Abstract
Structural alterations during atrial fibrillation (AF) not only lead to electrical dissociation within the epicardial layer, but also between the epicardial layer and the endocardial bundle network. The aim of the study was to investigate the role of transmural conduction in the stability of AF episodes using a dual-layer computer model.
A proof-of-principle dual-layer model was developed in which connections between the layers can be introduced or removed at any time during the simulation. Using an S1–S2 protocol, a spiral wave was initiated in one of the layers, which degenerated into a complex AF pattern after connection with the other layer at six randomly chosen sites. After 6 s, connections were either retained (dual-layer simulations) or removed (single-layer simulations). Dual-layer simulations were more complex, as indicated by the higher number of waves and phase singularities. Tracking waves through both layers revealed that the number of waves in dual-layer simulations was significantly higher than in the single-layer simulations, reflecting more opportunities for reentry and a concomitant increase in AF stability. In the dual-layer model, only 12% of the AF episodes died out within 6 s, while 59% died out in the single-layer model.
Atrial fibrillation patterns are more complex and AF episodes are more stable in a dual-layer model. This study indicates an important role for endo-epicardial conduction for the stabilization of AF.
Introduction
Atrial fibrillation (AF) is characterized by multiple fibrillation waves randomly propagating through the complex anatomy of the atrial wall. The development of the substrate for AF goes along with an increase in the incidence of conduction block and thus an increase in the number of fibrillation waves. Simultaneous mapping of the epicardial layer and the endocardial bundle network in goats revealed a surprisingly large degree of dissociation between these two layers. Goats with acutely induced AF showed 18% dissociation of electrical activity, while dissociation had increased to 40% after 3 weeks of AF and to 70% after 6 months of AF.1 This suggests that fibrillatory conduction acquires a much more three-dimensional character with progressive structural remodelling than was previously assumed. To address the question whether endo-epicardial dissociation increases AF stability, we have developed a proof-of-principle computer model of two interconnected layers.
Atrial fibrillation in itself and structural heart diseases cause structural alterations in the atria favouring local conduction heterogeneity. The most important structural alterations underlying these conduction disturbances are atrial fibrosis, hypertrophy of atrial myocytes, amyloidosis, and possibly gap junction alterations.2 Recently, the electrophysiological correlate of this process of structural remodelling was demonstrated to consist of electrical dissociation between muscle bundles.3,4 Electrical dissociation leads to more and narrower wavefronts, greatly increasing the stability of AF. Structural alterations do not only lead to electrical dissociation within the epicardial layer, but also between the epicardial layer and the endocardial bundle network.1,5 Differences in electrical activity between epi- and endocardium allow fibrillation waves to conduct transmurally, causing breakthroughs (BTs) in the contralateral layer. This hypothesis is supported by the observation that the incidence of BT has been shown to be much higher in structurally remodelled atria than in normal atria, both in patients6 and in animal models of AF.4 As endo-epicardial dissociation of electrical activity can be regarded as the conditio sine qua non for transmural conduction, this observation implies that endo-epicardial dissociation would be more pronounced in structurally remodelled atria. Indeed, differences in activation time and direction of conduction between the epicardium and endocardium increased progressively during 6 months of AF in goats.1
Despite strong evidence for the three-dimensional character of the conduction pattern during AF, computer models of transmural conduction and endo-epicardial dyssynchrony are lacking. We therefore developed a computer model allowing three-dimensional conduction during AF. In this model we have studied the effect of transmural conduction on the stability of the fibrillatory process and the dynamic properties of fibrillation waves.
Methods
Model
Electrical connections between the two layers were incorporated by adding resistances between opposing segments in a circular area with radius 0.1 cm (conductivity σz = 0.5 mS/cm). Connections can be introduced or removed at any time during the simulation.
Simulation protocol
To investigate the effect of transmural connections on fibrillatory behaviour and stability of AF episodes the following simulation protocol was applied:
In one of the layers, a spiral wave was initiated using an S1–S2 protocol,10 while the other layer was not stimulated (Figure 1B).
