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Abstract      

The goal of this paper is to use a simple cellular excitable medium to clarify the role of virtual electrodes and deexcitation during cardiac defibrillation and the induction of reentry.  This model can elucidate many phenomena predicted by numerical simulations and observed during experiments.

Key Words

virtual electrode, deexcitation, cellular excitable medium, defibrillation.

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Introduction

Although defibrillation of the heart is used widely to resuscitate patients, no one knows exactly how defibrillation works.  Recent research highlights the importance of virtual electrodes and deexcitation during electrical stimulation and defibrillation (Efimov et al., 2000; Trayanova, 2001).  These concepts arise from computer simulations and animal experiments designed to study the electrical behavior of cardiac tissue.  Unfortunately, the complexity of these simulations and experiments often hides the basic physical principles underlying virtual electrodes and deexcitation.  The goal of this paper is to illustrate these phenomena using the simplest possible model: a cellular excitable medium.  While this description lacks the realism of more detailed models, it has the virtues of clarity and simplicity; it highlights the fundamental mechanisms responsible for these complex events.

A Cellular Excitable Medium

Elementary models of cellular excitable media (sometimes called cellular automata) provide valuable insight into the electrical behavior of the heart (Kaplan et al., 1988; Saxberg and Cohen, 1991; or see the web-based book by Weimar ).  In his book, When Time Breaks Down (1987), Winfree described perhaps the simplest of such models.  A hexagonal array represents a sheet of cardiac tissue.  Each cell in the array is in one of three states: excited (yellow), refractory (red), or quiescent (purple).  A cell changes state by the following rules:

1.   If a cell is in the excited state, then at the next time step it changes to the refractory state,
2.   If a cell is in the refractory state, then at the next time step it changes to the quiescent state,
3.   If a cell is in the quiescent state, then at the next time step it remains in the quiescent state; unless one of its six nearest neighbors is in the excited state, in which case it changes to the excited state. 

Think of this cellular excitable medium as a game having a hexagonal playing board and three simple rules.  The aim of the game is to understand the heart better.  After describing this game, Winfree gives the sage advice:  Stop reading until you have played some.

Figure 1 shows how this medium behaves.

Figure 1

For the first time step (t = 1, the initial condition) the entire array is quiescent except for the central cell, which is excited.  Such a situation might arise if the central cell represents special tissue that is intrinsically oscillatory (such as the sinoatrial node) or if an electrical stimulus is delivered to the central cell by an electrode.  We assume an electrode stimulated the central cell; in Figure 1 the outline of this electrode is shown in bold.  In subsequent time steps, a closed wave front propagates outwards, followed by a concentric ring of refractoriness, and then a return to quiescence.  Once the wave front propagates off the edge of the array, the tissue remains quiescent until stimulated again. 

We can start this game with any initial condition, some of which are more interesting than others.  For instance, try starting with a broken wave front, as shown in the first frame (t = 1) of Figure 2.

Figure 2

On the left the wave front propagates upwards, but in the center the wave front just stops.  As time goes by, the wave front spirals in a clockwise direction, known as a spiral wave.  One of the interesting features of a spiral wave is that it keeps going around and around forever.  Once started, it persists.  A special point at the center of a spiral wave, called a phase singularity, is  where an excited cell, a refractory cell, and a quiescent cell all meet (Winfree, 1987; Gray et al., 1998).  In Figure 2, a green dot marks the phase singularity.  Spiral waves have been produced in the heart experimentally using cross-field stimulation (Frazier et al., 1989; Davidenko et al., 1990).  They may cause some cardiac arrhythmias

.In When Time Breaks Down, Winfree describes the pinwheel experiment (Figure 3).

Figure 3

Start (t = 1) with a planar wave front propagating upwards across the array (perhaps initiated by a line electrode along the bottom edge that delivers a initial stimulus, S1).  As this first wave front propagates by send a second stimulus, S2, through the central electrode (t = 2).  S2 produces two phase singularities.  The resulting pattern is called figure-of-eight reentry; two spiral waves rotate next to each other in opposite directions.  Researchers in Ideker's laboratory performed the pinwheel experiment in a dog heart and observed figure-of-eight reentry like that shown in Figure 3 (Shibata et al., 1988).  The timing of S2 is crucial.  If S2 is too early or too late, it does not induce figure-of-eight reentry.

Figure 3 demonstrates that two stimuli (S1 and S2) through different electrodes can initiate reentry.  Can two stimuli through the same electrode initiate reentry?  Try it, using the central electrode for both stimuli.  Regardless of the timing of S2, you cannot create phase singularities.  Yet, when Matta et al. (1976) performed this experiment in a dog heart, their S2 initiated an arrhythmia.  Apparently our simple cellular excitable medium lacks some key feature necessary to reproduce their results.

