Wessel Herzschrittmacher 2000

N. WesselA. VossH. MalbergC. ZiehmannH.U. VossA. SchirdewanU. MeyerfeldtJ. Kurths

Nonlinear analysis of complex phenomenain cardiological data

Herzschr Elektrophys 11:159–173 (2000)© Steinkopff Verlag 2000 MAIN TOPIC

H&

E287

Received: 6 April 2000Accepted: 26 July 2000

Dr. Niels Wessel (✉) · Christine ZiehmannJurgen KurthsNonlinear Dynamics Groupat the Institute of PhysicsUniversity of PotsdamAm Neuen Palais 10PF 60155314415 Potsdam, Germany

Andreas VossUniversity of Applied Sciences JenaGermany

Hagen MalbergUniversity of Karlsruhe, Germany

Henning U. VossUniversity of Freiburg, Germany

Alexander Schirdewan · Udo MeyerfeldtFranz-Volhard-Hospital Berlin, Germany

Nichtlineare Analyse komplexer Phanomenein kardiologischen Datenreihen

Zusammenfassung Das Hauptanliegen dieses Beitrages ist es, verschie-dene Ansa¨tze in der Herzfrequenz- und Blutdruckvariabilita¨t zu diskutierenund damit das Versta¨ndnis der kardiovaskula¨ren Regulation zu verbessern.Wir betrachten Komplexita¨tsmaße basierend auf der symbolischen Dyna-mik, die renormierte Entropie und die ,finite-time‘ Wachstumsraten. Weiter-hin werden die duale Sequenzmethode zur Bestimmung der Baroreflexsen-sitivitat sowie die Maximalkorrelationsmethode zur Scha¨tzung der nicht-linearen Kopplung in bivariaten Daten vorgestellt. Letztere stellt einegeeignete Methode zur Bestimmung der Kopplungssta¨rke und –richtungdar. Herzfrequenz- und Blutdruckvariabilita¨tsdaten einer klinischen Pilotstu-die und einer großangelegten klinischen Studie werden analysiert. Wir de-monstrieren in diesem Beitrag, dass Methoden der nichtlinearen Dynamiknutzlich sind fur die Risikostratifizierung nach Herzinfarkt, fu¨r die Vorher-sage von lebensbedrohlichen Rhythmussto¨rungen sowie fu¨r die Modellier-ung der Herzfrequenz- und Blutdruckregulation. Diese Ergebnisse ko¨nntenin der klinischen Diagnostik sowie fu¨r therapeutische und pra¨ventiveZwecke von implantierbaren Defibrillatoren der na¨chsten Generation vonBedeutung sein.

Schlusselworter Herzfrequenzvariabilita¨t – Blutdruckvariabilita¨t –Baroreflex – nichtlineare Kopplung – nichtlineare Dynamik –implantierbarer Defibrillator

Summary The main intention of this contribution is to discuss differentnonlinear approaches to heart rate and blood pressure variability analysisfor a better understanding of the cardiovascular regulation. We investigatemeasures of complexity which are based on symbolic dynamics, renorma-lised entropy and the finite time growth rates. The dual sequence method toestimate the baroreflex sensitivity and the maximal correlation method toestimate the nonlinear coupling between time series are employed for ana-lysing bivariate data. The latter appears to be a suitable method to estimatethe strength of the nonlinear coupling and the coupling direction. Heart rateand blood pressure data from clinical pilot studies and from very large clin-ical studies are analysed. We demonstrate that parameters from nonlineardynamics are useful for risk stratification after myocardial infarction, forthe prediction of life-threatening cardiac events even in short time series,

160 Herzschrittmachertherapieund Elektrophysiologie,Band11, Heft 3 (2000)© Steinkopff Verlag2000

and for modelling the relationshipbetweenheart rate and blood pressureregulation.Thesefindings could be of importancefor clinical diagnostics,in algorithmsfor risk stratification,and for therapeuticandpreventivetoolsof next generationimplantablecardioverterdefibrillators.

Key words Heartratevariability – blood pressurevariability – baroreflex– nonlinearcoupling– nonlineardynamics– implantabledefibrillator

Introduction

Annually, in the United Statesap-proximately400000 peopledie dueto sudden cardiac death (4, 11).Therefore,an accurateand reliableidentificationof patientswho are athigh risk of suddencardiacdeathisan important and challengingprob-lem. Heart rate variability (HRV)parameters,calculatedfrom the timeseries of the beat-to-beat-intervals,have beenusedto predict the mor-tality risk in patientswith structuralheartdiseases(26, 57). We havere-centlydemonstratedthata multivari-ate approachwith HRV parametersincluding nonlinearmethodsas wellas the combination of HRV mea-sures with clinical parameterslikethe ejectionfraction, the complexityof ventricular arrhythmias or thesignal-averaged electrocardiogramsignificantly improvesthe resultsofrisk stratification(59).

Physiological data very oftenshowcomplexstructureswhich can-not be interpreted immediately. Thedata we are analysing here aremainly heartrateandblood pressurevariability (BPV) time series. Ourmain intentionin this contributionisto discuss different approachestoobtain a betterunderstandingof theunderlying processes. In formerstudieson the one hand linear dataanalysis, especially correlation andspectral analysis (21), and on theother hand well-known nonlinearparameters,like correlation dimen-sion or Lyapunov exponents,wereused(2, 3, 17, 18, 27, 33). Amongthe variety of nonlinearapproachesthere are only a few studieswhichassessthe clinical usefulnessof non-linearmeasuresfor high risk stratifi-cation basedon heart rate or blood

pressurevariability analysis(25, 32,35, 59, 60). No extensiveprospec-tive study has been performed toshowthe efficacy of thesemethods.

The disadvantageof the linearparametersis the limited informa-tion about the underlying complexsystem, whereasthe nonlinear de-scription suffers from the curse ofdimensionality. Mostly, therearenotenoughpoints in the (often non-sta-tionary) time seriesto reliably esti-mate these nonlinear measures.Therefore,we favour the measuresof complexity which characterisequantitativelythe dynamicseven inrathershort time series(30, 59, 60,68). The measuresof complexityweareusing in this paperarebasedonsymbolic dynamics, renormalisedentropy and the finite time growthrates. Another main point in thiscontributionare methodsto analysebivariatedata.We describethe dualsequencemethod to estimate thebaroreflexsensitivity and the maxi-mal correlationmethod to measurethe coupling betweentime series.Afurther interestingfield are synchro-nous effects like cardiorespiratorycontrol (24, 52). The applicationofnew sophisticated synchronisationmethods(23, 50, 52, 55) seemstobe very promising to the heart rateand blood pressurevariability datasets,but this is beyondthe scopeofthis paper.

The analysis of heart rate orblood pressurevariability (BPV) isoften difficult due to excessivearte-facts and arrhythmias.While occa-sional ectopicbeatsare treatedsuc-cessfully by most preprocessingmethods,morecomplexarrhythmiasor arrhythmiaswhich are similar tonormal HRV fluctuations may re-main untreated.Therefore, prepro-

cessing of the data is presentedcomprehensively.

