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Wavelet transform on a rational dilation is proposed as a method of assessment of spectral power in low and high frequency (LF and HF, respectively) bands for heart rate variability (HRV) analysis. One of the unique properties of this method is a possibility to align the band limits of certain scales with the limits of ranges LF and HF used in HRV analysis. The method parameters are optimized for use in the context of HRV analysis. Suitable examples are tilt test recordings analyzed using the optimized rational-dilation wavelet transform method. Keywords: heart rate variability; non-stationarity; power spectral density; RADWT; rational dilation wavelet transform. predefined frequency bands: very low frequencies (VLF): 0.0030.04 Hz, low frequencies (LF): 0.040.15 Hz, and high frequencies (HF): 0.150.4 Hz. Each of these bands has its physiological interpretation related to the functional state of the ANS: LF is related to the sympathetic tone and HF to the parasympathetic tone . The theory is not unequivocally considered correct , which is not discussed in this paper. Furthermore, we denote these as `physiological bands'. Reliable measurement of the spectral power in LF and HF bands in a non-stationary signal is an important problem in cardiology diagnosis, as the powers in LF and HF and their ratio LF/HF are used as a marker in the status of a patient or to describe the response of the ANS to various changes of the internal and external environment of the body [1, 4]. Spectral decomposition of HRV may be performed using many methods; however, it must be kept in mind that signals exhibit strong non-stationarity, which makes the use of some methods problematic. Some of the most widely used methods are short time Fourier transform (STFT) and the autoregressive method . Other methods include evolutionary periodogram , bilinear transforms , discrete wavelet transform (DWT) , wavelet packets  or Gabor transform, which provides very good frequency resolution for predefined frequencies in the VLF band . Among these methods, some of the qualities of DWT include low computational effort (as compared with bilinear transforms) and better tolerance for non-stationarities, due to finite support, as compared with STFT. Note that in the fast Fourier transform (FFT)-based methods the frequency range is divided evenly, each frequency component has its well-defined resonant frequency in the middle of the band and the frequency distribution per component is (ideally) delta-shaped. On the contrary, in methods based on non-harmonic base functions, the time localization is much better, but according to the uncertainty principle, the frequency domain is partitioned into bands of non-zero width, related to different scales of the wavelet transform. The bands often overlap, the resonant frequency is not necessarily in the midpoint of the band and the frequency distribution within each band is rather broad (c.f. Figure 1A). *Corresponding author: Teodor Buchner, Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland, E-mail: email@example.com Grzegorz Grzyb: Faculty of Physics, Warsaw University of Technology, Warsaw, Poland Pawel Krzesiski: Military Institute of Medicine, Department of Internal Diseases and Cardiology, Warsaw, Poland Introduction The intervals between consecutive heart beats, socalled RR intervals, recorded over a prolonged time, exhibit rich dynamics. This is known as heart rate variability (HRV), and may be quantified using different analysis techniques of the time series of RR intervals (tachogram) . It is generally agreed that HRV reflects the functional state of the autonomous nervous system (ANS) and enables the quantification of the tone of its two branches: sympathetic and parasympathetic (vagal) . The quantification of the functional state of the ANS has diagnostic value in different groups of cardiology patients, for example, in heart failure patients [2 ], and is also used in other contexts . Many methods are used to give a quantitative measure of the ANS state from HRV . One of the most popular methods  requires spectral decomposition of the tachogram into a few 30Buchner et al.: Heart rate variability assessment with rational-dilation wavelet transform Figure 1Band structure for unoptimized (A) and optimal (B) sets of parameters. Discrete wavelet transform on dyadic and rational dilation DWT is calculated on a set of scales and time positions forming the time scale grid called a dilation. The most widely used is the dyadic dilation (Figure 2A), in which the frequency scale is divided by A = 2 to obtain resonant frequencies of the higher scales. It may be seen that in the frequency domain the time frequency atoms have width increasing with frequency and thus the frequency resolution decreases with increasing frequency. In the time domain, in turn, the density of points also changes by a factor of B = 2 between scales. For highest frequencies (lowest scales) the time density is the highest, whereas for lowest frequencies it is the lowest. As a result, the localization of signal components of the lowest frequencies is poor as the area of the atom is constant due to the uncertainty principle. The resolution of the DWT method may be improved with the use of overcomplete