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Am J Physiol Heart Circ Physiol 289: H1692-H1701, 2005. First published June 3, 2005; doi:10.1152/ajpheart.00241.2005
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Effects of Na+ and K+ channel blockade on vulnerability to and termination of fibrillation in simulated normal cardiac tissue

Zhilin Qu1,2 and James N. Weiss1,2,3

1Cardiovascular Research Laboratory and Departments of 2Medicine (Cardiology) and 3Physiology, David Geffen School of Medicine at UCLA, University of California, Los Angeles, California

Submitted 14 March 2005 ; accepted in final form 1 June 2005


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Na+ and K+ channel-blocking drugs have anti- and proarrhythmic effects. Their effects during fibrillation, however, remain poorly understood. We used computer simulation of a two-dimensional (2-D) structurally normal tissue model with phase I of the Luo-Rudy action potential model to study the effects of Na+ and K+ channel blockade on vulnerability to and termination of reentry in simulated multiple-wavelet and mother rotor fibrillation. Our main findings are as follows: 1) Na+ channel blockade decreased, whereas K+ channel blockade increased, the vulnerable window of reentry in heterogeneous 2-D tissue because of opposing effects on dynamical wave instability. 2) Na+ channel blockade increased the cycle length of reentry more than it increased refractoriness. In multiple-wavelet fibrillation, Na+ channel blockade first increased and then decreased the average duration or transient time (<Ts>) of fibrillation. In mother rotor fibrillation, Na+ channel blockade caused peripheral fibrillatory conduction block to resolve and the mother rotor to drift, leading to self-termination or sustained tachycardia. 3) K+ channel blockade increased dynamical instability by steepening action potential duration restitution. In multiple-wavelet fibrillation, this effect shortened <Ts> because of enhanced wave instability. In mother rotor fibrillation, this effect converted mother rotor fibrillation to multiple-wavelet fibrillation, which then could self-terminate. Our findings help illuminate, from a theoretical perspective, the possible underlying mechanisms of termination of different types of fibrillation by antiarrhythmic drugs.

arrhythmias; antiarrhythmic drugs; simulation

AS ANTIARRHYTHMIC DRUGS, Na+ and K+ channel blockers have been limited by proarrhythmic effects, leading to increased risk of sudden cardiac death due to ventricular fibrillation (VF). In anatomic reentry, the classic rationale for these drugs is straightforward: Na+ channel blockers reduce excitability in the reentrant circuit, so that unidirectional conduction block is converted to bidirectional conduction block; K+ channel blockers prolong refractoriness until the wavelength exceeds the length of the reentrant circuit, abolishing the excitable gap. However, cardiac fibrillation involves functional, rather than purely anatomic, reentry, and the effects of Na+ and K+ channel blockers in this setting are more complicated. Large-scale clinical trials (3, 55) and experimental studies (11, 47, 62) have convincingly demonstrated that Na+ and K+ channel blockers are sufficiently proarrhythmic after myocardial infarction to preclude their use as a primary strategy for preventing VF. Yet antiarrhythmic drugs can terminate VF (1, 28, 29, 52) and are fairly effective at terminating atrial fibrillation (AF) (9, 19, 21, 22, 31, 32, 51, 57, 58). To some extent, their antifibrillatory actions have been rationalized according to Garrey's "critical mass hypothesis" (16) and Moe's "multiple wavelet hypothesis" (30). The critical mass hypothesis posits that the persistence of fibrillation depends on tissue mass and form. The multiple-wavelet hypothesis states that "fibrillation is maintained by the irregular wandering of numerous wavelets generated by the fraction of wavefronts passing through tissue in a state of inhomogeneity with respect to excitability and conduction velocity." Class IA and III drugs prolong wavelength, so that the same-sized tissue can support fewer reentrant circuits, which may cause fibrillation to terminate (32). However, even this mechanism has been challenged by experimental studies (17, 57, 58) in which these drugs were shown to have limited effects on wavelength and refractory period, despite being very effective at terminating AF.

As a further complication, recent evidence indicates that multiple-wavelet fibrillation is not the only mechanism of fibrillation. Mother rotor fibrillation has been described; in mother rotor fibrillation, a stable high-frequency source, or mother rotor, drives fibrillation (4, 6, 7, 27, 59). In this mechanism, wave breaks are an epiphenomenon of fibrillatory conduction block in the surrounding tissue, which is unable to sustain 1:1 conduction of impulses arising from the mother rotor. It is not immediately obvious how antiarrhythmic drugs can terminate fibrillation driven by a stable high-frequency source. If fibrillation is driven by a mother rotor, which requires only a small tissue (larger than the spiral core) to sustain itself, it is also not clear why tissue size is important in AF therapies (5, 22, 23, 50). Recent studies by Kneller et al. (20) and Nattel et al. (32) showed in computer simulations that Na+ channel blockade destabilized the mother rotor, favoring AF termination.

