Phase defect lines during cardiac arrhythmias: from theory to experiment
Louise Arno, Jan Quan, Nhan Nguyen, Maarten Vanmarcke, Elena G. Tolkacheva, Hans Dierckx
PPhase defect lines during cardiac arrhythmias:from theory to experiment
Louise Arno , Jan Quan , Nhan Nguyen , Maarten Vanmarcke , Elena G.Tolkacheva , and Hans Dierckx Department of Mathematics, KULeuven Campus KULAK, E. Sabbelaan 53, 8500 Kortrijk, Belgium Biomedical Engineering Department, University of Minnesota, Minneapolis, MN, 55455 * L.A. and J. Q. contributed equally to this work
Abstract
During heart rhythm disorders, complex spatio-temporal patterns of electrical activationcan self-organize in vortices. Previously, it was assumed that at the center of such vortex, thereis a point singularity in the activation phase. We hypothesize that there is another possibility,namely a region of discontinuous phase that can exist during cardiac arrhythmias instead of apoint singularity. In this paper, we provide some theoretical insights into the existence of suchphase defect lines (PDL) and validate their presence in ventricular tachycardia in the wholeex-vivo heart using optical mapping experiments. We discuss the implications of phase defectlines and surfaces for the theory of arrhythmias, cardiac modeling and practical cardiology.
Significance statement
During cardiac arrhythmias, electromechanical vortices have been observed. Previously, the coresof such rotors were considered as point-like phase singularities; these are actively targeted in ar-rhythmia therapy such as catheter ablation. Our novel analysis of experimental data of ventriculartachycardia, however, shows that at the vortex core, wave block rather than a phase singularityoccurs, such that the fundamental entity is a phase defect line (PDL). The location of experimen-tally reconstructed phase defect lines coincides with the location where multiple short-lived phasesingularities were found. This observation could help to understand why stable rotors are seen insimulations, while clinical experiments usually reveal multiple short-lived rotors. Both theoreticalconsequences and clinical implications of the PDL concept should be investigated.
Introduction
About once per second, a wave of electrical depolarization is generated in our natural pacemaker,which then travels through our heart, coordinating its mechanical contraction. As such, the heartis a prime example of electrical activity with self-organisation across different scales. The flow ofions through the cell membrane of cardiac myocytes invokes various spatiotemporal patterns ofelectrical activity at the organ or tissue level, which correspond to either a normal or abnormalheart rhythm. If this abnormal rhythm occurs in the lower cardiac chambers (ventricles), whichpump blood to the body and lungs, the normal pumping is disturbed, leading to life-threateningventricular arrhythmias. If the abnormal electrical pattern, however, is situated in the uppercardiac chambers (atria), which pump blood into the ventricles, the arrhythmia is usually notdirectly lethal. However, atrial arrhythmias often evolve to a chronic disease, severely increasingthe risk for blood clot formation and stroke [1]. 1 a r X i v : . [ n li n . AO ] J a n t has been previously demonstrated, using cardiac surface electrodes, that electrical activityduring cardiac arrhythmias may travel around the heart in a closed circuit within the cardiac wall,and thus re-excite the heart [2]. Later, such rotating vortices of electrical depolarization haveexperimentally been observed in animal hearts during ventricular arrhythmias [3]. Depending onthe community, these vortices are also known as rotors, or as spiral waves in two-dimensional(2D) space and as scroll waves in three-dimensional (3D) media. More recently, rotors have beenmapped on the cardiac surface using high-density electrode arrays [4] and in 3D myocardiumusing ultrasound [5]. Since the rotation frequency of the rotors is generally higher than theheart’s natural rhythm, they take control of the heart’s excitation sequence and induce arrhythmia.Therefore, several therapies aim at localizing rotors in the heart and ablate the core of these rotorsto terminate the arrhythmia [6, 7]. This core of the rotor is traditionally found by applying a phaseanalysis and treating this the rotation center as a singularity point [8].However, rotors have been only observed indirectly in patients. For this reason, computa-tional modeling of the heart has been developed [9, 10, 11, 12], in which biophysically detailedor phenomenological equations are derived from observations at the level of the cell membrane.Next, homogenization theory was applied to find effective continuum models that sustain wavepropagation as is observed in the heart [13]. The numerical simulations can provide valuableinformation on the spatiotemporal dynamics of