Annals of the New York Academy of Sciences Volume 504 Issue None 1987 [Doi 10.1111_j.1749-6632.1987.Tb48733.x] ARY L. GOLDBERGER; BRUCE J. WEST -- Applications of Nonlinear Dynamics

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Applicationsof Nonlinear Dynamics toClinical Cardiology”

A RY L . GOL DBERGE R~ N D BRUCE J . WESTbCardiovascular Division

Beth I srael Hospitaland

Harvard Medical SchoolBoston, Massachusetts 02215

‘Division of Applied Nonlinear ProblemsLa J olla Institute

La J olla. California 92037

INTRODUCTION

What does nonlinear dynamics offer the clinician with regard to understanding

normal physiology and elucidating mechanisms of disease and sudden cardiac death?Over the past few years, in collaboration with others (includingV. Bhargava and A. J .Mandell), we have been interested in pursuing the relevance of nonlinear concepts tobedside medicine.’-” A lthough our initial focus was on cardiovascular dynamics, it

soon became clear that the models we were adapting had, potentially, more generalapplications, and the possibility of certain physiologic and pathologic universalities was

suggested. In this paper, we review our initial explorations in this area.The motivation for these investigations was a desire to address a variety of

questions whose only apparent unifying theme was their “resistance” to traditionalmodels. Among these questions were the following:

( 1) What are the mechanisms of cardiac electrical stabil ity?(2) How is a complex structure such as the His-Purkinje system organized and

does it share any similarities to other irregular branching systems seen, forexample, in the lung, the bil iary system, and the vascular network?

(3) How can one model the abrupt appearance of periodic behavior associated with

certain perturbations, for example, Wenckebach periodicity at a critical atrialpacing rate and electrical alternans with pericardial tamponade?

(4 ) A re there ordering principles to the apparently stochastic fluctuations inphysiologic variables such as cardiac interbeat intervals in active healthysubjects and how does one quantify such variability?

(5) A re there new approaches to monitoring patients at risk for cardiac death?

While answers to these questions are, at present, preliminary, we believe thatnonlinear dynamics does suggest new ways of approaching such problems. We begin in

‘This work was supported in part by a grant from NASA /A mes Research Center, MotTettField, California94035.

195



196 ANNAL S NEW YORK ACADEMY OF SCIENCES

the first section, therefore, by discussing nonlinear aspects of normal cardiac anatomyand physiology. In the second section, we describe nonlinear pathophysiology, and inthe third section, we discuss future directions in the clinical application of these

nonlinear strategies. The final section provides a brief summary of our findings.

NONL INEAR CARDIAC PHYSIOLOGY

Supraventricular Cond uction: SA-AV Node Interactions

Under physiologic conditions, the normal pacemaker of the heart is the SAnode-a collection of cells with spontaneous automaticity located in the right atrium.The impulse then spreads through the atrial muscle (triggering atrial contraction).

According to the traditional viewpoint, the depolarization wave then spreads throughthe AV node (junction) and down the His-Purkinje system into the ventricles. The keypremise of this model is that the AV node functions during normal sinus rhythm as apassive conduit for impulses originating in the SA node, and that the intrinsicautomaticity of the AV node is suppressed during sinus rhythm.

An alternative viewpoint, suggested by van der Pol and van der Mark in theirclassic description of the relaxation oscillator,” is that the AV node functions as anactive oscillator and not simply as a passive resistive element in the cardiac electricalnetwork. An active role of the AV node is supported by the clinical observation that,

under certain conditions, the sinus and AV nodes may become functionally dissociatedsothat independent atrial (P) and ventricular ( QRS)waves are seen on the electrocar-diogram (AV dissociation). Further, if theSA node is pharmacologically s~ppressed’ ~or ablated, then the AV node will assume an active pacemaker role. The intrinsic rateof this AV nodal pacemaker is about two-thirdsof that of the SA node in dogs” andpossibly in man.14

In contrast to the traditional passive conduit theory of the AV node, nonlinearanalysis suggests that the SA and AV nodes may function in an active and interactiveway, with the faster firing SA node appearing to entrain the AV node.‘ This

entrainment should be bidirectional, not unidirectional, with the SA node bothinfluencing and being influenced by the AV node. Previous nonlinear ofthe supraventricular cardiac conduction system did not explicitly incorporate thisbidirectional type of interaction.

To simulate bidirectional SA-A V node interaction, we4 adapted a computer modelof coupled nonlinear oscillator^'^The circuit includes two tunnel diodes--electroniccomponents with the same type of nonlinear voltage-current relationships found inphysiological pacemakers with hysteresisd properties (FIGURE ). In this circuit, theoscillators will be free running (uncoupled) whenR -0.Coupling occurs whenR #0.