The paper is organised as fol-lows. In the Dataprocessingsectionwe describethe preprocessingpro-cedure.In sectionMethodswe pre-sentparametersfrom nonlineardataanalysis based on symbolic dy-namics,renormalisedentropyandfi-nite-timegrowth rates.Additionally,the dual sequenceanalysis of thespontaneousbaroreflexand the con-cept of maximal correlationare in-troduced,followed by the resultsofthe data analysesand a discussionof our results.

Data preprocessing

The main objective in the analysisof heart rate and blood pressureisto investigatethe behaviourof thecardiovascularsystem.Therefore,itis necessaryto excludenot only ar-tefacts(e.g. double recognition, i.e.R-peak and T-wave recognisedastwo beats)but alsobeatsnot comingfrom the sinus node of the heart(ventricular prematurecomplexes–VPC). VPCs are not controlled bythe autonomous nervous systemand, thus, they are not part of theautonomousregulation. Practically,this exclusion meansa filtering ofthe time series. In this paper theoriginal time seriesare denotedasthe RR-series(derivedfrom the RR-intervals)andthe filtered time seriesas NN-series (normal-to-normal-beat-interval).

VPCsin the tachogramareusual-ly characterisedby a very short in-terval followed by a very long RR-interval (ventricularprematurebeat)or, respectively, only by a very shortinterval (superventricularprematurebeat). The 20% filter (19, 26, 72)considersthesefacts; if the currentvalue of the tachogramdiffers morethan 20% from its predecessorthecurrent value and its successoraremarkedas not normal. The advan-tageof this filter is the very simplerule; however, the disadvantagesare

rarely considered.VPCs with lessthan 20% difference are not re-movedfrom the seriesand may fal-sify almostall HRV or BPV parame-ters.The RR-intervalsrecognisedasnot normal are treated in differentways: either they are simply re-moved from the seriesor linear orspline interpolated(5, 31). The dis-advantageof simply removing thebeatsis the lossof time dependence.Interpolating linearly may lead tofalsedecreasedvariabilities, interpo-lating with splines often fails intime serieswith manyVPCs.

In the following a new adaptivefiltering algorithmis presentedwhichis basedon theideaof theintervalfil-ter describedin Wesselet al. (67).The new filtering algorithmconsistsof three sub-procedures: (i) the re-moval of obviousrecognitionerrors,(ii) the adaptive percent-filter, and(iii) the adaptivecontrollingfilter.

• Obvious misrecognitionsare RR-intervalsof lengthzero,beat-to-beatintervals less than 200ms (humanrefractory time) and pauses, i.e.when the heartdoesnot pump for acertain time. These pauses wereevaluatedwith every clinical Holtersystem;therefore, we do not consid-er them.• The adaptive filtering procedurewas developedbasedon the adap-tive meanvaluela and the adaptivestandarddeviationra. Firstly, to es-timatethe basicvariability in the se-ries a binomial-7-filtered series iscalculated.Given the tachogramx1,x2, . . . , xN, the binomial filtered se-ries is given by

The filtered series t1, t2, . . . , tN re-flects the global behaviourof sinusnode activity without the influenceof artefactsand VPCs.The adaptivemean value la and the adaptivestandarddeviation ra of the bino-mial filtered series t1, t2, . . . , tN aredefinedas:

la�n� :�la�nÿ 1� ÿ c�la�nÿ 1� ÿ tnÿ1�

�2�ra�n� :�

��la�n�2 ÿ ka�n�

q�3�

where c is a controlling coefficientc∈ [0,1] and ka(n) is the adaptivesecondmoment

ka�n� :� ka�nÿ 1�ÿ c�ka�nÿ 1� ÿ tnÿ1 � tnÿ1�

�4�

The exclusion rule of this filterreadsas follows: the RR-intervalxnis classifiedasnot normal,if

jxn ÿ xnÿ1j > p

100xnÿ1

� cf � �ra and

jxn ÿ xlvj > p

100xlv � cf � �ra

�5�

wherep is a proportionallimit (here10%), cf � �ra is a generalised3r-rule, xlv is the last valid RR-intervaland �ra is the averagedra. Valuesrecognisedas not normal are re-placedwith a randomnumberfrom�la�n� ÿ 1

2 ra�n�; la�n� � 12 ra�n�� to

avoid falsedecreasedvariabilities.• Finally, as a precaution,the adap-tive controlling procedurefollows.From the resulting time seriesx%

1 ; x%2 ; x

%3 ; . . . of the adaptiveper-

cent-filter, the binomial filtered se-ries and the respective adaptivemean value and standarddeviationare again calculated.The value x%

nis consideredto be not normalif

x%n ÿ la�n�

�� > cf1 � ra�n� � rb ;

�6�

where cf1 is the filter coefficient(here cf1=3.0) and rb standsfor abasic variability (for HRV hererb=20ms). This basicvariability rbwas introduced to reduce filteringerrorsfor time serieswith low varia-bility (near the accuracyof RR-in-terval detection).Not normal values

arereplacedwith the respectiveval-uesof the binomial filtered series.

The advantageof this adaptivefiltering procedure is the sponta-neousadaptationof the filter coeffi-cientsdue to suddenchangesin theseries (e.g. sudden heart rate in-crease).

Methods

In this sectionmethodsof nonlineardata analysis are presented.ThesubsectionsA–C refer to univariatedata records,whereasD and E areaboutbivariatedataanalysis.

Symbolicdynamics

Symbolic dynamicswas introduced,asfar aswe know, asearlyasHada-mard’s (20) ideasto analysecompli-catedsystemsin 1898.An importantpoint of this work was a simplede-scription of the possiblesequencesthat can arise in geodesicflows onsurfacesof negative curvature. Heintroduceda finite set of forbiddensymbol pairs and defined possiblesequencesas thosethat do not con-tain any forbidden symbol pair.Later on, Hedlund and Morse (40,41) used this method to prove theexistenceof periodic and other dy-namicsin differentclassicaldynami-cal systems.They showed that inmanycircumstancesa finite descrip-tion of a system’s dynamicsis pos-sible. Symbolic dynamicstoday re-presentsa rapidly growing and es-sential part of dynamical systemsanalysisand its applicationto phy-siology (7, 8, 14, 30, 60, 44, 47, 48,70).

Heart rate and blood pressurevariability reflect the complexinter-actions of many different controlloops of the cardiovascularsystem.In relation to the complexity of thesinus node activity modulation sys-tem, a more predominantly non-linear behaviourhasto be assumed.Thus, the detailed description and

161N. Wesselet al.Nonlinearanalysisof complexphenomenain cardiologicaldata

tn � xnÿ3 � 6xnÿ2 � 15xnÿ1 � 20xn � 15xn�1 � 6xn�2 � xn�3

64�1�

classification of dynamic changesusing time and frequencymeasuresis often not sufficient. Therefore,wehave introduced new methods ofnonlinear dynamics derived fromthe symbolic dynamics to distin-guishbetweendifferentstatesof the

autonomicinteractions(30, 60). Thefirst step of this approach is thetransformation of the time seriesinto symbolsequenceswith symbolsfrom a given alphabet.Some de-tailed informationis lost in this pro-cess,but the coarsedynamicbehav-iour can be analysed.Wackerbaueret al. (65) usedthe methodology ofsymbolic dynamicsfor the analysisof the logistic map wherea genericpartition is known. However, forphysiological time seriesanalysisamore pragmaticapproachis neces-sary. The transformations into sym-bols have to be chosenon a con-text-dependentbasis. For this rea-son, we developed measures ofcomplexity on the basis of suchcontext-dependent transformations,which have a close connectiontophysiological phenomenaand arerelativelyeasyto interpret.

