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Intracellular calcium cycling, early afterdepolarizations, and reentry in simulated long QT syndrome
ORIGINAL-EXPERIMENTAL
Intracellular calcium cycling, early afterdepolarizations, and reentry in simulated long QT syndrome Ray Huffaker, BS,a Scott T. Lamp, BS,c James N. Weiss, MD,b,c Boris Kogan, PhDa a
From the Departments of Computer Science, Medicine (Cardiology), and c Physiology, University of California, Los Angeles, and the David Geffen School of Medicine at UCLA, Los Angeles, California. b
OBJECTIVES The purpose of this study was to investigate interactions between early afterdepolarizations (EADs) and reentry in long QT (LQT) syndromes. BACKGROUND EADs, a characteristic feature of congenital and acquired LQT syndromes, are classically bradycardia dependent. Mechanisms by which they promote tachyarrhythmias such as torsades de pointes and ventricular fibrillation are not fully understood. Recent evidence suggests that EADs also may occur at rapid heart rates as a sequela of spontaneous sarcoplasmic reticulum (SR) Ca2⫹ release related to intracellular Ca2⫹ overload. Here, we performed computer simulations to explore the arrhythmogenic consequences of this phenomenon. METHODS We used a modified version of the Luo-Rudy dynamic model in one-dimensional and two-dimensional cardiac tissue with the time-dependent K⫹ currents IKr or IKs reduced by 50% to simulate acquired and congenital LQT syndromes. RESULTS (1) Spontaneous SR Ca2⫹ release prolonged action potential duration but did not induce overt EADs unless K⫹ current density was reduced to simulate acquired and congenital LQT syndromes. (2) In simulated LQT syndromes, EADs were capable of both terminating and reinitiating one-dimensional reentry. (3) A similar phenomenon in simulated 2D tissue led to reinitiation of spiral wave reentry that otherwise would have self-terminated. (4) Reentry reinitiation occurred only when the L-type Ca2⫹ current and SR Cai cycling were potentiated to simulate moderate sympathetic stimulation, consistent with the known arrhythmogenic effects of sympathetic activation (and protection by betablockers) in LQT syndromes. CONCLUSIONS These computer simulations suggest that EADs related to spontaneous SR Ca2⫹ release can enhance arrhythmogenesis in LQT syndromes by reinitiating reentry. KEYWORDS Long QT syndrome; Early afterdepolarizations; Potassium currents; Arrhythmias; Computer simulations; Cardiac electrophysiology; Sudden cardiac death; Ventricular tachycardia; Genetic channelopathies; Intracellular calcium cycling © 2004 Heart Rhythm Society. All rights reserved.
Early afterdepolarizations (EADs) are thought to be highly arrhythmogenic, especially in the settings of congenital and acquired long QT (LQT) syndromes. Experimentally, however, EADs are most readily observed at slow
The study was supported in part by NIH/NHLBI SCOR in Sudden Cardiac Death P50 HL52319. Supercomputing resources were provided by National Energy Research Scientific Computing Center, Office of Energy Research of the US Department of Energy, under contract no. DEAC0376SF00098. Address reprint requests and correspondence: Dr. Boris Kogan, Department of Computer Science, UCLA, Los Angeles, CA 90095. E-mail address: [email protected]. (Received March 16, 2004; accepted May 17, 2004.)
heart rates and are suppressed by fast heart rates.1 The mechanisms by which this bradycardia-dependent phenomenon contributes to rapid arrhythmias such as torsades de pointes and ventricular fibrillation (VF) leading to sudden cardiac death are not fully understood. EADs occur when the balance of the total membrane current during repolarization reverses from outward to inward. At slow heart rates, the action potential (AP) prolongs physiologically. If outward currents are further reduced or inward currents are potentiated by drugs or genetic defects, reactivation of the L-type Ca2⫹ current during repolarization can lead to EADs and triggered activity.2,3 EADs generated by this mechanism tend to be suppressed by fast heart rates, and triggered
1547-5271/$ -see front matter © 2004 Heart Rhythm Society. All rights reserved. doi:10.1016/j.hrthm.2004.06.005
Heart Rhythm (2004) 4, 441– 448
442 activity usually self-terminates. In the setting of impaired repolarization, however, it is possible that EADs may be induced at rapid heart rates by different ionic mechanisms that cause a temporal imbalance between inward and outward currents during repolarization. For example, very rapid heart rates may cause intracellular Ca2⫹ concentration (Cai) overload, leading to spontaneous (i.e., nonvoltage-gated) Ca2⫹ release from the sarcoplasmic reticulum (SR). Classically, spontaneous SR Ca2⫹ release due to Cai overload is thought to cause delayed afterdepolarizations (DAD).1 However, if SR Ca2⫹ release were to occur before repolarization was complete, activation of Ca2⫹-sensitive inward currents such as Na/Ca exchange and the Ca2⫹-activated nonselective cation channel could cause EADs, leading secondarily to reactivation of the L-type Ca2⫹ current and triggered activity. This possibility is supported by both computer simulations4,5 and experimental evidence.6 In canine ventricular myocytes exposed to a drug blocking the rapid component of the time-dependent K⫹ current (IKr), EADs were induced by sudden rate acceleration6 and were eliminated when spontaneous SR Ca2⫹ release was inhibited by ryanodine. Following spontaneous termination of atrial fibrillation, EADs caused spontaneous reinitiation of fibrillation6 and were also eliminated by ryanodine. In canine ventricular myocytes, isoproterenol has been reported to induce ryanodine-sensitive EADs as well as DADs.7 Comparing simultaneous optical maps of Vm and Cai during VF in porcine right ventricle, Omichi et al8 found that elevations in Cai often preceded voltage waves, consistent with spontaneous SR Ca2⫹ release events. The goal of the present study was to use computer simulations to explore the potential arrhythmogenic consequences of EADs induced by spontaneous SR Ca2⫹ release at rapid heart rates in the setting of simulated LQT syndromes. We performed simulations of one-dimensional (1D) and twodimensional (2D) reentry using the model proposed by Chudin et al,9 which is a modified version of the Luo-Rudy dynamic model.10 The model of Chudin et al was further modified as follows. Time-dependent K⫹ currents IKr or IKs were reduced to simulate acquired and congenital LQT syndromes, and the L-type Ca2⫹ current (ICa(L)) and the SR Ca2⫹ uptake pump (Ip(Ca)) were amplified to simulate a modest increase in sympathetic tone. Using simulated 1D reentry (ring-shaped tissue), we show conditions and mechanisms by which EADs can either block or reinitiate reentry, in the absence of complicating factors such as wave curvature and tissue border effects. Using 2D tissue simulations, we show that EADs can regenerate spiral waves that otherwise would have self-terminated in simulated LQT2 and LQT1 syndromes. These findings shed new light on why EADs, classically considered a bradycardiadependent phenomenon, are potentially so arrhythmogenic during tachyarrhythmias such as torsades de pointes and VF.
