Spatial Distribution of Phase Singularities inVentricular Fibrillation

Miguel Valderrábano, MD; Peng-Sheng Chen, MD; Shien-Fong Lin, PhD

Background—Multiple excitation wavelets are present during ventricular fibrillation (VF). The underlying waveletorganization of VF is unclear. Phase singularities (PSs)—locations of ambiguous activation state—underlie reentry andwavelet splitting and represent the sources of VF. Understanding the mechanisms of PS formation might be importantin the development of effective therapies for sudden death.

Methods and Results—We performed voltage, phase, and PS mapping in fibrillating ventricles, applying an automated PSdetection algorithm to optically recorded fibrillation signals. PS clustering was noted along epicardial vessels, ridges ofendocardial trabeculae, and papillary muscle insertions. Microscopically, these locations correlated with areas ofapposition of fibers with different angulations and intramural vessels. A total of 83.2% of PSs were formed at andmeandered about these anatomic structures, which acted as stabilizers: PSs colocalizing at anatomic substrates hadlonger life spans than nonanatomic PS (82.46�60.8 versus 40.5�31.9 ms, P�0.01). The RV endocardium had a higherPS incidence than the epicardium (42.3�9.2 versus 23.5�11.6 PS/s, P�0.01). Autocorrelation showed that irregularbehavior was spatially restricted to anatomic heterogeneities compared with other areas, which had nearly periodicbehaviors. Simple spatial PS distributions underlay complex and variable activation patterns attributable to variable PSbehaviors, life spans, and inter-PS interactions.

Conclusions—PSs occur in a nonrandom spatial distribution and colocalize with normal anatomic heterogeneities. VaryingPS behaviors and life spans but stable PS spatial distributions cause ever-changing activation patterns that characterizeVF. (Circulation. 2003;108:354-359.)

Key Words: fibrillation � arrhythmia � waves

Mapping studies of ventricular fibrillation (VF) haveshown multiple wavelets of excitation. These wavelets

can present as rotors but most commonly appear nonreen-trant.1,2 New wavelets can either arise from fragmentation ofpreexisting wavelets3 or emanate from a stable rotor4,5 ac-cording to the 2 leading hypotheses. Phase singularities(PSs)6,7 represent sites in which the activation state cannot bedetermined, surrounded by a continuum of activation statesranging from fully activated to fully recovered. PSs arebelieved to be crucial in VF: Wavelets are flanked by PSs,and PSs underlie the formation of rotors and wave splitting.Thus, PSs represent the sources of fibrillation.6 Althoughconceptually distinct, wavebreaks—fracture of propagationwavelets—and PSs are equivalent phenomenological terms.

PSs can arise at anatomic heterogeneities or originatefunctionally in purely homogeneous tissue on the basis offunctional, dynamic wavelength oscillations.3,8 The mecha-nisms of PS formation during VF remain poorly explored.Understanding the determinants of PS formation would pro-vide invaluable insight into the mechanisms of VF and mightlead to preventive or therapeutic strategies.

In this study we show a striking spatial colocalization ofPSs with normal anatomic structures in fibrillating healthyhearts. These results suggest a critical role of anatomicstructures in the maintenance of VF.

Methods

Isolated Swine Ventricle PreparationsThe experimental models have been previously described.9,10 Forright ventricular (RV) studies, the RV wall was excised, perfusedwith Tyrode’s solution, and placed in a tissue bath. Optical mappingduring VF was performed on the endocardial surface (n�12) as wellas the epicardial surface (n�9). For left ventricular (LV, n�9)studies, we used a modified wedge10 preparation: A rim of tissuesurrounding the left circumflex and the second obtuse marginalartery was excised and perfused, leaving an inverted L-shapedpreparation that contained at least part of the posteromedial papillarymuscle. The tissue was placed in the tissue bath with the transmuralcut surface up, which was the mapped surface. In both ventricles, VFdeveloped during tissue manipulation and persisted thereafter as longas perfusion was adequate. VF can persist in a stable fashion forhours in these models.9,10

Received February 24, 2003; revision received April 3, 2003; accepted April 4, 2003.From the Division of Cardiology, Department of Medicine, Cedars-Sinai Medical Center and David Geffen School of Medicine, University of

California Los Angeles, Calif.Movies are available in the online-only Data Supplement at http://www.circulationaha.org.Correspondence to Shien-Fong Lin, PhD, Cedars-Sinai Medical Center, 8700 Beverly Blvd, Annex, Los Angeles, CA 90048. E-mail linsf@cshs.org© 2003 American Heart Association, Inc.

