Method and system for dynamical systems modeling of electrocardiogram data

[0001] This application claims the benefit of provisional application 61/062,366 filed Jan. 25, 2008, the entirety of which is incorporated by reference herein.

[0002] 1. Field of the Invention

[0003] This invention pertains to a system for diagnosing a risk of cardiac dysfunction based on electrocardiogram data. Specifically, this invention pertains to the use of dynamical systems modeling techniques to identify metrics used to diagnose heart health.

[0004] 2. Background

[0005] An electrocardiogram (ECG) is a recording of the electrical activity of the heart over time. Cardiac cells are electrically polarized, that is, the insides of the cells are negatively charged with respect to their outside of the cells by means of pumps in the cell membrane that distribute ions (primarily potassium, sodium, chloride, and calcium) in order to keep the insides of these cells negatively charged (i.e. electronegative). Cardiac cells loose their internal electronegativity in depolarization. Depolarization is an electrical event which corresponds to heart contraction or “beating”. In depolarization, loss of electronegativity is propagated from cell to cell, producing a wave of electrical activity that can be transmitted across the heart.

[0006] Electrical impulses in the heart originate in the sinoatrial node (SA Node) and travel through the heart muscle where they impart electrical initiation of “systole” or contraction of the heart. The electrical waves can be measured by electrodes (electrical contacts) placed on the skin of a subject. These electrodes measure the electrical activity of different parts of the heart muscle. An ECG displays the voltage between pairs of these electrodes, and the muscle activity that they measure, from different directions. This display indicates the rhythm of heart contraction.

[0007] FIG. 1 depicts the peaks in an electrocardiogram signal. Electrocardiogram signals are comprised of three major structures which are used to characterize the health of a subject's cardiac system, the “QRS complex”, the “P wave” and the “T wave.” The P wave is a structure in the ECG signal which corresponds to the depolarization of the atria as the main electrical vector is directed from the SA node to the Atrioventicular Node (AV node). The QRS complex is a structure in the ECG signal that corresponds to the depolarization of the ventricles. Because the ventricles contain more muscle mass than the atria, the QRS complex is larger than the P wave. The T wave is a structure in the ECG signal that corresponds to the “repolarization” or recovery of electronegativity in the ventricles after depolarization.

[0008] Heart rate variability (HRV) refers to the beat to beat alteration in heart rate. Heart rate variability can be determined based on electrocardiogram (ECG) signals. The “RR Interval” is the distance between consecutive R peaks in an electrocardiogram signal. The heart rate for a given time period is defined as the reciprocal of an RR interval (in seconds) multiplied by 60. Healthy hearts exhibit a large HRV, whereas an absence of variability or decreased variability is associated with cardiac or systemic dysfunction. The term “cardiac dysfunction”, as used herein, refers to any type of abnormal functioning of the cardiac system including cardiac disease. Several studies have also shown that a reduction in heart rate variability is also predictive of a subject's likelihood of sudden death from cardiac dysfunction.

[0009] Another important interval used to diagnose heart health is the QT interval. The QT interval represents the total time needed for the ventricles to depolarize and regain electronegativity. The QT interval varies according to the heart rate and is typically corrected according to the heart rate. If the QT interval is abnormally lengthened or shortened, heart complications, including Torsade de Pointes (TDP) and sudden death can occur. Prolongation of the QT interval can be associated with certain metabolic and disease states, congenital disease states and adverse drug reaction.

[0010] Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, that is, systems whose states evolve with time in a manner which is difficult to predict over the long range. According to dynamical systems theory, systems may be characterized as being deterministic (meaning that their future states are, in theory, fully defined by their initial conditions, with no random elements involved) or non-deterministic (meaning the future states are random or undefined by their initial conditions). A periodic system is a system which deterministically returns to a same state over time. A random system is a system which is non-deterministic. Chaos theory is an area of dynamical systems theory which seeks to describe the behavior of certain dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions. As a result of this sensitivity, the behavior of chaotic systems appears to be random. This behavior happens even though chaotic systems are deterministic, meaning that their future dynamics are difficult to predict even though their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply “chaos”. This chaotic behavior is observed in natural systems, such as weather systems and is hypothesized to be observed in physiological systems including the cardiac system.

[0011] It has been proposed that physiological systems act as chaotic systems, even though this hypothesis is contrary to the classical paradigm of homeostasis. In homeostasis, physiological systems self-regulate through adjustments in order to maintain equilibrium and reduce variability. In contrast, this proposed hypothesis conjectures that a healthy physiological system exhibits characteristics of a chaotic system such as sensitivity to slight perturbations. This sensitivity and the associated responsiveness in the physiological system causes the system to produce a large variety of behaviors in the physiological system, such as the high variability/complexity in heart rate observed in subjects without cardiac disease or dysfunction. Conversely, the hypothesis proposes that unhealthy or dysfunctional biological systems are associated with a decreased sensitivity and have less variability in their behavior than the healthy systems. This corresponds to the low heart rate variability observed in subjects with poor heart health.

[0012] Early studies conducted by Dr. Chi-Sang Poon (Poon et al. (2001), Poon et al. (1997), Barhona and Poon (1996)) demonstrate that heart rate variability is not caused by random fluctuations but instead complex, deterministic patterns. Accordingly, a number of studies have applied different metrics traditionally used to study chaotic system to electrocardiogram data. Narayan et al. (1998) discovered that the times series RR interval data exhibit unstable periodic orbits (UPOs). A dense set of periodic orbits is indeed a criterion used to assert the deterministic chaotic dynamics of an underlying system. The Lyapunov Exponent is a metric used to characterize how chaotic a dynamic system is. Positive Lyapunov Exponents indicate that a system is chaotic. Unstable periodic orbits are chaotic and therefore are associated with positive Lyapunov Exponents. Similarly, Hashida and Takashi (1984) investigated the nature of the RR intervals during atrial fibrillation and determined that the distribution of the RR interval follows the Erlang distribution.

[0013] While these findings strengthen the hypothesis that the correspondence between heart rate variability and heart health is typical of a chaotic system, these studies have failed to provide a deeper understanding of the underlying dynamics of the cardiac system which cause the observed chaotic behavior. Accordingly, these estimates of chaotic behavior alone cannot reliably be used to predict heart health. Therefore, a deeper understanding of the role of chaos in cardiac dynamics is needed in order to develop accurate metrics of heart health and use these metrics in diagnostics.

[0014] The above and other needs are met by a computer-implemented method, a computer program product and a computer system for diagnosing a risk of cardiac disease based on a set of metrics that are derived from dynamical systems modeling of electrocardiogram data.

[0015] One aspect of the present invention provides a computer-implemented method for diagnosing cardiac dysfunction based on electrocardiogram data. Electrocardiogram data associated with a subject is received, the electrocardiogram data comprising a series of RR intervals and a series of QT intervals, wherein the series RR intervals corresponds, in part, to the series of QT intervals. A first value which indicates an amount by which uncertainty associated with the series of QT intervals is reduced given the series of RR intervals is generated. A second value which indicates an amount by which uncertainty associated with the series of RR intervals is reduced given the series of QT intervals is generated. The subject is determined to be associated with a low risk of cardiac dysfunction responsive to the first value exceeding the second value and a result of the determination is provided.

[0016] One aspect of the present invention provides a computer system for diagnosing cardiac dysfunction based on electrocardiogram data, the system comprising one or more computing devices and a memory. The system further comprises a reporting module stored in the memory and adapted to receive electrocardiogram data associated with a subject, the electrocardiogram data comprising a series of RR intervals and a series of QT intervals, wherein the series RR intervals corresponds, in part, to the series of QT intervals. The system further comprises a mutual information module stored in the memory and adapted to generate a first value which indicates an amount by which uncertainty associated with the series of QT intervals is reduced given the series of RR intervals and a second value which indicates an amount by which uncertainty associated with the series of RR intervals is reduced given the series of QT intervals. The system further comprises a diagnosis module stored in the memory and adapted to determine the subject to be associated with a low risk of cardiac dysfunction responsive to the first value exceeding the second value. The system further comprises a visualization module stored in the memory and adapted to provide a result of the determination.

[0017] Another aspect of the present invention provides a computer-readable storage medium encoded with computer program code for diagnosing cardiac dysfunction based on electrocardiogram data according to the above described method.

[0018] The features and advantages described in this summary and the following detailed description are not all-inclusive. Many additional features and advantages will be apparent to one of ordinary skill in the art in view of the drawings, specification, and claims hereof.

[0030] The figures depict an embodiment of the present invention for purposes of illustration only. One skilled in the art will readily recognize from the following description that alternative embodiments of the structures and methods illustrated herein may be employed without departing from the principles of the invention described herein.

[0031] As discussed above, a fundamental limitation of metrics used to assess chaotic behavior in dynamical systems, is that these metrics are simply “black box” metrics and do not provide insight into the underlying dynamics of the cardiac system. Therefore, these metrics cannot be accurately used to diagnose cardiac dysfunction. Accordingly, the focus of the present invention was to iteratively propose and validate hypotheses of the underlying mechanisms governing the dynamics of heart systems in order to develop metrics that can accurately be used to diagnose a risk of cardiac dysfunction. Based on these metrics, computational methods, computer program products and computer systems for diagnosing cardiac dysfunction based on electrocardiogram data associated with a subject are presented herein.

[0032] A hypothesis of heart dynamics based on Poincaré recurrence and strongly mixing theorems was proposed and validated in order to develop metrics used to diagnose cardiac dysfunction. In the proposed hypothesis, the RR interval corresponds to the compound Poincaré return time of the heart dynamical system.

[0033] The Poincaré recurrence theorem states that certain abstract dynamical systems will, after a sufficiently long time, return to a state very close to the initial state (i.e. an attractor state). FIG. 2 illustrates the concept of Poincaré recurrence. The Poincaré recurrence time is the length of time elapsed until the recurrence. The Poincaré recurrence theorem states:

For any ΕεΣ, the set of those points x of Ε such that fⁿ(x)εΕfor all n>0 has measure zero.

[0034] In other words, almost every point of Ε returns to Ε. In fact, almost every point returns indefinitely often; i.e. μ({xεΕ: there exists N such that fⁿ(x)εΕ for all n>N})=0. The Poincaré recurrence time is based on Ergodic hypothesis which states that, over long periods of time, the time spent by a particle in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equally probable over a long period of time.

[0035] The hypothesis presented herein proposes that the system of cardiac depolarization and repolarization which causes the heart to beat is a strongly mixing dynamic system. This hypothesis further proposes that the RR interval corresponds to the compound Poincaré recurrence time of this system.

