* Manipal Academy of Higher Education (Deemed University) Manipal 576 119,India
** Karnataka Regional Engineering College, Surathkal, India
*** Indian Institute of Science, Bangalore

Correspondence: Prof. B. M. Hegde, VC, MAHE University Manipal-576 119. India.
                         bmhegde@manipal.edu

Conventional electrocardiogram is the sum total of the millions of cardiac cell depolarization potentials. They are being recorded; using twelve surface leads (ECG), based on the Einthovan's triangle principle. The whole process represents just about a few seconds recording of the above mentioned potentials at any given time. Many times the procedure is done over a period of a day or two; the latter is called Holter monitoring. The graph is then interpreted using linear mathematical rules. This complicated wave structure is artificially split into parts, like the PR interval and ST segment etc. Linear calculations do not hold good in any dynamic structure like the human heart, as such these measurements are bound to go wrong when used to predict the future of the patient. "Doctors have been predicting the unpredictable" all these years was the observation of a physicist in a recent study.1
No organ of the human body works in isolation, although in reductionist science, organ based specialties have developed to the detriment of patient care! The various organs of the human body work in tandem having been "mode-locked" to one another. Mode locking makes it possible for the most dominant rhythm to control all other rhythms in the human body. The most dominant rhythm is the rhythm of breathing. Heart rhythm is, therefore, mode-locked to breathing. This could be made out in children even by the nurse, called sinus arrhythmia, wherein the pulse goes fast during inspiration and slows down during expiration. However, this becomes less marked as age advances. If one could analyze the heart rhythm more carefully using non-linear methods, one sees this happening even right up to the time of death. The intensity varies depending on the health of the whole system. Heart rate variability (HRV) is one such measure that gives a good indication of the health of the cardiovascular system.

We have gone a step further in this study. Instead of analyzing the whole wave pattern, we have studied the wavelet analysis. The latter is much more accurate measure of the ECG pattern. Coupled with mode locking, represented by HRV, wavelet analysis could assess the heart's function much more accurately. This enables better prediction of the future events possible, although predicting the future with one hundred per cent accuracy in a dynamic organism needs the total initial knowledge of the organism.2 The latter is hampered by our inability to assess the genotype and the consciousness of the organism with our present knowledge.3

Materials and Methods:
A total of 400 subjects, both apparently healthy ones and patients suffering from various cardiac diseases have been studied using the methods described below. The age and sex distribution is given in the table (1) below as also the disease distribution with the total number studied under each category is shown in the next table(2) The ECG data is stored in a holter monitor for the duration of 10-15 minutes. Then this data is sampled at a sampling rate of 131 sps? with a resolution of 12bits/sample and stored in a random access file. R-R interval is then found out.4, 5

Continous Time-Wavelet (CWT) Analysis
Wavelet analysis is essentially comparing the signal with a chosen wavelet; and recording the coefficients that indicate the correlation of the signal to the wavelet. The reference function is called the mother wavelet, which is appropriately dilated and translated to different scales before comparing with the time domain signal.6, 7 The wavelet coefficients are plotted against a two dimensional plane with one axis (y-axis) representing the dilation (scaling factor) of the wavelet, and the other (x-axis), its translation (shift along the time axis). The strength of the coefficients themselves is shown by the intensity of a chosen color, so that the distribution can be seen as color patterns in a two dimensional plane (scalogram). In the Continuous time Wavelet Transform (CWT)8, 9, 10 the wavelet coefficients are evaluated for infinitesimally small shifts of translation as well as scale factors. That is, color intensity of each pixel is separately evaluated, and therefore CWT scalogram is suitable to observe patterns in the distribution of coefficients. This approach provides a more accurate time localization of the abnormality or the defect in the signal. The duration of the abnormality can be surmised from the dilation factor, and its location by the translation factor. Thus the intensity and spread of the color pattern can be useful in classification of abnormalities.

