ORIGINAL ARTICLE

Self-affine fractal variability of human heartbeat interval dynamicsin health and disease

Accepted: 19 June 2003 / Published online: 27 August 2003� Springer-Verlag 2003

Abstract The complexity of the cardiac rhythm is dem-onstrated to exhibit self-affine multifractal variability.The dynamics of heartbeat interval time series wasanalyzed by application of the multifractal formalismbased on the Cramer theory of large deviations. Thecontinuous multifractal large deviation spectrumuncovers the nonlinear fractal properties in the dynam-ics of heart rate and presents a useful diagnosticframework for discrimination and classification ofpatients with cardiac disease, e.g., congestive heart fail-ure. The characteristic multifractal pattern in hearttransplant recipients or chronic heart disease highlightsthe importance of neuroautonomic control mechanismsregulating the fractal dynamics of the cardiac rhythm.

Keywords Cardiac disease Æ Heart rate variability ÆLarge deviation spectrum Æ Multifractals

Introduction

The beat-to-beat variation in the heart rate of humansand other mammals is generated by a complex processand displays inhomogeneous, nonstationary extremelyirregular temporal organization. The physiologicalmechanisms of cardiac control expected to result fromboth intrinsic and extrinsic factors operating at differ-ent time scales or resolution have not been identifiedclearly. A variety of mathematical methods have beendeveloped to characterize complex patterns in physio-logical time series. These methods include classicallinear stochastic techniques such as the mean and

standard deviation, as well as novel methods derivedfrom nonlinear systems theory including fractaldimension, scaling coefficients and multifractal spectra.The various ways in which these measures may beapplied to analyze physiological dynamics are brieflyreviewed with emphasis on techniques of assessmentand modeling of the multifractal properties of cardiactime series.

In this article we will first summarize standardmethods of time series analysis pointing at the pitfallsinherent in the application to a stream of heartbeatinterval data. Then some techniques in which conceptsfrom nonlinear systems dynamics have been applied willbe addressed. These include unifractal and multifractalapproaches of analysis. Finally, multifractal modeling ofcardiac time series complements this overview. However,since an extensive allusion to methods of nonlinear timeseries analysis is presented in several recent generaltextbooks (Bassingthwaighte et al. 1994; Kaplan andGlass 1995; Abarbanel 1996; Kantz and Schreiber 1997;Mandelbrot 1999; West 1999; Bunde et al. 2002;Doukhan et al. 2003), we will not attempt a compre-hensive review. In addition, the choice of methods is notexclusively based on objective criteria, rather there is astrong bias towards methods that we have found eitherconceptually interesting or useful in practical work, orboth.

Assessment of heart rate dynamics

Standard methods of time series analysis – when themean is meaningless

In research, quantitative analysis of physiological signalstypically starts (and often ends) with an analysis of themean and standard deviation. However it does not seemto be recognized that the most commonly used linear-stochastic statistical procedures provide an adequatecharacterization of the data only if the data have somevery specific mathematical properties.

Eur J Appl Physiol (2003) 90: 305–316DOI 10.1007/s00421-003-0915-2

M. Meyer Æ O. Stiedl

Dedicated to Paolo Cerretelli on the occasion of his 70th birthdayanniversary.

M. Meyer (&) Æ O. StiedlMax Planck Institute for Experimental Medicine, Hermann-ReinStr. 3, 37075 Gottingen, GermanyE-mail: meyer@em.mpg.deFax: +49-551-3899249

The arguments that follow are not restricted toheartbeat interval data, but may equally be leveled uponmany, if not all, biological time series, e.g., blood pres-sure, single neuron firing or channel gating dynamics.The characteristic features in observational heartbeatinterval time series of normal humans (and other speciesincluding rodents), for the purposes of illustration, arecompiled in Fig. 1. The left panels (from top to bottom)show an experimental RR-interval time series of a nor-mal subject (upper), a randomly shuffled surrogate of theoriginal time series (middle), and a genuine random dataset rescaled to the amplitude of the fluctuations of theoriginal record (lower). A difference in the appearance ofthe highly irregular, erratic and apparently randombehavior is readily apparent upon visual inspection. Intraditional linear statistics the measured values areconsidered as one representative set taken from theentire population having a well-defined arithmetic mean,the sample mean, standard deviation and variance.Hence no differences between the full-length data setsrepresented in the left panels become apparent,suggesting that the linear properties are the same.

However, the mean and standard deviationcalculated for cumulative subepochs of each data set(increasing data length), displayed in the middle-leftpanels, shows that this impression is wrong. While thestatistics for the random sets (middle and lower) rapidlystabilize, those of the original (upper) continue to fluc-tuate and, with increasing data length, would not con-verge to constant values. Thus, there is a sizable amountof uncertainty in the estimates of the sample mean and

the standard deviation, which would further increase ordecrease and never converge if more data were added tothe data stream. These findings suggest that a populationmean of the process that generates the data does not exist.There is thus no single value that can be identified as the‘‘correct’’ value of the mean. Ultimately, as the amountof data continues to increase the variance continues toincrease and becomes infinite.

