Bivariate phase-rectified signal averaging a novel technique for cross-correlation analysis in noisy nonstationary signals

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1 Available online at Journal of Electrocardiology 42 (2009) Bivariate phase-rectified signal averaging a novel technique for cross-correlation analysis in noisy nonstationary signals Axel Bauer, MD, a, Petra Barthel, MD, b Alexander Müller, MD, b Jan Kantelhardt, PhD, c Georg Schmidt, MD b a Medizinische Klinik III, Eberhard Karls Universität Tübingen, Tübingen, Germany b I. Medizinische Klinik und Deutsches Herzzentrum München, Technische Universität München, Munich, Germany c Institut für Physik, Martin-Luther-Universität, Halle, Germany Received 31 March 2009 Abstract Keywords: Signals generated by biologic systems are characterized by a high degree of nonstationarities and noise. Phase-rectified signal averaging (PRSA) was shown to be superior to conventional methods in detection of periodicities in biologic signals. Bivariate phase-rectified signal averaging (BPRSA) is an extension of the PRSA-method for analysis of interrelationships between 2 synchronously recorded biologic signals. Here, we review the methodology of the technique and demonstrate its performance in simulated data. As application to biologic data, we use BPRSA to analyze synchronously recorded time series of systolic arterial blood pressure, RR intervals, and respiratory activity Elsevier Inc. All rights reserved. Autonomic function; Baroreflex; Cross-correlation; Non-stationarity; Sinus arrhythmia Introduction Many natural systems generate periodicities at different time scales. Examples can be found in living nature such as in cardiovascular, endocrinologic, or neurologic systems but have also been described for phenomena in nonliving nature, for example, for the El-Niňo phenomenon, for sunspot numbers, and ice age periods. 1,2 In living systems, periodic modulations often reflect closed loop regulation processes and therefore provide important insights into the state of health of the system. However, the diagnostic approach to signals generated by biologic systems is limited. Nonstationarities are a major problem in the analysis of biologic signals particularly when recorded for a long period and not in a true experimental setting. Internal and external perturbations continuously affect the system causing interruptions of the periodic behavior. Moreover, almost all biologic signals contain a substantial amount of 1/f-noise. Recently, we introduced the method of phase-rectified signal averaging (PRSA) that is capable of detecting and quantifying periodic components in noisy nonstationary time series. 3 The PRSA-based deceleration capacity of heart rate has been Corresponding author. Medizinische Klinik III, Eberhard Karls Universität Tübingen, Tübingen, Germany. address: axel.bauer@med.uni-tuebingen.de shown to be a strong and independent predictor of late mortality after acute myocardial infarction. Its predictive power was shown to be superior to standard measures of heart rate variability and left ventricular ejection fraction. 4 Here, we review an extension of the PRSA method, the so-called bivariate PRSA (BPRSA), for the study of interrelationships of periodic behavior in 2 or more synchronously recorded biologic signals. 5 We briefly describe the methodology of BPRSA and use simulated data to demonstrate its performance. As application in natural data, we use BPRSA to quantify the interrelationships between periodic modulations of systolic arterial blood pressure, heartbeat intervals, and respiratory activity. Method of BPRS averaging Let X(i) and Y(i) be 2 synchronously recorded time signals, and let us assume that periodic modulations of X(i) cause periodic modulations of Y(i). X(i) might then be called the trigger signal and Y(i) the target signal. Examples for trigger and target signal might be the series of systolic arterial blood pressure values (SBPs) and the series of heartbeat intervals (RRIs) derived from a synchronously recorded electrocardiogram. Systolic arterial blood pressure values are temporally attributed to prior R waves. Series of /$ see front matter 2009 Elsevier Inc. All rights reserved. doi: /j.jelectrocard

2 A. Bauer et al. / Journal of Electrocardiology 42 (2009) Fig. 1. Application of BPRSA to simulated data with and without coupling. The upper 2 rows show trigger and target signal with (left) and without coupling (right). Only the first 500 of seconds are shown. The power spectra in the small boxes were calculated from the signals using Fourier transformation. For the sake of clarity, only the peaks of the power spectra are visualized by bars. The lower graphs show the resulting monovariate and bivariate PRSA signals. The time axis denotes the distance to the anchor (see text for explanations). heartbeat intervals are calculated as distances from the R wave to the following R wave. In the first step, RRIs are identified that occur when SBP increases. SBP ðþnsbp i ði 1Þ Such RRIs are called anchors. Typically, nearly one half of all RRIs are identified as anchors. Alternatively, one may define anchors by comparing averages of T values of SBP, 1 T X T 1 j =0 SBP ði + jþn 1 T X T 1 j =1 SBP ði jþ T can be used to filter out high frequent oscillations of SBP and RRI and thus works as low-pass filter. As it can be shown mathematically, BPRSA is most sensitive for detection of oscillations with frequencies of f = 1/ (2.5T). In the second step, segments of length 2L are defined around the anchors. Anchors close to the beginning or the end of the target signal, where no full surroundings of length 2L are available, are disregarded. L is freely definable and depends on the lowest frequency that shall be visualized. Note that most segments overlap. If we denote the positions (indices) of all regarded anchors by (i ν ), ν =1,, M, the points in segment number ν will be xi ð m LÞ; xi ð m L +1Þ; N ; xi ð m Þ; N ; Xði m + L 2Þ; Xði m + L 1Þ

