© 2005 The European Society of Cardiology. Published by Elsevier Ltd. All rights reserved.
Of circles and spirals: Bridging the gap between the leading circle and spiral wave concepts of cardiac reentry
aDepartment of Pharmacology and Therapeutics, McGill University McIntyre Medical Sciences Building 3655 Promenade Sir-William-Osler, Montréal, Québec, Canada H3G 1Y6; bThe Research Center, Montreal Heart Institute 5000 Belanger St. E., Montreal, Quebec H1T 1C8, Canada; cDepartment of Medicine and University of Montreal, Montreal Heart Institute 5000 Belanger St. E., Montreal, Quebec, Canada H1T 1C8
Manuscript submitted 10 February 2005. Revision received 27 July 2005. Accepted after revision 3 May 2005.
*Corresponding author. Department of Medicine and University of Montreal, Montreal Heart Institute, 5000 Belanger St. E., Montreal, Quebec, Canada H1T 1C8. Tel.: +1 514 376 3330; fax: +1 514 376 1355. E-mail address: stanley.nattel{at}icm-mhi.org (S.Nattel).
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Abstract |
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 Top Abstract IntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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The "leading circle model" was the first detailed attempt at understanding the mechanisms of functional reentry, and remains a widely-used notion in cardiac electrophysiology. The "spiral wave" concept was developed more recently as a result of modern theoretical analysis and is the basis for consideration of reentry mechanisms in present biophysical theory. The goal of this paper is to present these models in a way that is comprehensible to both the biophysical and electrophysiology communities, with the idea of helping clinical and experimental electrophysiologists to understand better the spiral wave concept and of helping biophysicists to understand why the leading circle concept is so attractive and widely used by electrophysiologists. To this end, the main properties of the leading circle and spiral wave models of reentry are presented. Their basic assumptions and determinants are discussed and the predictions of the two concepts with respect to pharmacological responses of arrhythmias are reviewed. A major difference between them lies in the predicted responses to Na+-channel blockade, for which the spiral wave paradigm appears more closely to correspond to the results of clinical and experimental observations. The basis of this difference is explored in the context of the fundamental properties of the models.
Key Words: reentry, leading-circle, spiral wave, pharmacological therapy
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Introduction |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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In 1887, MacWilliam provided one of the first known descriptions of wavefront re-entry in a previously-excited pathway, while observing what we would now call ventricular fibrillation [1]
. The beginning of the 20th century saw the first real description of reentrant activity by Mayer [2] in jellyfish tissue, in which the importance of unidirectional block was noted. Soon after the publication of Mayer's work, Mines described reentry around an obstacle as a mechanism of arrhythmia in rings of dog cardiac tissue [3]. Lewis made the first direct link between reentrant activity and clinical arrhythmias after describing atrial flutter in the dog as reentry in a cylinder of muscle around the vena cava and through the taenia terminalis [4], by identifying comparable behaviour in a patient with atrial flutter [5]. He subsequently analyzed electrocardiographic recordings from 2 patients and inferred that their atrial fibrillation was due to single discrete macroreentry circuits of variable location with fibrillatory conduction [6]. Numerous subsequent publications described reentrant activity, with a common underlying assumption of an anatomical basis that fixed a barrier around which reentry occurred. In fairness, Garrey's concept of activity in atrial fibrillation was clearly one of functional reentry with varying location and no fixed structural basis [7]; however, no clear theoretical formulation of functional reentry was presented until the 1970's.
Allessie et al. introduced what was probably the first detailed model of functional reentrant activity based on experimental observations [8–10]. They named this concept the "leading-circle model" because propagation was sustained by activity at the leading tip of a circuit located around a "circle" of critical dimensions. It should be noted that Allessie's work was preceded by studies of the wavelength concept and of the notion of functional reentry that did not necessitate the existence of an obstacle [11,12]. There is, however, no doubt that the leading-circle model was the first detailed explanation of the mechanisms underlying non-anatomical reentry from an experimental perspective.
Cardiologists and experimental physiologists appreciate the leading circle concept because of its obvious relation to electrophysiologically important and measurable variables, as well as its general applicability to clinically relevant determinants of arrhythmia. Relatively recent advances in biophysics and theoretical analysis point towards a somewhat different but related notion, that of "spiral wave" reentry. The objective of this review is to try to bridge the gap between the experimental/clinical view of reentry and the extensive physical/theoretical literature on spiral wave dynamics, in order to highlight their specific properties and predictions. We hope that in so doing, we will help clinicians and experimental physiologists to understand better the spiral wave concept and its implications, as well as to communicate to more basic biophysicists why, despite its limitations, the leading circle construct is still widely used by physiologists and cardiologists.
