A Novel Application of Wavelets to Real-Time Detection of R-waves

Transcription

1 A Novel Application of Wavelets to Real-Time Detection of R-waves Katherine M. Davis,* Richard Ulrich and Antonio Sastre I Introduction In recent years, medical, industrial and military institutions have shown interest in developing small, lightweight, reliable devices to continuously monitor Heart Rate Variability (HRV) of individuals during sleep, work, exercise, or under stress. Such devices, which ideally would be no larger than a modern digital music player, would provide life-saving information to hospital staff, to military commanders in the field, or to individuals at risk for heart or Such a device would require the accurate identification of some fiducal point in the EKG; for the purpose of this paper that point is the R-wave peak, representing a ventricular beat. The EKG s to be analyzed would be corrupted by motion artifacts, emf interference, and, typically, a strong electromyographic signal. The identification would have to be carried out in real time, using modern low-power microprocessors. To meet these requirements, we have developed R-wave detection software based on a novel application of wavelets. There is an extensive literature on the use of wavelets for R- wave identification; as noted in [], many of the techniques are able to attain over 99% accuracy on standard databases. Figure, reproduced from [2], illustrates a portion of the EKG from AHA Tape 29. The authors in [2] report an accuracy of 88.4% in recognizing R-peaks from this recording. However, many of these methods have been developed for analyzing clinical recordings, often made from resting subjects. The EKG signals we consider are taken from healthy young male and female volunteers, undergoing standard suites of exercises: abdominals, elliptical trainers, gauntlet, spin, stair, and taebo. Figure 2 shows a portion of the EKG of a healthy young female, Holter recorded at 8hz, while the subject undertook strenuous abdominal exercises. The R-peaks are contaminated with noise of amplitude comparable to the strength of the R-wave itself and the methods presented in this paper achieve an accuracy of 99.6% on this record (this being the worst performance in the suite of exercises tested). We believe this represents a significant improvement in the accuracy of peak identification for records of this type. II Review of Wavelets Wavelets are often used to provide a multi-resolution analysis, that is, to decompose a signal into averages, that is, low frequency components, and details, higher frequency components. This is an iterative process, which can be computed as efficiently as an FFT. Our approach does not require this formalism; we only need to resolve a signal until its QRS-wave is identified. We employ wavelets as simple linear filters, which can then be understood in the traditional language of digital filter design: convolution, FIR filters, phase and frequency response. Their application requires no sophisticated algorithms beyond convolution, and, in contrast to the multi-resolution approach, we do not decimate after applying the filter. This approach follows the maximal overlap discrete wavelet transform discussed in [3]. The wavelets considered here were constructed by Daubechies [4]. They are non-symmetric real FIR filters, with specified bandpass properties, optimized for phase response closest to linear, typically referred to as the least asymmetric compactly supported wavelets. They are indexed by two integers, L and j, where L is the length of the basic level filter, and j indexes successive levels of detail; that is, the bandpass frequency octave of the filter. L must be at least 4, and j at least one. The filter s(l)d(j) has length: L j = (2 j )(L ) Fig. A segment of AHA Tape 29. Dotted lines: cardiologist markings of R- onset; asterisk, identification of R-peak. After Kadamebe et. al., [2]. Fig 2. A segment of EKG from a healthy young female undertaking strenuous abdominal exercises.