Circulation is available at http://www.circulationaha.org DOI: 10.1161/01.CIR.0000080322.67408.B4

354 by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

Rabbit Langendorff PreparationNew Zealand rabbits (�3.5 kg, n�6) were obtained from a USDA-licensed commercial rabbit vendor in Southern California. Therabbits were anesthetized with sodium pentobarbital (60 mg/kg). Theheart was removed, mounted in a Langendorff apparatus, andperfused with Tyrode’s solution with a pressure of �70 mm Hg. Theheart was suspended in a vertical position from the perfusioncatheter, and gauze was sutured to the apex to drain the venous effluxand mitigate motion. The anterolateral epicardial surface of the LVwas mapped, with left anterior descending artery at the left edge ofthe mapping field. Pacing the RV at 5 times the diastolic thresholdwith decreasing cycle lengths (300 to 100 ms at 10-ms decrements)was performed until VF was induced.

Optical Mapping and Data ProcessingThe optical mapping system and spatiotemporal filtering methodshave been described previously.10 The tissues were stained with 1 to2 �mol/L di-4-ANEPPS. Light from a laser source (532 nm) wasdelivered to the tissue. The fluorescence was collected with a CCDcamera. No mechanical uncouplers were used. Recorded during eachacquisition were 2.3 to 11.5 seconds of data with a temporalresolution ranging from 255 to 420 frames per second. Phasemapping6 was performed to evaluate the location and evolution ofPSs in VF. For PS quantification and lifespan analysis, PSs wereidentified manually as sites where phase was ambiguous where allphases converged. PSs were counted in each frame, and their lifespanwas quantified as the number of frames for which individual PSspersisted. For cumulative PS display over long acquisition intervals(typically 200 frames), PSs were identified using our recentlydeveloped automated PS tracking algorithm.7 This was comparedwith a wavebreak tracking algorithm that initially identified thedepolarizing front and repolarizing back of the wavelet10 by detect-ing adjacent pixels whose values cross the median in either direction.The points where wavelet front and wavelet back meet are identifiedas wavebreaks (Figure 1). Phase portraits were generated by plottingfluorescence Fn against Fn��, where n is the frame number and � waschosen between 4 and 8 frames. To assess periodic behavior, voltagesignals were processed by autocorrelation: An individual signal wascorrelated with itself with progressive imbedded time delays. In thisanalysis, correlation coefficients (r) are calculated for a range ofdelays ranging from 0 to 200 frames: Beyond a delay of zero (r�1),the delay corresponding with the most common cycle length ofperiodicity, if present, would yield the highest r. Additional delayswould give variable r coefficients. In the presence of periodicity,subsequent peaks of high r coefficients will be present at delayscorresponding with multiples of the cycle length of periodicity (see

Figure 5). Data are presented as mean�SD. The proportions ofanatomic versus nonanatomic PSs were compared using �2 tests, andPS lifespans were compared using t tests. P�0.05 was consideredsignificant.

Histological StudiesAfter the mapping studies, 5-�m-thick transmural sections were cutparallel to the mapped surface from paraffin-embedded tissue blocks.The slides were stained routinely with trichrome stain.

ResultsPS LocalizationPSs and wavebreaks coincided spatially (Figure 1) and werenot randomly distributed. Cumulative PS display during VFshowed unequivocal alignment of PSs with certain anatomicstructures. In the epicardium, PSs formed along the course ofepicardial arteries, whereas in the endocardium, they formedalong ridges of endocardial trabeculae. Figure 2 showsexamples. Mapping of transmural surfaces revealed a non-random PS distribution. Superimposition of PS locations withlow-power histological cuts of the mapped tissue allowedgross estimation of the PS histological determinants. PSsclustered at the intramural insertion of the papillary muscles,intramyocardial arteries, and between subepicardial bundlesof myocyte fibers of different orientation (Figure 3). PSs wereeither generated at or attracted to anatomic structures. Once ina particular structure, they tended to meander within it untilextinguished, either by reaching a tissue boundary or bywavelet collision. Figure 4 shows an example of a PSmeandering on an epicardial artery (photograph shown inFigure 5). Movies of simultaneous voltage, phase, and PSmapping are available in the online Data Supplement.