Over a critical range of applied voltage V, withR #0,we observe that the oscillators

manifest 1:l phase-locking at a rate that is faster than the intrinsic frequency of theAV oscillator and slower than the intrinsic frequency of the SA oscillator (FIGURE).

To simulate the effects of driving the right atrium at increasing rates with an externalpacemaker, an external voltage of variable frequency was applied to the “SA” node

k mi l ar hysteresisloops,characteristic of dissipative nonlinear systems, describe the pressure-volume relations in the ventricles.



COLDBERCER & WEST NONLINEAR CARDIOLOGY 197

oscillator branch of the circuit (F IGURE ). Externally “pacing” the SA oscil latorresults in the appearance of a 3:2 Wenckebach-type periodicity over a critical rangeof

driving frequencies. Furthermore, when the system isdriven beyond a critical point,

2:1 “block” occurs with only every other SA pulse followed by an AV pulse.While this type of equivalent ci rcuit model is not unique, it does lend support to a

nonlinear c~ncept’ ~~’ ’ . ’ ~f cardiac conduction. In particular, the model is consistentwith the viewpoint that normal sinus rhythm involves a bidirectional interaction (1 : I

phase-locking) between coupled nonlinear oscillators that have intrinsic frequencies inthe ratio of about 3:2. Furthermore, the dynamics suggest that AV Wenckebach and

2:1 block, which have traditionally been considered purely as conduction disorders,

P0.6 1

0.2

0*4kI0.02. 00 0.0

CURRENT 1112 )

FIGURE I . ( A ) Analog circuit used to model bidirectional interaction of nonlinear pacemakers(tunnel diodes) with intrinsic frequencies in ratio of 3/2, similar to SA and AV nodes; ( B)Nonlinear (hysteretic) voltage-current responses across diodes when R =0 (uncoupled system);

(C) Voltage-current curves for coupled system ( R# 0); (D) Voltage-time pulses for twooscillators (resembling action potentials of SA and AV nodes) become 1: l phase-locked over acritical range of V,. In 1:l phase-locked state, the frequency of oscillators is faster than theintrinsic rate of the “AV” pacemaker and slower than that of the “ SA ” pacemaker. (Adaptedfrom reference 4; copyright Elsevier Science Publishers B.V.. North-Holland Physics PublishingDivision.)

may at least, under some conditions, relate to alterations in the nonlinear coupling ofthese two active oscil lators. A pparent changes in conduction, therefore, may under

certain circumstances be epiphenomenal. This type of model may also be relevant tothe study of so-called “electrotonic” interactions that are thought to underlie arrhyth-mias such as “modulated” parasystole.20

Ventricular Activation

Following the SA -A V node interaction, the cardiac electrical impulse spreadsrapidly through a branching network of specialized conduction fibers known as the



198 ANNALS NEW YORK ACADEMY OF SCIENCES

W(3

k.0>

a

A. 111 SA-AV COUPLING

0.6SA-dV

B. 3:2 AV WEWEBACH

n0.6 t

0.4

0.2

0.0

o,6 C. 2:l AV BLOCK

t - 7 10.4

0. 2

0. 0

TIMEFIGURE2. (A) With parameter values the same as in FI GURE.1:1 phase-locking persists when

theSA node is driven by an external voltage pulse train with pulse width0.5dimensionless timeunits and period 4.0. (B) Driver period reduced to 2.0, with emergence of 3:2 Wenckebachperiodicity. (C) Driver period reduced to 1.5, resulting in a 2:1 A V block. Closed brackets denoteSA pulse associated with AV response.Open brackets denote SA pulse without A V response(“nonconducted beat”). (Reprinted from reference4; copyright Elsevier Science PublishersB.V.,North-Holland Physics Publishing Division.)



GOLDBERGER & WEST NONLINEAR CARDIOLOGY 199

His-Purkinje system. Depolarization of the myocardial cells of the left and rightventricles via this His-Purkinje tree results in the QRS complex on the surface

electrocardiogram. T he normal QRS complex, therefore, represents the output of the

physiological depolarization sequence. What is the mechanism underlying this elec-trical stability?'

A central tenet of modern physiology is the link between structure and function.The ventricular conduction system may serve, in this regard, as a prototypical exampleof the link between nonlinear structure and nonlinear function. The His-Purkinje

network, schematically depicted in F IGURE 3, is an irregular branching, treelike

BUNDLE OF H IS

I LEFT BUNDLE BRA N C H

RIGHT BUNDLE BRANCH

\

PURKINJE FIBERS

\

FIGURE 3. Irregular, but self-similar pattern of branchings seen in the His-Purkinje network ischaracteristic of fractal structures. (Reprinted from reference3; copyright Biophysical Society.)

structure in which the smaller branches are self-similar to the larger branches. The

pattern of self-similar arborization is seen over multiple generations of branchingsdown to the microscopic level of the Purkinje fibers that conduct the impulses into themyocardium. This type of irregular, but self-similar architecture is characteristic ofthe non-Euclidian geometric forms that have been dubbedfractals by Mandelbrot."