In this study, we used computer simulation to investigate the mechanisms by which Na+ and K+ channel blockers affect vulnerability to and termination of fibrillation in structurally normal tissue. We simulated two-dimensional (2-D) tissue models with phase I of the Luo and Rudy (LR1) action potential model (26). Because our intention is to qualitatively delineate the different mechanisms by which multiple-wavelet fibrillation and mother rotor fibrillation are terminated by antiarrhythmic drugs, we chose to use relatively simple action potential and tissue models, allowing us to carry out a large number of simulations to statistically calculate the average fibrillation duration in a relatively large tissue. Our conclusions from these simple models provide a theoretical basis for future quantitative studies using more realistic action potential and tissue models as tools to illuminate experimental observations in intact atrial and ventricular tissue.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Mathematical Model

We used the LR1 ventricular action potential model to simulate 2-D tissue models (26). We simulated this model in homogeneous and heterogeneous tissues with the following differential equation

(1)
where V is the transmembrane potential, Cm is the membrane capacitance (1 µF/cm2), and D is the diffusion constant and was set to 0.001 cm2/ms. Iion is the total ionic current density of the membrane from the LR1 model, which is as follows: Iion = INa + Isi + IK + IK1 + IKp + Ib. INa = Nam3hj(V – ENa) is the fast inward Na+ current; Isi = sidf(V – Esi) is the slow inward current, assumed to be the L-type Ca2+ current; IK = Kxx1(V – EK) is the slow outward time-dependent K+ current; IK1 = K1K1{infty}(V – EK1) is the time-independent K+ current; IKp = 0.0183Kp(V – EKp) is the plateau K+ current; and Ib = 0.03921(V + 59.87) is the total background current. is mean conductance, E is reversal potential, and m, h, j, d, f, and x are gating variables satisfying the following type of differential equation

(2)
where y represents the gating variable and {tau} is the time constant. The ionic concentrations are set as intracellular Na+ concentration ([Na]i) = 18 mM, extracellular Na+ concentration ([Na]o) = 140 mM, intracellular K+ concentration ([K]i) = 145 mM, and extracellular K+ concentration ([K]o) = 5.4 mM, and intracelluar Ca2+ concentration ([Ca]i) obeys

(3)
where ICa is Ca2+ current.

Details of the LR1 action potential model were presented by Luo and Rudy (Table I in Ref. 26). By setting [K]o = 5.4 mM, the maximum conductances of IK and IK1 (K and K1) are 0.282 and 0.6047 mS/cm2, respectively. Luo and Rudy's values for Na and si are 23 and 0.09 mS/cm2, respectively. We used Na = 16 mS/cm2 and changed si and K to create different spiral wave behaviors. For control, we used si = 0.423 mS/cm2, and most of the simulations used si = 0.052 mS/cm2 [steep action potential duration (APD) restitution and spiral wave breakup] or si = 0.03 mS/cm2 (flat APD restitution and spiral wave meander). Other parameters are the same as in the original LR1 model. Tissue heterogeneity was simulated by a gradient of maximum conductance of the time-dependent K+ channel, i.e., K = K(x), in which x is the horizontal 2-D tissue coordinate. The specific sigmoidal functions are shown for each case. This gradient in K+ channel conductance resulted in an APD gradient similar to the gradient from apex to base on the epicardical surface (24, 41). Figure 1A shows APD distribution along the x direction at a slow heart rate.



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  		Fig. 1. A: action potential duration (APD) distribution in the horizontal direction (x-axis) of the tissue at pacing cycle length (PCL) of 500 ms for si = 0.052 mS/cm2 and K(x) = 0.282{(2.5 – 1.5)[1.25x2/(Lx/2)2 + x2]} mS/cm2, where is mean conductance, si and K denote slow inward (L-type Ca2+) channel and slow outward time-dependent K+ channel, respectively, and L is length. Pacing was applied at the left end. APD at the 2 ends was affected by boundary and stimulation. B: spiral tip detection. Black lines, isovoltage lines at –30 mV at time t; gray lines, isovoltage lines at another time t + {Delta}t; arrows, intersections of the 2 sets of isovoltage lines, which are defined as spiral tips.

 
Numerical Methods

We used an advanced numerical method (40) with a 0.02- to 0.2-ms adaptive time step and a fixed space step ({Delta}x = {Delta}y = 0.025 cm). This space step was justified by Courtemanche (12) for the Beeler-Reuter model, which is similar to the LR1 model. For the diffusion constant used in this study, we found little change in spiral wave dynamics if we used a smaller space step, such as 0.015 cm, as in our previous studies.

