dose groups (baseline, 160 mg, and 320 mg). The raw values of the MLEs were summarized by dose and by dose and time point. The MLEs were also summarized by the number and percent that were positive versus negative. The 24-hour mean Erlang values were summarized by dose with the typical statistics such as count, mean, minimum, maximum, and standard deviation. The same summary statistics were used to summarize the canonical correlations by time point and dose.

Analysis of variance (ANOVA) and Kruskal-Wallis tests were used to assess differences among the dose groups in terms of raw MLEs, overall, and at Cmax. Comparisons were also made among the 3 dose groups for the proportion of positive MLEs using a Chi-square test. The 3 dose groups were compared in terms of mean 24-hour Erlang values using ANOVA and Kruskal-Wallis tests. The ratio of MLEs and its logarithm were compared across dose groups for each algorithm, also using ANOVA and Kruskal-Wallis tests. Time-domain HRV analyses were compared between baseline and dosing using a paired t-test. P ≤ 0.05 was considered statistically significant for all comparisons. Further comparisons were made at the maximum concentrations (Cmax) of sotalol and the MLE. Because the Cmax of both doses occurred between 1 and 5 hours post-dose, comparisons were made after averaging the values over this time period.

Data are presented as the mean ± standard error.


A total of 96 Holter recordings from 39 subjects were of sufficient quality to be used in this analysis. Table 1 shows the number of recordings per dose and the reason for exclusions.

Callahan Table 1

Seventeen subjects did not have recordings at the 320-mg dose. Four recordings (1 at baseline and 3 at 320 mg) contained too much artifact to be reliable, so they were excluded from this analysis.

All variables were calculated for each 5-minute epoch after the recording began and averaged for the 24-hour period. Data are shown in Table 2. For this analysis, the data was compared around Cmax (1 to 5 hours post-dose).

Callahan Table 2

Lyapunov Exponent

The lowest average MLE (per subject) was higher at baseline than for the 160- and 320-mg doses (–0.1404, –0.1844, and –0.1721, respectively). The MLE (per subject) was also higher at baseline than for the 160- and 320-mg doses (0.2349, 0.0945, and 0.0296, respectively).

The average of the 5-minute MLE was higher at baseline than in the 2 doses of sotalol (P = 0.0042). Figure 1 shows the per-subject average of the 5-minute epochs for each dose. The mean 5-minute epoch was higher at baseline than with the 160- or 320-mg dose, showing that sotalol reduces the average maximum Lyapunov exponent in a dose-dependent manner. The slightly negative value at baseline indicates that the MLE values are slightly stochastic in this population.

Callahan Figure_1

Figure 1. Average of the 5-minute maximum Lyapunov exponents (MLEs) per dose.


The percentage of 5-minute epochs with MLE > 0 is shown in Figure 2. The number of positive exponents are greater at baseline than at 160- or 320-mg dosing (P = 0.0161). A reduction from baseline of 19.7% was seen in the percentage of positive Lyapunov exponents in the 160-mg dose and 27.3% from baseline in the 320-mg dose.

Callahan Figure_2

Figure 2. Percentage of the MLEs calculated in 5-minute epochs that were positive by dose.


The decrease in both the average maximum Lyapunov exponents and the number of times the MLE was positive indicates a trend toward stochasticity in the relationship between the RR intervals.

Erlang Distributions

As can be seen in Table 2, the average Erlang K and L values increased from baseline to 160 mg and 320 mg in both doses as compared with baseline. Statistical significance was shown in the K component (P = 0.0057) but not the L component of the Erlang distribution (P = 0.1407).

An increase in the K component of the Erlang distribution indicates the trend toward a more random process in RR-interval dynamicity. Although the trend is toward an increasing rate (L component) with increasing dosing of sotalol, this was not statistically significant.

As with the Lyapunov exponents, the change in the Erlang values showed a move to a more stochastic relationship in the RR intervals.

Although the 160- and 320-mg doses showed a slight decrease as compared with baseline, no statistical significance was found between baseline and the 160-mg dose, or baseline and the 320-mg dose when measuring the rMSSD (Table 2). For the SDNN, a slight decrease was seen from baseline in the 160- and 320-mg doses. These were statistically significant (P  = 0.028 and P = 0.044, respectively).

The mean ± standard error for the MLE during the Cmax period (1 to 5 hours post-dosing) was 0.0259 ± 0.0191 at baseline, –0.0706 ± 0.0141 at 160 mg, and –0.0825 ± 0.0218 at 320 mg. Statistical significance was reached when comparing the baseline and 160-mg dose (P < 0.0001), and baseline and 320-mg dose (P < 0.0009), but not when comparing the 160- and 320-mg dose. These results are shown in Figure 3.

Callahan Figure_3

Figure 3. Average MLEs 1 to 5 hours after dosing. The MLE is reduced at 160- and 320-mg dosing.



We have come to understand that homeostasis in the human body is the result of fluctuating dynamical systems. Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex systems. One area of dynamical systems theory is chaos theory.

Chaos theory seeks to describe the behavior of certain dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions. Paradoxically, chaos theory refers to systems that have repeating values over time or are ergodic in nature. A specific area of chaos theory, called fractal attractors, is characterized by unstable periodic orbits and is the basis for understanding entropy. Under normal conditions RR dynamics are based on the attractor model. Narayan et al12 describe 3 or 4 dominant unstable periodic orbits in normal, healthy subjects, whereas Lu and Chen13 describe 7 to 8 dimensions in a healthy, complex dynamical systems. The existence of chaos is the normal healthy state. When the attractor model changes to a more random, or stochastic process, and under certain conditions (such as prolonged QT), we hypothesize that the patient then becomes susceptible to the development of arrhythmias such as TdP.

