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1Department of Computer Science and 2Cardiovascular Research Laboratory, Department of Medicine (Cardiology), David Geffen School of Medicine at University of California, Los Angeles, Los Angeles, California
Submitted 29 November 2006 ; accepted in final form 15 February 2007
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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Torsades des pointes; spontaneous sarcoplasmic reticulum calcium release; ventricular tachycardia; mathematical modeling; parallel computing
Recent experimental findings [see Volders et al. (33) for review] suggest that repolarization reserve can also become reduced by rapid changes in heart rate or in response to 
-adrenergic stimulation. In this setting, EADs can occur in association with Ca accumulation in the sarcoplasmic reticulum (SR) and subsequent spontaneous SR Ca release. For example, enhancing SR Ca loading by -adrenergic stimulation elicited both delayed afterdepolarizations (DADs) and EADs (20, 32), whereas blockade of Ca release channels or the reduction of SR calcium load prevented their induction (29). Recent simultaneous optical recordings of membrane voltage and intracellular Ca in intact hearts showed that spontaneous SR Ca release preceded drug-induced EADs and Torsades des pointes (3). These findings suggest that, during established reentry, rapid rates that cause SR Ca overload leading to spontaneous SR Ca release may promote EADs, which influence the subsequent wave propagation.
We previously used simulations to show that such EADs potentiated by spontaneous SR Ca release during spiral wave reentry could reinitiate spiral waves, which would otherwise have spontaneously self-terminated by wandering off the tissue (12, 15). We proposed that this mechanism might be important in perpetuating Torsades des pointes. The regeneration of wave propagation was obtained under conditions similar to the long QT syndromes LQT1 and LQT2 (12), as well as under more general, yet currently less physiologically relevant, conditions by increasing some Ca-sensitive cell currents (15). Here, we have systematically explored the effects of EADs on wave propagation in tissue during reentry to address the following questions, which were not examined in our previous studies (12, 15) or in current literature. How do enough cells synchronously develop spontaneous SR Ca release to generate EADs and overcome the source-sink mismatch needed to trigger a new propagating wave? When EADs are successful in initiating a new wave, what modes of propagation and how many new waves can be observed? When EADs occur during reentry, how do they affect the meander pattern of rotors and break up into fibrillation? Do the same phenomena occur in three-dimensional (3D) tissue, where potential source-sink mismatches suppressing EAD formation and propagation are more pronounced?
To address these questions, we simulated one-dimensional (1D), two-dimensional (2D), and 3D cardiac tissue using a modified version of the Luo-Rudy dynamic ventricular AP model (5), tuned to elicit spontaneous SR Ca release at rapid heart rates. When reentrant wave propagation subjected this model to such rapid heart rates, and ionic currents were altered to further reduce repolarization reserve, the spontaneous SR Ca release could induce EADs. Nonuniformity in spontaneous SR Ca release throughout the tissue created regions of EADs next to regions without EADs. In 1D reentry, this allowed EADs to terminate reentry by prolonging local repolarization and initiate new wavefronts by propagating into adjacent repolarized tissue. EADs could initiate single waves in the same or opposite direction or double waves in the same or opposite directions. In 2D and 3D reentry, intermittent EADs caused sudden prolongation of local repolarization, resulting in rapid meander of rotors to new positions in the tissue, rapidly shifting the electrical axis as seen in Torsades des pointes. When this sudden meander pushed the rotor off the tissue, EADs often reinitiated new rotors, which could rotate in the same or opposite direction. These new findings recapitulate in silico many of the clinical features of Torsades des pointes.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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 | (1) |
is the Laplace operator in the relevant number of space dimensions, Iion is the net membrane ionic current, Ist is the external stimulating current, and Cm is membrane capacitance per unit membrane surface area. Equation 1 reflects a macroscopic approach, describing cardiac tissue as a continuous excitable medium (syncitium; see Ref. 18). A microscopic approach (24) to tissue simulation provides more realistic anatomic structure but is computationally intractable for large tissues. To make Eq. 1 closed, it is necessary to add the system of nonlinear ordinary differential equations (ODEs) that describes the behavior of all components of Iion and relevant processes in intracellular compartments. For this purpose, we have chosen the AP model proposed by Luo-Rudy (17) with the modifications of Ca dynamics introduced by Chudin et al. (5). The Chudin modifications introduced to the model several major properties observed in physiological experiments, including the graded response of Ca-induced Ca release (CICR) to the ICa,L (16), prevention of complete SR depletion during normal CICR (1), and spontaneous SR Ca release (implemented in the model by activations of Jspon, a Ca ion flux independent of CICR) dependent on the Ca concentration in both the SR and myoplasm (2, 25).