Methods The mathematical model of wave propagation in 1D tissue is described by the following partial differential equation:
Heart Rhythm, Vol 1, No 4, October 2004 ⭸V ⭸t
⫽D
⭸ 2V ⭸x
2
⫹ (Im ⫹ Ist(s, t))
1
(1)
Cm
with appropriate initial and boundary conditions. Here V is membrane potential, D is a diffusion coefficient, Ist is the external stimulating current, and Cm is a membrane capacity. To make Equation 1 closed, it is necessary to add the system of nonlinear ordinary differential equations that describes the behavior of all components of Im and processes in intracellular compartments. For this purpose we have chosen the AP model proposed by Chudin et al,9 which represents a modification of the Luo-Rudy AP model.10 The initial conditions and parameters were chosen to be the same as those of Chudin et al, but with slight modifications to Cai,tot (from 18.61 to 21.01 M) and [Ca2⫹]jsr,tot (from 8.44 to 6.9 mM). To simulate congenital (LQT1 and LQT2) or acquired LQT syndromes, we inhibited either IKr or IKs by decreasing ¯ or G ¯ ) by 50%. In the LQT the maximum conductivity (G kr
ks
cases we also simulated a modest increase in sympathetic tone by amplifying ICa(L) and Ip(Ca) by increasing the pa¯ rameters ¯I and ¯I by 31.8% in the case of decreased G p(Ca)
Ca
kr
¯ . LQT simulations and 34.8% in the case of decreased G ks were compared to simulations run with normal model parameters. For computer simulation, a ring was formed from a line of equidistant cell models (nodes) interconnected by diffusion. After stimulation of one end of this line and formation of the propagated wave of full length, the line was closed numerically, satisfying the condition of periodicity, to form a ring. To change the ring length, the same procedure was repeated but with a different number of initial nodes. The 1D simulation has mainly methodologic aims. We can study the conditions and mechanisms by which EADs may block or reinitiate reentry while avoiding the complicating effects of wave curvature and border conditions that occur in 2D and 3D tissues. The 1D simulation also allows us to easily modify the pacing rate of cells in the tissue by changing the length of the ring. Simulating very large rings first and then shorter rings permits us to follow the development of EADs as the pacing rate increases. Computer simulations were performed on an IBM SP RS/6000 massively parallel computer at Lawrence Berkeley National Laboratory. The simulations used the operator splitting algorithm.11 According to this algorithm, the integration (1) is split into two parts: integration of diffusion ⭸V ⭸2V equation ⫽ D 2 and integration of the system of ⭸t ⭸x ⭸V 1 ordinary differential equations ⫽ 共Im ⫹ Ist兲 . These ⭸t Cm integrations were executed in consecutive time cycles of predetermined duration ⌬t ⫽ 0.1 ms. The specific features of this algorithm are described in Chudin et al12 and Kogan et al.13 The operator splitting algorithm allows integration
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of the system of nonlinear ordinary differential equations at any point in space independently and with variable time steps. In all computer simulations, the time step for ordinary differential equation solutions varied from 0.005 to 0.1 ms and the space step ⌬x was fixed at 0.016 cm. These choices provided stability and accuracy with the chosen numerical integration method and did not disturb the conditions of medium continuity.14 Specifically, an explicit Euler numerical method was used to solve the diffusion equation, for ⌬x2 which the stability condition was ⌬t ⱕ . For ⌬x ⫽ 0.016 2D cm and D ⫽ 1 cm2/s, the critical value was ⌬t ⫽ 0.128 ms. In our computations, this equation was solved twice in one computational cycle, with ⌬t ⫽ 0.05 ms. The problem of computational error arises for solutions to the system of nonlinear ordinary differential equations describing the fast membrane processes during depolarization phase of the AP. To decrease these errors, we used the Euler explicit method with variable time steps, which were changed depending on the rate of the most rapid variable. For the fast sodium channel gate variable, we replaced the Euler method with the so-called hybrid method.15 The continuity condition proposed by Winfree14 requires that the chosen space step ⌬x satisfies the inequality D ⬎ ⌬x2⁄Tr. Here Tr is an activation rising time measured in a cell placed in tissue. This time is longer than in an isolated cell due to the effect of the local currents and incomplete recovery of INa during wave circulation in a comparatively short ring length. Estimating Tr ⫽ 2.5 ms, we found that for D ⫽ 1 cm2/s and ⌬x ⫽ 0.016 cm, this inequality was well satisfied. Individual nodes were distinguished by their position in the initial line of cells before it was closed. Node i is defined as the ith node in the line, for 0 ⱕ i ⱕ N ⫺ 1, where N is total number of nodes in the ring. The position of node i in the closed ring (in centimeters), after k turns of the circulating wave, was defined as (k ⫺ 1)*L ⫹ i*⌬x, where L is the length of the ring in centimeters. Action potential duration (APD) was calculated for each AP of each node in the ring and graphed as a function of the position in the ring. For long rings, APD was calculated at 90% repolarization (APD90). For shorter rings, when propagation was less regular, APD was defined as the time above ⫺50 mV. The error associated with this calculation of APD was approximately 2 ms, which is the amount of time required for a potential change of 1 mV in a cell near the tail of an elongated wave. Conduction velocity () was calculated as a function of position in the ring. The formula for velocity defined the wavefront at time Ti as the node Xi, where the maximum value of Vi ⫺ Vi⫹1 occurred. Conduction velocity was found by measuring the average of the time elapsed while the wave propagated from node Xi⫺k to node Xi and the time elapsed while the wave propagated from node Xi to node Xi⫹k, for some suitably chosen k. The formula is:
443
Ui ⫽
冉
1 Xi ⫺ Xi⫺k 2 Ti ⫺ Ti⫺k
⫹
Xi⫹k ⫺ Xi Ti⫹k ⫺ Ti
冊
(2)
Here Ui is the conduction velocity measured at node i. The error from this formula, defining as the true value of the conduction velocity, is given by the formula9: 1⫹
ⱕUⱕ
2kS
1⫺
(3)
2kS
Here S is defined as the quotient of the space step ⌬x and the time step ⌬t. In our simulation, ⌬x was 0.016 cm, ⌬t was 0.1 ms, and k was 5, giving a maximum error of |U ⫺ | ⬍ 3.2 cm/s. For 2D tissue simulation, we also used the operator splitting algorithm. The time step and space step for the diffusion equation were chosen as ⌬t ⫽ 0.05 ms and ⌬x ⫽ 0.025 cm, which satisfy the stability conditions and condition of continuity.14 The space step ⌬x was increased from 1D to 2D in order to shorten the overall computation time for 2D simulation. In solving the ordinary differential equations, we used a variable time step of 0.005 to 0.1 ms. For verification of the overall solution, we performed the computer simulation with the space step decreased to 0.0125 cm and the time step decreased to 0.0005 ms. No tangible differences were observed in simulation results. The tissue was simulated as a square grid of 256 ⫻ 256 diffusively coupled nodes (D ⫽ 1 cm2/s) with no-flux boundary conditions. The space step ⌬x ⫽ 0.025 cm yields a tissue size of 6.4 ⫻ 6.4 cm2. This tissue size is very large compared to most physiologic values, but it allows us to avoid the effects of the border conditions. Individual nodes were distinguished by their position in the grid with an ordered pair (column no., row no.). The upper left node was denoted as (0,0) and the lower right node denoted as (255,255). The parallel processors were distributed over entire grid in stripes, so each processor integrates only in its own part of the grid. The communication between processors was implemented using Message-Passing Interface. The velocity of a rectilinear front propagation was 55 cm/s.
Results Interactions between Cai cycling, EADs, and reentry in a 1D ring of cells Figure 1 illustrates reentry in a 1D a ring consisting of 3,200 nodes (or 51.2 cm) for the case of normal cardiac cell model parameters (A) and the case in which IKr has been reduced by 50% to simulate LQT2 and ICa(L) and Ip(Ca) have been increased by 31.8% to simulate an increase in sympathetic tone (B). The reentry cycle length (CL) was 960 ms, which corresponds roughly to a normal human heart rate. In both cases, propagation was stationary, with APD and
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Figure 1 Stationary propagation showing APD90 and conduction velocity () for a ring of length 51.2 cm (3,200 nodes), corresponding to a period of 960 ms. A: Normal model parame kr with mild sympaters. B: Simulated LQT2 syndrome (50% G thetic activation (131.8% ICa(L) and Ip(Ca)). Traces of membrane voltage (V) and [Ca2⫹]i are shown below for node 500. No spontaneous SR Ca release (Ispon) occurred.