Comparing different kinds ofsymbol transformations,we foundthat the useof four symbols,as ex-plained in Eq.(7), was appropriatefor our purpose.

The time seriesx1,x2,x3, . . . ,xN istransformed into the symbol se-quences1,s2,s3, . . . ,sN, si ∈A on thebasis of the alphabetA={0,1,2,3}.The transformation into symbolsre-fers to three given levels where ldenotesthe meanbeat-to-beatinter-val anda is a specialparameterthatwe havechosen0.05;we testedsev-eral valuesof a from 0.05 to 0.08,but the resulting symbol sequencesdid not significantly differ (seeFig. 2),

There are several quantities thatcharacterisesuch symbol strings.We analysethe frequencydistribu-tion of words of length 3, i.e. sub-strings which consistof three adja-centsymbolsleadingto a maximum64 different words (bins). This is a

compromisebetweenretaining im-portant dynamical information andof having robust statistics to esti-mate the probability distribution(compareFig. 2).

We consider3 measuresof com-plexity:• The Shannonentropy Hk calcu-lated from the distribution p ofwords is the classicmeasurefor thecomplexityin time series:

Hk � ÿX

x2Wk;p�x�>0

p�x� log p�x�

�8�whereWk is the set of all words oflength k. Larger valuesof Shannonentropy refer to higher complexityin the corresponding tachogramsandlower valuesto lower ones.• ‘Forbiddenwords’ in the distribu-tion of words of length 3 are thenumber of words which never (oralmostnever)occur. A high numberof forbiddenwords reflectsa ratherregularbehaviourin the time series.If the time seriesis highly complexin the Shannoniansense,only a fewforbiddenwordsarefound.• The parameter‘plvar10’ character-ising short phasesof low variabilityfrom successivesymbolsof anothersimplified alphabet B, consistingonly of symbols ‘0’ and ‘1’. Here‘0’ standsfor a small differencebe-tween two successiveRR-intervals(the resolution of the defibrillatorsusedin this study), whereas‘1’ re-presentscaseswhen the differencebetween two successiveRR-inter-

vals exceedsa certainlimit, specifi-cally

sn � 1 : jxn ÿ xnÿ1j � 10 ms0 : jxn ÿ xnÿ1j < 10 ms

��9�


Fig. 1 The procedureof datapreprocessing.From the original time series(a) a binomialfiltered time series(b) is calculatedto esti-mate the basicvariability (compareEq. (1)).From the filtered time series(b) the adaptivemeanvalue (c) and the adaptivestandardde-viation (d) are estimatedto quantify localchangesin variability (compareEqs.(2) and(3)). With the exclusionrule given in Eq. (5)basedon proportionaldeviationsfrom prede-cessorsmany VPCs and artefactscan be re-moved from the original time series(e). Fi-nally, using the adaptive controlling proce-durefrom Eq. (6) almostall remainingdistur-bancescanbe eliminated(f)

Abb. 1 Die Datenvorverarbeitung.Von deroriginalenZeitreihe(a) wird einebinomialge-filterte Reihe(b) zur Bestimmungder Grund-variabilitat berechnet(vgl. Formel (1)). Vonder gefiltertenZeitreihe(b) wird der adaptiveMittelwert (c) und die adaptiveStandardab-weichung(d) zur Quantifizierungvon lokalenVariabilitatsveranderungen bestimmt (vgl.Formeln (2) und (3)). Mit der Ausschlussre-gel, welche in Formel (5) gegebenist undwelcheauf prozentualenAbweichungenvomVorgangerbasiert,konnenviele Extrasystolenund Artefakte beseitigt werden (e). Absch-ließend,unterBenutzungdesadaptivenKon-trollfilters ausFormel (6) konnennahezuallerestlichenStorungenbeseitigtwerden(f)

si�xi� �0 : l < xi � �1� a�l1 : �1� a�l < xi < 12 : �1ÿ a�l < xi � l3 : 0 < xi � �1ÿ a�l

where i � 1; 2; 3; . . .

8>><>>: �7�

Wordsconsistingof uniquetypesofsymbols (either all ‘0’ or all ‘1’)were counted.To obtain a statisti-cally robust estimateof the worddistribution, we chose words oflength six, defining a maximum of64 different words. ‘Plvar10’ repre-sents the probability of the word‘000000’ occurrenceand thus de-tects even intermittently decreasedHRV.

Renormalised entropy

Another nonlinear method whichmight be capableof assessingcom-plex propertiesof cardiac periodo-gramsis the ‘renormalised entropy.’The basic idea is to determinethecomplexity of cardiacperiodogramsbasedon a fixed reference.Basedon generalconsiderationsin thermo-dynamics, Klimontovich suggestedcomparingthe relativedegreeof or-der of two differentdistributionsbyrenormalisingthe referencedistribu-

tion to a given energy. Saparin(51)proposeda procedurefor calculatingthis quantity from time series andappliedit to the logistic map.Appli-cations of renormalisedentropy tophysiological data were previouslyintroduced(28, 30, 60, 67). As theapplicationto heartratedatamethodsuffers from potentiallack of repro-ducibility, another method for thecomputationof renormalisedentro-py REAR basedon an autoregressivespectralestimationhasbeenrecentlydeveloped(66).

To comparethe relativedegreeoforder of two different distributions,the referencedistribution is renor-malisedto a given energy. The com-plexity of any distribution in rela-tion to a fixed referencedistributionis estimatedby solving an integralequation. Considering two tacho-grams (time series of beat-to-beatintervals) with the density distribu-tion estimatesf0�x� and f1�x� andusing the estimatef0�x� as a refer-ence, the renormaliseddensity dis-

tribution �f0�x� of f0�x� is definedas:

�f0�x� :� f0�x�TRf0�x�Tdx

�10�

whereT is the solution of the inte-gral equationZ

ln f0�x��f0�x�ÿf1�x��dx � 0

�11�The solution of Eq.(11) has to bedeterminednumerically. The renor-malisedentropyRE of the distribu-tion f1�x� is definedby the follow-ing interchangingalgorithm, whereS�f �x�� is the Shannonentropy ofdistributionf �x�, that is

S�f �x�� ÿZf �x� � ln f �x�dx

�12�Procedure:• CalculateD1 � S�f1�x��ÿS��f0�x�� with the distribution


Fig. 2 Symbolic dynamics– the dependenceon controlling parametera (see Eq.7). On the left side the original tachograms(notationsAS, AW, HV, NW) arepresentedfollowed by the respectiveword distributionsfor controllingparametersa equalto 0.05,0.06and0.08

Abb. 2 SymbolischeDynamik – die Abhangigkeit vom Kontrollparametera (vgl. Formel7). Links sind die Originaltachogramme(Bezeich-nungenAS, AW, HV, NW) dargestellt,gefolgtvon dendazugeho¨rigenWortverteilungenfur denParametera gleich0,05,0,06und 0,08

as the reference(f0�x� is renor-malised).The value of T is de-notedT1=T.