The ability to generate EADs requires a modification of the normal cell model to alter the balance of repolarization currents, favoring inward membrane currents and thereby reducing repolarization reserve. In our simulations, this requirement has been satisfied by decreasing the coefficient Km,ns(Ca) from the normal value of 1.2 to 0.9 µM to increase the sensitivity of the Ca-activated nonselective cation current (Ins(Ca)) to intracellular Ca. Although there is no physiological data showing the possible variation of Km,ns(Ca), altering K and Na currents instead could potentially contaminate spontaneous SR Ca release-induced EADs with bradycardia-induced EADs, and hence those currents were left unaltered. All other parameter values and initial conditions were the same as in Chudin et al. (5).
Simulations were performed on massively parallel Opteron and IBM SP clusters at Lawrence Berkeley National Laboratory. Communication between processors was implemented using message passing interface (MPI). Parallel computation used the operator-splitting algorithm (26). The operator-splitting algorithm allows integration of the system of nonlinear ODEs at any point in space independently and with a variable time step. A brief description of this algorithm and the considerations in choosing an appropriate time step (t) and space step (h) for numerical integration are given in APPENDIX A.
1D tissue was simulated as a ring of diffusively coupled, uniform, equidistant nodes (cell models). A ring of N nodes was formed from an initial open-ended line of N nodes. The first 20 nodes of the line were stimulated. After formation of the fully propagated wave, the line was closed numerically (satisfying the condition of periodicity) to form a ring. The ring length, L, was constant within each experiment (L = hN cm, h = 0.016 cm) but varied between experiments. Rings of different lengths were constructed by starting with a different number of nodes in the initial line and repeating the same procedure of ring formation. Individual nodes were distinguished by their position x in the initial line of nodes, 0 
x N – 1. The position of node x in the closed ring (in cm), after k turns of the circulating wave, is defined as (kN + x)h. The distribution of this tissue (as well as the 2D and 3D tissues) among the parallel processors is discussed in APPENDIX B.
2D tissue was simulated as a square grid of 256 x 256 diffusively coupled, uniform, equidistant nodes with no-flux boundary conditions. The space step h = 0.025 cm yields a tissue size of 6.4 x 6.4 cm2. This tissue size was chosen because it was large enough to allow self-sustaining stationary spiral waves (before intracellular Ca accumulation) in the square-form grid. Individual nodes were distinguished by their position in the grid with ordered pairs (x = column no., y = row no.) and the origin (0,0) located in the lower left corner of the tissue.
A spiral wave in the 2D tissue was induced using an S1-S2 protocol. The S1 stimulation was applied to a strip 20 nodes wide on the left edge of the tissue. This produced a rectilinear wave traveling from left to right. A premature S2 beat 30 nodes wide was then placed near the tail of the rectilinear wave. A 20-node gap was left at the bottom of the premature beat, allowing the new wavefront to turn counterclockwise into repolarized tissue in the wake of the rectilinear wave. This formed a counterclockwise-rotating spiral wave.
3D tissue was simulated as a right square prism grid of 256 x 256 x sizez diffusively coupled, uniform, equidistant nodes with no-flux boundary conditions. Two 3D experiments were performed, with thicknesses sizez = 30 and sizez = 64 nodes. The space step h = 0.025 cm yields a tissue size of 6.4 x 6.4 x 0.75 cm3 for sizez = 30 and 6.4 x 6.4 x 1.6 cm3 for sizez = 64. Individual nodes were distinguished by their position in the grid with ordered pairs (x = column no., y = row no., z = layer no.; Fig. 1A). The origin (0,0,0) is located in the bottom left, back corner of the tissue so x coordinates run from left to right, y coordinates from bottom to top, and z coordinates from back to front.
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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20 cm), we observed stationary propagation. For intermediate ring lengths (L 15–20 cm), we observed unstable irregular propagation, characterized by variability in the local period of excitation. At short ring lengths (L 9.728–15 cm), the rate of excitation was rapid enough to intermittently cause spontaneous late-diastolic SR Ca release (because of activation of Jspon) that, if sufficiently large, induced an EAD on the ensuing AP. This is shown in Fig. 2B, inset, which demonstrates that the EAD did not coincide temporally with the Jspon peak. Rather, spontaneous late-diastolic SR Ca release triggered the EAD during the ensuing AP, consistent with experimental observations of DADs preceding the upstroke of APs exhibiting EADs (20, 32). The spontaneous late diastolic Ca release summated with the ICa,L-induced SR Ca release produced by the ensuing AP, augmenting the Ca transient amplitude. The resultant larger Ca transient enhanced Ca-sensitive inward currents, specifically the Na/Ca exchange current (INaCa) and Ins(Ca), during the AP plateau phase. The enhanced inward currents decreased repolarization reserve and thereby established a more tenuous balance of repolarizing currents, such that window ICa,L reactivation was able to generate an EAD. If Jspon in the cell model was inactivated, EADs did not develop. Although every EAD was preceded by a corresponding late diastolic Jspon peak, not every Jspon peak was followed by an EAD (Fig. 2B). For very short ring lengths (L < 9.728 cm), the ring was not sufficiently long to sustain reentry. Here, we consider only those ring lengths for which Ca release-induced EADs occurred (L 9.728–15 cm).