Voltage Dynamics and Anatomy: Local PeriodicityVersus Irregular DynamicsVoltage and phase traces obtained from sites located inendocardial trabeculae ridges, the papillary muscle insertion,or epicardial arteries showed frequent instances of lowamplitude and double potentials with voltages close to themean value, interspersed with runs of fully developed poten-

Figure 1. Wavebreak-PS equivalence. a,Optical map snapshot showing depolar-ization wavefront (red line) and repolar-ization waveback (green line). The pointwhere they meet is identified as wave-break point (b). c, Simultaneous phase(�) map, showing a point where allphases converge (PS), which is identifiedusing the PS tracking algorithm (d) at thesame location as the wavebreak point. ethrough i, Simultaneous cumulative dis-plays of PS (top) and wavebreak (bot-tom) in 5 fibrillating Langendorff-perfusedrabbit preparations, showing equal spa-tial distributions.

Valderrábano et al Local Anatomy and VF Organization 355

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

tials. These unstable voltage dynamics led to frequent inde-terminate phases (close to the center in the phase portrait).Although this phenomenon was clearly associated withreadily identifiable anatomic structures, its occurrence wasstill unpredictable and subject to dynamic wavelet behavior.Whenever the PS was not visiting these structures, fullydeveloped potentials could be recorded. This effectively rulesout inadequate or artifactual signal recording. Autocorrelation(Figure 5) showed a first peak corresponding with the cyclelength of the occasional fully developed potentials but noother consistent tall peaks. In contrast, in locations whereanatomy is homogeneous (Figure 5), nearly periodic activa-tion patterns were detected, the phase portraits had hollowring-like patterns, and autocorrelation showed peaks corre-sponding with the periodicity cycle length, its double, triple,and so forth, suggesting a higher degree of regularity.

PS and Wavelet BehaviorPSs were continuously formed and extinguished, with vari-able chirality and unstable spatial locations. Despite theirmany variations, their cumulative spatial distribution was

stable, consistently clustered at anatomic structures. Thewavelet manifestations included reentry and wave split-ting10,12–15 but most commonly were simply a wavelet delimi-tator.6 Figure 6 shows an example where multiple PSs werepresent. Local anatomy (epicardial arteries in this example)generated PSs at consistent locations but led to widelyvariable activation patterns, depending on the PS chirality andinter-PS interactions, which were subject to dynamic behav-ior. Regardless of the varying activation patterns, the spatialdistribution of PSs remained relatively stable.

Dynamic Versus Anatomic PSsPrevious theoretical and experimental studies have suggestedthat dynamic instabilities of cardiac tissue are a sufficientsubstrate for sustaining fibrillation.8,16–18 However, the loca-tion of PSs was not determined in those studies. In this studywe found that PSs without a structural basis were rare (370 of2202 in 36 episodes, 16.8% of all detected; 175 of 1008 in 12

Figure 2. Nonrandom PS spatial distribution with PS clusteringat ridges of endocardial trabeculae and epicardial arteries. athrough c and g through i, Selected phase (�) maps, with PSsmarked with arrowheads, of endocardial and epicardial sur-faces, respectively. d and j, Raw optical pictures of the mappedendocardial and epicardial surfaces, respectively. Red squaresmark the locations of PSs. e and k, Cumulative PS displaysspanning 0.5 seconds of VF show a nonrandom PS spatial dis-tribution with PS clustering (counts in blue-to-white scale) alongridges of endocardial trabeculae (e) and epicardial arteries (k). f,Photograph of the mapped tissue, with endocardial trabeculaethat match the ones in the raw optical picture (d) and the PSclustering (e). l, Isochronal map showing reentry in 1 epicardialartery simultaneous with wave splitting in another artery.

Figure 3. PS localization in transmural LV wedges. A, PS loca-tion and chirality vary but remain restricted to underlying anato-my. A-a through A-d, Consecutive phase map snapshots (acqui-sition time in ms) showing PSs with complete (white arrows) andincomplete (black arrow) rotations. A-e through A-h, In the sameepisode, a widely different PS distribution. A-i and A-j, Schemat-ics of the PS locations (black lines) and chiralities (arrows) in theintervals shown in A-a through A-d and A-e through A-h,respectively. A-k, Low-power image of tissue histology, withapproximate locations of PSs. A-l through A-p, Anatomic corre-lates of PSs: intramural vessels (A-l and A-n), apposition offibers with different orientation (A-m and A-o), and separatemuscle bundles (A-p). B, PS clustering along the papillary mus-cle insertion. B-a and B-b, Examples of PSs in papillary muscle.B-c, Macroscopic appearance of mapped tissue. B-d, Cumula-tive display (over 0.4 seconds) with clustering of PSs along thepapillary muscle insertion and subepicardially.

