monitoring complexity response atrial...

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Monitoring the Complexity of VentricularResponse in Atrial Fibrillation

H. KA_SMACHER*, S. WIESE and M. LAHL

Clinic of Anaesthesiology, Medical Faculty of the University of Technology Aachen,Pauwelsstr 30, D-52074 Aachen, Germany

(Received 24 October 1998)

Atrial fibrillation does not present a uniform extent of variability of the ventricular responseexemplifying periodicities and more complex fluctuations, due to varying number and shapeof atrial wavelets and aberrant conduction in the AV-junction. It was sought to categorisedifferent degrees of complexity introducing an uncomplicated monitoring method for thatobjective.The fluctuation of RR-intervals was investigated in 66 patients presenting with atrial

fibrillation at different times of the day using conventional statistical methods as well asmethods derived from non-linear dynamics. Specifically Poincar-plots were employed toanalyse the data. One hundred and seventy data sets consisting of 5000-8000 intervals eachwere considered.

Statistical methods were shown to describe the observed dynamics not adequatelydue to background noise in the acquired data as well as owing to intrinsic qualities of the datasets. Namely non-uniformly distributed data and trends within the data sets constitutedlimitations of statistical methods.

Poincar-plots were proven to be an inexpensive and effective method to categorise thecomplexity of the ventricular response in atrial fibrillation. Periodical variation of intervallengths could be clearly differentiated from continuous variation in the considered data sets.The latter could be shown to exemplify either stochastic fluctuation or complex non-lineardynamics of RR-interval variation. Thus, different degrees of complexity could be clearlydistinguished.

It could be shown exemplary that the observed dynamics remained nearly constant withinthe observation period.

Poincar-plots, thus, provide a means to appreciate the fluctuation of RR-intervals semi-quantitatively and to distinguish different patterns of fluctuation dynamics of the ventricularresponse in atrial fibrillation without being affected by contaminated or inconsistent data.

For a complete visualisation of the concerned dynamics relatively small data sets suffice.Thus, it is possible to observe the pattern of fluctuation essentially in real time which makesthem ideally suited for future investigation.

Keywords: Atrial fibrillation, Ventricular response, Poincar6-plot, Complexity analysis

Corresponding author.

63

64 H. KSMACHER et al.

INTRODUCTION

Since the beginning of the century extremelyirregular ventricular activity known as pulsusirregularis perpetuus is recognised to possess a

supraventricular origin namely atrial fibrillation or-flutter with irregular AV-conduction (S6derstr6m,1950). The disorder is quite frequent and its inci-dence is as high as 1% of the general populationand even higher among the elderly (Cameron et al.,1988; Heisel et al., 1995).The chief complaint is the unpleasant sensation

of a highly irregular pulse, whereas the outcome is

particularly determined by the hemodynamic con-

sequences and thromboembolic complications ofthe disease (Ferguson and Cox, 1993).

Several investigations deal with the mechanismsof atrial fibrillation and -flutter itself whereas theensuing ventricular response was rarely considered

(Lammers and Allessie, 1993; Allesie et al., 1990).It was recognised early that distinct intervalswere more frequent than others, thus, constitutingone or more peaks in the frequency distribution

(Honzikova et al., 1973; S6derstr6m, 1950).The interbeat interval in atrial fibrillation T can

be legitimately considered to be the summation ofthe refractory period of the AV-junction r, theAV-conduction time c and the time in which theAV-junction integrates enough temporal-spatialdistinct influences of the atrial wavelets to reach a

threshold and trigger the propagation of depolari-sation through the AV-junction, thus, T-r + c +(Cohen et al., 1983; Zipes et al., 1973).

Several electrophysiological investigations de-scribe the dependence of a given refractory periodT of the preceding interbeat interval Ti_ as T

r[1--exp(--Ti-/roo)], thus the longer the pre-ceding interval is, the shorter the refractory periodwill be (Cohen et al., 1983; Mendez et al., 1956).However, the AV-conduction was shown to dependreciprocally on the preceding interval so that ci

Cmin _qt_ exp(-Ti-1) (Shrier et al., 1987).The time was explained to depend on the

number of circulating atrial wavetets, on their

velocity and the direction in which they reach theAV-junction (Chorro et al., 1990; Zipes et al., 1973;Janse, 1969).Whereas these electrophysiological properties

are far from a complete characterisation of theactual AV-conduction process, it seems legitimateto state:

There are influences of the AV-junction whichare essentially stochastic as the bombardmentby minute atrial excitations.There are influences of the AV-junction whichare essentially deterministic, e.g. governed byconduction times and refractory periods ofmembranes.The described mechanisms exert divergent influ-ences on a given interval.