• CalculateD2 � S�f0�x��ÿS��f1�x�� with the distributionf1�x� asthe reference(f1�x� is re-normalised).The resultingT val-ue is denotedT2=T.

• If T1>T2, the distributionf0�x� isfound to be the more disorderedone (in the senseof renormalisedentropy – i.s.r.e.) and the renor-malisedentropyRE is definedasRE=D1. Otherwise(T1<T2) f1�x�is the more disordereddistribu-tion (i.s.r.e.) and the renormalisedentropyis RE=–D2.

Calculationof the renormaliseden-tropy requiresestimatingthe tacho-gram distributions. Here we use anautoregressive spectralestimationofthe filtered and interpolatedtacho-gram. A known problem of autore-gressivespectralestimationsis thebias which might appear even inidealised circumstances.To over-come this problem a sinusoidalos-cillation with a fixed amplitudeandfrequencywasaddedto the time se-ries. The amplitude of 40ms waschosento obtaina dominantpeakinthe spectralestimation,and the fre-quencywas set to 0.4Hz, which isthe upper limit of the high fre-quencyband (21) (compareFig. 3).

A spectraldensityestimationin theinterval [0,0.42]Hz was usedto in-clude all physiological modulations,andthe calibrationpeak.Using a re-ference tachogramfrom a healthysubject with normal low and highfrequency modulations the REARmethodis designedso that either adecreasedHRV or a pathologicalspectrumleadsto positivevaluesofrenormalisedentropy.

Finite time growth rates

Lyapunovexponentsof a dynamicalsystemreflect effective growth ratesof infinitesimaluncertaintiesover aninfinite duration, yet, time seriesanalysisis restrictedto the analysisof finite time seriesandthusit is dif-ficult to determineLyapunovexpo-nents reliably (13, 29, 43, 53, 69).Therefore,we concentrated on quan-tifying the state-dependentshort-term predictability through finite-time growth rates. Note that thesediffer from the finite-time Lyapunovexponents(1, 10, 34, 71, 73) aswell as the finite-time growth ratesdescribedin (42), both of which re-quire the knowledgeof the tangentmaps thus the equationsgoverningthe dynamics. In Ziehmann et al.(73) the significant differencesbe-tweenthesequantitiesarediscussed.

Our finite-time growth ratesare ap-proximationsbasedon the idea ofWolf et al. (69). Firstly, pseudo-phasespacesof the systemarecon-structed using delay coordinates(54). Their dimensionis denotedbyn and the fixed delay by s. Next,for each point in this constructedphasespaceXk � �xk�1s; xk�2s; . . . ;xk��nÿ1�s�, k � 0; . . . ;N ÿ �nÿ 1�sof the measured tachogramX � �x1; x2; . . . ; xN � the nearestneighbour �Xk is determined. �Xk isdefined as that statewhich has theminimal Euclideandistanceto Xk.kXu ÿ Xvk denotes the Euclideandistanceof the stateXu to Xv, i.e.

kXu ÿ Xvk

��Xnÿ1

j�1

�xu�js ÿ xv�js�2vuut �13�

Then the minimal distancedk to theIk stateis given by

dk � min

fkXk ÿ Xikji � 0; . . . ;N ÿ �nÿ 1�s;ji ÿ kj > �nÿ 1�sg;k � 0; . . . ;N ÿ �nÿ 1�s

�14�

andthe nearestneighbourby

�Xk � XmjkXk ÿ Xmk � dkf g;k � 0; . . . ;N ÿ �nÿ 1�s

�15�Note that the time lag of thenearest neighbour has to beat least one window length, i.e.ji ÿ kj > �nÿ 1�s andwe only con-sider points as neighbours if theirdistance to the base point is lessthan 10% of the maximumdistancebetweenany two points. Next, weanalysethe evolution of the statesXk and �Xk during the time T. Afterthese T steps we obtain thestates XT

k � �xk�T�1s; xk�T�2s; . . . ;xk�T��nÿ1�s� and �XT

k respectively.The distance between both stateskXT

k ÿ �XTk k representsthe diver-

genceafter T evolution steps.Fromthe original distanceof both states


Fig. 3 The estimationof the tachogramdistribution is the basisfor calculatingrenormalisedentropy. Here,theautoregressivespectraldistribution(dotted)togetherwith thecalibrationpeakat0.4Hz wasused.Givenafixed referencedistribution,thecomplexityof all distributionswhichshouldbe analysedwill be determinedrelativeto this reference

Abb. 3 Die Schatzung der Tachogrammverteilungdient als Grundlagezur BerechnungderrenormiertenEntropie.Hier wurdeeineautoregressiveSpektralscha¨tzung(gepunktet)unterEin-satzeinesKalibrierungssignalsbei0,4Hz benutzt.Ist eineReferenzverteilungfestgelegt,sowirddie Komplexitat jederzu untersuchendenVerteilungrelativ zu dieserReferenzbestimmt

and the distanceafter T steps wecalculatethe finite-time growth ratek�n;s;T�k :

k�n;s;T�k

� 1

TlnkXT

k ÿ �XTk k

kXk ÿ Xkkk � 0; . . . ;N ÿ �nÿ 1�s

�16�

k�n;s;T�k quantifies the local short-

term predictabilityat the point Xk. Ifthesek�n;s;T�k are greaterthan zero,the distanceafter the evolution timeincreases;otherwise,it decreases.

Wecalculatethefinite-timegrowthratesfor eachpointof thedelayphasespace,which leadsto a growth ratetime seriesk�n;s;T�k . Its average,theaveragegrowth ratek�n;s;T�k ,

k�n:s;T�k � 1

N ÿ �nÿ 1�s� 1

�XNÿ�nÿ1�s

k�0

k�n:s;T�k

�17�

quantifies a global short-term pre-dictability. The dimensionn, i.e. thelength of the selected tachogram,part varied from 3 to 9. We chosethis rangeto coveran interval up to9, which is a typical order of anautoregressive model for short-termHRV tachograms(67). The evolu-tion time and the delay are definedas T=1,2,3 and s=1,2,3 respec-tively.

Additionally, to reduce randominfluences,we considera threeanda five nearestneighbourapproach.According to Eq. (15) we deter-mine the five nearest neighboursI1k ; . . . ; I5

k of the point Ik andevolveall neighbours over the evolutiontime T. The finite-time growth ratesfor the three and the five nearestneighbour approach are derivedfrom the averagedistancesbeforeandafter the evolutiontime.

Dual sequencemethod

The dual sequencemethod (DSM)(36, 37) was developedfor the esti-mation of the spontaneous barore-

flex sensitivity from the bivariateheart rate and blood pressuredata.Two kinds of RR-interval (beat-to-beat interval) responseswere ana-lysed: bradycardic (blood pressureincrease causes RR-interval in-crease)and tachycardic(blood pres-suredecreasecausesRR-intervalde-crease)fluctuations,whereprimarilythe bradycardic fluctuations repre-sent the vagal spontaneous barore-flex (12, 39). The analysisof the ta-chycardicfluctuationsallows the in-vestigationof the relation betweenthe vagally (bradycardic) and sym-pathetically (tachycardic) mediatedfluctuationsof heartrate.