356 Circulation July 22, 2003

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

endocardial, 81 of 414 in 9 epicardial, 64 of 504 in 9transmural, and 50 of 276 in 6 rabbit preparations), unstable(lifespan of 40.5�31.9 ms compared with 82.4�60.8 ms inthose with an anatomic determinant, P�0.01), and did notform reentry (the mean reentry cycle length of 74.3�16.4 mswas greater than the lifespan of all nonanatomic PSs). PSlifespan depended on the presence or absence of an underly-ing anatomic determinant (stabilizing) and interaction withother wavelets (destabilizing). The most common mechanismof PS extinction was the latter (91.2%), followed by mean-dering to a boundary.

Transmural PS GradientPrevious studies have suggested a critical role of the endocar-dium in the generation of fibrillation.19 An endocardial-to-epicardial activation rate gradient20 and more complex fre-quency distribution have been proven in the endocardiumrelative to the epicardium.5 We found a higher incidence of PSsin the endocardium than in the epicardium (42.3�9.2 PSs persecond, compared with 23.5�11.6 in the epicardium, P�0.01),which may explain the previous findings. The transmural surfacehad an intermediate incidence of 28.1�12.6 PSs per second(P�0.01 compared with the endocardium). The spatial densityof PSs was highest in the transmural surface (5.6 PSs/s per cm2),followed by the endocardium (4.7 PSs/s per cm2, P�NS), andlowest in the epicardium (2.6 PSs/s per cm2, P�0.01 comparedwith both endocardium and transmural surface). These differ-ences may be attributable to different degrees of histologicalcomplexity in the 3 preparations.

DiscussionThere are several major findings of our study. First, detailedquantitative analysis and cumulative display of PSs showedclose colocalization of PSs with underlying anatomic heter-ogeneities, suggesting that most have anatomic determinants.Second, PS meandering was determined by underlying ana-tomic heterogeneities. Third, spatial autocorrelation analysisdemonstrated spatially arranged local periodicity and irregu-lar dynamics. Fourth, spatial PS distribution was relativelystable in the presence of varying activation patterns.

Isolated episodes of reentry and wave splitting have beenshown to occur in certain anatomic heterogeneities.10,14,15

Despite their relevance, reentry and wave splitting are rela-tively rare phenomena during fibrillation. PSs, however, arethe necessary engines of fibrillation whether maintained by amother rotor or dynamic wavebreak and only lead to thesephenomena on a probabilistic basis. The colocalization ofmost PSs with these structures suggests an enhanced role ofanatomic substrates from mere anecdotal inducers of reentry

Figure 4. PS meandering along an epicardial artery in a rabbitLangendorff preparation. a, Consecutive phase maps during PS(arrows) meandering and schematic of PS trajectory (numbersrefer to PS locations). b, Local voltage signals (in fluorescence,F, units) at selected PS locations. Blue shaded area showsinterval of PS meandering at these locations. Arrows point tosegments of low-amplitude voltage, close to the mean, that cor-respond to PS visiting individual pixels. Although the PS loca-tion is unstable, it is restricted to the underlying epicardial artery(see red dots in Figure 5b). Some sites are visited more thanonce by the PS. Movies of simultaneous voltage, phase, and PSmapping in this tissue are available in the online DataSupplement.

Figure 5. Anatomic impact on voltage andphase: local periodicity. a, Photograph ofmapped epicardial surface (slightly rotat-ed). b, Corresponding cumulative PS dis-play over 0.4 seconds, with remarkabledelineation of underlying epicardial arter-ies. Red dots mark locations of PSsshown in Figure 4. Numbers refer to pixelswhose signals are shown in c. c, Opticalsignals, phase portraits, and autocorrela-tion graphs from locations on the epicar-dial arteries or away from them (1 through4 and 5 through 8, respectively). Irregularoscillations, segments of low amplitude,double potentials, and higher baseline arepresent in 1 though 4, indicative of proxim-ity to PSs as proven by the phase portraitwith frequent visits to the center. Autocor-relation shows an early peak at 70 to 90ms, but subsequent peaks have lower cor-relation coefficients. In contrast, locationsaway from arteries have regular signals,hollow ring phase portraits, and multiplepeaks of similar amplitude in autocorrela-tion, suggesting local periodicity.