For  a  given  wavelet ,  the  wavelet  coefficients  are  evaluated  as  given  below:

where  ;    a à scale  factor;   b à translation  factor  

Discrete Time - Wavelet (DWT) Analysis
The evaluation of Continuous time Wavelet Transform (CWT) coefficients is a highly computation intensive process . For each pixel in the scalogram represents a wavelet coefficient, and this coefficient is the result of a correlation algorithm. CWT is useful in signal analysis (to identify the patterns). It is possible to synthesize the original signal from the scalogram coefficients, but because of the high degree of computational intensity, the CWT is generally not suitable for data compression.

Discrete time Wavelet Transform (DWT), is a sampled version of the CWT in a dyadic grid. In CWT every pixel of the scalogram represents a wavelet coefficient, but in DWT, the wavelet coefficients are calculated for discrete values of scale factor and translation factors – the increments are in the dyadic scale.11, 12 Consequently, it is called dyadic scalogram, and therefore has fewer coefficients than CWT. In DWT, a whole tile in the scalogram represents a single coefficient, and hence would be shown with the same intensity. Hence the pattern visible in the CWT plot is conspicuous from its absence in DWT plot. The argument in favor of the DWT is that, though it makes use of fewer coefficients than CWT, it is possible to synthesize the original signal using these coefficients. The computational economy makes it suitable for data compression and storage.

Nonlinear Dynamical Analysis
Recent developments in the theory of nonlinear dynamics have paved the way for analyzing signals generated from nonlinear living systems. It is now generally recognized that these nonlinear techniques are able to describe the processes generated by biological systems in a more effective way. The nonlinear dynamical techniques are based on the concept of chaos and it has been applied to many areas including the areas of medicine and biology. The theory of chaos has been used to detect some cardiac arrhythmias such as ventricular fibrillation.13 Efforts have been made in determining nonlinear parameters like correlation dimension for pathological signals and it has been shown that they are useful indicators of pathologies. Methods based on chaos theory have been applied in tracking HRV signals and predicting the onset events such as ventricular tachycardia in many cardiac conditions.14
A novel method based on phase-space technique has been proposed for the analysis of cardiovascular signals.15

Phase- space
Phase space is any abstract space formed by position (linear or angular) and its associated speed. It is used by physicists to study the behaviour of physical systems including chaotic systems.

If x-axis indicates the position of an object (value of x indicates the position at any particular instant), and the y-axis indicates its velocity (y = dx/dt ) corresponding to the position, then the resulting plot indicates the nature of its motion.

About CD:
In the phase space plot shown X axis represents the heartrate x(k) and the Y axis represents the heartrate with a delay x(k+T). The choice of an appropriate delay T was calculated using the minimal mutual information technique.16, 17

It may be observed that the way in which the points are distributed depends on the structure of the signal which, in turn, depends on the condition of the subject.

The method of estimating the embedding dimension for measured signals was proposed by Grassberger and Procaccia.18 Other authors have verified that the embedding theorem restriction is only a sufficient, but not necessary condition for dynamic reconstruction.19 Nevertheless, the dimensionality of the attractor is usually unknown for experimental data and required embedding dimension is therefore unknown. In the present work an adequate embedding dimension of 2 was chosen. This method also allows us to determine the so called correlation dimension (CD) of the attractor which is computed from the equation:

where the correlation integral C (r ) is  given by

where  X_i  and X_j are   the points of the trajectory in the phase space, N is the number of data points in phase space, the distance r is a radius around each reference point X_i and

is the Heaviside function. The CD is a measure of complexity of the process being investigated.

Result :

Table 1 : The ranges of CD values for different types for various classes,
in each group a minimum of 10 subjects were analyzed.

