# Cardiac arrhythmia

 Flavio H Fenton et al. (2008), Scholarpedia, 3(7):1665. doi:10.4249/scholarpedia.1665 revision #121399 [link to/cite this article]
Post-publication activity

Curator: Leon Glass

Figure 1: Example of ventricular fibrillation, a deadly cardiac arrhythmia if not treated within minutes.

Figure 2: Heart structure.

Cardiac arrhythmia is the condition in which the heart's normal rhythm is disrupted. In this article, we describe the heart's normal sinus rhythm and a number of different types of disruptions of this rhythm and how dynamical systems can be used to understand the behavior of the heart in these circumstances.

The heart pumps blood containing oxygen, nutrients, immune cells, and regulatory molecules to the body organs. The rhythm of the heart is set by a small region of cardiac muscle cells in the right atrium called the sinoatrial (SA) node that acts as a spontaneous pacemaker, but is under the control of nerves and circulating hormones that affect the heart rate via a host of control circuits that maintain adequate blood pressure and oxygenation. The heart itself is composed of two upper chambers, the atria, and two lower chambers, the ventricles, as shown in Figure 2.

The normal heart rhythm is called sinus rhythm. In sinus rhythm ( Figure 3), each beat spontaneously generated from the SA node produces a propagating wave of bioelectricity that spreads throughout the four chambers of the heart in a coordinated fashion. Each impulse propagates throughout the atria before being channeled through the atrioventricular (AV) node to the ventricles. This electrical wave triggers intracellular calcium processes that produce the contractions of the cardiac muscle that pump the blood to the organs of the body. The slow (about 120-200 ms) conduction time through the AV node allows adequate time for atrial contraction and ventricular filling. Upon emerging from the AV node, the electrical impulse propagates through specialized conducting bundles called the His-Purkinje system and from there to the ventricles. The His-Purkinje system allows rapid conduction to all areas of the ventricles and therefore is responsible for ensuring effective ventricular contraction. Normally, the heart beats at a rate of approximately 75 beats per minute (although there is substantial individual variation) and pumps about 5 liters of blood per minute.

Figure 3: Simulation and electrocardiogram corresponding to the heart's normal sinus rhythm. The atrial contraction is initiated by the depolarization wave (shown in yellow), which originates from the sinoatial node and corresponds to the P wave on the ECG. After a delay passing through the atrioventricular node, the activation passes to the ventricles and produces a contraction. The QRS complex indicates ventricular depolarization (shown in yellow), while the T wave corresponds to ventricular repolarization.

The heart rhythm is typically monitored by an electrocardiogram (ECG), which measures the voltage differences between points on the surface of the body. Figure 3 shows the normal cardiac rhythm on an ECG, where the P wave is associated with the excitation of atria, the QRS complex is associated with the excitation of ventricles, and the T wave is associated with the relaxation of the ventricles. The duration of the time for an excitation to travel from the atria through the AV node to the ventricles is associated with the PR interval, and the duration of the excitation phase of the ventricles is associated with the QT interval.

## Overview of arrhythmias

Abnormal cardiac rhythms are called cardiac arrhythmias. Cardiac arrhythmias are associated with abnormal initiation of a wave of cardiac excitation, abnormal propagation of a wave of cardiac excitation, or some combination of the two. Cardiac arrhythmias can manifest themselves in many different ways, and it is still not always possible to determine the mechanism of an arrhythmia.

Arrhythmias can be classified in several ways. One useful classification that will be utilized here is reentrant versus non-reentrant arrhythmias. In reentry, cardiac tissue is repetitively excited by a propagating wave circulating around an obstacle (anatomical reentry) or circulating freely in the tissue as a spiral or scroll wave (functional reentry). Thus, there is a strong spatial component to reentrant arrhythmias: either a sufficiently large spatial extent is needed to support the initiation and continuation of the arrhythmia, or an appropriate geometry must be present to allow a reentrant circuit. Non-reentrant arrhythmias also may have a strong geometric component in which either propagation is blocked at particular anatomical sites or one or more pacemakers form at abnormal (ectopic) locations.

Arrhythmias also can be classified by the heart rate. Tachyarrhythmias are rhythms in which the heart rate is faster than normal, usually taken as greater than 100 beats per minute. These arrhythmias are further classified based on where they arise as either ventricular (within the ventricles) or supraventricular (anywhere above the ventricles, including the SA node, the atria, and the AV node). Tachyarrhythmias may lead to reduced blood flow to the organs of the body leading to reduced ability to exercise, faintness, or, in cases in which the blood flow is too low, death. Tachyarrhythmias can arise from an accelerated sinus rhythm, an accelerated rhythm from an abnormal ectopic site (as in tachycardias arising in the right ventricular outflow tract), or from the interactions of multiple ectopic sites (as in multifocal atrial tachycardia). However, more usually tachyarrhythmias are believed to arise from reentrant arrhythmias, in which the period of the oscillation is set by the time an excitation takes to travel in a circuitous path, rather than the period of oscillation of a pacemaker (Josephson, 2002). Bradycardias are rhythms in which the heart rate is decreased below 60 beats per minute. Bradycardias may arise because the sinus rhythm is abnormally slow, the sinus rhythm is absent and a secondary slower pacemaker (e.g. in the AV node) takes over, or the normal sinus rhythm is blocked in its passage through the heart. Since well-conditioned athletes may have normal heart rates below 60 beats per minute, not all bradycardic rhythms are associated with impaired cardiac function.