2 Fig 3. Frequency response of the Daubechies ʻleast asymmeyricʼ filters: (left to right) s4d2, s4d3, s4d4. The change in the second parameter corresponds to an octave narrowing of the passband of the filter. Choosing both L and j as small as possible maximizes computational efficiency. the choice L= 4 works well for Holter-recorded EKG s. To understand the meaning and consequences of choices of these parameters, we plot in Figure 3 the frequency response of Daubechies least asymmetric filters s4d2, s4d3 and s4d4, assuming a sampling rate of 8hz. The change in the second parameter corresponds to an octave narrowing of the passband of the filter, and to optimize our choice of filter we must examine the frequencies present in the EKG. Figure 4, taken from Thakor et. al. [5], shows the energy of major components of the EKG signal. The energy in the P and T waves is largely localized below 5hz and motion artifacts localized below about 7hz. Muscle noise and the QRS complex itself range from roughly 5 to 3 hz (see also Goldberger et. al. [6] for the higher frequency components of the QRS complex). The -4 hz frequency response of the s4d3 wavelet filter has also been superimposed on the powerdensity graph from [5]. The figure demonstrates that this wavelet filter is well-adapted to supressing motion artifacts as well as the P and T waves in the EKG; it is equally clear that a frequency filter alone cannot reduce the relative power of muscle noise. Wavelets are specifically designed to overcome this kind of difficulty; they resolve signals in time and frequency domains. The Daubechies least asymmetric filters are all finite-impulse response filters; one way to understand their effects is to think of them as measuring the degree of correlation between the signal and the filter. The s4 series of wavelets were chosen specifically because they roughly mimic the general shape of a QRS-wave. Figure 5 shows a segment of a QRS-wave, together with the impulse response s4d3 wavelet. A QRS-wave should have a high (negative) correlation with the s4d3 wavet, whereas electromyographic noise has a very different shape and is expected to correlate less well. The effect is then to pick out the QRS-wave from noise in the same frequency range; see Figure Samples Fig 5. Top: QRS-waves, from Holter recording of resting subject, sampled at 8hz. Bottom: Impulse response of the Daubechies s4d3 filter. The s4d3 response mimics the shape of the QRS-wave Fig 4. Power density of major components of the EKG, compared with frequency response of s4d3. Adapted from Thakor et. al. [5]. Fig 6. Top: Two QRS-waves, from Holter recording of subject performing abs exercises, sampled at 8hz. Bottom: QRS waves filtered by the Daubechies s4d3 filter. The s4d3 suppresses the electromyographic noise.