Valderrábano et al Local Anatomy and VF Organization 357

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

and wave splitting to key players in the maintenance offibrillation.

The origin of mapped multiple wavelets in VF is disputed.The multiple wavelet hypothesis proposed by Moe et al21

relied on preexisting dispersion of refractoriness to promotewavebreaks. Wavebreaks can also arise from dynamic oscil-lations in the recovery of excitability.8,16–18 Electrical resti-tution (the variation of action potential duration and conduc-tion velocity with the diastolic interval) has been shown to bea major determinant of dynamically induced wave-breaks.17,22–24 Pharmacological modulation of electrical res-titution may convert fibrillation into tachycardia in isolatedventricular tissues23,24 by eliminating spiral wave breakup.24

The present study suggests that normal anatomic heterogene-ities play a key role in either generating or attractingwavebreaks (PSs). Our findings are consistent with simula-tions showing that preexisting heterogeneities significantlyreduce the level of dynamic instability required to createPSs.25,26 However, it is impossible to discern whether PSs areprimarily formed at these locations or simply attracted tothem. The fact that PSs persist for longer periods of timewhen in a particular anatomic substrate suggests at least astabilizing effect and supports the relevance of thisfunctional-anatomic interaction. Functional dynamic hetero-geneities, which can determine spiral wave meandering insimulated cardiac tissues,8,16,17,22 are likely to be a determin-ing factor in PS meandering and inter-PS interactions. Ana-tomic heterogeneity, on the other hand, may exert a stabiliz-ing effect and lengthen the life span of these PSs.

The focal source hypothesis4,5,27 postulates that fibrillationis maintained by a stable, rapid reentrant circuit (the “motherrotor”) from which activation wavelets emanate but fail to

conduct 1:1 to the surrounding tissues because of preexistingheterogeneities. Fibrillatory conduction originates wavebreakand leads to multiple wavelets, which are considered anepiphenomenon rather than the origin of fibrillation. Alimitation of the focal source hypothesis has been the inabilityto identify a stable rotor in isolated pieces of tissue.10,18

Nevertheless, our data are compatible with this paradigm andmay explain situations in which dominant frequency bordersare stationary because of clustering of PSs at locationscorresponding to anatomic features.27 These frequency do-main boundaries have also been correlated with certainanatomic locations.10 Our study ties these 2 findings together.

The potential mechanisms for the colocalization of PSswith anatomic heterogeneity include the potential of relativeinexcitability. Pinning of scroll waves to unexcitable ele-ments is a well-documented phenomenon in excitable me-dia.28 Epicardial and intramural vessels can anchor reentrantexcitation.12,13 The papillary muscles and endocardial trabec-ulae may alter propagation by the additional current sink15

caused by the increased tissue thickness. Abrupt fiber orien-tation changes10 as seen transmurally and in the subepicar-dium can lead to anisotropic (resistive) discontinuities.29

LimitationsThe correlation of PS locations with the underlying anatomyis based on superimposition of PS locations and raw opticalpictures of mapped tissues. This provides enough detail tounequivocally identify the endocardial trabeculae, epicardialarteries, and papillary muscles. However, the correlation ofmapped data with microscopic structures identified post hocin histological preparations should be interpreted cautiouslyawaiting higher resolution mapping studies. Also, this studydoes not directly address the relative importance of anatomic