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Anatomical versus functional reentry |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Although anatomical reentry and functional reentry are considered quite distinct, the differentiation may sometimes be incomplete. For example, in animal models of atrial flutter, reentrant activity propagates around anatomical obstacles such as the venae cavae and the tricuspid annulus [11,
13,14], and the dimension of this obstacle controls the cycle length of the activity [13]. The role of obstacle size in determining the reentry cycle length is a defining feature of this type of reentry. On the other hand, Kim et al. [15] observed reentry that seemed to be anchored at a branch point of papillary muscle. Since the papillary muscle did not precisely define the reentry pathway, it may be argued that this does not represent true anatomical reentry. Nevertheless, the branch point did serve to anchor the circuit, limiting the degree of meander and drift. It is not yet clear how different types of reentry in normal tissue or in the context of specific pathological changes (e.g., in the presence of a myocardial infarction) should be classified, or whether it is always important to distinguish between different types of reentry. A key concept behind all types of reentrant activity is the source-sink relationship related to the propagating activation wavefront. A wavefront can propagate as long as unexcited but excitable cells (the "sink") have their sodium channels activated by the diffusion current moving forward from depolarized cells at the leading edge of the front (the "source"). The diffusion current acts as a drain on the source, so that if a relatively small source is attached to a larger sink, the loss of source current caused by the sink may reduce the current available for excitation to the point that propagation fails. There is thus a critical relationship between the source current for excitation and the mass of tissue being excited, which drains the source current electrotonically. This source-sink relation is key to understanding both the leading circle and spiral wave concepts.
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The leading-circle concept |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Allessie and co-workers studied reentrant activity in small segments of rabbit left atrium using intra- and extra-cellular electrodes [8]
. Based on their observations, they formulated the leading circle concept to account for reentrant activity in the absence of an anatomical obstacle. The principal characteristics of the concept are shown in Fig. 1A.
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Unidirectional block causes the impulse to travel in the direction indicated by the heavy arrow. The impulse can also travel centripetally, towards the centre (finer arrows), or centrifugally, away from the centre. Each revolution sends a wave of centrifugal activity emanating outwards, which activates the rest of the myocardium at the rapid rate of this reentry process. If one considers a circle with dimensions equal to the distance (the "wavelength", WL) travelled in one refractory period (RP), which is the product of RP and conduction velocity (CV), the impulse will propagate continuously through excitable tissue. Impulses attempting to transit through smaller circuits (such as the centripetal impulses shown by the finer arrows) will encounter refractory tissue and be extinguished. Impulses moving centrifugally into larger circuits would establish longer revolution times (the circuit dimension divided by CV), but are dominated by the faster activity emanating from the shortest possible circuit, the "leading circle", over a dimension equal to the WL. The impulse propagating at the leading edge of the leading circle excites tissue as soon as excitation is possible, only slightly behind its own wave tail in the relative refractory period (gray area in Fig. 1A), leaving no gap of full excitability. It thus follows that the length of the pathway is equal to the wavelength of the reentry and the revolution time is equal to the refractory period. The WL is a critical determinant of reentry arrhythmia maintenance in the leading circle model, because if the size of the tissue substrate is too small to accommodate the WL, reentry will be unable to sustain itself.
An important characteristic of the functional reentry emphasized by Allessie et al. [10] is that "the central area is activated by centripetal wavelets" (smaller arrows in Fig. 1A) that block. The concept states that the central zone is inexcitable because it is stimulated twice as fast as the leading-circle and is depolarized in the voltage range for sodium inactivation. The notion of centripetal wavelets was based on transmembrane potential recordings (recordings 3–4 of Fig. 2 in ref. [10]) which showed that the amplitude and duration of action potentials in the central area were markedly smaller than in outward regions, thus indicating that regions of the central area may have not been activated. Allessie et al. attributed this low-amplitude activity to propagation through incompletely recovered tissue, writing that "Here the impulse encounters fibres that are still in their refractory phase because they were already activated by another centripetal wavelet just half a revolution...". Regardless of their origin, these small-amplitude depolarizations and the time between these non-activated potentials are also compatible with an excitable core [16], a prediction of the spiral-wave concept for which there is now experimental evidence [17,18].
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A key feature of the leading circle concept is thus that the functional reentry is governed by activity in a limited region for which the wavelength occupies virtually the entire pathway of reentry. Knowing that wavelength is given by WL = RP × CV, the model can provide insights into the impact of pharmacological therapy and disease-induced remodelling on functional reentry, in terms of measurements that can readily be made by clinical and experimental electrophysiologists. A key question, of course, is whether this model adequately accounts for arrhythmia behaviour or whether the concept requires modification/updating.