This article is meant as a basic introduction to cardiac arrhythmias from a nonlinear dynamics perspective. The striking rhythms evidenced on an ECG have stimulated interest in understanding mechanisms of arrhythmia dating back to the early 1900s. Mechanisms have often been formulated using graphs or simple word models, and only subsequently have some of the deeper mathematical and theoretical contexts of the arrhythmia mechanisms become clear. First we summarize three non-reentrant arrhythmias—AV heart block, parasystole, alternans—whose basic properties can be understood using very simple finite difference equation mathematical models. Then we describe spatially-dependent reentrant rhythms, which require partial differential equations for appropriate theoretical modeling.

## Non-reentrant arrhythmias

If cardiac tissue is stimulated rapidly, two different scenarios can occur, depending on the circumstances. One is typified by AV block in which not all of the stimuli lead to action potentials that propagate through the tissue but are blocked during the progression through the tissue. The second is the development of alternans, in which each action potential is conducted through the tissue, but the duration of the action potentials alternates on a beat-to-beat basis. We give models for the simplest instances of these rhythms. A third scenario can arise when an abnormal (ectopic) ventricular pacemaker interacts with normal sinus beats.

### AV heart block

Figure 4: Example of Wenckebach 2:1 rhythm, in which only every other beat is conducted from the atria to the ventricles.
Figure 5: Example of Wenckebach map showing 3:2 rhythm arising from a stimulation period of 353 ms. Parameter values for this map correspond to those used by Shrier et al. (1987). Go to applet.

In Wenckebach rhythms, the sinus node generates a normal rhythm, but not all the excitations successfully traverse the AV node--some are blocked. It is usual to describe AV heart block using two integers. For example, 4:3 heart block indicates that every four sinus beats (P waves) produce only three ventricular excitations (QRS complexes). Figure 4 shows a 2:1 Wenckebach rhythm in which only every other atrial beat activates the ventricles. The original theoretical insight for understanding AV heart block goes back at least to Möbitz, who developed a graphical method for determining the dynamics. The basic idea, as proposed by Möbitz (1924) and subsequently discovered and rediscovered by several others (Guevara 1991), is to assume that the conduction time of the ith beat through the AV node, $$SH_i\ ,$$ is a function $$F$$ of the recovery time since the passage of the last excitation through the AV node, which is designated $${VA}_{i-1}\ .$$ Assuming that there is a periodic atrial stimulation, either from the sinus node or by an artificial pacemaker, with a period $$SP\ ,$$ we find (Shrier et al., 1987) that

$$SH_i = F(VA_{i-1}) = F(k SP - SH_{i-1})\ ,$$

where $$k$$ is the smallest integer such that $$k SP > \theta\ ,$$ and $$\theta$$ is the refractory period of the AV node. Typically, the recovery curve F is a monotonically decreasing curve. Figure 5 shows an example of a 3:2 rhythm obtained using this map in an interactive Java applet. In this case, it is possible to demonstrate mathematically that if the properties of the AV node are fixed, different types of N:M heart block can be calculated as the frequency of atrial activation is increased, where N is the number of sinus beats and M is the number of ventricular beats in a repeating sequence (see Figure 5). If there is N:M heart block at one stimulation frequency and N':M' heart block at a higher frequency, then N+N':M+M' heart block is expected at some intermediate stimulation frequency. Thus, as initially pointed out by Keener, the patterns of AV heart block generate a Devil's staircase if the conduction ratio is plotted as a function of stimulation frequency, as shown in Figure 5 (Roberge and Nadeau, 1969; Keener, 1981; Shrier et al., 1987; Guevara, 1991).

### Alternans

In alternans rhythms, the duration and possibly the morphology of some complex of the electrocardiogram alternates on a beat-to-beat basis. In recent years, T-wave alternans, in which the duration of the T-wave on the ECG alternates, has attracted attention, and there is some evidence that prominent T-wave alternans at about 110 beats per minute confers a higher risk of sudden cardiac death. However, other types of alternans, such as PR alternans, in which the conduction time through the AV node alternates, also may occur.

#### Electrical alternans

Figure 6: Example of action potential duration (APD) alternans in a canine Purkinje fiber paced at a cycle length (CL) of 155 ms measured using microelectrode recordings.

Alternans in the ECG signal has its origins in alternans in the action potential at the cellular level. Figure 6 shows how during fast pacing at a constant cycle length the action potential duration (APD) can alternate between long and short from one beat to the next in cardiac tissue.

Figure 7: Alternans and period-doubling bifurcation. APD as a function of DI. Go to applet. For an applet showing the same map but where $$APD_{n+1}$$ is a function of $$APD_{n},$$ visit this link.