Ventricular depolarization, therefore, appears to be mediated by the spread ofcurrent through a fractal network.' What are the dynamic consequences of this fractalanatomy? The effect of the irregular fractal structure will be to shatter a single cardiac



200 ANNAIS NEW YORK ACADEMY OF SCIENCES

impulse into a myriad of stimuli, with different arrival times at the Purkinje-myocardial interface. The frequency spectrum of the QRS complex, in turn, will

depend on the statistics of this decorrelation process. Because of the fractal structure, it

can be shown that the distribution of decorrelation rates will take the form of an inversepower law.3 The Fourier spectrum of the resultant voltage-time (QRS) waveformsshould also beof the inverse power-law form.

This hypothesis-that the fractal geometry of the His-Purkinje system should leadto a QRS waveform with an inverse power-law spectrum-was tested by analyzingelectrocardiographic data from a group of healthy adult males ( n- 21) using a bipolarchest lead.’ The Fourier spectrum is a plot of the component frequencies (harmonics)that comprise a waveform against their respective squared amplitudes (power).Power-law scaling will be evidenced by a straight-line graph when the logarithm of

power is plotted against the logarithm of frequency (harmonics). Inverse power-lawspectrawill have a negative slope, that is, the higher frequencies will have less power.This type of scaling can also be referred to as l/f-like because of the inverserelationship between frequency and power. (If the spectrum has unit slope, then theprocess is pure 1/ f . )

The frequency spectrum of the normal QRS ( FI GURE) shows a broadband profilewith a long, low amplitude high frequency tail (r100 Hz). When these data arereplotted in a log-log format, the predicted inverse power-law spectrum is observed(r = -0.99). Thus, the data are consistent with the conjecture that the fractal-likeHis-Purkinje anatomy serves as the structural substrate for the broadband, inversepower-law spectrum that characterizes the stable ventricular depolarization (QRS)waveform.

This nonlinear model also leads to the prediction that pathologies that disrupt thefractal depolarization mechanism shouldbeassociated with a relative decrease in highfrequency potentials.’ As a result, the QR S spectrum will appear to narrow (i.e., theslope’of the inverse power-law regression line will become more negative). Sucha shiftin the spectral level with a decrease in high frequency components would be predictedwith bundle branch blocks, as well as in certain cases of myocardial infarction withdamage to the conduction system. Studies have shown a decrease in high frequency

potentials in some cases of myocardial infarction” and a relative decrease in highfrequency potentials in the latter part of the QRS in certain patients at risk forventricular arrhythmias.”

The broadband spectrum of the QRS associated with physiologic fractal depolari-zation contrasts with the narrowband patterns seen in pathologic settings such asventricular tachyarrhythmias. The spectra of ventricular tachycardia and ventricularfibrillation are characterized by a few relatively low frequency peaks ( t25 H Z).~‘ .*~These pathologic conditions in which the normal fractal depolarization mechanism issubverted are discussed in the next section.

Fractal geometry in the cardiovascular system is not restricted to the conductionsystem of the ventricle^^^ The branching coronary artery tree” and the chordaetendineae are also examples of fractal structures. In addition, self-similar architectureis not only limited to a single organ system. Fractal morphogenesis appears to give rise

eThe slope of the regression line is related to the fractal dimension ( D )of the system.’




to structures as diverse as the biliary system, the urinary collecting tubes, and thetracheal-bronchial tree of the lungs. Gas exchange is mediated by the interfacing ofthree fractal networks: pulmonary arterial, pulmonary venous, and bronchial.

analogous to the His-Purkinje system, is a branchingnetwork of tubes, dichotomously bifurcating over multiple generations from the leftand right mainstem bronchi to thousands of tiny tubes called bronchioles. A ccording to

the classical model of bronchial scaling, the average diameters (and lengths) of these

The bronchial

QRS SPECTRUMrn

y =- 4 . 3 ~ 5.41

r =-0.99 p <0.001

-70

32HARMONICS

II

0.2 0.6 1.0 1.4 1.8

log(harmonic)