The situation 1s after the start of the simulation, was used as the starting condition for an additional 6 s of simulation time in which the layers were connected by six connections. To exclude a bias caused by a particular geometry of the connection points, eight separate simulations were performed with different sets of six randomly chosen connections. Connection points for each simulation were chosen such that two connections were at least 0.2 cm apart. Since clustering of connection sites may effectively reduce the number of connection sites, each configuration of the connection sites was visually inspected and rejected in case of clustering’. All simulations were continued for six more seconds.
Seven seconds after initiation, all simulations continued for another 6 s, either (i) without changing the connections or (ii) after removing all connections.
(A) Model structure with two layers (epi- and endocardium) and transmural connections (grey arrows). (B) Spiral wave initiation using S1–S2 protocol. (C) Dissociation of electrical activity between the two layers and transmural conduction resulting in a breakthrough wave (black arrow).
The simulations of step 3i formed a group of 8 dual-layer simulations and the simulations of step 3ii together with the continued simulation of step 1 formed a group of 17 single-layer simulations.
Analysis
Atrial fibrillation complexity and stability was analysed for all simulations. Stability was assessed by determining whether the tissue was still electrical active (stable AF) or quiescent (non-stable AF) after 6 s of simulation time. We determined four measures of complexity each 1 ms of simulation time:
Number of waves;
Number of phase singularities (PSs);
Lifespan of PSs;
Lifespan of wave chains.
Detecting waves
A wave was defined as a contiguous area in which all segments have membrane potentials above the excitation threshold of −60 mV. The number of waves was calculated for each 1 ms during the entire simulation time.
Detecting and tracking phase singularities
, phase θ is defined at simulation time t by
and time t, V* = −45 mV is the average of 
when the tissue is not quiescent, and τ is a time window of 10 ms. The resulting 
has a value between −π/2 and π/2.
The number of PSs was calculated by each 1 ms during the whole simulation time.
Phase singularities were tracked in time and space as follows. The distance was calculated between all PSs detected at simulation time t + 1 ms and all PSs detected at simulation time t. If the minimum distance was <1 mm, this PS obtained the same identification number (ID) as the PS with minimum distance. If for a PS the minimum distance was <1 mm, the PS was considered as ‘newborn PS’ and obtained a new ID. By choosing 1 mm as a threshold, it was assumed that a PS cannot travel faster than 1 m/s.
Detecting breakthroughs
A BT is a wave that appears in one layer and cannot be linked directly to the propagation of other waves in that layer (Figure 1C). Breakthroughs were detected as follows. Areas containing connection points were monitored each 1 ms. If a new wave appeared in one of these areas and could not be related to the propagation of other waves in that layer, and it had a size at most the size of the connection area, it was marked as a candidate BT. If a candidate BT increased in size within the next 2 ms, it was marked as a BT. The moment that the BT appeared as well as the location were saved for further analysis.
Wave chain tracking
The propagation of fibrillation waves in our model was analysed by tracking waves in time and space. After a wave passed a connection point, it propagated through this connection if the tissue in the opposing layer was excitable. Thus, a BT appears and the new wave is part of the same ‘wave chain’. The temporal dynamics of waves can be described by three events: In this analysis, waves appearing in the other layer as a BT were not considered ‘newly generated’ waves but rather as continuation of the same wave chain in the other layer. Waves were tracked in time by comparing AF patterns each 1 ms, similar to the approach reported by Ten Tusscher et al.12 and by Clayton and Holden.13 As described above, waves were defined as contiguous areas in which the membrane potential exceeded −60 mV. For all waves found at simulation time t, overlap (amount of segments) was computed with all waves found at simulation time t + 1. If a wave at simulation time t did not had overlap with any wave in simulation time t + 1, the wave was considered to be extinguished. If a wave at simulation time t + 1 had overlap with two or more waves at simulation time t, the waves were considered to be fused. If a wave at simulation time t had overlap with two or more waves at simulation time t + 1, a break-up occurred and a new wave was generated. Finally, if a wave appeared at simulation time t + 1 and has no overlap with any wave at simulation time t, it means that a BT occurred. In the latter case, overlap between the BT wave and all waves from the other layer is computed to determine the source of the BT.