The linear autocorrelation function (Cs) for increas-ing lag (s) is shown in the middle-right panels. WhereasCs rapidly decays to zero for the random data sets, thedecay for the original is much delayed and follows apower-law indicating the presence of long-time correla-tions that are absent in the random data. The rightpanels show the average mutual information Is asfunction of time lag (s). The information statistic Is actsas the nonlinear autocorrelation function telling, in anonlinear way, how the measurements at different timesare connected on average over all measurements. Thisquantity strictly connects the set of observed measure-ments (sn) with the same time-lagged set (sn+s) andestablishes a criterion for their mutual dependence basedon the criterion of information connection between thetwo. If sn is completely independent of sn+s, the amountof information between the two is zero. Hence, theaverage over all measurements of this information sta-tistic for s=1,2,3,... is zero for the random data setswhereas the amount of information gradually declineswith increasing s for the original and sn and sn+s willeventually become independent for larger s. Is=0 is thewell-known Shannon entropy of the signal. Stable linear

Fig. 1 Linear statistics ofexperimental and randomsurrogate data sets. Left panels:experimental RR-interval timeseries (@40-min recording) ofhealthy adult human subject(upper); randomly shuffledsurrogate of experimental timeseries (middle); rescaled genuinerandom surrogate (lower).Middle-left panels: RR-intervalmeans (MEANRR), standarddeviation (SDRR) and root-mean square of successiveinterval differences (RMSSD)as a function of accumulated100-point sub-epochs namelyincreasing data length. Middle-right panels: autocorrelation asa function of increasing lag.Note the shorter x-axis scale inmiddle and lower panels.Parallel thin lines around they-axis origin designate 95%confidence interval for whitenoise. Right panels: averagemutual information. See textfor details

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systems generate zero information, hence selectinginformation generation as the critical aspect of a non-linear system assures that the quantities derived (see‘‘Nonlinear time series analysis’’ and ‘‘Multifractal timesseries analysis’’) are a property of nonlinear dynamicsnot shared by linear evolution.

Thus, it becomes evident that, irrespective of the samelinear statistics, the dynamics of the original and therandom data sets do not share any element in common.The sluggish decay of both the autocorrelation functionand the average mutual information observed for theoriginal cardiac time series indicates the existence oflong-range correlations (‘‘memory’’) in the data set.Long memory in times series is hard to detect, but hasenormous effects on statistical quantities such as stan-dard errors and tests, and hence on the conclusionsdrawn.

In traditional analysis of irregular biological timeseries data, e.g., cardiac intervals, it is implicitly assumedthat the dynamics is inherently stochastic and producedby a linear Gaussian random process that is completelyspecified by its first and second moments and fullydescribed statistically by the mean, the variance and theautocorrelation function. Hence, if the sample data wereindeed representative of a Gaussian distribution, linearstatistical analysis focused on the fixed interval associ-ated with the mean would yield traditional variablesessentially independent of the interval selected foranalysis. Moreover, implicit in this assumption is theidea that the fluctuations around the mean are uncor-related, structureless random error with normal statistics(normal distribution of probability density function;in fact, much of biology and medicine is definitelynot ‘‘normal’’) containing no information about thedynamics.

However, as shown in Fig. 1, an artificial uncorre-lated random data set with normal statistics, rescaled tothe distribution of the values of the original with thesame linear properties but no further built-in determin-ism, shows that the cardiac interval time series has orderand structure embedded in the fluctuations that is notpresent in the uncorrelated time series. The deterministicstructure in the cardiac time series results from the factthat future events are causally set by past events(‘‘memory’’). This underlying complex structure incardiac interval variability presents a manifestation ofnonlinear control processes with many interacting com-ponents ultimately determining the cardiac rhythm.Hence, the present overview is concerned with nonlineartime series analysis for the detection and quantificationof possibly complicated structures in a signal.

Here, we emphasize that many universally acceptedlinear statistical measures would not adequately describethe complex dynamical properties of heartbeat intervaltime series, i.e., the correlated structure where each eventis statistically dependent on all past ones and resultsfrom the ordering of points in time. This is essentiallywhat happens for nonlinear so-called fractal randomprocesses (FRP) and is a consequence of a property of

fractals known as self-similarity. We will not review thetheory of fractals. Heuristically, a fractal signal producesa self-similar graph; ‘‘zooming’’ (in or out) yields apicture which is similar to the original. The signature ofself-similarity implies that: (1) the calculation of meanheartbeat interval or mean heart rate and associatedconventional linear statistics for discrimination aremeaningless; (2) before any statistical test of dependenceis used on heartbeat interval time series, its robustnesswith respect to infinite variance must be investigated;and (3) novel dimensionless measures may be used toreplace well-known but ambiguous linear-stochasticmeasures that generally encounter the commitment of atype 2 error in the assessment of heartbeat dynamics.