From a nonlinear dynamics perspective, alternans phenomena immediately suggest a period-doubling bifurcation in an appropriate mathematical model. A basic understanding of this using a graphical construction was provided by Nolasco and Dahlen (1968), and subsequently was rediscovered by Guevara et al. (1984), who placed the mechanism in a nonlinear dynamics context. The APD restitution curve describes the duration of the ith action potential $$APD_{i+1}$$ as a function $$G$$ of the time since the end of the preceding action potential. Typically, this is a monotonically increasing function, so that the longer the tissue has had to recover, the longer the action potential that can result. Representing this mathematically, we have $$APD_{i+1} = G(CL - APD_i) = G(DI_I)\ ,$$ where $$CL$$ is the time between sinus beats (or pacing cycle length in experimental settings) and $$DI_i$$ is the previous diastolic interval (see Figure 6). This is again a one-dimensional map and the stability can be obtained by linearizing around the fixed point $$APD^* = G(DI^*)\ .$$ Letting $$APD_n=APD^* + \delta APD_n \ ,$$ it follows that $$\delta APD_{n+1} = -G' (DI) \delta APD_n \ .$$

This equation shows a bifurcation when $$|G'(DI)| >1 \ .$$ Therefore, whenever the slope of the restitution curve is above 1 or below -1 (as may be the case for biphasic restitution curves), there is a period-doubling bifurcation leading to an alternation of the action potential duration, as shown in the interactive map of Figure 7. This slope 1 criterion is only valid for the one-dimensional map, where $$APD_{i+1}$$ depends only on $$DI_i\ .$$ However, in more general cases, $$APD_{i+1}$$ also depends on $$APD_{i}, DI_{i-1},$$ etc. For example, when $$APD_{i+1} = G(APD_i,Di_i)\ ,$$ the function $$G$$ is now a two-dimensional map and the criterion for alternans becomes $$| 1 - (1+ (1/S_{dyn}))*S_{12} | > 1$$ (Tolkacheva et al., 2003). Thus, the criterion now not only depends on the slope of the steady-state restitution curve $$S_{dyn}$$ but also on the slopes of the family of S1-S2 restitution curves $$S_{12}\ .$$ Furthermore, these maps are derived only to describe the dynamics of a single cell. In tissue, an electrotonic coupling term must be considered (Echebarria and Karma, 2002), which can change the alternans criterion so that even steep restitution curves with slopes much greater than 1 may not show alternans (Cherry and Fenton, 2004).

#### Calcium alternans

Figure 8: Alternans in intracellular calcium. Go to applet.

While electrical alternans can be attributed to cellular electrical currents and in some cases to dynamics associated with the APD restitution curve, alternans also can result from intracellular calcium dynamics. The calcium and membrane potential dynamics in cardiac cells are two coupled systems, as the shape of the action potential is determined by ionic currents across the cell membrane, and some of these currents are dependent on the intracellular calcium concentrations that may in turn depend on the membrane potential. It has been shown that during rapid pacing the intracellular calcium concentration can alternate even while the action potential shape, and thus its duration, is held constant (Chudin et al., 1999). One mechanism known to produce alternans in the intracellular cytoplasmic calcium concentration (calcium transient) is the nonlinear relation between the calcium fluxes across the sarcolemma and the sarcoplasmic reticulum (SR) content (Eisner et al., 2000). Figure 8 shows how alternans and other complex rhythms can appear when there is a steep relationship between the SR content and the calcium efflux, using the Eisner-Choi-Diaz-Neill-Trafford map of intracellular calcium dynamics:

$$SR_{i+1} = SR_i - \textit{Efflux} + \textit{Influx}$$

$$\textit{Efflux} = SR*SR^N / (50^N+SR^N)$$

.

This is a classical negative feedback system where $$N$$ is a Hill coefficient that represents the nonlinearity of the system. This relationship includes the dependence of calcium release on SR content, the calcium buffering properties of the cytoplasm, and the relationship between the calcium transient and calcium efflux. At low values of $$N$$ the system is stable; however, as N increases above 5 the relation becomes steep enough to produce alternans.

### Parasystole

An arrhythmia that is amenable to a simple mathematical analysis whose essence has been understand since the early 1900s is parasystole (Kaufman and Rothberger, 1917). In the “pure” case, the normal sinus rhythm occurs at a constant frequency, and an abnormal (ectopic) pacemaker in the ventricles beats at a second slower frequency (Glass et al., 1986). If the ectopic pacemaker fires at a time outside the refractory period of the ventricles, then there is an abnormal ectopic beat, identifiable on the electrocardiogram by a distinct morphology from the normal beat, and the following normal sinus beat is blocked. If the normal and abnormal beats occur at the same time, this leads to a fusion (F) beat. This simple mechanism has amazing consequences. These can be appreciated by forming a sequence of integers that counts the number of sinus beats between two ectopic beats. In general, for fixed sinus and ectopic frequencies and a fixed refractory period, in this sequence there are at most three integers, where the sum of the two smaller integers is one less than the largest integer. Moreover, given the values of the parameters, it is possible to predict the three integers. Clinical studies demonstrate the applicability of these results to patients with an artificial parasystolic pacemaker produced by periodically stimulating the ventricles with an intracardiac catheter at a fixed rate (Castellanos et al., 1991). The mathematics for this problem is related to the “gaps and steps” problem in number theory (Slater, 1967).

