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1Department of Medicine and Research Center, University of Montreal and Montreal Heart Institute, and 2Department of Pharmacology and Therapeutics, McGill University, Montreal, Quebec; and 3Department of Electrical Engineering, University of Calgary, Calgary, Alberta, Canada
Submitted 15 March 2005 ; accepted in final form 15 April 2005
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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heart arrhythmia mechanisms; phase singularity; spiral waves; bioelectricity computer modeling; cardiac action potential
Experimental support has been presented for both of these paradigms. AF persistence across species varies with their cardiac size, with AF being much longer lasting in larger hearts (28). On the other hand, the ability of the mouse heart to sustain fibrillation has been presented as a challenge to the critical mass hypothesis (41), and an analysis of dominant frequencies and phase singularity (PS) properties during AF in isolated sheep hearts has led to the suggestion that one or a small number of reentrant generators rather than multiple wavelet reentry underlies AF (4).
Testing the "critical mass" notion in biological tissues is difficult because paradigms that allow for the examination of a range of tissue sizes (e.g., different species, models producing atrial dilation, developmental changes, etc.) are also associated with discrepancies in atrial electrophysiological properties. Mathematical models of atrial activity do not share this limitation: the size of the computational substrate can be varied while the underlying electrophysiological properties are kept constant.
We (18) have previously analyzed the properties of AF in a mathematical model of cholinergic AF with a two-dimensional substrate having realistic cellular and tissue electrical properties for the dog. In a 5 x 10-cm substrate (taken to represent normal canine atrial dimensions), AF is generally maintained by a single dominant spiral wave with wavebreak in a heterogeneous substrate (18). The role of substrate dimensions was not addressed, and we are unaware of other studies that assessed the role of substrate dimensions specifically. In the present investigation, we analyzed the relationship between atrial substrate size and AF persistence, along with associated mechanisms, in the model. We evaluated activity under 12 discrete conditions of acetylcholine (ACh) concentration ([ACh]) and distribution in each of five rectangular atrial substrates over an 11.1-fold range of surface areas. Induced activity was followed for as long as it was maintained, to a maximum of 20 s.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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A two-dimensional rectangular atrial substrate was modeled as previously described (16, 18). The substrate consists of discreet cables representing muscle fibers, with each composed of a chain of myocytes that form a longitudinal syncytium. Fixed transverse resistors (10 M
) represent lateral coupling. Fiber resistivity and interfiber resistances were chosen to match experimental propagation characteristics. The discretization factor (segment length per length constant) was kept <0.1 to prevent numerical distortions (15, 36). Voltage-dependent expressions involving exponentials were precomputed with 50-µV resolution and stored in a table to increase computational efficiency (42). Activity was analyzed in substrates of 3 x 6, 4 x 8, 5 x 10, 7.5 x 15, and 10 x 20 cm, including two smaller and two larger substrates than in our previous studies (16, 18). To change tissue size, we varied both the number of cables (180, 240, 300, 450, and 600 cables, respectively) and the number of cells per cable (360, 480, 600, 900, and 1,200, correspondingly). The atrial conduction velocity (CV) anisotropy ratio was 2.4:1, and longitudinal CV was 80 cm/s, compatible with experimental measurements of CV in the canine atrium (21).
Two-dimensional simulations. Reentry was initiated with a cross-shock protocol (19). A basic (S1) stimulus was applied along the uppermost cable, creating a transversely propagating wavefront. Before the recovery front reached halfway, stimulus S2 was applied as used experimentally (46), creating a PS and initiating reentry. Stimuli were 1 ms and 180 µA/cm2. A previously described IKACh formulation was used (18). [ACh] was varied across the tissue as previously between a minimum and maximum according to a sine wave function (18). Four [ACh] distributions (1.67, 2.0, 2.4, and 3.3 cm peak to trough) and three mean [ACh] densities (1.875, 3.75, and 7.5 nM) were studied, based on values previously found to permit AF perpetuation (18). Concentration fluctuations were between 0 and twice the mean [ACh], such that for a given [ACh] distribution, larger mean [ACh] implied larger differences in maximum and minimum [ACh] and AP duration (APD) values. Note that the lowest [ACh] was 0 nM for all conditions, so that varying mean [ACh] corresponded to varying the maximum concentration and the [ACh] gradient, with minimum values fixed. To assess APD and APD distribution resulting from different [ACh] conditions, the 5 x 10-cm substrate was paced at 6 Hz. APD to 75% of full repolarization (APD75) of the 10th cycle was then determined at all sites in the grid with custom-written software. Simulations (20 s) were run for each concentration and spatial distribution in each substrate size, providing 12 conditions/size.