Scatter Plot Analysis

[0036] In order to validate the hypothesis that the RR interval corresponds to the Poincaré return time of the cardiac system, scatter plots of RR Interval Data were constructed based on ECG data obtained from 66 “normal” subjects (i.e. subjects without known cardiac dysfunction) and ECG data obtained from 12 subjects under several conditions in a clinical trial of a drug. These data are described in detail below in the section entitled “Clinical Data”. The Poincaré return time hypothesis is also consistent with the Erlang fit.

[0037] The scatter plot, a technique taken from nonlinear dynamics, portrays the nature of RR interval fluctuations in the heart rate dynamics. Scatter plot analysis is a quantitative-visual technique whereby the shape of the plot is categorized into functional classes that indicate the degree of cardiac dysfunction in a subject. FIG. 3 illustrates a scatter plot of RR intervals from a subject without known cardiac dysfunction. This type of plot provides summary information as well as detailed beat-to-beat information on the behavior of the heart. The geometry of the scatter plot is essential and can be described by fitting an ellipse to the graph. The ellipse is fitted onto the so called line-of-identity at 45 degrees to the normal axis. The standard deviation of the points which were perpendicular to the line-of-identity denoted by SD1 describes short-term heart rate variability which is mainly caused by respiratory sinus arrhythmia (RSA). The standard deviation along the line-of-identity denoted by SD2 describes long term heart rate variability.

[0038] FIG. 4 illustrates a scatter plot of RR intervals according to nonlinear dynamics. This “RR interval scatter plot” considers the scatter plot as the two dimensional (2-D) reconstructed RR interval phase spaces. In other words, each RR interval RR(i) in the series of RR intervals is plotted against the subsequent RR interval RR(i+1). In dynamical system theory, the reconstructed RR interval phase spaces describe the dynamics of the cardiac system by projecting the reconstructed attractor. Attractor reconstruction refers to methods of using geometrical and topological information about a dynamical attractor from observations. These methods have been developed as a means to reconstruct the phase space of the system, specifically for experimental and naturally occurring chaotic dynamical systems, which the phase space and a mathematical description of the system are often unknown. Since the evolution of a dynamical system can be described by its phase space, it is very important to reconstruct the phase space of the system. Herein, we proceed from the hypothesis that the dynamics of the SA node which generates the RR intervals is a dynamical attractor.

[0039] The RR interval scatter plot typically appears as an elongated cloud of points oriented along the line-of-identity. The dispersion of points in the scatter plot perpendicular to the line-of-identity reflects the level of short term heart rate variability. The dispersion of points along the line-of-identity is thought to indicate the level of long-term heart rate variability. The scatter plots were analyzed quantitatively by calculating the standard deviations of the distances of the RR(i) to the lines y=x and y=−x+2RR_m, where RR_m is the mean of all RR(i). The standard deviations are referred to as SD1 and SD2, respectively. SD1 related to the fast beat-to-beat variability in the data, while SD2 describes the longer-term variability of RR(i).

[0040] RR interval scatter plots of the type illustrated in FIG. 3 and FIG. 4 were generated for each subject using standard plotting functions available in the MATLAB™ numerical computing environment and programming language. The RR Interval Scatter plots for RR interval data associated with the normal subjects are included in pgs. 43-106 of provisional application 61/062,366. The RR Interval Scatter plots for RR interval data associated with Clinical Trial subjects are included in pgs. 110-180 of provisional application 61/062,366. The scatter plot of RR intervals (with lag one) for the subject data generally showed two morphologically distinct distributions: (I) an almost oval or “comet” shaped morphology and (II) non-oval shaped morphology. The scatter plots with the oval morphology were associated with subjects with a normal cardiac system (i.e. not associated with any known cardiac dysfunction). The scatter plots with the non-oval morphology were associated with subjects with abnormal or dysfunctional cardiac systems and cardiac systems of subjects administered with pharmaceutically active compounds. Overall, the “normal” subjects RR Interval scatter plots showed an oval morphology. The non-oval morphology was generally observed in the RR Interval Scatter Plots associated with the clinical trial patients. The results are consistent with the hypothesis that the RR interval is equal to the Poincare return time for subjects which are not known to have cardiac dysfunction.

Erlang Distribution Analysis

[0041] In conjunction with the scatter plot analysis, subject data was fit to Erlang distributions in order to validate the hypothesis that the RR interval corresponds to the compound Poincaré return time.

[0042] The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. The Erlang distribution was developed by A. K. Erlang in order to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queuing systems in general. The distribution is now used in the field of stochastic processes. The Erlang distribution also occurs as a description of the rate of transition of elements through a system of compartments. Such systems are widely used in biology and ecology. The Erlang distribution is the distribution of the sum of k independent identically distributed random variables each having an exponential distribution.

[0043] Histograms of RR interval data for the subjects were fit to the Erlang distribution using code developed in the MATLAB™ programming language and the SAS™ programming language. Parameters which described the best fit of the Erlang distribution to the RR Interval histograms using a “gamfit” function in MATLAB™ which provided maximum likelihood estimates (MLEs) for the parameters of a gamma distribution given the histogram data. The Erlang distribution is a special case of the gamma distribution in which the parameter K is an integer. For each RR interval histogram, an Erlang distribution was generated and the chi-squared error between the generated Erlang distribution and the RR interval histogram was generated to measure how well the RR interval histograms fit to the Erlang distribution. The least square error values calculated for the normal subjects are included in Appendix A. The least square error values calculated for the clinical trial subjects are included in Appendix C.

[0044] FIG. 5 illustrates a graph of the RR histogram data used to evaluate a fit to the Erlang distribution. Graphs of the RR interval histogram data and the associated Erlang distributions for the normal subjects are included in pgs. 399-463 of provisional patent application 61/062,366. Data for some, but not all of the “normal” subjects was observed to have an Erlang distribution based on the value of the least square error being below a threshold value. For these patients the least square error showed an unexpectedly good fit of which values less than 1e-05. Graphs of the RR interval histogram data and fitting parameters for the clinical trial subjects are included in pgs. 495-565 of provisional patent application 61/062,366.

[0045] It was further observed that there was weak correlation in the series of RR intervals associated with each of the subjects. This weak correlation within the series or sequence corresponds to work performed by N. Haydn (2004) which demonstrates that for a large class of mixing dynamical systems, the return event is a Poisson process.

[0046] The difference between the observed Erlang distribution in the RR Interval data and the hypothesis that the return event is a Poisson process is resolved by the hypothesis that the RR interval contains k>1 return times. The probability density function of the Erlang distribution is:

[0047] $f\left(x;k,\lambda \right)=\frac{{\lambda }^k{x}^{k-1}{e}^{-\lambda x}}{\left(k-1\right)!}\mathrm{for}x>0.$
Given a Poisson distribution with a rate of change λ, the distribution function indicates the waiting times until the k^th Poisson event. This distribution represents the sum of a series of exponential distributions. The parameter k is called the shape parameter and the parameter λ is called the rate parameter.

[0048] These results supported a second complementary hypothesis regarding the dynamics of the SA node. The SA node regulates the RR intervals by initiating depolarization. Proceeding from the initial hypothesis, it was hypothesized that the SA node acts as “dynamical attractor” in the cardiac system and a heartbeat is emitted when the state of the SA attractor enters some finite region A, so that the interval would have been the Poincare return time to A. Unfortunately, the theoretically exponential distribution of the Poincare return time did not fit the histograms generated from subject RR interval data. However, the theoretical Erlang distribution of the k-fold return time provided a coefficients which indicated a good fit to histograms generated from RR interval data, for some but not all of the normal subjects. Therefore, the coefficient which indicates the fit of the RR interval data to the Erlang distribution provides a stringent metric for assessing cardiac health.

[0049] Physiologically, the multiple return times may correspond to non-linear synchronization of the cells responsible for the depolarization which causes the heart to beat. We hypothesize that each R peaks or “beats” is emitted after k returns of a state w of the underlying abstract system to a state A. It is well known that the synchronization of the many cells responsible for depolarization at a rate consistent with the heart beat creates signals at frequency which is an integer multiple of the higher frequency (Michaels et al., 1987).

[0050] These results were further interpreted in view of the previous contradictory observation by Hashida and Takashi that the Erlang Distribution was observed for RR Interval data from subjects with atrial fibrillation (a type of cardiac dysfunction). It was noted that in the Hashida and Takashi study, the subjects were administered digitalis, a medication used to treat cardiac dysfunction. Specifically, digitalis is used to increase cardiac contractility and as an antiarrhythmic agent in atrial fibrillation. Therefore, the two seemingly contradictory observations may be resolved if it is assumed that the treatment using digitalis in the Hashieda and Takashi study restored cardiac contractility in the subjects producing a heart rate variability in the subjects which were similar to the heart rate variability observed in normal subjects in the present study.

Chaos Metrics

[0051] Based on the validation of this hypothesis, additional metrics used to assess chaos in dynamic systems were calculated based on subject data and correlated to data generated from the Erlang distribution and scatter plot analyses in order to gain further insight into the heart dynamics. The statistical properties of heart dynamics were further assessed using correlation functions and their long term behavior.

[0052] Lyapunov exponents were calculated based on subject RR intervals as a metric of chaotic behavior in dynamical systems using program code developed in MATLAB™ and code developed in the C programming language. FIG. 6 illustrates a plot of Lyapunov exponents over time. Plots for each of the subjects are included on pgs. 399-565 of provisional application 61/062,366. The observation of positive Lyapunov exponents among all subjects confirmed that the dynamics of the heart rate demonstrates characteristics of a chaotic system. The Lyapunov exponents were calculated for embedding dimension equal to 2. Kolmogorov-Sinai entropy, or KS entropy, was also calculated based on the subject RR intervals. The KS entropy is equal to 0 for non-chaotic systems and greater than zero for chaotic systems. The Lyapunov exponent and KS entropy values associated with the normal subjects are included in Appendix A.

[0053] Lyapunov exponents were also calculated for the sequences of RR intervals derived from clinical data representing a set of five electrocardiogram recordings from a group of 12 subjects treated with drugs. The Lyapunov exponent and KS entropy values associated with the clinical trial subjects are included in Appendix C. This data is described in detail below in the section entitled “Clinical Data.” Only one positive Lyapunov exponent observed for each of the five recording for each subject which was equivalent to the KS entropy for each subject. It was observed that the subjects that did not have Lyapunov exponents indicating chaotic behavior did not have a good fit to the Erlang distribution for the RR intervals. These results agree with the observation that the Erlang distributions are observed only in healthy cardiac systems which exhibit characteristics of chaotic systems.