The rules of pure parasystole are rarely followed for long times. In some patients, the parasystolic pacemaker is reset, or modulated by the beat originating at the sinus node (Moe et al., 1977). This situation is also amenable to detailed theoretical analysis (Courtemanche et al., 1989). One interesting aspect of parasystole is that it is normally considered to be a benign arrhythmia. Thus, despite a fascinating mathematical aspect, parasystole is not encountered frequently, and when it is, it is not appropriate to treat. In a review, Castellanos et al. (2004) described the situation: “ventricular parasystole appears to be only of historical and intellectual interest.”

### Other non-reentrant arrhythmias

Figure 9: Fitzhugh-Nagumo model phase space. Auto-oscillatory behavior results from changing $$\delta$$ from 0.03 to 0.1. Go to applet.

Along with AV block, alternans, and parasystole, other non-reentrant arrhythmias can occur in the heart. The SA node may beat more rapidly than usual (sinus tachycardia) or more slowly than usual (sinus bradycardia). While the details of SA node behavior are complicated, the auto-oscillatory dynamics of the SA node can be understood using the simple Fitzhugh-Nagumo model in phase space. Figure 9 shows that when the nullclines of the model intersect in the unstable region (for example, by varying $$\delta$$ between 0.03 and 0.1), auto-oscillatory behavior occurs, with the period depending on the intersection location. (Periodicity and duration also vary as a function of the parameter $$\epsilon\ ,$$ which is the scaling time between the $$u$$ and $$v$$ variables). Most other forms of bradycardia are also non-reentrant arrhythmias. These include sick sinus syndrome, in which SA node impulse formation becomes highly variable; escape rhythms, in which a long pause from atrial input induces latent pacemaker cells in the AV node or in the His-Purkinje system to fire; and premature atrial and ventricular beats, which often arise from ectopic beats occurring in parts of the atria or ventricles that do not normally serve as pacemaker sites.

Other sources of ectopic beats that do not arise from automaticity but instead are due to a response to a preceding impulse are the forms of triggered activity, which can take the form of either early afterdepolarizations (EADs) or delayed afterdepolarizations (DADs). EADs are oscillations in membrane potential that occur during the plateau phase of the action potential whenever an abnormally slow inactivation of the calcium or sodium currents is present, particularly when there is a prolongation of the APD. DADs are activations produced after the membrane potential has returned to its resting value. Pathological calcium dynamics are believed to be responsible for triggering a transient inward current that can depolarize the membrane. Triggered activity can be arrhythmogenic by initiating reentry.

## Reentrant arrhythmias

Reentrant arrhythmias can be confined to a single chamber of the heart, or can involve several chambers. In some instances, it is convenient to think of the underlying circuit for the reentrant excitation as a one-dimensional ring, as was initially proposed by Mines (1913). In other cases, the reentrant circuit might be taking place in two dimensions and the wave shape would be a rotating spiral wave. This notion was first made explicit by Wiener and Rosenblueth (1948). However, since the heart is three-dimensional, in other situations it is necessary to think of the reentrant circuit as a three-dimensional scroll wave as proposed by Winfree, who was the first to discover spiral waves experimentally in the context of excitable systems (Winfree, 1972). The first demonstration of a spiral wave in cardiac tissue occurred 20 years later (Davidenko et al., 1992).

### Anatomical reentry

Although the conceptualization of a wave traveling on a one-dimensional ring seems overly simplistic, from perspectives of both mathematics and medicine there are several interesting consequences (Rudy, 1995). Experimental systems, simulations, and theoretical analyses have demonstrated that waves circulating on one-dimensional rings may experience an instability such that the circulation is not constant. Instead, there can be a complex fluctuating propagation velocity that arises as a consequence of the interaction of the wavefront with its own refractory tail (Frame and Simpson, 1988; Quan and Rudy, 1991; Courtemanche et al., 1993; Vinet and Roberge, 1994). The analysis of the instability relies on employing both the APD restitution curve $$APD=G(DI)$$ and the conduction velocity (CV) restitution curve $$CV=F(DI)$$ for a pulse traveling in space. Now, however, at each point in space one must apply both the APD and CV restitution curve functions $$F(DI)$$ and $$G(DI)\ ,$$ which for simplicity can be considered to first order to depend only on the preceding diastolic interval $$DI\ .$$ When this is done it is possible to write an integral-delay equation for the diastolic interval as a function of space (Courtemanche et al., 1993):

$$DI(x) = \int_{x-L}^x \frac{ds}{(G(DI(s))} - F(DI(x-L)) .$$
Figure 10: Three wavelength modes possible in the Beeler-Reuter cardiac model during discordant alternans in a one-dimensional ring. From top to bottom: one node, three nodes, and five nodes. Left: voltage profiles of the propagating waves in space. Right: two successive action potential durations measured at all points in space. The zero node (no alternans) is also possible for this cycle length, but it is unstable and can only be reached using control algorithms. The one-node solution is the most stable mode (see Figure 10.) The three- and five-node modes were obtained using special initial conditions, and any perturbation can bring them back to the one-node mode.