Figure 6. PS variability but consistentlocation. a through f, Consecutive (5frames apart) phase maps from epicar-dial rabbit Langendorff preparation (top)with corresponding voltage maps (bot-tom). White arrows show direction ofactivations. a, Two adjacent PSs ofopposite chirality are present, but noreentry is completed. b, The wavelet onthe bottom (lower arrow) rotates clock-wise (contrary to the chirality of the initialPS, probably attributable to fusion withanother wavelet breaking through to theepicardium). This lower wavelet activatesthe lower portion of the tissue and com-pletes 1 rotation (b through f) around aPS that appears in b. Meanwhile, thewavelet on top activates the right supe-rior portion. On collision with the lowerwavelet, 2 PSs are generated (f). Later inthe same episode, a similar PS locationis seen (compare g with a). This time afigure-eight reentry is completed (gthrough k). Other wavelets are seen aris-ing from the lower portion of the tissuebut do not interfere with this rotation. k,Another wavelet collides with the lowerwavelet of the figure-eight circuit and

eliminates the counterclockwise PS (not present in l). The upper wavelet completes one more clockwise rotation alone. m and n,Cumulative PS display during intervals depicted in a through f (in m) and g through l (in n), showing very similar patterns despite thechanging activation patterns. o, Cumulative PS display of 1 second of fibrillation, outlining the epicardial vessels. p, Local electrogramsof sites visited by the PS in interval a through f (site 1) and the PS in interval g through l (site 2), as well as a site away from both (site 3).

358 Circulation July 22, 2003

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

factors versus dynamical restitution-based factors on PSformation. Future studies will be needed to test this idea byexamining the effects of flattening APD restitution on PSformation at anatomic structures.

AcknowledgmentsThis work was supported by a fellowship from the American HeartAssociation, Western States Affiliate; the Pauline and Harold PriceEndowment; and NIH grants R01HL58533, P50HL52319,R01HL66389, and R01HL 71140. The authors thank Avile Mc-Cullen, Nina Wang, and Elaine Lebowitz for their assistance andHrayr S. Karagueuzian, PhD, and James N. Weiss, MD, for criticalreview of the manuscript.

References1. Lee JJ, Kamjoo K, Hough D, et al. Reentrant wave fronts in Wiggers’

stage II ventricular fibrillation: characteristics, and mechanisms of termi-nation and spontaneous regeneration. Circ Res. 1996;78:660–675.

2. Rogers JM, Huang J, Smith WM, et al. Incidence, evolution, and spatialdistribution of functional reentry during ventricular fibrillation in pigs.Circ Res. 1999;84:945–954.

3. Weiss JN, Chen P-S, Qu Z, et al. Ventricular fibrillation: how do we stopthe waves from breaking? Circ Res. 2000;87:1103–1107.

4. Chen J, Mandapati R, Berenfeld O, et al. High-frequency periodic sourcesunderlie ventricular fibrillation in the isolated rabbit heart. Circ Res.2000;86:86–93.

5. Zaitsev AV, Berenfeld O, Mironov SF, et al. Distribution of excitationfrequencies on the epicardial and endocardial surfaces of fibrillatingventricular wall of the sheep heart. Circ Res. 2000;86:408–417.

6. Gray RA, Pertsov AM, Jalife J. Spatial and temporal organization duringcardiac fibrillation. Nature. 1998;392:75–78.

7. Bray MA, Lin SF, Aliev RR, et al. Experimental and theoretical analysisof phase singularity dynamics in cardiac tissue. J Cardiovasc Electro-physiol. 2001;12:716–722.

8. Karma A. Electrical alternans and spiral wave breakup in cardiac tissue.Chaos. 1994;4:461–472.

9. Kim Y-H, Garfinkel A, Ikeda T, et al. Spatiotemporal complexity ofventricular fibrillation revealed by tissue mass reduction in isolated swineright ventricle: further evidence for the quasiperiodic route to chaoshypothesis. J Clin Invest. 1997;100:2486–2500.

10. Valderrábano M, Lee M-H, Ohara T, et al. Dynamics of transmural andintramural reentry during ventricular fibrillation in isolated swine ven-tricles. Circ Res. 2001;88:839–848.

11. Deleted in proof.12. Davidenko JM, Pertsov AM, Salomonsz R, et al. Stationary and drifting

spiral waves of excitation in isolated cardiac tissue. Nature. 1992;355:349–351.

13. Pertsov AM, Davidenko JM, Salomonsz R, et al. Spiral waves of exci-tation underlie reentrant activity in isolated cardiac muscle. Circ Res.1993;72:631–650.

14. Wu T-J, Yashima M, Xie F, et al. Role of pectinate muscle bundles in thegeneration and maintenance of intra-atrial reentry: potential implicationsfor the mechanism of conversion between atrial fibrillation and atrialflutter. Circ Res. 1998;83:448–462.