These conditions seem ideally suited tocause extremely complex dynamics of intervalfluctuation.

Several investigations were sought to analyse theventricular response in atrial fibrillation by statis-tical means and considered the observed dynamicsto be mainly stochastic, even though positive auto-correlation was demonstrated in some studies

(Bootsma et al., 1970).Whereas phase-space-portraits have been intro-

duced as method for the analysis ofcardiac rhythmsthey have not been employed systematically for theanalysis of the ventricular response in atrial fibrilla-tion (Bigger et al., 1993; Denton et al., 1990).

In this investigation the Poincar6-plot as a speci-fic example ofphase-space-portraitwas used to anal-yse systematically the interval-fluctuation duringatrial fibrillation in order to distinguish differentdegrees ofcomplexity, to investigate whether differ-ent reproducible patterns could be discovered andpossibly to find a correlation with a given patient’sdiagnosis.

Furthermore the reliability of the method regard-ing background noise in the acquired data as well as

non-uniformly distributed data and trends withinthe data sets was appreciated.

VENTRICULAR RESPONSE 65

METHODS

Patients of both sexes with atrial fibrillation pre-senting prior to cardiac catherisation or implanta-tion of heart-valve prosthesis were included in theinvestigation after their informed consent. It was de-liberately avoided to discriminate patients accordingto their underlying diseases. Patients presentingwith acute cardiac decompensation and imme-

diately after cardiac surgery were excluded. Dueto the epidemiology of the disease the majority ofpatients with atrial fibrillation suffered from coro-nary heart disease or from a valvular heart defects.The underlying diseases were recorded as well ascardiac relevant drugs, the serum potassium con-

centration, and the time of the day of the particularregistration. Even though it was sought to investi-gate at different times of the day no registrationcould be made during the night. Of the patientsincluded for investigation 16 were considered fre-quently at different times of the day in order toanalyse the stability of the pattern of the ventricularresponse. Thus, a total of 182 registrations of 66patients were gathered.

Prior to each recording a conventional 12-1ead-ECG was used to verify the diagnosis of atrialfibrillation which was additionally confirmed bytwo experienced cardiologists. The patients were

required to rest in a supine position and refrainfrom all physical activities to reduce the chance ofartefacts during the heart rate recordings. Then aconventional data monitor (Hewlett Packard(C)) as

commonly employed for the monitoring of cardio-vascular parameters was connected to the patient.An interface (Hewlett Packard Careplane(C)) linkedthe monitor to a conventional 80368 personalcomputer transmitting the ECG with a samplingrate of 500 Hz. A software specially designed forthat purpose detected R-peaks with +2 ms preci-sion and calculated interval lengths which were

subsequently stored in the random access memoryof the computer. Thus, they could be accessedby software analysis immediately allowing for anon-line representation of the particular heart rate

dynamics. Typically 4000-7000 intervals wererecorded for each registration. The activity of a

particular patient (e.g. sleeping, reading) during therecording which typically lasted between 45 and60 rain was registered.

Usual statistical methods were used to charac-terise each registration such as the calculation ofmean, median, standard deviation, histograms andthe derived parameters of distribution and kurtosis.Furthermore, each interval length T; of a particu-lar registration was plotted against its sequentialnumber in order to detect non-stationarities in theobserved cardiac dynamics. Patients were groupedaccording to underlying disease, drugs, state (sleep-ing/non-sleeping) and an ANOVA was conductedto determine whether significant differencesbetween the subgroups exist.

Furthermore the data was examined for trendsand non-stationarity using Cox and Stuart’s trendtest.The above mentioned physiological background

suggests a certain dependence on a certain intervalof its immediate predecessor even though a uniforminfluence seems unlikely. To characterise the cor-relation between the length of one RR-interval andsubsequent RR-intervals the serial autocorrelationcoefficients were computed. Furthermore the num-ber of intervals elapsing until RR-intervals becomeindependent of their predecessors was determinedthat is where the autocorrelation curve reacheszero.