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From 2–5 s, spontaneous diastolic SR Ca release resulting from Jspon activation became large enough to induce EADs in various regions of the ring. These EADs could both terminate and regenerate wave propagation. EADs are seen in voltage traces from single nodes (Fig. 2B) and appear as bumps with positive dV/dt on the tail of the voltage wave, indicated in the spatial distribution of voltage (Fig. 2A) by the thick lines. EADs appeared slightly earlier in shorter rings. Multiple EADs were also observed during this time (Fig. 2B).
After sufficient time (5 s), the nodes in the ring entered a repolarization failure regime where EAD-like oscillations occurred indefinitely and voltage never returned to the rest potential (Fig. 2B), as has been observed experimentally in intact ventricular tissue (3, 35) and in other computer models (19). The oscillations gradually dampened over time until the entire range of voltage across the tissue was within a few millivolts (t = 8 s; Fig. 2A). This repolarization failure regime was avoided only if EADs terminated wave propagation before sustained depolarization developed. The repolarization failure regime, once established, did not rely on sustained Jspon. The maintained oscillations were the result of the balance between Ins(Ca), INaCa, and ICa,L even after Jspon ceased (Fig. 2B).
The evolution of EADs following the initiation of reentry is shown in more detail in Fig. 3. The peak value of Jspon is plotted at each node in the 1D ring during each turn of reentry, beginning with the sixth turn at which Jspon was first activated. Thick lines indicate Jspon activations that caused EADs (+dV/dt during repolarization) to appear in the AP traces of the corresponding nodes. For the sixth turn (orange), Jspon became activated at every node, but to a variable degree because of subtle regional differences in pacing history as a result of variability in the period of excitation during reentry. This variable degree of Jspon activation unloaded the SR Ca content nonuniformly so that on the seventh turn (yellow) Jspon activated more strongly in the central region where Jspon was previously small but did not activate at the two ends where Jspon had been large on the previous beat. The amplitude of Jspon was still too small to induce an EAD, but Jspon nevertheless prolonged repolarization nonuniformly, thereby causing head-tail interactions to amplify the variability in the period of excitation along the ring. The eighth turn (green) then produced a mirror image of Jspon activation that was slightly more fractionated, and so forth. On the ninth turn (cyan), Jspon reached sufficient amplitude (>15 mM/ms) to induce EADs in the region with large Jspon. These EADs failed to propagate retrogradely (in the opposite direction of the original reentrant wave) but succeeded in propagating antegradely (in the same direction as the original reentrant wave). Successful antegrade propagation is indicated by the downward projection of the thick line, showing that adjacent nodes, whose intrinisic Jspon was clearly below the threshold to generate an EAD, still exhibited EADs. Their EADs were generated by electrotonic current flow from the intrinsic EADs of nearby nodes with large Jspon.
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EADs affected wave propagation in several ways. A region of EADs could stop a propagating wave if the region was sufficiently large and arose just ahead of the wavefront, or it could regenerate wave propagation if it was sufficiently large and arose adjacent to a region of repolarized tissue into which the new wave could propagate. Alternatively, a region of EADs could occur in such a way that propagation was terminated and not regenerated. For example, a region of EADs could arise that prolonged refractoriness enough to block reentry of the original wave but was not large enough to trigger a new wave. Wave regeneration could be prevented manually by inactivating Jspon in all cell models just after the original wave stopped. Several different regions of EADs could also arise in the ring at the same time (t = 5 s; Fig. 2A).
Previous simulation experiments (12) described EAD-induced regeneration of wave propagation antegradely. This regeneration of a new antegrade wave was seen in the present study as well. We also observed the existence of the following new modes of EAD-induced wave regeneration: a new wave traveling retrogradely; two new waves traveling in the same direction, both antegrade (Fig. 4A); and (d) two new waves traveling in opposite directions, antegrade and retrograde (Fig. 4B). We also found a special case of the last mode, where two waves traveling in opposite directions arose from a single region of EADs, which then propagated in both directions.