For display and PS analysis purposes, the transmembrane potential (TMP) was subsampled at the center of each 6 x 6 cell square on the computational grid after each millisecond, which we showed caused no loss of resolution. The model was implemented in C language on a Sun Enterprise 10000 computer running on 16 parallel processors. Data was output every millisecond.
Data analysis. To analyze AF quantitatively, we used an automated algorithm for PS identification and tracking (51). The algorithm proved essential for analysis of complex activation in larger grids. PS candidates were located with Bray's convolution method (3) and an image-analysis approach. We identified PSs that satisfied both criteria, along with tracking considerations, followed by manual confirmation to exclude artifacts. To facilitate analysis of activation, we simultaneously displayed animated PS trajectories and TMP maps. Illustrative examples are provided in on-line movies as indicated (see http://ajpheart.physiology.org/cgi/content/full/00252.2005/DC1). Mechanisms underlying fibrillatory activity were classified as 1) single generator (SG), when one dominant rotor with consistent position (core tip remaining in a single low-[ACh] zone)-maintained activity; 2) a series of single generators (SSG), when activity was maintained by single dominant rotors but individual rotors eventually died out and were replaced by rotors arising in other positions; and 3) multiple generator zones (MGZ), when activity resulted from multiple simultaneous rotors of varying location and duration. To explore the frequency content of simulated AF, spectral analysis was performed with fast Fourier transformations (FFTs), with activity sampled at 1 kHz for 20,000 frames (20 s), providing a maximal resolution of 0.05 Hz. Algorithms were implemented with Matlab version 6.5 for Windows on a Pentium 4 personal computer.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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In our previous analysis of simulated AF in a 5 x 10-cm substrate (18), most cases of longer-lasting AF were maintained by single dominant rotors. In the present study, activity triggered by S2 was always initially maintained by a single clockwise rotor with its core tip located in a zone of low [ACh]. The fate of this rotor depended largely on [ACh]. For the highest mean [ACh] (corresponding to the largest [ACh] differences between peak and trough [ACh]), the initial rotor tended to remain strongly anchored in the same low-[ACh] zone. Figure 2 shows an example in a 4 x 8-cm grid. Figure 2A is an atrial TMP map at a single time point during sustained fibrillatory activity. A star indicates the position of the dominant rotor core tip, the dashed arrow shows the direction of rotation, and solid arrows show the directions of wavefront propagation. An animation of TMP and PS maps is shown in on-line movie 1, with an evolving TMP map at the top and corresponding PS trajectories at the bottom (see http://ajpheart.physiology.org/cgi/content/full/00252.2005/DC1). PS trajectories are color coded, with PSs lasting >1 s in green, PSs lasting >1 cycle but <1 s in blue, and PSs lasting <1 cycle in red. Instantaneous PS points are shown as bright color-coded dots, with the preceding 50 ms of each trajectory as a thick line and the complete previous PS trajectory history as thin lines. Running simulation time is shown in digital format. For the example illustrated, PS meander occurred but was never sufficient for the primary rotor core tip to escape its initial low-[ACh] zone location. Wavebreak occurred when the emanating impulse impinged on other low-[ACh] zones, generally producing two short-lived counterrotating daughter wavelets that mutually annihilated or drifted off the substrate surface. Mutual annihilation of counterrotating secondary waves resulting from wavebreak was strongly favored because of the sharp rotation of such waves, which restricted them to the same low-[ACh] zone.
