Heart Diseases Problem Diagnosis We discuss here wavelet methods to determine characteristics point of ECG signal, its compression, detection of its abnormalities, characterization of heart rate variability through the study of ECG signal, and probe a variety of arrhythmias.The Electrocardiogram:Muscular contraction is associated with electrical changes known as depolarization. The electrocardiogram (ECG) is measure at the body surface and results from electrical changes associated with activation first of the two small heart chambers, the atria, and then the two larger heart chambers, the ventricles.

Figure 1: A schematic of ECG exhibiting normal sinus rhyhum

Producing an algorithm for the detection of the P wave, QRS complex and T wave in an ECG is a difficult problem due to the time varying morphology of the signal subject to physiological conditions and the presence of noise.

ECG timing, distortions and noise:

Recently, a number of wavelet-based techniques have been proposed to detect these features. Senhadji et al compared the ability of wavelet transform based on three different wavelets (Daubechies, spline, and Morlet) to recognize and describe isolated cardiac beats. Sahambi et al used a first-order derivative of the Gaussian function Figure 2(a) as a wavelet for the characterization of the ECG waveforms.

They used modulus maxima-based wavelet analysis to detect and measure various parts of the signal, especially the location of the onset and offset of the QRS complex and P and T waves.

Note that for an anti-symmetric wavelet, such as the first derivative of the Gaussian, dominant peaks in the signal correspond to zero crossings in the wavelet transform. (For symmetric wavelets, dominant peaks in the signal correspond to extrema in the transform plot.) One of the ECG signals they analyzed, together with its wavelet transform at four consecutive scales, is shown in Figure 2(b).

The maxima and minima of the wavelet-transformed signal are used to determine the location and width of the QRS complex. This is shown in the lower two plots of the figure 2(b) containing, respectively, the signal and its corresponding wavelet transform at the largest scale.

The vertical lines above the ECG signal at the top of the plot show the location of the QRS complex determined from the zero crossings of the modulus maxima of the transformed signal.

Figure 2

Figure 3

Feature DetectionHeart Trouble

A Healthy Heart is a Fractal Heart

When you go to the doctors office one of the first thing they do is jot down your blood pressure and pulse rate. According to Ary Goldberg of Harvard University, they might do well to include another number: your heratbeat’s fractal dimension

Goldberg spoke on the role of fractal dynamics in physiology at the SIAM 2003 Annual Meeting, held June 16-20 in Montreal. Careful analysis of heartbeat time-series could give cardiologists new diagnosis tools in the battle against heart disease. Similar analysis of brain waves and stride length in walking could give researchers new insights into such conditions as epilepsy and Parkinson’s disease.The heart is part of a large feedback system whose dynamics are non-linear, non-stationary, and multi-scale Goldberger observes. As mathematicians well know, even simple nonlinear rules can lead to complex behavior –

and the rules the heart are hardly simple. “It’s an incredibly complicated system,” Goldberg says. Consequently, the healthy heartbeat “is one of the most complex signals in nature.”

All this runs counter to the view of the “regular sinus rhythm” – the comforting lub-dub we all associate with stress-free existence-as the gold standard of cardiology. To be sure, you don’t want your heart beating erratically, as in Don the first –page quiz: That patient has a condition known as atrial fibrillation. But neither do you want a steady or sinusoidally varying pulse rate of the types shown in A and C; those are from patients with severe congestive heart failure. The healthy heart rate is the one shown in B. Goldberg and colleagues, notably C.-K. Peng of Harvard, and Eugene Stanley and Plamen Ivanov of Boston University, have developed a number of methods for analyzing heart-beat and other physiologic data.

One is a method they call “detrended fluctuation analysis” (DFA). The basic idea is to calculate the average amount of fluctuation over bins of different sizes-that is, the root means square deviation between the signal and its trend in each bin-and then plot the result as a function of bin size. Remarkably DFA for a healthy heart form an almost perfect line of slope 1- a hallmark of Fractal behavior. (Its characteristic of what’s called 1/f noise and indicates the presence of long-range correlations in the signal. “White” and “brown” noise, by contrast, tend to line up with slopes of 0.5 and 1.5, respectively.) If all hearts produced slope -1 plots, the finding would be intriguing but not clinically valuable. Goldberger’s group has not found that to be the case, how ever: Recordings from the people with diseased hearts and from elderly people show marked departures from slope-1 straightness. This indicates a loss of complexity in the heartbeat as a result of illness or aging.