Na+ and K+ blockade

We modeled K+ channel blockade by reducing the time-dependent K+ channel conductance (K) in the LR1 model. We modeled the effects of Na+ channel blockade by reducing the maximum conductance (by the factor {alpha}) and slowing recovery from inactivation (by the factor {beta}) in the LR1 model, as done previously (42)

(4)
It has been shown that Na+ channel blockade reduces INa and slows its recovery from inactivation, especially for cells in the epicardial border zone after infarction (13, 38, 47).

APD Restitution

APD restitution is defined as the present APD as a function of the previous diastolic interval. APD is defined as V greater than –72 mV and diastolic interval (DI) as V less than –72 mV. APD restitution was measured in a 2-cm one-dimensional cable with a regular 500-ms S1 pacing train followed by a premature S2 at one end of the cable.

Effective Refractory Period

Effective refractory period (ERP) was measured in a 2-cm one-dimensional cable. A 500-ms S1 pacing train was applied and then a premature S2 was used to determine ERP. ERP is defined as the shortest S1-S2 interval that the S2 propagates successfully through the cable. The stimulation strength was 30 µA/cm2 and was applied in a 2.5-mm-long section at one end of the cable.

Fibrillation Duration

Initial conditions of multiple spiral waves were used and randomly perturbed to give rise to nonsustained fibrillation with different durations. We perturbed only V as follows

(5)
where V0(x,y) is transmembrane potential of the prestored spiral waves and {xi}(x,y) is a uniformly distributed random number in [0,1] for the location (x,y) and is applied only at t = 0. Because of the chaotic nature of spiral wave breakup, this perturbation results in very different fibrillation patterns (45), which then affect the duration of fibrillation. The fibrillation duration (Ts) was defined as the interval from the start of fibrillation to the time at which the tissue becomes quiescent; 20–40 Ts values were used to calculate the averaged duration of fibrillation (<Ts>).

Tips of Spiral Waves

Tips of spiral waves were defined as the points of intersection of successive isovoltage lines at –30 mV measured at 1-ms intervals. Figure 1B illustrates the detection of spiral tips. At time t, we record isovoltage lines at –30 mV (black lines); at time t + {Delta}t, another set of isovoltage lines is recorded (gray lines). Because of the rotation of the spiral waves, the two sets of isovoltage lines intersect at the points that are defined as spiral tips. Successive intersections form the tip trajectories shown in this study.

Vulnerable Window

We used rapid pacing to study the effects of Na+ and K+ channel blockade on vulnerability to reentry in heterogeneous 2-D tissue. We used 20 beats of fixed pacing cycle length (PCL) to induce reentry. The stimulus was applied in a 2.5 x 2.5 mm2 area at the lower-left corner of the tissue, where ERP is the shortest. The vulnerable window is defined as the difference between the longest and the shortest PCL at which reentry can be induced.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 GRANTS
 REFERENCES
 
Effects of Na+ Channel Blockers on Vulnerability to Fibrillation

We used computer simulation to study how Na+ channel blockers influence vulnerability to reentry in heterogeneous tissue during rapid pacing. Heterogeneity was created by regional variation in K+ conductance, with Na+ channel conductance remaining homogeneous. Figure 2, A and B, shows the vulnerable windows (see METHODS for definition) for different degrees of Na+ channel blockade, when APD restitution slope was steep (si = 0.052 mS/cm2) or flat (si = 0.03 mS/cm2). In both cases, the vulnerable window decreased as Na+ channel blockade increased. However, the vulnerable window was much larger in the case of steep APD restitution, because dynamical instability due to steep APD restitution further amplified dispersion of refractoriness (15, 41, 56). In Fig. 2C, voltage snapshots illustrate how reentry was induced by rapid pacing. In the presence of Na+ channel blockade, reentry also occurred at slower heart rates. For example, in the case of steep APD restitution ({alpha} = 1 and {beta} = 1) in Fig. 2A, reentry was induced when PCL decreased to 150 ms; in the case of {alpha} = 0.625 and {beta} = 4 (Fig. 2A), reentry occurred for PCL < 200 ms. Therefore, Na+ channel blockade decreased the vulnerable window, which is antiarrhythmic, but it also caused reentry at slower heart rates, which may be proarrhythmic, even in normal tissue.