A virtual two-dimensional plane was used tovisualise the information contained in the RR-interval fluctuation. In a Cartesian co-ordinate sys-tem a particular point P-is defined by the interval

T- and the intervals subsequently following T-+l,thus, Pi (Ti, Ti+I). Likewise, the whole data set isdepicted (see Fig. 1) forming more or less structuredshapes. Ideally these shapes will represent under-lying information about interval fluctuation andthe interdependence of intervals. In this investiga-tion was chosen as it has been done in similarinvestigations (Anan et al., 1990, Garfinkel et al.,1992). Described representation of the data is the


(a)

Pi Pi+l Pi+2

time

(b)

Pli+l ,,,, "hi+2

Pit3

Yi

FIGURE Construction of the Poincar&plots: (a) Interval lengths are measured in an ECG recording (T,., Ti+ 1, etc.). Adjacentintervals form a couple which define a particular point (P;, Pi+ , etc.) in a Cartesian co-ordinate system (b) the axis of which areTi and T+ respectively. Likewise the whole registration is depicted. The dotted line is the line of identity, i.e. T; T+ .Poincar&plot which is applicable for discrete dataas opposed to the conventional phase-space-plotwhere the concerned parameter is plotted againstits first derivative which requires continuous data(Denton et al., 1990). However, these maps are

closely related because the x-axes are identical(parameter amplitude), while the y-axes are mathe-matically related (the y-axis of the Poincar&plot isx + Ax/At, and the y-axis ofthe phase-plane- plot is

dx/dt).The inspection of the pattern of the resulting

plots immediately informs about:

the dispersion of Ti+ for a given Ti representedby the scattering of points along the abscissa,

the change of dispersion for different T;,the RR-variability as represented by the scatter-ing of points along ordinate and abscissa,the dynamics of a given RR-variability as therange of points vertical to the diagonal linerepresenting T;= T;+ represents the accelera-tions and decelerations within the observed ven-tricular responses (Kamen and Tonkin, 1995).

Morphology and dimension of the resultingplots were examined and they were classified ac-cording to morphological criteria. It was sought tofind reproducible common patterns among thePoincar&plots in order to facilitate visual recogni-tion of a given figure.


Ten independent test-person were asked toclassify 132 randomly selected plots according topreviously determined morphological subgroups inorder to verify the reproducibility of the classifica-tion. Furthermore, some indices were computed toquantify the morphology of each subgroup (e.g.number of points in each quadrant, relative dis-tance from zero, dispersion at certain T;-quantils).

RESULTS

Of 50 single registrations ofthe ventricular responseof patients suffering from atrial fibrillation fourhad to be disregarded due to technical difficulties orinterruption of the registration. Thus, 46 registra-tions were considered.

It was contrived to obtain at least ten registra-tions of each of the remaining 16 patients. Withfour patients the sequence of registration was cutoff because atrial fibrillation seized due to eitherspontaneous conversion to sinus rhythm or thenecessity for the implantation of a pace maker. Onepatient died before ten registrations could beobtained and one patient did not cooperate untilthe end of the investigation. Thus, with ten patientsa sequence of at least ten registrations could beobtained yielding a total of 136 registrations. Ofthese eight had to be disregarded due to technicaldifficulties or interruption of the registration.

Twenty-nine patients were male and 33 female.Forty-two patients suffered from valvular heartdefects, 17 had coronary heart disease, while therest presented with various other underlying dis-eases. All patients could be classified in more or lessadvanced states of congestive heart failure (NYHAcriteria II and higher). Twenty-nine patients hadcardiac surgery and all patients were antiarrhyth-mically treated with digitalis being used in 51patients.The mean interval lengths T of patients who

underwent cardiac surgery were significantly longerthan those of patients without previous surgicalintervention. Systematic statistical analysis withANOVA and parametric tests did not reveal any

more significant differences in mean interval length,standard deviation and the other above mentionedstatistic parameters between the different sub-groups of underlying diseases, used medication,and activity of the patients. Furthermore, it couldnot be shown that serum potassium concentrationcorrelated with the observed parameters.

It was shown that the majority of the recordeddata sets was not stationary, i.e. a significant trendwas detected in 76.5% of the sequential registra-tions. There was no favourite direction of a giventrend, in 53.5% the heart rate decelerated and in46.5% it accelerated in the course of registration.The inspection of the obtained interval histo-

grams showed relatively broad distributions andthe standard deviation s: was quite great in com-

parison with T, thus, the coefficient of variation

sT/T was generally greater than 0.2 which corre-sponds with literature data for this parameter(Cohen et al., 1983). Moreover, the shape of thehistograms was quite variable (Fig. 2). The dis-tribution could be unimodal with a single smoothpeak, or may have a single narrow peak super-imposed on a smooth background, or it may becharacterised by multiple peaks tributed. It couldbe shown, however, that the shape of the interval

histogram varied greatly within non-stationarydata sets which constituted the majority of registra-tions (see Fig. 3).