remaining fairly constant in space over the five cycles illustrated. As expected, APD was longer for the simulated LQT2 case. At this CL, SR Ca2⫹ accumulation was insufficient to cause spontaneous SR Ca2⫹ release. Similar findings were obtained when IKs was reduced by 50% to simulate LQT1. Figure 2 shows that when the ring was shortened to 1,024 nodes (or 16.364 cm), nonstationary propagation was observed. In both cases, spatial irregularities in the distribution of APD and developed but were more pronounced in the simulated LQT2 case (Figure 2B). Accumulation of Cai was sufficient in both cases to induce spontaneous SR Ca2⫹ release (bottom traces), which occurred more rapidly in the simulated LQT2 case because of its longer baseline APD. Only the APs preceded by spontaneous SR Ca2⫹ were significantly prolonged. APD prolongation was more prominent in the simulated LQT2 case. Similar findings were obtained when IKs was reduced by 50% to simulate LQT1. Further shortening of the ring increased these propagation irregularities. Figure 3 shows that, for the control case, shortening the ring to 584 nodes (or 9.344 cm) led to immediate onset of APD and CV alternans even before spontaneous SR Ca2⫹ release occurred, due to transient behavior associated with the sudden increase in heart rate. Prior to spontaneous SR Ca2⫹ release, however, the spatial
Heart Rhythm, Vol 1, No 4, October 2004 profile of voltage and Cai in a given reentrant cycle remained fairly uniform (Figure 3B(i)). Once spontaneous SR Ca2⫹ release occurred, however, the spatial profile of Cai developed bumps, which caused similar perturbations in voltage (Figure 3B(ii-iii)). The bumps in voltage reflected the local variation in APD along the ring and were not due to overt EADs in the control case, as can be seen from the voltage trace in Figure 3A. (Cai and voltage bumps due to spontaneous SR Ca2⫹ release were observed at longer ring lengths but were not as pronounced.) At short ring lengths, localized APD prolongation due to spontaneous SR Ca2⫹ release was sufficient to induce head-tail interactions, which impeded conduction of the next cycle of the wave. Instances can be seen in the graph of (Figure 3A), where transiently and repeatedly falls to approximately one third of its original value. From this graph, it is impossible to determine whether propagation was merely slowed or stopped momentarily and then restarted. This is because the calculated is not instantaneous but is an average velocity along several cells. However, if we look at the sign of dV/dt along the wave to determine whether propagation had truly stopped (Figure 3C), we see that a few cells in the wavefront still had positive dV/dt, indicating that the wave never actually stopped. Figure 4 shows the simulated LQT2 case. As expected, less ring shortening was required to produce highly irregular propagation because APD was longer. In a ring consisting
Figure 2 Onset of nonstationary propagation showing APD90 and conduction velocity () when ring length was shortened to 16.384 cm (1,024 nodes), corresponding to a period of 295 ms. A: Normal model parameters. B: Modified parameters (as in Figure 1B) simulating LQT2 syndrome with mild sympathetic stimulation. Traces of V, [Ca2⫹]i, and Ispon are shown for node 500.
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445
propagation but does play a significant role, along with Ca2⫹-sensitive inward currents, in generating the EAD. When Iks was reduced by 50% to simulate LQT1, EADs causing wave termination were similarly observed but occurred at even longer ring lengths, beginning at 1,088 nodes (or 17.408 cm). Figure 5 shows the effects of shortening the ring even further in the simulated LQT2 case. For a shorter ring of 702 nodes (or 11.232 cm), the phenomenon of wave regeneration by EADs was observed, illustrated in the graphs of and APD in Figure 5A. The fourth episode of spontaneous SR Ca2⫹ release (bottom trace) causes an EAD, after which decreases close to 0 (Figure 5A). However, and APD subsequently return to normal, indicating that propagation has been restored. Figure 5C shows the spatial profile of voltage along the ring when this EAD occurred. Again we see the appearance of a bump on the tail with positive dV/dt (t ⫽ 2.0 s), indicating the EAD, which becomes larger, causing the next wavefront to slow. The original wave then
Figure 3 Irregular propagation for normal model parameters when ring length was further shortened to 9.344 cm (584 nodes), corresponding to a period of 194 ms. A: APD (calculated using threshold of ⫺50 mV) conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 500 are shown below. No EADs are present. B: Distribution of membrane voltage (V, green line) and [Ca2⫹]i (purple line) in space along the ring at the different times indicated. In (iii), Ispon causes a bump on the tail of the wave, which has negative dV/dt. C: Distribution of V in space when the wavefront closely approaches the waveback. Region of positive dV/dt is shown in red. The wavefront has positive dV/dt ⬍ 10. In a normal propagating wavefront, typically dV/dt ⬎⬎10.
of 768 nodes (or 12.288 cm), Figure 4B shows that the fourth episode of spontaneous SR Ca2⫹ release caused an EAD that terminated reentry. The mechanism is illustrated in the spatial voltage profile shown in Figure 4C. Bumps in the spatial profile of voltage were much larger than observed for the control case (Figure 4C vs Figure 3C) and often had dV/dt ⬎ 0, indicating an overt EAD. A bump on the waveback with dV/dt ⬎ 0, indicating an EAD, first appeared at t ⫽ 2.13 s. As the next wavefront approached this EAD, it slowed and then stopped completely. Although the EAD produced a new wavefront (t ⫽ 2.24 s) that propagated for some distance, it eventually failed due to residual refractoriness from the old wave (t ⫽ 2.43 s), and reentry terminated. Figure 4B also shows selected membrane currents during the EAD. At the time of the EAD, there was no significant local diffusive current (IN), indicating that the EAD was self-generated and not an artifact of passive diffusive currents from surrounding cells in different phases of repolarization. Sodium current (INa) also is absent, indicating that it plays no role in the generation of the EAD. The L-type Ca2⫹ channel current ICa,L is smaller than in the case of
Figure 4 Appearance of early afterdepolarizations (EAD) for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B), at ring length ⫽ 12.288 cm (768 nodes), corresponding to a period of 248 ms. A: APD90 and conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 80 are shown below. An EAD (arrow) in the V trace is produced by the spontaneous Ca release seen in the Ispon trace, which terminates propagation. B: Various membrane currents for node 80 when the EAD occurred. C: Distribution of V in space at different times just before the early afterdepolarization terminates propagation. Regions of positive dV/dt are shown in red.
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Heart Rhythm, Vol 1, No 4, October 2004 block, and reinitiation of reentry in the 1D ring-shaped tissue. EADs were never observed with normal IKr and IKs, whether ICa(L) or Ip(Ca) was normal or enhanced to simulate modest sympathetic stimulation. When either IKr or IKs was reduced by 50% to simulate LQT2 or LQT1, respectively, EADs occurred with both normal and enhanced ICa(L) and Ip(Ca). However, reinitiation of reentry by EADs was seen only with enhanced ICa(L) and Ip(Ca). This finding is interesting in view of the clinical observation that beta-blockers reduce episodes of torsades de pointes in patients with LQT syndrome.
Arrhythmogenic consequences in simulated 2D tissue
Figure 5 Regeneration of wave propagation by an early afterdepolarization (EAD) for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B), at a ring length of 11.232 cm (702 nodes), corresponding to a period of 185 ms. A: APD90 and conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 50 are shown below. B: Distribution of V in space at different times just preceding wave regeneration due to the EAD. Regions of positive dV/dt shown in red.
stops completely and all of its cells repolarize, while the bump produces a new propagating wave front (t ⫽ 2.1 s). Unlike the case where EAD causes propagation to fail, the old wave has sufficiently repolarized by the time the new wavefront reaches that part of the ring (t ⫽ 2.2 s). The new wavefront is able to regenerate propagation (t ⫽ 2.3 s) and perpetuate reentry (t ⫽ 2.4 s). Propagation of this EAD-generated wave subsequently was terminated (as seen in Figure 5A) by an EAD in another region, which did not cause regeneration. Similar findings were observed when Iks was reduced by 50% to simulate LQT1 at a ring length of 832 nodes (or 13.312 cm).
Role of increased sympathetic tone Table 1 summarizes the conditions under which spontaneous SR Ca2⫹ release induced EADs, conduction Table 1
1 2 3 4
In 1D reentry, the consequences of EADs terminating and reinitiating reentry are ambiguous with respect to arrhythmogenesis in two or three dimensions. In these higher dimensions, conduction block must be global in order to terminate reentry. If conduction block is localized then rather than terminating reentry, it can further destabilize reentry, causing a spiral wave to break up into a VF-like state. To examine how our findings in 1D reentry relate to propagation behavior in higher dimensions, we simulated LQT2 with increased sympathetic tone in 2D tissue. For a 6.4 cm ⫻ 6.4 cm tissue, the case of normal cardiac cell model parameters produced nonstationary sustained spiral wave reentry. For the simulated LQT2 case (Figure 6), however, the baseline APD prolongation converted sustained spiral wave reentry to nonsustained reentry, a stop of propagation, due to the mismatch between the now longer wavelength and the fixed tissue size (t ⫽ 2.800 s). If spontaneous SR Ca2⫹ release was disabled in the cell AP model at the time just before the propagation stopped (Figure 6B), EADs were less prominent and incapable of causing wave regeneration. Reentry always terminated when the meandering spiral wave wandered off the tissue. However, when spontaneous release was allowed (Figure 6A), the following scenario was observed: as the wave wandered to the border to self-extinguish (t ⫽ 2.751 s), spontaneous SR Ca2⫹ release in a region of the tissue generated an EAD (Figure 6C, in the vicinity of point b). When the region of tissue near the region of EAD became fully repolarized, the EAD began to propagate (t ⫽ 2.860 s) and, in the face of inhomogeneous repolarization, regenerated a new spiral wave (t ⫽ 2.935 s). Thus, EADs reinitiated reentry in a tissue in
The conditions of EAD and wave regeneration appearance in 1D ring-shaped tissue K⫹ conductivity
Presence of increased sympathetic tone in Ca2⫹ dynamics
EAD observed?