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  		Fig. 2. Effects of Na+ channel blockade on vulnerability to reentry. A: vulnerable window (VW) size for different degrees of Na+ blockade, with steep APD restitution slope. si = 0.052 mS/cm2 and K(x) = 0.282{(2.5 – 1.5)[1.25x2/(Lx/2)2 + x2]} mS/cm2. B: same as A, but with shallow APD restitution slope. si = 0.03 mS/cm2 and K(x) 0.282{(2.5 – 1.75)[1.25x2/(Lx/2)3 + x3]} mS/cm2. C: voltage snapshots showing induction of reentrant arrhythmias by rapid pacing with steep APD restitution slope. Tissue size was 10 x 10 cm2.

 
Effects of Na+ Channel Blockers on Maintenance of Fibrillation

Spiral wave reentry. Na+ channel blockers reduce conduction velocity and prolong the ERP, slowing the cycle length (CL) of reentry. Figure 3, A and B, shows the average CL (<CL>) vs. ERP during unstable spiral wave breakup and stable spiral wave reentry in homogeneous tissue. In both cases, CL was proportional to ERP, but with slopes >1. In other words, CL increased more than ERP, resulting in a large increase in temporal excitable gap. These computer simulations agree with the experimental AF data of Wijffels et al. (58) with class I drugs. In Fig. 3C, we replotted their experimental data, which demonstrated a linear relation with a slope close to 2.5 for all three class I drugs used in their experiments.



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  		Fig. 3. Simulated effects of Na+ channel blockade on cycle length (CL) of spiral wave reentry. A: average CL (<CL>) vs. effective refractory period (ERP) during simulated multiple-wavelet fibrillation (spiral wave breakup regime, si = 0.052 mS/cm2). Numbers in parentheses indicate {alpha} and {beta} in Eq. 4. B: CL vs. ERP for a stable spiral wave with use of a model with flat APD restitution. Parameter changes from control are as follows: si = 0.12 mS/cm2, {tau}d -> 0.25{tau}d, and {tau}f -> 0.25{tau}f, where {tau}d and {tau}f are activation and inactivation time constants of the L-type Ca2+ channel in phase I of the Luo-Rudy (LR1) model. Numbers in parentheses indicate {alpha} and {beta} in Eq. 4. C: atrial fibrillation CL (AFCL) vs. ERP (RPAF) during atrial fibrillation (AF) in a goat. Data were digitized and replotted from Wijffels et al. (58).

 
Multiple-wavelet fibrillation. We next studied the effects of Na+ channel blockade on the average duration, or average Ts (<Ts>), of nonsustained multiple-wavelet fibrillation. Figure 4A shows that, in homogeneous tissue, <Ts> increased as ERP was prolonged by mild Na+ channel blockade but then decreased as ERP was prolonged further with stronger Na+ channel blockade. In heterogeneous tissue (as a result of a K+ conductance gradient), Na+ channel blockade was more effective in termination of multiple-wavelet fibrillation. Figure 4B compares <Ts> for homogeneous tissue with that for heterogeneous tissue with or without Na+ channel blockade. With normal Na+ current amplitude, <Ts> was much longer in heterogeneous than in homogeneous tissue, because average K+ channel conductance was increased, which shortened wavelength relative to tissue size and flattened APD restitution slope (see Fig. 7). However, for the same degree of Na+ channel blockade, <Ts> became even shorter than in homogeneous tissue. Figure 4C shows the number of spiral wave tips vs. time in control homogeneous tissue, homogeneous tissue with Na+ channel blockade, and heterogeneous tissue with Na+ channel blockade. Fewer waves were present with Na+ channel blockade.



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  		Fig. 4. Effects of Na+ channel blockade on average duration of multiple wavelet fibrillation (<Ts>): si = 0.052 mS/cm2. A: <Ts> vs. ERP in 8.75 x 8.75 cm2 homogeneous tissue. B: <Ts> for 10 x 10 cm2 homogeneous (Hm) tissue with normal Na+ kinetics [{alpha} = 1, {beta} = 1, Hm(1,1)] and Na+ channel blockade [{alpha} = 0.5, {beta} = 8, Hm(0.5,8)] and for 10 x 10 cm2 heterogeneous (Ht) tissue (K = 0.282{2 – 0.5[1.25x2/(Lx/2)2 + x2]} mS/cm2; i.e., K changed from 0.564 to 0.423 mS/cm2) with no Na+ channel blockade [Ht(1,1)] and Na+ channel blockade [{alpha} = 0.5, {beta} = 8, Ht(0.5,8)]. C: spiral tip number vs. time for Hm(1,1) (black), Hm(0.5,8) (gray), and Ht(0.5,8) (light gray).