3 604 A. Bauer et al. / Journal of Electrocardiology 42 (2009) In the third step, the single segments are aligned (ie, phase-rectified) and averaged. The BPRSA signal x (k) is xk ðþ= 1 X M M xi ð m =1 m + kþ for k = L; L +1; N ; 0; N ; L 2; L 1 Modulations of the BPRSA signal can be solely attributed to periodicities in the trigger signal. If there are no interrelationships between trigger and target signal, then the BPRSA signal shows no periodic patterns. A detailed and illustrated description of the BPRSA method can be found elsewhere. 5 The original (monovariate) PRSA method equals the BPRSA with exception of step 1. For computation of the monovariate PRSA method, the anchors are derived from the target signal itself (here: the RRI signal). The monovariate PRSA signal contains all periodicities of the target signal while noise and nonstationarities are eliminated. 3 Application of the BPRSA method to simulated data The performance of BPRSA was tested in simulated data. The simulated trigger signal was composed of 1/f-noise (generated with the Fourier filtering method) as well as of 2 additional quasiperiodicities P 1 and P 2 with frequencies of f 1 = 0.03 Hz and f 2 = 0.11 Hz. To simulate a high degree of nonstationarities, phase jumps were randomly inserted after an average of 0.3 and 0.9 periods, respectively. The simulated target signal 1 also contained 1/f-noise as well as (quasi)periodicity P 1 that was coupled to P 1 of the trigger signal using a fixed time lag of 2 seconds. We also generated a target signal 2 with same statistical properties, but P 1 was not coupled to the trigger signal. Fig. 1 shows BPRSA computation (T = 1) for target signals 1 (left) and 2 (right) using the same trigger signal. The BPRSA transformation of target signal 1 shows a clear oscillation of 0.11 Hz but no other oscillations (left lower panel of Fig. 1). The BPRSA transformation of target signal 2 shows no oscillations at all (right lower panel). The monovariate PRSA transformation of the trigger signal shows 2 oscillations of 0.03 Hz and 0.11 Hz (blue dashed curve in the lower panels of Fig. 1). The lower frequent oscillation in the monovariate PRSA signal precedes the oscillation in the BPRSA signal by 2 seconds. As for monovariate PRSA signals, we suggest quantifying the central part of the BPRSA signal using Haar wavelet analysis with the scale s = T From these analyses and other experiments, 5 it can be concluded that (1) Similar to the monovariate PRSA method, the bivariate PRSA method is capable of detecting (quasi)periodicities in presence of nonstationarities and 1/f-noise. (2) BPRSA only visualizes (quasi)periodicities in the target signal that are coupled to (quasi)periodicities in the trigger signal. Non-coupled periodicities are disregarded. (3) The time lag between the (monovariate) PRSA transformation of the trigger signal and the (bivariate) PRSA transformation of the target signal can be directly seen and allows for conclusions about causalities of regulation processes. A time lag of zero might suggest a higher ranking mechanism causing periodic changes in trigger and target signal as well, whereas a positive time lag might suggest a cause and effect relationship. Application of the BPRSA method to natural data As first application in natural data, we studied the interrelationships between RRI, SBP, and respiratory activity (RespA) in survivors of acute myocardial infarction. Time series were derived from simultaneous 30-minute recordings of high-resolution electrocardiogram (1.600 samples per second), blood pressure, and thorax excursion. Blood pressure was recorded noninvasively using a finger photoplethysmographic device (Finapres, Ohmeda, Madison, WI). Thorax excursion was measured using an elastic belt. Respiratory phase (inspiration/exspiration) was calculated from thorax excursion using Hilbert transformation. The studies were performed within the second week after infarction in supine position and under resting conditions. Fig. 2 shows a typical example of BPRSA analysis in a postinfarction patient. The left panels show the monovariate and bivariate PRSA signals; the right panels show the corresponding power spectra of the PRSA signals. Interrelationship between SBP and RRI Systolic arterial blood pressure value was used as trigger signal and RRI was used as the target signal. Phase-rectified signal averaging and BPRSA were calculated for T = 1 (panels A and B) and T = 4 (panels C and D) to focus on periodicities in the high and low-frequency band. Using T = 1, monovariate PRSA transformation of SBP (red curve of panel A) and bivariate PRSA transformation of RRI (black curve of panel A) both showed a dominant periodicity with a frequency of approximately 0.23/RRI and a time lag of zero. The monovariate PRSA signal of SBP also showed a low amplitude oscillation with a frequency of 0.05/RRI. Using T = 4, the dominant frequency changed to 0.05/RRI in both the monovariate PRSA transformation of SBP and the bivariate PRSA transformation of RRI. The high frequent periodic component disappeared in both signals. Time lag was 1 RRI (Fig. 2; panels A-D). Interrelationship between RespA and RRI Respiratory activity was used as trigger signal, and RRI was used as target signal (T = 1). Both the monovariate PRSA signal of RespA and the bivariate PRSA transformation of RRI showed a periodicity with a frequency of 0.23 Hz. Time lag was 1 RRI (Fig. 2; panels E-F). Discussion Bivariate PRSA is a useful tool for the analysis of interrelationships between 2 or more synchronously