In this study, the MLE was larger, showing a higher degree of chaotic behavior at baseline than after dosing with sotalol. This was shown by both of the long-term averages, but also in the number of times the MLEs were positive.

Poon and Merrill14 detected chaotic segments in both congestive heart failure patients and normal subjects. In their study, fewer chaotic segments were seen in the congestive heart failure patients, supporting the hypothesis that chaotic dynamics are a sign of health. Therefore, we can postulate that a reduction in chaos is a movement toward a reduction in healthy dynamics.

Both the K and L components of the Erlang distribution were increased by dosing with sotalol. Hashida and Tasaki15 described the RR intervals in patients with atrial fibrillation as following an Erlang distribution. The stochastic process of atrial fibrillation is opposite of healthy, chaotic behavior. We can then postulate that sotalol disrupts the normal chaotic dynamics seen in sinus rhythm. In the present study this happened in a dose-dependent manner, with the 320-mg dose of sotalol having the highest Erlang values (both K and L).

In recent years the determination of cardiac safety in clinical trials has been mainly focused on prolongation of the QT interval as a biomarker for the development of TdP, a sometimes fatal polymorphic ventricular tachycardia. Prolongation of the QT interval can be seen in patients of all ages and genders with congenital long QT syndrome or acquired long QT syndrome. Patients can develop acquired long QT syndrome due to comorbidities such as hypocalcemia, coronary artery disease, or congestive heart failure, or by taking certain drugs that block specific ion channels in the myocardium.

Regulatory agencies have developed guidelines for testing pharmaceutical compounds for the ability to prolong the QT interval. The relevant one for this study is International Conference on Harmonization guideline E14,16 which calls for a single dedicated clinical trial to detect the ability of the pharmaceutical compound to prolong the QT interval. However, it is well understood that not all QT prolongation leads to TdP. Some drugs, such as ranolazine and verapamil, prolong the QT interval without causing TdP. Therefore, by itself, prolongation of the QT interval is not entirely prognostic. Ongoing efforts are underway to help determine what other factors may be involved in the development of cardiac arrhythmias. The current study may lead to a better understanding of the conditions in which TdP occur.

It has been shown that certain disease states can alter HRV. For example, increased mortality is correlated with a decrease in HRV in heart failure patients,17 and diabetic patients also show a decrease in HRV.18

To acquire the data for HRV analysis, a long-term ECG (Holter) is recorded. The typical Holter recording usually lasts for 24 hours, but can be as short as 5 minutes. The longer recording has advantages over the shorter recording, as the theory relies on the state of the system. Also the longer recording can quantify the changes in the phase space that occur over time.

Traditional HRV analyses have not proven to be robust enough to predict sudden cardiac death or arrhythmic events. This is most likely due to the adaptive processes in cardiac system. We feel that the current analysis is a step beyond traditional HRV and can provide further insights by using dynamical systems modeling.

The data presented here show that sotalol decreases both SDNN and rMSSD. Although the rMSSD changes were not significant, a slight decrease from baseline was seen. The reduction from baseline in SDNN was significant at both dosing levels. It is important to note that the values for both time-domain indices were higher than the normal values or within 1 standard deviation of the normal values set forth by the European Society of Cardiology.2

The current study utilized long-term ECG recordings to quantify the nonlinear dynamics of the ECG. Because both chaotic and stochastic processes have been described previously, parsing the data into 5-minute segments offered an opportunity to see both types of processes.

Sotalol is a class III antiarrhythmic with potassium-blocking capabilities. Prolongation of the QT interval due the delay in ventricular repolarization is thought to be important in the development of TdP. Although it is unknown whether the results of this study are due mainly to the antiarrhythmic effect of sotalol or the beta-blocking action, Zweiner et al19 showed that the beta-blockers mediated chaotic processes. It is possible that both actions have a role in the reduction of the MLE.

All subjects showed both chaotic and stochastic processes. The amount (time and magnitude) of the chaotic processes was reduced with sotalol dosing.


There are limitations to this study that should be understood as we further investigate the use of nonlinear dynamics when describing the RR/QT relationship. The data were sampled at 180 Hz. Because chaotic systems are sensitive to initial conditions, it is unknown how the sample rate will affect the analyses. The sample rate in this study, however, was consistent with that of Poon and Merrill,14 who used sample rates of between 128 and 250 Hz. Because there are relatively few studies using these values, no normal values exist. Our conclusions are based on changes from baseline with the assumption the baseline values are normal.

A further limitation of this study is that these analyses have been used in relatively few subjects under strict conditions. Therefore, it would be inappropriate to extrapolate these limited results for useful clinical information.

Also, we did not look at the circadian rhythm of the MLE or Erlang distributions, as that was beyond the scope of this analysis.


We have shown that dosing with sotalol changes the mean MLE and the K and L components of the Erlang distribution in this study. Higher doses of sotalol have a larger impact than lower doses. This could help lead to understanding the proarrhythmic potential of certain pharmaceutical compounds. These are noninvasive analyses that can be conducted relatively inexpensively and with a minimum inconvenience to the patient.


The authors would like to thank Dr. Dick Kovacs for help in reviewing the manuscript, Mr. Luc Devriendt for his help in writing the computer code, Dr. Nenad Sarapa for lending the data for these analyses, and Mr. Brian Mitchell for his statistical help.

Conflict of Interest Statement

Timothy Callahan, PhD, is an employee of Biomedical Systems Corp., St. Louis, MO. William P. Stuppy, MD, has no conflict of interest to declare.



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