First autocorrelation coefficients rl were usuallyrelatively small albeit significant in the data sets ofthe sequential registrations. 91.7% of registrationshad significant rl typically -0.2 < rl < 0.2. Eachcoefficient r denotes the degree of correlationbetween one RR-interval and the RR-intervaloccurring j beats later. The autocorrelation curvethat is r# considered as a function of j typicallyshows that the RR-intervals during atrial fibrilla-tion are essentially statistically independent of eachother, except for a slight correlation between theduration ofimmediate subsequent beats (see Fig. 4).However, only 50% of the autocorrelation func-tions derived from sequential registrations dis-played the above criteria. The remainder showedstrangely distorted autocorrelation functions where


(a)

750

Z

0

200 700 1200Interval [ms]

1700

(c)

200 700 1200 1700Interval [ms]

750

200 700 1200 1700Interval [ms]

FIGURE 2 Different interval length histograms: (a) singlesmooth peak, (b) single narrow peak superimposed on asmooth background, (c) multiple peaks.

(a) Intervals 1- 1000

200

200 1200Interval [ms]

(b) Intervals 3001 -4000

200

200 1200Interval [ms]

FIGURE 3 Shape of histogram of interval changes within anon-stationary registration: (a) depicts the histogram of inter-val lengths 1-1000 as bimodal, whereas in (b) the histogramof interval 3001-4000 is unimodal with a steeper left slope.

the whole autocorrelation curve seemed set offfrom the abcissa. The occurrence of "strange"autocorrelation functions was highly significantlylinked to non-stationarity of a given data set with rbeing highly significantly higher in non-stationarydata.

Poincarh-plots were inspected and classifiedaccording to morphological criteria. These classeswere distinguished with capital Latin letters asshown in Table I. Note that roughly 400 intervalssufficed to yield the same pattern as the total data


0,8

0,6

0,4

0,2

-0,2

10 15 20

FIGURE 4 Typical autocorrelation curve: The dotted linesrepresent the treshold beyond which r./ is significant(p < 0.001). r# is highly significant only for j= 1.

set (see Fig. 5). The overall reproducibility of theclassification as verified with ten test persons wasas high as 76.5% which is comparable to otherinvestigations involving visual pattern recognition(Reidborg and Redington, 1992).The patterns were distributed as shown in

Table II among the investigated collective. Theycould not be significantly correlated to underlyingdisease, state, or employed medication.The patterns of any individual of whom sequen-

tial registrations were available remained nearlyconstant during the observation period with theexception of the occurrence of a separate "angle" inthe type A or B, thus, yielding an example oftype C.As will be discussed below, this does, however,probably not constitute a fundamental change inthe observed dynamics. Two patients experiencedcircumstances which were apt to modify thephysical cardiac entity within the observationperiod: cardiopulmonal resuscitation in one caseand cardiac surgery in the other. These occurrencesaltered the observed pattern as shown in Fig. 6.

DISCUSSION

The majority of the considered data sets was non-Gaussian distributed and non-stationary. Theseproperties must be inherent to biological processesas the system will never be stable over a longerperiod of time. Thus, to reach stationary biologicaldata registrations ought to be infinitesimally smallwhich on the other hand is not feasible because thetime-scale of the process in question is primarilyunknown. The direction of a trend within the datais not known, therefore, they cannot be excluded byfiltering from the data sets. These features consti-tute limitations for any kind of statistical analysisas previously recognised (Vibe-Reymer el al., 1996;Webber and Zbilut, 1994).With these limitations in mind we might regard

the significantly longer T in patients who had beensubmitted to cardiac surgery prior to the investiga-tion as the result of the denervation of the cardiacentity due to surgical preparation. This in turn leadsto a decreased influence of vagal tone which other-wise shortens the refractory period in the atria

(Schuessler et al., 1992; Allesie et al., 1984). Thus,lesser mean heart frequencies and longer intervallengths ensue.The same restrictions as for statistical analysis

apply for the calculation of the autocorrelationcoefficient as shown in Fig. 7. The larger thenumber of intervals included in the computationof the autocorrelation function the more theinfluence of non-stationarity increases in increas-ingly damping the curve and shifting it to largerautocorrelation coefficients. The shortcoming ofautocorrelation analysis for biological data isdescribed elsewhere (Webber and Zbilut, 1994;Zbilut et al., 1991).

