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Am J Physiol Heart Circ Physiol 292: H180-H189, 2007. First published August 4, 2006; doi:10.1152/ajpheart.00944.2005
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Short-term cardiac memory and mother rotor fibrillation

Ali Baher, Zhilin Qu, Ashkan Hayatdavoudi, Scott T. Lamp, Ming-Jim Yang, Fagen Xie, Stephen Turner, Alan Garfinkel, and James N. Weiss

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

Submitted 2 September 2005 ; accepted in final form 18 June 2006


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Short-term cardiac memory refers to the effects of pacing history on action potential duration (APD). Although the ionic mechanisms for short-term memory occurring over many heartbeats (also called APD accommodation) are poorly understood, they may have important effects on reentry and fibrillation. To explore this issue, we incorporated a generic memory current into the Phase I Luo and Rudy action potential model, which lacks short-term memory. The properties of this current were matched to simulate quantitatively human ventricular monophasic action potential accommodation. We show that, theoretically, short-term memory can resolve the paradox of how mother rotor fibrillation is initiated in heterogeneous tissue by physiological pacing. In simulated heterogeneous two-dimensional tissue and three-dimensional ventricles containing an inward rectifier K+ current gradient, short-term memory could spontaneously convert multiple wavelet fibrillation to mother rotor fibrillation or to a mixture of both fibrillation types. This was due to progressive acceleration and stabilization of rotors as accumulation of memory shortened APD and flattened APD restitution slope nonuniformly throughout the tissue.

ventricular fibrillation; mathematical modeling; electrical restitution; action potential accommodation

RECENTLY, TWO DISTINCT MECHANISMS of fibrillation have been experimentally characterized and replicated in computer simulations. Multiple wavelet fibrillation is initiated and maintained by constant dynamic generation of new wavebreaks and does not require underlying tissue heterogeneity for its maintenance. Tissue heterogeneity, on the other hand, is central to mother rotor fibrillation, which is driven by a single, rapid, and relatively stationary "mother rotor," with fibrillatory conduction block occurring in the surrounding electrophysiologically heterogeneous tissue that is unable to follow the mother rotor with 1:1 conduction due to its longer refractory period.

Although both forms of fibrillation have been documented experimentally, a paradox exists about the initiation of mother rotor fibrillation. In previous computer simulations, initiation of mother rotor fibrillation used cross-field stimulation to induce the mother rotor in a region of tissue with a short refractory period, critical for the rapid rate of the mother rotor. Although cross-field stimulation, which requires simultaneous depolarization of a large region of tissue, may be relevant to initiation of fibrillation by a large premature electrical (e.g., 50–100x threshold, as used in ventricular fibrillation threshold testing) or mechanical stimulus (as in commotio cordis), it has limited physiological relevance to how fibrillation occurs spontaneously. Spontaneous initiation of fibrillation is triggered by rapid heart rates and/or premature extrasystoles, which are barely suprathreshold. In this case, wavebreak occurs when an impulse propagates from a region of short refractory period into a region of longer refractory period. Thus the first conduction block initiating reentry inevitably forms a rotor in a region of the tissue with a longer refractory period. The paradox arises because simulations show that, in electrophysiologically heterogeneous tissue, rotors drift toward regions with prolonged refractory periods and away from areas with short refractory periods (23). This drift occurs because the core of reentry makes a tighter arc as it travels through the short refractory region than through the long refractory period region, causing it to precess in the direction of longer refractoriness. How, then, does the initial rotor ever migrate to the region with the shortest refractory period, as required for a fast-turning mother rotor with peripheral conduction block due to a prolonged refractory period in the surrounding tissue?

It occurred to us that short-term cardiac memory, also called action potential accommodation, might provide a resolution to this paradox. The influence of short-term memory on cardiac reentry and fibrillation dynamics is poorly understood but potentially important. In the heart, short-term memory refers to the effects of pacing history on the action potential duration (APD) and should be distinguished from long-term memory related to changes in gene expression causing electrical remodeling. Short-term memory clearly exists in cardiac tissue, since pacing history influences properties such as APD restitution (APDR) (7, 8, 12, 15, 19, 24, 26) and continues to modulate APD over a variable range of several beats to hundreds of beats (3, 10, 26). Here, we show that, by altering APD and APDR slope, a quantitatively realistic degree of short-term memory, based on human ventricular monophasic action potential data, is theoretically capable of converting multiple wavelet fibrillation (4, 11, 18, 22, 25, 27, 29, 30, 32) into mother rotor fibrillation in heterogeneous tissue. This resolves the initiation paradox because the initial presence of multiple wavelet fibrillation naturally generates wavebreaks not only in regions with long refractory periods but also in regions with short refractory periods. One of the rotors thus formed can evolve into a mother rotor because its dynamics are stabilized and its rate is accelerated by the accumulation of short-term cardiac memory.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Action potential model. Phase I of the Luo and Rudy (LR1) ionic model was used in both two-dimensional (2D) tissue and in the anatomic heart simulations. This model contains a Na+ current, a slow, inward (L-type) Ca2+ current, a time-dependent K+ current (IK), a time-independent K+ current (IK1), a plateau K+ current, and a background current. The detailed formulation of each current can be found in the original paper by Luo and Rudy (17). We set the maximum conductance of L-type Ca2+ channel (Gsi) to 0.06 mS/cm2 and reduced the maximum conductance of Na+ channel to 16 mS/cm2, according to Qu et al. (21).

