INTRODUCTION

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SPIRAL WAVE BREAKUP HAS ATTRACTED interest among cardiovascular researchers because of its potential clinical relevance to ventricular fibrillation (VF) and sudden cardiac death, the leading cause of cardiovascular mortality in industrialized countries. Spiral wave reentry as a cause of cardiac arrhythmias was first proposed on the basis of theoretical simulations in a cardiac tissue model (6, 7, 10, 20) and was subsequently documented experimentally in real cardiac tissues (1-3, 8, 11, 18). Simulations have demonstrated that spiral wave reentry can spontaneously break up into a fibrillation-like state, even in completely homogeneous cardiac tissue, if the electrophysiological properties of the cardiac cell model have certain properties, such as a steeply sloped action potential duration (APD) restitution curve (6, 16, 22). This type of spiral wave breakup arises solely from the inherent dynamical instability of cardiac electrical wave propagation and does not require any fixed heterogeneity in the tissue. However, real cardiac tissue has both regional electrophysiological and anatomical heterogeneities (4, 5), which are exacerbated by disease processes such as acute myocardial ischemia. Acute myocardial ischemia is associated with a variety of factors altering the regional electrophysiological characteristics of the ischemic zones, but arguably the single most important factor implicated in the development of VF is extracellular potassium (concentration) ([K⁺]_o) accumulation (12-15). [K⁺]_o accumulation depolarizes the resting membrane potential, reduces membrane excitability, shortens APD, slows conduction velocity, and prolongs recovery of excitability after an action potential (25). In this study, we simulated regional ischemia by elevating [K⁺]_o locally, with or without other alterations simulating acute ischemia, and studied the effect on spiral wave stability in two-dimensional (2-D) cardiac tissue. To model cardiac tissue, we used the phase-1 formulation of the Luo-Rudy cardiac action potential model (19), which contains detailed physiological formulations of most of the important cardiac ionic currents.

	METHODS

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Ignoring microscopic cell structure, cardiac tissue can be treated as a continuous system in which 2-D tissue is modeled by the partial differential equation

(1)

where V is membrane voltage (in mV) and t is time; C_m = 1 µF/cm² is membrane capacitance; D = 1 cm²/s is the diffusion coefficient; I_ion = I_Na + I_si + I_K + I_K1 + I_Kp + I_b is total transmembrane ionic current density; I_Na = _Nam3hj(V  54.4) is the fast inward Na⁺ current; I_si = _sidf(V  E_si) is the slow inward Ca²⁺ current; I_K = _Kx(V  77) is the time-dependent outward K⁺ current; I_K1, I_Kp, and I_b, which are solely functions of V, are the time-independent outward current, plateau K⁺ current, and background current, respectively; where is maximal conductance; E_si is the reversal potential of calcium; and m, h, j, d, f, and x are gating variables, all governed by the same type of ordinary differential equation. For details of the equations and functions in the Luo-Rudy model, see Ref. 19. In our simulations we set _Na = 23 mS/µF, _K = 0.705 mS/µF, and _si = 0.05 mS/µF. Because of the stiffness of the equations, a very small time step is required to integrate Eq. 1 during the action potential upstroke. To overcome this disadvantage, we developed an advanced method (21) to integrate Eq. 1. First, we split the reaction operator and the diffusion operator in Eq. 1 using the operator-splitting method. Then, we integrated the reaction term using a second-order Runge-Kutta method with adaptive time step and the diffusion term with an alternating-direction implicit method to guarantee numerical stability. In this study, the adaptive time step varied from minimum time step (t_min) = 0.01 ms to maximum time step (t_max) = 0.1 ms, and the diffusion term was integrated with the maximum time step, t = t_max = 0.1 ms, to keep all cells synchronized. With this approach, the integration speed increased >10-fold, with the relative error not exceeding 2% (21). Tissue size was fixed at 80 × 80 mm² (400 × 400 nodes) in all 2-D simulations, with no-flux boundary condition. To mimic regional ischemia, [K⁺]_o in a square section of variable size at the center of the tissue was raised from its control value of 5.4 mM to various higher levels.

Spiral wave reentry in 2-D tissue was initiated by two successive perpendicular rectilinear waves. Tip trajectories of spiral waves were traced using the intersection point of successive contour lines of voltage corresponding to 30 mV measured every 2 ms. The intersection points of these successive contour lines form a tip trajectory.