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We performed 20 simulations with ring lengths within the range where EADs occurred (9.728–15 cm). Of these simulations (n = 20), 50% (n = 10) ended with all nodes in the repolarization failure regime, whereas the other 50% (n = 10) ended with all nodes returning to resting potential, since wave propagation self-terminated before the repolarization failure regime could develop. In the former group, EAD-induced wave regeneration occurred an average of 6.5 times before repolarization failure developed, whereas in the latter group, wave regeneration occurred an average of only 3.2 times before termination. In 10% of the simulations (n = 2), the first incidence of EAD terminated propagation, and no wave regeneration occurred. Antegrade regeneration occurred in the other 90% of the simulations (n = 18) at an average of 4.1 times per simulation. Regeneration of two waves in opposite directions occurred in 60% of the simulations (n = 12) but only occurred more than once in two of those simulations. Retrograde regeneration and regeneration of two waves in the same direction each occurred in 15% of the simulations (n = 3), and neither occurred more than once in the same simulation. In 5% of the simulations (n = 1), all four modes of wave regeneration occurred. Antegrade regeneration was the most common, since an EAD on the tail of a propagating wave will naturally face repolarizing tissue in the antegrade direction and be blocked retrogradely by the next incoming wavefront. Two waves in opposite directions was the second most common mode of regeneration, since it required only a single region of repolarized tissue into which two simultaneous EADs could propagate from either direction. Two waves in the same direction was relatively rare, since it required two simultaneous regions of repolarized tissue and two simultaneous EADs. Retrograde regeneration was relatively rare, since its occurrence required two simultaneous EADs, one of which must stop the incoming wavefront but not propagate antegradely itself.
Spiral wave propagation and EADs in 2D tissue. In previous simulations (12, 15), we found that EADs induced by spontaneous SR Ca release could cause regeneration of spiral waves. After a counterclockwise-rotating spiral wave meandered off the tissue, a region of EADs subsequently regenerated a new wave in the counterclockwise direction (12). This is analogous to the 1D case in which a new wave was regenerated in the antegrade direction. We also found that a region of EADs could regenerate a new spiral wave in the opposite direction (15), analogous to the 1D case, in which a new wave was regenerated in the retrograde direction. In the present study, we observed new effects of EADs on spiral wave reentry and more rigorously examined the mechanism behind them.
Figure 5 shows that the time dependency of wave propagation and EAD appearance in 2D tissue is qualitatively similar to the 1D ring. After a short transient period following spiral wave initiation, we observed stationary spiral wave propagation from 0.66 to 1.38 s (Figs. 5Ai and 5Bi). Between 1.38 and 2.25 s, no frank EADs were observed, but irregularities appeared in the wavefront (Fig. 5Aii), indicating inhomogeneous spatial distribution of SR Ca accumulation and Jspon along the tissue. The spiral wave also began to meander during this time (Fig. 5Bii).
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When the spiral wave was forced off the tissue by the peninsula of EADs, the rest of the tissue, excluding the peninsula, repolarized. However, the nodes in the peninsula continued to exhibit multiple EADs (Fig. 6E). If Jspon was disabled in the cell model at this time to prevent multiple EADs from continuing in the peninsula (Fig. 6E), the tissue fully repolarized (Fig. 6A). If Jspon remained enabled, however, the continuing multiple EADs in the peninsula were able to regenerate wave propagation. As shown in Fig. 6B, a region near the bottom of the peninsula (t = 3.91 s) generated multiple EADs. Because this region was adjacent to fully repolarized tissue, the voltage gradient was sufficient to create a new wavefront (Fig. 6B, t = 3.93 s). This peninsula region of maintained EADs produced a wave that propagated both clockwise and counterclockwise and attempted to form figure-eight reentry (t = 3.96; Fig. 6B). This event is analogous to the 1D ring case in which a single region of EADs caused two waves in opposite directions in the 1D ring.
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After these two waves, the bulk of the tissue repolarized (t = 4.50 s; Fig. 6D), but no regions of EADs in the peninsula, remained strong enough to form new waves (t = 4.77 s; Fig. 6D). The peninsula shrunk, and the entire tissue returned to the resting potential, terminating propagation (4.89 s; Fig. 6D). There are two possible explanations for the disappearance of the peninsula where multiple EADs were occurring. The multiple EADs may have finally ended on their own, or, alternatively, the voltage sink from the repolarized tissue around the peninsula gradually eroded the edges of the peninsula by electrotonically pulling those nodes with ongoing multiple EADs back down to rest potential.
These new findings demonstrate that, during spiral wave reentry, spontaneous diastolic SR Ca release can synchronize over a sufficiently large area in 2D tissue to generate EADs, which then cause the spiral wave to rapidly meander to a new location in the tissue. This rapid meander causes the electrical axis to shift suddenly, in the characteristic feature of Torsades des pointes, as seen in the simulated electrocardiogram shown in Fig. 6F. At the new location, if EADs are sufficiently strong, they can perpetuate sustained membrane potential oscillations and repolarization failure (3, 35), which acts as a source for generating new waves focally.