A New Rule for Strong Stimuli

My goal is to introduce a new rule governing the stimulus through the central electrode, which allows us to understand Matta et al's experiment, as well as several others.  In Figure 1, we assumed that a stimulus excites one cell directly under the electrode.  This is correct for weak stimuli, but for strong stimuli the response is more complicated.  Sepulveda et al. (1989) showed that the region depolarized (excited) by a strong stimulus has a dog-bone shape, with its long axis perpendicular to the direction of the myocardial fibers.  Moreover, regions of hyperpolarization (called virtual anodes) exist adjacent to the electrode along the fiber direction.  This response is anisotropic; it depends on direction.  A virtual anode is an example of a virtual electrode"; a region of polarization that does not surround a stimulus electrode.  Sepulveda et al. (1989) found hyperpolarization near to, but not surrounding, a cathode.  No anodal electrode (which one normally associates with hyperpolarization) was nearby.

Many researchers have observed these regions of depolarization and hyperpolarization experimentally (Knisley, 1995; Neunlist and Tung, 1995; Wikswo et al., 1995).   Depolarization can excite a cell; it forces a quiescent cell into the excited state.  Conversely, hyperpolarization can deexcite a cell; it forces an excited or refractory cell into the quiescent state. 

Figure 4

We account for this behavior by assuming that a strong stimulus has the effect shown in Figure 4 (the fibers are horizontal, and the stimulus is superimposed on an otherwise uniformly refractory tissue).  The yellow cells are the dog-bone shaped depolarization, and the two purple cells are the virtual anodes.  

We can sum up the affect of the stimulus by postulating a fourth rule governing our excitable medium:
4.   During a cathodal stimulus, the state of the cell directly under the electrode and its four nearest neighbors in the direction perpendicular to the fibers change to the excited state, and the two remaining nearest neighbors in the direction parallel to the fibers change to the quiescent state, regardless of their previous state.

The rule for an anodal stimulus is the same, except the excited and quiescent states are interchanged.

Quatrefoil Reentry

Try applying two consecutive stimuli through the same central electrode using our new rule.  The first stimulus (S1) is weak and applied at t = 1; the second (S2) is strong and applied at t = 4   (Figure 5).

Figure 5

S1 initiates an outwardly propagating wave front, like that in Figure 1.  S2 excites the cells in the dog-bone region under the electrode, and deexcites cells to the left and right of the electrode (t = 4).  Deexcitation at the virtual anodes is crucial for the subsequent dynamics.  (I tried once to account for the dog-bone distribution of excitation during S2 without accounting for deexcitation, but I was not very successful (Roth, 1998).)  The S2 wave front has nowhere to propagate except through the deexcited cells at the virtual anode (t = 5).  Such break excitation was predicted theoretically by Roth (1995) and was observed experimentally by Wikswo et al. (1995).  S2 produces four phase singularities (green dots), and initiates two back-to-back figure-of-eight reentry patterns.  Reentry with four phase singularities is called quatrefoil reentry.  Winfree (1990) first predicted quatrefoil reentry, Saypol and Roth (1992) first simulated it numerically, and Lin et al. (1999) first observed it experimentally in a rabbit heart.

Quatrefoil reentry can occur without an S1 gradient of refractoriness.  Traditionally, researchers emphasize the spatial distribution of S1 refractoriness (the red cells in Figures 1-4) during the initiation of reentry (Frazier et al., 1989; Shibata et al., 1988).  However, Winfree (2000) and Roth (2000) demonstrated that even if the S1 refractoriness is uniform before the S2 stimulus (as in Figure 4), S2 can still induce quatrefoil reentry.  To understand how, note that four phase singularities are present in Figure 4, even though the S1 refractoriness is uniform.  The S2 itself creates the phase singularities by exciting some cells and deexciting others.  Efimov et al. (1998) observed such virtual electrode induced phase singularities, and found that they are nearly independent of the distribution of S1 refractoriness (Cheng et al., 2000).  They may play an important role during defibrillation (Skouibine et al., 1999; Efimov et al., 2000; Trayanova, 2001).

A stimulus of opposite polarity can also initiate quatrefoil reentry.  The stimulus in Figure 4, with depolarization directly under the electrode, is called a cathodal stimulus, whereas the opposite polarity is called an anodal stimulus.   A weak anodal stimulus has no effect, but a strong anodal stimulus deexcites the cell under the electrode and adjacent cells in the direction perpendicular to the fibers, and excites adjacent cells along the fiber direction (the virtual cathodes). 