4 A. Bauer et al. / Journal of Electrocardiology 42 (2009) Fig. 2. Application of BPRSA to synchronously recorded time series of RRI, SBP, and RespA (see text for explanations). recorded signals generated by biologic systems. In past, several methods have been proposed for the study of interrelationships between 2 signals including cross-correlation or cross-spectral analysis. However, BPRSA differs in many aspects from standard methods. Bivariate PRSA transforms the target signal into a condensed version only showing (quasi)periodicities that are coupled to the trigger signal. Noncoupled periodicities, artifacts, or noise are eliminated. Conclusions about interrelationships between trigger and target signal can be drawn by comparing the BPRSA transformation of the target signal with the monovariate PRSA transformation of the trigger signal. Of note, PRSA signals have same units as original signals which is in striking contrast to other methods and which has important implications for physiologic interpretations such as estimation of baroreflex sensitivity. Secondly, BPRSA

5 606 A. Bauer et al. / Journal of Electrocardiology 42 (2009) does not require stationarity of the signal. Because nonstationarities are a characteristic of biologic signals, this is a significant advantage over standard methods. In previous work, we have shown that BPRSA is significantly superior to cross-correlation in presence of nonstationarities and noise. 5 Sensitivity of BPRSA for detection of periodicities of certain frequencies can be enhanced by appropriate setting of the parameter T that acts as low-pass filter. 5 Thirdly, the time lag between coupled periodicities in trigger and target signal can be directly seen and calculated (eg, by estimating the maximum cross-correlation for any time shift) that allows for conclusions about causal links between coupled periodicities. Fourthly, compared to other methods (eg, crossspectral analysis), BPRSA is a rather simple mathematical algorithm that makes the method particularly suitable for implementation in devices with limited calculating capacity. As first application of BPRSA to biologic data, we analyzed the interrelationship between RRI, SBP, and RespA. From clinical and experimental studies, it is known that RRI and SBP are coupled by oscillations in the high as well as in the low-frequency band. 6 Oscillations in the highfrequency band have been closely linked to respiratory activity that influences both heart rate and arterial blood pressure. Periodicities in the low-frequency band are believed to be primary due to vasomotor oscillations that modulate heart rate via the baroreflex. Bivariate PRSA visualizes coupled oscillations of RRI and SBP in both high and low-frequency range. For high-frequency oscillations, we observed a time lag of zero suggesting a tertiary mechanism, namely respiratory activity. Low-frequency oscillations of SBP and RRI had a time lag of 1 beat interval that is compatible with a baroreflex mechanism. It should be noted that there was no need for extensive preprocessing of raw data because BPRSA is robust to artifacts. In a recent study assessing baroreflex sensitivity by the transfer function method in heart failure patients, meaningful results could only be assessed in 72% of patients due to high number of ectopic beats. 7 However, it has still to be shown in clinical studies whether BPRSA allows for meaningful assessment of baroreflex sensitivity in presence of noise and artifacts. We foresee a broad spectrum of potential applications for BPRSA. Assessment of baroreflex sensitivity and sinus arrhythmia by BPRSA might be useful for risk stratification after myocardial infarction. Moreover, BPRSA might also be useful for analysis of couplings between other physiologic signals, for example, couplings between RRI and QT interval, rhythms of hormone releases, and many more. References 1. Tyson JJ. Biochemical oscillations. In: Fall C, Marland E, Wagner J, Tyson J, editors. Computational cell biology: an introductory text on computer modelling in molecular and cell biology. New York: Springer; Glass L. Synchronization and rhythmic processes in physiology. Nature 2001;410: Bauer A, Kantelhardt JW, Bunde A, Malik M, Schneider R, Schmidt G. Phase-rectified signal averaging detects quasi-periodicities in nonstationary data. Physica A 2006;364: Bauer A, Kantelhardt JW, Barthel P, et al. Deceleration capacity of heart rate as a predictor of mortality after myocardial infarction: cohort study. Lancet 2006;367: Schumann AY, Kantelhardt JW, Bauer A, Schmidt G. Bivariate phaserectified signal averaging. Physica A 2008;387: Robbe HW, Mulder LJ, Ruddel H, Langewitz WA, Veldman JB, Mulder G. Assessment of baroreceptor reflex sensitivity by means of spectral analysis. Hypertension 1987;10: Pinna GD, Maestri R, Capomolla S, et al. Applicability and clinical relevance of the transfer function method in the assessment of baroreflex sensitivity in heart failure patients. J Am Coll Cardiol 2005;46:1314.