representation, which resembles the overcomplete set of atoms used in the matching pursuit method . One method for constructing such overcomplete representation, proposed by Bayram and Selesnick , is to introduce the rational dilation, in which BA0 BA0 f/A Frequency f/A f/A2 f/A3 f/A4 BA2 BA3 BA4 Time BA1 BA2 BA3 BA4 BA1 f/A2 f/A3 f/A4 Time Figure 2Dyadic dilation (A) and rational dilation for s = 3, q = 25, p = 18 (B). Buchner et al.: Heart rate variability assessment with rational-dilation wavelet transform31 the scale and shift scaling parameters are A=q/p and B=s, respectively. Here p, q and s being small natural numbers are the parameters of the method. Note that A=2 and B=2 define a dyadic dilation being the special case of a rational dilation. The increased time resolution in lower frequencies is well visible in Figure 1B for B=s=3 and A=q/p=25/18=1.38. Note that in Figure 1A and 1B only the first five scales are shown: the empty space at lower frequencies in Figure 1B is filled by the higher scales and the density of the band on a frequency scale is obviously higher there. The DWT calculated on such rational dilation is thus called rational-dilation wavelet transform (RADWT) . The hallmark of the RADWT method is the constant Q factor (being the ratio of the resonant frequency and the bandwidth), which is constant over all scales . Note that DWT or wavelet packets used also for HRV analysis do not have the constant Q property. The value of Q may be increased by a suitable choice of parameters . One of the qualities of the RADWT method is that the frequency bands structure (the frequency range occupied by each scale) can be altered by changing the parameters p, q and s. This property is especially important in the context of HRV analysis, where the power spectral components must be calculated for the physiological bands that are predefined. The frequency bands in this method are defined by the following equations. The resonant frequency of j-th band: RFj = p j- z 1 1 p p 1- + q j- z 2 s q q stress introduced by this maneuver demands a response from the ANS in order to maintain homeostasis. A consequence of ANS failure is an episode of central ischemia with the possibility of subsequent fainting, syncopies, heart arrest and other undesirable phenomena. The tilt test is performed in the diagnosis of unexplained syncope, where it has to be determined whether the primary reason for fainting is neurological or cardiological. Because different hypotheses exist concerning the balance between the activation and its withdrawal for each of the branches of the ANS under orthostatic stress , it is important to reliably assess the spectral power in LF and HF bands during the tilt test and the time sequence of their changes. Materials and methods As a test case we used a single tilt test table recording obtained from a patient diagnosed with unexplained syncope. The signal was acquired using the Porti system at sampling frequency of 1.6 kHz. Custom software was used to detect the QRS complexes using the Pan-Tompkins algorithm . After obtaining the tachogram, spline interpolations of the segments of tachogram were performed and the resampled tachogram (with even sampling) was obtained by resampling the spline interpolation at 16 Hz. The above procedure is typically used to avoid the problem of uneven sampling (the point process such as the beating of the heart `samples itself '). The sampling frequency is fairly high more commonly used are frequencies of 4Hz or even 2 Hz. Of course increasing the resampling frequency does not give us any better information about the signal (note that the HRV itself is a result of a point process with a mean frequency as low as 1.2 Hz), but it was used to facilitate the subsampling in other channels used in our study (frequency reduction from 1.6 kHz to 16Hz does not require the anti-aliasing filter). In fact, such a high frequency proved to be beneficial, which is discussed below. We used the canonical implementation of RADWT (provided by the authors) by Selesnick, which is available online (version 2) . In this implementation, the Daubechies family of wavelets is used . All calculations of the band power were performed using a convenient combination of GNU Octave and Python scripting language. The bandwidth of the j-th band: p j- z 1 1 p p BWj = j-z 1- + 1- q 2 s q q And finally, the left and the right band limit (derived from (43) in ): p j -1 1 p j -1 supp G j ( w) = j-1 1- , j-1 q s q In the above equations denotes the sampling (or resampling) frequency. The functions Gj(), mentioned above, define the frequency characteristics of each band. An example of such a function set is shown in Figure 1. The aim of this paper is to apply the RADWT method to HRV analysis of tilt test table recordings. In such recordings, a significant non-stationarity is introduced as the body position undergoes a passive (i.e., with no muscle effort) change from supine to standing. The orthostatic Optimization of RADWT Figure 1A shows the frequency bands typical for the dyadic dilation. Two important features are observed. 