FIGURE 4. The power spectrum (inset) of normal ventricular depolarization ( QRS)waveform(mean data of 21 healthy men) shows a broadband distribution with a long, low amplitude highfrequency tail. I nverse power-law distribution is demonstrated by replotting the same data inlog-log format. Fundamental frequency = 7.81 Hz. (A dapted from reference 3; copyrightBiophysical Society.)

tubes are thought to decrease as exponential functions of generation number (wherethe first generation consists of the left and right mainstem bronchi).26 However, if thelung is a fractal structure, it should not be characterized by a single scaling factor;’.”just as the fractal conduction along the His-Purkinje system cannot be represented by a

single decorrelation rate.’ Instead, an infinite sequence of scales should contribute tothe variability in bronchial dimensions. The relationship between each contributing




scale and its immediate neighboring scales and the relative weighting of these scales forthis type of fractal structure can be derived using renormalization groupThis theory, which has been used to model a variety of so-called “critical phenomena”

in physical systems?* leads to the prediction that mean bronchial dimension shouldscale as an inverse power law in generation number modulated by a harmonicvariation.

To test the renormalization (fractal) versus classical (exponential) model ofbronchial scaling, we” reanalyzed data obtained by Weibel and Gomez.26When these

A. 0.

-0.8

0.2 0.6 1.0 1.4 1.e 2.2 2.6 3.0

I n GENERATION

I I IOIICnlALQCNCRAWN

FIGURE 5. Mean bronchial diameters for successive bronchial generations [a)]s tradition-ally assumed to follow exponential scaling.” However, when data of Weibel and Gomez” from

single human lung cast are plotted on log-linear graph (A ), systematic deviation of datadnts orhigher bronchial generations is observed where the straight line gives exponential fit [d(z) e“]for generations1-10. Replotting the same data points on log-loggraph (B) reveals an excellent fitof the entire data set to the inverse powdaw (l/z’) relationship with apparent harmonicmodulation [A (z ) ]around regression line:d(z)- A ( z ) /z ” .The pattern in B is consistent with theprediction of renormalization group theory for fractal morphogenetic processes in which there isno single characteristic scale. Fractal scaling also appears important in the structure and functionof multiple other systems. (Adapted from reference 1 1.)

data from a single lung cast are plotted (F IGUREA), it is apparent that the simpleexponential scaling model only adequately accounts for mean bronchial diameters forthe first ten generations of branchings. The next ten generations show a systematicdeviation from the predicted regression. However, when the data are replotted in alog-log format (FIGURE B) on which an inverse power law will appear as a straightline, an excellent fit to the data is seen. Furthermore, the data points also appear tovary harmonically around this inverse power-law line, which is in keeping withrenormalization predictions. I n addition, when data from other species (dog, rat, and

’



GOLDBERCER & WEST NONLINEAR CARDIOLOGY 203

ham~ter)’~re plotted in a similar manner, the same kind of harmonically modulatedinverse power-law dependenceon bronchial generation number is observed.”

Based on these data and the observation of fractal structure in other systems, we

have conjectured that fractal scaling is a universal organizing principle in morphogene-sis. A fractal code based on self-similarity may provide an optimal mechanism for

generating complex structures with minimal instr~ctions.’*~*”ractal morphogenesis,in a wider sense, can be viewed as a kind of cri tical phenomenon’ evolving over time.Uncovering the ways in which fractal scaling information is encoded and processed isa

major challenge to nonlinear biology. The mechanism appears to require more than the‘‘linear’’ transcription of nucleotide sequences.

Up to this point, we have focused on the apparent universality of f ractal geometryin physiology, which is evidenced by self-similar structures. In the case of the

His-Purkinje system, we have also suggested that fractal structure is associated withthe broadband, inverse power-law spectrum of the QRS waveform. Fractal (renormali-

zation) scaling evidenced by 1/f-like distribution^^also appears to be an organizingprinciple underlying apparently stochastic fluctuations i n other physiological systems.For example, K obayashi and Musha’l reported that a 1 f-like spectrum characterizes

the variability in cardiac interbeat intervals in healthy subjects (with a superimposedspike at the breathing frequency). Previous data have apparently shown chaoticfluctuations i n day-to-day counts of neutrophils in the peripheral blood of normal

vol~nteers.’~pectral analysis of these data also reveals a 1 &like spectrum.’ Theseobservations support the hypothesis that regulation of complex processes such as

heart-rate variability and neutrophil dynamics is governed by self-similar (fractal)

mechanisms operating over multiple time scales.’ This type of scaling suggests a basisfor the “constrained type of randomness”’*’0hat appears characteristic of physiolog-ical variability and flexibility.

