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International Journal of Bifurcation and Chaos, Vol. 17, No. 10 (2007) 37093713 c World Scientific Publishing Company

TIME SERIES ANALYSIS OF ECG: A POSSIBILITY OF THE INITIAL DIAGNOSTICS
ALEXANDER LOSKUTOV and OLGA MIRONYUK Physics Faculty, M.V. Lomonosov Moscow State University, Moscow 119992, Russia Loskutov@chaos.phys.msu.ru Received Octob er 13, 2005; Revised March 29, 2006

The metho ds of nonlinear dynamics are applied to reveal the pathologies of patients with different heart failures. Our approach is based on the analysis of the correlation and embedding dimensions of the RR-intervals of ECGs. We demonstrate that these characteristics are quite convenient to ols for the initial diagnosis. Advantages and disadvantages of the metho d are discussed. Keywords : Correlation dimension; ECG; RR-intervals.

1. Introduction
Cardiovascular diseases (CVDs) are resp onsible for more than 4 million deaths p er year, and so the investigations of physiological features of the cardiac function attract considerable interest [Europ ean Cardiovascular Disease Statistics, 2004]. A ma jor group of CVDs refer to the disturbances of the normal cardiac rhythm (arrhythmias). It app ears that some arrhythmias can b e analyzed by the contemp orary methods develop ed in the frameworks of nonlinear dynamics. These are mainly based on the analysis of electrocardiograms (ECGs) as time series. It is exp ected that these methods may help to provide an initial diagnosis and to determine in certain cases the typ e of the cardiac pathology. Investigations of the heart rhythm disturbances have b een carried out for more than 50 years. Nevertheless, only recently, new methods of nonlinear dynamics have found applications in the clinical practice [Kurths et al., 1995; Peng et al., 1995; Govindar et al., 1998; Wessel et al., 2000a; Wessel et al., 2000b; Schulte-Frohlinde et al., 2001; Janson et al., 2002; Loskutov et al., 2004; Loskutov et al., 2005]. The basic goal of the present study

is to reveal dynamical characteristics of ECG as a function of different physiological pathologies of the heart tissue and investigate its p ossible application in clinical practice. In the medical practice, the normal palpitation is commonly describ ed by so-called "normal sinus rhythm". However, in spite of this term the intervals b etween the heart b eats p erform certain fluctuations. Namely, even at rest a healthy heart has certain rhythm variations. Moreover, the exact p eriodic dep endencies in the heart rhythm indicate the evidence of the severe cardiac pathologies [Goldb erger & Rigney, 1988]. The extensive clinical investigations of 80's manifested that a normal heart rate exhibits some features, sp ecific for deterministic chaos (see, e.g. [Goldb erger, 1990; Goldb erger et al., 1990]). The results of subsequent studies have shown that dynamic characteristics found in the ECG analysis of healthy and cardiac patients were different [Peng et al., 1995; Kurths et al., 1995]. Moreover, the imp ortance of the deterministic chaos in the heart rate has also b een revealed (see, e.g. [Govindar et al., 1998] and refs. cited therein). However, an alternative p oint of view is also supp orted (see, e.g. [Kanters et al., 2002]). Hence, a lot of effort has

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b een recently devoted to reveal the role of chaos in the development of heart diseases. In the present pap er, we analyze ECGs as time series of RR-intervals and study the complexity of the heart rate. We analyze correlations and emb edding dimensions of the RR-sequences for a quite large group of patients with heart failures such as stenocardia, AV-block, cardiac infarction, etc. (see Sec. 4). The aim of our investigations is to obtain clear evidence that ECGs of patients with different pathologies may b e quantitatively classified using the dimensional characteristics.

chaotic processes inherent for the initial ECG (e.g. to estimate the emb edding dimension).

3. Dimension of ECG
It is known that an app earance of pathology in the cardiac tissue leads to certain changes in the ECG. As shown b elow, these changes can b e recognized by the analysis of such quantitative characteristics of the attractor as the correlation dimension and the emb edding dimension. As an attractor characteristic, the correlation dimension may b e obtained from time series of RR-intervals of the ECG by the well-known method [Grassb erger & Procaccia, 1983]. According to it, the reconstruction procedure of the phase space and the system attractor is reduced to the analysis of pseudoattractor. The time delay was chosen where the autocorrelation function of the series has its first zero. The correlation dimension d and the emb edding dimension m carries the information ab out the system complexity. However, we wish to stress that estimates of these characteristics may b e correctly obtained on the physical assumption that the dynamical processes are statistically stationary, i.e. the considered system has already passed a transient state and reached the asymptotic regime where its prop erties are indep endent of the initial conditions. This also implies that the system parameters must remain constant. In addition to that, the reasonable estimates of d and m require ab out O(N 2 ) op erations: This method of the time series analysis loses its validity when a short sequence (less than 104 values) is addressed [Loskutov et al., 2002]. Thus, to estimate correctly the emb edding and the correlation dimensions, the following restrictions are to b e taken into account: (i) the time series should b e stationary [Schuster, 1984; Kennel, 1997; Ivanov et al., 2002]; (ii) The sample length should not b e less than Nmin 10d/2 . According to the first condition, the estimate of the HRV may b e incorrect. However, for our analysis the exact value of the correlation dimension d is not of primary imp ortance, since we need to find a clear-cut distinction b etween the ECG dimension d for patients with different diseases. For this formulation of the problem, it is sufficient to determine d for several fixed (quite large) values of the emb edding dimension m. One can exp ect that at large m the dep endence d(m) should b e different for various heart failures. To get more accurate results, we considered the groups of patients, that is, a statistical