rotors as they allow to control different factorsindependently, including anisotropy, wall shape and thickness, heterogeneity and physiological pa-rameters. Also, (semi-)analytical approaches can provide a description of the formation of rotorsand their dynamics [14, 15, 16, 17].It was furthermore found that the analysis of complex 3D patterns can be significantly simpli-fied by only considering the dynamically moving rotation axis of each rotor, called the filament [8].The dynamics of such filaments in the 3D heart could then be related to a perturbation problemof 2D spiral waves in the same medium [18, 19]. Numerical evidence suggests that these rotorsare mostly sensitive to stimuli applied near their core [20, 19, 21, 22], allowing to consider rotorsas localized objects or quasi-particles that drive the arrhythmia [7].In this paper, we reconstruct the activation phase of rotors using both numerical and experi-mental data, and demonstrate the absence of a singularity point, which was traditionally associatedwith rotors [8]. Instead we demonstrate a line of conduction block at the center of these vortices,along which the excitation phase is changing abruptly. We call these structures ‘phase defect lines’(PDL) in 2D and ‘phase defect surfaces’ (PDS) in 3D. This manuscript is organized as follows.We will first set up a mathematical framework for these phase defects and describe methods toidentify them in data. Thereafter, we demonstrate the presence of PDLs in numerical simula-tions and optical mapping experiments. Finally, we discuss implications of of our findings forarrhythmogeneity at the conceptual and applied levels. Theoretical observations
Let us assume that the physiological state of cells in every point of the medium can be representedby a state vector u ( t ) = [ u ( t ) ... u N ( t )] T of N state variables, where t is time and u is usuallythe transmembrane potential (and denoted as V )[9, 23, 12]. Remarkably, in many oscillatory andexcitable media, the local dynamics of the system implies that over time, only a small part of allpossible states u ( t ) are visited. A plot of these states typically reveals a closed loop in state space,see Fig. 1b. In electrophysiology of the heart, this closed loop in state space corresponds to anaction potential.The term ‘phase’, as used in phase singularity or phase defect, here refers to different parts ofthe action potential: a cardiac cell can be in the resting state, upstroke, plateau or repolarizationphase and transitions between them. There are different ways to translate such definition inmathematics, and Fig. 1 shows two of these ways, in its top and bottom row, respectively. Wewill demonstrate below that in either case, a phase defect region can be found; the first phase( φ act ) is the classical one, while the second phases ( φ arr ) shows a direct relation between PDLsand conduction block lines. 2 DLPDLDATA STATES PHASE SPATIAL PHASE PHASE DEFECTS a) b) c) d) e)
Figure 1: From numerical or experimental data to phase defect lines, demonstrated for state spacephase φ act (Eq. 1) on the top row, and the arrival time phase φ arr (Eq. 3) on the bottom row. (a)Starting point is a numerical simulation of a rotor in 2D in the Fenton-Karma (FK) MLR-I model[11]. (b) The state of a cell can be represented either in the plane of excitation-recovery variables(top) or using the elapsed time since last activation (bottom). We use for V the normalizedtransmembrane potential ( u in the FK model) and R = − v with v the second variable in theFK model. (c) Using either Eq. 1 or 3, a phase can be defined. (d) When coloring the originalpattern according to phase, a phase defect line (PDL) can be observed (e).The first, classical, definition of phase starts from plotting two suitable different observables ofthe system in a plane, say V ( t ) and R ( t ) . In cardiac context, V ( t ) is usually the transmembranepotential (or proxy thereof), and R ( t ) can either be a recovery variable, or transformation of the V ( t ) time-series in a point (e.g. the delayed signal V ( t − τ ) or its Hilbert transform H ( V ( t )) ) [24].Without loss of generality, we can assume that the activation loop is followed counterclockwise (ifnot, one can replace R ( t ) by − R ( t ) ). One then shifts the origin to a value ( V ∗ , R ∗ ) within thetypical activation loop and defines activation phase as the polar angle in this system: φ act = atan2 ( R − R ∗ , V − V ∗ ) + C. (1)Here, atan2 ( y, x ) is the 2-argument inverse tangent function, such that the result points to thecorrect quadrant in the XY-plane. In simulations, we choose the constant C such that φ act = V ( t ) the optical intensity and R ( t ) its Hilbert transform [24]. From the state space plots in Fig.1, it is seen that values of φ act that differ by an integer multiple of 2 π from each other representthe same state. Therefore, throughout this