PATHOLOGICAL ALT ERATIONS I N CARDIOVASCULAR DYNAMI CS

Disruption of the physiologic nonlinear mechanisms discussed so far may lead to a

variety of different types of pathologic dynamics in cardiology. Moreover, such cardiacdisturbances may serve as a paradigm for understanding nonlinear syndromes in otherorgan systems. We will briefly review selected aspects of nonlinear pathophysiology.

Pat hol ogi cal Bi f urcat i ons

As noted earlier, i n a nonlinear system, variations i n some value of a parameterover a critical range may lead to different behavioral patterns. Such bifurcations willbe reflected in a marked change in the frequency spectrum. One generic mechanism

underlying the appearance of certain periodicities in a variety of apparently unrelatednonlinear systems is the period-doubling or subharmonic bifurcation phenomenon. As

implied by the name, period-doubling is manifested by an abrupt change in thebehavior of a system such that there is a halving of the frequency response ( A -fi/2)or,equivalently, a doubling of the period. Subharmonic bifurcations have been noted inavariety of mathematical and physical systems in which period-doubling behavior maybe a route to chaotic dynamic~.~’.’‘




Recent work has suggested the possible relevance of period-doubling mechanismsto biological dynamics. For example, Guevara, Glass, and Shrier3’ describe period-doubling behavior when the spontaneous rhythm of chick embryo heart cells is

perturbed by injection of current pulses. Ritzenberg et ~ 7 1 . ~ ~eport cardiac interbeatinterval oscillationssuggestive of period-doubling when norepinephrine is administeredto dogs.

We hypothesized that perturbation of a system of cardiac oscillators (pacemakers)might be associated with period-doubling behavior in the clinical setting. FIGUREshows a heartbeat time series’ obtained from the Holter record of a 60-year-old manwith the sick sinus syndrome. The graph plots consecutive heartbeat number againstR-R interval. The data clearly show spontaneous shifts in the dynamics of the R-Rinterval through quasi-stable steady states (labeled S, corresponding to either A V

junctional escape rhythm or sinus bradycardia), initial bifurcation to periodicity(period 2;A ), subsequent bifurcations to longer periods (periods4 and8; B, C,D), andreturn to relative apericdicity (S), ollowed by further bifurcations (A’, C).

The electrocardiogram at the time of these bifurcations showed complex combina-tions of supraventricular beats of varying origin (FIGURE ). These apparent bifurca-

ve . 4 t A s

S A’ C’

I 1

50 I08 I5 8 200 258 300

HEARTBEATSFIGURE 6. Representative heartbeat series from a patient with “sick sinus syndrome”,encompassing342 consecutive data points (-500 s). Periodic state (A ) is followed by apparentsubharmonic bifurcations (B,C, D) and then a return to a relatively aperiodic steady state. Asimilar pattern is seen throughout the record with the reappearance of periodic bursts (A , C‘).Note the reverse bifurcation sequence in the right-hand section (arrow) in which period 2 (A’)follows longer cycle length oscillations. (Adapted from reference 5 ; copyright Elsevier SciencePublishersB.V. ,North-Holland Physics Publishing Division.)




A

6

C

D

S

FIGURE 7. Electrocardiographic ecords corresponding to portions of data identified in FIGURE

6.Period 2corresponds to atrial bigeminy in which two supraventricular (i.e., sinus, atrial, or A Vjunctional) beats are clearly coupled together. Period-4 and period-8 sequences correspond tomore complex patternsof supraventricular beats. I n the bottom panel (S),he steady staterhythmcorresponds either to A V junctional escape rhythm (with no apparent P waves) or to sinusrhythm. (Adapted from reference 5 ; copyright Elsevier Science PublishersB.V., North-HollandPhysics Publishing Division.)

tion sequences are not accounted for by traditional models in electrophysiology.However, the patterns are consistent with the general notion of an attractor of the R-R

interval undergoing bifurcations due to spontaneous changes in one or more of theparameters modulating the interaction of the SA node and other pacemakers. Thiskind of observation suggests that the clinical term “sick sinus syndrome” is a misnomer

because it implies impairment of a single physiologic component (sinus node). Thegeneral nature of the conduction disturbances depicted in FI GURES6and 7 is consistent

with electrophysiological studies showing a high prevalence of abnormalities in otherparts of the cardiac conduction system in patients with sinus node disease, which isindicative of a global conduction di ~order .~

Bifurcation behavior of this sort is not confined to the cardiac electrical system.