Wave regeneration observed?
Decreased G kr or G ks Decreased G kr or G ks Normal Normal
Present, increased ICa and Ip(Ca) Absent Present, increased ICa and Ip(Ca) Absent
Yes Yes No No
Yes No No No
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Figure 6 Spiral wave regeneration by an early afterdepolarization (EAD) caused by spontaneous Ca2⫹ release in simulated two-dimensional tissue, for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B). A: Snapshots of membrane voltage (V) at the times indicated. The spiral wave initiated at t ⫽ 0.273 s had fully depolarized the available tissue at t ⫽ 2.800 s, when an EAD emerged and regenerated propagation. B: Eliminating the EAD by disabling Ispon at time t ⫽ 2.725 s prevented spiral wave regeneration. C: Superimposed traces of membrane voltage (V) at every third node between sites a (77,2) and b (95,2) indicated in A. Tissue size is 256 ⫻ 256 nodes (6.4 cm ⫻ 6.4 cm).
which reentry otherwise would have self-terminated (Figure 6B, t ⫽ 2.935 ms). Similar findings were obtained when IKs was reduced by 50% to simulate LQT1.
Discussion Although EADs classically are thought to be promoted by slow heart rates, recent experimental evidence suggests that EADs also may be induced by rapid heart rates as a result of Cai overload and spontaneous SR Ca2⫹ release that occurs before repolarization is complete.6,16 This mechanism could explain how EADs, in addition to providing bradycardiadependent triggers involved in the initiation of reentry, continue to be proarrhythmic after reentry is induced, en-
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hancing the likelihood of VF and sudden cardiac death in the setting of LQT syndrome. In the present study, we used 1D and 2D computer simulations to investigate the arrhythmogenic consequences of Cai-induced EADs at rapid heart rates. Our findings can be summarized as follows. (1) Spontaneous SR Ca2⫹ release prolonged APD under control conditions but did not induce overt EADs unless K⫹ current density was reduced to simulate acquired and congenital LQT syndromes. (2) In simulated LQT1 and LQT2 syndromes, EADs were capable of both terminating and reinitiating 1D reentry. (3) A similar phenomenon in simulated 2D tissue led to reinitiation of spiral wave reentry that otherwise would have self-terminated. (4) Reentry reinitiation occurred only when the L-type Ca2⫹ current and SR Cai cycling were potentiated to simulate moderate sympathetic stimulation, which is consistent with the known arrhythmogenic effects of sympathetic activation (and protection by beta-blockers) in LQT syndromes. This novel mechanism, which regenerates a spiral wave after the wavefront has stopped, may provide additional insight into why triggered activity occasionally becomes sustained and degenerates to VF in the settings of acquired and congenital LQT syndromes. Although we simulated LQT1 and LQT2 syndromes, spiral wave regeneration appears to be a general consequence of EADs related to spontaneous SR Ca2⫹ release, regardless of the specific underlying cellular mechanism. We previously found that EADs generated by increasing the Ca2⫹ sensitivity of Ins(Ca) produced the same effect.5 Thus, we speculate that the same phenomenon likely is observed when EADs are generated by other mechanisms, such as Na⫹ channel defects in LQT3 syndrome, and possibly when Cai-induces DADs.
EADs and intracellular Ca2ⴙ dynamics From a physiologic and clinical standpoint, it is appealing that Cai should play a role in the maintenance of triggered activity and/or reentry, because at fast heart rates characteristic of torsades de pointes, both diastolic and systolic Cai increase substantially relative to normal heart rates.9 The elevated Cai levels will potentiate Ca2⫹-sensitive inward currents and make them more effective at resisting repolarization. In addition, increasing Cai enhances Ca2⫹ uptake by the SR, leading to SR Ca2⫹ overload. The latter effect is well known to promote spontaneous Ca2⫹ release in a variety of experimental situations.17 Moreover, recent optical mapping experiments in which membrane voltage and Cai were mapped simultaneously provided evidence that nonvoltage-gated Cai release occurs in fibrillating porcine ventricle, which is consistent with spatially localized spontaneous SR Ca2⫹ release.8
Study limitations We performed our 1D and 2D tissue simulations in homogeneous isotropic tissue, whereas the real heart is three
448 dimensional and contains preexisting electrophysiologic and anatomic heterogeneities. Although there may be differential regional sensitivities to spontaneous SR Ca2⫹ release and EADs in the real heart, they likely do not affect our basic conclusions. In addition, nonuniform anisotropy and transmural action potential gradients may promote drift of scroll waves toward tissue borders, perhaps amplifying the likelihood of termination and regeneration by the mechanism we have proposed. Finally, direct experimental evidence that rapid heart rates induce spontaneous SR Ca2⫹ release before repolarization is complete, thus causing EADs, is limited. However, our findings may serve as an impetus for experimentalists to carefully investigate this issue, given its potential importance to arrhythmogenesis in LQT syndrome.
Conclusion The mechanism of spiral wave regeneration by Cai-induced EADs could be clinically important in prolonging the duration of reentry initially triggered by EADs, thereby enhancing the risk of VF and sudden cardiac death in patients with congenital or acquired LQT syndromes. Although there is no direct experimental evidence supporting this hypothesis in real hearts, this simulation study provides an incentive to look for this mechanism in the experimental setting. The regeneration of the spiral wave in the original or opposite direction5 might be the most recognizable indication, although the 3D nature of real cardiac tissue may make the experimental mapping data difficult to interpret unequivocally. Intermittent reversal of the spiral wave might contribute to the variable electrical QRS axis characteristic of torsades de pointes. Our simulation conditions may not be that far off from clinically relevant ones, because spiral waves that drift without breaking up have been documented experimentally,18 and acquired and congenital LQT syndromes are characterized by many nonsustained episodes of unstable VT that self-terminate. This implies that the prolonged wavelength from APD lengthening is at the edge of sustainable reentry, similar to the conditions under which spiral wave regeneration was observed here. However, the relevance of this mechanism of spiral wave regeneration to the real heart awaits simulation on realistic 3D geometries and experimental confirmation. Our findings may also be relevant to failed defibrillation shocks, in which postshock EADs due to spontaneous SR Ca2⫹ release might reinitiate reentry.
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References 1. Peters NF, Cabo C, Wit AL. Arrhythmogenic mechanisms: automaticity, triggered activity and reentry. In: Zipes DP, Jalife J, editors. Cardiac Electrophysiology: From Cell to Bedside, 3rd ed. Philadelphia: WB Saunders, 2000:348 –349. 2. Viswanathan PC, Rudy Y. Pause induced early afterdepolarizations in the long QT syndrome: a simulation study. Cardiovasc Res 1999;42: 530 –542. 3. Wehrens XH, Abriel H, Cabo C, Benhorin J, Kass RS. Arrhythmogenic mechanism of an LQT-3 mutation of the human heart Na(⫹) channel alpha-subunit: a computational analysis. Circulation 2000; 102:584 –590. 4. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. 2. Afterdepolarizations, triggered activity, and potentiation. Circ Res 1994;74:1097–1113. 5. Kogan B, Lamp S, Weiss J. Role of intracellular Ca2⫹ dynamics in supporting spiral wave propagation. In: Bekeg G, Kogan B, editors. Modeling and Simulation. Dordrecht, The Netherlands: Kluwer Academic Publishers, 2003:177–193. 6. Burashnikov A, Antzelevitch C. Acceleration-induced action potential prolongation and early afterdepolarizations. J Cardiovasc Electrophysiol 1998;9:934 –948. 7. Priori SG, Corr PB. Mechanisms underlying early and delayed afterdepolarizations induced by catecholamines. Am J Physiol 1990;258(Pt 2):H1796 –1805. 8. Omichi C, Lamp ST, Lin SF, Yang J, Baher A, Zhou S, Attin M, Lee MH, Karagueuzian HS, Kogan B, Qu Z, Garfinkel A, Chen PS, Weiss JN. Intracellular Ca dynamics in ventricular fibrillation. Am J Physiol Heart Circ Physiol 2004;286:H1836 –1844. 9. Chudin E, Goldhaber J, Garfinkel A, Weiss J, Kogan B. Intracellular Ca(2⫹) dynamics and the stability of ventricular tachycardia. Biophys J 1999;77:2930 –2941. 10. Luo CH, Rudy Y. A dynamic model of the cardiac ventricular action potential. 1. Simulations of ionic currents and concentration changes. Circ Res 1994;74:1071–1096. 11. Strang G. Numerical analysis. SIAM J 1969;5:526 –517. 12. Chudin E, Garfinkel A, Weiss J, Karplus W, Kogan B. Wave propagation in cardiac tissue and effects of intracellular calcium dynamics (computer simulation study). Prog Biophys Mol Biol 1998;69:225–236. 13. Kogan BY, Karplus WJ, Chudin EE. Heart fibrillation and parallel supercomputers. Paper presented at the Proceedings of International Conference on Information and Control, 1997, St. Petersburg, Russia. International Conference on Informatics and Control Proceedings 1997;3:857– 865. 14. Winfree AT. Heart muscle as a reaction-diffusion medium: the roles of electrical potential diffusion, activation front curvature, and anisotropy. Int J Bifurc Chaos 1997;7:487–526. 15. Chudin E. Instability of cardiac excitation wave propagation and intracellular calcium dynamic. PhD thesis. Los Angeles, CA: Biomathematics, University of California, Los Angeles, 1999. 16. Burashnikov A, Antzelevitch C. Reinduction of atrial fibrillation immediately after termination of the arrhythmia is mediated by late phase 3 early afterdepolarization-induced triggered activity. Circulation 2003;107:2355–2360. 17. Bers D. Excitation-contraction coupling and cardiac contractile force, 2nd ed. Boston: Kluwer Academic Publishers, 2001. 18. Davidenko JM, Pertsov AV, Salomonsz JR, Baxter W, Jalife J. Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 1992;355:349 –351.