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Single rotor-based AF, as shown in Fig. 2, occurred typically at the highest mean [ACh] (7.5 nM). Of 17 simulations at 7.5 nM mean [ACh], AF was sustained by single rotors for the full simulation in 14 simulations (82%; Table 2). In 10 of these, a single stable rotor maintained AF throughout the simulation (SG-type activity, as in Fig. 2), whereas in 4 the initial rotor eventually terminated but was immediately replaced by another long-lasting rotor generated by a daughter wave. Fibrillatory activity was then maintained by SSG. Figure 3 and on-line movie 2 (see http://ajpheart.physiology.org/cgi/content/full/00252.2005/DC1) show an example in the 10 x 20-cm substrate. Figure 3A shows APs in the low-[ACh] core of the predominant rotor. Activation is clearly irregular. The PS trajectory map (Fig. 3B) shows the principal generator rotor in the upper center as well as two transient rotor zones lasting >1 s (green trajectories) in other low-[ACh] regions. Figure 3C shows TMP maps at four time points corresponding to the extinction and replacement of a primary rotor. The clockwise rotor is shown at 8,401 ms, with its core tip PS at the star. At 8,891 ms, two counterrotating daughter wavelets have formed with core tip PSs (small stars) in the same low-[ACh] zone. The primary rotor collided with one of these wavelets (dashed red line), causing mutual annihilation. However, the remaining daughter wave formed a stable rotor that took over primary generator function, as shown at 9,076 and 9,401 ms. The sequence of events can be visualized in corresponding running time in on-line movie 2. SSG activity tended to occur with larger [ACh] distributions, which allowed for the transient formation of multiple wavefronts with core tips within individual (relatively large) low-[ACh] zones.
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In contrast to the well-anchored primary rotors observed at the highest mean [ACh], at lower mean [ACh]s (with smaller spatial [ACh] gradients), primary rotors tended to drift away from their initial locations. Primary rotor drift favored annihilation against a boundary or a daughter wave. In smaller sheets, primary rotor extinction due to such drift usually terminated AF. Figure 4 and on-line movie 3 (see http://ajheart.physiology.org/cgi/content/full/00252.2005/DC1) show an example of this type of activity in a 4 x 8-cm substrate with the lowest mean [ACh] (1.875 nM) and a 2-cm [ACh] distribution distance. APs from a low-[ACh] zone are shown in Fig. 4A, and the PS trajectory map is shown in Fig. 4B. TMP maps showing the progression of activity are provided in Fig. 4C. S2 initiated a single clockwise-rotating rotor toward the middle of the substrate (Fig. 4C, top left). The PS at the core tip of this spiral showed much more meander (green trajectory in Fig. 4B) than for the example shown in Fig. 2, moving from the initial low-[ACh] area toward a low-[ACh] area to the upper left of the substrate at 
880 ms, returning toward the middle of the substrate (Fig. 4C, top right and bottom left), and then moving to another low-[ACh] area to the lower right part of the substrate at 2,300 ms (Fig. 4C, bottom right). The latter area is adjacent to a border of the grid, and AF terminated at 2,400 ms when the primary rotor drifted to the lower right corner and extinguished. In this example, several intermediate-lasting rotors established themselves temporarily in the lower left and far left portions of the grid, as reflected by the blue trajectories in Fig. 4B, but were annihilated by collisions with other waves or with borders. Because no longer-lasting secondary rotors were available to assume generator function at the time of primary generator rotor extinction, AF terminated.
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TMP maps during this MGZ type of activity, as illustrated in Fig. 5A, resembled classical multiple circuit reentry. PS trajectory maps (Fig. 5B) showed numerous zones of long-lasting (green) and intermediate-lasting (blue) generator activity. The PS histogram in this case (Fig. 5C) showed 39 PSs lasting >1 s, reflecting a large number of generator rotors over the course of 20 s of activity. However, the longest-lasting PS was present for 1.79 s (in contrast to 20 s for examples of sustained SG activity), indicating the tendency of even long-lasting generators to be annihilated and the need for multiple generators to ensure continued activity. APs showed highly irregular activity (Fig. 5D), and the power spectrum was much more broad based (Fig. 5E) than for activity maintained by one or successive single generators (e.g., compare with Fig. 2, E and G). Sustained MGZ activity clearly did not depend on the persistence of a single primary generator but rather on the presence of multiple shorter-lasting generators able to maintain activity at all times.