Applying the DSM, the barore-flex sensitivity slopes were deter-mined from threeconsecutivebloodpressureandRR-intervalvalues(seeFig. 4). Sequencesof more thanthree valueswere proved not to beuseful,becauseof the shortdurationof spontaneous heartrate and bloodpressurefluctuations, in contrasttopharmacologically induced changes(46). The calculated slopes weregroupedinto defined slope classes.Additionally, the total number andthe percentageto the total numberof all slopeswerecalculated.

Former studies showed that theheartrate doesnot respondsimulta-neously to the blood pressurefluc-tuation.Obviously, the heartratere-sponsedependson the different ve-locity of vagaland sympatheticreg-

ulation (38, 39). In contrastto othersequencemethods(6, 45), the timeseriesareshiftedto registerthe syn-chronous and the postponedheartrate responseon the sameBP fluc-tuation.

The following parametergroupsarecalculatedby DSM:• The total numberof slopesin the

different sectorswithin the timeseries,

• The percentageof the slopes tothe total numberof slopesin thedifferentsectors,

• The numbersof bradycardicandtachycardicslopes,

• The shift operationfrom the firstto the third heartbeattriple and

• The averageslopesof all fluctua-tions.

Theseparametersare calculatedforbradycardicaswell asfor tachycardicfluctuationsup to a delayof 80 val-ues to analyse the long-term andmaximal responseof the heart rateto the sameblood pressureoscilla-tion. Additionally, parametersfromHRV andBPVanalysisarecalculatedto investigatea possiblecommonap-plicationto risk stratification.

Nonlinearregressionandoptimal transformations

Optimal transformations and the as-sociatedconceptof maximal corre-


Fig. 4 The baroreflexregulatorybehaviour:the points reflect the correspondingdiscretesys-tolic blood pressureand RR-intervals,the regressionlines demonstratethe estimationof therealslopes(curves)

Abb. 4 Die Baroreflexregulation:die Punktereprasentierendie entsprechendendiskretensys-tolischenBlutdruck- und RR-Intervall-Werte,die Regressionsliniedemonstriertdie Schatzungder realenAnstiege

RR-interval [ms]

lation provide a nonparametric pro-cedureto detectand determinenon-linear relationshipsin bivariatedatasets.Let X and Y denotetwo zero-meandatasetsand

R�X ;Y� � E�XY ��E�X2�E�Y2�p �18�

their (normalised)linear correlationcoefficient, whereE[.] is the expec-tation value. The basic idea of thisapproachis to find suchtransforma-tionsH�Y� andU�X� that the abso-lute value of the correlationcoeffi-cient betweenthe transformedvari-ables is maximised. This leads tothe maximal correlation (16, 22,49).

W�X ;Y�:� supH;UjR�H�Y�;U�X��j

�19�The functionsH�Y� and U�X� forwhich the supremumis attainedarecalledoptimal transformations. Thisconceptgeneralisesthe linear corre-lation, where the linear correlationcoefficient R(X,Y) quantifies onlylineardependencies

Y � aX � g �20�while W�X;Y� quantifiesnonlineardependenciesof the form

H�Y� � U�X� � g �21�Especially, if there is completesta-tistical dependence(49), i.e. Y is afunction of X or vice versa, themaximal correlation attains unity.This is alsotrue for the relation(21)with g � 0. Here we are mainly in-terestedin the estimationof the op-timal transformations for the multi-variateregressionproblem

H�Y� �U1�X1� � . . .

�Um�Xm� � g�22�

This is an additive model for the(not necessarilyindependent)inputvariables Y,X1, . . . ,Xm. The regres-sion functionsinvolved can be esti-mated as the optimal transforma-tions for the multivariate problemanalogousto Eq. (19). To estimatethem nonparametrically, we use the

Alternating Conditional Expectation(ACE) algorithm (9). This iterativeprocedureis nonparametric becausethe optimal transformations are esti-mated by local smoothing of thedatausingkernelestimators. We usea modified algorithm1 in which thedataare rank-orderedbeforethe op-timal transformations are estimated.This makesthe result less sensitiveto the datadistribution.It shouldbementionedthat optimal transforma-tions for multiplicativecombinationsof variables ~X1; . . . ; ~Xl can be esti-matedby forming

Xi � hi� ~X1; . . . ; ~Xl� �23�wherethe hi arearbitraryfunctions.

The maximal correlationand op-timal transformationshavebeenap-plied to identify delay (61) and par-tial differential equations (63). Inthe applicationto differential equa-tions, time derivativeshaveto be es-timatedfrom the data.The effect ofnoise on the estimated optimaltransformations is not yet complete-ly understood.On the one hand,forneglectableamounts of noise thisapproachhas successfullybeen ap-plied to experimentalphysical dataof differentorigin (62, 64), yieldingeven quantitativelyaccurateresults.On the otherhand,if noisecontami-nation is strong,this methodshouldbe used as an exploratory tool inthe processof model selection.Tominimise the influenceof the noise

on the results,the variablewith thebest signal-to-noise ratio, usuallythe undifferentiated time series,shouldbe chosenasY.

Results

Preprocessing

To test the accuracyof the filteringprocedure, tachograms of normalhealthy personswere superimposedby simulated ventricular prematurecomplexes(VPC) of differentkinds.Applicablepreprocessing proceduresshould be able to remove thesesuperimposedsignals to obtain ap-proximately the sameseriesas be-fore. BesidessingularVPCs,severalcomplexesof trigeminus and bige-minuswerealsosuperimposed.Nextthe HRV parameterfrom the timeand frequency domain as well asfrom nonlineardynamicswere cal-culated.The parametervaluesof theoriginal unfiltered time serieswerecomparedwith the valuesof the fil-tered superimposed series. Themeanerror, given in Table1, is cal-culatedas the meanvalueof the re-lative errors of all parameters.Therelativeerror is the deviationof theparametercalculated from the fil-tered to the original unfiltered pa-rametervalue.As one cansee,evenfor trigeminusperiods the adaptivefiltering procedureseemsto workreliably. On the contrary, the resultsof the 20% filter demonstratethatthis filter is not sufficient for anaccuratetime series analysis. The


Table 1 The meanerror of HRV parametersfor two different filtering procedures(seetext)anddifferentsuperimposedsignals

Tab. 1 Der mittlere Fehlerder HRV-Parameterfur zwei verschiedeneFilterprozeduren(vgl.Text) und verschiedeneuberlagerteSignale

Meanerror [%] Adaptivefilter 20% filter

VPC 100ms premature 5.99 13.55VPC 200ms premature 5.27 18.68VPC 300ms premature 6.34 16.43Trigemini 300ms premature 8.10 57.89Bigemini 300ms premature 17.64 92.36

1 A MATLAB – and a C-programfor theACE algorithm can be obtained from theauthors or from the web page http://www.fdm.uni-freiburg.de/*hv/hv.html.

meanerror of the adaptivefilter in-creasesrapidly for episodesof bige-minus – the filter should not beused in such cases.However, withthis filtering procedurewe obtainmuch better results; therefore, theadaptivefiltering procedureis usedhere.