Poincar6-plots

Poincar6-plots as employed in the registrations ofRR-intervals in this investigation are used widely to

investigate the complex dynamics of non-linearprocesses (Denton et al., 1990). They were origin-ally designed to detect complex periodic rhythms in


TABLE

Type Description Example

A 100o"Cluster"Round shape centeredaround the Ti= Ti+l(mainly stochastic)

"Angle"Cluster with apparentshortest interval length(stochastic with shortestinterval)

"Cluster and detachedangle"

Similar shapes as abovewith detached angularshape (mainly stochasticwith ventricular ectopy)

"Points"Discontinuous point-shapedcluster on and on bothsides of Ti-- Ti +(complex periodic)

"Shapes"Linear shapes which cross

Ti= T+(non-linear deterministic)

1000

1000

1000

oo

10ooT [msecl

oooT, [msec]

1000T, [msec]

Ti [msec] 1000

" 1000

1000

T1 [msec]

The dotted line represents the line of identity where T; T,- + .


(a) Intervals 1 -4000_.

1000

’"m,

Ti 1500

(b)

Ti Ti(all units are [ms])

FIGURE 5 Excerpts of a full registration yield similar pat-tern as the full registration: (a) depict 4000 intervals of theregistration, whereas (b) represents interval 500-1000 and(c) characterises interval 2000-2500. The smaller portionsexemplify the same behaviour as the full registration.

TABLE II

Type Number ofregistrations

A "Cluster" 39B "Angle" 52C "Cluster and.detached angle" 30D "Points" 26E "Shapes" 23

Total 170

discrete data (Anan et al., 1990; Crutchfield et al.,1986; Packard et al., 1980). Unlike phase-space-portraits they are quite apt to withstand noisy data.They were appreciated as suitable means to detect

(a) 04 Nov 1993

1000

Ti 1200

(b) 20 Nov 1993

1000

0 Ti 1200

(all units are [ms])

FIGURE 6 Occurrences which change the cardiac entityalter the appearance of the Poincar&plots: (a) is a registrationprior to cardiac surgery which the patient underwent on the12 Nov 1993; (b) shows the different aspect of the resultingplot. Both times the ECG diagnosis was atrial fibrillation.

structural properties which would escape statisticalanalysis (Denton et al., 1990). As method integrat-ing time they provide a means to characterisefeatures of numerous sequential data (Kamen andTonkin, 1995; Woo et al., 1992). It allows toconsider a given data set qualitatively and semi-quantitatively in compact visual patterns.

72 H. KJSMACHER et al.

-0,6- j

FIGURE 7 Dependence of the autocorrelation function ofthe size of the data set: The larger the extract of the consid-ered data used for the computation of the autocorrelationfunction the more evident becomes the influence of the non-stationarity present in the registration. The autocorrelationfunction with larger excerpt seems damped and shifted to lar-ger autocorrelation coefficients. Only for./’= the autocorrela-tion seems quite resistent against the trend contaminateddata.

function of AV-node and the conduction below itas well as their particular refractory periods.

Type A (Mainly Stochastic)

This type represents most likely the predominanceof stochastic processes which lead to a clustering ofintervals to all sides of a mean interval length. Theinterval fluctuation does not seem to have a par-ticular direction. This type was thought to be thearchetype of ventricular behaviour in atrial fibrilla-tion in numerous investigations. The resulting inter-val lengths seem to depend solely of the randomarrival of minute excitations by atrial wavelets. Itwas sought to find out whether these registrationswere comparable to white noise data. Thus, the dataof type A registrations were exemplary submittedto the calculation of the largest Lyapunov exponent(LLE) (Wolf et al., 1985) using the SANTIS-algorithm (Vandenhouten et al., 1996). It couldbe shown that this type possesses a LLE whichis positively different from those of a computergenerated series of random numbers.

Poincar6-Plots were previously employed for theappreciation of atrial fibrillation (Anan et al.,1982). These investigators mentioned the possibilityto describe those properties of the disease whichconcern the coupling interval of the precedinginterval which they consider in the functionalrefractory period of the AV-node in atrial fibrilla-tion (Suyama et al., 1993; Chishaki et al., 1991).Systematical analyses of the dynamics of theventricular response in particular patients over thetime, however, were not undertaken previously.