In contrast to the highest mean [ACh] substrate, for which MGZ activity occurred in only three simulations, MGZ was observed for seven simulations at the lowest mean [ACh] and nine simulations for the intermediate mean [ACh] (Table 2). The duration of activity maintained exclusively by MGZ reentry showed a strong dependency on substrate size at the two lower mean [ACh] values, with such activity lasting under 1 s in the 4 x 8-cm substrate, <4 s in the 5 x 10-cm substrate, up to 10 s in the 7.5 x 15-cm substrate, and between 15 and 20 s in six of seven cases in the 10 x 20-cm substrate.
Properties associated with different mechanisms of AF maintenance. Two important contributors to perpetuation of activity were noted. The occurrence of stable SG activity was closely related to mean [ACh], with longer-lasting SG and SSG activity being very likely at high [ACh]s irrespective of substrate size. Low mean [ACh]s tended to be associated with unstable, meandering primary generators, with AF persistence relying on MGZ mechanisms linked to substrate size.
Given the critical importance of mean [ACh] in governing the mechanisms underlying persistent activity in our model, we assessed the effects of [ACh] and [ACh] distribution on APD values and heterogeneity. Figure 6 shows the relationships between mean [ACh] and spatial [ACh] distribution on one hand and mean APD75 and APD gradient (difference between largest and smallest APD75 at a frequency of 6 Hz). The primary determinant of mean APD75 was clearly mean [ACh] (Fig. 6A). Larger spatial [ACh] gradients were associated with slightly larger mean APD75 values, likely because of reduced electrotonic influences of short APD75 (high [ACh]) regions on the longer APD75 zones. The APD gradient was predominantly determined by [ACh] distribution (Fig. 6B), presumably because larger peak to trough [ACh] distances resulted in reduced APD smoothing by electrotonic interaction. Mean [ACh] was a smaller determinant, tending to produce larger APD gradients as mean concentration increased. Mean [ACh] was also a significant determinant of activation rate during AF. The dominant frequency in the FFT histogram in the low-[ACh] core zone and in the high-[ACh] periphery of spiral wave generators averaged 8.7 ± 0.1 and 10.4 ± 0.1 Hz, respectively, at 1.75 nM mean [ACh]; 9.3 ± 0.1 and 11.4 ± 0.1 Hz at 3.5 nM; and 9.6 ± 0.1 and 12.1 ± 0.1 Hz at 7.5 nM. The larger dominant frequencies at higher [ACh]s reflect the shorter APD values, resulting in sharper rotation of reentrant wavefronts and smaller rotor cores.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Relationship to previous conceptual models of AF. There are essentially three conceptual schemas of AF, all rooted in notions developed in the early 20th century: 1) multiple circuit reentry, 2) single circuit reentry; and 3) sustained rapid ectopic activity (31). Garrey (8) and Mines (25) were the first proponents of multiple circuit reentry as the mechanism underlying AF. Moe (27) refined the multiple reentry circuit schema with his "multiple wavelet hypothesis" and investigated the conditions promoting fibrillatory activity in a computer model. Lewis (20) held that AF is due to a single primary rapidly rotating wave, with wavefront break-up against variably refractory tissue producing fibrillatory activity. For many years, the multiple circuit model was preeminent, with substantial support from clinical and experimental evidence (reviewed in Refs. 30 and 31). Lewis' mother wave concept has recently received support from a range of experimental observations that point to single dominant rotors underlying AF in a variety of experimental paradigms (4, 5, 14, 24, 45).