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Properties of spiral wave reentry |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Spiral wave phenomena have been observed in multiple systems including chemical reactions and electrophysiological models [19]
. The development of research on spiral wave dynamics derived from two main approaches: 1) experimental study of the Belouzov-Zhabothinsky (BZ) chemical reaction [19,20] and 2) numerical/theoretical work on mathematical models of cellular electrical activity, particularly the Fitzhugh-Nagumo (FHN) model [21]. Spiral waves were also reproduced in models of cardiac tissue [12,22–25]. In general, the activity induced by a spiral wave (shown in Fig. 1B) follows from the inside (from the thick dashed circle) to the outside of the excitable medium. In the simplest case, the spiral wave will keep its shape and rotates around a core of constant size with unvarying angular velocity. The shape can be described by the activation front (black continuous curve) starting from the inner area of the spiral and propagates away from this centre zone. Note that the front is curved, an important characteristic of the spiral wave that is not included in the leading circle concept. Curvature of the activation front modulates the velocity of propagation through a change in the source term as illustrated in Fig. 2. For a wavefront that is convex as seen along the activation front of the spiral wave (Fig. 2, left), electrical impulses will be radiating outward, and each cell on the propagating wavefront will be activating more than one cell downstream. Thus, the ratio of excited (source)/resting (sink) cells is less than unity, the value for a planar wave (Fig. 2, middle). Consequently, the source-sink mismatch will limit the current that depolarizes each cell ahead of the front. Decreasing the current leads to a slower rate of voltage rise ahead of the front and a longer time before the sodium channels are activated. This process results in a slowing in propagation by decreasing the stimulating efficacy of the activation front. The conduction velocity (CV) of a convex wavefront will therefore be less than for the value (CV0) for a planar wave. The opposite holds for a concave wavefront (Fig. 2, right).
The repolarization front at the end of the refractory period (red continuous curve in Fig. 1B) follows the activation front. There exists a point where both fronts "coincide". This point is called "the phase singularity" of the spiral wave (the trajectory of this point over time is depicted as a thin dotted circle in Fig. 1B). Since all phases of activity meet at the phase singularity, it corresponds to a non-excited point at the inner tip of the revolving wavefront. Biophysical simulations suggest that points within the thin circle traced by the phase singularity are excitable but not excited during spiral-wave activity (they correspond in a sense to the eye in a hurricane). A second circle (heavier dotted circle in Fig. 1B and dotted circle in Fig. 3B) can be defined corresponding to the movement of the point on the wavefront from which the line perpendicular to the wavefront is tangential to a circle parallel to that traced by the phase singularity. Inside this outer circle depolarization does not reach the all-or-nothing INa threshold and is electrotonic. Thus, the circle delimits the wave-tip of the spiral wave. The behaviour of wave-tip dynamics is of primary importance, since it is a crucial determinant of spiral wave dynamics and reentry properties in spiral wave systems [26], as discussed below.
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In cardiac tissue, a spiral wave results when spontaneously occurring events result in the formation of a phase singularity around which the wave rotates. A classic scenario is provided by an ectopic activation that initiates a wavefront which crosses the recovery front of a previous sinus beat. The advancing wavefront propagates through non-refractory tissue but blocks on the edge of the refractory tissue. As the recovery front continues to advance, tissue regains excitability and is activated by the ectopic wavefront, which curves in the direction of the newly excitable cells. With further recovery, this curving continues until the advancing wavefront completes a complete revolution. In this scenario, the phase singularity represents the interface between fully excited tissue at the leading edge of the advancing wavefront and refractory tissue at the trailing edge of the recovery front.
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Duration of the action potential |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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The action potential duration (APD) is linked to the refractory period, which is the time from the upstroke of the action potential until the point in phase 3 repolarization at which there is sufficient Na+-channel recovery to permit re-excitation. The leading circle concept specifies that the pathway taken by functional reentry circuits is directly related to tissue wavelength. However, the relationship between APD and reentrant properties may be complex for the spiral wave. Interaction between the wavefront and wave-tail can induce a transition to more complex trajectories of the wave tip (meandering, as discussed below and depicted by the black curve in Fig. 5A) [27–
31] and to wave breakups [31–36]. These complex behaviours make it difficult to predict in a simple way the effect of APD changes on spiral waves. At a simple level, decreased APD should diminish the distance between the black and red curves in Fig. 1B, and by so doing, lower the interaction between the spiral wavefront and the wave-tail. The core should then decrease in size through an increase in tissue excitability. |
Electrotonic effects |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Currents flow between neighbouring cells whenever a voltage difference between them exists. When this occurs during the repolarization phase of the action potential, any APD differences will tend to be reduced (a smoothing effect). This phenomenon modifies local APD and is very sensitive to the action potential morphology [37]
. Recently published studies show that this effect of coupling, in conjunction with the velocity restitution relation, can stabilize the dynamics of anatomical reentry [38,39].
The actions of electrotonic effects on spiral waves were considered recently, and a stabilizing effect similar to that for anatomic reentry was found [40]. Changes in the shape (accelerated repolarization) of the action potential at the core could stabilize a spiral wave, preventing wave breakup. An important role for electrotonic phenomena related to the amplitude and rectification of the inward rectifier current IK1 in determining ischaemic ventricular tachycardia properties was recently shown by Samie et al. [41].