2. ECG as a Sequence of RR-Intervals
Physically, ECG is a record of an electric signal produced by a heart. In a normal state each p eak of ECG (p ositive or negative) corresp onds to the excitation or rep olarization of the different parts of the cardiac tissue. The interval b etween the two neighboring peaks of R-waves is equal to the total cardiac cycle and is called RR-interval. To analyze the heart rate variability (HRV), an ECG is represented in a form of a time series. Usually this time series is an array of N numb ers, b eing certain values of a dynamical variable x(t) taken after equal time intervals. In the application to the ECG analysis, there are two different ways of the time series formation: classical [Schuster, 1984; Mikhailov & Loskutov, 1995; Cutler & Kaplan, 1997] when the variable is measured after equal time intervals, and a series comp osed of the intervals b etween R-waves [Sauer, 1994, 1997; Racicot & Longtin, 1997]. Presently, we apply the second method. Let us explain it in more detail: Each R-p eak of ECG app eared in time ti is replaced by a single impulse, which is approximated by a Dirac delta function (t - ti ). Thus, the total ECG is replaced by the sequence of RR-intervals: x(t) i (t - ti ). Therefore, the desired time series V (i) is formed by the values of intervals b etween the single pulses RR(i): RRi = V (i), V = (V (1),V (2),... ,V (N )), where N is the total numb er of elements in the time series. Note, that this method is widely used to analyze biological systems characterized by p eak signals. Although such representation is not a classical sequence obtained via equal time intervals, it may b e treated as a typical realization of some nonlinear system [Sauer, 1994]. Within this approach, it is p ossible not only to reveal latent regularities of the heart rate but also quantify characteristics of

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approach has b een applied. The analysis has b een carried out using the data obtained from a few to tens of ECGs. Then the results have b een averaged. Such an approach allows to make a presumptive diagnosis (see Sec. 4). To comment on the second restriction we note the use of mobile devices, such as Holter monitor, allows to record the ECG almost few days without interruption. This is quite sufficient to receive time series of RR-intervals consisting of more than 105 elements.

Table 1. 1. Patients with cardiac vessel diseases in various range (21 ECGs). 2. Patients with coronary artery bypass grafting and myocardial infarction (12 ECGs). 3. Patients with resting angina and cardiac vessel diseases (17 ECGs). 4. Patients with resting angina and myocardial infarction (9 ECGs). 5. Patients with congestive heart failure (21 ECGs). 6. Patients with resting angina and coronary artery involvement (13 ECGs). 7. Patients with effort angina and myocardial infarction (16 ECGs). 8. Patients with mixed angina and different vessel diseases (32 ECGs). 9. Patients with AV-block (6 ECGs). 10. Patients endured ventricular fibrillation (7 ECGs). 11. Patients with atrial flutter (18 ECGs). Group of ECGs

d for m = 3 2.51 2.13 2.68 2.74 2.83 2.57 2.32 2.53 1.02 2.98 2.61 ± ± ± ± ± ± ± ± ± ± ± 0.04 0.02 0.04 0.05 0.03 0.03 0.03 0.04 0.03 0.04 0.03

d for m = 4 3.12 2.60 3.29 3.45 3.70 3.08 2.81 2.98 1.36 3.81 3.19 ± ± ± ± ± ± ± ± ± ± ± 0.05 0.03 0.07 0.06 0.06 0.08 0.05 0.05 0.04 0.06 0.04

d for m = 5 3.61 2.78 3.66 3.89 4.44 3.78 3.57 3.75 1.42 4.23 3.51 ± ± ± ± ± ± ± ± ± ± ± 0.1 0.03 0.08 0.1 0.09 0.1 0.1 0.1 0.06 0.1 0.08