Symbolicdynamicsin a large clinical study

In an extensiveclinical studya mul-tiparametric heart rate variabilityanalysiswas performed(59) to testif a combinationof HRV measuresimprovesthe resultof risk stratifica-tion in patientsafter myocardialin-farction. Standardtime domain,fre-quency domain and symbolic dy-namics measuresof HRV assess-ment were appliedto 572 survivorsof acute myocardial infarction. Ofthem 43 died in the follow-up timeof two years. Besidessome linearparameters almost all parametersfrom symbolic dynamics showedsignificant differencesbetween thefollow-up survivors and the de-ceased.A correlation analysis wasperformed (58) and showed thatHRV analysisbasedon symbolicdy-namics is an independentassess-mentandhasonly weakcorrelationswith time (0.38±0.13) and fre-quency(0.27±0.14) domainparam-eters.Finally, for this populationwecould show that a multiparametricapproachwith the combination offour parametersfrom all domainsisa betterpredictorof high arrhythmiarisk than the standardmeasurementof globalheartratevariability.

Renormalised entropyin a clinical pilot study

In a clinical pilot studythe renorma-lised entropy REAR was applied todata of 18 cardiacpatientsand 23healthysubjects.The cardiacpatientgroup consisted of patients aftermyocardial infarction with docu-mented life threateningventricular

arrhythmias;10 of themweresurvi-vors of a suddencardiacarrestwhoreceivedautomaticimplantable defi-brillators.From the groupof healthysubjects,the most disorderedtacho-gram(i.s.r.e.) wasdetermined as thereference for REAR calculation(REAR=0 for healthy person no.16). Figure 5 shows the result ofthis clinical pilot study. Renorma-lised entropy REAR correctly classi-fied 13 of 18 high risk patients(greater than zero or under thedashed line at –0.33). The pre-viously introduced method (60)classifiedcorrectly only 7 high riskpatients.

Finite-timegrowth ratesandsymbolicdynamicsin forecastingcardiacarrhythmias

Recentstudiesshowedthat HRV as-sessesthe autonomic nervous sys-tem dysfunctionand is able to iden-tify patientsat risk of suddencardi-ac death (15, 21); however, thesemethodswerenot developedto per-form short-termpredictions of ma-lignant ventriculararrhythmias.Theobjective of this study was to findearly signs of sustainedventriculartachycardia/ventricular fibrillation(VT/VF) in patients with an im-planted cardioverter-defibrillator(ICD). These devices are able tosafeguardpatientsby returningtheirheartsto a normalrhythmvia strong

defibrillatory shocks; additionally,they store the 1000 beat-to-beatin-tervalsimmediatelybeforethe onsetof a life-threatening arrhythmia.Weanalysedthese1000 beat-to-beatin-tervals of 17 chronic heart failureICD patientsbefore the onset of alife-threatening arrhythmiaand at acontrol time, i.e. without VT/VFevent (68). To characterise theserathershort datasets,we calculatedheart rate variability parametersfrom time and frequency domain,from symbolic dynamicsas well asthe finite-time growth rates. Wefound that neither the time nor thefrequencydomain parametersshowsignificant differencesbetween theVT/VF and the control time series.As already visible in Fig. 6, onlythe mean beat-to-beat intervalshoweda remarkablebut nonsignifi-cantdifferencebetweenthe groups.

Two parametersfrom symbolicdynamics,however, as well as thefinite-time growth ratessignificantlydiscriminateboth groups(seeTable2). The Shannon entropy of theword distributionHk is significantlyhigher in the control group,whereasthe low-variability measure‘plvar10’ is larger in the VT/VFgroup. Both parametersindicate apartly decreasedheart rate variabil-ity in the VT/VF group which can-not be detectedwith the standarddeviation or other variability mea-suresfrom time domain. It is inter-esting to note that the forbidden


Fig. 5 Resultsof renormalisedentropyin a clinical pilot study, black barsrepresentthe con-trol groupwhereasthe white barsrefer to the cardiacpatients

Abb. 5 Die Ergebnisseder renormiertenEntropie in der klinischen Pilotstudie, schwarzeBalkenreprasentierendie Kontrollgruppe,weißedie Herzpatienten

word statistics fail to distinguishboth groups.The finite-time growthrateswerecalculatedfor a five-near-est-neighboursapproach. The pa-rameterk�3;1;1� denotesthe averagegrowth rate with a dimension of3, lag 1, and evolution time 1;k�6;1;1�, k�9;1;1� respectively. For di-mensionn=3 therewere no signifi-cant differences,whereasthe aver-agegrowth ratesfor n=6 and9 dif-fered significantly in both groups.For evolution times T=2,3 signifi-cant differences disappeared for

n=6. Interestingly, for all dimen-sionsthe growth rateswerelarger inthe group of the control tachogramsthan in the VT/VF group, indicatingan enhancedpredictability or lesscomplex dynamics in the VT/VFgroup. Additionally, we calculatedthe averagegrowth ratesfor differ-ent delay times s=2,3, evolutiontime T=1, and n=3,. . . ,9. For s=2we found a significantdifferenceforn=6 (p=0.016). For s=3 we onlyrecogniseda considerablebut non-significantdifferencefor (p=0.052);

no further growth ratesshowedsig-nificant differencesbetweenboth in-vestigatedgroups.

Dual sequencemethodin patientswith dilatedcardiomyopathy

The baroreflex sensitivity (BRS) isan important parameterto classifypatientswith reducedleft ventricularfunction. This study aimedat inves-tigating the BRS to differentiatebe-tween patientswith dilated cardio-myopathy (DCM) and healthy per-sons (controls) as well as to com-pare the BRS with heart rate andblood pressurevariability.

In 27 DCM patientsand 27 ageand gender matched controls theelectrocardiogram, continuousbloodpressure, and respiration curveswere recorded.The dual sequencemethod(DSM) includesthe analysisof the spontaneousblood pressurefluctuations and the correspondingRR-intervals of heart rate to esti-mate the bradycardic baroreflexfluctuationsas well as the oppositetachycardicfluctuations.We found areducednumber of heart rate andblood pressurefluctuations in DCMpatients compared to the controls(DCM: male: 154.4±93.9, female:93.7±40.5, controls: m: 245.5±112.9, f: 150.6±55.8, p<0.05). Theaverageslope in DCM patientswaslower than in controls (DCM:5.2±1.9,controls:8.0±5.4,p<0.05).

A correlation analysis was per-formed to investigatethe differentHRV, BPV and BRS parametersforlinear dependence(21, 60). In gen-eral, the average correlation waslow betweenthe differentparametersets (r =0.22). However, severalhighly correlated parameterswerecalculatedbetweenBRS and HRV(time domain, frequency domain,nonlinear dynamic), and betweenBRS and BPV, respectively (onlyfrequencydomain).