32Buchner et al.: Heart rate variability assessment with rational-dilation wavelet transform Firstly, it may be seen that the frequency bands overlap, which manifests the low Q factor of the transform. In the RADWT method, the degree of redundancy may be regulated by a suitable choice of dilation parameters and quantified using parameter R (see  for definition). Secondly, the borders between the bands are not aligned with the LF and HF band limits. Owing to this feature some studies redefine LF and HF for the purpose of DWT analysis , which makes the results incomparable with any other method and seems rather unfavorable. From a methodological point of view it seems reasonable to use the rational dilation and optimize it in such a way that it is rather the borders of the scales that become aligned with the limits of the physiological bands and not vice versa. We found that the optimal set of parameters for sampling frequency 16Hz was: s=3, p=18, q=25. Values of the dilatation and redundancy parameters  for this optimal set are D=1.3889 and R=1.1905, respectively. The structure of frequency bands is shown in Figure 1B. It may be seen that the optimal parameter set provides good alignment between the scales and the physiological bands. Note that the frequency distributions of all the scales have flat tops. This would raise a problem in the context of frequency localization but is not important for the power spectrum assessment in predefined bands (as in the current context). Table 1 shows the numerical values of the band limits calculated using the formulae from . The LF band is composed of scales 1316, whereas the HF of scales 10, 11 and 12. The LF and HF powers are obtained by summing up powers of the relevant scales. Another parameter of the method is the total number of scales J in which the calculations must be performed, determined by the lowest frequency we wish to observe. In the case of HRV analysis, it depends on the maximum scale required to cover the physiological band. A lesser number of scales results in a shorter computational time. It may be seen that the maximum number of scales required for current calculation in J = 16, which requires only reasonable computational effort. With some parameter sets we tested the number of scales with up to J = 80 and the computational burden was higher (discussed below). Table 1Frequency range of scales for s = 3, p = 18, q = 25, f0 = 16 Hz. Scale 1 2 3 4 5 6 7 8 9 10 (HF) 11 (HF) 12 (HF) 13 (LF) 14 (LF) 15 (LF) 16 (LF) 17 Lower limit, Hz 5.4583 3.9300 2.8296 2.0373 1.4669 1.0561 0.7604 0.5475 0.3942 0.2838 0.2044 0.1471a 0.1059 0.0763 0.0549 0.0395a 0.0285 Upper limit, Hz 7.6153 5.4830 3.9478 2.8424 2.0465 1.4735 1.0609 0.7639 0.5500 0.3960a 0.2851 0.2053 0.1478a 0.1064 0.0766 0.0552 0.0397 Bold font denotes where power bands overlap (details further in text). Results Figure 3A shows the fragment of the tachogram with a length of 600 s, centered at the moment of tilting. The previously mentioned non-stationarity is well visible. The spectral power measurements (Figure 3B) show a significant increase of the spectral power in the LF band. The spectral power in the HF band (Figure 3C) also rises, but not to the same extent. The average power in the LF band before and after tilting is 63 ± 83 vs. 688 ± 473 and in the HF band 7.7 ± 5.8 vs. 18.5 ± 19.3. The rise in the LF band is thus five times larger than that in the HF band and definitely statistically significant. Next we were looking for a way to compare our results with those obtained with the use of another standard method. As the signal is strongly non-stationary, the short time windows had still to be used. STFT seemed to be an obvious choice; however, at least in the reference method we wanted to avoid the potential error related to the resampling procedure. Therefore, we decided to use the Lomb evolutionary periodogram method , which is suitable for non-uniformly sampled signals. The use of the averaged Lomb evolutionary periodogram is also advocated by other researchers in this field . To maintain a good standard, we used the classical implementation by Moody, published in the WFDB package on Physionet . The shortest possible time window capable of handling the 0.04-Hz frequency is as long as 25 s. As we used the short time approach, the Lomb method becomes parametric and the free parameter is the window length. As we assume no a priori knowledge of the signal, we start from the shortest window length which enables us to reliably measure the power spectrum in the LF band (25 s). Then we repeated the calculations for window length T = 25, 50 and 100s with time step 1 s. As a method of verification, we calculated the relative change in band pass power using both methods. The results are given in Table 2. Buchner et al.: Heart rate variability assessment with rational-dilation wavelet transform33 Figure 3Results of band pass power calculations for a test patient. Panel (A) (top) shows the fragment of tachogram. Panel (B) (middle) shows the integrated spectral power in the LF band. Panel (C) (bottom) shows the integrated spectral power in the HF band. The marker in the middle shows the moment of tilting. Discussion First, note that it proved beneficiary to increase the resampling frequency to 16 Hz. The properties of the RADWT method can be utilized much better when the range of the signal in the frequency domain moves down from = towards = 0, where the scales are numerous and dense the possibilities to optimize the transform are