 


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  		Fig. 7. Effects of K+ channel blockade on vulnerability and duration of multiple-wavelet fibrillation. A: APD restitution curves (si = 0.052 mS/cm2) for control (K = 0.423 mS/cm2), 10% increase in K (K = 0.465 mS/cm2), and 20% increase in K (K = 0.338 mS/cm2). B: vulnerable window size for different degrees of K+ channel blockade. si = 0.052 mS/cm2 and K = K0. Control: K0= 0.282 mS/cm2; 20% increase: K0= 0.338 mS/cm2; 10% blockade: K0= 0.254 mS/cm2; 20% blockade: K0= 0.226 mS/cm2. C: same as B, but si = 0.03 mS/cm2 and K = 0.282{2.5 – 1.75[1.125x3/(Lx/2)3 + x3]} mS/cm2. D: <Ts> for different degrees of K+ channel blockade in homogeneous tissue. si = 0.052 mS/cm2 and K = 0.423 mS/cm2 for control. E: <CL> of fibrillation in homogeneous tissue vs. baseline APD for +10%, control, –10%, and –20% K+ channel blockade. si = 0.052 mS/cm2 and K = 0.423 mS/cm2 for control. Tissue size was 10 x 10 cm2. DI, diastolic interval.

 
The likely explanation for these observations is as follows. We previously showed that reducing Na+ channel conductance tended to prevent spiral waves from breaking up, but it also substantially increased meander of the spiral tip (46). Figure 5A compares the tip trajectories of a meandering spiral wave in normal homogeneous tissue and homogeneous tissue with Na+ channel blockade. Na+ channel blockade enlarged the area of tip meander. Figure 5B compares spiral wave drift in heterogeneous tissue with and without Na+ channel blockade. The spiral wave drifted even faster when Na+ channel was blocked (~0.9 cm/s in blockade vs. ~0.3 cm/s in control). On the basis of these observations, we can explain the results shown in Fig. 4 as follows. The stabilizing effect of Na+ blockade tends to prolong <Ts>, but the enhanced meander increases the chance of waves colliding with each other or a border, which tends to shorten <Ts>. This competition caused <Ts> to increase first and then decrease as Na+ channel blockade was enhanced. In heterogeneous tissue, spiral wave drift was even more potentiated by Na+ channel blockade, making Na+ channel blockade more effective in termination of fibrillation.



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  		Fig. 5. Effects of Na+ channel blockade on spiral wave meander and drift (si = 0.03 mS/cm2). A: tip trajectories of meandering spiral waves in tissue with normal Na+ current (INa) parameters (left, {alpha} = 1, {beta} = 1) vs. Na+ channel blockade (right, {alpha} = 0.5, {beta} = 8). B: tip trajectories of spiral wave drift in a heterogeneous tissue (K = 0.282{2.5 – [1.25x2/(Lx/2)2 + x2]} mS/cm2) with normal Na+ channel (top, {alpha} = 1, {beta} = 1) and Na+ channel blockade (bottom, {alpha} = 0.5, {beta} = 8). Spiral wave drifted from left to right. Tissue size was 10 x 7.5 cm2.

 
Mother rotor fibrillation. Figure 6 illustrates the effects of Na+ channel blockade on mother rotor fibrillation. Mother rotor fibrillation was induced in a heterogeneous tissue in which heterogeneity was created by a gradient of K+ channel conductance. From 0 to 3 s, Na+ channel properties were normal, and sustained mother rotor fibrillation was present in the tissue (Fig. 6A). From 3 to 4 s, Na+ channel blockade was applied. Because of the slowing of the mother rotor, fibrillatory conduction block at the periphery of the mother rotor resolved, leaving several meandering and drifting spiral waves in the tissue (Fig. 6B). Because of frequency competition, only the fastest spiral wave survived (Fig. 6C). However, the single remaining spiral wave itself drifted violently because of tissue heterogeneity and Na+ channel blockade and finally drifted to the tissue border, resulting in self-termination. In our simulation, we used a monotonic APD gradient in space, which caused the spiral wave to drift off the border. If the APD gradient was nonmonotonic, the spiral wave could become sustained, resembling fast ventricular tachycardia or atrial flutter.



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  		Fig. 6. Effects of Na+ blockade on mother rotor fibrillation. A: tip trajectories of spiral waves during mother rotor fibrillation in a heterogeneous tissue (si = 0.03 mS/cm2 and K = 0.282{2.5 – 1.5[1.125x3/(Lx/2)3 + x3]} mS/cm2) from 0 to 3 s (left) and a voltage snapshot at 3 s (right). Gray arrow, tip trajectory of mother rotor; black arrow, fibrillatory conduction block. B: tip trajectories of spiral waves from 3 to 4 s and a voltage snapshot at 4 s. Na+ conductance was linearly deceased by 50% ({alpha} from 1 to 0.5), and recovery was slowed by a factor of 8 ({beta} was linearly increased from 1 to 8). C: tip trajectories of spiral waves from 4 to 10 s and a voltage snapshot at 8 s. Na+ channel parameters were fixed ({alpha} = 0.5 and {beta} = 8). Open arrows, drift directions. Tissue size was 10 x 7.5 cm2.