To understand the role of electrotonic effects in spiral-wave reentry, consider the diagram showing two coupled cells in Fig. 3A. For illustration purposes, the two cells shown have, when uncoupled, short and long APDs (respectively cell 1 and cell 2). If the cells are coupled with a finite resistance, cell 1 will tend to repolarize before cell 2, inducing current to flow from cell 2 to cell 1 (losing inward current from cell 2 by flow of inward current to cell 1). This current will delay the repolarization of cell 1 while at the same time accelerating repolarization of cell 2. The combined effect decreases the intrinsic dynamic difference in APD between the cells. The amplitude of the electrotonic current is a function of the difference in membrane potential and gap junction resistance. The effect on APD will depend on how the resulting currents alter the voltage-time trajectory resulting from the complement of channels in each cell.
In the case of a spiral wave, the unexcited (fully-polarized) core serves as a current sink created by the electrotonic effect. The voltage difference between the core and the depolarized wavefront accelerates repolarization in the active wave, thus decreasing APD in a zone around the core, as shown by the reduced APD area in Fig. 3B[42,43]. Reduced APD will tend to accelerate and stabilize reentry. Because the outward current portion of IK1 may be considerable at voltages just below that of the action potential plateau, electrotonic effects between the core and depolarized tissue in the spiral wave underlying a ventricular tachycardia rotor can move voltage into a critical zone causing rapid repolarization, thus greatly accelerating the reentry rate of ventricular tachycardias [41].
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Wave-tip dynamics and spiral wave reentry |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Numerous type of wave-tip trajectories have been described for spiral waves [19,
44–46], from circular to various forms of meandering, as shown in Fig. 4. The transition from a circular core to meandering has been shown to occur when a second frequency appears through a Hopf bifurcation [27,47]. The new frequency introduces a second type of spiral wave behaviour, with different core dimension and angular period co-existing with that of the initial spiral. This manifests as a single spiral wave that exhibits a complex compound rotation showing characteristics of both spiral types. Numerical simulations with cardiac action-potential ionic models exhibit patterns of meandering similar to those obtained with the simpler FHN model [44] when changing either the conductance or kinetics of currents [25,31,42]. The trajectories traced by the spiral tip can vary as shown in Fig. 4, producing irregularities in local activation frequency and pattern. However, it is important to note that the interdependence of the different currents yields a complex relationship between the type of trajectories and the tissue parameters, making it difficult to understand simply the impact of different ionic current alterations in realistic cardiac ionic models on core meandering.
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Curvature of the spiral wavefront and its consequences |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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Zykov first established the foundations of the "kinematic" theory governing the morphology and behaviour of steady spiral waves, by describing the local curvature along the wavefront [48]
. Knowing that there exists a point on the activation front always propagating tangentially to the core circle (Q point), the radius of this circle and the angular velocity of the spiral wave can be found if the curvature at the Q point is assumed to correspond to the critical curvature. Analysis and numerical integration of Zykov's model show that a smaller core is associated with a shorter rotation period, i.e. faster reentry [48]. The change in the velocity of propagation with curvature is essential to the sustainability of a spiral wave in excitable tissue.
Spiral wavefront curvature is greater near the centre and decreases away from the centre, as illustrated in Fig. 1[48,49]. The curvature is limited by the ability of a convex wave to excite the tissue ahead of it, as explained above. From the source/sink relation, a critical convex curvature (responsible for wave-tip dynamics) exists, corresponding to the limit where the wavefront can no longer propagate because the diffusive current cannot activate downstream cells [48–52]. The greater the source current, the greater will be the degree of convexity that sustains activity and, therefore, the greater the critical curvature. The notion of critical curvature, not included in the leading-circle concept, provides the notion that a decreased source current can lead to block of propagation and terminate functional reentry [48,50]. Furthermore, the critical curvature may control the minimal pathway of functional reentry with a full excitable gap. The curvature of the activation front has long been thought to be essential to spiral formation and propagation, since curvature is understood to modulate the velocity of propagation by changing the diffusive current available to drive conduction [48,53]. A geometrical deduction can show that the sink of diffusive current is greater ahead of the activation, meaning that the ratio (cells to excite/exciting cells) is less than unity [53], causing the decrease in velocity. Wavefront geometry thus dictates that the relationship of observed conduction velocity to the theoretical velocity with linear transmission: for planar waves, the value is 1:1, convex <1:1, concave >1:1, with values deviating increasingly from unity as curvature increases [54]. The modulation of the velocity by increasing convexity is essentially linear (called the eikonal relation) until the minimum diffusive current needed for excitation of the downstream cell is approached [48,55].
In addition to the curvature per se of the activation front, the rapidly varying curvature and position of the wavefront near the core produces a twisting effect, reducing the diffusive current [42,43]. These twisting effects can lead to electrotonic propagation in the core (see discussion of electrotonic effects and related Fig. 3), even if curvature is less than the critical curvature. Wave-tip twisting and electrotonic effects are important to spiral wave behaviours, since relatively large excitable gaps have been documented in some experiments [41,56] and are probably important to the linear core exhibited by functional reentries [57–59].
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Predicted effects of pharmacological therapy according to leading circle and spiral wave concepts |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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We have explained above some of the differences between the properties of the spiral wave and leading circle concepts. In this section of the review, we will consider differences in the expected impact of different ion channel blockers or activators to appreciate better the practical consequences of these differences. In some cases, we will use for illustration purposes simulations in a 2-dimensional sheet of cells based on an ionically realistic model of canine atrial action potentials (for details see refs. [60,
61]).