Intracellular calcium cycling, early afterdepolarizations, and reentry in simulated long QT syndrome Ray Huffaker, BS,a Scott T. Lamp, BS,c James N. Weiss, MD,b,c Boris Kogan, PhDa a
From the Departments of Computer Science, Medicine (Cardiology), and c Physiology, University of California, Los Angeles, and the David Geffen School of Medicine at UCLA, Los Angeles, California. b
OBJECTIVES The purpose of this study was to investigate interactions between early afterdepolarizations (EADs) and reentry in long QT (LQT) syndromes. BACKGROUND EADs, a characteristic feature of congenital and acquired LQT syndromes, are classically bradycardia dependent. Mechanisms by which they promote tachyarrhythmias such as torsades de pointes and ventricular fibrillation are not fully understood. Recent evidence suggests that EADs also may occur at rapid heart rates as a sequela of spontaneous sarcoplasmic reticulum (SR) Ca2⫹ release related to intracellular Ca2⫹ overload. Here, we performed computer simulations to explore the arrhythmogenic consequences of this phenomenon. METHODS We used a modified version of the Luo-Rudy dynamic model in one-dimensional and two-dimensional cardiac tissue with the time-dependent K⫹ currents IKr or IKs reduced by 50% to simulate acquired and congenital LQT syndromes. RESULTS (1) Spontaneous SR Ca2⫹ release prolonged action potential duration but did not induce overt EADs unless K⫹ current density was reduced to simulate acquired and congenital LQT syndromes. (2) In simulated LQT syndromes, EADs were capable of both terminating and reinitiating one-dimensional reentry. (3) A similar phenomenon in simulated 2D tissue led to reinitiation of spiral wave reentry that otherwise would have self-terminated. (4) Reentry reinitiation occurred only when the L-type Ca2⫹ current and SR Cai cycling were potentiated to simulate moderate sympathetic stimulation, consistent with the known arrhythmogenic effects of sympathetic activation (and protection by betablockers) in LQT syndromes. CONCLUSIONS These computer simulations suggest that EADs related to spontaneous SR Ca2⫹ release can enhance arrhythmogenesis in LQT syndromes by reinitiating reentry. KEYWORDS Long QT syndrome; Early afterdepolarizations; Potassium currents; Arrhythmias; Computer simulations; Cardiac electrophysiology; Sudden cardiac death; Ventricular tachycardia; Genetic channelopathies; Intracellular calcium cycling © 2004 Heart Rhythm Society. All rights reserved.
Early afterdepolarizations (EADs) are thought to be highly arrhythmogenic, especially in the settings of congenital and acquired long QT (LQT) syndromes. Experimentally, however, EADs are most readily observed at slow
The study was supported in part by NIH/NHLBI SCOR in Sudden Cardiac Death P50 HL52319. Supercomputing resources were provided by National Energy Research Scientific Computing Center, Office of Energy Research of the US Department of Energy, under contract no. DEAC0376SF00098. Address reprint requests and correspondence: Dr. Boris Kogan, Department of Computer Science, UCLA, Los Angeles, CA 90095. E-mail address: [email protected]. (Received March 16, 2004; accepted May 17, 2004.)
heart rates and are suppressed by fast heart rates.1 The mechanisms by which this bradycardia-dependent phenomenon contributes to rapid arrhythmias such as torsades de pointes and ventricular fibrillation (VF) leading to sudden cardiac death are not fully understood. EADs occur when the balance of the total membrane current during repolarization reverses from outward to inward. At slow heart rates, the action potential (AP) prolongs physiologically. If outward currents are further reduced or inward currents are potentiated by drugs or genetic defects, reactivation of the L-type Ca2⫹ current during repolarization can lead to EADs and triggered activity.2,3 EADs generated by this mechanism tend to be suppressed by fast heart rates, and triggered
1547-5271/$ -see front matter © 2004 Heart Rhythm Society. All rights reserved. doi:10.1016/j.hrthm.2004.06.005
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442 activity usually self-terminates. In the setting of impaired repolarization, however, it is possible that EADs may be induced at rapid heart rates by different ionic mechanisms that cause a temporal imbalance between inward and outward currents during repolarization. For example, very rapid heart rates may cause intracellular Ca2⫹ concentration (Cai) overload, leading to spontaneous (i.e., nonvoltage-gated) Ca2⫹ release from the sarcoplasmic reticulum (SR). Classically, spontaneous SR Ca2⫹ release due to Cai overload is thought to cause delayed afterdepolarizations (DAD).1 However, if SR Ca2⫹ release were to occur before repolarization was complete, activation of Ca2⫹-sensitive inward currents such as Na/Ca exchange and the Ca2⫹-activated nonselective cation channel could cause EADs, leading secondarily to reactivation of the L-type Ca2⫹ current and triggered activity. This possibility is supported by both computer simulations4,5 and experimental evidence.6 In canine ventricular myocytes exposed to a drug blocking the rapid component of the time-dependent K⫹ current (IKr), EADs were induced by sudden rate acceleration6 and were eliminated when spontaneous SR Ca2⫹ release was inhibited by ryanodine. Following spontaneous termination of atrial fibrillation, EADs caused spontaneous reinitiation of fibrillation6 and were also eliminated by ryanodine. In canine ventricular myocytes, isoproterenol has been reported to induce ryanodine-sensitive EADs as well as DADs.7 Comparing simultaneous optical maps of Vm and Cai during VF in porcine right ventricle, Omichi et al8 found that elevations in Cai often preceded voltage waves, consistent with spontaneous SR Ca2⫹ release events. The goal of the present study was to use computer simulations to explore the potential arrhythmogenic consequences of EADs induced by spontaneous SR Ca2⫹ release at rapid heart rates in the setting of simulated LQT syndromes. We performed simulations of one-dimensional (1D) and twodimensional (2D) reentry using the model proposed by Chudin et al,9 which is a modified version of the Luo-Rudy dynamic model.10 The model of Chudin et al was further modified as follows. Time-dependent K⫹ currents IKr or IKs were reduced to simulate acquired and congenital LQT syndromes, and the L-type Ca2⫹ current (ICa(L)) and the SR Ca2⫹ uptake pump (Ip(Ca)) were amplified to simulate a modest increase in sympathetic tone. Using simulated 1D reentry (ring-shaped tissue), we show conditions and mechanisms by which EADs can either block or reinitiate reentry, in the absence of complicating factors such as wave curvature and tissue border effects. Using 2D tissue simulations, we show that EADs can regenerate spiral waves that otherwise would have self-terminated in simulated LQT2 and LQT1 syndromes. These findings shed new light on why EADs, classically considered a bradycardiadependent phenomenon, are potentially so arrhythmogenic during tachyarrhythmias such as torsades de pointes and VF.