If different types of patterns of interval fluctua-tion can clearly be distinguished within the same

diagnosis of atrial fibrillation it can be assumed thatthe observed irregularities of ventricular responsepossess different degrees. Intraoperative mappingstudies indicate that a wide range of distinct cardiacbehaviour is subsumed to that diagnosis (Cox et al.,1991). In this investigation it was sought to pointout different degrees of complexity of ventricularbehaviour which is mainly determined by the

Type B (Stochastic with Shortest Interval Length)

This type is characterised by a histogram whichpossesses a steep left slope and which smoothestowards longer interval lengths. A limiting shorterinterval length seems to exist from which the suc-

ceeding interval is more likely to become succes-

sively longer. This could be the result of a cut-offinterval length determined by the refractory periodof the AV-node. This assumption was also madeby other authors which could also simulate theobserved ventricular dynamics by random pacingof dog atria (Chishaki et al., 1991). These and otherauthors (Suyama et al., 1993; Anan et al., 1982;1990) regard this type as typical pattern of ven-tricular response in atrial fibrillation but fail tomention other types. In this investigation this typeis the most frequently occurring type and possessesthe largest T among the unimodal distributedregistrations. Possibly these registrations are theresult of a particular state dependent aspect (Lydic,1987) in which a high vagal tone prevails thereby


lengthening the functional refractory period of theAV-node. However it could not be demonstratedthat this type is predominant among sleepingpatients who ought to have a quite high vagal tone.

Type C (Stochastic with Ventricular Ectopy)

This type is not clearly distinct from the abovementioned characteristics with the exception of the"angle" which additionally appears towards smallerinterval lengths. Most likely this "angle" is thesubstrate of ventricular extrasystoly with a con-secutive recovery pause.

Type D (Complex Periodic)

This type generally displays bi- or multimodalhistograms. Different from the previously men-tioned types here discontinuous patterns are foundin the Poincar6-plots which are separated by"forbidden zones" (Denton et al., 1990). The inter-vals cluster to certain distinct interval lengths andsuccessive intervals follow discrete interval differ-ences instead of continuous interval variation.Furthermore, from a certain interval length thenext interval can only have a finite number of dif-ferent durations which are multiples of the previousinterval length. It appears, thus, that the succeedinginterval length is strongly determined by the lengthof the previous interval. Not surprisingly this typepossessed the largest rl. This behaviour suggests a

dependence ofinterval lengths of certain AV-blockswhich permit only conduction of atrial impulses infixed conduction ratios (e.g. 2: 1-, 3 1-, 3:2-, [...conduction ratio) (Shrier et al., 1987). Theseblockades of the AV-junction usually occur at highatrial fibrillation frequencies so that high atrialstimulation frequencies are inversely related to theresulting ventricular heart rate (Chorro et al., 1990).Elsewhere it was shown that an infinite numberof conduction ratios exist for each given ratioallowing for extremely complex interval fluctua-tion dynamics (Keener, 1981). However, note thatcertain stable interval lengths exist along the "line ofidentity" where T Ti + 1.

Type E (Non-linear Deterministic)

This type produces Poincarb-plots which have lessdefined point-shaped cluster as the previouslymentioned dynamic. Instead we find linear elementswhich cross the "line of identity" temporarily equalsuccessive interval lengths occur. The developmentof interval lengths along certain trajectories isusually associated with directions which approacheach other infinitesimally in certain fixed points inorder to diverge along other directions. Elsewherethese directions were called stable and unstablemanifolds to describe the local geometry arounda local recurrence point (Garfinkel et al., 1992;Liebert, 1991). Consider Fig. 8 where certainexemplary interval lengths display a typical devel-opment: Until P908 interval lengths stay suffi-ciently close to a local recurrence point (RP) toremain stable. Interval 909 is somewhat shorter andthe resulting P909 is already outside the RP. Thesuccessive interval lengths now diverge from theRP along an unstable manifold until P912 comesclose to the stable manifold and the system ap-proaches the RP along P913. This behaviour hasbeen regarded to be typical of non-linear determin-ism (Denton et al., 1990,1991; Glass and Mackey,1988). It was mentioned previously that the direc-tions of mentioned manifold vary over the time

(Garfinkel et al., 1992). However, in the investiga-tion the employed Poincar6-plots integrated ven-tricular dynamics over a period of approximatelyone hour or in the case of several successive regis-trations even over several days. Thus, it can bepostulated that a given dynamics remains stable fora considerable period of time.

Poincar6-plots are not subjective to non-Gaus-sian distributions and quite resistant against ran-

dom noise in the acquired data. Furthermore, theregistration of interval lengths is fairly effortlessand little time-consuming using a common moni-toring device in a single session. Moreover, theycould be used in certain experimental settings.Thus, they represent an inexpensive method of datavisualization.