A K+ memory current (Imem) was introduced into the model as

Formula 1(1)
and

Formula 2(2)
where Gmem is maximum conductance, Xmem is the activation gate and Formula 2mem is the rate of change of Xmem, r is the inactivation gate, and Emem is the equilibrium potential. The formulation of these variables was based on rapid delayed rectifier K current (IKr) in Phase II of the Luo and Rudy model (16), except that the activation time constant ({tau}mem) was adjusted to delay activation/deactivation kinetics to simulate the slow kinetics of short-term cardiac memory. We chose an IKr-like formulation for the memory current because it has been implicated as a contributor to short-term memory because of its slow activation/deactivation kinetics (9).

APD was defined as the interval between the action potential upstroke and repolarization crossing –72 mV, which is roughly equivalent to APD at 90% repolarization. APDR was measured by the S1-S2 extrastimulus method, in which a one-dimensional cable was paced at the S1 cycle length until APD equilibrated and then an S2 extrastimulus was introduced at progressively shorter S1-S2 coupling intervals. APD of the S2 beat was plotted as a function of the preceding diastolic interval to obtain the APDR curve.

2D tissue model. Cardiac cells were resistively coupled to each other in a 12 x 12-cm2 sheet according to the following equation:

Formula 3(3)
where Vm is the transmembrane potential, Cm is the total capacitance, t is time, and D is the diffusion coefficient determined by gap junction resistance. Here, we use Cm = 1 µF/cm2 and D = 0.001 cm2/s. Iion is the total ionic current from the LR1 model. We used a space step (dx and dy) of 0.015 cm for all simulations. Integrations were done with an adaptive integration method, with dt varying from 0.02 to 0.2 ms (20, 21).

Anatomic heart model. We used rabbit ventricular anatomy and fiber-orientation data obtained from the Cardiac Mechanics Research Group at the University of California (San Diego, CA). The differential equation used for our anatomic rabbit heart is

Formula 4(4)
where Formula 4 is the anisotropic diffusion tensor (31). For better resolution in our numerical integration, we reduced the space steps from the original rabbit heart data set (dx, dy, and dz = 0.025 cm) to finer space steps (dx, dy, and dz = 0.0125 cm), resulting in a total of more than 3,600,000 computational cells. We used an operator-splitting method (20, 21) to integrate the diffusion term and an adaptive second-order Runge-Kutta method (with a maximum dt of 0.1 ms and a minimum dt of 0.01 ms) to integrate the ordinary differential equations. All simulations were carried out on a 32-node Beowulf Cluster. With these methods, a 1-s simulation took ~3.5 h.

Pseudo-ECG calculation. A pseudo-ECG ({Phi}e) was computed from Vm using the following integral expression:

Formula 5(5)

Formula 6(6)
where {nabla}Vm is the spatial gradient of Vm and r is the distance from a source point (x,y,z) to a field point (x',y',z'). {Phi}e was computed for an "electrode" located 2.0 cm above the center of the 2D tissue (13).

Dominant frequency maps and space-time plots. Voltage traces were analyzed by a fast Fourier transform to detect the dominant frequency (DF) to construct spatial DF maps, as described previously (25). Space-time plots were also constructed from voltage traces along a diagonal line in the tissue, as described previously (25).


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Mother rotor fibrillation in an anatomic model of the ventricles: the paradox of initiation. To create mother rotor fibrillation, we simulated the previously reported gradient in IK1 densities between the left ventricle (LV) and right ventricle (RV) in the rabbit heart (22) by doubling the magnitude of IK1 in the LR1 model in the LV but not in the RV. With Gsi in the LR1 model set at its control value, a rotor initiated by cross-field stimulation broke up into multiple wavelet fibrillation because of its steep APDR slope (>1), regardless of whether a gradient in IK1 was present. However, if Gsi was reduced from 0.06 to 0.03 mS/cm2 to flatten APDR slope to <1, a rotor initiated in the LV by cross-field stimulation remained in the LV, and 2:1 conduction block occurred at the septum where IK1 density abruptly changed (Fig. 1A). Thus, in agreement with previous simulations in 2D tissue (22), in the setting of the greater structural complexity of the three-dimensional (3D) anatomic ventricles, a rotor initiated in the LV by cross-field stimulation produced mother rotor fibrillation when a gradient in IK1 was present. In contrast, when reentry was initiated in the RV, a stable, slow rotor conducted 1:1 across the septum into the LV without conduction block, producing tachycardia (data not shown).