Figure 6

Figure 6 shows anodal quatrefoil reentry (S1 at t = 1 is weak and cathodal, S2 at t = 4 is strong and anodal).  The main difference between cathodal and anodal quatrefoil reentry is that the S2 wave fronts rotate in opposite directions.  Roth (1997) predicted anodal quatrefoil reentry, and Lin et al. (1999) observed it experimentally.

The Pinwheel Experiment Revisited

Lindblom et al. (2000) revisited the pinwheel experiment to determine the effect of virtual electrodes on the wave front dynamics.  Their results depended on the stimulus polarity and on the direction of the  S1 wave front.  Consider an S1 wave front propagating in the longitudinal direction (parallel to the fibers) and a cathodal S2 stimulus applied at t = 2.  (Lindblom et al. refer to this case as S1L, S2C, meaning S1 Longitudinal and S2 Cathodal).  Figure 7 shows the dynamics, which are analogous to Figure 8 of Lindblom et al.'s publication.

Figure 7	Figure 8

After S2, wave fronts propagate through both virtual anodes, but the rightward-propagating wave front encounters refractory or excited cells and dies  (t = 3).  The leftward-propagating wave front contains two phase singularities and initiates figure-of-eight reentry.Figure 8 illustrates the dynamics with S1 propagating transversely (perpendicular to the fibers) and S2 cathodal (S1T, S2C).  (Compare our Figure 8 to Figure 9 in the Lindblom et al. publication.)  Again, two phase singularities form and figure-of-eight reentry begins.  Figures 9 and 10 show two cases of figure-of-eight reentry caused by an anodal S2 (S1L, S2A and S1T, S2A).

Figure 9	Figure 10

 (Compare our Figures 9 and 10 to Figures 10 and 12 in the Lindblom et al. publication).

Figure-of-eight reentry is not the only outcome of the pinwheel experiment.  If the timing of S2 changes, quatrefoil reentry can occur.  Figure 11 shows quatrefoil reentry for S1L, S2A.  I chose this case because we can compare it to Figure 11 of Lindblom et al., which also shows quatrefoil reentry.

Figure 11

All four cases (S1L, S2C; S1T, S2C; S1L, S2A; S1T, S2A) may result in quatrefoil reentry if S2 is timed appropriately.The dynamics in Figures 7-11, produced by a strong S2 that accounts for deexcitation and virtual electrodes, are more complicated than the dynamics in Figure 3, produced by a relatively weak S2.  Which behavior is observed experimentally?  In their experiment, Shibata et al. (1988) used extracellular electrodes to track the wave front position; they did not record transmembrane potential directly.  Also, they could not record data during and immediately after S2, because of shock artifacts.  No one has repeated this experiment, measuring the transmembrane potential during and immediately after S2.  Figures 7-11 and the simulations of Lindblom et al. (2000) suggest that the dynamics may be more complicated than originally thought.

Limitations

Although our simple excitable medium reproduces many results of more sophisticated numerical simulations, it nevertheless has significant limitations.  For instance, anisotropy comes into the model only through rule 4 for strong stimuli.  In cardiac tissue, the propagation speed also depends on direction, but this effect is absent in our model.  Furthermore, we neglect the variation of the fiber direction throughout the heart.

Propagation speed also depends on factors such as wave front curvature and the degree of refractoriness of the tissue just ahead of the wave front.  I have not incorporated these effects into the excitable medium, although they can be (Gerhardt et al., 1999).  Moreover, in some cases, the discrete cellular structure of cardiac tissue may be important (Ito and Glass, 1991).  Finally, I have neglected any heterogeneity that may exist in cardiac tissue (Bardou et al., 1996; Bub et al., 2002).   We could overcome most of these limitations by making the model more complex.

Conclusion

This extremely simple cellular excitable mediumÑwhich is nothing more than a toy model, stripped down to contain only the essential featuresÑcan, with one simple modification for strong stimuli, predict many interesting and important phenomena.  Much of what we have learned about virtual electrodes and deexcitation is predicted correctly by the model (Efimov et al., 2000; Trayanova, 2001).  I am astounded that this simple model can reproduce the complex results obtained by Lindblom et al. (2000).  The model provides valuable insight into the essential mechanisms of electrical stimulation without hiding the important features behind distracting details.

Acknowledgments

This research was supported by grants from the National Institutes of Health (RO1 HL57207) and the American Heart Association-Midwest Affiliate.

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