Scroll wave propagation and EADs in 3D tissue. In our previous studies, we did not examine whether EADs were capable of producing comparable or novel effects on reentry in 3D tissue, in which the source-sink mismatches affecting the ability of EADs to propagate may be larger. Accordingly, we investigated homogeneous 3D tissue with two different thicknesses: sizez = 30 nodes (0.75 cm) and sizez = 64 nodes (1.6 cm). For the thinner tissue (sizez = 30), we observed no scroll wave instabilities arising in the z dimension (Fig. 7). Every layer of the tissue appeared to behave identically. EAD appearance, wave termination, and wave regeneration occurred synchronously in every layer, equivalent to a 2D tissue and qualitatively equivalent to our 2D simulation (compare Fig. 7A with Fig. 5A and Fig. 7, B–D, with Fig. 6, B–D). As in 1D and 2D experiments, wave regeneration could be prevented by inactivating Jspon.
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24 nodes (near a, observe the 6 overlapping voltage traces). However, the voltage gradient was not large enough to cause a new wave because the nearby nodes (near b) were still repolarizing. In the back layer, EADs occurred over roughly the same region of 24 nodes (near a'). However, they appear staggered so that the peak EAD voltage of 0 mV was maintained somewhere in the region for nearly 50 ms. The EAD-induced depolarization persisted long enough for the nearby nodes (near b') to fully repolarize, and the resultant gradient was large enough to create a new wave. These findings demonstrate that, in 3D tissue, the timing, as well as the amplitude, of regional EADs plays a key role in whether they elicit a propagated response. |
DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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Localized desynchronization of spontaneous diastolic SR Ca release in adjacent cells is required for EAD propagation in tissue. Although EADs are readily induced in isolated myocytes, the mechanism by which cellular EADs overcome source-sink mismatches to trigger propagated responses and arrhythmias at the tissue level is poorly understood. In tissue, EADs can only emerge from clusters of neighboring cells. Presumably, if only a single cell were on the verge of undergoing an EAD, the prevailing inward current would not be large enough to counteract the diffusive current from its repolarizing neighbors to generate an EAD because of the source-sink mismatch. This has been demonstrated experimentally with EADs in an isolated myocyte (exposed to inhibition of IKr, the rapid component of the delayed rectifier K current) suppressed by electrical coupling to a second, normal myocyte (37). Thus some minimum size of a group of cells must synchronously develop an imbalance in repolarizing currents for an EAD to emerge. In our tissue simulations, this minimum size necessary for an EAD was larger than the size of a single computational node. Because our simulated tissue was homogeneous, the initial spontaneous diastolic SR Ca release occurred synchronously (i.e., during the same turn of reentry) at every point in the tissue, since the pacing history following initiation of reentry was relatively uniform up to that point. However, because of subtle differences in the period of excitation at different points in the tissue, the amplitude of Jspon was nonuniform (e.g., see Fig. 3, turn 6), leading to larger spontaneous diastolic SR Ca release in some regions compared with others. This affected repolarization reserve nonuniformly, further potentiating local variation in the period of excitation during reentry and desynchronizing the regional development of EADs. This desynchronization produced regions with large EADs adjacent to regions with no EADs, creating a heterogenous substrate that allowed EADs to initiate propagation and generate new waves in antegrade and/or retrograde directions. Regenerated waves further desynchronized repolarization reserve, creating a complex spatial pattern. This scenario of local desynchronization of EADs leading to the generation of new waves was most clearly evident in the 1D ring (Fig. 3) but is also likely to underlie the mechanism by which even greater source-sink mismatches are overcome in 2D and 3D tissue to allow EADs to initiate propagation.
Although synchronous spontaneous diastolic SR Ca release among adjacent cells is required to generate an EAD in tissue, our findings demonstrate that other factors, such as local gradients in repolarization, play a critical role in whether the EAD can propagate to generate a new wave. For example, consider the 3D simulation where EADs regenerated a wave in the back layer but not the front (Fig. 8). In both layers, the region of EADs was roughly the same size, and the EADs at the nodal level had roughly the same duration. In the back layer, the EADs were staggered in time, allowing the region of EADs to persist while nearby tissue repolarized. In the front layer, however, where the EADs were more closely synchronized, the region of EADs disappeared before the nearby tissue could repolarize. Thus greater synchronization does not necessarily translate into greater likelihood of propagation. Localized repolarization also plays a key role.
In contrast to our simulated tissue, real cardiac tissue is inherently heterogeneous with respect to both repolarization reserve and intracellular Ca cycling properties (13, 21). If the threshold for spontaneous SR Ca release were markedly heterogeneous on a cell-to-cell basis, then an initially synchronous spontaneous diastolic SR Ca release event might be difficult to achieve, preventing EADs from developing. Even so, if the threshold of all cells in the tissue were exceeded, the same scenario outlined for homogeneous tissue would likely occur. Moreover, if the threshold for spontaneous SR Ca release varies regionally (e.g., base to apex or transmurally in the endo-, mid-, and epicardial layers), this would promote the development of interfacing regions with and without EADs, facilitating EAD propagation and wave regeneration. These issues will be important to study in future simulations.