Figure 1
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  		Fig. 1. Failure of rapid pacing to induce mother rotor fibrillation in the anatomic rabbit ventricle model. Time-independent K+ current (IK1) in Phase I of the Luo and Rudy (LR1) model was kept normal in the right ventricle (RV) and doubled in the left ventricle (LV) to simulate experimental data (22). Surface voltage snapshots are shown at the times indicated. A: with the LR1 model with a flat action potential duration restitution (APDR) slope [L-type Ca2+ channel conductance (Gsi) = 0.03 mS/cm2], cross-field stimulation induced a rotor in the LV that led to mother rotor fibrillation as wave fronts from the fast LV mother rotor developed 2:1 conduction block while propagating across the septum into the RV, whose refractory period was longer because of reduced IK1 density. B: under the same conditions, rapid LV pacing led to wavebreak and initiation of a rotor in the RV, but the rotor failed to migrate to the LV to form a fast, stable mother rotor, as predicted theoretically (23). C: rapid RV pacing, on the other hand, failed to induce rotor formation. Conduction was 1:1 throughout the RV and LV until 2:1 block developed at the pacing site. Imem, memory current.

 
Although cross-field stimulation is a convenient method of initiating a rotor at any desired location in tissue, it is not physiologically relevant as a model of spontaneous tachycardia initiation, since it requires simultaneous depolarization of a large region of tissue. Spontaneously occurring premature beats or rapid heart rates that induce rotors under physiological conditions are barely suprathreshold. Rotors form when these impulses block as they propagate from regions of short to long refractoriness. This is problematic for mother rotor formation, however, because theoretical studies have shown that rotors in heterogeneous tissue drift toward the regions with prolonged refractory periods (23). If so, how does a slow rotor formed in a region of long refractory period ever migrate into a region of short refractory period to become a fast, stable mother rotor?

To address this issue, we attempted to induce reentry by rapid pacing in the anatomic ventricle model with an IK1 gradient between the RV and LV. RV pacing did not induce reentry, since impulses propagated 1:1 from RV to LV in the direction of long-to-short refractory period, until 1:1 capture at the pacing site was lost (Fig. 1B). Rapid LV pacing, on the other hand, led to induction of reentry due to wavebreak as the impulse propagated across the septum into the RV region with longer refractory period (Fig. 1C). However, because the initial rotor always formed in the RV and drifted away from the LV, as predicted from 2D simulations (23), this precluded the formation of a fast, stable mother rotor in the LV. Fibrillatory conduction block never developed, and the ECG showed tachycardia that never transitioned to fibrillation. We therefore conclude that, in the 3D anatomic ventricle model, a gradient in IK1 density between the LV and RV cannot produce mother rotor fibrillation if reentry is initiated physiologically by rapid pacing.

We hypothesized that short-term memory might resolve this paradox. Short-term cardiac memory broadly refers to everything in the pacing history besides the last diastolic interval that affects APD, and its usual effect is to shorten APD progressively as CL decreases, especially at rapid heart rates (3, 6, 10, 14, 26). If memory-induced APD shortening were also to flatten APDR slope, then, when a rotor is initiated before short-term memory has accumulated, steep APDR slope leads to rotor breakup into multiple wavelet fibrillation. Importantly, rotors are continuously created in regions of both short and long refractoriness because of the dynamical instabilities resulting from steep APDR. As short-term memory accumulates, however, APD shortens and APDR slope flattens, causing rotors to accelerate and become more stable. The region of short refractory period containing a rotor thus has the chance to evolve into the fast, stable mother rotor, whereas regions with long refractory periods develop fibrillatory conduction block.

Effects of short-term cardiac memory on rotor dynamics. Because the ionic determinants of short-term memory are complex and not fully understood, we formulated a generic K+ memory current (Imem), the amplitude and time dependence of which could be adjusted arbitrarily to simulate experimentally measured short-term memory effects. We added Imem to the LR1 model, which exhibits no short-term cardiac memory and has dynamics that are well characterized (11, 21, 31). The properties of Imem, IK, and the slow, inward (L-type) Ca2+ current were adjusted to simulate the experimental data of Franz et al. (10) from in vivo human ventricular myocardium. To replicate these results, Gsi was changed to 0.07 mS/cm2, K+ conductance (GK) was reduced by 80%, and Gmem and {tau}mem were set to 0.40 mS/cm2 and 200 s, respectively. With these parameters, steady-state APD at pacing CL (PCL) of 750 ms decreased in two phases when the PCL was abruptly shortened to 410 ms (Fig. 2A). The first immediate shortening was due to APDR, followed by a slow 20% additional shortening (time constant of ~30 s) because of further accumulation of Imem, quantitatively similar to the human ventricular monophasic action potential data superimposed in Fig. 2A. Figure 2B compares the changes in Xmem, the gating variable of Imem, when pacing was initiated at a PCL of either 1,000 or 300 ms. Xmem accumulated to a higher steady-state value at the faster PCL compared with the physiological CL because it had less time for deactivation during diastole. Figure 2, C and D, compares the time course of APD changes when pacing was initiated at either 1,000 ms PCL vs. 300 ms PCL. At 1,000 ms, no APD alternans occurred. At 300 ms, APD alternans occurred initially but then gradually disappeared as pacing continued, consistent with the flatter steady-state APDR slope at 300 ms PCL compared with 1,000 ms (Fig. 2E), measured using an S1-S2 protocol in a one-dimensional cable. For these Imem parameter settings, maximum APDR slope remained >1 at 1,000 ms PCL but <1 at 300 ms PCL. In contrast, in the absence of Imem, APDR curves were nearly identical for pacing at 1,000 vs. 300 ms (Fig. 2F), confirming that the native LR1 model displays no significant short-term cardiac memory.