Na+ channel blockade
Current passing through sodium channels depolarizes the cell, moving transmembrane potential towards positive values. Changing the density of channels and/or their activation/inactivation properties will directly affect the propagating wave. The impact of Na+ channel blockade on functional reentry according to leading circle and spiral wave models is summarized in Table 1.
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Fig. 5A shows a snapshot of spiral wave activity in a 2-dimensional sheet of canine atrial tissue with realistic action potential properties [60,
61], following initiation by a single extrastimulus (cross-shock protocol). Simulated electrogram activity was fibrillatory, although activity was maintained by a single stable spiral rotor. The trajectory of the wave-tip at the core of the rotor over 1.5 seconds of activity is indicated by the thin black line. Fig. 5B shows the changes in wave-tip trajectory induced by acetylcholine, which stabilizes the rotor and reduces meander [61], and by Na+-channel blockade in the presence of acetylcholine. The spiral wave model predicts an increase in core size with INa inhibition, whereas the leading circle predicts a decrease because the wavefront is assumed always to equilibrate so as to be the closest to its own wave-tail. In the spiral wave, a decrease in INa reduces the maximum curvature by decreasing source current and increasing sensitivity to the twisting effect. This diminution of diffusive current produces slowing and an increase in the circular core, as confirmed experimentally [65]. The consequence is not so simple when there is an interaction between the wavefront and wave-tail, but the trend seems to be in the same direction. Although the wave-tip seems to cover a greater area [31] (shown in Fig. 5B), the transition in the meandering pattern can exert some variations in this tendency [64]. Overall, the leading circle concept predicts that reduced Na+-current should promote reentry by reducing the wavelength, whereas the spiral wave model predicts an antiarrhythmic action, by virtue of increased meander, increased core size and decreased critical curvature, all of which can lead to termination (depending on substrate properties). These predictions are consistent with experimental observations of the effects of Na+-channel blockers in reentrant atrial fibrillation [66–68].
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K+-current enhancement
Cellular K+-current can be increased in a variety of ways. Cholinergic activation of muscarinic M2 receptors, inducing acetylcholine-dependent K+-current (IKACh), and activation of ATP-dependent potassium current (IKATP), most typically by acute myocardial ischaemia, are two ways to increase the net repolarizing current and decrease APD (and consequently the RP). These interventions also increase background current at rest. Table 2 summarizes the predicted effects of this type of increase in K+-current.
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Similar predictions are made by both concepts for interventions causing an increase in only repolarizing current (e.g. increased delayed rectifier current). The only differences between the models are in the detailed dynamics of functional reentry, since meandering is not considered within the leading circle concept, whereas core meandering is an important component of spiral wave behaviour. An example of decreased core meander with activation of IKACh is provided in Fig. 5B. The other difference is that increased K+ inward-rectifier current might increase electrotonic effects in a spiral core, permitting faster reentry than would be predicted based on the RP of a plane wave [40,
42]. These predictions have been experimentally confirmed [69].
K+-channel blockers
Blockade of K+-channels involved in cellular repolarization (most specifically the delayed rectifiers) leads to APD prolongation and increased RP. The predicted effects of such action are provided in Table 3.
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The comparison shows again similar results for both concepts, although the altered meandering behaviours documented in experiments [70–
72] are not reproduced by the leading circle concept. Increasing the RP can induce a transition from a stable circular to meandering spiral wave. Blocking K+-channels can also lead to more complex rhythms by creating numerous spiral waves similar to fibrillatory-like activity [46,73–75]. The restitution curve, which depicts the relationship between APD and inter-excitation interval, is an important determinant of the type of spiral wave behaviour exhibited in cardiac tissue. A discussion of the role of restitution goes beyond the scope of the present paper – interested readers are referred to excellent reviews on the subject [76,77]. By altering the shape of the repolarization phase of the action potential, and by affecting excitability during and following repolarization, interventions that alter K+-currents can have important effects on restitution properties and thereby on spiral wave behaviour. The diversity of response to change in currents indicates different sensitivity to the type of K+-channel that is blocked. |
Conclusions |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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In this review, we have attempted to present the main properties of leading circle and spiral wave concepts of reentry. We have also discussed some of the predictions of the two concepts with respect to pharmacological responses of arrhythmias. The major differences between them lie in the responses to Na+-channel blockade, for which the predictions of spiral wave theory appear to correspond more closely to the results of clinical and experimental observations. The major advantage of the leading-circle model lies in the fact that it is framed in terms of simple electrophysiological properties that are readily understood and measured with conventional electrophysiological techniques. The most important limitation of the leading circle model is that it does not consider key biophysical properties like electrotonic interaction, complex properties of the medium, source-sink relations and dynamic core behaviours. From a biophysical point of view, the spiral wave notion is much more realistic and this is born out by the fact that wherever mapping studies have been performed with sufficient resolution and definition, the behaviours observed have been consistent with spiral wave theory. The great challenge for biophysicists will be to frame the properties of spiral wave reentry in terms of variables and predictions that can be understood and tested by clinical and classical basic electrophysiologists, so that the powerful biophysics of spiral wave theory can be used to gain insights into clinically-relevant mechanisms of arrhythmias and their treatment.