EADs create rapid shifts in electrical axis as seen in Torsades des pointes by promoting meander, self-termination, and reinitiation of reentry. In our model, when a rotor was initiated in 2D or 3D tissue, its core remained relatively stable until a region of EADs following upon a spontaneous SR Ca release event developed. By prolonging refractoriness, the region of EADs displaced the tip of the wave off its prior course so that it rapidly meandered to another area of the tissue. As shown by the simulated electrocardiogram in Fig. 6F, this feature provides a possible explanation for the rapidly shifting electrical axis that is characteristic of Torsades des pointes.
If the displacement of the rotor pushed it to a tissue border, then the rotor self-terminated, as frequently occurs with Torsades des pointes. However, if EADs persisted over a large enough area, they subsequently reinitiated new waves, perpetuating the arrhythmia. Perpetuation of the arrhythmia could be achieved by two mechanisms. The new waves could generate new rotors, which could sustain reentry if the tissue dimensions were large enough. Alternatively, EADs themselves could become sustained and induce local repolarization failure, as shown in Fig. 5D. These sustained EADs could generate new focal waves, even if these waves fail to generate new rotors or the new rotors self-terminate. Sustained repolarization failure generated by repetitive EADs, as seen in Fig. 2B, has been reported previously in real cardiac tissue (3, 35). Thus the rapidly shifting electrical axis could be caused either by the rapid EAD-induced meander of rotors or by shifting foci of sustained EADs. Both mechanisms have been proposed to underlie the rapidly shifting electrical axis during Torsades des pointes, and experimental evidence favoring both mechanisms has been reported (3, 10). Our simulations suggest that both patterns can result from EADs.
EADs and degeneration of Torsades des pointes into ventricular fibrillation. Our 1D ring simulations identified several new observed modes of EAD-induced regeneration of waves. In addition to regenerating a new antegrade wave, EADs could induce a new retrograde wave, two new waves traveling in the same direction (Fig. 4A), or two new waves traveling in opposite directions (Fig. 4B). In each case, the original wave first collided with a region of EADs and terminated. The mode of wave regeneration depended on the distribution of EADs and repolarized tissue along the ring. If there happened to be two regions of EADs that arose simultaneously and bordered repolarized tissue, then two waves were regenerated (Fig. 4). It is theoretically possible for more than two waves to be regenerated simultaneously. However, more than two simultaneous regions of EADs bordering repolarized tissue is very unlikely, especially at such short ring lengths. If the ring were lengthened to create more room for additional EADs, the corresponding decrease in the rate of stimulation of the original wave would prevent EADs from occurring at all.
In 2D tissue, wave regeneration is more complicated because of wavefront curvature and larger effects of electrotonic currents. Previous 2D simulation results showed that EADs can stop and regenerate a spiral wave in the same (12, 15) or opposite (15) direction. In those studies, regeneration occurred only from the edge of the tissue, and only a single wave was regenerated at a time. In this study, we present the following new 2D tissue findings: regeneration of the wave from the middle of the tissue and regeneration of multiple new waves simultaneously (Fig. 6). The region from which a new wave is regenerated depends on the distribution of EADs and repolarization across the tissue. Wherever a sufficiently large region of EADs interfaces with repolarized tissue, a new wavefront will form. As observed in previous experiments, this is most likely to occur near the border of the tissue where there is less of an electrical sink to affect the region of EADs. The observed regeneration from the middle, however, provides definitive proof that the regeneration is not an artifact of the border conditions. The regeneration of multiple new waves simultaneously in the 2D tissue may provide a mechanism by which Torsades des pointes may suddenly degenerate into ventricular fibrillation, as seen clinically.
In 3D tissue, EAD formation and propagation is yet more complex because of even greater electrotonic current effects, rotor filament twist, and other instabilities introduced by the third dimension. For 3D tissue, Figs. 7 and 8 shows a breakdown of scroll wave propagation into multiple waves, similar to the 2D results (Fig. 6). This is not a trivial result, since we did not simply stack multiple copies of our 2D spiral wave simulations into a larger 3D tissue. In that case, each 2D layer in the 3D tissue would be identical, so diffusion between the layers would be zero, and they would all behave exactly like the 2D tissue. Instead, we delivered the S1 beat to only the top layer to introduce a slight perturbation in the initial conditions of the spiral waves in each layer of the 3D tissue that join to form the scroll wave. Thus our 3D tissue simulations show the robustness of the EAD-induced wave regeneration mechanism.