The difference between healthy and diseased heart signal is even more striking in a multi-fractal analysis. The signal from a beating shows signs of being a multifractal, which means that it is characterized by a range of exponents, of which the DFA slope is only the first.

Indeed, a wavelet-based analysis of a healthy heart shows non-zero Fratcal dimension over a broad range of exponents, whereas the same analysis from a subject with congestive heart failure has a much more restricted range.

“You actually see very reproducible degradation of the Fractal scaling pattern,” Goldberg says. “ That also raises the possibility that there are diagnostics and prognostic uses-besides application-of some of these newer monitoring techniques.”

Peng, Goldberger, and Madalena Costa of the University of Lisbon have devised another application technique, called multi-scale entropy, which measures the complexity of physiologic signal like the heartbeat and demonstrates a consistent loss of information with aging and disease.

Their paper on the technique appeared last year in Physical Review Letters.

For fractals and other complexity-related measures to reach the bedside will require close collaboration among researchers from different disciplines, Goldberger stresses. In particular, he says “ physiologists are not used talking about Fractals and nonlinear mechanisms and complex systems, although they deal with those every day.”

The concept of homeostasis-the idea that organisms strive to maintain an ideal steady –has been a staple of physiology for the last seventy years and still has strong appeal. The fractal patterns observed by Goldberger and colleagues indicate an adaptive variability, so that they don’t locked into one mode of behavior.

Progress in the field will also depend on researchers having ready access to data. “Data that are generated in an intensive care unit may be of huge interest of people who have been looking at complex systems in a very different guise,” Goldberg points out. Toward that end, his group has set up physioNet (www.phsionet.org), with funding from National Center of Research Resources of the National Institutes of Health. “It’s like a GenBank for physiology,” Goldberger says. “It allows people around the world to look at data sets that they otherwise wouldn’t have access to.”

PhysioNet includes an archive of physiologic data sets, such as the “Fan-tasia Database” : EKG and respiration records of 40 adults (20 young and 20 elderly) who spent two hours watching the classical Disney film. It also provides a toolkit of open-source software, plus a discussion site, tutorials, publications, and set of challenges for all comers.Golberger is enthused by the possibilities. “ Once there’s good data and a communications network, then anything and every thing can happen,” he says “ The potential for discovery is enormous.”

Time Series Challenge: Opening the 2003 SIAM Annual Meeting with an invited talk titled “ Fractal Dynamics in Health and Their Breakdown with Disease and Aging,” Ary Goldberger of Beth Israel Deaconess Medical Center and Harvard Medical School showed the audience these four rate recordings and invited them to choose the one they would most like to have. As hands went up for each possibility, Goldberger pointed out that only one was from a healthy patient; the other three came from “ people at risk of some thing really bad.” (When he presented the same challenge to medical people, he said , a cardiologist was among those who made the worst possible choice.) Which one would you hope to have?

Contribution of the Wavelet Analysis to the Non-Invasive Electro-cardiology 

Today, various digital-signal-processing methods are applied to the ECG to identify, extract and analyze the different ECG signal components.

In this large set of signal-processing tools, a new technique called wavelet transformation appears to be a promising method describing time and frequency characteristics of ECG waves.

Here we present an overview of the wavelet technique applied to the area of quantitative electro-cardiology without describing mathematical details of the wavelet theories.

In the first place some rationale for the utilization of new ECG processing tools are discussed. In the second place will describe the contribution of the wavelet transformation in quantitative electro-cardiology.

This technique will be discussed and compared to the classical techniques using time-domain and frequency-domain measurements

The frequency representation of a signal can be obtained using different techniques including the Fourier transformation, and the autoregressive method. The most frequently used in electro-cardiology is the Fast-Fourier

Transform (FFT) that is able to decompose any temporal signal (theoretically this signal should be deterministic and periodic) in an infinite set of sinusoid functions.