Furthermore,the stepwisediscri-minant function analysis was usedto find the optimal combinationsofHRV, BPV and BRS parametersto


Fig. 6 The last 1000 beat-to-beatintervalsbeforea sustainedVT (a) and the respectivecon-trol time series(b) from the samepatient.To analysethe dynamicsjust beforean arrhythmia,all beat-to-beatintervalsof the VT itself at the end of time series(a) are removedfrom thetachograms

Abb. 6 Die letzten 1000 RR-Intervallevor einer anhaltendenVT (a) und die zugeho¨rigeKontrollzeitreihe(b) desselbenPatienten.Um die Dynamik kurz vor der Arrhythmie zu bes-timmen,wurdenalle RR-Intervalleder VT selberam Endeder Zeitreihe(a) geloscht

Table 2 Resultsof finite-time growth ratesand of parametersfrom symbolic dynamicsindiscriminatingthe controlandVT/VF series

Tab. 2 Ergebnisseder ,finite-time‘ Wachstumsratenund der Parameterder symbolischenDy-namikbei der Trennungvon Kontroll- und VT/VF-Zeitreihen

VT/VF Control p

k�3;1;1� 0.37± 0.31 0.51± 0.36 n.s.k�6;1;1� 0.22± 0.08 0.26± 0.05 <0.05k�9;1;1� 0.11±0.03 0.13± 0.02 <0.05Forbiddenwords 32.26± 10.50 32.16± 9.30 n.s.Hk 2.13± 0.59 2.43± 0.43 <0.05Plvar10 0.12± 0.18 0.04± 0.05 <0.05

differentiatebetweenDCM patientsand healthy persons.Thesecombi-nationsof parametersare presentedin Fig. 7.

A correct classification of 74–86% was obtainedwith setsof sixparametersfrom HRV, BPV andBRS. Theseparametersets containmainly threetypesof informationtodiscriminatebetweenDCM patientsandhealthypersons:1) the high fre-quencyin HRV, 2) the meanvaluesand low frequencyof BPV and 3)the largestslopesandthe numberofBR fluctuations.

A setof six parametersfrom HRVandBPV increasedthecorrectclassi-fication up to 88%.Furthermore,theaccuracyof classification could beimproved using additionalBRS pa-rameters.Using a setof six parame-tersof BPV andBRS a correctclas-sification of 94% was calculated,whereasusinga setof six parametersof HRV and BRS the classificationaccuracycould be improved up to96%. The use of a six parametersset of HRV, BRV and BRS did notfurtherimprovethecorrectclassifica-tion greaterthan96%.

Optimal transformationsfor revealingcouplingdirections

To learn somethingabout the cou-pling between blood pressureand

heartrate,we adoptedthe view thatwe do not know anythingaboutthedynamical description of the non-linear processesthat describe thetime seriesof heart rate, xt, and ofblood pressure,yt, and the couplingbetweenthem. In other words, weusedthe statisticalprinciple of max-imum a-priori uncertaintywhich isthe basisof manyparameterestima-tion methods.

However, we need at least arough model assumption.Since thedataareheavily high-pass-filtered toinvestigateonly vagally (respiratory)inducedfluctuations,long-rangecor-relations are filtered out and wetherefore assume that the systemdescribing the change of heartrate can be written as a mapxt�1 � F1�xt� � g, where F1 is anarbitraryfunction g andsomenoise.Furthermore,we assume that thevariable x is a scalar quantity; allthe following considerationscan beeasilyextendedto the higher-dimen-sional case. Since the method, asappliedto heartrateandblood pres-sure works well for the simplestcaseof scalarstatevariables,we donot go into detail with respect tothis extension. Including the cou-pling betweenx and y, the overallsystemis assumedto be

xt�1 � F1�xt� � F2�xt ÿ yt� � gx

�24�yt�1 � G1�yt� � G2�yt ÿ xt� � gy

�25�The functions F2 and G2 are cou-pling functions that dependon thedifferencebetweenthe two systems,an assumptionthat is often met inmodelling synchronising systems.Now, using the principle of maxi-mum a-priori uncertainty, the func-tions F1, F2, G1 and G2 are esti-mated from pairs of time series,using the maximalcorrelationmeth-od. We use a modified ACE-algo-rithm where the function H in Eq.(21) is fixed to be the identity func-tion sincein the model (24,25) it isnot needed.

For each of the four analysedpairs of heart rate and blood pres-sure,we find a maximal correlationvery close to unity (W>0.98) withthe chosensmoothing kernel. Thisshows that the model can explainmostof the varianceobservedin thedata; physiologicallyspeakingthereis nearly a one-to-onerelationshipbetween respiratory induced varia-bility in blood pressureand in heartrate. The results for the four ana-lysed pairs of time series are de-pictedin Fig. 8.

The functions F1, and G1, de-scribing the dynamics of the un-coupledsystems,are close to linearand are not shown. The couplingfunctions F2�xt ÿ yt�, however,show a strong nonlinear behaviourwhereas the functions G2�yt ÿ xt�arenearlyflat. This is a strongindi-cation that, assumingthe crude as-sumption of Eqs. (24, 25) whichshould be in first order a goodapproximation to the observeddynamics,the heart rate is stronglyinfluencedby the blood pressurebutnot vice versa.Therefore,the totalsystem is highly skewed, or, inother words, a drive-responsesys-tem whereblood pressureis the dri-ver and heartbeatthe response.Thisis in agreementwith the widelyacceptedexplanationfor respiratorysinusarrhythmiain heartrate,whichmostly reflects arterial baroreflexbuffering of respiration-induced ar-terial pressurefluctuations(56).

Discussion

We have introduced different non-linear approachesto heart rate andblood pressurevariability analysisand their application to differentclinical studies.

Symbolic dynamics is a usefultool in severalfields of complexityanalysisin science.We haveappliedsymbolic dynamicsmainly to char-acterisethe dynamicsof heart ratevariability. Symbolsrepresentlevelsof time differencesbetweensucces-


Fig. 7 Resultsof the discriminant functionanalysisusing different setsof 6 parameters.The black markedbarscontainDSM parame-ters

Abb. 7 Die Ergebnisseder Diskriminanza-nalyseunter Verwendungvon verschiedenenSatzen mit je 6 Parametern.Die schwarzmarkierten Balken enthalten ParameterderdualenSequenzmethode

sive beats. We showed that sym-bolic dynamics is an independentmethod(low correlationsto standardtime and frequency measures)andincreasesthe predictiveaccuracyinhigh risk stratification. Complexitymeasuresbased on nonlinear dy-namicsseparatebestbetweendiffer-ent patient groups becauseof thepartly nonlinear behaviour of theheart rate generation.The combina-tion of parametersof time and fre-quency domain and nonlinear dy-namicsleadsto an optimaldetectionrate of patientsafter myocardialin-farction with high risk for sudden

cardiac death. Moreover, symbolicdynamicsis a method with a veryclose connection to physiologicalphenomenaand is relatively easytointerpret.