Nonlinear time series analysis

Early studies have attributed the complex dynamics ofthe cardiac rhythm to be the result of an autonomouslow-dimensional deterministically chaotic system but thisconcept has remained enigmatic (Lefebvre et al. 1993;Yamamoto et al. 1993; Kanters et al. 1994; Sugihara etal. 1996; Poon and Merrill 1997). Physiological systemsare typically high-dimensional. For example, heart rateand its variability are modified by variables other thanneuroautonomic outflow, such as respiration, arterialblood gases, and circulating hormones. Each of thesevariables in turn presents the output of dynamics coupledthrough nonlinear time-delayed feedbacks. Although fortruly deterministic systems the entire system’s dynamicsmay be determined from a single measured variable, thefeasibility of this approach for noisy biological timeseries has not been established. It does not seem to berecognized that conventional algorithms for calculationof correlation dimension or Lyapunov exponents haveserious limitations in the analysis of physiological dataand a convincing demonstration of the utility for high-dimensional systems has not been provided. Nonlinearstatistics can however be useful for describing thedynamics of time series even though these systems maynot be chaotic in the strict mathematical sense. This canbe demonstrated when the value of the statistic changesconsistently as some physiological condition changes orby differences between populations in distinctly differentphysiological states. In many instances the justificationfor a given statistic is founded on assumptions that maynot be appropriate in physiology but nonetheless provesto have physiological significance though the assump-tions do not hold.

Recent studies have identified scale invariance or self-affinity in the fluctuations of cardiac rate which arecharacterized as 1/ƒ-noise (strictly, ƒ)Bnoise, with a Fou-rier spectral density given in terms of the frequency ƒ bythe power law ƒ–B and B@1) and quantified by a globalroughness statistic (Thurner et al. 1988; Peng et al. 1995,1996; Ivanov et al. 1996, 1998a;Meyer at al. 1998a, 1998b;Amaral et al. 1998). Of particular interest has been thecalculation of fractal dimension and scaling properties.

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Fractal dimension

The fractal dimension, D, which presents the mostimportant measure of the roughness or irregularity of atime series is calculated from the log-log plot of therelative dispersion, D, versus the number of systemati-cally aggregated data points, n (Fig. 2, lower left). The

relative dispersion is given by the ratio of the standarddeviation to the mean. The procedure is to form groupsfrom the original consisting of m consecutive datapoints. The relative dispersion is calculated for eachgroup that is obtained as the aggregates are enlarged tocontain increasingly more points (coarse graining). Therenormalization group operation of heartbeat intervalsbeyond their immediate neighborhood yields groups oftwos, then groups of threes, then groups of fours and soon. For a simple fractal, D decreases as n increases andyields a rectilinear line with power-law slope, b. D iscalculated from the relationship: D=1–b. The straight-line relationship implies that cardiac interbeat dynamicsfollow an inverse power-law and exhibit scaling whichindicates the presence of long-time self-similar correla-tions extending over hundreds of beats. The serialdependence or ‘‘memory’’ observed in cardiac time seriesis a reflection of different regulatory mechanisms that,although acting mutually independently on differenttime scales, are tied together and their effects on heartrate dynamics are interconnected by scaling. Thisbehavior which operates against lawlessness is essen-tially what is called a fractal random process (FRP). Inturn, impairment of one individual component ofcardiac regulation is expected to influence the otherregulatory mechanisms by interdependence. The fractaldimension calculated from aggregated dispersion anal-ysis for realizations of sequence-randomized surrogates(DSURR), representing uncorrelated random behavior(white noise), is also displayed in Fig. 2 (lower left). Thefractal dimension of normal cardiac interbeat time seriesis 1.0>D>1.5, indicating that the underlying dynamics

Fig. 2 Non-linear time series analysis. Upper: RR-interval timeseries of healthy subject in supine posture. Fractal dimension –lower left: Logarithm, base 2, of relative dispersion (D) versus log,base 2, number of aggregated data points (n). The upper line is thebest fit to the data with slope )0.25(D(n)�n)0.25) yielding a fractaldimension of D=1.25 for the experimental time series. The lowerline is the average best fit for an ensemble of 10 realizations ofsequence-randomized data sets, i.e., shuffling the original timeseries data points to random positions in the sequence, with slope –0.50 (0.01) yielding a mean fractal dimension of the surrogates ofDsurr=1.50 (0.01). The probability that the difference in the slopesbetween the two lines or fractal dimensions can be explained by alinear, additive, uncorrelated process is p<10)6. Scaling coefficients– lower right: Log-log plot of root-mean-square fluctuations [F(m)]versus size, namely number of points (m) in window. The points ofline in the middle follow different regression slopes, b1 and b2,separated at a cross-over breakpoint (evident at higher magnifica-tion, here indicated by upward arrow). The scaling coefficient forshort-range correlations in the experimental time series is b1=1.03;for the long-range correlations it is b2=1.07. The lower line ofpoints depicts the average of 10 realizations of sequence-random-ized surrogate data sets representing white noise with mean slopebsurr=0.49 (0.02). The upper line of points depicts the average of 10realizations of integrated sequence-randomized surrogate data setsrepresenting Brownian noise (integration of white noise) with meanslope bsurr=1.50 (0.04). The differences of bsurr suggest that thescaling in the experimental time series cannot be explained by whiteor Brownian noise. See text for details

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is different from periodic behavior (D=1.0) and differ-ent from white noise (DSURR@1.5), i.e., is fractal.