Figure 2
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  		Fig. 2. Rate-dependent effects of Imem on action potential duration (APD) and APDR for Imem parameter values of memory conductance (Gmem) = 0.04 mS/cm2 and memory activation time constant ({tau}mem) = 200 s. A: normalized APD shortening (APD/APD at 750 ms) in the LR1 model containing Imem during steady-state pacing at 750 ms when the pacing CL (PCL) is suddenly shortened to 480 ms (red trace) or 410 ms (green trace) and then returned to 750 ms. The APD traces generated by the model are compared with the experimental data of Franz et al. (10) (bullet). B: traces of Imem activation gate (Xmem) illustrating the lesser accumulation at PCL = 1,000 ms (red trace) than at 300 ms (black trace). C: corresponding traces of membrane voltage following the onset of PCL at 1,000 ms (red trace) or 300 ms (black trace). D: APD changes with PCL at 1,000 ms (red trace) and 300 ms (black trace). E and F: APDR curves measured by the S1-S2 extrastimulus method for PCL = 1,000 ms (red trace) and PCL = 300 ms (black trace), using the LR1 model with (E) or without (F) Imem. DI, diastolic interval. Dashed line indicates slope of 1.

 
To examine how these Imem-induced changes in APD and APDR slope affected the behavior of rotors, we incorporated Imem into 2D isotropic tissue (Fig. 3). Under control conditions without Imem in the LR1 model, reentry initiated by cross-field stimulation during pacing (at any CL) broke up spontaneously into multiple rotors resembling multiple wavelet fibrillation (Fig. 3A), and the simulated ECG showed a fibrillation-like tracing. With Imem present, however, the pattern was different and depended on the initial PCL. For the same parameter settings as in Fig. 2 (Gmem = 0.40 mS/cm2 and {tau}mem = 200 s), when reentry was initiated during steady-state physiological pacing (CL = 1,000 ms), so that Imem was not sufficiently activated to flatten APDR to slope <1, the rotor spontaneously broke up into multiple rotors, as in the case without Imem. As Imem further accumulated, however, APD shortened, CL decreased, and the multiple rotors became progressively more stable, presumably due to progressive flattening of the APDR to slope <1 (Fig. 3B). This coalesced into a "spiral glass" pattern, in which multiple rotors all rotated at the same frequency, producing a Torsades-like ECG.


Figure 3
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  		Fig. 3. Effects of Imem on rotor dynamics in two-dimensional (2D) homogeneous tissue. A: LR1 model without Imem, showing voltage snapshots at 1 and 200 s after initiation of a rotor. The rotor spontaneously broke up into multiple wavelet fibrillation because of steep APDR slope. The simulated ECG showed a fibrillation-like pattern. B: pattern was the same for the LR1 model with Imem (Gmem = 0.4 mS/cm2 and {tau}mem = 200 s), when reentry was initiated during PCL of 1,000 ms for a total of 200 s. Because Imem had not preaccumulated enough during pacing to flatten APDR to slope <1 (Fig. 1A), the rotor initially broke up into multiple wavelet fibrillation as in A. As Imem subsequently accumulated, however, the multiple rotors stabilized into a "spiral glass" pattern (multiple rotors with identical frequency). The simulated ECG showed a Torsades-like pattern. C: pattern for the LR1 model with Imem, when reentry was initiated during PCL of 300 ms. Because Imem accumulated during pacing to flatten APDR to slope < 1 (Fig. 1B), the rotor never broke up and remained single. The simulated ECG showed a tachycardia-like pattern. In A–C, reentry was initiated by cross-field stimulation.

 
In contrast, when reentry was initiated during steady-state rapid PCL at 300 ms, so that Imem was sufficiently activated to flatten APDR slope to <1, the induced rotor remained intact and stable as sustained monomorphic tachycardia (Fig. 3C). This was presumably due to the effects of Imem on APDR slope, since flattening APDR to the same degree by increasing IK in the LR1 model without Imem produced the same pattern of a stable rotor (data not shown).