By converting this equation into a neutral delay-differential equation by taking a derivative with respect to $$x$$ (Courtemanche et al., 1993) and linearizing around the steady state, it is possible to obtain the ring lengths below which an instability will arise. This analysis, however, leads to an infinite number of eigenvalues or wavelength modes, which is physiologically impossible. This infinite number of possible wavelengths is due to the absence of a coupling term between cells to account for electrotonic effects. By including this term (Echebarria and Karma, 2002}, the number of possible wavelengths becomes finite. Figure 10 shows the three possible wavelength modes of a one-dimensional ring using the Beeler-Reuter model (1977). These are discordant alternans with one node, three nodes, and five nodes. Discordant alternans (Qu et al., 2000; Watanabe et al., 2001) is discussed further in the restitution and the alternans sections. Figure 11 shows the propagation of an action potential on a one-dimensional ring. Using the accompanying applet, it is possible to decrease the size of the ring (which decreases the period of rotation) and thus observe the initiation of the primary wavelength mode, which is discordant alternans with one node. In addition, if a single stimulus is delivered to the medium during the course of the reentrant propagation (by clicking in the applet window), the propagating wave will either be reset or annihilated (Quan and Rudy, 1991; Glass and Josephson, 1995; Sinha et al., 2002; Comtois and Vinet, 2002). Further, periodic stimulation can lead to the entrainment or annihilation of the propagating wave (Glass et al., 2002; Sinha et al., 2002; Comtois and Vinet, 2002). Finally, a sequence of premature stimuli delivered to the heart during normal sinus rhythm can often lead to the initiation of tachycardia. In some clinical settings, analysis of the resetting, entrainment, and initiation of tachycardias offers clinicians important clues about the arrhythmia mechanism, and consequently can help the cardiologist choose an appropriate therapy (Stevenson and Delacretaz, 2000; Josephson, 2002). The ability to induce monomorphic ventricular tachycardia using a sequence of up to three premature stimuli is often taken as an indication of anatomical reentry as a mechanism for the tachycardia. Since at least part of the reentrant circuit is assumed to be one-dimensional, this can provide a target for ablation therapy.

Figure 11: Propagation in a one-dimensional ring. Discordant alternans develop as the ring size is decreased. Go to applet. ]

Several types of reentrant arrhythmias can be explained clearly using one-dimensional rings and cables. One example is AV nodal reentrant tachycardia (AVNRT). When this arrhythmia is present, the AV node has two distinct pathways, one fast, with a rapid velocity but relatively long refractory period, and the other slow, with a slower velocity but shorter refractory period. Normally, an impulse from the atria comes from a common pathway and is conducted by both pathways. However, while the impulse from the fast pathway reaches the His-Purkinje system, the impulse in the slow pathway takes longer to propagate and finds the common pathway in the His-Purkinje system refractory from the fast pathway impulse. Certain circumstances (such as an atrial premature beat) may produce an impulse that arrives at the AV node while the fast pathway is still refractory, so that only the slow pathway conducts the impulse. However, upon reaching the common pathway at the His-Purkinje system, the fast pathway may no longer be refractory, and the impulse may propagate retrogradely back to the atria. In turn, this impulse may propagate to the slow pathway, setting up a reentrant ring in the AV node. A similar phenomenon occurs in preexcitation syndrome (Wolff-Parkinson-White Syndrome). In this arrhythmia, an abnormal accessory pathway provides a secondary means for impulses to propagate from the atria to the ventricles. An impulse that travels along this accessory pathway avoids the AV nodal delay and therefore reaches the ventricles before the impulse that travels through the AV node and His-Purkinje system, leading to ECG abnormalities during normal sinus rhythm including a shortened PR interval, a widened QRS complex and a broader QRS upstroke (called the delta wave) that arises because the impulse begins to propagate slowly through the ventricles before the specialized conduction system is invoked and initiates rapid conduction. The presence of this pathway can lead to a reentrant loop. Another anatomically based reentrant tachycardia is atrial flutter. This rhythm is characterized by circulating waves, most usually confined to the right atrium. During atrial flutter, the excitation typically passes through a narrow isthmus between the tricuspid valve and the coronary sinus. Atrial flutter is usually associated with a regular conduction pattern through the AV node, most usually 2:1 or 4:1 conduction, although in some instances there are fluctuations in the conduction ratio.