 
Effects of K+ Channel Blockers on Vulnerability to Fibrillation

The major action of the class III antiarrhythmic drugs is prolongation of ERP by blocking time-dependent K+ currents. K+ channel blockade prolonged APD more during slow than during fast pacing (reverse use dependence), causing the APD restitution curve to become steeper (Fig. 7A). Figure 7, B and C, shows the vulnerable windows in heterogeneous tissue for different degrees of K+ blockade strengths: si = 0.052 and 0.03 mS/cm2 for steep and flat APD restitution, respectively. In both cases, K+ channel blockade increased the vulnerable window for reentry and caused reentry to occur at slower heart rates. Because, in our simulation, the K+ channel was uniformly blocked by the same percentage, the dispersion of refractoriness in the tissue was increased. In addition, K+ channel blockade steepens APD restitution, which promotes APD alternans and spatially discordant APD alternans (35, 41, 56), which further increases the dispersion of refractoriness.

Effects of K+ Channel Blockers on Maintenance of Fibrillation

Multiple-wavelet fibrillation. Under control conditions in a 10 x 10 cm2 tissue, the APD restitution slope was steep enough to cause spiral wave breakup and nonsustained multiple-wavelet fibrillation, with <Ts> = 3.5 s. If K+ channel conductance was decreased by 20% in this same tissue, the baseline APD at 2-Hz pacing increased from 190 to 210 ms, the APD restitution curve was steeper, and <Ts> decreased to 1.6 s (Fig. 7D). If K+ channel conductance was increased by 10%, baseline APD decreased from 190 to 180 ms, the APD restitution curve was shallower, and <Ts> increased to 6.5 s. If we further increased K+ conductance, then spiral wave breakup no longer occurred because of flat APD restitution and spiral wave reentry became sustained, causing <Ts> to become infinite. Therefore, K+ channel blockade substantially shortened the transient time of multiple-wavelet fibrillation because of the longer wavelength and increased dynamic instability as a result of steepening APD restitution. By prolonging APD, K+ channel blockade also increased reentry CL. Figure 7E shows <CL> vs. APD for different K+ channel conductances, showing that <CL> is proportional to APD.

Mother rotor fibrillation. K+ channel blockade could cause or terminate mother rotor fibrillation, depending on the degree of block. In Fig. 8, A and B, moderate K+ channel blockade caused a drifting spiral wave into mother rotor fibrillation by enhancing electrophysiological heterogeneity and inducing peripheral fibrillatory conduction block. However, with further K+ channel blockade, mother rotor fibrillation was converted to multiple-wavelet fibrillation as a result of steepening of the APD restitution slope, which then self-terminated. In the simulation shown in Fig. 8, C and D, multiple wavelets disappeared before the last wave drifted off the border. The mechanism was the result of the steepening of APD restitution, which first destabilized all spiral waves in the tissue, converting mother rotor fibrillation to multiple-wavelet fibrillation, and then self-terminated because of increased dynamical instability. Dynamical instability due to steep APD restitution has been shown to promote wave break, leading to fibrillation (44, 48). However, we also showed that this instability promotes spiral wave annihilation and meander, leading to self-termination of fibrillation in a finite-sized tissue (39).



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  		Fig. 8. Effects of K+ channel blockade on mother rotor fibrillation. si = 0.03 mS/cm2 and K = 0.282{2.5 – [1.25x2/(Lx/2)2 + x2]} mS/cm2, at which a spiral wave drifts without resulting in mother rotor fibrillation. Voltage snapshots (A) and tip trajectories (B) show that 50% K+ channel blockade resulted in mother rotor fibrillation. Gray arrows, tip trajectories of mother rotors. C and D: same as A and B, respectively, but further K+ channel blockade to 75% block then converted mother rotor fibrillation to multiple wavelet fibrillation, which then self-terminated. Open arrow in D, point at which last wave drifted off the border, after which tissue became quiescent. Tissue size was 10 x 7.5 cm2.