To explore the range of possible effects of Imem on rotor behavior, we varied the values of parameters Gmem and {tau}mem over a wide range. In this set of simulations, we set Xmem = 0 as initial condition in all cells, which allowed Imem to accumulate on its own after the initiation of reentry, rather than preequilibrate during pacing. Because it was not computationally feasible to explore a wide parameter space with a {tau}mem of 200 s, as was used to model the experimental data (9), we used different parameters (Gsi = 0.06 mS/cm2, GK = 0.282 mS/cm2) to replicate the same APDR properties (slope > 1 with slow pacing and slope < 1 with fast pacing), but with {tau}mem values ranging from 0 to 20 s. Figure 4 shows that the final rotor pattern depended on the amplitude (Gmem) and activation/deactivation time constant ({tau}mem) of Imem. When Gmem and {tau}mem were both small, rotors broke up into multiple wavelet fibrillation (region A in Fig. 4) because APDR remained steep after short-term memory was engaged. When Gmem and {tau}mem were both large, the rotor initially broke up because of steep APDR but then evolved into a spiral glass pattern as APDR slope flattened (region B in Fig. 4). When Gmem was large and {tau}mem was small to intermediate, however, Imem activation flattened APDR slope too quickly for the initial rotor to break up, and the single rotor persisted throughout, with its CL decreasing as Imem increased (region C in Fig. 4).


Figure 4
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  		Fig. 4. Phase diagram of rotor behavior as a function of Imem amplitude (Gmem) and kinetics ({tau}mem) for reentry initiated by cross-field stimulation with initial Xmem set to 0. Different parameter ranges produced a total of 3 types of behavior: continuous multiple wavelet fibrillation (region A), spiral glass (region B), or single rotor without breakup (region C). Insets: representative voltage snapshots of rotor behavior from the 3 regions.

 
Effect of short-term memory gradients on rotor dynamics in 2D tissue. The spiral glass pattern in homogeneous 2D tissue with short-term memory described above is not equivalent to mother rotor fibrillation, since all rotors have the same frequency, and the simulated ECG shows Torsades rather than fibrillation. Mother rotor fibrillation is characterized by the following features (32): 1) a relatively stationary fast rotor (the mother rotor); 2) spatial domains with different DFs, as detected by fast Fourier transform analysis. The domains are stationary over space and time, with the highest DF corresponding to the location of the mother rotor; surrounding areas exhibit lower DF values, corresponding to discrete ratios (e.g., 3:2, 2:1, etc.) reflecting fibrillatory conduction block; and 3) space-time plots showing evidence of Wenckebach conduction. To produce these features requires heterogeneous tissue, so we next investigated whether short-term memory, combined with a spatial gradient in IK1, could produce mother rotor fibrillation in 2D tissue.

In Fig. 5A, we created a tissue with three distinct zones by decreasing time-independent GK from top right to bottom left in a step-wise manner, motivated by experimental findings that IK1 density differs between the LV and RV in the rabbit heart (22). Imem was constant throughout the tissue such that all regions were in the spiral glass regime (Fig. 4); i.e., the APDR slope was initially >1 everywhere in the tissue, but Imem accumulation at fast rates strong enough to flatten APDR slope to <1 everywhere. When a rotor was initiated during pacing at a long CL to avoid preactivation of Imem, the rotor broke up into multiple wavelet fibrillation, which then converted to mother rotor fibrillation as Imem accumulated. DF maps were stationary over time and space, and DF values exhibited whole number ratios (3:2 between the fast region and intermediate region and 4:3 between the fast region and the slow region, respectively). Space-time plots also showed 3:2 and 4:3 Wenckebach conduction block along the diagonal of the tissue. By adjusting the locations of the step-wise Imem gradients, the spatial dimensions of different DF regions could be adjusted to match those observed experimentally (22, 32).


Figure 5
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  		Fig. 5. Pure mother rotor fibrillation produced by step-wise spatial electrophysiological gradients in 2D heterogeneous tissue. A: stepwise gradient in IK1, with constant Imem (Gmem = 0.4 mS/cm2 and {tau}mem = 10 s). B: step-wise gradient in Imem. The tissue was divided into 3 regions (dashed lines in A). Top: voltage snapshots at times indicated after initiation of a rotor by cross-field stimulation. Multiple wavelet fibrillation occurred initially; however, over time, a fast, stable mother rotor formed in the top right corner (arrow) where the magnitude of Imem (A) or IK1 (B) was maximal. Middle: spatial dominant frequency (DF) maps obtained from successive 2-s epochs of fibrillation. DF domains were largely stationary in both time and space, as indicated by the DF difference map between the 2 epochs in bottom left panels. Color bars indicate DF in hertz. Bottom right panels: space-time plot of membrane voltage along the diagonal line in A (middle), showing regular Wenckebach conduction block in the lower portion of the tissue as well as the simulated ECG-resembling fibrillation. Parameter settings for A, from top right to bottom left corner regions of the tissue are as follows: time-independent GK = 1.8141, 1.2094, and 0.6047 mS/cm2, respectively; for B: Gmem = 2.0, 1.0, and 0.4 mS/cm2; {tau}mem = 20.0, 12.0, and 6.2 s, respectively.