[0054] The Lyapunov or Kaplan-Yorke dimension is a measure of the complexity of the system which is based on the Lyapunov exponents generated for the system. Lyapunov dimensions are described in detail below in the section entitled “Lyapunov Dimensions”. Lyapunov dimensions typically range from D to D+1, where D represents the number of the Lyapunov exponents whose summation provides a positive value. Lower values of the Lyapunov dimension indicate less complexity and high values of the Lyapunov exponent indicate more complexity.

Canonical Correlation Analysis

[0055] Linear Canonical Correlation Analysis, Non Linear Canonical Correlation Analysis, Linear/Nonlinear Canonical Correlation Analysis of the Aggregated Data were performed on the electrocardiogram data from the normal and clinical trial subjects. These results are described in detail in the respectively titled sections below.

[0056] Mutual Information Analysis was performed on the series RR intervals and the corresponding series of QT intervals for each of the 66 normal subjects and the 12 subjects in the clinical trail under 5 different conditions. Mutual information metrics indicate how much the uncertainty in one variable is reduced by knowing another variable. In these analyses, the Kolmogorov-Sinai mutual information was used to generate values indicating how much the uncertainty in future QT interval data in the series of QT intervals is reduced by knowing the past RR interval data (MI(RR->QT)) and how much the uncertainty in future RR interval data in the series of RR intervals is reduced by knowing the past QT interval data (MI(QT->RR)). The mutual Kolmogorov-Sinai information is defined as I(u−, y+)=KS(y+)−KS(y+lu−), where KS(y+lu−) is the conditional entropy. The “lag” is an embedding dimension parameter used to generate the mutual information metrics. Although the value of the lag is arbitrary, best results are achieved when a lag is selected such that the data is separated as much as possible. Accordingly, mutual information values were generated using lag values of 25, 50, 100 and 500. Results from the normal subjects are tabulated in Appendix B. Results from the clinical trial subjects are tabulated in Appendix C. It was determined that the lag of 500 produced maximal separation in the RR and QT interval data.

[0057] Based on the generated values, it was observed that in the normal subjects the mutual information metric MI(RR->QT) generally exceeds the metric MI(QT->RR) and in clinical trial subjects at baseline the mutual information metric MI(QT->RR) generally exceeds the metric MI(RR->QT). Based on these results and hypotheses, the comparison of the two mutual information values MI(RR->QT) and MI(QT->RR) provide valuable metrics used to diagnose cardiac dysfunction.

[0058] The observed results correspond to another complementary hypothesis that the RR and QT intervals are metrics of a feedback loop between the SA node and the AV node. This feedback loop is illustrated in FIG. 11. In healthy systems, the depolarization starting at the SA node controls this feedback and in cardiac dysfunction, the SA node looses its control and the AV node controls the system. These complimentary hypotheses further demonstrate that information theory metrics and dynamical modeling metrics may be used in conjunction to model functional cardiac systems and develop metrics that can predict a risk of cardiac dysfunction.

[0059] Based on the correlation between the observed results and the feedback loop hypothesis, another complimentary hypothesis regarding the role of the autonomic nervous system (ANS) and other factors extrinsic to the cardiac system in regulation of the RR-QT feedback loop may be proffered. It is well known that cardiac dysfunction can occur due to two distinct causes: intrinsic dysfunction and extrinsic dysfunction. In general, intrinsic cardiac dysfunction includes dysfunction due to an inherent, purely dynamical, aspect of the cardiac system which is internal to the cardiac system whereas extrinsic cardiac dysfunction includes dysfunction due to other factors external to the cardiac system which are not part of the cardiac system but cause a change in the function of the cardiac system such as pharmaceuticals and effect of the Autonomic Nervous System (ANS). Intrinsic cardiac dysfunction and extrinsic cardiac dysfunction by definition are mutually exclusive.

[0060] Proceeding from the above discussed hypothesis that the SA node acts as an “attractor” in healthy cardiac systems and the AV node acts as an attractor in unhealthy cardiac systems, it is hypothesized that the disruption to the feedback loop between the SA node and the AV node which causes cardiac dysfunction differs between intrinsic causation of cardiac dysfunction and extrinsic causation of cardiac dysfunction. It is further proposed that the causation of the dysfunction may be distinguished by further characterizing relationships within RR intervals and between the RR intervals and the QT intervals, which respectively can be used to characterize attractor behaviors of the SA node and the AV node.

[0061] Specifically, it is proposed that the degree of stationarity observed within the RR intervals data can be used to distinguish whether cardiac dysfunction is due to extrinsic or intrinsic factors. It is proposed that a higher degree of stationarity can be observed in extrinsic cardiac dysfunction due to the depleted effect of the Autonomic Nervous System (ANS) on the cardiac function via the vagus nerve and the SA node. For example, clinical trials have revealed such an increase of stationarity, as revealed by the Poincare meta-recurrence plots, concurrent with an increased dosage of some drug (see pgs. 560-566 of provisional patent application 61/062,366). As such a higher degree of stationarity would be an extrinsic cardiac dysfunction.

[0062] Information theory metrics such as mutual information metric may be further used to characterize the RR and QT interval data, thus providing distinction between intrinsic and extrinsic cardiac dysfunction. Since the entropy KS(QT⁺|RR⁻) is conditioned upon the past RR interval data, the effect of the ANS via the SA node is removed, thus providing a window on the intrinsic heart function. Conversely, the second entropy KS(RR⁺|QT⁻) is conditioned on the past QT interval data, dynamics based on factors intrinsic to the cardiac system are removed allowing for the assessment of extrinsic effects. Additionally, a higher degree of chaotic behavior within the RR interval data as indicated by Lyapunov coefficients or other chaos metrics such as entropy may indicate a higher likelihood of normal intrinsic cardiac function. The presence of deterministic chaos within RR interval data, as defined in dynamical system theory, is consistent with the hypothesis that the intrinsically heart function corresponds to a electro-hydro-dynamical attractor.

[0063] The distinction between intrinsic and extrinsic causation of cardiac dysfunction can be used to determine the relative effect of pharmaceuticals and the ANS system in subjects based on electrocardiogram data. Further, the ability to quantitatively determine the relative causation of extrinsic factors allows for the evaluation of pharmaceuticals and the ANS system in subjects with known cardiac dysfunction due to intrinsic (e.g. congenital) disorders such as congenital long QT syndrome.

Heart Health Prediction Server

[0064] Based on the observed associations between the above described metrics and cardiac dysfunction, a heart health prediction sever 110 is presented herein. FIG. 7 is a high-level block diagram of a computing environment 100 according to one embodiment. FIG. 7 illustrates a heart health prediction sever 110 and three clients 150 connected by a network 114. The clients 150 transmit electrocardiogram data associated with a subject to the heart health prediction server 110. The heart health prediction server 110 received the electrocardiogram data from the clients 150 and generates metrics used to assess the risk of cardiac dysfunction according to the dynamical modeling techniques described above. In alternate embodiments, the heart health prediction server 110 receives the electrocardiogram data directly. The heart health prediction server 110 uses the metrics of heart health to determine a risk of cardiac dysfunction associated with the subject and transmit an indication of the risk to the clients 150. In some embodiments, the heart health prediction server 110 displays the indication of the risk directly to the clients 150. Only three clients 150 are shown in FIG. 7 in order to simplify and clarify the description. Embodiments of the computing environment 100 can have thousands or millions of clients 150 connected to the network 114.

[0065] The network 114 represents the communication pathways between the heart health prediction sever 110 and clients 150. In one embodiment, the network 114 is the Internet. The network 114 can also utilize dedicated or private communications links that are not necessarily part of the Internet. In one embodiment, the network 114 uses standard communications technologies and/or protocols. Thus, the network 114 can include links using technologies such as Ethernet, 802.11, integrated services digital network (ISDN), digital subscriber line (DSL), asynchronous transfer mode (ATM), etc. Similarly, the networking protocols used on the network 114 can include the transmission control protocol/Internet protocol (TCP/IP), the hypertext transport protocol (HTTP), the simple mail transfer protocol (SMTP), the file transfer protocol (FTP), etc. The data exchanged over the network 114 can be represented using technologies and/or formats including the hypertext markup language (HTML), the extensible markup language (XML), etc. In addition, all or some of links can be encrypted using conventional encryption technologies such as the secure sockets layer (SSL), Secure HTTP and/or virtual private networks (VPNs). In another embodiment, the entities can use custom and/or dedicated data communications technologies instead of, or in addition to, the ones described above.

[0066] FIG. 8 is a high-level block diagram illustrating a typical computer 200 for use as a heart health prediction server 110 or client 150. Illustrated are a processor 202 coupled to a bus 204. Also coupled to the bus 204 are a memory 206, a storage device 208, a keyboard 210, a graphics adapter 212, a pointing device 214, and a network adapter 216. A display 218 is coupled to the graphics adapter 212.

[0067] The processor 202 may be any general-purpose processor such as an INTEL x86 compatible-CPU. The storage device 208 is, in one embodiment, a hard disk drive but can also be any other device capable of storing data, such as a writeable compact disk (CD) or DVD, or a solid-state memory device. The memory 206 may be, for example, firmware, read-only memory (ROM), non-volatile random access memory (NVRAM), and/or RAM, and holds instructions and data used by the processor 202. The pointing device 214 may be a mouse, track ball, or other type of pointing device, and is used in combination with the keyboard 210 to input data into the computer 200. The graphics adapter 212 displays images and other information on the display 218. The network adapter 216 couples the computer 200 to the network 114.

[0068] As is known in the art, the computer 200 is adapted to execute computer program modules. As used herein, the term “module” refers to computer program logic and/or data for providing the specified functionality. A module can be implemented in hardware, firmware, and/or software. In one embodiment, the modules are stored on the storage device 208, loaded into the memory 206, and executed by the processor 202.

[0069] The types of computers 200 utilized by the entities of FIG. 7 can vary depending upon the embodiment and the processing power utilized by the entity. For example, a client 150 that is a mobile telephone typically has limited processing power, a small display 218, and might lack a pointing device 214. The heart health prediction server 110, in contrast, may comprise multiple blade servers working together to provide the functionality described herein.

[0070] FIG. 9 is a high-level block diagram illustrating the heart health prediction server 110 according to one embodiment. As shown in FIG. 9, the heart health prediction server 110 includes multiple modules. Those of skill in the art will recognize that other embodiments of the heart health prediction server 110 can have different and/or other modules than the ones described here, and that the functionalities can be distributed among the modules in a different manner.