Generation: appearance of a new wave because a wave breaks up into two or more waves.
Fusion: the fusion of two or more waves into one wave as they merge with each other.
Extinction: the disappearance of a wave, because it hits a boundary or runs into unexcitable tissue.
Identification numbers were assigned as follows. At simulation time t = 1, all waves in both layers were detected and obtained an ID. If a wave was extinguished, the ID of that wave was not used again. If, because of break-up, one or more new waves were newly generated, the size of all daughter waves was calculated. The largest one retained the parent's ID, whereas the remaining waves obtained new IDs. If fusion occurred, the size of all waves that merged was calculated and the fused wave obtained the ID of the largest one. If a BT occurred, the BT wave obtained the ID of the wave in the contralateral layer that was determined to be the source of the BT.
Numerical methods and implementation
The model used for the present simulation study was based on our previously published bi-domain model.14 The mono-domain equation was solved assuming no-flux boundary conditions using an explicit numerical scheme with time steps of 0.01 ms as previously described.15 Gating variables and intracellular ion concentrations were updated with time steps of 0.01 ms during the action potential upstroke and otherwise with time steps of 0.1 ms.16 Gating variables were integrated using the Rush–Larsen method.7,17 The model was implemented in C++ and executed on a normal workstation with Intel i7 processor and 6 GB memory. It took 72 h to simulate an AF episode of 6 s on the dual-layer model. Up to six simulations could run simultaneously on the multi-core processor without increase in computation time.
Statistical analysis
Statistical tests were performed to compare the two groups of simulations (single-layer n = 17 and dual-layer n = 8). For each group, the average number of waves and the average number of PSs during the whole simulation time was calculated. In addition, the average of all PS lifespans and the average of all wave chain lifespans were calculated. Each data set was tested for normal distribution using the Kolmogorov–Smirnov test. An unpaired Student t-test was performed to compare normally distributed data sets and a non-parametric test (Mann–Whitney) was performed otherwise.
Results
Stability
The total number of simulations in the single-layer group was 17 and 8 in the dual-layer group. Membrane potentials of one of the simulations described in steps 2, 3i, and 3ii of the simulation protocol are presented in Figure 2A–C. In these simulations, AF persisted when the connections were retained (Figure 2A and C), while AF terminated in both layers when the connections were removed (Figure 2B). Figure 2D illustrates a Kaplan–Meier curve showing the persistence of AF episodes in both groups. Atrial fibrillation terminated in only one out of eight dual-layer simulations, while it terminated in 10 of 17 single-layer simulations.
(A) Membrane potential of a simulation at t = 1 s (introduction of connections) and t = 7 s. (B) Continuation of the simulation in panel A after removing all connections (step 3ii in methods section 2.2). (C) Continuation of the simulation in panel A without changing connections (step 3i in methods section 2.2). (D) Kaplan–Meier curve showing atrial fibrillation persistence in single and dual-layer simulations (n = 17 for single layer and n = 8 for double layer).
Number of waves and phase singularities
In all simulations, the number of waves and PSs was determined in both layers each 1 ms. An overview of the analysis results is presented in Figure 3. The average number of simultaneously existing waves and PSs was significantly larger in the dual-layer model compared with the single-layer model (Figure 3C).
(A) Phase singularity lifespan in dual-layer (red lines) and single-layer (blue lines) simulations and linear fit (black line), analysed in the simulations shown Figure 2B and C. (B) Wave chain lifespan in dual-layer (red lines) and single-layer (blue lines) and linear fit (black line) from the simulations shown in Figure 2B and C. (C) Average number of waves and phase singularities during the time that atrial fibrillation persisted for all simulations in each group. (D) Average of phase singularities and wave chain lifespan. (E) Average of phase singularity and wave generation rates.