3 III The Daubechies s4d3 Wavelet The s4d3 wavelet is optimized for recognizing a QRS-wave of a certain duration. One expects exercise to narrow the QRS complex, possibly affecting recognition. Goldberger et.al. [7] showed that exercise shortened the QRS duration, typically by only 8%, which we do not expect to affect recognition. The s4d3 filter is therefore likely to be appropriate for the recognition of QRS-waves in healthy young individuals. It is well known, however, that various illness can substantially alter the duration of the QRS-wave; see [8], [9] for example. In general it is difficult to use wavelets to provide a collection of finely tuned filters which can accomodate a variety of QRS durations. Wavelets are constructed to provide ʻoctave-band ʼ resolution, that is, with scale varying exponentially rather than linearly, as one sees in the formula for the length of the s(l)d(j) wavelet: L j = (2 j )(L ) + Wavelets derived from a multiresolution cannot, in general, be interpolated between octaves. However, if the wavelet filters s4d2, s4d3 etc are viewed as simple linear filters, they can be interpolated in the obvious way (that is, by means of simple linear interpolation of the frequency response, adjusted for phase shift). Pick a distance t, < t <; the interpolated wavelet at distance t from s4d2 is denoted s4d2.t, and is called a ʻfractletʼ For example, Figure 7 shows the fractlet s4d2.5, between s4d2 and s4d3. Note in particular the narrowing of the QRS part in the fractlet. As one would expect, the frequency response is an interpolation between the two outer wavelets, as well. Fractlets allow a finer tuning of the wavelet to the QRS duration. A final consideration in the choice of filter for QRS-wave recognition is the phase behavior of the filter. As noted in [], [], a filter with non-linear phase can distort the shape of a QRS-wave. Even if the R-wave peak were not distorted, the phase causes a peak in the filtered signal to be shifted in time..8 Interpolation of Wavelets: s4d2 -- s4d s4d3 For a linear filter, this shift is constant. Our application requires the computation of RR-interval lengths; as this involves a difference, linear phase shifts have no effect. However, Daubechies [2] showed that the only compactly supported wavelet with linear phase is the Haar wavelet, and the possibility thus arises that the wavelet filters might shift different R-wave peaks, by different amounts, corrupting the computation of the RR intervals. The effects of non-linear phase must therefore be considered. There is no simple closed form expression for the phase of the s(l)d(j) wavelets (asymptotic estimates such as [3] are of no help). The simplest approach is to compute the group delay, that is, the slope of the phase, for the s4d3 filter. As Figure 4 shows, the -3db points for the power in the QRS-wave range from roughly 5 hz to 2 hz. As our subjects are exercising, we must also account for the possible effects of exercise on QRS duration. Goldberger et. al. [4] examined the effect of exercise ob QRS duration for healthy young men; they found a mean resting QRS duration of 58.3 msec, with change in exercise being a mean 8.3%, which is within the 3db range specified. The group delay for s4d3 is plotted in Figure 8, in the region from 5 to 2 hz, together with the average value (deviation from average indicates non-linearity). In this range, the phase varies about 2.7 samples, which could introduce an error in estimating RR of some 5.4 samples. To test this variation, we hand-computed the time lag between R-wave peaks and the peaks of the filtered signal. The time lag computations were reproducible to an accuracy of.46 msec. Analysis showed an apparently random fluctuation in the lag, ranging from 23.5 samples to 2.8 samples, with a mean lag of samples, and a standard deviation of 2.56 msec. This is well within the predicted range, and the variation of.25 samples is significantly smaller than the predicted fluctuation. The random fluctuation in phase lag is equivalent to clock jitter. For these exercise files, the average interbeat interval length ranges from 359 msec to 548 msec. Jitter within one standard deviation of average accounts for less than one-tenth of one percent of the interbeat interval length, that is, a S/N ratio of roughly -43db Samples to 5 hz Fig 7. Interpolation of wavelets; impulse response. Open circles and dashed lines, s4d2. Open circles and solid lines, s4d2. The fractlet s4d2.5 is interpolated between these, and is shown with asterisks and dotted lines. Fig 8. Solid line: group delay of the s4d3 filter, from 5 hz to 2 hz, measured in samples. Dashed line: average value; deviation from this line indicates non-linear phase.

4 s4d3 TP # FP # FN SE SP abdadd abs ellip gauntlet stair taebo Fig 9. Sensitivity and predictability results for the s4d3 filter applied to six exercise files. IV R-Wave Detection The software we employ for R-wave detection is a modification of the technique developed by Pan and Tompkins [5], and involves a three-stage process. The EKG signal is initially filtered with a Daubechies s4d3 filter, which supresses the P and T-waves and reduces electromyographic noise. The filtered signal is then passed though a sharpening proceedure, which emphazises peaks; finally, an adaptive peak detector is used. The Pan-Tompkins peak detector requires initialization of basic parameters for each file; here the intended application requires a self-starting and restarting proceedure, which we implement using an adaptive autocorrelation technique. The performance of R-wave detection techniques is evaluated not through percentage of peaks detected, but rather by two indices: sensitivity and predictability. These indices take into account not only the number of true peaks (TP) correctly identified, but also the number of spurious peaks produced (false positives: FP) as well as the number of peaks which were missed (false negatives, FN). These are computed as proportions: S e = T P T P + F N ; S p = T P T P + F P The results are shown in Figure 9; we applied the s4d3 detection scheme outlined above to six exercise files, ranging from upper body to abdominal exercises, to lower body, and to mixed exercise files. It is not surprising that the best results are for lower body exercises; muscle noise in this case was well away from the EKG leads. It is also not surprising that the worst performance was on taebo exercises.... What is surprising is that the sensitivity and specificity in all cases exceeded 99%. This is comparable to or better than best performance of other techniques on clinical databases, where typically much less noise is present.

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