	RESULTS

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Effects of global [K+]_o elevation on spiral wave reentry. Figure 1 shows that global elevation of [K⁺]_o in the tissue had significant electrophysiological effects on spiral wave dynamics. For the control value [K⁺]_o = 5.4 mM, we chose the parameters of the LR1 model to produce a single hypermeandering spiral wave (6, 22) (Fig. 1A) in which the trajectory of the spiral wave tip showed chaotic motion (Fig. 1E). Increasing [K⁺]_o to 8.0 mM (Fig. 1, B and F) shortened APD and mildly depolarized membrane potential, thereby speeding conduction velocity (CV) (25). This combination of shorter APD and increased excitability caused the tip motion to become more regular and shortened the cycle length (CL), respectively. With further increases in [K⁺]_o to 11 mM (Fig. 1, C and G) and 13.2 mM (Fig. 1, D and H), membrane depolarization caused CV slowing and reduced excitability as the action potential upstroke velocity was depressed by progressive inactivation of the Na⁺ current (25). These changes increased CL and further regularized tip meander as the slope of APD restitution became progressively more shallow (26).

View larger version (42K): [in this window] [in a new window]   Fig. 1.   Effects of global hyperkalemia on spiral wave behavior. A-D: voltage snapshots (voltage values decreasing from white to black) for extracellular K+ concentration ([K+]o) = 5.4, 8, 11, and 13.2 mM, respectively. E-H: tip trajectories of the spiral waves in A-D, respectively. Tissue size is 80 × 80 mm2.

View larger version (14K): [in this window] [in a new window]   Fig. 2.   The average cycle length (CL; solid line), action potential duration (APD; dashed line), and diastolic interval (DI; dashed-dotted line) along the arm of the spiral wave as [K+]o was globally elevated in homogeneous 2-dimensional cardiac tissue. Dotted line shows the critical DI = 26 ms, at which point the slope of APD restitution exceeds 1 in normal [K+]o = 5.4 mM.

Effects of regional [K+]_o elevation on spiral wave reentry. To simulate the hyperkalemia accompanying regional ischemia, we elevated [K⁺]_o within a central square defect [length (L) × L] in the tissue and maintained [K⁺]_o at the control value of 5.4 mM everywhere else. The effects on spiral wave stability depended on both the degree of elevation of [K⁺]_o and the size of the hyperkalemic area, as shown in Fig. 3. For small defects (1.2 × 1.2 mm²; Fig. 3, A-C), elevation of [K⁺]_o to 8, 12, or 15 mM had no significant effect on the spiral wave, which continued to hypermeander as in the homogeneous case (Fig. 1A). For an intermediate defect (4 × 4 mm²; Fig. 3, D-F), however, the spiral wave broke up into a fibrillation-like state at all three values of [K⁺]_o. For a large defect (8 × 8 mm²; Fig. 3, G-I), the spiral wave broke up into a fibrillation-like state only at [K⁺]_o = 8 but not at 12 or 15 mM. Figure 4, A-D, shows the respective corresponding tip trajectories for 600 ms, starting from first wavebreak, in relation to APD dispersion in the tissue. Although the APD dispersion differed for each spiral wave breakup case, all of the first as well as the subsequent new wavebreaks occurred far from the ischemic region, away from the area in which APD dispersion gradient was largest.

View larger version (77K): [in this window] [in a new window]   Fig. 3.   Effects of a regional hyperkalemia on spiral wave reentry. Voltage snapshots of spiral wave reentrant activity at time (t) = 0.2, 0.5, 1.0, and 2.0 s after initiation of spiral wave in tissue containing a central region of hyperkalemia [length (L) × L, in mm2]. Values of [K+]o in the defect are 8, 12, or 15 mM, respectively, as indicated, with defect sizes of 1.2 × 1.2 mm2 in A-C, 4 × 4 mm2 in D-F, and 8 × 8 mm2 in G-I, respectively. SW, spiral wave.

View larger version (32K): [in this window] [in a new window]   Fig. 4.   Spiral wave tip trajectories starting at the time of the first new wavebreak through the subsequent 600 ms. Panels A-D correspond to Fig. 3, D-G, respectively. Top graphs: contour maps of APD over the tissue surface. Bottom graphs: tip trajectories. The tip trajectory in the center of each graph is from the original spiral wave. Note that for all 4 cases, the initial wavebreak as well as subsequent new wavebreaks occurred in the normal tissue well away from the maximum APD dispersion gradient.