The renormalisedentropymethodREAR is designedin sucha way thattachogramswith normal variabilityand typical periodogramshaveneg-ative values of REAR. Either a de-creasedHRV (under stressor evenat rest) or pathologicalspectraleadto positivevaluesof REAR. Patholo-gical spectraare periodogramswithdominant ULF, VLF or LF peaks.Dominant ULF or VLF peaks are

seenin tachogramswithout any res-piratory modulations and withoutany influenceof blood pressurereg-ulation. DominantLF peaksare re-cognisable in tachogramswithoutany long-term regulation and de-creasedvagalactivity. The resultsofthe St. George’s hospitalpostinfarc-tion databasestudy(66) (not shownhere) confirmed the hypothesisthatthe survivorson averagehavenega-tive and the deceasedpositive val-ues of renormalisedentropy. Calcu-lating renormalisedentropyassumesa fixed referencedistribution.In thisstudy the referencedistribution was


Fig. 8 Top row: For the four analyseddatasetsthe estimatedcoupling functionF2�xt ÿ yt� from Eq. (24), describingthe influenceof bloodpressureon heart rate. Bottom row: The samefor the other coupling direction, i.e., the function G2�yt ÿ xt� from Eq. (25), describingtheinfluenceof heartrateon blood pressure.For a bettercomparison,the plots in eachcolumn,correspondingto the samedataset,arenormalisedso that a quantitativecomparisonof the two plots in eachcolumncanbe performed.Onenoticesthat in the bottomrow the couplingfunctionsarealmostcompletelyflat; it seemsto be that thereis no influenceof heartrateon blood pressure

Abb. 8 ObereReihe:Fur die vier analysiertenDatensa¨tze sind die gescha¨tztenKopplungsfunktionenF2�xt ÿ yt� ausFormel (24) dargestellt,sie beschreibenden EinflussdesBlutdrucksauf die Herzfrequenz.UntereReihe:DieselbeDarstellungfur die andereKopplungsrichtung,d.h.die FunktionG2�yt ÿ xt� ausGl. (25), welcheden Einflussder Herzfrequenzauf den Blutdruck darstellen.Damit man die Ergebnissebesservergleichenkann,sind die Grafikenwelchezu einerPersongehoren normiert.Aus den Grafiken ist zu ersehen,dassdie unterenKopplungs-funktionenalle nahezuflach sind,dasbedeutet,dasseswenig Einflussder Herzfequenzauf denBlutdruckzu gebenscheint

selectedusing the interchangingal-gorithm applied to the data of 23healthypersons.In futureHRV stud-ies, it needsto be investigated if ageneralor a diseasedependentrefer-encedistributionshouldbe chosen.

The aim of the forecastingstudywas to find heart rate variabilitychanges just before the onset ofventriculartachycardiaor ventricularfibrillation (VT/VF), i.e., to look forsomeprecursor-like activitiesbeforethis qualitative change. Two ap-proachesfrom nonlinear dynamicsindeedexhibit significantchangesinheart rate variability: methods ofsymbolic dynamics and finite-timegrowth rates.It is importantto notethat standardlinear techniquesarenot able to discriminate betweentheseboth groups.From finite-timegrowth rates we obtain significantdifferencesbetweenthe VT/VF andthe control time seriesonly for rela-tively high dimension (n>5). Thisindicates that the changes aremainly causedby sympatheticactiv-ities of the autonomous nervoussys-tem. Vagal activities would affectsmaller dimensions. These resultsarein goodagreementwith the phy-siological fact that the sympatheticactivity increasesbeforethe onsetofVT. The significantly lower valuesof the finite-time growth ratesin theVT/VF group provide a possibilityto predict a VT/VF. The calculationwith different delayss for the em-bedding has shown the importanceof analysingbeat-to-beatvariability;no successiveheartbeatsshouldberemoved. For delays greater thanone, significant differences in theaverage growth rates decreased,which showsthat the differencesbe-tween the VT/VF time series andthe controls are causedby beat-to-beatregulation. Moreover, this studydemonstratedthe advantageof usingmethods from nonlinear dynamics.The evolution of points in phasespaceprovides deeper insight into

dynamical aspects of the cardiacsystem.Limitationsof this studyarethe relatively small numberof timeseries and the reduced statisticalanalysis(no subdivisionsconcerningage,sex,andheartdisease).For thisreason,theseresultshaveto be vali-datedon a larger database.

The analysis of the baroreflexsensitivity (BRS) is an establishedmethod for improved classificationof differentcardiacdiseases.In con-trast to standardBRS methods,thedual sequencemethod (DSM) con-sidersshiftedbradycardicandtachy-cardic fluctuations. The biosignalsof patientsare recordedat rest andwithout any additional challenge(e.g. applicationof drugs and otherstimuli). The analysis showed thatthere are significant differencesbe-tween DCM patients and healthypersonswith respectto the calcu-lated parameters.In DCM patientsand healthy personswe calculatedthe mean values of heart rate andblood pressureto determinethe dif-ference in the active range of theblood pressure.The determinationof the active range of the bloodpressure,the value of the averageRR-interval of DCM patients,wascalculated to be 795.9± 107.7ms.This value is considerably lowerthan in the control group (862.4±130.5ms, p<0.05). The values ofthe averageblood pressurein DCMpatients and in healthy personsbehave inversely to the heart ratevalues (DCM: 101.5±19.6mmHg,Controls: 112.9±25.7mmHg,p<0.05). Hencewe canseethat theautonomousregulationin DCM pa-tients seemsto be more ineffectivethan in healthypersons.This behav-iour is characterisedby the lowernumber of the baroreflex fluctua-tionsandby the lower averageslopeof BRS. In comparisonto the stan-dard sequencemethod, the DSMobtained an improved accuracyofclassification (84% versus 76%).

The considerationof DSM parame-ters in combinationwith HRV andBPV parametersleads to a signifi-cant increaseof discriminantpower(96%). Summarising, the dual se-quencemethod increasesthe infor-mationaboutthe cardiovascularsys-tem.

Revealingthe coupling directionand the strengthof couplingvia op-timal transformations is a new andpromisingapproach.No a priori as-sumptionsabout the dynamicalsys-tem that regulates heart rate andblood pressurehas to be made. Inthis contributionfirst the respiratoryinduced fluctuations were investi-gated and long-range correlationswere filtered out. The resultsof thefour bivariate data sets analysedhere strongly indicate and confirmthe knowledgeof respiratorysinusarrhythmia in heart rate: it reflectsbaroreflex buffering of respiration-induced arterial pressure fluctua-tions. In Taylor et al. (56) it wasalsoshownthat thereis an influenceof heart rate on blood pressurevariability. Therefore,we next haveto investigatethe influenceof high-passfiltering to coupling directions.However, the approach introducedhere seemsto be useful for deter-mining nonlinear coupling betweenheartrateandblood pressure. More-over, on a larger databasethe cou-pling directionshave to be investi-gatedmoreintensively.

In conclusion, this paperdemon-strated that parametersfrom non-linear dynamicsare meaningful forrisk stratification after myocardialinfarction, for the predictionof VT/VF eventseven in short-termHRVtime seriesandfor modellingthe re-lationship between heart rate andblood pressure variability. Thesefindings seemto be of importancefor clinical diagnostics, in algo-rithms for risk stratification and toimprove the therapeuticand preven-tive tools of next generationICDs.


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