Methods The mathematical model of wave propagation in 1D tissue is described by the following partial differential equation:
Heart Rhythm, Vol 1, No 4, October 2004 ⭸V ⭸t
⫽D
⭸ 2V ⭸x
2
⫹ (Im ⫹ Ist(s, t))
1
(1)
Cm
with appropriate initial and boundary conditions. Here V is membrane potential, D is a diffusion coefficient, Ist is the external stimulating current, and Cm is a membrane capacity. To make Equation 1 closed, it is necessary to add the system of nonlinear ordinary differential equations that describes the behavior of all components of Im and processes in intracellular compartments. For this purpose we have chosen the AP model proposed by Chudin et al,9 which represents a modification of the Luo-Rudy AP model.10 The initial conditions and parameters were chosen to be the same as those of Chudin et al, but with slight modifications to Cai,tot (from 18.61 to 21.01 M) and [Ca2⫹]jsr,tot (from 8.44 to 6.9 mM). To simulate congenital (LQT1 and LQT2) or acquired LQT syndromes, we inhibited either IKr or IKs by decreasing ¯ or G ¯ ) by 50%. In the LQT the maximum conductivity (G kr
ks
cases we also simulated a modest increase in sympathetic tone by amplifying ICa(L) and Ip(Ca) by increasing the pa¯ rameters ¯I and ¯I by 31.8% in the case of decreased G p(Ca)
Ca
kr
¯ . LQT simulations and 34.8% in the case of decreased G ks were compared to simulations run with normal model parameters. For computer simulation, a ring was formed from a line of equidistant cell models (nodes) interconnected by diffusion. After stimulation of one end of this line and formation of the propagated wave of full length, the line was closed numerically, satisfying the condition of periodicity, to form a ring. To change the ring length, the same procedure was repeated but with a different number of initial nodes. The 1D simulation has mainly methodologic aims. We can study the conditions and mechanisms by which EADs may block or reinitiate reentry while avoiding the complicating effects of wave curvature and border conditions that occur in 2D and 3D tissues. The 1D simulation also allows us to easily modify the pacing rate of cells in the tissue by changing the length of the ring. Simulating very large rings first and then shorter rings permits us to follow the development of EADs as the pacing rate increases. Computer simulations were performed on an IBM SP RS/6000 massively parallel computer at Lawrence Berkeley National Laboratory. The simulations used the operator splitting algorithm.11 According to this algorithm, the integration (1) is split into two parts: integration of diffusion ⭸V ⭸2V equation ⫽ D 2 and integration of the system of ⭸t ⭸x ⭸V 1 ordinary differential equations ⫽ 共Im ⫹ Ist兲 . These ⭸t Cm integrations were executed in consecutive time cycles of predetermined duration ⌬t ⫽ 0.1 ms. The specific features of this algorithm are described in Chudin et al12 and Kogan et al.13 The operator splitting algorithm allows integration
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of the system of nonlinear ordinary differential equations at any point in space independently and with variable time steps. In all computer simulations, the time step for ordinary differential equation solutions varied from 0.005 to 0.1 ms and the space step ⌬x was fixed at 0.016 cm. These choices provided stability and accuracy with the chosen numerical integration method and did not disturb the conditions of medium continuity.14 Specifically, an explicit Euler numerical method was used to solve the diffusion equation, for ⌬x2 which the stability condition was ⌬t ⱕ . For ⌬x ⫽ 0.016 2D cm and D ⫽ 1 cm2/s, the critical value was ⌬t ⫽ 0.128 ms. In our computations, this equation was solved twice in one computational cycle, with ⌬t ⫽ 0.05 ms. The problem of computational error arises for solutions to the system of nonlinear ordinary differential equations describing the fast membrane processes during depolarization phase of the AP. To decrease these errors, we used the Euler explicit method with variable time steps, which were changed depending on the rate of the most rapid variable. For the fast sodium channel gate variable, we replaced the Euler method with the so-called hybrid method.15 The continuity condition proposed by Winfree14 requires that the chosen space step ⌬x satisfies the inequality D ⬎ ⌬x2⁄Tr. Here Tr is an activation rising time measured in a cell placed in tissue. This time is longer than in an isolated cell due to the effect of the local currents and incomplete recovery of INa during wave circulation in a comparatively short ring length. Estimating Tr ⫽ 2.5 ms, we found that for D ⫽ 1 cm2/s and ⌬x ⫽ 0.016 cm, this inequality was well satisfied. Individual nodes were distinguished by their position in the initial line of cells before it was closed. Node i is defined as the ith node in the line, for 0 ⱕ i ⱕ N ⫺ 1, where N is total number of nodes in the ring. The position of node i in the closed ring (in centimeters), after k turns of the circulating wave, was defined as (k ⫺ 1)*L ⫹ i*⌬x, where L is the length of the ring in centimeters. Action potential duration (APD) was calculated for each AP of each node in the ring and graphed as a function of the position in the ring. For long rings, APD was calculated at 90% repolarization (APD90). For shorter rings, when propagation was less regular, APD was defined as the time above ⫺50 mV. The error associated with this calculation of APD was approximately 2 ms, which is the amount of time required for a potential change of 1 mV in a cell near the tail of an elongated wave. Conduction velocity () was calculated as a function of position in the ring. The formula for velocity defined the wavefront at time Ti as the node Xi, where the maximum value of Vi ⫺ Vi⫹1 occurred. Conduction velocity was found by measuring the average of the time elapsed while the wave propagated from node Xi⫺k to node Xi and the time elapsed while the wave propagated from node Xi to node Xi⫹k, for some suitably chosen k. The formula is:
443
Ui ⫽
冉
1 Xi ⫺ Xi⫺k 2 Ti ⫺ Ti⫺k
⫹
Xi⫹k ⫺ Xi Ti⫹k ⫺ Ti
冊
(2)
Here Ui is the conduction velocity measured at node i. The error from this formula, defining as the true value of the conduction velocity, is given by the formula9: 1⫹
ⱕUⱕ
2kS
1⫺
(3)
2kS
Here S is defined as the quotient of the space step ⌬x and the time step ⌬t. In our simulation, ⌬x was 0.016 cm, ⌬t was 0.1 ms, and k was 5, giving a maximum error of |U ⫺ | ⬍ 3.2 cm/s. For 2D tissue simulation, we also used the operator splitting algorithm. The time step and space step for the diffusion equation were chosen as ⌬t ⫽ 0.05 ms and ⌬x ⫽ 0.025 cm, which satisfy the stability conditions and condition of continuity.14 The space step ⌬x was increased from 1D to 2D in order to shorten the overall computation time for 2D simulation. In solving the ordinary differential equations, we used a variable time step of 0.005 to 0.1 ms. For verification of the overall solution, we performed the computer simulation with the space step decreased to 0.0125 cm and the time step decreased to 0.0005 ms. No tangible differences were observed in simulation results. The tissue was simulated as a square grid of 256 ⫻ 256 diffusively coupled nodes (D ⫽ 1 cm2/s) with no-flux boundary conditions. The space step ⌬x ⫽ 0.025 cm yields a tissue size of 6.4 ⫻ 6.4 cm2. This tissue size is very large compared to most physiologic values, but it allows us to avoid the effects of the border conditions. Individual nodes were distinguished by their position in the grid with an ordered pair (column no., row no.). The upper left node was denoted as (0,0) and the lower right node denoted as (255,255). The parallel processors were distributed over entire grid in stripes, so each processor integrates only in its own part of the grid. The communication between processors was implemented using Message-Passing Interface. The velocity of a rectilinear front propagation was 55 cm/s.