View larger version (18K): [in this window] [in a new window]   Fig. 5.   The critical boundaries of spiral wave breakup vulnerability window in [K+]o-ischemia size parameter space. A: control conditions (hyperkalemia), with si = 0.05. B: conditions incorporating the effects of anoxia and acidosis in addition to hyperkalemia in the "ischemic" region. See RESULTS for details. SS, stable spiral wave; SB, spiral breakup; AS, anchored spiral wave; UAS, unanchored spiral wave.

Mechanism of spiral wave breakup with moderate [K+]_o elevation. To understand the mechanism of spiral wave breakup induced by small hyperkalemic regions, we examined the role of APD restitution steepness, originally shown by Karma (16) to be the major determinant of spiral wave stability in homogeneous cardiac tissue. As shown in Fig. 6, the APD restitution curve (solid line) for the LR1 model with _si = 0.05 has a slope >1 only when the DI becomes <26 ms. To achieve such a short DI, the spiral wave CL must be <60 ms (Fig. 2), which is lower than the actual CL selected by a spiral wave in homogeneous tissue at [K⁺]_o = 5.4 mM. However, as is apparent from Fig. 2, for [K⁺]_o = 5.8-10.4 mM, the spiral wave CL decreases to <60 ms. Thus if a spiral wave is initiated in the hyperkalemia region, and this region is large enough to support the core of the spiral wave, the rapid CL <60 ms drives the outlying normal tissue at DIs <26 ms, at which level the APD restitution slope is >1. The spiral arm therefore becomes unstable, and wavebreak occurs, leading to spiral breakup and a fibrillation-like state.

View larger version (13K): [in this window] [in a new window]   Fig. 6.   APD restitution curves for the LR1. Dotted line shows a slope of 1 for reference.

View larger version (86K): [in this window] [in a new window]   Fig. 7.   The effects of regional hyperkalemia on spiral wave behavior for si = 0.03. Voltage snapshots at t = 0.2, 0.5, and 1.5 s are shown. Right panel: the tip trajectories (Tip Traj). A: no defect, [K+]o = 5.4 mM throughout. B and C: [K+]o = 8 mM, 4 × 4 and 8 × 8 mm2 defect, respectively. D and E: [K+]o = 15 mM, 4 × 4 and 8 × 8 mm2 defect, respectively.

Mechanism of spiral wave breakup with [K+]_o elevations causing unexcitability in ischemic tissue. For defects with [K⁺]_o > 10.4 mM, however, the spiral wave CL in homogeneous tissue exceeded 60 ms, so that the above-mentioned mechanism of an accelerated spiral wave in the hyperkalemic region could not explain the wavebreak outside this region. Instead, the mechanism of spiral wave breakup was related to the significant degree of membrane depolarization within the hyperkalemic region. This membrane depolarization, which reached 65 mV at [K⁺]_o = 13.4 mM, reduced the current sink faced by the tip of the spiral wave circulating around the circumference of the unexcitable ischemic defect. Because the cycle length of reentry is the time it takes for the impulse to circumvent the ischemic region, this time is shortened when the electrotonic effect of the unexcitable tissue partially depolarizes the active tissue at the border, thereby bringing it closer to threshold and speeding conduction velocity around the unexcitable core. The net effect was to again shorten the CL of the spiral wave to <60 ms, so that the spiral arm further out in the normal tissue was subjected to short DIs (<26 ms), at which the APD restitution slope exceeded one. This led to spiral breakup until the size of the defect was made large enough so that the time required for the spiral tip to travel around its circumference exceeded 60 ms, at which point the spiral arm became dynamically stable (because of the longer DI at which APD restitution slope <1). For [K⁺]_o > 13.4 mM, at which point the hyperkalemic region became unexcitable, the spiral wave anchored to the defect. For intermediate values of [K⁺]_o from 10.4 to 13.4 mM, the hyperkalemic defect retained low excitability, and the tip did not anchor but continued to hypermeander because of the partial excitability of the hyperkalemic region.

View larger version (87K): [in this window] [in a new window]   Fig. 8.   Effects of regional hyperkalemia on spiral wave behavior when the spiral is initiated far from the hyperkalemic defect. Voltage snapshots at t = 0.2, 0.5, and 1.5 s are shown. Right panel: spiral tip trajectories for defect sizes of 12 × 12 (A), 8 × 8 (B), and 4 × 4 mm2 (C). [K+]o = 15.0 mM, with si = 0.05. The location of the wavebreak and wave fusion around the defect paralleled the quadrant in which the spiral wave was initiated, as expected.