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
Our major findings, which relate to normal cardiac tissue, are as follows: 1) Na+ channel blockade decreases, whereas K+ channel blockade increases, the vulnerable window of reentry in heterogeneous 2-D tissue. 2) Na+ channel blockade increases CL of reentry more than it increases refractoriness, causing widening of the excitable gap. In multiple-wavelet fibrillation, weak Na+ channel blockade prolongs <Ts>. With stronger Na+ channel blockade, however, <Ts> shortens as a result of increased spiral wave meander, enhancing wave collision. Heterogeneity further promotes spiral wave drift and, thus, can make <Ts> even shorter. In mother rotor fibrillation, Na+ channel blockade causes resolution of peripheral fibrillatory conduction by prolonging CL more than ERP and causes the mother rotor to drift. The drifting spiral waves can then self-terminate or become sustained, depending on the spatial profile of heterogeneity. 3) K+ channel blockade increases dynamical instability by steepening the APD restitution slope. In multiple-wavelet fibrillation, this shortens <Ts> because of enhanced dynamical instability and prolonged wavelength. In mother rotor fibrillation, K+ channel blockade causes a conversion to multiple-wavelet fibrillation by increasing dynamical instability. However, K+ channel blockade could also convert a heterogeneity-induced drifting spiral wave to sustained mother rotor fibrillation, depending on the strength of K+ channel blockade. Thus Na+ and K+ channel blockers affect vulnerability and maintenance of fibrillation, but the underlying mechanisms are different and vary for different types of fibrillation. These findings are important in conceptualizing novel antiarrhythmic drug development, because they emphasize that, even under simplified conditions of 2-D normal tissue with a relatively simple cardiac action potential model, the interactions of Na+ and K+ channel-blocking drugs with tissue heterogeneity, dynamic wave stability, and excitability are complex. Not only do they cause anti- and proarrhythmic effects, but the balance between these features is sensitive to the underlying mechanism of fibrillation. Incorporation of additional physiological reality, such as 3-D tissue structures, intracellular Ca2+ cycling dynamics, and disease-related electrical and structural remodeling, is likely to complicate the situation further.

Termination of Fibrillation by Na+ Channel Blockers

A classic mechanism for terminating fibrillation is an increase in the wavelength of reentry, which effectively reduces tissue size (32). However, in contrast to K+ channel blockers, pure Na+ channel blockers shorten, rather than prolong, wavelength. Consistent with this prediction, experimental studies (17, 57, 58) have demonstrated limited effects on wavelength and ERP by class I drugs. How, then, do Na+ channel blockers terminate fibrillation in normal tissue? Our computer simulations showed that, during reentry, Na+ channel blockade had a greater effect on CL than on the refractory period, which caused the excitable gap to widen, in agreement with experimental observations (19, 58). When Na+ channel blockade was mild, it tended to stabilize reentry during multiple-wavelet fibrillation, because <Ts> increased. However, with stronger Na+ channel blockade, low excitability caused spiral waves to meander over a larger area, enhancing wavelet-wavelet and wavelet-border interaction, thereby reducing wave number (i.e., effectively reducing the tissue size available) and shortening <Ts>. Our simulation in Fig. 5 shows that, with Na+ channel blockade, spiral waves meandered over a larger territory and made sudden pivoting turns, in agreement with the observation by Wijffels et al. (58) of pivot point delay, which prolongs CL. In mother rotor fibrillation, the greater effect on CL than on refractoriness caused fibrillatory conduction block to disappear as the mother rotor slowed. The mother rotor also began to drift and could drift off the border, causing fibrillation to terminate, or stabilize, in the tissue, causing tachycardia, such as atrial flutter. These findings may help explain why Na+ channel blockers sometimes induce conversion to atrial flutter, instead of termination (2, 19, 32).

Termination of Fibrillation by K+ Channel Blockers

In contrast to Na+ channel blockade, K+ channel blockade increased dynamical instability by steepening the APD restitution slope, promoting spiral wave meander and breakup (46). This shortened the duration of multiple-wavelet fibrillation by increasing wave competition. The same mechanism was operative in terminating mother rotor fibrillation. Mother rotor fibrillation is inherently resistant to self-termination if the mother rotor is stable and does not drift. By steepening APD restitution, K+ channel blockade induced spiral wave drift and breakup, converting mother rotor fibrillation to multiple-wavelet fibrillation, which then could self-terminate (39).