paper, we take φ act in [ , π ) .When a propagating wave travels through a point of the medium, its phase is changingmonotonously from 0 to 2 π . This state, however, is again the unexcited state since phase dif-ferences of 2 π only imply that another action potential or activation loop has been completed.For this reason, we consistently use a cyclic colormap in this publication, where blue denotes both φ act = φ act = π . If a spiral wave is colored according to φ act , one notes that if its center iscircumscribed clockwise once at large radius, the total phase difference along that path expressedin multiples of 2 π will be the topological charge of the spiral wave: Q = ∮ dφ π = { − ( clockwise rotation ) , + ( counterclockwise rotation ) . (2)Eq. (2) is used in practice as a rotor detection algorithm [8, 25]. Another algorithm for rotorlocalization, the isosurface method, [11] looks for the intersections of the curves V = V ∗ , R = R ∗ which also reveal the phase singularities. 3e here introduce a new, second definition of the phase as follows: for every point in themedium, the local activation time (LAT) of the wave is the most recent time when the statechanged from V < V ∗ to V > V ∗ . With t elapsed the elapsed time difference with most recentLAT, we define the new phase as φ arr ( t elapsed ) = π tanh ( t elapsed / τ ) , (3)where τ is the typical action potential duration in the medium, and the factor 3 makes thatat φ arr ( τ ) ≈ π . The difference between φ act and φ arr is mainly a reparameterization, φ act = f ( φ arr ) . Note that the use of φ arr avoids the rapid change in phase during the depolarization andrepolarization that is typical for φ act , as can be seen from Fig. 1 by comparing top and bottomimages in panels d-e.When the state variables are initialized around a given point in space, such that the phaseincreases (spatially) from 0 to 2 π , a rotating spiral pattern will develop from this state as in Figs.1a and 2a,c. Each point far away from the central point will execute its excitation cycle, but at thecenter the phase is necessarily undefined since the periodic interval [ , π ) cannot be continuouslymapped onto a single value. Three different possibilities exist at the center of rotation.A first possibility is a phase singularity, i.e. a point where all different phases converge, asusually observed in many circular-core spiral waves, and traditionally also assumed to occur withline-core spiral waves. A second option is that phase becomes spatially disorganized in a so-calledchimera state [26]. Surprisingly, a third state, simpler than a chimera, is also possible in 2D space:the phase singularity extends along a curve (in 2 spatial dimensions), thus forming a PDL, asshown in Fig. 1c.Our explanation of PDL existence is based on biological rather than mathematical arguments.Consider the activation cycle of an excitable cell, i.e. its action potential, as shown in Fig. 1b.Now, if one expects a phase singularity in the center of a vortex, one assumes that the cells canalso be in a state that lies somewhere in the middle of the cycle in state space. However, thissituation may be biologically impossible: a chain of biochemical reactions will push the cell alongits activation cycle, not necessarily allowing it to occupy the middle state.Hence, we hypothesize that of the many PSs that have been identified in cardiac measurements,a significant fraction could be unrecognized PDLs. Numerical simulations have shown that, if theaction potential duration is increased, a wave break evolves to a so-called linear core, where thespiral wave tip travels along the edge of the region with previously excited tissue. This edge isprecisely the PDL, and the fact that PDLs occur in regions of conduction block should be clearfrom their definition in terms of discontinous φ arr .In the systems with long action potentials and linear spiral cores, we argue that the PDL isa more fundamental entity than the PS, for the following reasons in addition to the biologicalargument above. First, if we represent the data V ( t ) , R ( t ) as a continuous field, and apply a PSdetection algorithm, then by the assumed continuity the algorithm using isolines [11] will indeeddetect a point satisfying V = V ∗ , R = R ∗ and attribute a PS to it. So, even if there is a PDL,a PS may be detected. However, if one changes the thresholds V ∗ , R ∗ , the PS will shift alongthe PDL, such that the PS bears no true physical meaning, except from being located on a PDL.Second, the outcome of the circle integral method (Eq. (2)), depends on the implementation ofthe method and spatiotemporal resolution of the data [25]. For, if the contour intersects the PDL,the assigned topological charge will depend on how that jump in phase is accounted for. Below, wedemonstrate that this method generates multiple PSs coinciding with the PDL. Third, the PDLis more robust, since one can also find it from pure LAT measurements as demonstrated in Fig. 1.The situation in Fig. 1 is reminiscent of a branch in mathematics called complex analysis, seeFig. 