[0071] The reporting module 910 functions to receive electrocardiogram data associated with subjects from the clients 150 comprising a series of QT intervals and a series of RR intervals. According to the embodiment, the reporting module 910 may correct the series of QT intervals using correction formulae such as Fridericia's and Bazett's correction formulae. The reporting module 910 is adapted to transmit the electrocardiogram data to the diagnosis module 920 for diagnosis. The reporting module 910 is further adapted to receive a value indicating the risk of cardiac dysfunction from the diagnosis module and transmit an indication of this value to the client 150. In some embodiments, the reporting module 910 is adapted to display an indication of the value indicating risk of other data used to generate the value indicating risk of cardiac dysfunction such as mutual information values, Erlang distribution values.

[0072] The diagnosis module 920 functions to generate values which indicate a risk of cardiac dysfunction based on the electrocardiogram data. The diagnosis module 920 communicates with the chaos metric module 960, the recurrence model 970, the Erlang fitting module 940 and the mutual information module 950 to receive metrics which indicate a subject's risk of cardiac dysfunction based on the electrocardiogram data. The diagnosis module 100 combines these metrics to determine the subject's risk of cardiac dysfunction.

[0073] The diagnosis module 920 may combine the metrics to generate a continuous value or a binary value. In one embodiment, the diagnosis module 920 generates a binary value indicating either a high risk of cardiac dysfunction or a low risk of cardiac dysfunction. In a specific embodiment, the diagnosis module 920 generates the binary value based on a first mutual information value which indicates an amount by which the uncertainty in future QT intervals of the series of QT intervals is reduced given the past RR interval data and a second mutual information value which indicates an amount by which the uncertainty in future RR intervals in the series of RR intervals is reduced given the past QT interval data. In this embodiment, the diagnosis module determines whether the first mutual information value is greater than the second mutual information value. If the diagnosis module 920 determines that the first mutual information value is greater than the second mutual information value, the diagnosis module 920 generates a value indicating a low risk of cardiac dysfunction. If the diagnosis module 920 determines that the second mutual information value is greater than the first mutual information value, the diagnosis module 920 generates a value indicating a high risk of cardiac dysfunction.

[0074] In some embodiments, the binary value may also be based on additional metrics and the mutual information metrics. In one embodiment, the diagnosis module 920 only generates a value indicating a high risk of cardiac dysfunction if a coefficient that indicates a least square fit of a histogram of the subject's RR interval data to an Erlang distribution is above a defined threshold value. Likewise, the diagnosis module 920 only generates a value indicating a low risk of cardiac dysfunction if a coefficient that indicates a fit of a histogram of the subject's RR interval data to an Erlang distribution is below a defined threshold value. In another embodiment, the diagnosis module 920 only generates a value indicating that a subject is associated with high risk of the cardiac dysfunction if one or more chaos metrics derived from the subject's electrocardiogram data are beneath a threshold value indicating a low quantity of chaos in the data. In this embodiment the diagnosis module 920 only generates a value indicating that a subject is associated with low risk of the cardiac dysfunction if one or more chaos metrics derived from the subject's electrocardiogram data are above a threshold value and indicate a high quantity of chaos in the data.

[0075] In some embodiments, the diagnosis module 920 further functions to generate values for subjects associated with a high risk cardiac dysfunction indicating whether the risk of cardiac dysfunction is due to intrinsic cardiac dysfunction or extrinsic cardiac dysfunction. In these embodiments, the diagnosis module 920 provides a determination of intrinsic and extrinsic cardiac dysfunction based additional metrics and/or additional threshold values. In some embodiments, the additional metrics may include stationarity metrics generated by the recurrence metric module 970 such as Poincare meta-recurrence metric or Dickey-Fuller Root of Unity metrics. In some embodiments, additional threshold values may be applied to conditional entropy metrics generated by the mutual information module 950 such as KS(RR⁺|QT⁻) and KS(QT⁺|RR⁻) in order to, respectively, quantify intrinsic and extrinsic effects in order to provide a determination whether the risk of cardiac dysfunction is due to intrinsic cardiac dysfunction or extrinsic cardiac dysfunction. According to the embodiment, other additional metric and/or threshold values may include, for example, Erlang fit. Since the Erlang fit can be justified on the ground of compounded Poincare return time, it is a purely dynamical feature, hence relevant to intrinsic cardiac function.

[0076] The chaos metric module 960 functions to generate values which quantify chaotic behavior in the subject's cardiac system based on the electrocardiogram data associated with a subject. The chaos metric module 960 generates chaos metrics including: Lyapunov coefficients, Lyapunov (Kaplan-Yorke) dimensions and Komolgorov Sinai Entropy values based on RR Interval data derived form the electrocardiogram data. The chaos metric module 960 transmits the chaos metrics to the diagnosis module 920.

[0077] The mutual information module 950 generates information theory metrics such as entropy, conditional entropy and mutual information based on the electrocardiogram data associated with a subject. The mutual information module 950 can generate any type of information theory metric using correlation analyses such as the canonical correlation analyses described above. In some embodiments, the mutual information module 950 generates Shannon differential entropy values based on the series of RR intervals and the series of QT intervals. In a specific embodiment, the mutual information module 950 generates Kolmogorov-Sinai mutual information metrics based on the series of RR intervals and QT intervals derived form the electrocardiogram data. In this embodiment, Komolgorov-Sinai mutual information metrics are calculated in order to generate a value MI(QT->RR) which quantifies an amount by which the uncertainty in future RR intervals in the series of RR intervals is reduced based on the past QT interval data and a value MI(RR->QT) which quantifies an amount by which the uncertainty in the future QT intervals in the series of QT intervals is reduced based on the past RR interval data. The mutual information module 950 transmits the information theory metrics to the diagnosis module 920.

[0078] The Erlang fitting module 940 generates coefficients which describe how well a histogram of RR intervals derived from a subject's electrocardiogram data fit an Erlang distribution. The Erlang fitting module 940 can generate any value that describes the fit of the histogram to the Erlang distribution. In a specific embodiment, the Erlang fitting module 940 generates a chi-squared value that describes the likelihood that the RR Interval histogram would fit the Erlang distribution by chance as a coefficient. In another specific embodiment the Erlang fitting module 940 further generates parameters to describe the best Erlang distribution for the data as described above. The Erlang fitting module 940 transmits the coefficients to the diagnosis module 920.

[0079] The recurrence module 970 functions to generate stationarity value which characterize the stationarity of the electrocardiogram data. The recurrence module 970 generates stationarity values based on RR interval data according to stationarity metrics such as Poincare meta-recurrence metrics and Dickey-Fuller Root of Unity. The recurrence module 970 transmits the stationarity value to the diagnosis module 920.

[0080] The visualization module 930 generates interfaces for displaying the electrocardiogram data, the generated metrics and the risk of cardiac dysfunction on the display 218 of the client 150 and/or the heart health prediction server 110. The visualization module 930 generates plots of the generated metrics as described above. The visualization module 930 further generates displays of the generated metrics as compared to the threshold values used by the diagnostic module 920 to determine whether the subject is associated with a high or low risk for cardiac dysfunction. These displays allow a user of the visualization module 930 to qualitatively assess a subject's risk of cardiac dysfunction based on the generated metrics.

[0081] FIG. 10 is a flow chart illustrating steps performed by the heart health prediction server 110 to diagnose a risk of cardiac dysfunction based on electrocardiogram data.

[0082] The heart health prediction server 110 receives 1010 electrocardiogram data, including a series of RR intervals and a series of QT intervals. The heart health prediction server 110 generates 1012 a coefficient value which quantifies the fit of the RR intervals to the Erlang distribution. The heart health prediction server 110 generates 1014 a first value MI(RR->QT) which indicates an amount by which uncertainty in future QT intervals of the series of QT intervals is reduced by knowing the past RR interval data. The heart health prediction server 110 generates 1016 a second value MI(QT->RR) which indicates an amount by which the uncertainty in future RR intervals of the series of RR intervals is reduced by knowing the past QT interval data. The heart health prediction server 110 determines 1018 whether the fitting coefficient is below a threshold value X and whether the first value MI(RR->QT) is greater than the second value MI(QT->RR). If the coefficient is below the threshold value X and the first value MI(RR->QT) is greater than the second value MI(QT->RR), the heart health prediction server 110 determines that the subject is associated with a lower risk of cardiac disease and/or dysfunction and provides 1020 an indication of a low risk of cardiac dysfunction. If the coefficient exceeds the threshold value X and/or the first value MI(RR->QT) is not greater than the second value MI(QT->RR), the heart health prediction server 110 determines 1022 whether the coefficient is above a threshold value and whether the second value MI(QT->RR) is greater than the first value MI(RR->QT). If the coefficient exceeds the threshold value and the second value MI(QT->RR) is greater than the first value MI(RR->QT), the heart health system 110 determines that the subject is associated with a high risk of cardiac disease and/or dysfunction and provides 1024 an indication of a high risk of cardiac dysfunction.

[0083] In some embodiments, the heart health prediction server 110 generates 1023 a stationarity value responsive to determining that the subject is associated with a high risk of cardiac dysfunction. The heart health prediction server 110 determines 1025 whether the stationarity value exceeds a threshold value y. If the heart health prediction server 110 determines 1025 that the stationarity value exceeds a threshold value y, the heart health prediction server 110 provides 1026 an indication of risk of extrinsic cardiac dysfunction. If the heart health prediction server 110 determines 1025 that the stationarity value does not exceed the threshold value y, the heart health prediction server 110 provides 1028 an indication of risk of cardiac dysfunction due to intrinsic dysfunction.

[0084] The above description is included to illustrate to a client 150 according to one embodiment. Other embodiments the operation of certain embodiments and is not meant to limit the scope of the invention. The scope of the invention is to be limited only by the following claims. From the above discussion, many variations will be apparent to one skilled in the relevant art that would yet be encompassed by the spirit and scope of the invention.

Clinical Data:

[0085] The data analyzed in this research is from randomized, double blind, 5-way crossover. There were two data sets incorporated into the analysis, data from a phase 1 clinical drug trial and data from normal subjects (i.e. subjects with no known cardiac dysfunction). There were 12 subjects included in the phase 1 clinical drug trail data set. All 12 subjects gave informed consent to the pharmaceutical company that allowed the data for use. 11 of the 12 subjects had 5 electrocardiogram recordings, one subject had 4 electrocardiogram recordings. The electrocardiogram recordings were taken in randomized order at baseline (untreated with drugs), placebo, low-dose, medium dose and high-dose drug consumption. The original use of the data was to evaluate a sodium-channel blocker for arrhythmias and cardiac dysfunction which induced RR interval prolongation. The sodium-channel blocker is no longer being developed. Data sets include the 24-hour measurement of RR and QT intervals. There were also 66 data sets from normal subjects.