Effects of including other components of ischemia in addition to hyperkalemia. To address whether these findings are still valid when other components of acute ischemia are included along with regional hyperkalemia, we simulated anoxia and acidosis, using the approach described by Shaw and Rudy (24). Low intracellular ATP (3 mM) due to anoxia was simulated by incorporating the ATP-sensitive K⁺ current and a 12% reduction in _si into the LR1 model, and acidosis (pH 6.5) was simulated with a 25% reduction of _si and _Na, changes typical after 10-15 min of acute ischemia (17). Figure 5B demonstrates that the critical boundaries of vulnerability to spiral wave breakup in the [K⁺]_o-"ischemic" area parameter space were qualitatively similar for this case, because the net effect of these changes was to shorten the CL of the spiral wave in the ischemic region over a wide range of [K⁺]_o.

	DISCUSSION

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Major findings. This study presents novel insights into the mechanisms of spiral wave breakup caused by regional ischemia that are relevant to VF and sudden cardiac death in the setting of acute myocardial ischemia. In contrast to the standard concept that an ischemic defect promotes arrhythmogenesis mainly by slowing conduction and altering refractoriness, we show that regional membrane depolarization can destabilize spiral wave reentry by accelerating the spiral wave CL in the normal tissue, causing spiral wave breakup. Importantly, the spiral wave in the ischemic region that drives the normal tissue can be either dynamically stable or unstable; the only requirement is that it have a sufficiently short CL to engage steep APD restitution in normal tissue. This is particularly germane to the experimental observation that ischemia promotes VF despite flattening APD restitution slope in the ischemic region (9, 17). Our findings suggest it is the fact that membrane depolarization and APD shortening accelerate the CL of spiral wave reentry in the ischemic region, rather than APD restitution slope in the ischemic region, that is important. This acceleration of the spiral wave frequency, when transmitted to normal tissue, drives the normal tissue at short diastolic intervals associated with steep APD restitution slope, generating wavebreak and VF.

Limitations. Several limitations of this study should be recognized. First, we simulated ischemia primarily by hyperkalemia, without including many other factors with important electrophysiological effects (for review see Ref. 15). However, hyperkalemia is recognized as the major factor causing membrane depolarization during acute ischemia, which is the main factor responsible for the mechanism of spiral breakup that we have described. In addition, the inclusion of other components of the ischemic environment such as anoxia and acidosis did not substantially alter the results (Fig. 5B). Nevertheless, we did not simulate in our model the important phenomenon of postrepolarization refractoriness (i.e., effective refractory period > APD) that characterizes the acutely ischemic ventricle (13). For the case in which the hyperkalemic ischemic defect is still excitable, postrepolarization refractoriness will slow the rate of spiral wave reentry, thereby preventing wavebreak in the normal tissue, which leads to VF. However, postrepolarization refractoriness typically is only observed after 5-10 min of acute ischemia (for review see Ref. 15), before which [K⁺]_o can be already markedly elevated (15). In addition, postrepolarization refractoriness affects the central ischemic area more than the border zone (12), which also becomes significantly hyperkalemic (15). Finally, the issue of postrepolarization refractoriness does not apply to the situation in which the hyperkalemic ischemic area has become unexcitable, yet can still promote VF by its electrotonic effect of accelerating an anchored spiral wave in normal tissue. Second, we modeled regional hyperkalemia as uniform over the ischemic area, whereas in real ischemic tissue there is considerable heterogeneity in the level of [K⁺]_o, especially at the border zone of the infarct (15). Third, we only examined a square defect, whereas the shapes of real ischemic defects are highly variable. Fourth, we have simulated 2-D cardiac tissue under the assumption that similar mechanisms will apply to the three-dimensional case. Finally, the acceleration of the spiral wave by the ischemic defect depends on the spiral wave being initiated very near to the ischemic zone. Experimental studies support this scenario, however, as the premature ventricular beats initiating reentry commonly occur very close to the edge of the ischemic border zone (14). In addition, APD dispersion is greatest at the border zone between normal and ischemic tissue, making this a likely place for the first wavebreak-initiating spiral wave reentry to occur.