Fluffy clouds. craggy coastlines. and branching coral formations are all examples of fraetals. ..... M. R. Bucnaurtrt at A. J. MAHDELL. 1954. Nonlinear dynamics in ...
Fractal Electrodynamics of the Heartbeat" A R Y L. GOLDBERGER Cardiovascular Division Beth Israel Hospital and Harvard Medical School Boston, Massachusetts 0221 5
INTRODUCTION We have proposed that the physiological function of the heart on a number of different structural and functional levels of organization is related to its fractal geometry and fractal Fractal objects are most simply defined by their self-similar ar~hitecture.'.~Inspection of a fractal object reveals multiple layers of detail such that the small-scale structure resembles the larger scale form. As a consequence, fractal structures do not have a single (characteristic) scale of length. Fluffy clouds, craggy coastlines, and branching coral formations are all examples of fractals. Many anatomic structures also appear to be fractal.'-' Examples include the folds of the brain and bowel, the tracheobronchial tree, branching vascular and neural networks, and the pancreatebiliary duct system. The fractal-like structure of the heart has been described previously.I4 The coronary arterial and venous systems are self-similar networks. The mitral and tricuspid valves are anchored to the papillary muscles by branched connective tissue structures, the chordae tendineae. Certain of the cardiac muscle bundles also have a fractal-like branching structure. Of particular interest to electrophysiologists is the fractal-like appearance of the His-Purkinje network that transmits the depolarization wave from atria to ventricles.' To review briefly, the normal pacemaker of the heart is the sinoatrial (sinus) node, located in the upper part of the right atrium. The heart rate is regulated by the firing rate of this spontaneous pacemaker. Sinus-node depolarization, in turn, activates the atria, represented by the P wave on the electrocardiogram. Electrical stimulation of the atria is followed by activation of the AV node, located at the anatomic junction between atria and ventricles. AV nodal activation is followed by rapid depolarization of the specialized conduction fibers that make up the ramifying His-Purkinje network. The trunk of this tree is the His-bundle that bifurcates into the left and right main bundle branches. The bundle branches undergo a series of further bifurcations down to the level of the microscopic Purkinje fibers (FIG. 1). The final phase of this sequence is depolarization of the ventricular myocardium, represented by the QRS complex on the electrocardiogram. 'This work was supported in part by National Heart, Lung and Blood Institute Grant R01 HLI 72, in part by National Aeronautics and Space Administration Grant NAG2-514, and in part by the G. Harold and Leila Y. Mathers Charitable Foundation. 402
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FIGURE 1. A, the His-Purkinje system is a fractal-like tree. Depolarization of the myocardium via this arborizing network generates the QRS waveform of the electrocardiogram. B, the power spectrum (inset) of the QRS waveform is broad, with a Iff-like (inverse power-law) distribution. Fundamental frequency = 7.8 Hz.(Adapted From Goldberger et al.' Reproduced by permission.)
FRACTAL DEPOLARIZATION OF THE HEART We have previously described the effect of this fractal-like conduction network on the electrodynamics of normal cardiac depolarization.' This subject is important because electrical activation of the ventricular myocardium via the His-Purkinje tree is the structural substrate of cardiac electrical stability. This physiological stability is paradoxically achieved by a process involving the subtle desynchronization (decorrelation) of electrical impulses. What happens to the cardiac electrical impulse as it traverses the His-Purkinje
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tree? At each branch of the tree, the impulse splits, giving rise to two impulses. As a result, the single stimulus a t the beginning of the network (His-bundle) will be shattered into multiple impulses a t the Purkinje-myocardial interface. Furthermore, because the fractal-like conduction tree is asymmetric and irregular, the arrival times of these impulses a t the myocardium will be decorrelated. What are the statistics of this decorrelation process? Because the depolarization process is effected by a fractal network having no single length scale, there will not be any characteristic decorrelation rate. Instead, to describe the decorrelation cascade, an infinite series of terms is needed.’ Each term in this series gives the probability of a higher decorrelation rate contributing to the overall process. Mathematically, it can be shown that for a fractal process, the distribution of decorrelation rates based on this infinite series will take an inverse power-law form. Since the frequency spectrum of the electrocardiographic QRS complexes will depend in part on the statistics of the arrival times of impulses a t the myocardium, we predicted that the frequency spectrum of normal Q R S complexes would have an inverse power-law (l/f-like) distribution. The fractal model of cardiac activation also predicts a loss of higher frequency QRS components in circumstances (e.g., bundle branch blocks or ventricular ectopic beats) where the fractal pattern of ventricular depolarization is disrupted. Support for this fractal hypothesis of ventricular activation is provided by two sets of observations. First, by subjecting Q R S complexes of healthy individuals to Fourier analysis, we showed that the power spectrum of the normal Q R S does, in fact, have the anticipated broad-band, inverse power-law spectrum’ (FIG.1). The second line of support for the fractal hypothesis of ventricular depolarization comes from mathematical modeling studies performed by S. Abboud and coworkers at Tel Aviv U n i ~ e r s i t y These .~ investigators have developed a discrete-element model of the ventricular myocardium and cardiac conduction system. The most recent version of the model consists of about 560,000 cubic “muscle” cells arranged in two prolate spheroids. Conduction occurs from each cell to its six nearest neighbors. A crucial feature of the model is the presence of a self-similar conduction system. Activation of the myocardial network occurs via this fractal-like conduction system, which has a higher conduction velocity than the “muscle” cells. A simulated electrocardiogram is then obtained by measuring the dipole potential generated from adjacent “excited” and “resting” elements. Several important observations based on simulations with this fractal conduction system have been reported by the investigators. I . Activation of the system via a fractal conduction tree produces physiologic “QRS” complexes with a broad-band, 1/f-like spectrum (S. Abboud, personal communication), 2. In contrast, activation of the system without a fractal conduction system produces “pathologic” widening of the Q R S complex with marked attenuation of higher frequency components. This pathology is reminiscent of the patterns seen clinically with bundle branch blocks or ventricular ectopic beats. 3. Slowing of propogation velocity in selective regions of the conduction system or “myocardium” results in the appearance of low-amplitude, high-frequency QRS components that resemble so-called lare potentials. Late potentials have been reported in patients with myocardial infarction, and appear to be a marker of increased risk of ventricular tachyarrhythmias.’
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Computer models of this kind may help elucidate the mechanism of “reentrant” tachyarrhythmias and the role, if any, of the His-Purkinje system in initiating and sustaining ventricular tachycardia and ventricular fibrillation. These models may also suggest new diagnostic measurements to detect subtle alterations in QRS spectral content as a marker of myocardial damage or conduction system dysfunction. The role of the fractal His-Purkinje network in generating electrocardiographic pulses with a broad spectrum exemplifies the notion of fractal dynamics (fractal time). Just as a fractal structure (e.g., the His-Purkinje tree) lacks a well-defined length scale, a fractal process (ventricular depolarization) cannot be characterized by a single-frequency component (time scale). Instead, fractal processes are broad band, that is, they are comprised of many frequencies. Furthermore, as illustrated previously, the Fourier spectrum of a fractal process typically has a l/f-like distribution in which there is an inverse relationship between spectral power and frequency.
FRACTAL HEART-RATE DYNAMICS Up to now, this review had focused on the purported fractal mechanism of normal ventricular depolarization. W e have also proposed that the fractal concept may be
NORMAL SINUS RHYTHM TIME SERIES
FIGURE 2. Normal sinus rhythm in healthy subjects, even at rest, is not metronomically regular, but shows erratic fluctuations. There is no characteristic scale of time for this healthy variability: fluctuations on different time scales are self-similar. As a consequence, the spectrum is broad, with a l/f-like distribution (see FIG. 3A). R-R interval = interbeat interval. (Adapted from Goldberger and West.3 Reproduced by permission.)
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FIGURE 3. Fractal heart-rate variability may be altered in disease states Top panels show representative heart-rate time series. Bottom panels give amplitude spectra. A, data set from healthy subject shows considerable variability represented by a broad, llf-like spectrum. B and C, data sets from patients with severe left ventricular dysfunction and congestive heart failure show loss of normal fractal variability with either low-frequency oscillations (B) or a flat pattern characterized by overall loss of variability (C). (Adapted from A.L. Goldberger. 1987. Nonlinear dynamics, fractals, cardiac physiology and sudden death. I n Temporal Disorder in Human Oscillatory Systems, L. Rensing, U. an der Heiden, and M.C. Mackey, Eds. SpringerVerlag. New York/Berlin. Reproduced by permission.)