Scaling coefficients

The calculation of scaling coefficients of a time series bya technique referred to as Detrended Fluctuation Analysis(DFA) is derived from principles of random walk theory(Peng et al. 1995; Meyer et al. 1998a; Stiedl and Meyer2002, 2003). In the DFA analysis the original time series,Yj, length N, is first integrated. The integrated timeseries, Y(k), is self-similar if the fluctuations at differentobservations windows, F(m), scale as a power-law withthe window size, m, i.e., the number of cardiac intervalsin the window of observation. The root-mean-squarefluctuation, F(m), is calculated for all window sizesnamely time scales, m:

F ðmÞ ¼ 1

N

XN

k¼1Y kð Þ � Ym kð Þ½ �2 ð1Þ

where Ym(k) denotes the local trend in each window. Alinear relationship in the log F(m) versus log m graphindicates that F(m) @ mb, where b (slope of the log F(m)versus log m relationship) is the scaling exponent (orself-similarity parameter). In the log-log plot, F(m)typically increases upon recurrent operations withincreasing window size m yielding straight-line power-law relationships eventually separated by a cross-overbreakpoint (Fig. 2, lower right). The different slopes, b1

and b2, are interpreted to reflect the coefficients of dis-tinct ranges of the beat increment: short-range and long-range scaling, respectively. For normal cardiac timeseries b1 @ 1.0 and b2 @ 1.0. A key issue in the analysisof non-stationary physiological times series data is thatfluctuations driven by extrinsic uncorrelated stimuli canbe interpreted as a systematic ‘‘drift’’ or ‘‘trend’’ inrelation to the frequency of the stimuli and distin-guished from the intrinsic correlation properties of thedynamics. These nonstationarities which do not scalethe time series are treated by removing the least-squareslinear regression trend in each window. For clarity,scaling so interleaves the data that no procedure fordifferencing can remove its effect, i.e., a fractal timesseries cannot be detrended systematically. Here, thefluctuations scale themselves and the longest and theshortest time scales contributing to the process are tiedtogether so that anything that affects one time scaleaffects all of them.

Nonstationarity and fractal dynamics

Physiological time series under free-running conditionsmay be visualized as ‘‘badly behaved’’ data originatingfrom nonstationarities driven by uncorrelated extrinsicfactors and/or from the intrinsic nonlinear correlateddynamics of a fractal process. In a formal way, the

signal generated by a fractal process is nonstationary,i.e., it varies all over as a result of its ‘‘built-in’’ long-range power-law properties and its moments depend ontime. A fractal-generating mechanism with fixed com-ponents can generate a time series with moments thatvary in time. A self-similar fractal process typicallyproduces a time series with long-run dependence of thefluctuations that display the kind of ‘‘drift-like’’appearance at all time scales. It can be demonstratedthat the characteristics of a genuine (computer-gener-ated) self-similar fractal time series demonstrating non-stationarity with ‘‘drift-like’’ appearance at all timescales due to ‘‘built-in’’ long-range power-law correla-tions (b1,2=1.0, 1/ƒ)noise) display similar fractaldynamics compared to the experimental data ofhumans. The situation is further complicated by the factthat a fractal process may not be uniscaling or unifractaland hence may not be fully described by a unique fractaldimension or unique scaling coefficients because it maypresent a time-dependent multiscaling or multifractalprocess with varying dimensions and scaling properties(see below).

On the other hand, the intrinsic dynamics of a com-plex nonlinear system may be biased by extrinsic sour-ces, i.e., non-steady state physiological or environmentalconditions, that could give rise to highly nonstationarybehavior. Although strain-related variations may bephysiologically important, their correlation propertieswould be expected to be related to the stimulus anddifferent from the long-range correlations (‘‘memory’’)generated by a dynamical system. Unlike the calculationof fractal dimension, D, from the relative dispersionwhich presumes stationarity of the data stream, theestimation of scaling coefficients, b, does not rely on thisassumption. A key issue of the DFA-analysis is that theextrinsic fluctuations from uncorrelated stimuli can beinterpreted as ‘‘trends’’ and decomposed (by detrendingacross time scales) from the intrinsic dynamics of thesystem itself.