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Acknowledgements |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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This work emanates from a research programme supported by the Canadian Institutes of Health Research and the Mathematics of Information Technology and Complex Systems (MITACS) Network of Centers of Excellence. Dr Comtois is a research fellow of the Natural Science and Engineering Research Council of Canada (NSERC).
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References |
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TopAbstractIntroductionAnatomical versus functional...The leading-circle conceptProperties of spiral wave...Duration of the action...Electrotonic effectsWave-tip dynamics and spiral...Curvature of the spiral...Predicted effects of...ConclusionsAcknowledgementsReferences |
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[1] MacWilliam JA. Fibrillar contraction of the heart. J Physiol 1887; 8: 296–310.
[2] Mayer AG. Rhythmical pulsation in scyphomedusae 1906; Washington DC Carnegie Institute of Washington 47: pp. 1–62.
[3] Mines GR. On circulating excitation in heart muscle and their possible relation to tachycardia and fibrillation. Trans Roy Soc Can 1914; 4: 43–53.
[4] Lewis T, Feil H, Stroud W. Observations upon flutter and fibrillation ii. Nature of auricular flutter. Heart 1920; 7: 191–244.
[5] Lewis T, Drury AN, Iliescu CC. A demonstration of circus movement in clinical flutter of the auricles. Heart 1921; 8: 341–355.
[6] Lewis T, Drury AN, Iliescu CC. A demonstration of circus movement in clinical fibrillation of the auricles. Heart 1921; 8: 361–369.
[7] Garrey WE. Auricular fibrillation. Physiol Rev 1924; 4: 215–250.
[8] Allessie MA, Bonke FI, Schopman FJ. Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circ Res 1973; 33: 54–62.
[9] Allessie MA, Bonke FI, Schopman FJ. Circus movement in rabbit atrial muscle as a mechanism of tachycardia. II. The role of non-uniform recovery of excitability in the occurrence of unidirectional block, as studied with multiple microelectrodes. Circ Res 1976; 39: 168–177.
[10] Allessie MA, Bonke FI, Schopman FJ. Circus movement in rabbit atrial muscle as a mechanism of tachycardia. III. The "leading circle" concept: a new model of circus movement in cardiac tissue without the involvement of an anatomical obstacle. Circ Res 1977; 41: 9–18.
[11] Wiener N and Rosenblueth A. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiologia Mexico 1946; 16: 205–265.
[12] Moe GK, Rheinboldt WC, Abildskov JA. A computer model of atrial fibrillation. Am Heart J 1964; 67: 200–220.
[13] Frame LH, Page RL, Boyden PA, Fenoglio JJ Jr, Hoffman BF. Circus movement in the canine atrium around the tricuspid ring during experimental atrial flutter and during reentry in vitro. Circulation 1987; 76: 1155–1175.
[14] Daoud EG and Morady F. Pathophysiology of atrial flutter. Annu Rev Med 1998; 49: 77–83.
[15] Kim YH, Xie F, Yashima M, et al. Role of papillary muscle in the generation and maintenance of reentry during ventricular tachycardia and fibrillation in isolated swine right ventricle. Circulation 1999; 100: 1450–1459.
[16] Janse MJ. Functional reentry: leading circle or spiral wave? J Cardiovasc Electrophysiol 1999; 10: 621–622.
[17] Ikeda T, Uchida T, Hough D, et al. Mechanism of spontaneous termination of functional reentry in isolated canine right atrium. Evidence for the presence of an excitable but non-excited core. Circulation 1996; 94: 1962–1973.
[18] Karagueuzian HS, Athill CA, Yashima M, et al. Transmembrane potential properties of atrial cells at different sites of a spiral wave reentry: cellular evidence for an excitable but non-excited core. Pacing Clin Electrophysiol 1998; 21: 2360–2365.
[19] Winfree AT. Spiral waves of chemical activity. Science 1972; 175: 634–636.
[20] Winfree AT. When time breaks down: the three-dimensional dynamics of electrochemical waves and cardiac arrhythmias 1987; Princeton, NJ Princeton University Press.
[21] Panfilov AV and Pertsov AM. Mechanism of the origin of the helical waves in active media associated with the phenomenon of critical curvature. Biofizika 1982; 27: 886–889.
[22] van Capelle FJ and Durrer D. Computer simulation of arrhythmias in a network of coupled excitable elements. Circ Res 1980; 47: 454–466.
[23] Grenadier AK and Panfilov AV. Spiral waves in heart. Biofizika 1981; 26: 1107–1108.
