The processing of beat-to-beat time intervals by multipoles

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Copyright © IFAC Modelling and Control in Biomedical Systems, Melbourne, Australia, 2003

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THE PROCESSING OF BEAT-TO-BEAT TIME INTERVALS BY MULTlPOLES

M. Lewkowicz(l), J. Levitan(l), K. Saermark(2) and P.E. Bloch Thomsen(J)

(/)Dept. ofPhysics, College ofJudea and Samaria, Ariel 44837, Israel (2) Dept. ofPhysics, The Technical University ofDenmark, DK 2800 Lyngby, Denmark (3) Dept. ofCardiology, Gentofte Amtssygehus, Copenhagen University Hospital, DK 2900 Hellerup, Denmark

Abstract: We present a new method to describe time series with a highly complex time evolution as the beat-to-beat time series. The time series is projected into a two dimensional phase space plot which is quantified in terms of a multipole expansion where every data point is assigned an unit mass. The multipoles provide an efficient characterization of the original time series. Copyright © 2003 IFAC Keywords: Complex systems, Data handling systems. Geometrical approaches. Information analysis, Phase space, Pulse sequences, Pulse signals.

I. INTRODUCTION prognosis and risk of subsequent heart disease. Recognition of early dysfunction is therefore important. In overt heart disease, autonomic imbalance is of significant importance in the pathophysiology of sudden cardiac death. Abnormal autonomic balance is an important prognostic factor. In heart failure this control system may be significantly deranged. Heart Rate Variability has been in use for more than two decades as a prognostic indicator of risk associated with a large variety of diseases. For example, independent of other established risk factors, depressed HRV has been shown to be a powerful predictor of cardiac events after myocardial infarct. It is therefore important to establish a measure of HRV and to classify the HRV of different pathological cases in order to discriminate the healthy HRV from that of patients at risk (Moelgaard, 1995; Malik, 1998).

Measurement of heart rate (HR) and evaluation of its rhythmicity have been used for a long time as a simple clinical indicator (Moelgaard, 1995). The main adaptive regulation of the sinus node function, and thereby the HR, is exerted by the autonomic nervous system. The sinus node of the heart is a major organ in the integrated control of cardiovascular function. HR abnormality may therefore be an early or principle sign of disease or malfunction. Research from the last decade indicates that a quantification of the discrete beat-to-beat variations in HR - heart rate variability (HRV) may be used more directly to estimate efferent autonomic activity to the heart and the integrity of this cardiovascular control system (Furian et al., 1990). The finding that power spectral analysis of HRV could be used as a marker of cardiac autonomic outflow to the heart was considered a breakthrough for clinical research (Akselrod et al., 1981; Pomeranz et aI., 1985).

Up to now, two different measures of HRV analysis have been applied in the medical research: a) ScaIedependent measures as SDNN (the standard deviation of the beat-to-beat interval time series); the standard deviation of the multiresolution wavelet transform as used by Turner, et al. (1998a); the

Autonomic dysfunction is an important factor in a number of conditions. In diabetes, an abnormality in autonomic nervous function signals an adverse 199

are measured along and vertical to the diagonal of the original phase-space plot. In this principal coordinate system the off-diagonal terms of the quadrupole tensor vanish. Assuming to a first approximation Gaussian distributions for the projection of the data points along the x and y axis with cr. and cry as standard deviations, the two diagonal quadrupole moments are

standard deviation of the detrended time series as defined by Ashkenazy, et al. (1998), and b) scaleindependent measures (as the scaling exponent of the detrended fluctuations (DFA) as introduced by Peng, et at. (1995». Both methods have shown limited applicability individually (Numes Amaral et al., 1998; Thumer et al., 1998b; Saermark et al., 2000), but rather a combination of both is necessary (Ashkenazy et al., 2001).

= 2a x 2 - a ... 2 Qyy = 2a y 2 -a x 2 Qrc

In clinical medicine, the dynamics of the beat-to-beat (RR) time series is commonly represented by a phase-space (or recurrence) plot, where each R-R interval is plotted against the previous one. Huikuri et al. (1993) used among other methods phase portraits in order to investigate abnormalities in beatto-beat dynamics of the heart beat before spontaneous onset of life-threatening ventricular tachycardia in patients with prior myocardial infarction. The classification of the phase-space plots is traditionally performed by visual inspection and semi-quantitative analysis of the various measures describing the features of the plot, as the length or width (Huikuri et al., 1993; Kamen et al., 1995; Kamen et aI., 1996). But, as pointed out by Malik (1998), this approach ignores the varying density of points leading to similar plots due to hearts with very different dynamics.