2. Herein, the field of real numbers is extended by the imaginary unit i such that i = − z = x + iy , with x , y real; then, z is a complex number. Asa consequence, the real axis is extended to a complex plane, in which the polar angle associatedwith z is called (not coincidentally) the complex phase of z and denoted arg ( z ) . (In fact, theHilbert method of [24] makes use of this property.) Next, one can consider functions of a complexvariable: w = f ( z ) with both w and z complex numbers. To make a visual representation of the4hase singularities Phase defect lines / branch cutsAP model (rigidly rotating) arg z FK model (linear core) arg (√ z − ) Figure 2: Phase singularities versus phase defect lines in cardiac models and complex anal-ysis. Phases are rendered in-plane (top row) and in 3D, as a Riemannian surface (bottomrow). (a) Rigidly rotating spirals, as in the Aliev-Panfilov (AP) reaction-diffusion model [10]correspond to a phase singularity, similar to φ ( x, y ) = arg ( z ) . (b) Linear-core cardiac models,such as Fenton-Karma (FK) MLR-I kinetics [11] exhibit a phase defect line or branch cut, like φ ( x, y ) = arg (√ z − ) . Grey areas denote a jump in the phase over a quantity not equal to aninteger multiple of 2 π , i.e. a phase defect line (physics) or branch cut (mathematics).function value, one can draw colormaps of the phase arg ( w ) . In case of polynomial or rationalfunctions, such phase map will reveal point singularities (see Fig. 2a), similarly to circular-corespiral waves. When the phase of the simple function w = f ( z ) = z is shown as a graph of ( x, y ) ,the phase surface resembles a helicoid. If one walks around the PS once in the XY-plane, thephase will change by 2 π , but since phase is measured disregarding terms of 2 π , one ends in thesame state. This effect is more easily seen using a cyclic colormap in 2D than on the Riemanniansurface, where phase is represented as height above the XY-plane, see Fig. 2b.In contrast to polynomial and rational functions, functions f which include roots exhibit adiscontinuity in their complex phase, also called ‘branch cut’ in mathematics. In particular, forthe function w = f ( z ) = √ z −
1, there is a line of discontinuous phase for ( x, y ) between ( − , ) and ( , ) . This line is easily noticed in the representation as a color map as a sudden transitionin color, even in a cyclic colormap (see Fig. 2). This situation is in our opinion very similar to thephase discontinuity in the rotor simulation of Fig. 1.To localize a PDL, it is natural to look for large phase differences between nearby pairs ofpoints. Here, one should take into account that phase differences of 2 π should be neglected, suchthat we take along an edge between adjacent measurement points (in experiment or simulation):∆ φ = mod ( φ − φ + π, π ) − π. (4)This way, if the wave pattern is sampled with a fine resolution, most differences between adjacentgrid points will be small, while large differences are rounded towards [ − π, π ] . One can thenthreshold on ∆ φ between adjacent pixels in the image and connect the regions where the phasejumps to locate phase defects.We now continue with demonstrating PDLs in numerical simulations and optical mappingexperiments. Numerical results
A first result in a reaction-diffusion system with Fenton-Karma MLR-I kinetics (3 variables) [11]was given in Fig. 1. Both φ arr and φ act show a phase defect line.5econd, we remark that PDLs can also exist in media that sustain spiral waves with PSs. Thisis illustrated in simulations where application of S1S2 pacing induce a PDL at the wave back ofthe first stimulus, see Fig. 3a. Since Aliev-Panfilov reaction kinetics were used [10], the PDL willdisappear after both sides have reached full recovery, resulting in a PS at the core of the spiralwave, see Fig. 3b. Hence, we conclude that PDLs and PS can coexist in the same system.We further note that phase discontinuities not only occur at the core of vortices, but wheneverconduction block happens, i.e. when a wave front hits a portion of non-recovered tissue. Fig. 3ashows a wave impeding on a non-recovered portion of tissue, leading to the temporary formationof a PDL, which vanishes after repolarization. Hence PDLs can have Q ∈ { − , , } , and perhapsalso other integers.It is well-known that the concept of PS can be extended to dynamical filament curves when thethickness of the medium is also considered. Since PDLs have one dimension less than the ambientspace, one expects in 3D to observe a phase discontinuity surface (PDS), and this is confirmed bynumerical simulation in a ventricular geometry in Fig. 3c.a) b) c)Figure 3: Relation of phase defect lines to other phase