It can be assumed that a single registrationof interval lengths suffices to characterise the


(a)

%-, 1000

0

890 94O

(b)

1000P911 ]

P910

"i=Ti+l

P909

500

500 T 1000

(all units are [ms])

FIGURE 8 Example of a non-linear deterministic registra-tion with evidence of a stable (SM) and unstable manifold(UM): (a) shows the concerned part of the registration.Note that relatively stable regions are interspersed by inter-val lengths which undergo rapid changes. (b) is the specificPoincar-plot of the data.

complexity of ventricular response using Poincar6-plots. Apparently the observed dynamics are quiteheterogeneous ranging from mainly stochasticprocesses to determined non-linear dynamics.The implications of the differences in complexity

could be relevant for a differential therapy as well asfor the prognosis as future investigations haveto show.

References

Allesie, M., Lammers, W., Bonke, I.M. and Hollen, J. (1984).Intra-atrial reentry as a mechanism for atrial flutter induced byacetylcholine and rapid pacing in the dog. Circulation 71), 123.

Allessie, M.A., Rensma, P.L., Brugada, J., Smeets, J.L., Penn, O.and Kirchhof, H.J. (1990). Pathophysiology of atrial fibrilla-tion. In: Zipes, D., Jalife, J. (Ed.) W.B. Saunders, Philadelphia,London, p. 548.

Anan, T., Nagagaki, O. and Nakamura, M. (1982). An appli-cation ofLORENZ and multi-LORENZ plot to computerizedanalysis of arrhythmia. Jpn. Heart J. 23 (Supplement), 695.

Anan, T., Sunagawa, K., Araki, H. and Nakamura, M. (1990).Arrhythmia analysis by successive RR plotting. J. Electro-cardiology 23(3), 243.

Bigger, J.T., Fleiss, J.F., Rolnitzky, L.M. and Steinman, R.C.(1993). The ability of several short-term measures of RR-variability to predict mortality after myocardial infarction.Circulation 88, 927.

Cameron, A., Schwartz, M.J., Kronmal, R.A. andKosinski, A.S. (1988). Prevalence and significance of atrialfibrillation in coronary artery disease (CASS Registry). Am. J.Cardiol. 61,714.

Chishaki, A.S., Sunagawa, K., Hayashida, K., Sugimachi, M.and Nakamura, M. (1991). Identification of the rate-depen-dent functional refractory period of the atrioventricular nodein simulated atrial fibrillation. Am Heart J. 121,820.

Chorro, F.J., Kirchhof, H.I., Brugada, J. and Allessie, M.A.(1990). Ventricular response during irregular atrial pacing andatrial fibrillation. Am. J. Physiol. 259, H 1015.

Cohen, R.J., Berger, R.D. and Dushane, T.E. (1983). A quanti-tative model for the ventricular response during atrialfibrillation. IEEE Trans. Biomed. Eng. 3t), 769.

Cox, J.L., Canavan, T.E., Schuessler, R.B., Cain, M.E.,Lindsay, B.D., Stone, C., Smith, P.K., Corr, P.B. and Boineau,J.P. (1991). The surgical treatment of atrial fibrillation. II.Intraoperative electrophysiologic mapping and description ofthe electrophysiologic basis of atrial flutter and atrialfibrillation. J. Thorac. Cardiovasc. Surg. 101,406.

Crutchfield, J.P., Farmer, J.D., Packard, N.H. and Shaw, R.S.(1986). Chaos. Sci. Am. 255, 46.

Denton, T.A., Diamond, G.A., Helfant, R.H., Khan, S. andKaragueuzian, H. (1990). Fascinating rhythm: A primer onchaos theory and its application in cardiology. Am. Heart J.120, 1419.

Denton, T.A., Diamond, G.A., Khan, S.S. and Karagueuzian,H.S. (1991). Can the techniques of nonlinear dynamics detectchaotic behavior in electrocardiographic signals? J. Electro-cardiol. Suppl. 24, 84.

Ferguson Jr., T.B. and Cox, J.L. (1993). Surgical therapy foratrial fibrillation. Herz 18, 39.

Garfinkel, A., Spano, M.L., Ditto, W.L. and Weiss, J.N. (1992).Controlling cardiac chaos. Science 257, 1230.

Glass, L. and Mackey, M.C. (1988). From Clocks to Chaos. TheRhythms of Life. Princeton University Press, NJ.