In the CWT plot colors, yellow and white colors elicit high frequency variation and black indicates low frequency variations.
The phase space plot of typical Normal heart rate is shown in Fig.2 and its CWT is shown in Fig.3. In the CWT plot there is a continuous variation in the colors indicating that the heart rate is continuously varying. And its phase space plot is restricted to a range. The phase space plot of Ectopics is shown in Fig. 4 and its CWT is shown in Fig.5. The ectopic beat is a sudden surge of white line in the CWT plot. In the phase space plot the range is stretched on one side. The phase space plot of SSS 1 (Sick Synus Syndrome) (BRADY) heart rate is shown in Fig.6 and its CWT is shown in Fig.7. There are black patches indicating the brady syndrome. In the phase space plot there is more variation. The phase space plot of Atrial Fibrillation (AF) heart rate is shown in Fig.8 and its CWT is shown in Fig.9. The variation of colors is sudden in the CWT indicating there is sudden changes in the heart rates. The phase space plot is highly varying. The phase space plot of complete Heart Block (CHB) heart rate is shown in Fig.10 and its CWT is shown in Fig.11. The color variation in CWT is very subtle, indicating that the heart rate variation is very less. And in the phase space plot, we get a almost a point. The phase space plot of SSS 111 (BRADY-TACHY) heart rate is shown in Fig.12 and its CWT is shown in Fig.13. The color variation takes place in sudden patches of black (brady) and colored (tachy). And the phase space plot is highly chaotic. The phase space plot of Ischemic /Dilated cardiomyopathy heart rate is shown in Fig.14 and its CWT is shown in Fig.15. The color variation is slow and periodic, indicating that the heart rate varies slowly. And the phase space plot is a very small domain.

Discussion:

The present study brings out the following important facets of the heart's function hitherto not studied.

• Heart rate variability is present in all situations even in adults; it is dampened depending upon the gravity of the disease process.
• Study of the "CHAOS" pattern of HRV gives a better visual image of the disease state, making it easier to recognize even for an untrained eye.
• Wavelet analysis gives classical patterns that are unique to different diseases.
• Fractal structure of the heart is elegantly shown by the self-similarity seen in normal ECG wavelet analysis, which differentiates the normal from the abnormal at a glance.
• If the patterns arrived at in this study are shown to be consistent in a larger study of the patients, cardiac diagnosis could be made even by a novice in far off village hospitals without hi-tech gadgets available.

Converting the linear data to non-linear format needs the help of complicated mathematical formulae that might look complicated to most medical readers. Since this is the first attempt to understand the working of the dynamic system one has to try to understand the basics that are described in the section on materials and methods. These are all time tested methods and do not require further authentication.

Conclusions:
Heart rate variability (HRV) is a better measure of the working of the heart, as it measures many other aspects of the cardiac function compared to chronotrophicity alone. Non-linear fractal assessment of the HRV represented graphically gives unique patterns of cardiac abnormalities that makes diagnosis of the latter easier than by the conventional methods. Pattern recognition has been made easier still by our study using colour imaging methods. When this is done on still larger numbers the patterns could be standardized.

BIBLIOGRAPHY.

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13) Kaplan DK and Cohen JR . "Searching for chaos in fibrillation", Annals NY Acadamic Science. 1991; 11: 367-374.

14) Cohen ME, Hudson DL, and Deedwania PC. "Applying continuous chaotic modeling to cardiac signal analysis", IEEE Engg. Med. Biol.,1996; 15: 97-102.

15). Dutt ND and Krishnan SM. “Application of Phase space techniques to the analysis of cardiovascular signals”, Proceedings of IEEE EMBS Conference, Atlanta, U.S.A.,1999 and Signal Processing, Singapore, U.S.A.,1999.

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17) Fraser AM, "Information and entropy in strange attractors", IEEE transactions on Information Theory, 1989;35: 245-262.
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19) Ding M, Grebogi E, Ott E, Sauer T, Yorke JA, "Estimating correlation dimension from a chaotic time series: When a plateau occur ?", Physica D, 1993; 69: 404-424.

DIAGRAMS.

Fig.2	Fig.3 CWT plot of Fig.2
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Fig.4	Fig.5 CWT plot of Fig.4
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Fig.6	Fig.7 CWT plot of Fig.6
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Fig.8	Fig.9 CWT plot of Fig.8
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Fig .10	Fig.11 CWT plot of Fig.10
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Fig .12	Fig.13 CWT plot of Fig.12
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Fig .14	Fig.14 CWT plot of Fig.14
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