This set of sinusoid functions is then represented in the frequency space using the amplitude and the phase of each of these functions. The FFT thus provides a link between the time representation of a signal (in seconds) and the frequency representation (in Hertz or cycle/second).

As the digitized ECG is a finite signal, its boundaries are usually abrupt. These abrupt cuts of the signal make it discontinuous. This introduces a smearing (decrease and spread) of all the estimated frequency peaks. In order to avoid this, the calculation of the FFT is applied to the windowed ECG.

The windowing aims to smoothly decrease the boundary of the ECG signal to zero, removing its discontinuity. The limitation of this approach is that windowing reduces the frequency resolution and therefore lowers the quality of the estimation of the ECG signal frequencies

Another unavoidable limitation of the Fourier transformation for the ECG analysis is that this technique does not provide insight into exact location of frequency components in time. The frequency content of the ECG varies in time; the QRS complex is a high frequency wave whereas the T wave contains low-frequency components.

Therefore, the need for an accurate description of the ECG frequency contents according to their location in time is essential. Utilization of time-frequency representation in quantitative electrocardiology is thus justified.

This kind of representation provides insight into three dimensions of the ECG signal: the time, the frequency and the amplitude (figure 1)

Figure 1: Time-frequency domain represents a combination of time-domain and frequency-domain characteristics of the ECG signal

Among these various time-frequency transformations, the the wavelet transformationswavelet transformations also called also called time-scale time-scale transformationstransformations, is gaining a particular interest in quantitative electro-cardiology.

Figure 2: The wavelet-coefficient calculation is illustrated using a set of analyzing wavelet from the ‘Mexican hat’ wavelet and the ECG signal from a healthy subject. The analyzing wavelets are first multiplied by the ECG signal. Then the wavelet coefficients are calculated using the area under the resulting curves. The area values are then plotted in the time-scale domain providing the three-dimensional representation of the signal.

Thus, when the wavelet analyzes slow waves as the T wave, longer wavelets are needed and frequency resolution is good. Whereas with rapid waves, like the QRS complex, shorter wavelets provide better signal time description: time-resolution is good but frequency resolution is poor.

As a microscope can focus on specific details of a slide, the wavelet shape can be adapted to focus the analysis on specific components of the ECG.

Increasing use of computerized ECG processing systems requires effective ECG data compression techniques which aim to enlarge storage capacity and improve methods of ECG data transmission over phone and internet lines.

The wavelet compression methods described in 1992 provide a robust technique suited for detecting and removing redundancy in the signal. The few publications available on this topic suggest that the ECG data compression using wavelets could decompose the ECG without any redundancy and provide high compression ratio and high quality reconstruction of ECG signal.

According to these preliminary reports, wavelet-based compression seems to be more efficient than the classical compression methods.

One of the crucial steps in the ECG analysis is to accurately detect the different waves forming the entire cardiac cycle. Some studies aim to design effective methods for detection and classification of ECG waves. For instance, Li et al. developed a wavelet-based classification method that correctly identifies 99.8% of ECG waveforms from the MIT/BIH arrhythmia database.

Some authors have also shown that the wavelets, configured following a wavelet network, provide efficient extraction for discriminating between normal and abnormal cardiac pattern. Even if many algorithms have already been defined for ECG pattern recognition, the wavelet transformation seems to offer a new approach worth investigating, especially in areas of limited performance of current techniques, like P and T-wave recognition.

The wavelet technique has also been used for the evaluation and monitoring of ischemic ECG changes. Mac Leod et al. used wavelet technique for identification of the ECG changes resulting from acute coronary artery occlusion and reperfusion observed during PTCA procedure.

This study demonstrates that the wavelets are able to identify specific detailed time-frequency components of ECG signal, which are sensitive to transient ischemia and eventual restoration of electrophysiological function of the myocardial tissue.