Multifractal time series analysis

Implicit in the calculation of fractal dimension andscaling coefficients (see above) is the assumption that thecardiac interbeat time series is fractionally homoge-neous, i.e., unifractal or uniscaling, and can appropri-ately be described by a single fractal dimension orscaling coefficient. Further generalization of the analysisbeyond the property of uniscaling leads to multiscalingor multifractals allowing the scaling exponents to dependon time and to be chosen from an infinity of possibledistinct values. Multifractals are therefore characterizedby a plethora of scaling relations or power-laws withcorrespondingly many exponents. Multifractals, origi-nally introduced by the seminal papers of Frisch andParisi (1985), Halsey et al. (1986) and the work ofMandelbrot (1999), exemplify extreme variability andbelong to the broad and unified notion of self-affine

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fractal variation. Evidence for multifractality in humanheartbeat dynamics has been suggested earlier (Meyeret al. 1998a, 1998b) and the multifractal formalismbased on the Legendre spectrum (see below) has recentlybeen invoked in the analysis of the variability of heartrate (Ivanov et al. 1999).

Multifractal spectrum

Biological signals typically: (1) exhibit significant long--range dependence (LRD), but display short-termcorrelations and scaling behavior inconsistent with strictself-similarity; (2) the scaling behavior of moments is anon-trivial (nonlinear) function of the moment order asthe signal is aggregated; (3) the increments in the signalare inherently positive and hence non-Gaussian. Signalswith these properties fall naturally into the class ofmultifractal processes. Hence, multifractal analysisretrieves positive measures or distributions exhibitingself-similarity but nonhomogeneous scaling.

The multifractal analysis of irregular but otherwisearbitrary one-dimensional signals is first demonstratedon a classical multifractal signal (trinomial measure,weights: 0.1, 0.3, 0.6; Fig. 3a). The analysis yields amultifractal spectrum [ƒ(a)] that may be viewed as ameasure of spikiness or global characterization of thesingularity structure of the data. It provides informationas to which singularities occur in the signal and whichare dominant (Falconer 1990; Jaffard 1997; Vehel andVojak 1998). More precisely, ƒ(a) is a graph where the

abscissa represents the Holder or regularity exponent (a)in the signal and the ordinate is the fractal co-dimensionwhich measures the extent by which a given singularity isencountered (coarse-grained multifractal spectrum). Thelower the exponent a, the more irregular is the signal.

Large deviation spectrum

The large deviation spectrum [ƒg(a)] which is based onthe Cramer Theory of Large Deviations (Holley andWaymire 1992; Vehel and Vojak 1998) is defined as:

fg að Þ ¼ lime!0

limn!1

lnN nc að Þ

� ln dn; ð2Þ

where

N ne að Þ ¼ # an

e a� ani

�� ��\e� �

: ð3Þ

The function ƒg(a) reflects the exponentially decreas-ing rate of Nn

e að Þ which is the number of intervals havinga coarse grain Holder exponent, an

i , close to a Holderexponent a up to a precision � when the resolution napproaches ¥. The exponent an

i is the logarithm of somequantity measuring the variation e.g., max)min, of thesignal in a given interval divided by the logarithm of theinterval (Fig. 3b, c). The term dn is related to the partitiondefined by the measure at the resolution n. Hence, the setof an

i reflects the scaling behavior of the data at finiteresolution. For each resolution n, equispaced intervalsare considered. ƒg(a) is equal to a Large Deviation

Fig. 3a–d Multifractality ofgenuine multinomial measure. aTime series of multinomialmeasure (6561 data points). bThe coarse grain Holderexponent (a) denotes the rangefrom regularity (high a) toirregularity (low a). The broadrange of coarse grain Holderexponents associated with thesignal reflects the complextemporal organization ofsingularities characterized bythe range of irregularfluctuating Holder exponents. cColor coding of b. The colorspectrum goes from blue to cyanto yellow to orange and to red.Blue indicates small values of awhile yellow-orange indicateslarge values of a. d Themultifractal spectra, f að Þ of theHolder coefficients (a) showsmooth concave functions andclose agreement of the Legendreand continuous large deviationspectrum with the theoreticalspectrum. See text for details

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Principle (LDP) rate function (Ellis 1985; Riedi 1995;Vehel and Vojak 1998) which measures how frequently orhow likely the observed an

i deviates from the ‘‘expectedvalue’’ of a. More generally, the multifractal formalism,assessing the singularity spectrum of the function ƒg(a),yields information about the statistical behavior of theprobability of finding a point with a given Holder expo-nent in the signal under changes of resolution.