 
We also examined whether a gradient in Imem alone, in otherwise homogeneous tissue, could produce mother rotor fibrillation. Figure 5B shows 2D tissue with a step-wise gradient in Imem. When a rotor was induced, it broke up into multiple wavelet fibrillation due to the initially steep APDR slope. As Imem accumulated, however, all the wavelets formed stable rotors, with the fastest rotor in the region with the highest Imem density. This rotor became the mother rotor, and fibrillatory conduction occurred as it propagated into the adjacent regions with longer APD due to lower Imem density. DF maps showed stationary domains similar to Fig. 5A, and the space-time plots showed Wenckebach conduction. Linear gradients in the strength of Imem produced either mixtures of mother rotor fibrillation (where memory was strongest) and multiple wavelet fibrillation (where memory was weakest) or true mother rotor fibrillation (see APPENDIX). However, the latter case required very large tissue size (at least 12 x 12 cm). Thus, with either a step-wise gradient in IK1 and constant Imem throughout the tissue or a step-wise gradient in short-term memory, true mother rotor fibrillation could be obtained in a physiologically realistic tissue size.

Mother rotor fibrillation in the simulated 3D anatomic rabbit ventricles. To test whether short-term memory has similar effects and can resolve the initiation paradox in the more complex setting of the 3D anatomic heart, we introduced the LR1 model containing Imem into the anatomic rabbit ventricle model with the same IK1 gradient between LV and RV as in Fig. 1. In this simulation, Imem was constant throughout the heart. For Imem, we used the same parameter settings that reproduced experimental data of Franz et al. (GSi = 0.07 mS/cm2, GK = 0.226 mS/cm2, Gmem = 0.4 mS/cm2, and {tau}mem = 200 s) as in Fig. 2. Under these conditions, when the PCL in the LV was suddenly shortened from 1,000 to 300 ms, wavebreak occurred as the impulse propagated across the septum into the RV, leading to rotor formation as in Fig. 1 (Fig. 6). However, because APDR slope was initially >1 before Imem further accumulated, the rotor broke up into multiple wavelet fibrillation, creating new rotors in both the RV and LV. As time passed and Imem accumulated further, however, a fast rotor in the LV stabilized to form a mother rotor. Fibrillatory conduction block (2:1) occurred in the RV as the impulse crossed the septum. The pattern was similar to when cross-field stimulation was used to initiate the rotor in the LV without Imem present (Fig. 1A).


Figure 6
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  		Fig. 6. Effects of Imem on induction of mother rotor fibrillation by rapid pacing in the anatomic rabbit ventricle model. As in Fig. 1, IK1 in the LR1 model was kept normal in the RV and doubled in the LV to simulate experimental data (22). Surface voltage snapshots are shown at the times indicated. With Imem present (Gmem = 0.4 mS/cm2 and {tau}mem = 200 s) and steep APDR slope (Gsi = 0.06 mS/cm2) in the LR1 model, rapid LV pacing led to wavebreak at the septum, which initiated multiple wavelet fibrillation because of steep APDR slope. Subsequently, rotors formed in both RV and LV. As Imem accumulated, the rotors stabilized, allowing a fast, stable mother rotor (arrows) to form in the LV, with fibrillatory conduction block in the RV.

 
Thus, with Imem present, rapid pacing superimposed on a physiological heart rate was now able to induce multiple wavelet fibrillation, which, in the presence of an IK1 gradient between LV and RV, subsequently converted to mother rotor fibrillation.


   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 GRANTS
 REFERENCES
 
In the present study, we explored the effects of short-term memory on spiral and scroll wave reentry during simulated fibrillation. We found that a physiologically realistic degree of short-term memory is theoretically capable of resolving a paradox about mother rotor fibrillation, i.e., its spontaneous initiation by sudden increases in heart rate (e.g., rapid burst pacing) and, presumably, also by premature extrasystoles. Without memory, the wavebreak initiating reentry always occurred in a region of long refractoriness, and the rotor formed at this location naturally drifted away from regions of short refractoriness (23). Short-term memory, however, allowed multiple wavelet fibrillation to create wavebreaks in areas of short refractoriness, one of which is subsequently converted into a fast, stable mother rotor as memory accumulated. Thus, in electrophysiologically heterogeneous tissue, short-term memory could spontaneously convert multiple wavelet fibrillation to mother rotor fibrillation or to a mixture of both fibrillation types (see APPENDIX). This was presumably due to progressive acceleration and stabilization of rotors as memory accumulation shortened APD and flattened APDR slope.