(I) Whereas in three dimensions a homogenous sphere has only zero quadrupole moments, its projection on the plane has a Gaussian mass distribution along the radial direction. If the projection is on the x-y plane one has a phase-space plot with a circular shape where the density of points decreases with increasing distance from the origin with cr. =cry. In this case both principal quadrupole moments are positive.

A vanishing quadrupole moment, for example

ax

obtained

for

reference

ellipse

=hay in

,

two

3 2 Q-_ = -a 2 x .Varying the

Qyy,

is

which defines a dimensions;

,yo V x

j,yo y 'V

here

ratl·o , one

obtains more or less elongated ellipses along the x axis or y axis, depending on

2. THE MULTIPOLE METHOD

a xI

lay

> "2 or < "2.

The quadrupole moments are hence measures of the overall distributions of the data points and thus of the beat-to-beat intervals. In the literature 'healthy' HRV phase-space plots are described as cigar shaped along the diagonal (for example (Kamen et al., 1996; Malik, 1998», hence a negative Qyy and a positive Q.. are anticipated.

We recently presented a new method (Lewkowicz et aI, 2002), that we named the Multipole Method, which draws its data from the phase-space plot. This method is effective in describing time series with a highly complex time evolution. The time series is projected into a two-dimensional phase-space plot which is interpreted as a two-dimensional body where each data point is assigned a unit mass. The distribution of points is expressed by the various moments known from potential theory, where they are called the multipoles. These multipole moments serve as measures or parameters describing the features of the plot, and hence yield information about the underlying time series. By calculating the various moments for recordings from a healthy control group the normal ranges of the various moments are established.

For a symmetrical distribution as the Gaussian distribution the octupole moments vanish. They are related to the skewness of the distribution. A positive skewness along the diagonal reveals that the heart beats for extended periods with a low pulse rate (large RR-intervals) and vice versa. A positive skewness along the normal to the diagonal (positive y-skewness) indicates a slow decrease and a fast increase in the heart rate which (a sign of impaired sympathetic and\or parasympathetic nervous system). Negative skewness on the y-axis, implies a slowly increasing and fast decreasing pulse.

Every single multipole parameter describes different features in the distribution of points in the phasespace plot and thus has its specific consequence on the evaluation of the state of the cardiac function. Explicit expressions for the various moments can be found in the literature; here we outline a possible interpretation.

The hexadecapole moments are rather cumbersome to interpret, hence we choose to employ the kurtosis, which is related to the hexadecapole moments. The kurtosis is negative for a flat topped distribution, vanishes for a Gaussian distribution and is positive for a sharp peaked distribution. This implies that two phase-space plots both with similar values for the quadrupole moments and for the octupole moments may have very different values for the kurtosis and their hexadecapole moments. For increasing positive values of the kurtosis the distribution has increasing sharper peaks and more enhanced tails. There is a

The monopole represents the total 'mass' of the plot, i.e. the number of data points. The gravitational dipole moment vanishes by choosing the origin of the coordinate system in the centre of mass. The quadrupole moments are thus the first significant moments. The x and y coordinates of the data points

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large concentration of data points near the average value indicating that the heart beats for extended periods with a low variability and also a large concentration of data points far away from the average indicating that the heart beats for extended periods with a very high variability.

to the octupole moment Tyyy on the y-axis. The main result is that a high concentration of phase space points on the positive y-axis (positive y-skewness) indicates a slow decrease and a fast increase in the heart rate. Negative skewness on the y-axis, signifying a slowly increasing and fast decreasing pulse, is indicative for a well functioning heart function. Fig. 3 shows the separating power of the octupole along the x-axis.

3. APPLICATION TO THE DIAMOND STUDY We shall in the following demonstrate how the multipoles perform as predictors for mortality in the group of survivors after acute myocardial infarction in the Diamond study (Huikuri et al., 2000). In this study 446 survivors of acute myocardial infarction (AMI) were enrolled. Different methods of analyzing HRV were compared with respect to predictive power of death after AMI. HRV was obtained from consecutive R-R interval from 24 hours ECG recording 5-10 days after AMI. The mortality was 25.6 % after a follow - up of 685 ± 360 days (114 died). By calculating the quadrupole moments for all 446 recordings we achieve a first separation into a high risk group and into a low risk group. This first separation has a better overall predictive accuracy than the standard deviation of the RR intervals (SDNN), which is the conventional HRV marker. In Fig. I we show that the mortality is decreasing with increasing values of Qyy.

~---.---r------.--------,----,

2S

Fig. I The quadrupole Qyy as a risk factor. The patients are grouped into groups of 50 according to decreasing Qyy.

Fig. 2. Phase-space plot for survivor (2a, upper) and non survivor (2b, lower). Both plots have similar quadrupole moments.