Phase singularities and wave chain tracking
Figure 3A and B show PSs and wave chain lifespans of single and dual-layer simulations. Over time, new PSs and waves are continuously formed, while old PSs and waves disappeared. Most of the PSs and waves were present for only a short period of time (many short lines), while a few PSs and waves lasted for a longer period (few long lines). The slopes of the lines fitted through the starting points of lines representing PS and wave chain lifespan represent PS and wave generation rate. The average slope of the fitted lines for PSs and wave chain lifespan (black lines in Figure 3A and B) is larger in the dual-layer model compared with the single-layer model, indicating that the generation rate of wave chains and PSs in the dual-layer was higher (Figure 3E). Phase singularities and wave chains were tracked in time for both layers. Figure 3D shows that the average lifespan of PSs in the dual-layer simulation was shorter than in the single-layer while there was no difference in average lifespan of the wave chains.
Atrial fibrillation cycle length
Atrial fibrillation cycle length (AFCL) is the time interval between two consecutive activations, which are defined as the moment during the action potential upstroke when the membrane potential crosses −60 mV. In all simulations AFCL was calculated and averaged over time and space. Average AFCL in single- and dual-layer simulations were 148.5 ± 4.6 and 143.5 ± 3.6 ms, respectively, and were not significantly different.
Discussion
A novel dual-layer computer model was developed to study the effect of endo-epicardial dissociation of electrical activity and transmural conduction on stability and complexity of AF. To the best of our knowledge, this is the first model simulating dyssynchronous electrical activity in two separate layers—representing epicardium and endocardium—coupled by electrical connection points enabling transmural conduction and ‘BT’ of fibrillation waves from one layer to the other. The study demonstrates that adding the third dimension to the substrate of AF greatly increases the stability of the simulated arrhythmia, which was explained by an increase in the number of coexisting waves due to a higher rate of wave generation.
Radial spread of activation as new type of simulated fibrillation wave
Most computer models of AF treat the atrial wall as a two-dimensional layer.18 Most ‘three-dimensional’ atrial models are three-dimensional in the sense that a two-dimensional atrial sheet is folded into the shape of the atria, with two main cavities, atrial appendages, and connections between the two atria by a septal ring.19–21 As a first step towards more realistic computer models of AF, a few computer models incorporated heterogeneities in atrial thickness.16,22–25 These studies have shown how spatial differences of anisotropy in specific muscle bundles affect normal conduction in the atria and that the heterogeneity of the atrial wall thickness significantly contributes to the stability of AF. However, these models did not take disruption of transmural connections and endo-epicardial electrical dissociation (EED) into account. The novelty of our model is that it describes truly ‘three-dimensional’ or ‘transmural conduction’ between an endocardial and an epicardial layer. Connection points were randomly distributed and chosen such that two connections were at least 0.2 cm apart. Each configuration of connection points was visually inspected and rejected in case of clustering. The important conceptual consequence is that this model introduces a new type of simulated fibrillation waves. After transmural conduction from one layer to the other, conduction showed radial spread of activation away from the electrical connection point, thereby resembling BTs occurring during experimental and clinical AF.
So far PSs represent the exclusive type of reentrant fibrillation wave in most, if not all, computer models of AF. The correlate of PSs in experimental AF is spiral wave reentry with a rotor-shaped fibrillation wavefront conducting around an excitable yet unexcited core. Spiral wave reentry has been documented in animal models of AF with homogenous substrates and short refractory periods.26,27 Heterogeneities in electrophysiological properties like fibrosis cause wave break and largely reduce the lifespan of spiral waves. In dogs with heart failure, atrial fibrosis results in the existence of multiple unstable rotors—a conduction pattern very similar to multiple wavelets.28 Importantly, spiral wave reentry in patients with AF is a rare phenomenon.3 In contrast, numerous authors have documented BT with radial spread of activation of fibrillation waves in experimental4,23,29 and clinical investigations.6,30,31 As BTs contribute significantly to the perpetuation of AF, realistic computer models should be able to represent this type of conduction pattern. In the computer model described here, >40% of all waves are BTs and the existence of these BTs greatly increases the stability of the simulated fibrillatory process. Introduction of the third dimension can therefore be regarded as an important step towards more realistic computer models of AF.