Results Interactions between Cai cycling, EADs, and reentry in a 1D ring of cells Figure 1 illustrates reentry in a 1D a ring consisting of 3,200 nodes (or 51.2 cm) for the case of normal cardiac cell model parameters (A) and the case in which IKr has been reduced by 50% to simulate LQT2 and ICa(L) and Ip(Ca) have been increased by 31.8% to simulate an increase in sympathetic tone (B). The reentry cycle length (CL) was 960 ms, which corresponds roughly to a normal human heart rate. In both cases, propagation was stationary, with APD and
444
Figure 1 Stationary propagation showing APD90 and conduction velocity () for a ring of length 51.2 cm (3,200 nodes), corresponding to a period of 960 ms. A: Normal model parame kr with mild sympaters. B: Simulated LQT2 syndrome (50% G thetic activation (131.8% ICa(L) and Ip(Ca)). Traces of membrane voltage (V) and [Ca2⫹]i are shown below for node 500. No spontaneous SR Ca release (Ispon) occurred.
remaining fairly constant in space over the five cycles illustrated. As expected, APD was longer for the simulated LQT2 case. At this CL, SR Ca2⫹ accumulation was insufficient to cause spontaneous SR Ca2⫹ release. Similar findings were obtained when IKs was reduced by 50% to simulate LQT1. Figure 2 shows that when the ring was shortened to 1,024 nodes (or 16.364 cm), nonstationary propagation was observed. In both cases, spatial irregularities in the distribution of APD and developed but were more pronounced in the simulated LQT2 case (Figure 2B). Accumulation of Cai was sufficient in both cases to induce spontaneous SR Ca2⫹ release (bottom traces), which occurred more rapidly in the simulated LQT2 case because of its longer baseline APD. Only the APs preceded by spontaneous SR Ca2⫹ were significantly prolonged. APD prolongation was more prominent in the simulated LQT2 case. Similar findings were obtained when IKs was reduced by 50% to simulate LQT1. Further shortening of the ring increased these propagation irregularities. Figure 3 shows that, for the control case, shortening the ring to 584 nodes (or 9.344 cm) led to immediate onset of APD and CV alternans even before spontaneous SR Ca2⫹ release occurred, due to transient behavior associated with the sudden increase in heart rate. Prior to spontaneous SR Ca2⫹ release, however, the spatial
Heart Rhythm, Vol 1, No 4, October 2004 profile of voltage and Cai in a given reentrant cycle remained fairly uniform (Figure 3B(i)). Once spontaneous SR Ca2⫹ release occurred, however, the spatial profile of Cai developed bumps, which caused similar perturbations in voltage (Figure 3B(ii-iii)). The bumps in voltage reflected the local variation in APD along the ring and were not due to overt EADs in the control case, as can be seen from the voltage trace in Figure 3A. (Cai and voltage bumps due to spontaneous SR Ca2⫹ release were observed at longer ring lengths but were not as pronounced.) At short ring lengths, localized APD prolongation due to spontaneous SR Ca2⫹ release was sufficient to induce head-tail interactions, which impeded conduction of the next cycle of the wave. Instances can be seen in the graph of (Figure 3A), where transiently and repeatedly falls to approximately one third of its original value. From this graph, it is impossible to determine whether propagation was merely slowed or stopped momentarily and then restarted. This is because the calculated is not instantaneous but is an average velocity along several cells. However, if we look at the sign of dV/dt along the wave to determine whether propagation had truly stopped (Figure 3C), we see that a few cells in the wavefront still had positive dV/dt, indicating that the wave never actually stopped. Figure 4 shows the simulated LQT2 case. As expected, less ring shortening was required to produce highly irregular propagation because APD was longer. In a ring consisting
Figure 2 Onset of nonstationary propagation showing APD90 and conduction velocity () when ring length was shortened to 16.384 cm (1,024 nodes), corresponding to a period of 295 ms. A: Normal model parameters. B: Modified parameters (as in Figure 1B) simulating LQT2 syndrome with mild sympathetic stimulation. Traces of V, [Ca2⫹]i, and Ispon are shown for node 500.
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445
propagation but does play a significant role, along with Ca2⫹-sensitive inward currents, in generating the EAD. When Iks was reduced by 50% to simulate LQT1, EADs causing wave termination were similarly observed but occurred at even longer ring lengths, beginning at 1,088 nodes (or 17.408 cm). Figure 5 shows the effects of shortening the ring even further in the simulated LQT2 case. For a shorter ring of 702 nodes (or 11.232 cm), the phenomenon of wave regeneration by EADs was observed, illustrated in the graphs of and APD in Figure 5A. The fourth episode of spontaneous SR Ca2⫹ release (bottom trace) causes an EAD, after which decreases close to 0 (Figure 5A). However, and APD subsequently return to normal, indicating that propagation has been restored. Figure 5C shows the spatial profile of voltage along the ring when this EAD occurred. Again we see the appearance of a bump on the tail with positive dV/dt (t ⫽ 2.0 s), indicating the EAD, which becomes larger, causing the next wavefront to slow. The original wave then
Figure 3 Irregular propagation for normal model parameters when ring length was further shortened to 9.344 cm (584 nodes), corresponding to a period of 194 ms. A: APD (calculated using threshold of ⫺50 mV) conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 500 are shown below. No EADs are present. B: Distribution of membrane voltage (V, green line) and [Ca2⫹]i (purple line) in space along the ring at the different times indicated. In (iii), Ispon causes a bump on the tail of the wave, which has negative dV/dt. C: Distribution of V in space when the wavefront closely approaches the waveback. Region of positive dV/dt is shown in red. The wavefront has positive dV/dt ⬍ 10. In a normal propagating wavefront, typically dV/dt ⬎⬎10.
of 768 nodes (or 12.288 cm), Figure 4B shows that the fourth episode of spontaneous SR Ca2⫹ release caused an EAD that terminated reentry. The mechanism is illustrated in the spatial voltage profile shown in Figure 4C. Bumps in the spatial profile of voltage were much larger than observed for the control case (Figure 4C vs Figure 3C) and often had dV/dt ⬎ 0, indicating an overt EAD. A bump on the waveback with dV/dt ⬎ 0, indicating an EAD, first appeared at t ⫽ 2.13 s. As the next wavefront approached this EAD, it slowed and then stopped completely. Although the EAD produced a new wavefront (t ⫽ 2.24 s) that propagated for some distance, it eventually failed due to residual refractoriness from the old wave (t ⫽ 2.43 s), and reentry terminated. Figure 4B also shows selected membrane currents during the EAD. At the time of the EAD, there was no significant local diffusive current (IN), indicating that the EAD was self-generated and not an artifact of passive diffusive currents from surrounding cells in different phases of repolarization. Sodium current (INa) also is absent, indicating that it plays no role in the generation of the EAD. The L-type Ca2⫹ channel current ICa,L is smaller than in the case of
Figure 4 Appearance of early afterdepolarizations (EAD) for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B), at ring length ⫽ 12.288 cm (768 nodes), corresponding to a period of 248 ms. A: APD90 and conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 80 are shown below. An EAD (arrow) in the V trace is produced by the spontaneous Ca release seen in the Ispon trace, which terminates propagation. B: Various membrane currents for node 80 when the EAD occurred. C: Distribution of V in space at different times just before the early afterdepolarization terminates propagation. Regions of positive dV/dt are shown in red.
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Heart Rhythm, Vol 1, No 4, October 2004 block, and reinitiation of reentry in the 1D ring-shaped tissue. EADs were never observed with normal IKr and IKs, whether ICa(L) or Ip(Ca) was normal or enhanced to simulate modest sympathetic stimulation. When either IKr or IKs was reduced by 50% to simulate LQT2 or LQT1, respectively, EADs occurred with both normal and enhanced ICa(L) and Ip(Ca). However, reinitiation of reentry by EADs was seen only with enhanced ICa(L) and Ip(Ca). This finding is interesting in view of the clinical observation that beta-blockers reduce episodes of torsades de pointes in patients with LQT syndrome.
Arrhythmogenic consequences in simulated 2D tissue
Figure 5 Regeneration of wave propagation by an early afterdepolarization (EAD) for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B), at a ring length of 11.232 cm (702 nodes), corresponding to a period of 185 ms. A: APD90 and conduction velocity (). Traces of V, [Ca2⫹]i, and Ispon for node 50 are shown below. B: Distribution of V in space at different times just preceding wave regeneration due to the EAD. Regions of positive dV/dt shown in red.
stops completely and all of its cells repolarize, while the bump produces a new propagating wave front (t ⫽ 2.1 s). Unlike the case where EAD causes propagation to fail, the old wave has sufficiently repolarized by the time the new wavefront reaches that part of the ring (t ⫽ 2.2 s). The new wavefront is able to regenerate propagation (t ⫽ 2.3 s) and perpetuate reentry (t ⫽ 2.4 s). Propagation of this EAD-generated wave subsequently was terminated (as seen in Figure 5A) by an EAD in another region, which did not cause regeneration. Similar findings were observed when Iks was reduced by 50% to simulate LQT1 at a ring length of 832 nodes (or 13.312 cm).