In order to appreciate the significance of the octupole moment we show in Fig. 2 the phase space plot from two of the recordings from the Diamond study. They have approximately the same value for the quadrupole moment but very different octupole moments. Fig. 2a shows the phase space plot of a survivor with the highest concentration of data points on the positive part of the x-axis, with the interpretation that the heart beats for extended periods with a low pulse rate (large RR-intervals), where as Fig. 2b shows the plot of a deceased individual with the larger part of data points concentrated on the negative part of the x-axis, i.e. the heart is beating with a high pulse rate for extended periods. A similar analysis can be applied

The additional octupole moments add very little to the overall predictive accuracy in the Diamond study and therefore are excluded here. Instead of seeking out the significant ones among the 16 hexadecupole moments, we used the kurtosis as mentioned above which is a powerful risk marker in the Diamond study. Combining by optimization the various moments into one parameter we obtained an improved predictive accuracy. Fig.4 shows the predictive potential.

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System by a New Analysis of Heartbeat Intervals. Fractals 6, 197 Ashkenazy Y. et a!. (2001). Scale Specific and Scale Independent Measures of Heart Rate Variability as Risk Indicators. Europhysics Lett. 53, 709 Furlan R. et a!. (1990). Continuous 24-hour assessment of the neural regulation of systemic arterial pressure and RR variabilities in ambulant subjects. Circulation 81, 537 Huikuri H.V. et a!. (1993). Abnormalities in Beat-to-. Beat Dynamics of Heart Rate Before the Spontaneous Onset of Life-Threatening Ventricular Tachyarrhythmias in Patients With Prior Myocardial Infarction. Circulation, 93, 1836 Huikuri H.V., Makikallio T.H., Peng C.K., Goldberger A.L., Hintze U. and M011er M. (2000). Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after acute myocardial infarction. Circulation, 101,47 Kamen P.W. and Tornkin A.M. (1995). Application of the Poincare plot to Heart Rate Variability: A New Measure of Functional Status in Heart Failure. Aust NZ J Med, 25, 18 Kamen P.W. and Tornkin A.M. (1996). Poincare plot of heart rate variability allows quantitative display of parasympathetic nervous activity in humans. Clin Sci, 91, 201 Lewkowicz M., Levitan J., puzanov N., Schnerb N.and Saermark K. (2002). Description of complex time series by multipoles. Physica A, 311,260 Malik M. (1998). Heart rate variability. Current Opinion in Cardiology, 13, 36 Moelgaard H. (1995). "24-hour Heart Rate Variability. Methology and Clinical Aspects". Doctoral Thesis, University of Aarhus. Nunes Amaral L.A. et al. (1998). Scale-Independent Measures and Pathologic Cardiac Dynamics. Phys. Rev. Lett.81,2388 Peng C.K. et a!. (1995). Quantification of scaling exponents and crossover phenomena In nonstationary heartbeat time series. Chaos 5, 82 Pomeranz B. et a!. (1985). Assessment of autonomic function in humans by heart rate spectral analysis. Am. J. Physiol., 248, 151 Saermark K. et a!. (2000). Comparison of recent methods of analyzing heart rate variability. Fractals 8, 4 Thumer S. et a!. (I 998a). Multiresolution Wavelet Analysis of Heartbeat Intervals Discriminates Healthy Patients from Those with Cardiac Pathology. Phys. Rev. Lett. 80, 1544 Thumer S. et al. (1998b). Receiver-OperatingCharacteristic Analysis Reveals Superiority of Scale-Dependent Wavelet and Spectral Measures for Assessing Cardiac Dysfunction. Phys. Rev. Lett. 81, 5688

Fig. 3 The octupole Txxx as a risk factor. The patients are grouped into groups of 50 according to decreasing Txxx.

Fig. 4 The combination of the multipoles as a risk factor. The patients are grouped into groups of 50 according to the optimized parameter.

4. CONCLUSION The series of RR intervals is an excellent example of a non-stationary and non-linear time series with a very complex behaviour. It seems reasonable to expect that the regulation of the heart rhythm which is a very complex mechanism due to its dependence on many subsystems in the body can be described optimally only by a method which has a diversity of different parameters describing partly different behaviours of those subsystems. The outlined method performs better than the SDNN and the DFA which were superior among the methods compared by Huikuri, et al. (2000). We conclude that the Multipole Method extracts information both in the frequency domain as well as in the time domain, and therefore performs better in prognostics than the traditional HRV methods, which are imbedded in one of the two domains. The multipoles moments differ crucially from the SDNN which does not include any time-ordering (shuffeling the RR intervals will result in almost the same value for SDNN), while the multipoles due to the very construction of the phase-space plot bear intrinsic time dependence. REFERENCES Akselrod S. et af. (1981). Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to-beat cardiovascular control. Science 213, 220 Ashkenazy Y. et af. (1998). Discrimination of the Healthy and Sick Cardiac Autonomic Nervous 202