[24] Courtemanche M and Winfree AT. Re-entrant rotating waves in a Beeler-Reuter based model of two-dimensional cardiac electrical activity. Int J Bifurcation Chaos 1991; 1: 431–444.
[25] Leon LJ, Roberge FA, Vinet A. Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics. Ann Biomed Eng 1994; 22: 592–609.
[26] Biktasheva IV and Biktashev VN. Wave-particle dualism of spiral waves dynamics. Phys Rev E 2003; 67: 026221.
[27] Karma A. Meandering transition in two-dimensional excitable media. Phys Rev Lett 1990; 65: 2824–2827.
[28] Hakim V and Karma A. Spiral wave meander in excitable media: the large core limit. Phys Rev Lett 1997; 79: 665–668.
[29] Golubitsky M, LeBlanc VG, Melbourne I. Meandering of the spiral tip: an alternative approach. Non-linear Sci 1997; 7: 557–586.
[30] Otani NF. A primary mechanism for spiral wave meandering. Chaos 2002; 12: 829–842.
[31] Qu Z, Xie F, Garfinkel A, Weiss JN. Origins of spiral wave meander and breakup in a two-dimensional cardiac tissue model. Ann Biomed Eng 2000; 28: 755–771.
[32] Panfilov A and Hogeweg P. Spiral breakup in a modified FitzHugh-Nagumo model. Phys Lett A 1993; 176: 295–299.
[33] Karma A. Electrical alternans and spiral wave breakup in cardiac tissue. Chaos 1994; 4: 461–472.
[34] Qu Z, Weiss JN, Garfinkel A. Cardiac electrical restitution properties and stability of reentrant spiral waves: a simulation study. Am J Physiol 1999; 276: H269–H283.
[35] Qu Z, Garfinkel A, Chen PS, Weiss JN. Mechanisms of discordant alternans and induction of reentry in simulated cardiac tissue. Circulation 2000; 102: 1664–1670.
[36] Xie F, Qu Z, Garfinkel A, Weiss JN. Electrical refractory period restitution and spiral wave reentry in simulated cardiac tissue. Am J Physiol 2002; 283: H448–H460.
[37] Laurita KR, Girouard SD, Rudy Y, Rosenbaum DS. Role of passive electrical properties during action potential restitution in intact heart. Am J Physiol 1997; 273: H1205–H1214.
[38] Cytrynbaum E and Keener JP. Stability conditions for the traveling pulse: modifying the restitution hypothesis. Chaos 2002; 12: 788–799.
[39] Comtois P and Vinet A. Stability and bifurcation in an integral-delay model of cardiac reentry including spatial coupling in repolarization. Phys Rev E 2003; 68: 051903.
[40] Cherry EM and Fenton FH. Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory, and conduction velocity restitution effects. Am J Physiol 2004; 286: H2332–H2341.
[41] Samie FH, Berenfeld O, Anumonwo J, et al. Rectification of the background potassium current: a determinant of rotor dynamics in ventricular fibrillation. Circ Res 2001; 89: 1216–1223.
[42] Beaumont J, Davidenko N, Davidenko JM, Jalife J. Spiral waves in two-dimensional models of ventricular muscle: formation of a stationary core. Biophys J 1998; 75: 1–14.
[43] Beaumont J and Jalife J. Rotors and spiral waves in two dimensions. In Zipes DP and Jalife J (Eds.). Cardiac Electrophysiology From Cell to Bedside 2000; 3rd ed. Philadelphia, PA W.B. Saunders pp. 327–335.
[44] Winfree AT. Varieties of spiral wave behavior: an experimentalist's approach to the theory of excitable media. Chaos 1991; 1: 303–334.
[45] Fenton F and Karma A. Vortex dynamics in three-dimensional continuous myocardium. Chaos 1998; 8: 20–47.
[46] Fenton FH, Cherry EM, Hastings HM, Evans SJ. Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 2002; 12: 852–892.
[47] Barkley D. Linear stability analysis of rotating spiral waves in excitable media. Phys Rev Lett 1992; 68: 2090–2093.
[48] Zykov VS and Winfree AT. Simulation of wave processes in excitable media 1987; Manchester Manchester University Press.
[49] Comtois P and Vinet A. Curvature effects on activation speed and repolarization in an ionic model of cardiac myocytes. Phys Rev E 1999; 60: 4619–4628.
[50] Zykov VS. Kinematics of stationary circulation in an excitable medium. Biofizika 1980; 25: 319–322.
[51] Cabo C, Pertsov AM, Baxter WT, Davidenko JM, Gray RA, Jalife J. Wave-front curvature as a cause of slow conduction and block in isolated cardiac muscle. Circ Res 1994; 75: 1014–1028.
[52] Fast VG and Kleber AG. Block of impulse propagation at an abrupt tissue expansion: evaluation of the critical strand diameter in 2- and 3-dimensional computer models. Cardiovasc Res 1995; 30: 449–459.
[53] Zykov VS. Analytic evaluation of the relationship between the speed of a wave of excitation in a two-dimensional excitable medium and the curvature of its front. Biofizika 1980; 25: 888–892.