Continuous large deviation multifractal spectrum

The computation of ƒg(a) is based on kernel-densityestimation of the coarse grain Holder exponents an

i . Theestimator of ƒg(a) is optimized for (1) implicit depen-

dency between precision � and resolution n, (2) spatialadaptation of precision �, (3) stable resolution within therange nmin £ n £ nmax, yielding a new definition (Canuset al. 1998), the continuous large deviation multifractalspectrum, f c

g að Þ (Fig. 3d). Interval discretization is line-arly spaced between amin and amax which are the mini-mum and maximum values of the coarse grain Holderexponents. f c

g að Þ yields the large deviations from the‘‘most frequent’’ singularity exponent and thus displaysinformation about the occurrence of rare events such asbursts (small a). Figure 3d reveals a rich multifractalspectrum indicating that there is ‘‘burstiness’’ in thesignal everywhere. Contrary to the more convenient andgenerally more robust (though at the expense of severeloss of information) Legendre spectrum (Ivanov et al.1999), ƒe(a), which is based on the Legendre transfor-mation of the Renyi exponents [s(q)] of the qth momentsof the measure and is always shaped like a \ (concaveand thus continuous, and almost everywhere differen-tiable), the continuous large deviation multifractalspectrum, f c

g að Þ, is superior in that it does not need to beconcave and hence is more appropriate in a generalsetting. It has been demonstrated, that the partitionfunction namely moment exponent function s(q) andsingularity spectra ƒ(a) are mutually related functionsand ƒ(a) is the concave hull of ƒg(a) (Brown et al. 1992;Vehel and Vojak 1998). For the truly multifractaldeterministic measure (Fig. 2a) ƒg(a)=ƒ(a), both show-ing close correspondence with the theoretical multifrac-tal spectrum. The multifractal spectrum addresses thelong-range dependence particularly on large scales or

Fig. 4a–h Multifractality in heartbeat interval time series in health(left panels) and cardiac disease (right panels). Successive heartbeatintervals (ms) as a function of time (@40 min) from a healthysubject (a) and a patient with ischemic heart disease (b), both insupine posture. The cardiac time series exhibits wild irregularfluctuations and nonstationary behavior in the healthy subjectwhich is much attenuated in cardiac disease. c, d The coarse grainHolder exponents (a) distributed over the interval [0,1] reflects theheterogeneity that is present in the cardiac time series irrespectiveof the fact that the subjects were in steady-state resting supineconditions. e, f Color coding of c, d. g, h The continuous largedeviation multifractal spectrum, f c

g að Þ, of the Holder coefficients (a)shows a smooth concave function supported by amin to amax withmode, amode [0.33] for the healthy subject and amode [0.50] forcardiac patient, indicated by filled circles. The extracted parame-ters, amode, amin, and amax, respectively, are used to quantify thespectra for the purpose of discrimination and classification ofheartbeat interval time series of normal subjects and patients withcardiac disease

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aggregation levels as well as events such as bursts. Giventhe strength of a burst with exponent a, the spectrumƒ(a) indicates how frequently this strength a is encoun-tered. The larger ƒ(a) the more often is a observed.

Experimental data sets

The cardiac interval time series (length @105 beats) ofhealthy subjects [n=9, 7 males, 2 females; mean age (SD),33.7 (4.6) years] and patients with congestive heart failure[n=11, 9 males, 2 females; mean age (SD), 56.4(7.2) years]were obtained by 24-hHoltermonitoring. Theelectrocardiographic data base was obtained using a dual-channel ambulatory ECG recorder (Model R6, CustoMed, Munich, Germany) provided with RAM-memoryand advanced data compression technology (samplingrate 500 Hz, resolution 2 ms). Additional short-termsteady-state episodes (40 min) in healthy control subjects,

in patients with nonsustained ventricular tachycardia,and in heart transplant recipients (<2 years after trans-plantation) were recorded by a custom-made dual-chan-nel battery-powered miniature ECG recorder amplifierinterfaced to a PC by a fiber-optical link (sampling rate1200 Hz, resolution @ 0.8 ms).

The digitized ECG was automatically analyzed usingan adaptive QRS-template pattern-matching algorithmto obtain a discrete cardiac time series or function,x(t)=ti+1–ti, i.e., the time interval between successiveR-wave maxima of the ECG. Ectopic beats or outlierswere identified by fitting a third-order autoregressivemodel to the interbeat interval data points using multi-ples (3.5) of the interquartile distance as detectionthreshold and corrected by linear-spline interpolation.

Cardiac dynamics in health and disease

Figure 4 shows the continuous large deviation multi-fractal spectrum for a healthy adult human subject (leftpanels) and a patient with ischemic heart disease (rightpanels). The difference in the dynamics between the twobecomes readily apparent by eye, as the human eye is anexcellent pattern-recognition device. The quantitativeaspect of the difference is reflected in the coarse grainHolder exponents (Fig. 4c, d) or its colored transfor-mation (Fig. 4e, f) and the disparate shapes of themultifractal spectra (Fig. 4g, h).