From a mathematical perspective, all the anatomically-based reentrant arrhythmias mentioned here as a first approximation can be thought of as a wave circulating on a one-dimensional ring (at least for part of the circuit). A better approximation would be to consider propagation in an annulus (Comtois and Vinet, 2005), where interactions between the periods associated with the inner and outer edges can occur along with source/sink effects from curvature. Clinically, the localization of a reentrant circuit is useful to the cardiologist, who can change the topology by interrupting the ring or disk using ablation, which destroys tissue that forms part of the anatomical circuit and thereby eliminates the anatomical basis for the arrhythmia.

### Functional reentry: spiral and scroll waves

Figure 12: Initiation of counter-rotating spiral waves by a properly timed stimulus during the vulnerable window occurring as the wave repolarizes.

Single or double spiral waves or scroll waves are often generated in excitable cardiac tissue or models of cardiac tissue by a single impulse delivered in the wake of a propagating wave during the vulnerable period, as shown in Figure 12. These reentries do not rotate around obstacles; instead, they are called functional as they rotate around a "functional" obstacle called the core of the spiral or scroll wave.

A single spiral or scroll wave with a fixed repetitious motion (which may be anchored to some anatomical feature such as a blood vessel or scar) likely would lead to a monomorphic tachycardia. A meandering spiral or scroll wave likely would be associated with a polymorphic tachycardia or perhaps fibrillation. Polymorphic tachycardias and fibrillation also may be associated with "fibrillatory" conduction, in which a rotating spiral or scroll wave fractionates as it propagates throughout the cardiac tissue, or with multiple spiral or scroll waves. In the latter case, the spiral and scroll waves may not be located around stationary cores, but may migrate. Spirals and scrolls may disappear by collision with boundaries or by collision with rotating waves of opposite chirality, and they may be regenerated as a consequence of fibrillatory conduction.

The mathematics underlying the generation, stabilization, migration, and destruction of spiral and scroll waves is a rich topic and has been subject to extensive investigation. If an initiated spiral wave is itself unstable, it may quickly break up into multiple waves. Clinical evidence exists for this, especially in the case of ventricular fibrillation, which is usually preceded by a short-lived ventricular tachycardia. The transition from tachycardia to fibrillation can occur either by a single relatively stable wave with breakup far from the core, or by waves that continually form and annihilate. Many different mechanisms have been proposed to explain the transition from a single spiral wave to multiple waves (Bar and Eiswirth, 1993; Biktashev et al., 1994; Fenton and Karma, 1998; Fenton et al., 2002; Bernus et al., 2003).

Since real hearts are three-dimensional, and there is still no good technology to image excitation throughout the heart depth (as opposed to the surface), the actual geometry of excitation waves in cardiac tissue associated with some arrhythmias is not as well understood and is now the subject of intense study. From an operational point of view, it seems likely that any arrhythmia that cannot be cured by a small localized lesion in the heart will best be described by rotating spiral or scroll waves. Such rhythms include atrial and ventricular fibrillation. In these rhythms, there is evidence for strong fractionation (breakup) of excitation waves giving rise to multiple small spiral waves and patterns of shifting blocks (Fenton et al., 2002). Ventricular tachycardias also can occur in patients other than those who have experienced a previous heart attack, and perhaps even in hearts with completely normal anatomy. In these individuals, it is likely that spiral and scroll waves are the underlying geometries of the excitation. A particularly dangerous arrhythmia, polymorphic ventricular tachycardia (in which there is a continually changing morphology of the electrocardiogram complexes), is probably associated with meandering spiral and scroll waves (Gray et al., 1995).

#### Ventricular tachycardia

Figure 13: Ventricular tachycardia. Link to 3D applet.

Ventricular tachycardia (VT) refers to an arrhythmia characterized by rapid ventricular rates generally around 100 to 200 beats per minute and may be either sustained (lasting longer than 30 seconds) or nonsustained. Causes include reentry (see Figure 13) and, less commonly, abnormal automaticity. In addition, VT sometimes is associated with anatomical reentry around scar tissue. It is common for VT to precede ventricular fibrillation, and it has been posited that this transition may arise from the rapid fractionation of reentrant waves that formed at the onset of tachycardia. VT can be described as monomorphic or polymorphic depending on its appearance on the ECG. In monomorphic VT, the ECG appears as an undulating signal with broad QRS complexes that all have similar shapes. In polymorphic VT, the shapes of the QRS complex are different and may be indicative of multiple ectopic foci. Torsades de pointes ("twisting of the points") is an important type of polymorphic VT in which the amplitudes of the QRS complexes vary in a sinusoidal pattern. This arrhythmia often results from triggered activity in the context of other electrophysiological abnormalities, especially a prolonged QT interval, which may result not only from electrolyte imbalances and congenital prolongation but also from the use of many drugs, including a number of antiarrhythmic drugs. Prolongation of cardiac action potentials, especially when the prolongation primarily affects the plateau phase, can increase the likelihood of early afterdepolarization development.

#### Ventricular fibrillation

Figure 14: Ventricular fibrillation. Link to 3D applet.