wider there. The computational burden increases with the number of scales, but it is a matter of choice where to set the working point to obtain a compromise between the computational burden and accuracy. We also tested the optimal parameters for resampling the tachogram at 4Hz and the results were not so promising. An optimal set was p = 15, q = 16, s = 16 with maximum scale equal to J = 80. Such a value of the maximum scale is highly computationally Table 2Relative change of power in physiological bands calculated using RADWT and averaged Lomb periodogram. /Band name LF HF RADWT, % 807 57 Lomb T = 25 s, % 131 58 Lomb T = 50 s, % 125 62 Lomb T = 100s % 103 64 demanding. Another set was p = 3, q = 4 and s = 4, where the maximum scale was equal to J = 20, but the lower limit of the LF band and the higher limit of the HF band was then located in the middle of the range of some scales, which could make the calculations complicated and would increase the error. Increasing the resampling frequency definitely improved the situation at a relatively low cost. Analyzing Table 1 we can see that the power bands overlap (numbers in bold font in Table 1) but the amount of this overlapping is negligible: 0.0007Hz in the case of the most important LF/HF border. Also, the higher limit of the HF band is shifted by 0.0016 Hz. Overlapping or shift of such a magnitude is not likely to seriously affect the results. Note also that the limit between the LF and HF set at 0.15Hz is somewhat arbitrary  (discussed below). The non-stationarity of the HF signal in Figure 3C results from a leakage from the LF band, which is unfortunately a result of the weakness in the definition of the LF and HF bands. The strong 0.1-Hz drive which appears mostly in diastolic blood pressure is not likely to be purely harmonic. The LF band, however, ends as early as 0.15Hz leaving no space for aliquot frequencies, which arise from the non-harmonic (in general) waveform of the driving signal. Consequently, some of the LF power inevitably leaks to the HF band. When we experimentally 34Buchner et al.: Heart rate variability assessment with rational-dilation wavelet transform added the power from scale 12 not to the LF band but rather to the HF band, the non-stationarities in the HF band completely disappeared. In fact, the problem is caused by overlapping of the physiological bands themselves and it cannot be simply avoided. Proper handling of this situation requires the use of exogenous signals (such as the instantaneous lung volume signal) and may be solved, for example, using the complex demodulation method . Regarding the results of verification, described in Table 2, the result seems fairly convincing. RADWT gives eight times more pronounced relative change of power in the LF band, whereas it performs equally well in the HF band. Further verification of the capabilities of RADWT would require either the use of an artificial signal (which is an art in itself in the case of the self-sampled tachogram) or verification on a larger number of patients, or both. The plots obtained with the use of the Lomb method are not shown as they demonstrate no substantial difference, compared with those in Figure 3. It was also rather interesting to find that the rate of change of the HF power in the transition region was approximately of the same order of magnitude as for RADWT. This is clearly a merit of the Lomb method, which, despite the use of the basis of harmonic functions, is still able to approach the same order of the time frequency resolution which is provided by the wavelet transform. resampling frequency of the tachogram, the better the accuracy of alignment between the physiological bands and the band limits for scales. We advise to use 16Hz resampling. Analysis of tilt test table recordings reveals reasonable accuracy of determination of the onset of orthostatic stress in the time domain. The method seems to work rather well despite the strong non-stationarity introduced by orthostatic stress. Limitations To show the superficiality of the RADWT method it would be desirable to increase the number of analyzed patients or to use a customized test signal. Statistical significance of the LF/HF band power changes before and after tilting was not tested, although it must be acknowledged that at least in the LF band the effect is so clear that it does not show a huge need for deeper statistical analysis. Acknowledgments: This work was partially published as part of the MSc thesis of G.G. at Warsaw University of Technology, Faculty of Physics, Warsaw, and presented at the Neurocard Conference, Belgrade, Serbia in October 2010. This work was supported by the European Social Fund implemented under the Human Capital Operational Program (POKL): project `preparation and realization of medical physics specialty'. The authors thank the referees for their valuable remarks. J. Piskorski and P. Guzik are thanked for an interesting discussion. Conclusions We conclude that for an optimized set of parameters RADWT may be used for HRV analysis. The higher the Received October 4, 2012; revised January 23, 2013; accepted January 24, 2013; previously published online February 23, 2013
Bio-Algorithms and Med-Systems – de Gruyter
Published: Mar 1, 2013
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