For all healthy subjects, the spectrum, f cg að Þ, is a

smooth concave function [group mean (SD), amode 0.36(0.02); amin 0.09 (0.01); amax 0.50 (0.03)] over a broadrange of Holder exponents a (Fig. 5a). The broad range

Fig. 5a–c Multifractality in health (a) and cardiac disease (b).Group averages (SD) of the continuous large deviation multifractalspectrum f c

g að Þ versus Holder exponents (a) in healthy subjects andpatients with congestive heart failure. The broad shape of themultifractal spectra exemplifies multifractal behavior in the cardiacinterbeat rhythm which is different between the healthy group andthe heart-failure group. The spectra of the cardiac-failure groupdisplay a markedly left-sided bimodal shape and shift of amode

towards higher values. c Stratification of healthy subjects andpatients with congestive heart failure by continuous large deviationmultifractal spectrum estimation. Mode of spectrum (amode) versusdegree of multifractality (amax)amin) based on full-length 24-h timeseries records. The multifractal approach clearly discriminateshealthy from heart-failure subjects. The results demonstrate that asingle quantity, amode, may be used successfully for discretization ofnormal subjects and patients with cardiac disease

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spectrum indicates that heart rate of healthy subjectsexhibits multifractal dynamics, i.e., the normal cardiacrhythm displays self-affine multifractal variability. In

contrast (Fig. 5b), the spectrum of heart rate dynamicsfrom patients with congestive heart failure consistentlydisplays a marked departure from concavity showing a

Fig. 6a–f Circadian heart ratedynamics. a, b Mean RR-interval (SD) of 1-h sub-epochsversus the time of day in ahealthy subject and a patientwith congestive heart failure.The heartbeat interval timeseries of normal subjects exhibita circadian rhythm withlengthening of RR-intervals(lower heart rate) during night-time episodes which is almosteliminated in chronic cardiacfailure. c, d Group average (SD)of a versus f c

g að Þ of 1-h sub-epochs. The multifractal spectradisplay a characteristicdifference of shape betweenhealth and cardiac failure butexhibit a remarkable stabilityover a circadian 24-h cycle. e, fThe mode of the multifractalspectra is lower in normalsubjects and higher in cardiacpatients but would not undergosystematic circadian-relatedchanges

Fig. 7a, b Multifractality andneuroautonomic cardiaccontrol. Multifractal spectrafrom heart transplant recipients(a), and patients with non-sustained ventriculartachycardia (b). The differentpatterns of the multifractalspectra exemplify theimportance of neuroautonomiccardiac control in generatingthe broad-range multifractalspectrum of healthy cardiacdynamics. Cardiac disease isassociated with a characteristicpathological shape of themultifractal spectrum

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pronounced trough on the increasing part of the spec-trum and amode is shifted to larger Holder exponents[group mean (SD), amode 0.45 (0.03); amin 0.09 (0.02);amax 0.53 (0.04)]. The shift of the amode to larger valuesand the ‘‘removal’’ of low singularity strength(0.1 £ a £ 0.4) render ‘‘smoothing’’ and indicate that thecardiac time series in chronic cardiac disease is moreregular while the degree of multifractality given by theamax–amin difference is not materially different betweenthe two groups. The pattern of the spectrum estimate incardiac patients appears to uncover a superposition oftwo ‘‘basic’’ spectra reminiscent of those which areobserved by synthesis of theoretical multinomial mea-sures, e.g., mixing or sum of two binomial measures.This can be taken as an indication for (at least) twodifferent phenomena underlying perturbation of thedynamics in a pathological condition – congestive heartfailure. Less prevalent parasympathetically mediatedcardiac control whereby the heart operates in a sympa-thetically dominated regime signifies the pathophysiol-ogy of advanced chronic heart disease.

The calculation of the f cg að Þ spectrum clearly dis-

criminates healthy subjects from heart-failure patientswhen based on a single quantity: amode (Fig. 5c). Fur-ther analysis of the data base by randomly selectingsubsets of varying data lengths reveals that statisticaldiscrimination of healthy and diseased individuals maybe achieved on the basis of any arbitrarily selected seg-ment of 8192 data points which corresponds to an @3-hrecord.

Circadian cardiac dynamics

In order to analyze the circadian stability of the f cg að Þ

spectrum and to assess the impact of extrinsic factorssuch as physical activity, f c

g að Þ was calculated for 1-hsub-epochs of the full-length 24-h data base. While theheart rate of a healthy subject typically shows a circadianrhythm with shorter interbeat intervals (higher heartrates) during day-time episodes and longer intervals(lower heart rates) during night-time epochs (Fig. 6a),cardiac failure patients demonstrate little, if any, circa-dian changes (Fig. 6b). Indeed, the absence of a circadianrhythm in the clinical setting is interpreted as evidencefor chronic heart failure. The multifractal spectrum ofthe cardiac time series does not undergo appreciablecircadian changes in healthy subjects or cardiac patients(Fig. 6c, d) and amode, amax and amin remain essentiallyunchanged in the course of the day (Fig. 6e, f). Thus, thephysiological mechanisms controlling the dynamics ofheart rate appear to be independent of regulatory circa-dian factors and the form of f c

g að Þ is solely a function ofintrinsic factors affecting cardiac activity.