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applied, not only to depolarization of the ventricles, but also the beat-to-beat regulation of heart A common misconception among dynamicists and clinicians is that the normal heart beats with clocklike regularity. This notion that normal sinus rhythm is strictly regular is incorrect, however. When beat-to-beat heart rate is carefully measured, it is apparent that the healthy heartbeat is quite erratic, even under resting conditions (FIGS.2 and 3). This variability, however, is subjectively imperceptible and is difficult to assess by routine clinical examination. These fluctuations of the normal heartbeat on short and longer time scales are related to the competing influences of the two branches of the autonomic nervous system. The sympathetic branch is acceleratory, while parasympathetic (vagal) stimulation has a slowing effect on heart rate. The two branches of the autonomic nervous system appear to interact in a nonlinear manner both under healthy conditions and with pathologic states. Most analyses of heart-rate variability have focused on relatively periodic oscillations associated with breathing, blood pressure control, and other physiological control systems. These oscillations appear as spectral peaks when a heart-rate time series is subjected to Fourier analysis. Under physiologic conditions, however, these peaks only account for a fraction of overall heart-rate variability. In fact, as first reported by Kobayashi and Musha," these peaks are superimposed on a broad-band-type spectrum with a 1/f-like (inverse power-law) distribution (FIG. 3A). To explain this type of 1/f-like spectrum, we have proposed that normal heart-rate variability is regulated by a nonlinear feedback network that generates fluctuations across multiple orders of temporal magnitude, ranging from hours or longer to seconds or less. Furthermore, these fluctuations are self-similar. That is, the chaotic-appearing variations apparent on longer time scales are similar to the fluctuations on shorter time scales, although the amplitude of the higher frequencies is lower (FIGS. 2 and 3A). The hypothesis that the healthy heartbeat is fractal raises several questions. What are the nonlinear equations of motion that actually govern the control system that regulates heart-rate variability? Presumably, the 1/f-like spectrum arises as a consequence of sympathetic-parasympathetic interactions. The detailed mechanism of this fractal process remains undefined, however. Additional questions concern the effects of aging and pathologic states on fractal heart-rate dynamics. Preliminary data from our laboratory indicate that aging is associated with a reduction in fractal dimensionality."
LOSS OF FRACTAL DYNAMICS IN SUDDEN DEATH SYNDROMES We have also reportedI2the effects of cardiac pathologies on heart-rate dynamics in subjects at high risk of sudden death. We have analyzed data from patients with severe congestive heart failure and from patients who actually sustained a fatal or near-fatal arrhythmia while wearing a portable electrocardiographic monitor. Two abnormal heart-rate patterns are observed in these high-risk patients that differ markedly from the fractal pattern just described.'* One dynamical pattern we term the oscillatory pattern because it is characterized by relatively low-frequency (usually 0.01-0.06 Hz) oscillations in heart rate. These oscillations are represented by a sharply peaked spectral pattern. The other dynamical pattern we term the p a t pattern because it is
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characterized by a marked reduction in beat-to-beat variability, leading to an overall reduction in spectral power. These patterns are reminiscent of the abnormal heart-rate dynamics previously described in fetal distress ~ y n d r o m e . ' ~ Why do certain patients at high risk of sudden death show abnormal sinus-rhythm heart-rate dynamics? Patients with severe congestive heart failure are a t particularly high risk of sudden cardiac death. Congestive heart failure, in turn, is associated with increased resting levels of sympathetic tone and with decreased parasympathetic tone, factors that would simultaneously reduce heart-rate variability and decrease the threshold for ventricular fibrillation, the arrhythmia most commonly associated with sudden death. Loss of heart-rate variability in certain high-risk patients may therefore be an epiphenomenon, a secondary effect of other factors such as autonomic perturbations that may be the primary causes of electrical instability. Low-frequency oscillations in heart rate may also be secondary markers of increased risk of sudden cardiac death. These low-frequency heart-rate oscillations are commonly recorded in patients with severe congestive heart failure, a syndrome that also is associated with low-frequency oscillations in respiratory rate and amplitude (Cheyne-Stokes breathing). We have reported a correlation between Cheyne-Stokes breathing oscillations and the low-frequency heart-rate oscillations observed in some patients at high risk of sudden death.I4 Whether the periodic heart-rate dynamics are actually entrained by respiration in these patients or whether pathologic low-frequency oscillations generated in the brainstem simultaneously entrain both heart rate and breathing at the same frequency remains uncertain. Regardless of their precise mechanism, oscillations in heart rate in patients with organic heart disease may also be associated with oscillations in neuroautonomic, ionic, bioelectric, or other parameters that could be primarily responsible for destabilizing cell membranes, promoting sustained arrhythmias. Of interest, recent data from our laboratory indicate that ferrets given toxic doses of cocaine, a drug with prominent autonomic effects, show patterns of heart-rate dynamics (loss of variability and low-frequency oscillations) similar to those previously described in high-risk patients with heart d i ~ e a s e . ' ~
CONCLUSIONS This paper has briefly reviewed evidence supporting the relevance of fractals to cardiac electrodynamics. The fractal concept has been applied specifically at two levels of cardiac function: ventricular depolarization and heart-rate regulation. In the first instance, that of ventricular depolarization, a direct link can be established between a fractal structure (the His-Purkinje conduction tree) and fractal function (depolarization pulses with a broad, llf-like spectrum). In the second instance, that of heart-rate variability, there is no evident physical fractal structure that underlies the healthy variability of sinus rhythm?.'6 As noted, the mechanism responsible for the self-similar fluctuations in heart rate across many time scales remains to be defined. Of considerable clinical interest are observations that disruption of physiologic fractal function (perturbation of His-Purkinje depolarization or alterations in beat-to-beat heart-rate control) may be associated with increased risk of sudden death.