The FFT method, frequently used to evaluate heart rate variability (HRV), has inevitable limitations related to spectral leakage caused by abrupt changes at the boundary of the HRV signal. In addition, FFT introduces inconsistent spectral components of the tachogram

Because the efficiency of the time-domain late-potential detection is limited to the terminal portion of the QRS complex and is also affected by inaccuracies of QRS-end detection, frequency-domain methods have been investigated

The HRECG analysis is the field of research most actively seeking to benefit from the wavelet signal-processing technique. In 1989, Meste et al. applied for the first time, the wavelet transform to 5 KHz sampled ECGs. Subsequently, they used the Meyer wavelet for the detection of the late potentials. The first quantitative analysis of the HRECG using wavelet transformation was described by Dickhaus et al., who identified significant differences in HRECG between post-infarction patients with ventricular tachycardia and healthy subjects.

A different approach was used by Shinnar and Simson who examined the local scaling behavior of the ECG wavelet transformation. Patients without ventricular tachycardia produced ECG wavelet transformation with relatively constant slope, while patients prone to ventricular tachycardia produced ECG wavelet transformation with multifractal behavior. Morlet et al. designed a wavelet-based method for the detection of irregular structure, or singularities in the HRECG, consisting of an algorithm for tracking the evolution of the so-called local maxima of the wavelet transform across scales.

This method is based on the detection, the connection and the acknowledgment of the connected maxima as signal singularities.

It provides an increase in sensitivity and specificity for detecting late potentials in comparison with the results from the time-domain approach. Couderc et al. and Rubel et al. reported studies using non-redundant wavelet-decomposition of the HRECG for the accurate description of the time-frequency components of late potentials without the need of QRS-endpoint localization.

However, no specific wavelet techniques have yet demonstrated a clinically relevant advantage (of wavelet) over the classical Simson's method for late potential detection. The major contribution of the wavelet to late potential detection is that the wavelets also seem to be able to detect abnormal intra-QRS potentials. An example of wavelet transformation of HRECG is presented in the figure 3. This approach is of particular value in the detection of late potentials in patients with anterior infarction and those with bundle branch blocks.

Figure 3: Examples of redundant wavelet transformations of the vector magnitude from two post-infarction patients without (left panel) and with (right panel) sustained ventricular tachycardia. The two vertical lines mark the onset and the offset of the unfiltered QRS complex. The ellipse emphases the time-frequency area with abnormal high-frequency components between 89 and 230 Hz and between 80 and 130 ms after the onset of the QRS complex.

The wavelet ECG quantifiers could thus be a new alternative to time-domain analysis of the repolarization segment, especially for diagnosing the long QT syndrome in patients with borderline QT duration and for quantifying heterogeneity of repolarization44.

Time-domain analysis of the ECG is the simplest approach for detecting and quantifying the different ECG waves. However, efficacy of this approach is often limited by the inaccurate definition of ECG wave endpoints (end of the QRS, T wave location). More complex signal-processing tools, i.e. simultaneous time and frequency domain techniques, were developed to provide more accurate representation and identification of the ECG potentials.

The wavelet transformation is a new promising technique in non-invasive electrocardiology providing improved methods for late-potential detection, HRV analysis and evaluation of the repolarization segment abnormalities.

The benefit of the wavelet transformation lies in its capacity to highlight details of the ECG signal with optimal time-frequency resolution. Since the application of wavelet transformation in electrocardiology is relatively new fields of research, many methodological aspects (choice of the mother wavelet, values of the scale parameters) of the wavelet technique will require further investigations in order to improve the clinical usefulness of this novel signal processing technique.

Simultaneously diagnostic and prognostic significance of wavelet techniques in various fields of electrocardiology needs to be established in large clinical studies.

Feature Detection

Heart Trouble

Here we shall described a standard method for detecting feature within a complicated signal. This method, known as correlation, is a fundamental part of Fourier analysis. We shall describe here some of the ways in which wavelet analysis can be used to enhance the basic correlation method for feature detection.

Lets begin by examining feature detection for ID signals.Lets begin by examining feature detection for ID signals. Feature detection is important in seismology, where there is a need to identify characteristics features that indicate, say, earthquakes tremors within a long seismological signal. Or, in an electrocardiogram (ECG), it might be necessary to identify portions of the ECG that indicate an abnormal heartbeat.

f~

f

At the top of the Figure (1) we show a simulated ECG, which we shall refer to as signal C. The feature that we wish to locate within signal C is shown in the middle of Figure (1); this feature is meant to simulate an abnormal heartbeat. It is, of course easy for us to visually locate the abnormal heartbeat within the Signal C, but that is a far cry from an algorithm that a computer could use for automatic detection.