[54] Fast VG and Kleber AG. Role of wavefront curvature in propagation of cardiac impulse. Cardiovasc Res 1997; 33: 258–271.
[55] Tyson JJ and Keener JP. Singular perturbation theory of traveling waves in excitable media (a review). Physica D 1988; 32: 327–361.
[56] Davidenko JM, Kent PF, Chialvo DR, Michaels DC, Jalife J. Sustained vortex-like waves in normal isolated ventricular muscle. Proc Natl Acad Sci 1990; 87: 8785–8789.
[57] Frazier DW, Wolf PD, Wharton JM, Tang AS, Smith WM, Ideker RE. Stimulus-induced critical point. Mechanism for electrical initiation of reentry in normal canine myocardium. J Clin Invest 1989; 83: 1039–1052.
[58] Davidenko JM, Pertsov AV, Salomonsz R, Baxter W, Jalife J. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 1992; 355: 349–351.
[59] Efimov IR, Sidorov V, Cheng Y, Wollenzier B. Evidence of three-dimensional scroll waves with ribbon-shaped filament as a mechanism of ventricular tachycardia in the isolated rabbit heart. J Cardiovasc Electrophysiol 1999; 10: 1452–1462.
[60] Ramirez RJ, Nattel S, Courtemanche M. Mathematical analysis of canine atrial action potentials: rate, regional factors, and electrical remodeling. Am J Physiol 2000; 279: H1767–H1785.
[61] Kneller J, Zou R, Vigmond EJ, Wang Z, Leon LJ, Nattel S. Cholinergic atrial fibrillation in a computer model of a two-dimensional sheet of canine atrial cells with realistic ionic properties. Circ Res 2002; 90: E73–E87.
[62] Cabo C, Pertsov AM, Davidenko JM, Jalife J. Electrical turbulence as a result of the critical curvature for propagation in cardiac tissue. Chaos 1998; 8: 116–126.
[63] Hakim V and Karma A. Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications. Phys Rev E 1999; 60: 5073–5105.
[64] Efimov IR, Krinsky VI, Jalife J. Dynamics of rotating vortices in the Beeler-Reuter model of cardiac tissue. Chaos Solitons Fractals 1995; 5: 513–526.
[65] Kawase A, Ikeda T, Nakazawa K, et al. Widening of the excitable gap and enlargement of the core of reentry during atrial fibrillation with a pure sodium channel blocker in canine atria. Circulation 2003; 107: 905–910.
[66] Kalifa J, Kneller J, Zaitsev AV, et al. Mechanism of atrial fibrillation termination by sodium channel blockers: Insights from computational and experimental data. Circulation 2003; 108:Suppl. IV IV–148 [abstract].
[67] Wijffels MC, Dorland R, Mast F, Allessie MA. Widening of the excitable gap during pharmacological cardioversion of atrial fibrillation in the goat: effects of cibenzoline, hydroquinidine, flecainide, and d-sotalol. Circulation 2000; 102: 260–267.
[68] Kneller J, Kalifa J, Zou R, et al. Mechanisms of atrial fibrillation termination by pure sodium channel blockade in an ionically-realistic mathematical model. Circ Res 2005; 96: e35–e47.
[69] Uchida T, Yashima M, Gotoh M, et al. Mechanism of acceleration of functional reentry in the ventricle: effects of ATP-sensitive potassium channel opener. Circulation 1999; 99: 704–712.
[70] Gray RA, Jalife J, Panfilov A, et al. Non-stationary vortex-like reentrant activity as a mechanism of polymorphic ventricular tachycardia in the isolated rabbit heart. Circulation 1995; 91: 2454–2469.
[71] Ikeda T, Czer L, Trento A, et al. Induction of meandering functional reentrant wave front in isolated human atrial tissues. Circulation 1997; 96: 3013–3020.
[72] Ikeda T, Wu TJ, Uchida T, et al. Meandering and unstable reentrant wave fronts induced by acetylcholine in isolated canine right atrium. Am J Physiol 1997; 273: H356–H370.
[73] Riccio ML, Koller ML, Gilmour RF Jr. Electrical restitution and spatiotemporal organization during ventricular fibrillation. Circ Res 1999; 84: 955–963.
[74] Garfinkel A, Kim YH, Voroshilovsky O, et al. Preventing ventricular fibrillation by flattening cardiac restitution. PNAS 2000; 97: 6061–6066.
[75] Swissa M, Qu Z, Ohara T, et al. Action potential duration restitution and ventricular fibrillation due to rapid focal excitation. Am J Physiol 2002; 282: H1915–H1923.
[76] Franz MR. The electrical restitution curve revisited: steep or flat slope–which is better? J Cardiovasc Electrophysiol 2003; 14:Suppl. 10 S140–S147.
[77] Weiss JN, Chen PS, Qu Z, Karagueuzian HS, Lin SF, Garfinkel A. Electrical restitution and cardiac fibrillation. J Cardiovasc Electrophysiol 2002; 13: 292–295.
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