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REFERENCES 1. GOLDBERGER, A. L., V. BHARGAVA, B. J. WEST & A. J. MANDELL. 1985. On a mechanism of cardiac electrical stability: The fractal hypothesis. Biophys. JJ 4 8 525-528. 2. GOLDBERGER A. L. & B. J. WEST. 1987. Applications of nonlinear dynamics to clinical cardiology. Ann. N.Y. Acad. Sci. 504: 195-212. A. L. & B. J. WEST.1987. Fractals in physiology and medicine. Yale J. Biol. 3. GOLDBERGER, Med. 6 0 421-435. 1987. Physiology in fractal dimensions. Am. Sci. 4. WEST, B. J. & A. L. GOLDBERGER. 7 5 354-365. B. B. 1982. The fractal geometry of nature. Freeman. New York. 5. MANDELBROT, 6. FEDER,J. 1988. Fractals. Plenum. New York. O., D. SADEH & S. ABBOUD.1990. Simulation of late potentials using a 7. BERENFELD, computerized three dimensional model of the heart’s ventricles with fractal conduction system. Computers in Cardiology. In Press. M. G. 1981. Identification of patients with ventricular tachycardia. Circulation 8. SIMSON, 6 4 235-242. A. L. & D. R. RIGNEY.1988. Sudden death is not chaos. In Dynamic 9. GOLDBERGER, Patterns in Complex Systems. J. A. S. KELSO,A. J. MANDELL, and M. F. SHLESINGER, Eds.: 248-264. World Scientific Publishers. Teaneck, N.J. M. & T. MUSHA.1982. l/f fluctuation of heartbeat period. IEEE Trans. 10. KOBAYASHI, Biomed. Eng. BME 2 9 456457. 1989. Spectral 11. LIPSITZ,L. A,, J. MIETUS,D. R. RIGNEY,G. MOODY& A. L. GOLDBERGER. characteristics of heart rate variability before and during postural tilt: Relationships to aging and risk of syncope. Circulation. In press. A. L., D. R. RIGNEY,J. MIETUS,E. M. ANTMAN& S. GREENWALD. 1988. 12. GOLDBERGER, Nonlinear dynamics in sudden cardiac death syndrome: Heartrate oscillations and bifurcations. Experientia 4 4 983-987. R. K. & T. J. GARITE.1981. Fetal heart rate monitoring. Williams & Wilkins. 13. FREEMAN, Baltimore. A. L., L. J. FINDLEY, M. R. BLACKBURN & A. J. MANDELL.1984. Nonlinear 14. GOLDBERGER, dynamics in heart failure: Implications of long-wavelength cardiopulmonary oscillations. Am. Heart J. 107 612-615. B. S., J. P. MORGAN,J. MIETUS,G. B. MOODY,D. R. RIGNEY& A. L. 15. STAMBLER, GOLDBERGER. 1989. Distinctive heart rate dynamics precede cocaine-induced sudden death. J. Mol. Cell. Cardiol. Suppl. I1 21: S16. A. L., D. R. RIGNEY& B. J. WEST. 1990. Chaos and fractals in human 16. GOLDBERGER, physiology. Sci. Am. 262: 42.