Subharmonic bifurcations may also underlie the abrupt appearance of certain changes

in cardiac hemodynamic variables. For example, when fluid accumulates around theheart (pericardial effusion), the heart begins to swing back and forth with a frequencyinitially equal to that of the heartbeat ( f , ) . hen the hemodynamic state calledpericardial tamponade supervenes, the frequency of cardiac oscillation in the pericar-dial sac appears to abruptly change to a frequency that is half of that of the heartbeat

(f , /2).*This bifurcation is reflected on the surface electrocardiogram by the beat-

to-beat alternation in waveform vector (2:1 electrical alternans). Whether this




bifurcation relates to a critical change in heart rate, cardiac mass, or some othervariable(s) is not certain. Efforts are currently under way to quantitatively study thisphenomenon using a model of a swinging nonlinear spring.” Such models may

eventually prove helpful in understanding other alternans-type phenomena (e.g.,pulsus alternans in heart failure).

Scaling Pathologies

The spectrum associated with the output of a variety of physiological processesgoverned by self-similar (fractal) mechanisms is characterized by a broadband, inversepower-law spectrum. We have that avariety of pathologic perturbationsmight be associatedwith alterations in such scaling patterns and that the spectral slope

might be of diagnostic and prognostic utility.One generic marker of pathologic dynamics appears to be a narrowing of the

physiologic broadband spectrum, which reflects a decrease in healthy variability(spectral reserve loss hypothesis)? A loss of variability in cardiac interbeat intervalfluctuations, for example, has been noted in a variety of settings, including aging,”diabetes mellitus,”’ multiple s~lerosis,~’nd bedrest deconditioning,q perhaps reflect-ing dysautonomic changes.

We have also speculated that certain pathologic alterations may lead to a selectiveloss of high frequency fluctuations (FIGURE).’*9The loss of spectral reserve associated

with this decrease in physiologic variability may be of help in quantifying andeventually monitoring response to therapy in a variety of conditions such as aging. Inthis regard, changes seen in cardiac dynamics may be analogous to the selective loss ofhigh frequency responses that characterizes presbycusis (hearing lossin the elderly).

In the most severe pathological syndromes, loss of spectral reserve may lead to astate characterized by a few, relatively low frequency bands.’ This narrowband typespectrum, in contrast to the physiological broadband spectrum, should be representedby highly periodic behavior. Such pathological periodicities are widespread, not onlyin cardiovascular dynamics, but in multiple organ systems. For example, Cheyne-

Stokes syndrome“ is characterized not only by relatively low frequency (-.02 Hz)oscillations in respiratory rate and amplitude, but also by oscillations in interbeatintervals at the same frequency in some patients.’ This type of global oscillatorydynamic, in which an entire system appears to become “slaved” by a single dominantfrequency, should be contrasted with physiological oscillations that can bedetected asspikes riding on the “noisy” broadband 1 f-like spectrum that characterizes healthyvariability.’

Periodic behavior in cardiac electrophysiologic dynamics isalso apparent duringfatal ventricular tachyarrhythmias associated with sudden cardiac death.6 For exam-ple, torsades de pointes-type of ventricular tachycardia is characterized by periodicoscillation of the QRSvector (FIGURE). The mechanism of torsades is traditionally‘*thought to relate to two simultaneously firing ectopic pacemakers. However, spectralanalysis of surface electrocardiographic waveforms (FIGURE ) reveals a narrowbandspectrum that is remarkably consistent from one subject to another and from one leadto another in the same subject. Based on these spectral data, have speculated thattorsades is due to a unitary process, either a single automatic focus that periodicallymigrates or to periodic alternation in ventricular conduction with a fixed site of




stimulation. Such a process appears to involve some kind of nonlinear wave mecha-nism. A lthough the search for the traditional mechanismsof ventricular tachyarrhyth-

mias has focused exclusively on reentry or automaticity, these pathologies could be

related to a scroll-waveu or some other periodic, nonlinear wave process.Finally, a counterintuitive example of pathologic periodicity isventricular fibrilla-

tion: which is the arrhythmia most commonly associated with sudden cardiac death. A

prevalent view is that ventricular fibrillation represents cardiac chaos (turbulence).45s*However, multiple s t ~d i es ~~. ~~. ~’ndicate that fibrillatory oscillations are represented

-LOG FREQUENCY

FIGURE 8. The spectral reserve hypothesis predicts that a variety of apparently unrelatedpathologic states will be characterized by the loss of physiological variabil ity, thus leading to aloss of spectral power. This decrease may be most apparent in high frequency bands. Such arelative decrease in high frequency components would shift the power-law regression line morenegatively.

by a relatively narrowband spectrum, which is a finding inconsistent with the

turbulence theory. Similarly, in studies6of canine fibrillation utilizing epicardial andbody surface electrocardiographic data, we observed only a few relatively discretefrequency bands during the first minute. Further narrowing of the spectrum wasobserved during the second minute of electrically induced fibrillation. These spectralfindings are consistent with epicardial mapping studies4*and endocardial recordings4’in canine fibrillation that show unexpected evidence of spatial organization and

temporal periodicity. Whether the mechanism of ventricular fibrillation will be related




Average: Leads I, II, 1110’

-10 *

/ 1

- 5 0 .