Whether our generic representation of short-term memory as a K+ current, even though experimental APD data were quantitatively reproduced from human ventricle (10), is sufficient to capture all of the essential qualities of short-term memory in real heart is difficult to evaluate. All of the available experimental data indicate that short-term memory shortens APD as heart rate increases, consistent with the properties of our generic short-term Imem; however, the underlying ionic mechanisms are multifactorial and not well understood. The time scale over which the short-term Imem activates plays a critical role. Specifically, Imem has to activate with a time constant of at least several seconds (Fig. 4), so that APDR slope remains steep long enough for the initial rotor to break up into multiple rotors. Experimentally, the time constant of short-term memory activation can be estimated from the number of beats over which APD stabilizes with a sudden change in heart rate. Although most of the APD shortening in response to a rate change occurs over several beats, full equilibration to a new steady-state APD can take tens to hundreds of beats (3, 10, 26). We formulated our Imem to match quantitatively the experimental data from human ventricle (10), to ensure that short-term memory-induced APD changes in our model were in a physiologically realistic range. We used the earlier generation LR1 cardiac action potential model, whose dynamics are well characterized (21), to study the effects of short-term memory. Although this model does not include detailed intracellular Ca2+ cycling or as many ionic currents as later generation models, to our knowledge only two of these later generation models (9, 16) exhibit significant short-term cardiac memory. Short-term memory in the Luo-Rudy dynamic model (16) is mainly related to gradual pacing-dependent changes in intracellular Na+ and Ca2+. However, short-term memory present in Luo-Rudy dynamic reflects that of guinea pig myocytes and not human myocytes. Short-term memory in guinea pigs is much smaller in magnitude, reflecting the physiological differences between the two species. For the same cycle length changes as shown in Fig. 2A, the effect on APD was much smaller (2–5% compared with the 20% shortening observed experimentally in human ventricle), and the time constant was considerably longer (data not shown). Short-term memory in the model by Fox et al. (9) is exclusively due to IKr, whose activation accumulates over a time scale of several beats to shorten APD during fast pacing. This was the rationale for basing our formulation of Imem on IKr. However, the standard kinetics of IKr cannot account for the experimentally observed memory effects that accumulate over tens or hundreds of beats (3, 10, 26); therefore, we tuned the kinetics of Imem to produce these slower effects. Until a better mechanistic understanding of short-term memory is available and can be incorporated into models, there is no clearly superior alternative.

Although our memory current was tuned to match experimental data on human ventricular APD shortening in response to increased heart rate (10), the effects of memory-induced APD shortening on APDR slope were not available from that study. In other studies (6, 14), APDR slope has been reported to be flatter when measured by dynamic pacing (i.e., incremental shortening of CL) compared with S1-S2 pacing. This suggests that short-term memory, which accumulates more during dynamic than S1-S2 pacing, tends to flatten APDR slope, consistent with our simulations. Although one study (15) reported the opposite, the S1 PCL at which the S1-S2 APDR slope was measured was very rapid (300 ms). In this case, short-term memory may have already accumulated and flattened APDR slope.

The physiological basis of short-term memory effects on the action potential, which accumulates over a time course of several seconds or more (also known as accommodation), is not well understood. The intrinsic recovery kinetics of the classic ionic currents regulating APD are all shorter than several seconds, but heart rate may also modify these ionic currents extrinsically. For example, recent experimental evidence indicates that intracellular Ca2+ cycling may contribute to short-term cardiac memory (14). Rapid pacing alters diastolic, systolic, and sarcoplasmic reticulum Ca2+ levels gradually over the course of many beats (2). Rate-dependent changes in intracellular Na+ and Ca2+ cycling can directly influence currents such as Na+/Ca2+ exchange and L-type Ca2+ current inactivation, which affect APD (14). Intracellular Ca2+ cycling can also modify Ca2+/calmodulin kinase activity, leading to phosphorylation and altered activity of various ion channels affecting APD (1, 28). Finally, in intact ventricle, rate-dependent accumulation of extracellular K+ may also influence APD over the course of many heartbeats (5). Unfortunately, these processes have not yet been simulated to produce a physiologically realistic version of short-term cardiac memory matching experimental data.

On the other hand, the primary goal of this study was not to discern what causes memory but what memory causes, i.e., its potential effects on fibrillation dynamics. Our findings offer a potential compromise over the long-debated question of whether fibrillation is driven by intrinsic dynamical instabilities or a single mother rotor in heterogeneous tissue. We show that short-term cardiac memory provides a rationale for the coexistence of both mechanisms, while resolving the paradox for initiation of mother rotor fibrillation. Future experiments will be necessary to establish the detailed ionic basis for short-term cardiac memory, as it exists in real hearts, and to test whether these detailed mechanisms provide the same outcomes reported in this study.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
GRANTS
 REFERENCES
 
Figures 7 and 8 show the effects of linear gradients of short-term memory, with minimum memory increasing diagonally to maximum memory in the 2D tissue. In Fig. 7A, the bottom left was assigned no short-term memory and thus remained in the rotor breakup regime (APDR slope >1). Figure 7A, top right, on the other hand, had strong enough memory to stabilize rotors into a spiral glass pattern (APDR slope <1 after activation of memory). When reentry was initiated during pacing at a long CL to prevent preaccumulation of short-term memory, rotor breakup occurred and evolved into a fibrillation-like pattern. As short-term cardiac memory accumulated in the top right portion of the tissue, this region developed a fast stationary rotor as APD shortened and APDR slope became <1. However, except for the top right corner, spatial DF domains were not stable in either time or space, and DF values in different regions of the tissue were not whole number ratios of each other (Fig. 7B). In addition, space-time plots in different regions of the tissue showed intermittent conduction block, but not regular Wenckebach patterns (Fig. 7C). The absence of stationary DF domains and Wenckebach conduction block led us to conclude that this type of fibrillation represented an intermediate form, with mixed features of mother rotor and multiple wavelet fibrillation.