Lyapunov Exponents:

[0086] For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents. The Lyapunov exponent is a quantitative measure of separation the trajectories that diverge widely from their initial close positions. There are as many Lyapunov exponents as there are dimensions in the state space of the system, but the largest is usually the most important. The magnitude of this exponent is proportional to how chaotic the system is. For periodic signals, the Lyapunov exponent is zero. A random signal will also have an exponent of zero. A positive Lyapunov exponent indicates sensitive dependence on the initial conditions and is diagnostic of chaos, although these exponents are not easily measured. The flow map:

• • [0087] Φ^t: ⁿ→
    • [0088] xΦ^t(x)
      describing the dynamical system acts on the n-dimensional state space M=ⁿ and is generate by vector field v:
  • [0089] {dot over (x)}=v(x), xεⁿ, tε
    To gather information about the time evolution of infinitesimally small perturbed initial states, the linearized flow map has to be considered.
  • [0090] D_xΦ^t: T_xM→T_Φt(x)M
    • [0091] uD_xΦ^tu

[0092] The linearized flow map D_xΦ^t is given by an invertible n×n matrix describing the time evolution of a vector u in tangent space. For ergodic systems the Lyapunov exponents are defined as the logarithms of the eigenvalues μ_i(1≦i≦m) of the positive and symmetric limit matrix.

[0093] ${\Lambda }_x=\underset{t\to \infty }{\lim }{\left[D_x{\Phi }^{t*}{D}_x{\Phi }^t\right]}^{\frac{1}{2t}}$
as given by the theorem of Oseledec. The Lyapunov exponents are the logarithmic growth rates

[0094] ${\lambda }_i=\underset{t\to \infty }{\lim }\frac{1}{t}\ln D_x{\Phi }^t{e}_i,\left(1\le i\le m\right)$
where {e_i: 1≦i≦m} are basis vectors that span the eigenspaces of Λ_x.

[0095] Any continuous time-dependent dynamical system without a fixed point will have at least one zero exponent, corresponding to the slowly changing magnitude of a principal axis tangent to the flow. The sum of the Lyapunov exponents is the time-averaged divergence of the phase space velocity; hence any dissipative dynamical system will have at least one negative exponent, the sum of all of the exponents is negative, and the post transient motion of trajectories will occur on a zero volume limit set, an attractor. The exponential expansion indicated by a positive Lyapunov exponent is incompatible with motion on a bounded attractor unless some sort of folding process merges widely separated trajectories. Each positive exponent reflects a direction in which the system experiences the repeated stretching and folding that decorrelates nearby states on the attractor. Therefore, the long-term behavior of an initial condition that is specified with any uncertainty cannot be predicted; this is chaos. An attractor for a dissipative system with one or more positive Lyapunov exponents is said to be strange or chaotic.

[0096] For time series produced by dynamical systems, the presence of a positive characteristic exponent indicates chaos. Recognizing that the length of the first principal axis is proportional to e^λ1t the area determined by the first two principal axes is proportional to e^(λ1+λ2)t; and the volume determined by the first k principal axes is proportional to: e^{(λ1+λ2+. . .+λk)t}. Thus, the Lyapunov spectrum can be defined such that the exponential growth of a k-volume element is given by the sum of the k largest Lyapunov exponents. Note that information created by the system is represented as a change in the volume defined by the expanding principal axes.

[0097] In a geometrical point of view, to obtain the Lyapunov spectra, imagine an infinitesimal small ball with radius dr sitting on the initial state of a trajectory. The flow will deform this ball into an ellipsoid. That is, after a finite time t all orbits which have started in that ball will be in the ellipsoid. The i^th Lyapunov exponent is defined by:

[0098] ${\lambda }_i=\underset{t\to \infty }{\lim }\frac{1}{t}\left(\frac{d{l}_i\left(t\right)}{dr}\right)$
where dl_i(t) is the radius of the ellipsoid along its i^th principal axis 2.4)).

[0099] The separation must be measures along the Lyapunov directions which correspond to the principal axes of the ellipsoid previously considered. These Lyapunov directions are dependent upon the system flow and are defined using the Jacobian matrix, i.e., the tangent map, at each point of interest along the flow. Hence, one must preserve the proper phase space orientation by using a suitable approximation of the tangent map. This requirement, however, becomes unnecessary when calculating only the largest Lyapunov exponent. If we assume that there exists an Ergodic measure of the system, then the multiplicative Ergodic theorem of Oseledec justifies the use of arbitrary phase space directions when calculating the largest Lyapunov exponent with smooth dynamical systems. This is due to the fact that chaotic systems are electively stochastic when embedded in a phase space that is too small to accommodate the true dynamics.

[0100] In Ergodic systems most trajectories will yield the same Lyapunov exponent, asymptotically for long times. The computation of the full Lyapunov spectrum requires considerably more effort than just the maximal exponent. An essential ingredient is some estimate of the local Jacobians, i.e., of the linearized dynamics, which rules the growth of infinitesimal perturbations. One either finds it from direct fits of local linear models of the s_n+1=a_ns_n+b_n, such that the first row of the Jacobian is the vector a_n, and (J)_i,j=δ_i−1,j for i=2, . . . m, where m is the embedding dimension. The a_n is given by the least squares minimization

[0101] ${\sigma }^2=\sum _l{\left(s_{l+1}-a_n{s}_l-b_n\right)}^2$
where {s_i} is the set of neighbors of s_n. Or one constructs a global nonlinear model and computes its local Jacobians by taking derivatives. In both cases, one multiplies the Jacobians one by one, following the trajectory, to as many different vectors u_k in tangent space as one wants to compute Lyapunov exponents. Every few steps, one applies a Gram-Schmidt orthonormalization procedure to the set of u_k, and accumulates the logarithms of their rescaling factors. The average of these values, in the order of the Gram-Schmidt procedure, give the Lyapunov exponents in descending order. The routine used in this research, “lyap_spec”, uses this method of employing local linear fits. This routine is described in detail on pgs. 329-334 of Provisional Application 61/062,366.
Kolmogorov-Sinai Entropy:

[0102] The Kolmogorov-Sinai Entropy metric measures how chaotic an experimental signal is. In the case of deterministic chaos, K is positive and measures the average rate at which the information about the state of the system is lost over time. In other words, K is inversely proportional to the time interval over which the state of the system can be predicted. Moreover, K is related to the sum of the positive Lyapunov exponents. The Kolmogorov-Sinai entropy can be evaluated quantitatively and is diagnostic of chaos, whereas some other methods such as spectral analysis, time autocorrelation function and scatter plot construction are qualitative methods.

[0103] As an approximation, the sum of the corresponding exponents (i.e., the positive exponents), equals the Kolmogorov entropy (K) or mean rate of information gain:

[0104] $K=\sum _{{\lambda }_i>0}{\lambda }_i$
Lyapunov (Kaplan-Yoke) Dimension:

[0105] Lyapunov dimension is another Fractal dimension, introduced by Kaplan and Yorke based on the Lyapunov exponents. If Φ^t is a map on ⁿ and O_Φ⁺(x₀) is a bounded forward orbit having Lyapunov exponents λ_j=λ_j(x₀; Φ) with, for the integer k such that:

• • [0106] λ₁+λ₂+ . . . +λ_k≧0
  • [0107] λ₁+λ₂+ . . . +λ_k+1<0
    then, the Lyapunov Dimension of the orbit is

[0108] ${\dim }_L\left(O_{\Phi }^{+}\left(x_0\right)\right)=k+\frac{\sum _{i=1}^k{\lambda }_i}{{\lambda }_{k+1}}$
Notice that λ₁+λ₂+ . . . +λ_k<|λ_k+1| so dim_L(O_Φ⁺(x₀))<k+1. If the attractor has a positive Lyapunov exponent, then k≧1. The fractal dimensions of chaotic flows are shown to be given D=m⁰+m^+{1+|λ⁺/λ⁻|}, where m⁰ and m⁺ are the numbers of zero and positive Lyapunov characteristic exponents λ_α and λ^± are the mean values of positive and negative λ_α respectively.
Canonical Correlation Analysis:

[0109] Canonical Correlation Analysis (CCA) is a second moment technique. Therefore, it is not suitable for the systems with infinite variance, such as self-similar signals or highly noisy data, in its linear version; however, in the nonlinear version, this is not the issue. In the nonlinear CCA, since the variance analysis is applied to a nonlinear distortion of the original process, which is restricted to result in a finite variance process.

[0110] Assume that time series {y(k): k= . . . , −1, 0, 1, . . . } is a centered process, bounded and viewed as weakly stationary process with finite covariance E(y(i)y(j))=Λ_i-j defined over the probably space (Ω, A, μ). If the process is not stationary, we can simply compute z(k)=y(k)−y(k−1), which is usually stationary. The past and the future of the process are defined, respectively as:
y₋(k)=(y(k), y(k−1), y(k−2), . . . , y(k−L+1))^T
y₊(k)=(y(k+1), y(k+2), y(k+3), . . . , y(k+L)^T
where L is the lag. Interrelation between past and the future is as a preliminary study of whether a recipe of the form
y₊=f(y₋)
is likely to work. The ability to devise a good model can be gauged from the Kolmogorov-Sinai, or Shannon, mutual information between the past and the future. The mutual information between the past y₋ and the future y₊ is the amount by which the Shannon entropy of the future decreases the past is given; that is,

[0111] $I\left(y_{-},y_{+}\right)=h\left(y_{+}\right)-h\left(y_{+}|y_{-}\right)=\int \int \log \frac{p\left(y_{-},y_{+}\right)}{p\left(y_{-}\right)p\left(y_{+}\right)}p\left(y_{-},y_{+}\right)d{y}_{-}d{y}_{+}$
In the above equation, h(y₊) is the Shannon entropy of the future and h(y₊|y₋) is the conditional entropy of the future given the past.
Linear Canonical Correlation Analysis:

[0112] In the linear version of CCA, the best linear regression between the past and the future data is sought. In this regard and to proceed from a numerical algebra point of view, the covariance of the past and the future are factored as
E(y₋(k),y₋^T(k))=C₋₋=L₋^TL₋
E(y₊(k),y₊^T(k))=C₊₊=L₊^TL₊
where L₋ and L₊ are Cholesky factorization of the past and future, respectively. C₋₋ and C₊₊ are Toeplitz matrices and measures of strength of the past and the future, respectively. These quantities are used for normalization to get the information interface, independently of the strength of the signals.