When we initially developed the model described in the present study, we noted that AF was generally maintained by a single primary generator rotor (18). In the present study, we extended this work to a range of atrial substrate dimensions and to a longer simulation time, applying a recently developed method for automated PS identification and tracking (51) that permits the analysis of the complex activity that occurs in larger atrial substrates. This allowed us to refine the ideas that emerged from our previous analysis, by indicating that either dominant rotors (similar to Lewis' mother wave) or multiple unstable reentrant generators (essentially multiple wavelet reentry) can underlie fibrillatory activity in the model, depending on the functional properties and dimensions of the atrial substrate. To our knowledge, the present analysis is the first to identify in a mathematical tissue model a spiral wave counterpart to Moe's multiple wavelet hypothesis of AF.
We found that the MGZ mechanism, corresponding to multiple circuit reentry, was less likely than single rotor mechanisms (SG and SSG) to maintain fibrillatory activity for all conditions but the largest substrate. This finding supports the evolving notion that single rotor reentry with fibrillatory conduction is an important, and possibly predominant mechanism, for the maintenance of AF (4, 14, 24, 40). However, in larger, diseased atria, mechanisms corresponding to MGZ activity may become more important and may account for greater difficulty in AF suppression and prevention.
Relationship to experimental evidence regarding AF mechanisms. Detailed experimental evidence for multiple wavelet reentry in cholinergic AF was first presented in 1985 (1). Subsequent studies provided epicardial mapping evidence supporting multiple circuit reentry in vagal (47) and tachycardia remodeling AF (9). The role of atrial wavelength as a determinant of AF provided further support for multiple wavelet reentry (35). However, other evidence has called into question the role of multiple circuit reentry. Rapid left atrial activity, often with an apparent dominant generator zone, underlies cholinergic AF in isolated sheep hearts (4, 14, 24, 40). In these studies, most wavelets were generated by passive wave break (4). In the present analysis, either single dominant rotors (Figs. 2 and 3 and on-line movies 1 and 2) or continuous unstable generator activity (Fig. 5 and on-line movie 4) sustained fibrillatory activity. Our observation of SG AF in very small substrates supports observations of fibrillation in mouse hearts (41) and the notion that such fibrillation may not require multiple circuit reentry.
Several of our observations parallel findings in experimental studies of cholinergic AF. ACh alters cardiac electrophysiology primarily by activating IKACh (18). IKACh is particularly important in atrial tissue and, when activated, can produce atrial hyperpolarization (removing voltage-dependent Na+ current inactivation) and cause striking AP shortening. These effects provide greater source current for impulse conduction and allow for more rapid spiral wave rotation, accelerating and stabilizing atrial reentry (18). The importance of ACh actions are compounded by the spatial heterogeneity of effects associated with parasympathetic stimulation of the heart (22), which produces zones of lower [ACh]/effect that serve as functional barriers, and therefore anchoring points, in close proximity to areas of higher [ACh]/action. These properties are analogous to the changes induced by altered amplitude of K+ current, IK1, which appears to be a significant determinant of the stability of reentry and fibrillation in both atrial (33) and ventricular (37) tissues. We noted in the present study that lower mean [ACh]s were associated with meandering unstable rotors, whereas at higher concentrations single rotors stabilized and dominated activity. Schuessler et al. (39) observed multiple reentry circuits at moderately high [ACh] but single, dominant rotors that controlled the rhythm at the highest [ACh]s in canine atrial preparations. Ikeda et al. (12) noted meandering unstable wavefronts in the canine right atrium exposed to ACh. They subsequently described rapid nonstationary wavefronts during cholinergic AF in the canine right atrium, but stationary wavefronts associated with the shortest APDs, <30 ms (48).
Role of tissue size in AF maintenance. Garrey (8) first pointed to tissue size as a determinant of the likelihood of fibrillation, a notion supported by the persistence of AF in animals with different cardiac sizes (28). Atrial size is a significant determinant of clinical AF (10). Surgical procedures that reduce left atrial size may prevent recurrent AF (7, 38), and some of the benefit from linear ablation in AF may be due to decreased left atrial size (2). Smaller left atria are associated with higher conversion rates from AF (6). Our study supports the role of atrial size as a determinant of AF maintenance and provides potential mechanistic insights.