Role of increased sympathetic tone Table 1 summarizes the conditions under which spontaneous SR Ca2⫹ release induced EADs, conduction Table 1
1 2 3 4
In 1D reentry, the consequences of EADs terminating and reinitiating reentry are ambiguous with respect to arrhythmogenesis in two or three dimensions. In these higher dimensions, conduction block must be global in order to terminate reentry. If conduction block is localized then rather than terminating reentry, it can further destabilize reentry, causing a spiral wave to break up into a VF-like state. To examine how our findings in 1D reentry relate to propagation behavior in higher dimensions, we simulated LQT2 with increased sympathetic tone in 2D tissue. For a 6.4 cm ⫻ 6.4 cm tissue, the case of normal cardiac cell model parameters produced nonstationary sustained spiral wave reentry. For the simulated LQT2 case (Figure 6), however, the baseline APD prolongation converted sustained spiral wave reentry to nonsustained reentry, a stop of propagation, due to the mismatch between the now longer wavelength and the fixed tissue size (t ⫽ 2.800 s). If spontaneous SR Ca2⫹ release was disabled in the cell AP model at the time just before the propagation stopped (Figure 6B), EADs were less prominent and incapable of causing wave regeneration. Reentry always terminated when the meandering spiral wave wandered off the tissue. However, when spontaneous release was allowed (Figure 6A), the following scenario was observed: as the wave wandered to the border to self-extinguish (t ⫽ 2.751 s), spontaneous SR Ca2⫹ release in a region of the tissue generated an EAD (Figure 6C, in the vicinity of point b). When the region of tissue near the region of EAD became fully repolarized, the EAD began to propagate (t ⫽ 2.860 s) and, in the face of inhomogeneous repolarization, regenerated a new spiral wave (t ⫽ 2.935 s). Thus, EADs reinitiated reentry in a tissue in
The conditions of EAD and wave regeneration appearance in 1D ring-shaped tissue K⫹ conductivity
Presence of increased sympathetic tone in Ca2⫹ dynamics
EAD observed?
Wave regeneration observed?
Decreased G kr or G ks Decreased G kr or G ks Normal Normal
Present, increased ICa and Ip(Ca) Absent Present, increased ICa and Ip(Ca) Absent
Yes Yes No No
Yes No No No
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Figure 6 Spiral wave regeneration by an early afterdepolarization (EAD) caused by spontaneous Ca2⫹ release in simulated two-dimensional tissue, for modified parameters simulating LQT2 with mild sympathetic stimulation (as in Figure 1B). A: Snapshots of membrane voltage (V) at the times indicated. The spiral wave initiated at t ⫽ 0.273 s had fully depolarized the available tissue at t ⫽ 2.800 s, when an EAD emerged and regenerated propagation. B: Eliminating the EAD by disabling Ispon at time t ⫽ 2.725 s prevented spiral wave regeneration. C: Superimposed traces of membrane voltage (V) at every third node between sites a (77,2) and b (95,2) indicated in A. Tissue size is 256 ⫻ 256 nodes (6.4 cm ⫻ 6.4 cm).
which reentry otherwise would have self-terminated (Figure 6B, t ⫽ 2.935 ms). Similar findings were obtained when IKs was reduced by 50% to simulate LQT1.
Discussion Although EADs classically are thought to be promoted by slow heart rates, recent experimental evidence suggests that EADs also may be induced by rapid heart rates as a result of Cai overload and spontaneous SR Ca2⫹ release that occurs before repolarization is complete.6,16 This mechanism could explain how EADs, in addition to providing bradycardiadependent triggers involved in the initiation of reentry, continue to be proarrhythmic after reentry is induced, en-
447
hancing the likelihood of VF and sudden cardiac death in the setting of LQT syndrome. In the present study, we used 1D and 2D computer simulations to investigate the arrhythmogenic consequences of Cai-induced EADs at rapid heart rates. Our findings can be summarized as follows. (1) Spontaneous SR Ca2⫹ release prolonged APD under control conditions but did not induce overt EADs unless K⫹ current density was reduced to simulate acquired and congenital LQT syndromes. (2) In simulated LQT1 and LQT2 syndromes, EADs were capable of both terminating and reinitiating 1D reentry. (3) A similar phenomenon in simulated 2D tissue led to reinitiation of spiral wave reentry that otherwise would have self-terminated. (4) Reentry reinitiation occurred only when the L-type Ca2⫹ current and SR Cai cycling were potentiated to simulate moderate sympathetic stimulation, which is consistent with the known arrhythmogenic effects of sympathetic activation (and protection by beta-blockers) in LQT syndromes. This novel mechanism, which regenerates a spiral wave after the wavefront has stopped, may provide additional insight into why triggered activity occasionally becomes sustained and degenerates to VF in the settings of acquired and congenital LQT syndromes. Although we simulated LQT1 and LQT2 syndromes, spiral wave regeneration appears to be a general consequence of EADs related to spontaneous SR Ca2⫹ release, regardless of the specific underlying cellular mechanism. We previously found that EADs generated by increasing the Ca2⫹ sensitivity of Ins(Ca) produced the same effect.5 Thus, we speculate that the same phenomenon likely is observed when EADs are generated by other mechanisms, such as Na⫹ channel defects in LQT3 syndrome, and possibly when Cai-induces DADs.
EADs and intracellular Ca2ⴙ dynamics From a physiologic and clinical standpoint, it is appealing that Cai should play a role in the maintenance of triggered activity and/or reentry, because at fast heart rates characteristic of torsades de pointes, both diastolic and systolic Cai increase substantially relative to normal heart rates.9 The elevated Cai levels will potentiate Ca2⫹-sensitive inward currents and make them more effective at resisting repolarization. In addition, increasing Cai enhances Ca2⫹ uptake by the SR, leading to SR Ca2⫹ overload. The latter effect is well known to promote spontaneous Ca2⫹ release in a variety of experimental situations.17 Moreover, recent optical mapping experiments in which membrane voltage and Cai were mapped simultaneously provided evidence that nonvoltage-gated Cai release occurs in fibrillating porcine ventricle, which is consistent with spatially localized spontaneous SR Ca2⫹ release.8
Study limitations We performed our 1D and 2D tissue simulations in homogeneous isotropic tissue, whereas the real heart is three
448 dimensional and contains preexisting electrophysiologic and anatomic heterogeneities. Although there may be differential regional sensitivities to spontaneous SR Ca2⫹ release and EADs in the real heart, they likely do not affect our basic conclusions. In addition, nonuniform anisotropy and transmural action potential gradients may promote drift of scroll waves toward tissue borders, perhaps amplifying the likelihood of termination and regeneration by the mechanism we have proposed. Finally, direct experimental evidence that rapid heart rates induce spontaneous SR Ca2⫹ release before repolarization is complete, thus causing EADs, is limited. However, our findings may serve as an impetus for experimentalists to carefully investigate this issue, given its potential importance to arrhythmogenesis in LQT syndrome.
Conclusion The mechanism of spiral wave regeneration by Cai-induced EADs could be clinically important in prolonging the duration of reentry initially triggered by EADs, thereby enhancing the risk of VF and sudden cardiac death in patients with congenital or acquired LQT syndromes. Although there is no direct experimental evidence supporting this hypothesis in real hearts, this simulation study provides an incentive to look for this mechanism in the experimental setting. The regeneration of the spiral wave in the original or opposite direction5 might be the most recognizable indication, although the 3D nature of real cardiac tissue may make the experimental mapping data difficult to interpret unequivocally. Intermittent reversal of the spiral wave might contribute to the variable electrical QRS axis characteristic of torsades de pointes. Our simulation conditions may not be that far off from clinically relevant ones, because spiral waves that drift without breaking up have been documented experimentally,18 and acquired and congenital LQT syndromes are characterized by many nonsustained episodes of unstable VT that self-terminate. This implies that the prolonged wavelength from APD lengthening is at the edge of sustainable reentry, similar to the conditions under which spiral wave regeneration was observed here. However, the relevance of this mechanism of spiral wave regeneration to the real heart awaits simulation on realistic 3D geometries and experimental confirmation. Our findings may also be relevant to failed defibrillation shocks, in which postshock EADs due to spontaneous SR Ca2⫹ release might reinitiate reentry.
Heart Rhythm, Vol 1, No 4, October 2004
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