[0113] Therefore the canonical correlation is defined as:
Γ(y₋,y₊)=L₋^−TE(y₋(k),y₋^T(k))L₊⁻¹
which asymptotically is a Hankel matrix in the case of large data records. The Singular Value Decomposition (SVD) of the canonical correlation matrix is given as Γ(y₋(k),y₊(k))=UΣV^T where U and V are orthogonal matrices:

[0114] $\Sigma \left(\begin{array}{ccc}{\sigma }_1& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & {\sigma }_L\end{array}\right),1\ge {\sigma }_1\ge \dots \ge {\sigma }_L\ge 0$

[0115] The σ_i's are called canonical correlation coefficients (CCC's) and are invariant under the nonsingular linear transformations of the past and the future. For the Gaussian processes it is well known that:

[0116] $I\left(y_{-},y_{+}\right)=-\frac{1}{2}\log \left(\det \left(I-{\Sigma }^2\right)\right)$
wherein the right hand side of the question represents the maximum information that can be achieved.

[0117] It is claimed that there are only restricted number D≦L of significant CCC's, grouped in Σ₁. Hence, matrices Σ, U, and V are partitioned. The canonical past and the canonical future are defined as:
y₋(k)=U₁L₋^−Ty₋(k)
y₊(k)=V₁L₊⁻¹y₊(k)

[0118] The state is defined as the minimum collection of past measurable random variables necessary to predict the future, that is, E( y₊(k)| y₋(k)). The state space model can be defined as:
x(k+1)=Fx(k)+w(k)
where w(k) is desired to be white noise and x(k)=Σ₁ y₋(k). F is the regression matrix of x(k+1) on x(k) and for the best prediction it is defined as:
F=E{x(k+1)x^T(k)}(E{x(k)x^T(k)})⁻¹
and the residual error is:
E{[x(k+1)−Fx(k)][x(k+1)−Fx(k)]^T}˜(I−Σ₁²)
Non-linear Canonical Correlation Analysis:

[0119] In nonlinear canonical correlation, a nonlinear processing is done on the past and the future. This is the case in which the maximum possible mutual information is attempted to reach. In general (non-Gaussian setup):

[0120] $\underset{f,g}{\sup }I\left(f\left(y_{-}\right),g\left(y_{+}\right)\right)\le I\left(y_{-},Y_{+}\right)$
where f,g: ^L→^L are measurable, objective functions such that E(f)=E(g)=0, E(ff^T)<∞×I, where I is the identity matrix. Equality is achieved if and only if f(y₋), g(y₊) can be made jointly Gaussian. In this case the linear estimation ĝ(y₊)=Af(y₋) is optimum. In fact, components of f(y₋), g(y₊) can be expressed as linear combination of functions p_j(y₋), q_j(y₊), j=1, 2, . . . , such that E₋(p_j)=E₊(q_j)=0, E₋(pp^T)<∞×I and E₊(qq^T)<∞×I, and forming basis of the Lebesgue spaces of zero mean measurable functions such that E₋ff^T<∞ and E₊g^Tg<∞, respectively. For L<∞, the same procedure as the linear case is followed and canonical correlation matrix Γ(p(y₋), q(y₊)) is computed.

[0121] The motivation for the nonlinear processing of the data is to gauge in comparison with the full dimensional linear case, how much increase in CCC's is gained by going to the nonlinear analysis. Observation of increased information indicates the nonlinearity of the process.

[0122] Consider that f(y₋), g(y₊) are jointly Gaussian functions, it can be found that:

[0123] $\underset{A}{\min }E{g\left(y_{+}\right)-Af\left(y_{-}\right)}_{C_{++}^{-1}}^2=L-\mathrm{Trace}({\Gamma }^T\Gamma \left(f\left(y_{-}\right),g\left(y_{+}\right)\right)$
for
A=Σ_{f(y−),g(y+)}.

[0124] It is claimed that:
f(y₋)=U₁^TL₋^−Tp(y₋)
g(y₊)=V₁L₊⁻¹q(y₊)
where U₁ and V₁ are computed from factorization of the SVD of the nonlinear canonical correlation matrix. The state space model is constructed following the same procedure as in the linear case.

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[0185]

APPENDIX A
Normal Subject Metrics
		Estimated	Estimated
	Lyapunov	Kolmogorov	Kaplan-Yorke
Index (Normal No.)	Exponents	Entropy	Dimension	Goodness of Erlang Fit (LSE)
1	0.4626789	0.466206	2.715300	2.67E−06
	0.0035272
	−0.6517636
2	0.2895753	0.289575	2.289766	2.12E−05
	−0.0888379
	−0.6927569
3	0.4603707	0.460371	2.606088	2.14E−05
	−0.0324804
	−0.7059873
4	0.4656253	0.465625	2.633640	8.87E−06
	−0.0236436
	−0.6975281
5	0.4693237	0.471095	2.724245	1.94E−05
	0.0017717
	−0.650464
6	0.4437319	0.443732	2.675322	8.11E−06
	−0.0079419
	−0.6453073
7	0.4294281	0.429659	2.671062	4.86E−06
	0.0002305
	−0.6402668
11	0.4412964	0.441296	2.607837	6.11E−06
	−0.02450989
	−0.6856878
12	0.4110762	0.411076	2.550688	1.97E−05
	−0.0404962
	−0.6729402
13	0.296361	0.296361	2.254169	5.77E−05
	−0.1145268
	−0.7154075
14	0.3736964	0.373696	2.449392	7.54E−06
	−0.06247597
	−0.692537
15	0.4061888	0.406189	2.547120	1.14E−05
	−0.0357784
	−0.6770191
16	0.3776069	0.377607	2.409464	1.58E−05
	−0.0789358
	−0.7294197
17	0.4122025	0.412203	2.563903	1.23E−05
	−0.0352633
	−0.6684468
18	0.3881749	0.388175	2.469609	1.25E−06
	−0.0635190
	−0.6913329
19	0.4429376	0.442938	2.656145	4.16E−06
	−0.0078992
	−0.6630221
20	0.4855205	0.485521	2.665932	1.57E−05
	−0.0188687
	−0.7007498
21	0.2955212	0.295521	2.303651	2.86E−05
	−0.0914378
	−0.6720979
22	0.4507159	0.450716	2.626018	6.80E−05
	−0.0272657
	−0.6764186
23	0.4257192	0.425719	2.642946	Inf
	−0.0170116
	−0.6356794
24	0.4064944	0.406494	2.558870	1.39E−05
	−0.0329571
	−0.6683800
25	0.3205683	0.320568	2.319766	1.92E−05
	−0.0992338
	−0.6921756
26	0.4248038	0.424804	2.619001	2.57E−06
	−0.0174760
	−0.6580406
27	0.4383994	0.438399	2.585760	2.29E−05
	−0.0359339
	−0.6870821
28	0.4483487	0.448349	2.616728	1.99E−05
	−0.0266034
	−0.6838431
29	0.4935920	0.49359	2.723977	6.25E−07
	−0.0063782
	−0.6729682
30	0.4301225	0.43012	2.630674	3.19E−05
	−0.0157002
	−0.6571100
31	0.4838183	0.48633	2.742830	2.05E−06
	0.0025148
	−0.6547033
32	0.4495922	0.44959	2.667396	4.09E−06
	−0.0104394
	−0.6580090
33	0.5209759	0.53443	2.812011	9.17E−06
	0.0134498
	−0.6581510
35	0.4993124	0.53346	2.867934	2.83E−06
	0.0341490
	−0.6146335
36	0.4430250	0.44303	2.594182	4.78E−06
	−0.0331921
	−0.6897429
37	0.5016889	0.53180725	2.838693	1.13E−05
	0.0301184
	−0.6340905
38	0.4992861	0.52375215	2.820210	2.95E−06
	0.0244661
	−0.6385589
39	0.4380424	0.4380424	2.681344	2.19E−06
	−0.0036062
	−0.6376166
40	0.5040903	0.53091445	2.841540	6.73E−05
	0.0268242
	−0.6308844
41	0.4820569	0.50799932	2.809463	4.21E−06
	0.0259424
	−0.6275755
42	0.4493009	0.4493009	2.687264	1.26E−06
	−0.0026156
	−0.6499468
43	0.5159200	0.53466111	2.817934	4.15E−06
	0.0187411
	−0.6536726
44	0.4867633	0.50240156	2.767732	3.99E−06
	0.0156383
	−0.6543973
45	0.5617581	0.61518794	2.984875	1.94E−05
	0.0534298
	−0.6246357
46	0.5403758	0.5698395	2.891160	2.47E−06
	0.0294637
	−0.6394360
47	0.5037656	0.51879166	2.796302	8.71E−06
	0.0150261
	−0.6515009
48	0.4261795	0.4261795	2.664762	9.70E−07
	−0.0050270
	−0.6335384
49	0.5019840	0.52225343	2.818195	4.12E−06
	0.0202694
	−0.6382995
50	0.4603309	0.47351243	2.759773	3.83E−06
	0.0131815
	−0.6232286
51	0.4924856	0.52279164	2.847047	1.55E−05
	0.0303060
	−0.6171928
52	0.4739981	0.477319276	2.709982	4.28E−06
	0.0033212
	−0.6722974
53	0.5014370	0.52811035	2.841481	1.34E−05
	0.0266734
	−0.6275965
54	0.4136532	0.421062986	2.682217	1.34E−05
	0.0074098
	−0.6171979
55	0.4187179	0.4187179	2.653716	1.42E−05
	−0.0031114
	−0.6357596
56	0.4599587	0.47031936	2.724315	1.44E−06
	0.0103607
	−0.6493294
57	0.5225741	0.5611117	2.897044	5.98E−06
	0.0385376
	−0.6255116
58	0.6037701	0.6678791	3.000000	4.39E−06
	0.0641090
	−0.6158282
59	0.4336755	0.4336755	2.641207	6.60E−06
	−0.0156398
	−0.6519512
60	0.4231674	0.4231674	2.669113	Inf
	−0.0018768
	−0.6296249
61	0.4893115	0.51763451	2.832247	1.09E−05
	0.0283230
	−0.6219722
62	0.4627798	0.4627798	2.653340	1.90E−05
	−0.0176291
	−0.6813463
63	0.4560998	0.4560998	2.701962	3.53E−05
	−0.0001225
	−0.6495752
64	0.4923906	0.49494146	2.753499	8.27E−06
	0.0025509
	−0.6568578
65	0.4663177	0.4663177	2.687289	7.03E−06
	−0.0073879
	−0.6677395
66	0.4973051	0.499582069	2.754085	6.42E−07
	0.0022770
	−0.6625014

[0186]