Figure 7
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  		Fig. 7. A mixture of mother rotor and multiple wavelet fibrillation caused by a linear spatial gradient in Imem in 2D heterogeneous tissue. Gmem and {tau}mem were increased linearly from 0 to 0.3 mS/cm2 and from 1 to 3,000 ms, respectively, along the diagonal line shown in B, left. A: voltage snapshots at various times after initiation of a rotor by cross-field stimulation. Multiple wavelet fibrillation occurred initially (0.8-s panel); however, over time (2.5- and 4-s panels), a fast, stable mother rotor formed in the top right corner (arrows) where the magnitude of Imem was maximal. B: spatial DF maps obtained from 2 successive 2-s epochs of fibrillation from 4 to 6 and 6 to 8 s, respectively. DF domains shifted in both time and space, as indicated by the DF difference map between the two epochs (right). Color bars indicate DF in hertz. C, left: space-time plot of membrane voltage along the diagonal line in B, left, showing complex conduction block in the lower portion of tissue. C, right: action potential traces from points a and b on the diagonal line in B and the simulated ECG (bottom trace) resembling fibrillation.

 

Figure 8
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  		Fig. 8. Pure mother rotor fibrillation caused by a linear spatial gradient in Imem in 2D heterogeneous tissue. Gmem and {tau}mem were increased linearly from 0.2 to 2.0 mS/cm2 and from 3 to 20 s, respectively, along the diagonal line shown in B, left. A: voltage snapshots at various times after initiation of a rotor by cross-field stimulation. Multiple wavelet fibrillation occurred initially (0.8-s panel); however, over time (10- and 25-s panels), a fast, stable mother rotor formed in the top right corner (arrows) where the magnitude of Imem was maximal. B: spatial DF maps obtained from 2 successive 2-s epochs of fibrillation, from 22 to 24 and 24 to 26 s, respectively. DF domains were largely stationary in both time and space, as indicated by the DF difference map between the 2 epochs (right). Color bars indicate DF (in Hz). C, left: space-time plot of membrane voltage along the diagonal line in B (left), showing a Wenckebach conduction block pattern in the lower portion of the tissue. C, right: action potential traces from points a and b on the diagonal line in B and the simulated ECG (bottom trace) resembling fibrillation.

 
Figure 8 shows a simulation in which short-term memory was strong enough to decrease APDR slope to <1 everywhere after Imem accumulated. In this case, all rotors in the tissue eventually became faster and more stable, but the fastest was located in the top right corner, where Imem was largest. When reentry was initiated during pacing at a long CL to prevent preaccumulation of Imem, rotor breakup occurred and a fibrillation-like pattern again developed. It is important to note that initial wave breakup still occurred because of dynamic wave instability from the initially steep APDR slope. As rotors stabilized because of Imem accumulation, the fastest rotor formed in the top right corner where Imem was largest. Peripheral conduction block occurred in the lower left corner where Imem was weakest (and APD longest), but only in a very small region of the tissue (Fig. 8A). Spatial DF maps showed a large domain (DF = 30 Hz) and two small regions at the bottom left corner (DFs = 14.65 and 7.62 Hz, respectively), which remained stable over time and space (Fig. 8B). Space-time plots showed Wenckebach-like conduction block (Fig. 8C), and the simulated ECG showed a fibrillation-like pattern (Fig. 8C). These results show that a linear gradient in short-term cardiac memory current can convert multiple wavelet fibrillation into true mother rotor fibrillation. However, for a linear gradient of Imem, a tissue size larger than 12 x 12 cm2 was required to have stable regions with different DFs and peripheral conduction block at their boundaries characteristic of mother rotor fibrillation.


   GRANTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
REFERENCES
 
This work was supported by National Heart, Lung, and Blood Institute Grants P50 HL-53219 and P01 HL-078931, American Heart Association Scientist Development Grant 0130171N, and Laubisch and Kawata endowments.


   ACKNOWLEDGMENTS
 
We thank the National Biomedical Computation Resource at the University of California, San Diego, for the anatomy and fiber vectors of the rabbit ventricles.


   FOOTNOTES
 

Address for reprint requests and other correspondence: J. N. Weiss, David Geffen School of Medicine at UCLA, Division of Cardiology, 47-123 CHS, 10833 LeConte Ave., Los Angeles, CA 90095-1679 (e-mail: jweiss{at}mednet.ucla.edu)

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


   REFERENCES
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 

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