structures in simulations. (a) A propagatingwave to the right (S1) was followed by stimulation of the top-left quarter of the medium (S2) inthe Aliev-Panfilov model [10], showing a PDL separating the newly excited S2 zone from the waveback of the S1 stimulus. This system later evolves to a PS, see Fig. 2 top left, showing that PDLand PS can occur in the same medium. (b) S1 stimulation followed by a rectangular S2 stimulationin the Bueno-Orovio-Fenton-Cherry (BOFC) model [27] for human ventricular tissue. A PDL isformed without a rotor attached to it; its topological charge from Eq. 2 is zero. The wave frontson the left will approach each other along the right-hand side of the PDL and fuse without spiralformation. (c) 3D simulation in the BOFC model in a biventricular human geometry, showing3 phase defect surfaces (red) in the left and right-ventricular free walls and the interventricularseptum. Colors of a & b as in Fig. 2. Experimental results
Electrical activity of the Langendorff-perfused rat heart was visualized during ventricular tachy-cardia via optical mapping experiments [28]. In the image sequence of 80 ×
80 pixels, the opticalintensity I ( t ) , which varies linearly with the transmembrane potential, was normalized between 0and 1, and after smoothing, the phase was computed using the Hilbert transform [24]. In additionto the PDL detection method (see Eq. (4)), we conducted traditional PS detection using themethod of intersecting isolines [11] and by computing circle integrals of the phase using a 2 × × ⧫ ). PS detected by the circle integral methodare drawn as ▲ for counterclockwise rotation ( Q =
1) and as ▼ for clockwise rotation ( Q = − Discussion
We have started this study with defining a new concept to describe meandering spiral waves, andhypothesized that a phase defect line is present in the core rather than a point singularity, as istraditionally assumed. From the experimental observations in Fig. 4-5 and Table 1, we note thatthe phase singularities from the integral method [24, 25] nearly always lie near a phase defect line.Our results are in line with recent experiments where PS were seen to co-localize with conductionblock lines [29] in patients with atrial fibrillation, but these authors did introduce the concept of aPDL. Our results further show that PDLs had an average lifespan about four times longer than PS(Tab. 1), and the wave front was seen to revolve around them in Fig. 5a. Note that for both PDLand PS, the lifespan is still shorter than the inverse of dominant frequency (dominant period),which may be due to incomplete filtering of short-living structures persisting for 1-5 ms. Fromthe spatial collocation and longer persistence in time, we conclude that PDLs are fundamentalentities that occur during ventricular tachycardia in the observed rat hearts. Given the ongoingdiscussion on the difficulty to observe rotors in mapping experiments or in patients, we suggest toalso look for PDLs instead of PSs there.One question is whether the phase is truly discontinuous across the PDL, as in the mathematicalbranch cuts in Fig. 2. From our experiments this cannot be answered, since spatial resolution islimited. In simulations, Fig. 1, φ act seems to suggest a continuous transition, while φ arr showsa clear discontinuity, inherited from the discontinuity in LAT. Here, we keep in mind that thereaction-diffusion paradigm emerges from a continuity approximation, and that the question ofdiscontinuity should essentially be answered by physiological observation, e.g. detailed opticalmapping with a conduction block in the field of view. We expect that, as with other interfacesbetween phases in nature, a boundary layer will form with a thickness related to fundamentalconstants of the problem. For this reason, we name the conduction block region ‘phase defects’7 xp Table 1: PS and PDL properties in 6 optical mapping experiments, for PS computed using theintegral method. Phase was computed for both PSs and PDLs with V ( t ) the normalized opticalintensity and R ( t ) its Hilbert transform.a) t y x Wave FrontPSPDL b) po i n t s i n P D L , pe r f r a m e Figure 5: Post-processing of experimental data. (a) Position of the wave front, phase defectline and phase singularities, where time has been added as the vertical dimension. (b) Positivecorrelation between number of detected PS and number of points in the PDL for the experimentshown in Fig. 4.rather than ‘phase discontinuities’.There exist several open questions, to which the concept of a PDL may offer a partial answer.A first issue, which inspired this work, is how a meandering spiral can be described in terms of aPS. This is now clear: there is a line defect of phase, rather than a singularity point, and this PDLis located near the classical tip trajectory, i.e. the path followed by the apparent phase singularitylocated at V = V ∗ , R = R ∗ . For so-called