0 5 10 15 20 25

Frequency ( Hz)

FIGURE 9. Torsadesde pointes-type of ventricular tachycardia is characterized by a periodicoscillation of theQRS vector. The power spectrum shows a series of discrete, relatively lowfrequency peaks, including peak (E) that represents “envelope” frequency of waveform spindles,

and peaks at heart-rate frequency (F,) and its higher harmonics (F2.F3). A similar spectralpattern was observed in other cases of tor~ades,‘~hereby suggesting a unitary process that ispossibly related to a nonlinear wave mechanism. Spectra were derived by hand-digitization (5 1.2samples/sec) using optically magnified electrocardiographic data from a previously publishedcase (K rikler,D M. & P. V. L. Curry. 1976.Br. Heart J . 38: 117).



GOLDBERGER & WEST NONL INEAR CARDIOLOGY 209

to that of rorsades remains unknown. However, any successful model of fatalventricular tachyarrhythmias will have to account for their periodic spectral represen-

tations.

The narrowband spectra associated with the sudden cardiac death contrast withthe broadband spectra associated with both normal sinus rhythm and normal ventricu-lar dep~larization.~n this way, ventricular fibrillation may serve as a paradigmaticexample of the bifurcation from broadband fractal stability to pathological periodicity’(FIGURE0)!A potentially analogous example of this transition isseen in white bloodcell dynamics in certain cases of granulocytic where relatively low

NSR VF

VFtSRt

0 5 10 0 5 10

FREOUENCY (HZ ) FREQUENCY ( Hz)

FIGURE 10. Transition from slow normal sinus rhythm ( NSR) oventricular fibril lation (VF) isassociated with bifurcation from a broadband spectrum to a narrowband pattern (“inversebifurcation”). The amplitude is in arbitrary units. (A dapted from reference 24; copyrightInstitute of Electrical and Electronics Engineers, 1977.)

frequency cyclic oscillations may replace the apparently erratic fluctuations that areseen normally and that are characterized by a broadband ]//-like spectrum.

NONLINEAR CARDIOLOGY: FUTURE DIRECTIONS

These preliminary findings related to fractal structure, inverse power-law spectra

and distributions, and bifurcation behavior suggest the potentially varied applicationsof nonlinear models to clinical cardiology. Perhaps most exciting is the prospect thatthese theoretical concepts will suggest new ways of monitoring patients at high risk for

’We have referred to this transition as an “inverse bifurcation” in order to distinguish it fromthe classic bifurcation sequence seen in physical systems in which period-doubling behavior is aroute to chaotic (fractal) dynamic '* .



210 ANNA IS NEW YORK ACADEMY OF SCIENCES

sudden cardiac death. We’.9 have speculated that spectral analysis of Holter monitordata may provide a useful means of identifying high risk subsets evidenced by thepresence of pathologic low frequency oscillations or a relative loss of physiologic high

frequency components (F IGURE) when the subject is still in sinus rhythm. This kindof analysis may be helpful not only in adults,M but in monitoring fetal distresssyndrome^ *^ as well. A t the present, Holter analysis is usually confined to portions ofthe record showing actual arrhythmias.

The same approach may also be generalized to other systems. For example, wehave proposed that spectral analysis of serial blood cell aunts may help monitorhematologic status: the transition from physiologic fluctuations havingabroadband,inverse power-law spectrum to periodic behavior may be of diagnostic and prognosticimportance in certain leukemias.’ Spectral analysis may also enable us to quantify the

loss of reserve that appears to characterize the aging process in different organsystems.9*’**”Another field, not yet mentioned here, is that of nonlinear pharmacology.

Nonlinear models of drug action may be essential in understanding the mechanismofaction of agents such as bretylium tosylate and digitalis glycosides, which havemultiple “direct” (electrophysiologic) and “indirect” (autonomic) actions. Spectralanalysis may also offer new insights into their pharmacologic action and toxicity. Thereported54 ncrease in interbeat interval variability following exercise with therapeuticdigitalis administration in atrial fibrillation is of interest in this regard. In contrast,toxic doses of digitalis in atrial fibrillation may lead to periodic behavior in the form ofregularization of the ventricular response, Wenckebach phenomenon, ventricularbigeminy, or bidirectional ventricular tachycardia. The potential applications ofnonlinear models to basic electrophysiologic processes at the cellular and subcellularlevel (e.g., delayed after-depolarizations associated with digitalis glycosides) lieoutside the scope of this review.