Consideration of the model. Compared with Moe's original computer model of AF (27), the model we used is much more realistic. Moe's model used a 992-unit array, with each unit having 5 discrete excitability states governed by an algebraic expression. The relative refractory period was much longer than in intact tissue, and there were no electrotonic interactions between functional units. Our model is based on cells of physiological size with APs controlled by realistic mathematical representations of canine ionic currents, with APs responding to various conditions similarly to experimental observations (34). Coupling, conduction, and anisotropic properties are appropriate for the dog (18).
On the other hand, the cellular substrate geometry is limited to a two-dimensional rectangular array. Three-dimensional complexities of in vivo atria, such as holes (venae cavae, pulmonary veins, and mitral and tricuspid valves), structural complexity (pectinate muscles and crista terminalis), and tissues with specific properties (septum and pulmonary veins) are not included. A more geometrically realistic three-dimensional model of the atria has been described (43) and provides insights into the potential involvement of various geometric structures in AF. However, other recent modeling studies of AF have similarly used two-dimensional planar arrays (32, 50), which remain instructive for asking specific questions such as the ones we posed in the present study. It will be interesting to determine in future work whether the principles defined in the present analysis continue to hold in more geometrically realistic models. Despite the simplifications inherent in our model, it has been shown in previous studies to agree closely with a range of experimental observations (16, 18) and with the results of more complex three-dimensional models (44). Of interest, the frequency peaks in FFT spectra in the present study are quite comparable to previous observations during vagal AF in the anesthetized dog (22, 46, 47).
We have previously observed that high, homogeneous [ACh]s stabilize reentry and greatly reduce meander of the spiral wave core tip (18). In the present study, we note that this anchoring effect of high [ACh]s results in very stable dominant generator-based AF in a heterogeneous substrate. It is likely that similar stabilization could occur with other factors that anchor primary generator spiral waves, such as anatomic obstacles and pectinate muscles, as suggested by previous experimental observations (13, 49).
We used a sinusoidal distribution to govern local [ACh] and produce spatially heterogeneous vagal action. Spatial heterogeneity in vagal effects is well recognized and is an important contributor to AF promotion (22, 46). The distribution of extracellular [ACh] during vagal stimulation in vivo has not been defined, but a sinusoidal distribution is clearly an oversimplification. During vagal AF in vivo, there are substantial differences in local frequency over distances of 1–2 cm (22, 46), of the order of [ACh] distributions used in the present study. Our model has discrete boundary conditions. Boundary conditions do exist in the intact cardiac atria, notably at the septum, great veins, and atrioventricular valves. Ikeda et al. (11) noted spontaneous AF termination in isolated canine right atrial preparations by movement to the tissue border. Similar to findings in the present study, Skanes et al. (40) reported rotor termination upon collision with other wavefronts during AF in isolated sheep hearts.
Our study was designed to address a very specific question: how substrate dimensions affect the persistence of fibrillatory activity in a two-dimensional model of canine atrium. Many of our findings are similar to those observed in experimental contexts, as outlined above. However, we are in no way claiming that our simplified model reflects the full biological complexity of AF, particularly as observed clinically with the complicating features of cardiac disease, geometric complexities, autonomic factors, etc. In addition, our model deals only with fibrillatory activity of the reentrant type and is not relevant to forms of AF that may be due predominantly to rapid ectopic activity as believed to arise from the pulmonary veins in some patient populations (31).
In conclusion, we found that atrial substrate size is a potentially important determinant of the maintenance of fibrillatory activity in our model. Unlike our previous study, which found a predominance of single rotor mechanisms in a substrate of fixed, 5 x 10-cm dimensions, the present analysis shows that either single dominant rotors or multiple unstable rotor-based reentry can underlie persistent fibrillatory activity in an atrial model. Although AF can be maintained in relatively small substrates by single dominant rotor mechanisms, larger substrates allow for persistent AF more easily because they can support a sufficient number of meandering rotors that the probability of all being extinguished simultaneously is very low, a mechanism that cannot occur in smaller substrates. These observations support the notion that larger substrates favor AF persistence but indicate that the importance of this phenomenon depends on the mechanisms underlying AF. Furthermore, these findings speak to recent debates regarding the relative importance of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain AF depending on atrial size and electrophysiological properties.
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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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