linear cores, a PDL occurs, and we conclude that thedifferent tip trajectories are located in the aforementioned boundary layer around the PDL. Thus,the question on how a rotor’s core is structured seems to be answered by the concept of PDLs:there is not a single cell in its middle with undefined phase. Rather, there is an elongated regionof cells which are in a boundary layer of ill-defined phase at the center of a rotor.The recognition that there is a PDL at the center of linear-core rotors is likely to start a newchapter in the mathematical description of cardiac arrhythmias. From a mathematical physics per-spective, it seems that the string-like dynamics [17] need to be generalized to brane-like dynamics(Fig. 3d), since already for a 2D medium, a worldsheet is traced out over time by the phase defect(Fig. 5a). Many previous results on rigidly rotating spirals [14, 16, 17] including the concepts offilament tension [15] and response functions [19] will need to be revisited and generalized. Oneoutstanding question is why PDL in numerical simulations usually precess, while this phenomenonis not observed in our experiments. Further investigations could target to quantify PDL and PDSgrowth and rotation in experiment, and crank the computational models to behave in the samemanner. This research question has implications for pharmaceutical therapy too, since one way inwhich drugs have pro- or anti-arrhythmic effects could be via the modification of PDL properties.At the fundamental level, the concept of PDLs gives a new handle on how to interpret cardiacpatterns during complex arrhythmias. These should be reviewed in terms of wave fronts, wavebacks and PDLs separating them, and could shed new light on the onset and perpetuation ofcomplex arrhythmias, including atrial and ventricular fibrillation.8mportantly, the possible role of rotors in clinical management of arrhythmias is still a pointof debate in the scientific community [30, 31]. Examples have been given when a simple conduc-tion block is detected as a rotor [29] and the role of regions of conduction block is surging in thedescription of cardiac activation patterns [29, 32]. Here we argue that, if one adjusts one’s termi-nology that rotors can rotate also around a finite conduction block line (PDL) and adapts analysismethods to also detect those ‘rotors’, a more coherent picture is obtained, as in our Fig. 4. In thisview, there are different types of conduction block or PDLs, depending on the integral of phase ifone circumscribes the conduction block line (Eq. (2)): if Q = ±
1, a rotor is attached to the PDL,if Q =
0, it is only a local conduction block. Thus, all conduction blocks are PDLs, but not all ofthem are at the center of rotation of a spiral wave. Note that our observations pertain to a set ofoptical mapping experiments in rat hearts only, and we do not claim that in other systems, all PSshould be interpreted as PDLs. Still, since we worked with epicardial recordings, the conductionblock lines found at the rotor cores are not due to endocardial structures, as suggested in [29] inthe case of atrial fibrillation.Even at this stage, one can start thinking of consequences for arrhythmia management in theclinic. First, if PDLs are present rather than point singularities, this may be a reason why rotorshave been so elusive in experimental or clinical recordings. It is sometimes quoted that rotorsonly last for part of a period and then disappear; usually, such short living rotors are filtered away[33]. This observation is consistent with the camera or electrode covering only one end of thePDL. Further analysis is therefore recommended, using novel algorithms to pick up end pointsof a PDL (i.e. the branch point of the phase surface) or the middle part of the region, i.e. aconduction block line. Another option is to actively develop mapping methods with a wider fieldof view, to detect the entire PDL on the cardiac surface. Second, with better knowledge of therotor core structure, one can actively reconstruct how the associated electrogram would look likeand look for those signatures in recordings, which may depend on the orientation of the electrodewith respect to the PDL. Finally, if PDLs have a typical orientation with respect to anatomicalfeatures (e.g. endocardial structure, heterogeneity, myofiber direction, scars), this may affect theoptimal ablation lesion.In conclusion, we find that the name of mathematician Bernhard Riemann is inscribed twicein our hearts: the heart is not only a Riemannian manifold [16, 17, 34], but also features phasedefect lines showing non-trivial Riemannian surfaces that organize the electrical patterns duringarrhythmias.
Acknowledgements
E.G.T. was supported by National Science Foundation DCSD grant 1662250. H.D. received mo-bility funding from the FWO-Flanders, grant K145019N.
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