4. Analysis of Real Data
We have analyzed 159 sequences of RR-intervals obtained from ECGs of patients with different deviations in the cardiac rhythms. All the ECGs have b een taken from the source of National Center for Research Resources (P41 RR13622) [Goldb erger, 2000; PhysioNet, 2003]. This database is sub divided into groups of ECGs according to the patient's cardiac diseases. The age of patients and their gender have not b een taken into account. The length of each time series was on average, 105 elements. To exclude random pulses (they always occur in contemp orary mobile electrocardiographs) the chosen ECGs were sp ecially processed, i.e. false pulses were filtered by standard methods. In accordance with them, we have eliminated the false p eaks which were identified in the PhysioNet Data as artefacts. After computer analysis of each ECG, the obtained values in each group were averaged. The results of the computed correlation dimension of ECGs for the values of the emb edding dimension m = 3, m = 4, m = 5 with corresp onding standard deviations are shown in Table 1. One can see that the correlation dimension of the groups of patients with various cardiac pathologies differ from each other. However, this does not hold for all groups. For example, for a certain value of the emb edding dimension, m = 3, the range of values of d may overlap. This happ ens for the third (patients with the rest angina and the cardiac vessel involvement) and the fourth (patients with the rest angina and myocardial infarction) groups, as well as for the first (patients with the cardiac vessel involvement of various extent) and the eighth (patients with mixed angina and cardiac vessel involvement) groups (see Table 1). Nevertheless, the diagnoses of the corresp onding patients partly coincide, which explains the overlap in the correlation dimensions.

1 2 3 4 5 6 7 8 9 10 11

With increasing emb edding dimension (m = 4 and m = 5) the intervals of d for the same groups (3 and 4, 1 and 8) do not overlap any more. Increasing the dimension m it makes it p ossible to recognize the pathology group, i.e. with the further growth of the emb edding dimension the correlation dimensions for each patient group demonstrate strong divergency. These results are shown in Fig. 1.

Fig. 1. The dep endence of the correlation dimension of the patient ECG with different pathologies from the emb edding dimension. -- resting angina and myocardial infarction, -- congestive heart failure, · -- resting angina and cardiac vessel diseases, -- coronary artery bypass grafting and myocardial infarction, -- AV-block.

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They are supp orted also by the one factor analysis of variance [Altman, 1991] with the significance level of 0.05. We can summarize the obtained results as follows. The correlation dimension of ECGs for various pathologies may b e clearly discriminated for large values of the emb edding dimension. This means that starting at some m it makes it p ossible to sub divide the patients into groups. Thus, we may come to the conclusion that, even despite the fact that we used the database where the age of patients and their gender have not b een taken into account, methods of nonlinear dynamics can play a key role in the detection of certain cardiac diseases.

References
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5. Concluding Remarks
In the present study we have analyzed the sequences of RR-intervals of ECGs obtained from patients, who suffer some cardiac diseases and time series corresp onding to the normal cardiac rhythm of healthy p ersons. As a result of this analysis it has b een shown that the dimensional characteristics used allow to resolve the inverse problem, that is, to divide patients into groups according to their cardiac diseases. The increase and/or decrease of the degree of chaos has b een observed in the ECG according to the disease. We wish to stress that this conclusion has b een made on the basis of the analysis of sufficiently long time series and a quite large numb er of patients from each group. The use of dimensional characteristics of ECGs makes it p ossible to automatize the process of the initial diagnosis for patients with several cardiac pathologies. Solution of this and similar questions would allow us to find the b oundary b eyond which chaotic processes (that are inherent in the cardiac rhythm) do not already corresp ond to the healthy state but uniquely indicate to pathologies. Finally, we would like to emphasize the following. It is well known that cardiac pathologies are not necessarily caused by the cardiovascular system dysfunction; they may also b e a result of other diseases. This makes the investigation of so-called latent cardiac pathologies and indirect disturbances in the cardiac rhythms of a sp ecial imp ortance. Usually it is not p ossible to detect such pathologies by the standard approaches. Application of nonlinear dynamical methods seems to b e rather promising [Loskutov et al., 2005].

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Peng, C. K., Halvin, Sh. et al. [1995] "Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series," Chaos 5, 8287. PhysioNet, A public service of the research resources for complex physiologic signals (PhysioNet, P41 RR13622). Racicot, D. M. & Longtin, A. [1997] "Interspike interval attractors from chaotically driven neuron mo dels," Physica D 104, 184204. Sauer, T. [1994] "Reconstruction of dynamical systems from interspike intervals," Phys. Rev. Lett. 72, 3811 3814. Sauer, T. [1997] "Reconstruction of integrate-and-fire dynamics,"Nonlinear Dynamics and Time Series,eds. Cutler, C. & Kaplan, D., Fields Inst. Communications, Vol. 11, pp. 6375.

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