The analysis of short-term data from healthy subjectsrecorded in steady-state conditions (@40 min) comparingdifferent postures (supine versus sitting) and the effectsof breathing frequency (spontaneous versus pacedbreathing over a range of 6–48 breaths/min) did notreveal any systematic changes in the f c

g að Þ spectrum(data not shown). These findings along with the absence

Fig. 8 Modeling of heartbeatinterval time series. Originaltime series of healthy subject(upper left) and continuouslarge deviation multifractalspectrum (upper right).Synthesis of cardiac time seriesby multifractal wavelet model(lower left) and continuouslarge deviation multifractalspectrum (lower right). Note theclose correspondence of visualappearance and multifractalproperties of real and simulatedheartbeat interval time series

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of circadian changes suggest that the multifractal spec-trum of cardiac dynamics is unlikely to result fromextrinsic factors such as physical activity, posture orrespiration. This feature of the multifractal spectrumfacilitates the comparison of cardiac dynamics acrossspecies, e.g., humans versus mice, humans and miceexhibiting identical spectra irrespective of the fact thatheart rate in mice is about 8 times higher than inhumans (data not shown).

Physiological mechanisms of multifractal variability

The physiological mechanisms underlying the multi-fractal variability in the cardiac rhythm have not beenidentified. Intrinsic instability, extrinsic stochastic per-turbations and a changing environment act together toproduce the intriguing irregular patterns. In healthysubjects, an important mechanism is the control of sinusnode activity by the autonomous nervous system. Adirect approach as to the significance of autonomiccardiac control for the multifractal dynamics of heartrate is facilitated by studies in recipients of a cardiactransplant (Fig. 7). The data base of heart transplantrecipients (<2 years after transplantation) and patientswith nonsustained ventricular tachycardia was obtainedfrom 40-min records. The f c

g að Þ spectrum in hearttransplant recipients (Fig. 7a) shows a preservation ofamode but cut-off of the upper-range Holder exponentsindicating break-down of multifractality strength whichis given by the amax–amin difference. Thus, the trans-planted denervated allograft in transplant recipientswhich is effectively deprived of its neuroautonomiccontrol is operating over a narrowed multifractalregime. In contrast, patients with episodes of ventriculartachycardia (Fig. 7b) demonstrate enhanced broadeningof the spectrum and a bimodal shape similar to thatobserved in congestive heart failure (cf. Figs. 5b, 6d).These findings emphasize the differential effects of theabsence (in cardiac transplantation) or degradation (inventricular tachycardia) of autonomous nervous systeminfluences on the multifractal spectrum of the cardiacrhythm. However, the precise pathology of neuro-cardiac control mechanisms that is uncovered bychanges in shape of the multifractal spectrum remains tobe determined.

Multifractal modeling of heartbeat interval time series

Studying real world physiological data such as heartbeatinterval fluctuations is generally restricted to the use oftime series analysis techniques aimed at deriving asuitable quantitative statistic in order to discriminatebetween physiological conditions or populations inphysiologically distinct states. Little attention has beengiven to the modeling of heart rate dynamics. Theparticular challenge in modeling cardiac traces resides inthe ‘‘spiky’’ and ‘‘bursty’’ behavior caused by long-range

dependence in the data. A multifractal wavelet modeladapted from internet traffic modeling (Riedi et al. 1999)was used for synthesizing coarse-to-fine multiscale posi-tive-valued data with long-range dependence. The modelis based on the Haar wavelet transform and a multiplica-tive cascade structure. Starting from a set of training data,the set synthesized by the multifractal wavelet modelmatches the mean, standard deviation, variance, auto-correlation function and the power spectrum of the ori-ginal time series. Figure 8 compares the real data of ahealthy adult with the result of one realization of themultifractal wavelet model synthesis (left panels). Themultifractal continuous large deviation spectra (rightpanels) reflect the accuracy of the synthesis in terms of themultifractal statistical measure, f c

g að Þ. The quality of thematching indicates that the multifractal model convinc-ingly captures and synthesizes the dynamics across large(global behavior) as well as small time scales (rare events,burstiness). The results suggest that the mechanismsinteracting in the control of heart rate that operate overtime scales from seconds (the parasympathetic nervoussystem) to hours (renal fluid volume) may exhibit aninherent multiplicative structure.

Concluding remarks

The multifractal properties of heartbeat dynamics elic-ited by control mechanisms that regulate the cardiacrhythm may be associated with the behavior ofdynamical systems typically operating far from equilib-rium near a critical point of phase transition (Meneveauand Streenivasan 1987; Ivanov et al. 1998b). Multifrac-tality or multiscaling in cardiac time series signifies theinteraction of coupled mixed-feedback systems operat-ing over a wide range of time scales. These propertieshelp provide responsiveness to environmental stimuli(elasticity) while preserving a relative insensitivity toerrors (error tolerance). From a practical point of view,the continuous large deviation multifractal spectrum ofheartbeat interval time series is expected to provide auseful diagnostic framework for discretization andclassification of patients with cardiac disease and may beapplied to other biological time series data.

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