SUMMARY AND CONCLUSIONS

(1) Nonlinear mechanisms may apply both to the understanding of SA -AV nodeinteractions and to bifurcations leading to certain types of AV block.

(2 ) The fractal His-Purkinje system serves as the structural substrate for thegeneration of the broadband, inverse power-law spectrum of the stableventricular depolarization (QRS) waveform.

(3) Fractal anatomy is also seen in multiple other systems: pulmonary, hepatobil-iary, renal, etc. Fractal morphogenesis may reflect a type of critical phenome-non that results in the generation of these irregular, but self-similar struc-tures.

(4 ) Self-similar (fractal) scaling may underlie the l/f-like spectra seen inmultiple systems (e.g., interbeat interval variability, daily neutrophil fluctua-tions). This fractal scaling may providea mechanism for the “constrainedrandomness” that appears to underlie physiological variability and adaptabil-ity.

(5) Behavior consistent with subharmonic bifurcations is seen in cardiac electro-physiology (e.g., sick sinus syndrome) and hernodynamic perturbations (e.g.,swinging heart phenomenon in pericardial tamponade).



GOLDBERGER 81WEST: NONLINEAR CARDIOLOGY 21 1

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21.22.

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26.27.28.29.

(6) Ventricular tachyarrhythmias associated with sudden cardiac death (e.g.,rorsades de poinres, ventricular fibrillation) appear to reflect relatively

periodic, not chaotic (turbulent) processes resulting from disruption of the

physiologic fractal depolarization sequence.(7) Spectral analysis of Holter monitor data may help in the detection of patients

at high risk for sudden death.

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DISCUSSION OF THE PAPER

M.SCHOMAKER:notice that the regular QRS complex is associated with very fasttime scales. I wonder if the broadband power spectrum feature of that is just related to

the fact that the spike issosharp that you would get a broadband feature independentof whether there wasachaotic component in the dynamics. I also wonder if you aresimply losing the very fast time scales in the pathological case.

A. L. GOLDBERGER:or the QRS complex, ventricular depolarization takes placeover about 100milliseconds, but one can see asubtle widening, a smoothing of thatcomplex, in pathologic settings. In our analysis of one or two preliminary cases ofbundle branch block, there does appear to bea lossof the higher frequencies. Whenyou get these pathologies, theQRSwill broaden out by a variable amount (20,40,60,80milliseconds); thus, there is a loss of those sharp features, but those are generated bythe high frequencies. Therefore, it becomes a kind of a circular process. If you alter the

depolarization mechanism and change the way the cardiac sequence is activated, thereis slowing and the QRS will widen concomitantly. Tosome extent, what you say iscorrect, but I do not think it is just an artifact of the fast time scale.

J .M LTON:had a question that may be a little naive for this audience. I recallfrom my days in undergraduate laboratories that one way that I could salvage anexperiment was to do a log-log plot and get a straight line. I was wondering if thismethod of analysis is particularly sensitive to deciding that this isan inverse square law




as opposed to some other distribution that one might not pick. I n your plots for the

QRSand the bronchial tree, it did seem to me that near the zero axis the points did notfit the straight line as well as they did over to the right. In the QRS figure, it seemed

the fit was better to a straight line around the higher harmonics than at the lower ones.I just wondered i f you had looked at alternative methods of testing the data.

GOLDBERGER:n the case of theQRScomplex, the correlation to that straight-linefit is very high. 1 said it was consistent with an inverse power law; I did not say that itwas diagnostic, and we did not attempt to look at other curve fits. There is that bowingupward, and whether that is a characteristic feature and relates to something in the

dynamics, I do not know at the moment. With regard to the bronchial tree data, thosedata are very nicely fit by the harmonically modulated inverse power law. One can seethe variation around the regression line and that accounts for virtually all the deviation

from the pure inverse power law.F. ABRAHAM:ow would a fractal physician’s use of an artificial pacemaker differfrom the way that a periodic physician would use one?

GOLDBERGER:he older pacemakers that were in use were ventricular pacemakers

that left you with one heart rate; obviously, that was not very adaptable and indeed ledto major restrictions in activity. New pacemakers are dual chamber and can adaptphysiologically; thus, they keep the interbeat interval spectrum broader, and people in

fact do better with them. I think a fractal physician would be inclined to maintain thevariability in heart rate as much as possible. Even nonfractal physicians would agreewith this.

Documents

Annals of the New York Academy of Sciences Volume 504 Issue None 1987 [Doi 10.1111_j.1749-6632.1987.Tb48733.x] ARY L. GOLDBERGER; BRUCE J. WEST -- Applications of Nonlinear Dynamics