We characterized two different tissue thicknesses, since linear vortex filaments in homogeneous excitable media are unstable only beyond some critical thickness determined by the intrinsic dynamics of the system (22). When tissue thickness is less than the critical thickness, the filaments become synchronized in the z dimension, and every 2D cross section of the 3D tissue along the x-y plane is identical. The synchronization between layers is maintained regardless of how chaotic the 2D wave propagation becomes. If tissue thickness is greater than the critical thickness, however, filaments become unstable, and small initial perturbations in the z dimension can lead to irregular z-dimensional propagation. In the present study, the thin tissue sizez = 30 nodes (0.75 cm) was less than the critical thickness for filament instability, and the simulations were equivalent to a 2D case (notice in Fig. 7 that all layers are identical). In this case, the 3D simulation mirrors the 2D results (compare Fig. 7A with Fig. 5A, and Fig. 7, B–D, with Fig. 6, B–D). Increasing the tissue thickness to sizez = 64 nodes (1.6 cm) satisfied the critical thickness requirement for instability in the z dimension, demonstrated by a wave filament that does not traverse the entire z dimension of the tissue (t = 4.254 s; Fig. 8C). The major result is that the appearance of EADs and regeneration of wavefronts did not occur identically in each layer of the tissue. For example, in Fig. 8, we observed a new wave originating from the back layer but not the front layer. The wave then propagated via diffusion through the z dimension.
In summary, despite the complexity of 3D tissue and its greater potential for source-sink mismatches, spontaneous diastolic SR Ca release was able to occur synchronously over a large enough region of tissue to allow EADs to emerge and trigger new waves. Although we have so far characterized only the homogeneous 3D tissue case, the ability of waves to originate in a single layer and then propagate to other layers is likely to be very important in real ventricular muscle with transmural and/or base-to-apex AP and intracellular Ca cycling heterogeneity (13, 21), especially if the propensity to develop spontaneous SR release differs between these layers.
Limitations. In extrapolating these simulation results to EAD-induced arrhythmias in real cardiac tissue, several important limitations should be recognized. We did not consider bradycardia-dependent EADs, which potentially could interact with tachycardia-dependent EADs to produce even more novel effects. However, we felt it important to understand theoretically the effects of EADs occurring during reentry in their own right, before considering the combination. For this reason, we did not directly simulate long QT syndromes caused by decreased K currents or incomplete Na channel inactivation, even though these cases are very clinically relevant. Nevertheless, our previous work in 1D and 2D tissue (12, 15) demonstrated that EADs and wave regeneration resulting from increased Ca sensitivity of Ins(Ca) also occurred when EADs arose from simulated LQT1 (decreased IKs, the slow component of the delayed rectifier K current) or LQT2 (decreased IKr). Thus we anticipate that the present findings will be generalizable to these other settings, although this needs to be confirmed in future work.
Spontaneous SR Ca release in our model was also treated phenomenologically, whereas the physiological basis of this event is much more complex, involving regenerative CICR from a complex SR network spatially distributed throughout the cytoplasm of the cell. Although spatially detailed models are under development (11), they are computationally intractable for tissue-level simulations at the present time. Contraction-excitation feedback could also play an important role in modulating EAD responses but was not incorporated into the model.
Finally, we studied only homogeneous tissue and did not include known transmural or base-to-apex tissue heterogeneities in the AP or intracellular Ca cycling that are present in real tissue (13, 21). Nor did we include heterogeneous cell-to-cell electrical coupling, which directly influences source-sink relationships that are critical in allowing EADs to emerge and propagate. How tissue heterogeneities will influence the findings is an important question, but a key finding of our study is that desynchronization of repolarization reserve sufficient to permit EAD emergence and propagation in tissue can occur even with normal cell-to-cell coupling. Thus, as a first step, homogeneous tissue provides the ability to analyze how intrinsic dynamics of cell properties allow EADs to emerge and influence wave propagation. Although EADs are readily induced in isolated myocytes, there are no solid criteria in real cardiac tissue to distinguish EADs generated by intrinsic cellular membrane currents from afterdepolarizations resulting from diffusive electrotonic currents from nearby cells, especially when only APs on the surface layers of tissue can be mapped. In computer simulations, however, the distinction is unequivocal. Inclusion of 3D tissue heterogeneity will exacerbate the already great challenges in computation, data storage, and visualization that arise even with modern parallel supercomputers. The results here using homogenous 3D tissue provide a valuable framework for future studies investigating how additional interactions with tissue heterogeneity impact EAD generation and wave propagation.