15. Kim Y-H, Xie F, Yashima M, et al. Role of papillary muscle in thegeneration and maintenance of reentry during ventricular tachycardia andfibrillation in isolated swine right ventricle. Circulation. 1999;100:1450–1459.

16. Courtemanche M. Complex spiral wave dynamics in spatially distributedionic model or cardiac electrical activity. Chaos. 1996;6:579–600.

17. Qu Z, Weiss JN, Garfinkel A. Cardiac electrical restitution properties andstability of reentrant spiral waves: a simulation study. Am J Physiol.1999;276:H269–H283.

18. Valderrábano M, Yang J, Omichi C, et al. Frequency analysis of ventric-ular fibrillation in swine ventricles. Circ Res. 2002;90:213–222.

19. Janse MJ, Kleber AG, Capucci A, et al. Electrophysiological basis forarrhythmias caused by acute ischemia: role of the subendocardium. J MolCell Cardiol. 1986;18:339–355.

20. Worley SJ, Swain JL, Colavita PG, et al. Development of an endocardial-epicardial gradient of activation rate during electrically induced sustainedventricular fibrillation in dogs. Am J Cardiol. 1985;55:813–820.

21. Moe GK, Rheinboldt WL, Abildskov JA. A computer model of atrialfibrillation. Am Heart J. 1964;64:200–220.

22. Qu Z, Xie F, Garfinkel A, et al. Origins of spiral wave meander andbreakup in a two-dimensional cardiac tissue model. Ann Biomed Eng.2000;28:755–771.

23. Riccio ML, Koller ML, Gilmour RFJ. Electrical restitution and spatio-temporal organization during ventricular fibrillation. Circ Res. 1999;84:955–963.

24. Garfinkel A, Kim Y-H, Voroshilovsky O, et al. Preventing ventricularfibrillation by flattening cardiac restitution. Proc Natl Acad Sci U S A.2000;97:6061–6066.

25. Fenton F, Karma A. Vortex dynamics in three-dimensional continuousmyocardium with fiber rotation: filament instability and fibrillation.Chaos. 1998;8:20–47.

26. Xie F, Qu Z, Garfinkel A, et al. Electrophysiological heterogeneity andstability of reentry in simulated cardiac tissue. Am J Physiol Heart CircPhysiol. 2001;280:H535–H545.

27. Samie FH, Berenfeld O, Anumonwo J, et al. Rectification of the back-ground potassium current: a determinant of rotor dynamics in ventricularfibrillation. Circ Res. 2001;89:1216–1223.

28. Vinson M, Pertsov A, Jalife J. Anchoring of vortex filaments in 3Dexcitable media. Physica D. 1993;72:119–134.

29. Spach MS, Miller WT Jr, Dolber PC, et al. The functional role ofstructural complexities in the propagation of depolarization in the atriumof the dog: cardiac conduction disturbances due to discontinuities ofeffective axial resistivity. Circ Res. 1982;50:175–191.

Valderrábano et al Local Anatomy and VF Organization 359

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from

Miguel Valderrábano, Peng-Sheng Chen and Shien-Fong LinSpatial Distribution of Phase Singularities in Ventricular Fibrillation

Print ISSN: 0009-7322. Online ISSN: 1524-4539 Copyright © 2003 American Heart Association, Inc. All rights reserved.

is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231Circulation doi: 10.1161/01.CIR.0000080322.67408.B4

2003;108:354-359; originally published online June 30, 2003;Circulation. 

http://circ.ahajournals.org/content/108/3/354World Wide Web at:

The online version of this article, along with updated information and services, is located on the

http://circ.ahajournals.org/content/suppl/2003/07/21/01.CIR.0000080322.67408.B4.DC1.htmlData Supplement (unedited) at:

  http://circ.ahajournals.org//subscriptions/

is online at: Circulation Information about subscribing to Subscriptions: 

http://www.lww.com/reprints Information about reprints can be found online at: Reprints:

  document. Permissions and Rights Question and Answer available in the

Permissions in the middle column of the Web page under Services. Further information about this process isOnce the online version of the published article for which permission is being requested is located, click Request

can be obtained via RightsLink, a service of the Copyright Clearance Center, not the Editorial Office.Circulation Requests for permissions to reproduce figures, tables, or portions of articles originally published inPermissions:

by guest on June 22, 2015http://circ.ahajournals.org/Downloaded from