Ventricular fibrillation (VF) is an immediately life-threatening arrhythmia that requires immediate medical intervention. During VF, many small waves propagate throughout the ventricles (see Figure 14), most likely caused by meandering reentrant waves, spiral or scroll wave breakup, fibrillatory conduction, or some combination of these. As a result, electrical waves in the ventricles are uncoordinated and cannot produce an effective contraction. Blood cannot be pumped properly, and blood pressure drops to zero. The only effective therapy is electrical defibrillation delivered either from external defibrillators or from internally positioned devices implanted into patients diagnosed as at risk for severe cardiac arrhythmias. Defibrillators attempt to reset the heart's electrical activity by delivering a high-energy electrical shock, following which sinus rhythm resumes. Because the heart's inability to pump blood deprives body tissues of oxygen, defibrillation must be performed within minutes for the body to recover.

#### Atrial flutter

Figure 15: Atrial flutter. Link to 3D applet.

Atrial flutter is a reentrant supraventricular arrhythmia (see Figure 15) characterized by a rapid "sawtooth" appearance of the ECG owing to the presence of multiple P waves between QRS complexes. Atrial rates typically fall between 250 and 350 beats per minute, but the ventricular rate can be substantially slower. Atrial flutter can be paroxysmal or persistent and may degenerate to atrial fibrillation. Treatment options for atrial flutter include direct electrical cardioversion and catheter ablation of the underlying reentrant circuit, which often is located in the right atrium but also may be in the left atrium or may involve both atria. Pharmacologic therapy sometimes is used as well, although care must be taken to avoid adverse effects on ventricular rhythm. Because many of the rapid atrial beats associated with flutter block at the AV node, decreasing the atrial rate may paradoxically increase the ventricular rate if the increased time between atrial beats allows more of them to be conducted through the AV node to the ventricles.

#### Atrial fibrillation

Figure 16: Atrial fibrillation. Link to 3D applet.

Atrial fibrillation (AF) is a complex arrhythmia characterized by an extremely rapid atrial rate (350 to 600 beats per minute). Like ventricular fibrillation, during AF multiple wavefronts are present in the atria (see Figure 16) and compromise the ability of the atria to contract in a coordinated fashion. However, because the atria serve primarily as filling chambers, their contraction is not as necessary for cardiac function, and for this reason AF is not immediately life-threatening. Nevertheless, AF can lead to other cardiac problems, including an increased and more irregular ventricular rate along with a significantly increased risk of stroke, which can result from blood stasis in the atria that promotes the development of clots. Like atrial flutter, AF can be paroxysmal or persistent. It also has been documented that even short bouts of atrial fibrillation lead to electrophysiological changes that over time tend to make atrial fibrillation more frequent and longer-lasting (Wijffels et al., 1995). AF can be treated by direct cardioversion but more frequently is treated with antiarrhythmic and/or anti-coagulation drug therapy.

## References

Bar M, Eiswirth M. (1993) Turbulence due to spiral breakup in a continuous excitable medium. Phys Rev E 48: R1635-R1637.

Beeler GW, Reuter H. (1977) Reconstruction of the action potential of ventricular myocardial fibres. J Physiol (london) 268: 177-210.

Bernus O, Verschelde H, Panfilov AV. (2003) Spiral wave stability in cardiac tissue with biphasic restitution. Phys Rev E 68: 021917.

Biktashev VN, Holden AV, Zhang H. (1994) Tension of organizing filaments of scroll waves. Philos Trans R Soc London A 347: 611-630.

Castellanos A, Fernandez P, Moleiro F, Interian A, Myerburg RJ. (1991) Symmetry, broken symmetry, and restored symmetry of apparent pure ventricular parasystole. Am J Cardiol 68: 256-259.

Cherry EM, Fenton FH. (2004) Suppression of alternans and conduction blocks despite steep APD restitution: electrotonic, memory and conduction velocity effects. Am J Physiol 286: H2332-2341.

Chudin E, Goldhaber J, Garfinkel A, Weiss J, Kogan B. (1999) Intracellular Ca(2+) dynamics and the stability of ventricular tachycardia. Biophys J 77: 2930-41.

Comtois P, Vinet A. (2002) Resetting and annihilation of reentrant activtiy in a model of a one-dimensional loop of ventricular tissue. Chaos 12: 903-923.

Comtois P, Vinet A. (2005) Multistability of reentrant rhythms in an ionic model of a two-dimensional annulus of cardiac tissue. Phys Rev E 72: 051927.

Courtemanche M, Glass L, Rosengarten MD, Goldberger AL. (1989) Beyond parasystole: Promises and problems in modelling complex arrhythmias. Am J Physiol 257: H693-H706.

Courtemanche M, Glass L, Keener JP. (1993) Instabilities of a propagating pulse in a ring of excitable media. Phys Rev Lett 70: 2182-2185.

Davidenko JM, Pertsov AM, Salomonsz R, Baxter WT, Jalife J. (1992) Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355: 349–351.

Echebarria B, Karma A. (2002) Instability and spatiotemporal dynamics of alternans in paced cardiac tissue. Phys Rev Lett 88: 208101.

Eisner DA, Choi HS, Díaz ME, O'Neill SC, Trafford AW. (2000) Integrative analysis of calcium cycling in cardiac muscle. Circ Res 87: 1087-94.