Relevance to Class I and Class III Antiarrhythmic Drugs

In the experimental study by Wijffels et al. (58), class I drugs (cibenzoline, hydroquinidine, and flecainide) prolonged CL of AF (79%, 57%, and 48%, respectively) and temporal excitable gap (191%, 157%, and 189%, respectively) to a much greater extent than the class III drug sotalol (24% in CL of AF and 102% in excitable gap). Wijffels et al. found lower cardioversion rates with class I drugs (80%, 100%, and 40%, respectively) than with sotalol (100%). Our simulation results showed that Na+ channel blockers had greater effects on widening the excitable gap than K+ channel blockers (compare slopes in Fig. 3, A and B, with slope in Fig. 7E). However, Na+ channel blockers tended to suppress dynamical instability, whereas K+ channel blockers increased dynamical instability. This may explain why the class III drug sotalol had a much smaller effect on prolonging CL of AF and excitable gap than the class I drugs but had a greater cardioversion rate (58). This also suggests that dynamical instability is an important factor influencing the efficacy of drugs in termination of fibrillation. In addition, our simulations showed that K+ channel blockade increased and Na+ channel blockade decreased the vulnerable window for reentry. This indicates that class III drugs may be less effective than class I drugs in preventing the recurrence of fibrillation, although they may be more effective in terminating fibrillation. In a previous study, we simulated heterogeneous Na+ channel blockade in a heterogeneous diseased tissue model mimicking myocardial infarction. In that setting, Na+ channel blockade promoted dispersion of refractoriness, which increased vulnerability to reentry, in agreement with the CAST trial (3) and experimental studies (11, 47, 62) of infarcted hearts. Heterogeneous effects of class I drugs in ischemia have been demonstrated by many experimental studies (37, 38, 47, 62) and may add significantly to the proarrhythmic effects after infarction compared with normal tissue.

Limitations

Primarily because of computational constraints, we used relatively simple action potential and tissue models, rather than a physiologically detailed late-generation action potential model (20, 25, 33) or an anatomically realistic tissue model (18, 49, 54, 61). These simplifications may affect the results and conclusions drawn from our simulations. 1) The interactions between the Na+ current and other ionic currents during the action potential are important for spiral wave dynamics; therefore, blocking the Na+ channel may result in different spiral wave dynamics in different models. However, our results showing that Na+ channel blockade increases spiral core size and meander and decreases the number of spiral waves agree with a recent study by Kneller et al. (20) using an ionically realistic model. 2) The LR1 model lacks the detailed formulation of the specific K+ channels, such as IKs, IKr, and Ito. Class III antiarrhythmic drugs may have different effects on these channels; thus blocking these channels may result in different spiral wave behaviors. However, most of the class III antiarrhythmic drugs have an effect called "reverse use dependence," which increases APD much more at slow than at fast heart rate. This effect is equivalent to steepening of the APD restitution curve, an effect that we found to be important for termination of spiral waves by blocking the nonspecific K+ channel in the LR1 model. 3) The LR1 model does not take into account Ca2+ cycling dynamics. Ca2+ cycling dynamics can also create dynamical instabilities, which may be very important in fibrillation (8, 10, 34, 36). Na+ and K+ channels couple with intracellular Ca2+ through membrane potential. How Na+ and K+ channel blockade affects spiral wave dynamics when dynamical instabilities are caused by Ca2+ cycling is critically important and needs to be assessed in future studies. 4) We used a simple 2-D monodomain tissue model with a simple spatial gradient in electrical heterogeneity, whereas structural and electrophysiological complexities of 3-D ventricular and atrial tissue are undoubtedly important factors in the genesis and termination of fibrillation. For example, fiber rotation (14, 43, 61) and geometry (49) in the ventricles and the complex anatomic structures in the atrium (18, 53, 54, 60) affect spiral wave dynamics, which may also affect the ability of Na+ and K+ channel blockade to terminate fibrillation. However, simulations at this level of detail are computationally costly, making it impractical to evaluate <Ts> statistically. The heterogeneity used in our model caused an almost linear drift of the spiral wave, but different spatial profiles of electrical heterogeneity will result in different drifting patterns, thus affecting the chance that a spiral wave will drift to the tissue border and self-terminate. Finally, we also have not explicitly simulated fibrillation maintained by high-frequency sources arising from the thoracic veins (6, 63), although the mother rotor mechanism might be considered a relevant analogy, if it is assumed that the focal source is reentrant. Despite these limitations, our conclusions may help illuminate, from a theoretical perspective, possible underlying mechanisms of drug-induced termination of fibrillation, which need to be further validated in tissue experiments and by simulation studies with more realistic action potential and tissue models.


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This study was supported by National Heart, Lung, and Blood Institute Specialized Center of Research in Sudden Cardiac Death Grant P50 HL-53219, American Heart Association Scientist Development Grant 0130171N, and Laubisch and Kawata endowments.


   FOOTNOTES
 

Address for reprint requests and other correspondence: Z. Qu, David Geffen School of Medicine at UCLA, 47–123 CHS, 10833 Le Conte Ave., Los Angeles, CA 90095 (E-mail: zqu{at}mednet.ucla.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


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