Effect of three-dimensional conduction on atrial fibrillation complexity and stability
We demonstrate that AF was more stable in dual-layer simulations than in single-layer simulations. After 6 s simulation time, AF terminated in only 1 of 8 dual-layer simulations (12.5%), but in 10 of 17 (59%) single-layer simulations. The explanation for the increased stability of the AF episodes lies in an altered dynamic behaviour of the AF waves. Both wave and PS numbers and lifespan were analysed by monitoring BT, splitting, fusion, and extinction of individual waves. The number of both waves and PS were significantly larger in the dual-layer model, which might well explain the enhanced AF stability. Most waves and PSs were short lived (shorter than 450 and 1000 ms in 90% of the wave chains and PSs, respectively), while <5% of PSs lived longer than 1300 ms. Surprisingly, the average PS lifespan during AF episodes in dual-layer simulations was significantly shorter compared with single-layer simulations. The average lifespan of wave chains in the dual-layer simulations was not altered. A possible explanation for the smaller survival rate of PSs in dual-layer simulations might be that BTs reduce the space for PSs to meander and thereby disrupt the stable reentrant pattern of PS as has been described in an optical mapping study on fibrillating sheep atria.23 On the other hand, many BTs appear in an area with heterogeneous distribution of excitability and as a result evolve into PSs. Indeed, we found that the rate of generation of new PSs was twice as large in the dual-layer as in the single-layer model. This increase in the wave generation rate is the main mechanism of the increase in the number of waves in the double-layer model.
Implications and possible applications of the model
Endo-epicardial electrical dissociation and BT during AF have been described already in the 1990s.29,30,32 However, only very recently it was recognized that endo-epicardial dissociation increases with the degree of remodelling1,5 and enhanced EED in remodelled atria offers an explanation of enhanced BT rates in complex substrates for AF. As BTs significantly contribute to the stability of AF, mathematical models investigating the perpetuation of AF or therapeutic interventions in AF need to implement this phenomenon. For example, a recent experimental study has demonstrated that flecainide greatly reduces the rate of BT, which was attributable to a strong decrease in endo-epicardial dissociation.24 Our model offers the opportunity to verify whether the antiarrhythmic action of flecainide can be explained by its effect on endo-epicardial dissociation and transmural conduction or whether other mechanisms related to changes in ectopic activity mediated by enhanced phosphorylation of the ryanodine receptor might play a role.33 Likewise, our model might be used to study mechanisms of AF perpetuation. By introducing changes in coupling between the two layers but also within the epicardial and endocardial layer, the effect of electrical dissociation on the three-dimensional fibrillation pattern during AF can be studied. Clinical observations indicate that disruption of transmural connections leads to more BT events,6 although the number of possible pathways for BT waves is decreased. Our model provides the ideal research environment to increase mechanistic insights in this phenomenon. With our model, changes in BT rate during different stages of structural remodelling can be investigated and compared with clinical observations.
Study limitations
Our dual-layer model clearly is a simplified representation that does not reflect the anatomical complexities of the atria, the role of specialized structures like the pulmonary veins or heterogeneity in ionic membrane currents. It should be regarded as a proof-of-principle study, demonstrating that adding endo-epicardial dissociation of electrical activity and ‘truly’ three-dimensional conduction strongly increases complexity and stability of AF. Future studies will have to address the contribution of a more realistic anatomy and pathological changes in the atria to the perpetuation of AF.
Conflict of interest: none declared.
Funding
This work was funded by the Leducq Foundation (07 CVD 03), the Dutch Research Organization (NWO, VIDI-grant 016.086.379), and the European Network for Translational Research in AF (EUTRAF, FP7 collaborative project, 261057).