APPENDIX B
Normal Subject Mutual Information Metrics
	Lag = 25	50	100	500
No.	RR->QT	QT->RR	RR->QT	QT->RR	RR->QT	QT->RR	RR->QT	QT->RR
1	15.494	15.523	29.832	31.438	53.797	62.634	Inf	Inf
2	21.542	21.139	28.95	30.965	38.353	37.997	Inf	75.867
3	15.152	14.409	28.136	29.754	57.01	55.707	Inf	Inf
4	17.442	14.73	33.804	35.58	62.51	67.738	Inf	Inf
5	15.308	14.371	28.614	27.839	62.234	63.077	Inf	Inf
6	15.069	13.361	26.51	26.979	57.697	61.65	Inf	Inf
11	14.333	15.603	28.76	29.251	55.762	60.377	Inf	Inf
12	14.297	14.833	29.634	30.203	59.385	57.915	Inf	Inf
13	29.493	31.514	35.93	39.696	51.198	56.266	Inf	Inf
14	13.888	14.226	26.668	28.524	58.862	59.221	Inf	Inf
15	15.988	14.697	29.137	29.764	57.545	61.634	279.49	Inf
16	14.986	13.677	50.659	57.023	72.731	79.507	238.28	245.48
17	17.187	13.657	34.307	32.542	62.039	61.796	Inf	Inf
18	15.157	15.207	29.013	35.504	58.481	68.675	Inf	Inf
19	27.585	31.309	49.565	39.698	52.016	58.674	167.81	218.92
20	16.606	15.783	32.699	19.927	64.8	53.081	Inf	Inf
21	17.65	15.874	34.963	31.032	58.756	71.279	Inf	Inf
22	13.473	14.867	25.146	30.104	51.323	58.519	Inf	Inf
23	15.052	14.81	30.699	27.967	56.789	67.423	Inf	Inf
24	13.98	14.477	26.9	26.726	53.245	56.105	Inf	Inf
25	15.427	17.456	31.82	29.071	65.263	62.13	Inf	Inf
26	16.063	16.939	28.185	26.55	55.842	57.396	Inf	Inf
27	12.874	13.797	30.468	30.838	71.212	83.572	Inf	Inf
28	14.31	14.942	27.641	27.597	60.383	55.061	Inf	Inf
29	12.636	14.629	26.895	27.023	54.9	54.374	Inf	Inf
30	14.232	14.667	28.629	29.023	53.35	56.553	Inf	Inf
31	14.127	22.248	27.517	34.603	48.933	63.713	304.25	290.55
32	13.093	14.176	31.072	28.786	61.055	57.016	Inf	263.93
33	14.635	14.501	26.413	28.015	56.384	53.871	Inf	Inf
34	16.475	16.161	29.314	29.01	51.882	53.227	Inf	280.95
35	14.337	14.717	28.683	26.948	61.841	57.41	278.26	274.36
36	16.198	13.306	31.21	31.34	61.19	60.076	Inf	Inf
37	15.24	14.08	27.27	27.102	54.756	57.952	291.98	276.74
38	15.522	13.675	29.125	29.003	55.232	57.727	Inf	269.38
39	14.546	14.227	30.42	27.801	59.506	61.142	270.9	281.31
40	15.091	15.775	27.915	28.759	57.502	46.809	256.59	254.44
41	13.148	21.738	25.291	33.532	65.151	49.582	301.65	251.94
42	15.152	15.267	29.549	27.189	58.781	57.221	Inf	Inf
43	12.495	14.222	29.525	30.967	50.791	61.361	Inf	Inf
44	13.608	14.021	27.7	29.729	56.018	59.324	Inf	Inf
45	15.113	15.598	26.734	29.258	55.469	56.838	284.32	Inf
46	15.855	15.561	27.344	27.487	51.35	53.21	271.15	279.09
47	14.843	15.669	28.196	29.255	57.016	56.563	Inf	Inf
48	14.899	14.908	29.835	28.999	53.696	60.256	Inf	Inf
49	14.811	14.111	27.747	26.504	57.092	53.148	264.81	278.99
50	14.147	16.173	28.863	29.533	57.028	58.192	277.14	267.94
51	14.938	15.368	28.612	28.937	58.228	55.5	Inf	274.82
52	16.001	14.453	27.072	27.207	57.724	57.67	Inf	Inf
53	19.037	11.095	31.796	32.642	47.693	71.819	300.13	228.89
54	17.05	14.457	30.427	28.56	53.744	61.836	Inf	Inf
55	12.256	14.909	26.809	24.92	52.461	49.294	290.02	311.59
56	16.325	14.763	26.088	27.069	52.5	56.791	298.24	Inf
57	17.102	16.33	33.43	28.348	62.142	53.741	Inf	Inf
58	16.477	14.299	27.903	25.256	49.492	54.879	Inf	294.02
59	15.716	17.071	28.234	29.441	52.727	59.839	284.74	279.24
60	14.521	15.364	29.87	30.657	61.904	59.515	Inf	Inf
61	14.304	17.143	26.97	31.799	55.888	60.799	Inf	Inf
62	16.941	16.164	29.141	33.96	55.575	56.431	Inf	285.21
63	15.042	15.421	24.862	30.471	48.008	54.162	278.47	Inf
64	16.776	14.112	24.935	26.753	53.299	64.475	Inf	Inf
65	15.554	15.305	27.797	29.078	57.719	62.719	Inf	Inf
66	17.868	13.64	33.006	29.076	66.693	59.174	Inf	Inf

[0187]