As noted above, the standard method used for feature detection is correlation. The correlation of a signal f with a signal g, both having lengths of N values, will be denoted by (f:g). It is also a signal of length N, and its kth value (f:g)k

is defined by

1 2 1 1( : )k k k N k Nf g f g f g f g (1)

In order for the sum in (1) to make sense, the signal g needs to be periodically extended, i.e., we assume that

for each k. When computing the correlation (f:g), the signal f is the feature that we wish to detect within the signal. Usually the signal f is similar to the abnormal heartbeat signal shown in the middle of Figure (1), in the sense that the values of f are 0 except near the central portion of the signal. This reduces the distortion that results from assuming that g is periodic. We will describe their use in feature detection

The rationale behind using the correlation (f:g) to detect the location of f within g is the following. If a portion of g matches the form of the central portion of f – where the significant, non-zero values are concentrated - then for a certain value of k, the terms in (1) will all be squares.

k N kg g

This produces a positive sum which is generally larger than the sums for the other values of (f:g). In order to normalize this largest value so that it equals 1, we shall divide the values of (f:g) by the energy of f. That is, we denote the normalized correlation normalized correlation of f with g by (f:g), and the kth value of (f:g) is

1 2 1 1( : ) k k N k Nk

f

f g f g f gf g

(2)

where

2 2 21 2f Nf f f

We can show that under the right conditions, the maximum value for <f:g> is approximately 1.

As an example of this ides we show at the bottom of Figure (1) the graph of the squares of the positive values of the normalized correlation <f:g> for the abnormal heartbeat and Signal C. Notice how the maximum for this graph clearly locates the position of the heartbeat with in Signal C.The value of this maximum is 1, thus proving the following simple criterion for locating an abnormal heartbeat: if a normalized correlation value is near 1, then an abnormal then an abnormal heartbeat is probably present at the location of this value. heartbeat is probably present at the location of this value. We have ignored the negative values of normalized correlation because a negative value of <f:g> indicates a preponderance of oppositely signed values, which is a clear indication that the values of f and g are not matched.

The squaring serves to emphasize the value near 1. It is not necessary , but produces a more easily interpretable graph – a graph for which the maximum value near 1 more clearly stands out from smaller values.

Notice that there are smaller peaks in the bottom graph of Figure (1) that mark the locations of the normal heartbeat in Signal C. These smaller peaks are present because the abnormal heartbeat was created by forming a sum of a normal heartbeat plus a high frequency “noise” term. Consequently, these peaks reflect a partial correspondence between the normal heartbeat term and each of the normal heartbeats in signal C.We shall now describe a wavelet based method for suppressing these peaks in the detection signal resulting from the normal heartbeats. While this may not be necessary for the case of Signal C, it might be desired for other signals.

As we know in the previous section, the higher the level k of the averaged signal Ak , the nearer to zero are the low frequencies which make up the non-zero values of the averaged signal’s spectrum . Hence by subtracting away from Signal C an averaged signal Ak for high enough k, we can remove the low frequency values from the spectrum, of f that rise from the normal heartbeat.

Each normal heartbeat in Signal C has a spectrum that has significant values only for very low frequencies in comparison to high frequency oscillations that are clearly visible in the abnormal heartbeat. Our method consists, therefore, of subtracting away an averaged signal from Signal C.

2kAF

For example at the top of Figure 1(b) we show the signal that is the difference between Signal C and its fourth averaged signal A4. Comparing this signal with signal C we can see that the normal heratbeats have been removed, but there is till a residue corresponding to the abnormal heartbeat. In the middle of Figure 1(b) we show the signal graph of all the squares of positive values of the normalized correlation of the middle signal with the top signal . This signal clearly locates the position of the abnormal heartbeat, at the same location as before, but without the secondary peaks for the normal heartbeats.