In summary, despite these limitations, the major novel results regarding the effect of EADs on reentrant excitation wave propagation in cardiac tissue can be summarized as follows. 1) During reentry in 1D, 2D, and 3D homogeneous tissue, the first episode of spontaneous late diastolic SR Ca release occurred synchronously (i.e., during the same turn of reentry) throughout the tissue but at different amplitudes, reducing repolarization reserve nonuniformly. However, the resulting disturbance of propagation caused this synchrony in SR Ca release to break down, eventually creating regions of EADs next to regions without EADs. This allowed EADs to initiate new wavefronts propagating into repolarized tissue. 2) New wavefronts thus initiated by EADs could propagate in either the original or opposite direction not only as single new waves, as shown previously (12, 15), but as two new waves simultaneously traveling in the same or opposite directions, depending on the refractoriness of tissue bordering the EAD region. The creation of multiple new waves could be a factor in the breakdown of Torsades des pointes to ventricular fibrillation. 3) By suddenly prolonging local refractoriness, EADs caused rapid displacement of reentrant rotors to new regions, rapidly changing the electrical axis as observed electrocardiographically in Torsades des pointes. 4) Rapid displacement of rotors also promoted rotor self-termination by collision with tissue borders, but persistent EADs causing local repolarization failure could then regenerate single or multiple focal excitations that reinitiated reentry.
The simulation findings presented in this study are helpful in explaining many of the features of Torsades des pointes, such as perpetuation by focal excitations, rapidly changing electrical axis, frequent self-termination, and occasional degeneration to fibrillation. As such, they provide a useful framework for future experimental studies to test and validate the underlying predicted cellular and tissue mechanisms.
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APPENDIX A |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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V/t = DV and integration of the system of ODEs dV/dt = (Iion + Ist)/Cm. These integrations were executed in consecutive time cycles of predetermined duration, 0.1 ms.
During each 0.1-ms cycle, at each point in space, the system of ODEs was integrated either once with time step t = 0.1 ms or 20 times consecutively with t = 0.005 ms. The smaller of the variable time steps (t = 0.005 ms) was used when dV/dt 
5 mV/ms. For such large values of dV/dt, rapidly changing ion channel gate variables presumably require the smaller t for the calculations to be accurate. ODE integration was performed using an explicit Euler method, except for the equation describing the fast Na channel gate variable m, which was integrated using the hybrid method (30). More specific features of the numerical methods are described (4, 14).
The diffusion equation was integrated using an explicit Euler method. During each 0.1-ms time cycle, the diffusion equation was integrated two times with t = 0.05 ms. The space step h was fixed at 0.016 cm for 1D simulations and 0.025 cm for 2D and 3D simulations. The value of h was increased in 2D and 3D to shorten the overall computation time for higher-dimensional simulation.
An important consideration in choosing t and h was to satisfy the conditions of numerical stability for integration of the diffusion equation. The explicit Euler method used for solving diffusion has stability condition t h2/2jD, where D = 1 cm2/s is the diffusion coefficient and j is the number of space dimensions of the grid. For 1D (j = 1 and h = 0.016 cm), the inequality is satisfied by t 0.128 ms. For 2D (j = 2 and h = 0.025 cm), the inequality is satisfied by t 0.15625 ms. For 3D (j = 3 and h = 0.025 cm), the inequality is satisfied by t 0.078125 ms. The chosen value of t = 0.05 ms for the diffusion calculation satisfies these conditions.
A further constraint is placed on the space step h by the continuity condition proposed by Winfree (34). The diffusion coefficient D must "generously exceed" the value h2/Tr, where Tr is the risetime of the activation front. Estimating Tr = 2.5 ms, the inequality D > h2/Tr is satisfied by h < 0.05 cm. Our space steps h = 0.016 cm and h = 0.025 cm are well within this bound. Decreasing the space step to 0.125 mm and minimal time step to 0.0005 ms did not significantly affect simulation results.
Our chosen h yields grid nodes considerably larger than anatomical cells, particularly in the 2D and 3D cases. However, we do not feel this interferes with the results. Similar values of h are common in simulations of propagation in tissue, e.g., Courtemanche (9), and our conduction velocity is similar to physiological values (roughly 50 cm/s). The EAD events we observe (appearance, wave termination, and regeneration) require naturally arising synchronization of a region of tissue the size of multiple nodes, and hence enforcing artificial synchronization of a region of tissue the size of a single node should not have an effect.
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APPENDIX B |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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i m) receives nodes
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For the 2D tissue, the parallel processors were distributed over the entire grid in horizontal strips such that each processor pi (1 i m) receives nodes
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For the 3D tissue, the parallel processors were distributed over the entire grid in horizontal slabs such that each processor pi (1 i m) receives nodes
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GRANTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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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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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BGRANTSREFERENCES |
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2.5 s). The wavefront rapidly meanders away from c as a result of the EADs, and subsequently no more EADs occur. D: traces of V, Cai, and Jspon in node d (x,y = 134,237) where multiple EADs delay repolarization and force the spiral wavefront off the tissue. The color scale for V snapshots (not shown) is the same as in the 3D case.