Fenton F, Karma A. (1998) Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos 8: 20-47.

Fenton FH, Cherry EM, Hastings HM, Evans SJ. (2002) Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12: 852-892.

Frame LH, Simpson MB. (1988) Oscillations of conduction, action potential duration, and refractoriness: A mechanism for spontaneous termination of reentrant tachycardias. Circulation 78: 1277-1287.

Glass L, Goldberger A, Bélair J. (1986) Dynamics of pure parasystole. Am J Physiol 251: H841-H847.

Glass L, Josephson ME. (1995) Resetting and annihilation of reentrant abnormally rapid heartbeat. Phys Rev Lett 75: 2059-2063.

Glass L, Nagai Y, Hall K, Talajic M, Nattel S. (2002) Predicting the entrainment of reentrant cardiac waves using phase resetting curves. Phys Rev E 65: 021908.

Gray RA, Jalife J, Panfilov A, Baxter WT, Cabo C, Davidenko JM, Pertsov AM (1995) Nonstationary vortexlike reentrant activity as a mechanism of polymorphic ventricular tachycardia in the isolated rabbit heart. Circulation 91: 2454-2469.

Guevara MR, Ward G, Shrier A, Glass L. (1984) Electrical alternans and period doubling bifurcations. Comput Cardiol: 167–170.

Guevara MR. (1991) Iteration of the human atrioventricular (AV) nodal recovery curve predicts many rhythms of AV block. In: Glass L, Hunter P, McCulloch A, eds. Theory of Heart: Biomechanics, Biophysics, and Nonlinear Dynamics of Cardiac Function, New York: Springer-Verlag, pp. 313-358.

Josephson ME. (2002) Clinical Cardiac Electrophysiology: Techniques and Interpretations, Philadelphia; London: Lippincott Williams and Wilkins.

Kaufman R, Rothberger CJ. (1917) Beitrage zur Kenntnis der Entstehungsweise extrasystolischer Allorhythmien. Z Ges Exp Med. 5: 349-370.

Keener JP. (1981) On cardiac arrhythmia: AV conduction block. J Math Biol 12: 215-225.

Mines GR. (1913) On dynamic equilibrium in the heart. J Physiol (London) 46: 349–383.

Möbitz W. (1924) Uber die unvollstandige Storung der Erregungsuberleitung zwischen Vorhof und Kammer des menschlichen Herzens. Z Ges Exp Med 41: 180-237.

Moe GK, Jalife J, Mueller WJ, Moe B. (1977) A mathematical model of parasystole and its application to clinical arrhythmias. Circulation 56: 968-979.

Nolasco JB, Dahlen RW. (1968) A graphic method for the study of alternation of cardiac action potentials. J Appl Physiol 25: 191–196.

Qu Z, Garfinkel A, Chen PS, Weiss JN. (2000) Mechanisms of discordant alternans and induction of reentry in simulated cardiac tissue. Circulation 102: 1664-1670.

Quan WL, Rudy Y. (1991) Termination of reentrant propagation by a single stimulus: A model study. Pacing Clin Electrophysiol 14: 1700-1706.

Roberge FA, Nadeau RA. (1969) The nature of Wenckebach cycles. Can J Physiol Pharmacol 47: 695-704.

Rudy Y. (1995) Reentry: Insights from theoretical simulations in a fixed pathway. J Cardiovasc Electrophysiol 6: 294-312.

Shrier A, Dubarsky H, Rosengarten M, Guevara MR, Nattel S, Glass L. (1987) Prediction of complex atrioventricular conduction rhythms in humans with use of the atrioventricular nodal recovery curve. Circulation 76: 1196-1205.

Sinha S, Stein KM, Christini DJ. (2002) Critical role of inhomogeneities in pacing termination of cardiac reentry. Chaos 12: 893-902.

Slater NB. (1967) Gaps and steps for the sequence n mod 1., Proc Camb Phil Soc 63: 1115-1123.

Stevenson WG, Delacretaz E. (2000) Strategies for catheter ablation of scar-related ventricular tachycardia. Curr Cardiol Rep 2: 537-544.

Tolkacheva EG, Schaeffer DG, Gauthier DJ, Krassowska W. (2003) Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics. Phys Rev E 67: 031904.

Vinet A, Roberge FA. (1994) The dynamics of sustained reentry in a ring model of cardiac tissue. Ann Biomed Eng 22: 568-591.

Watanabe MA, Fenton FH, Evans SJ, Hastings HM, Karma A. (2001) Mechanisms for discordant alternans. J Cardiovasc Electrophysiol 12: 196-206.

Wiener N, Rosenblueth A. (1946) The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiol Mex 16: 205–265.

Wijffels MC, Kirchhof CJ, Dorland R, Allessie MA. (1995) Atrial fibrillation begets atrial fibrillation. A study in awake chronically instrumented goats. Circulation 92: 1954-68.

Winfree AT. (1972) Spiral Waves of Chemical Activity. Science. 175: 634-636.

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