APPENDIX C
Clinical Trail Subject Data
R104:
			Estimated	Estimated	Goodness
Case		Lyapunov	Kolmogorov	Kaplan-Yorke	of Erlang
No.	Index	Exponents	Entropy	Dimension	Fit (LSE)
R104	Baseline	0.4257192	0.4257192	2.642946	Inf
		−0.0170116
		−0.6356794
	Placebo	0.4117576	0.4117576	2.609185	Inf
		−0.02004995
		−0.6430027
	Low	0.4020163	0.4020163	2.576478	Inf
	Dose	−0.0246642
		−0.6545819
	Medium	0.3728832	0.3728832	2.516684	Inf
	Dose	−0.03939239
		−0.6454437
	High	0.3858872	0.3858872	2.540707	Inf
	Dose	−0.0396689
		−0.6403063
Average	0.3996527	2.5772
STD	0.0208427	0.050843
	Lag = 25	50	100	500
Baseline	15.052	14.810	30.699	27.967	56.789	67.423	Inf	Inf
Placebo	15.808	14.017	31.859	30.733	58.897	60.310	Inf	Inf
Low Dose	14.513	16.289	31.113	29.346	57.369	56.251	Inf	Inf
Medium	16.215	18.367	64.036	59.299	64.036	59.299	Inf	Inf
Dose
High Dose	16.803	16.214	34.137	31.680	56.526	59.293	Inf	Inf
R105:
	Index		Estimated	Estimated
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Goodness of Erlang Fit
No.	No.)	Exponents	Entropy	Dimension	(LSE)
R105	Baseline	0.4383994	0.4383994	2.585760	2.20E−05
		−0.0359339
		−0.6870821
	Placebo	0.3808797	0.3808797	2.477324	Inf
		−0.0550283
		−0.6826629
	Low	0.4504898	0.4504898	2.635047	1.73E−05
	Dose	−0.0226678
		−0.6736853
	Medium	0.44423	0.44423	2.6400	1.56E−05
	Dose	−0.022082
		−0.65952
	High	0.41724	0.41724	2.5614	Inf
	Dose	−0.031641
		−0.68679
Average	0.42624778	2.5799062
STD	0.0282759	0.0662413
	Lag = 25	50	100	500
Baseline	13.854	15.075	28.530	29.149	73.504	69.372	Inf	Inf
Placebo	16.673	14.378	29.228	31.386	60.029	58.656	Inf	Inf
Low Dose	17.061	17.766	29.581	31.072	66.954	64.683	Inf	Inf
Medium	13.824	16.628	30.737	32.338	61.782	56.897	Inf	Inf
Dose
High Dose	23.381	13.109	48.778	32.229	58.119	60.108	319.168	265.123
R106:
	Index		Estimated	Estimated
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Goodness of
No.	No.)	Exponents	Entropy	Dimension	Erlang Fit (LSE)
R106	Baseline	0.4064944	0.4064944	2.558870	1.24E−05
		−0.0329571
		−0.6683800
	Placebo	0.4333435	0.4333435	2.582881	7.14E−06
		−0.0383438
		−0.6776672
	Low	0.449674	0.449674	2.661384	1.06E−05
	Dose	−0.0139489
		−0.658808
	Medium	0.4433273	0.4433273	2.622548	4.10E−06
	Dose	−0.023964
		−0.6736245
	High	0.4571653	0.4571653	2.645643	Inf
	Dose	−0.0173126
		−0.6812629
Average	0.4380009	2.6142652
STD	0.0196612	0.0427988
	Lag = 25	50	100	500
Baseline	13.980	14.477	26.900	26.726	53.245	56.105	Inf	Inf
Placebo	16.019	16.136	28.791	27.599	64.869	64.492	Inf	Inf
Low Dose	16.043	13.888	33.348	30.029	61.330	58.183	Inf	Inf
Medium Dose	15.224	15.989	31.130	30.764	57.209	61.457	Inf	Inf
High Dose	14.639	15.773	33.686	30.595	69.401	68.378	Inf	Inf
R107:
	Index		Estimated	Estimated	Goodness of
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Erlang Fit
No.	No.)	Exponents	Entropy	Dimension	(LSE)
R107	Baseline	0.3205683	0.3205683	2.319766	1.85E−05
		−0.0992338
		−0.6921756
	Placebo	0.3066369	0.3066369	2.301670	NaN
		−0.09702648
		−0.6948343
	Low	0.3368838	0.3368838	2.365772	6.21E−06
	Dose	−0.0839962
		−0.6913801
	Medium	0.318106	0.318106	2.305732	1.38E−06
	Dose	−0.101117
		−0.709736
	High	0.3489985	0.3489985	2.396179	6.98E−06
	Dose	−0.077165
		−0.6861377
Average	0.3262387	2.3378238
STD	0.0166872	0.0413808
	Lag = 25	50	100	500
Baseline	15.427	17.456	31.820	29.071	65.263	62.130	Inf	Inf
Placebo	16.035	15.031	28.974	25.695	60.983	65.712	Inf	Inf
Low Dose	15.848	16.783	31.524	27.809	60.704	41.956	298.597	Inf
Medium	14.132	14.188	28.407	31.088	57.907	61.577	Inf	Inf
Dose
High Dose	13.523	21.860	21.920	29.631	82.596	46.164	269.247	361.590
R108:
	Index		Estimated	Estimated	Goodness of
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Erlang Fit
No.	No.)	Exponents	Entropy	Dimension	(LSE)
R108	Baseline	0.3776069	0.3776069	2.409464	1.62E−05
		−0.0789358
		−0.7294197
	Placebo	0.3218944	0.3218944	2.272803	3.90E−06
		−0.1189825
		−0.7438030
	Low	0.3743164	0.3743164	2.425119	7.09E−06
	Dose	−0.07587542
		−0.7020173
	Medium	0.4127410	0.4127410	2.507844	1.54E−05
	Dose	−0.0566170
		−0.7012472
	High	0.4332701	0.4332701	2.590228	1.16E−05
	Dose	−0.0324193
		−0.6791455
Average	0.38396576	2.4410916
STD	0.04255976	0.1186107
	Lag = 25	50	100	500
Baseline	15.234	13.798	71.057	78.088	71.057	78.088	233.488	241.924
Placebo	25.385	24.492	37.375	36.088	60.458	60.483	Inf	215.364
Low	17.119	16.955	28.385	31.744	91.713	Inf	243.071	Inf
Dose
Medium	14.508	16.720	30.891	32.514	58.030	59.426	Inf	Inf
Dose
High	15.543	15.673	31.098	31.343	56.614	57.213	Inf	Inf
Dose
R201:
	Index		Estimated	Estimated
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Goodness of Erlang Fit
No.	No.)	Exponents	Entropy	Dimension	(LSE)
R201	Baseline	0.2284109	0.2284109	2.141849	1.58E−05
		−0.1270624
		−0.7144813
	Placebo	0.3951689	0.3951689	2.529324	1.64E−05
		−0.0347777
		−0.6808515
	Low	0.4579045	0.4579045	2.681081	6.01E−06
	Dose	−0.0079399
		−0.6606626
	Medium	0.3941604	0.3941604	2.530400	1.74E−05
	Dose	−0.0430239
		−0.6620226
	High	0.3985903	0.3985903	2.538195	2.71E−05
	Dose	−0.0332158
		−0.6788887
Average	0.374847	2.4841698
STD	0.086156	0.2018987
	Lag = 25	50	100	500
Baseline	13.077	14.279	19.856	24.002	71.264	68.311	156.200	149.486
Placebo	17.379	16.375	30.394	30.148	62.555	64.297	Inf	Inf
Low	16.427	13.964	28.665	31.015	64.732	60.448	Inf	Inf
Dose
Medium	15.372	16.555	57.589	61.522	57.589	61.522	Inf	Inf
Dose
High	17.066	16.858	28.900	31.493	57.652	59.239	Inf	Inf
Dose
R202:
	Index		Estimated	Estimated
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Goodness of Erlang Fit
No.	No.)	Exponents	Entropy	Dimension	(LSE)
R202	Baseline	0.4248038	0.4248038	2.619001	2.03E−05
		−0.01747596
		−0.6580406
	Placebo	0.4291282	0.4291282	2.616737	3.38E−06
		−0.02000201
		−0.6633723
	Low	0.4239012	0.4239012	2.598675	5.11E−06
	Dose	−0.0201440
		−0.6744179
	Medium	0.4100861	0.4100861	2.588417	3.50E−06
	Dose	−0.0234355
		−0.6571032
	High	0.4102783	0.4102783	2.585815	3.73E−06
	Dose	−0.02448697
		−0.6585543
Average	0.4196395	2.601729
STD	0.0088567	0.015519
	Lag = 25	50	100	500
Baseline	16.099	15.315	28.010	28.789	58.636	57.767	Inf	Inf
Placebo	16.261	15.468	29.944	33.227	66.219	62.698	Inf	Inf
Low Dose	18.026	15.744	32.099	32.397	62.654	64.756	Inf	Inf
Medium	15.551	15.454	30.444	30.176	68.613	60.013	Inf	Inf
Dose
High Dose	15.568	16.903	29.208	32.349	62.404	61.188	Inf	Inf
R203:
				Estimated
			Estimated	Kaplan-	Goodness
	Index	Lyapunov	Kolmogorov	Yorke	of Erlang
Case No.	(Patient No.)	Exponents	Entropy	Dimension	Fit (LSE)
R203	Baseline	0.3881749	0.3881749	2.469609	1.12E−06
		−0.0635190
		−0.6913329
	Placebo	0.4147708	0.4147708	2.521725	4.73E−06
		−0.0491116
		−0.7008653
	Low Dose	0.4244452	0.4244452	2.560483	2.65E−06
		−0.0362641
		−0.6925834
	Medium	0.3980816	0.3980816	2.504774	1.23E−06
	Dose	−0.04892012
		−0.6917190
	High Dose	0.4028135	0.4028135	2.510517	1.89E−05
		−0.0502408
		−0.6906193
Average	0.4056572	2.5134216
STD	0.0142052	0.0327356
		Lag = 25	50	100	500
	Baseline	15.266	15.222	28.638	31.876	Inf	Inf
	Placebo	15.598	15.824	29.816	33.919	Inf	Inf
	Low Dose	14.420	14.696	60.012	60.110	Inf	Inf
	Medium	14.126	16.378	29.839	32.319	Inf	Inf
	Dose
	High	14.303	14.935	30.472	29.939	Inf	Inf
	Dose
R204:
	Index		Estimated	Estimated	Goodness
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	of Erlang
No.	No.)	Exponents	Entropy	Dimension	Fit (LSE)
R204	Baseline	0.4412964	0.4412964	2.607837	5.18E−06
		−0.02450989
		−0.6856878
	Placebo	0.4244762	0.4244762	2.575992	1.49E−06
		−0.03239431
		−0.6807070
	Low	0.4278505	0.4278505	2.535785	4.62E−06
	Dose	−0.0502191
		−0.7048186
	Medium	0.4284803	0.4284803	2.595601	7.80E−06
	Dose	−0.0252411
		−0.6770293
	High	0.4069436	0.4069436	2.507459	1.16E−05
	Dose	−0.0521102
		−0.6.992356
Average	0.4258094	2.5645348
STD	0.0123374	0.04199897
	Lag = 25	50	100	500
Baseline	14.348	15.404	28.778	29.519	55.589	60.832	Inf	Inf
Placebo	16.411	15.492	32.058	29.449	66.022	60.961	Inf	Inf
Low Dose	15.709	15.536	32.447	30.625	54.379	62.877	Inf	Inf
Medium	14.747	14.818	29.562	30.434	57.064	62.936	Inf	Inf
Dose
High Dose	15.893	15.207	28.111	31.188	63.109	54.819	Inf	Inf
R205:
	Index		Estimated	Estimated	Goodness	RR-QT
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	of Erlang	or
No.	No.)	Exponents	Entropy	Dimension	Fit (LSE)	QT-RR
R205	Baseline	0.4122025	0.4122025	2.563903	1.40E−05	QT-RR
		−0.0352633
		−0.6684468
	Placebo	0.4339110	0.4339110	2.627903	3.35E−05	RR-QT
		−0.01929673
		−0.6603159
	Low	0.4715819	0.474600	2.744306	2.71E−05	RR-QT
	Dose	0.00301865
		−0.6376417
	Medium	0.4397894	0.4397894	2.643012	2.86E−05	QT-RR
	Dose	−0.0145817
		−0.6612745
Average	0.4401257	2.644781
STD	0.0258645	0.074687
	Lag = 25	50	100	500
Baseline	17.187	13.657	Inf	Inf	Inf	Inf	Inf	Inf
Placebo	14.773	15.574	Inf	Inf	Inf	Inf	Inf	Inf
Low Dose	13.778	14.225	27.288	29.050	58.494	61.879	Inf	Inf
Medium	14.452	13.857	30.453	32.576	58.996	70.487	Inf	Inf
Dose
R207:
	Index		Estimated	Estimated	Goodness	RR-QT
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	of Erlang	or
No.	No.)	Exponents	Entropy	Dimension	Fit (LSE)	QT-RR
R207	Baseline	0.3736964	0.3736964	2.449392	6.11E−06	QT-RR
		−0.06247597
		−0.692537
	Placebo	0.4432160	0.4432160	2.635272	7.17E−06	QT-RR
		−0.0210113
		−0.6646048
	Low	0.3709137	0.3709137	2.417337	5.83E−06	QT-RR
	Dose	−0.0790944
		−0.6992408
	Medium	0.2479515	0.2479515	2.130739	NaN	QT-RR
	Dose	−0.1497657
		−0.7510070
	High	0.4.392516	0.4392516	2.615425	2.39E−05	RR-QT
	Dose	−0.02.93135
		−0.6.661058
Average	0.3750058	2.449633
STD	0.0789642	0.202907
	Lag = 25	50	100	500
Baseline	13.888	14.226	26.668	28.524	58.862	59.220	Inf	Inf
Placebo	14.608	15.955	29.947	30.609	66.690	59.010	Inf	Inf
Low Dose	13.594	14.203	27.581	27.773	53.302	52.973	Inf	Inf
Medium	37.236	39.076	39.019	41.077	39.019	41.077	132.386	130.343
Dose
High Dose	13.823	14.479	30.999	31.879	59.453	56.495	Inf	Inf
R208:
	Index		Estimated	Estimated	Goodness of	RR-QT
Case	(Patient	Lyapunov	Kolmogorov	Kaplan-Yorke	Erlang Fit	or
No.	No.)	Exponents	Entropy	Dimension	(LSE)	QT-RR
R208	Baseline	0.2955212	0.2955212	2.303651	3.16E−05	QT-RR
		−0.09143779
		−0.6720979
	Placebo	0.4108713	0.4108713	2.580940	Inf	QT-RR
		−0.0307061
		−0.6543966
	Low	0.4094391	0.4094391	2.591571	1.41E−05	QT-RR
	Dose	−0.019539
		−0.6590926
	Medium	0.4555751	0.4555751	2.674842	Inf	RR-QT
	Dose	−0.0117224
		−0.6577136
	High	0.4461778	0.4461778	2.659203	Inf	RR-QT
	Dose	−0.0153413
		−0.6535721
Average	0.4035169	2.5620414
STD	0.0638009	0.1501325
	Lag = 25	50	100	500
Baseline	17.544	15.788	34.829	30.903	58.684	71.490	Inf	Inf
Placebo	14.415	15.830	31.508	30.604	66.267	67.670	Inf	Inf
Low Dose	16.504	14.502	32.871	28.643	57.326	55.636	Inf	Inf
Medium	16.087	17.558	32.280	35.208	64.244	70.436	Inf	Inf
Dose