Heisel, A., Jung, J., Stopp, M., Bay, W. and Schieffer, H. (1995).Vorhofflimmern. Bedeutendste Arrhythmie beim Alterspa-tienten. Therapiewoche 24, 1426.

Honzikova, N., Fiser, B. and Semrad, B. (1973). Ventricularfunction in patients with atrial fibrillation. Cot Vasa 15, 257.

Janse, M.J. (1969). Influence of the direction of the atrial wavefront on AV nodal transmission in isolated hearts of rabbits.Circ. Res. 25, 439.

Kamen, P.W. and Tonkin, A.M. (1995). Application of thePoincarh plot to heart rate variability: a new measure offunctional status in heart failure. Aust. NZ J. Med. 25, 18.

Keener, J.P. (1981). On cardiac arrhythmias: AV conductionblock. J. Math. Biol. 12, 215.

Lammers, W.J. and Alessie, M.A. (1993). Pathophysiology ofatrial fibrillation: current aspects. Herz 18, 1.


Liebert, W. (1991). Chaos und Herzdynamik: Rekonstruktionund Charakterisierung seltsamer Attraktoren aus skalarenchaotische Zeitreihen. Thun (Reihe Physik; Bd. 4) Frankfurtam Main.

Lydic, R. (1987). State-dependent aspects of regulatory physiol-ogy. FASEB J. 1, 6.

Mendez, C., Gruhnit, C.C. and Moe, G.K. (1956). The influenceof cycle length upon the refractory period of auricles, ventri-cles, and the AV-node in the dog. Am. J. Physiol. 184, 287.

Packard, N., Crutchfield, J.P., Farmer, J.D. and Shaw, R.S.(1980). Geometry from a time series. Physical Review Letters45/9, 712.

Reidborg, S.P. and Redington, D.J. (1992). Psychophysiologicalprocesses during insight-orientated therapy. J. Nerv. Ment.Dis. 180, 649.

Schuessler, R.B., Grayson, T.M., Bromberg, B.I., Cox, J.L. andBoineau, J.P. (1992). Cholinergically mediated tachyarryth-mias induced by a single extrastimulus in the isolated canineright atrium. Circ. Res. 71, 1254.

Shrier, A., Dubarsky, H., Rosengarten, M., Guevara, M.R.,Nattel, S. and Glass, L. (1987). Prediction of complexatrioventricular conduction rhythms in humans with use ofthe atrioventricular nodal recovery curve. Circulation 76, 1196.

S6derstr6m, N. (1950). What is the reason for the ventriculararrhythmia in cases of auricular fibrillation?. Am. Heart J.40, 212.

Suyama, A.C., Sunagawa, K., Sugimachi, M., Anan, T.,Egashira, K. and Takeshita, A. (1993). Differentiation

between aberrant ventricular conduction and ventricularectopy in antrial fibrillation using RR interval scattergram.Circulation 88(1), 2307.

Vandenhouten, R., Rasche, M., Tegtmeier, H. and Goebbels, G.(1996). SANTIS a tool for signal analysis and timeseries processing. Version 1.1. Biomedical System Anal-ysis, Institute for Physiology, RWTH Aachen, Germany.ftp.Physiology.RWTH-Aachen.de:/pub/santis.

Vibe-Reymer, K. and Vesin, J.M. (1996). Using statisticalparameters for chaos detection. Procceedings of the 1996IEEE Digital Signal Processing Workshop, Loen, NorwaySeptember 1-4, p. 510.

Webber, C.L. and Zbilut, J.P. (1994). Dynamical assesment ofphysiological systems and states using recurrence plot strate-gies. J. Appl. Physiol. 76(2), 965.

Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A. (1985).Determining Lyapunov exponents from a time series.Physica D 16, 285.

Woo, M.A., Stevenson, W.G., Moser, D.K., Trelease, R.B. andHarper, R.M. (1992). Patterns of beat-to-beat heart ratevariability in advanced heart failure. Am. Heart J. 123, 704.

Zbilut, P.Z., Koebbe, M., Loeb, H. and Mayer-Kress, G. (1991).Use of recurrence plots in the analysis of heart beat intervals.In: Proc. IEEE Computers in Cardiology, Muray, A., Ripley,K.L. Los Alamitos (Ed.), CA, p. 263.

Zipes, D.P., Mendez, C. and Moe, G.K. (1973). Evidence forsummation and voltage dependency in rabbit atrioventricularnodal fibers. Circ. Res. 32, 170.

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