Unified mechanism of local drivers in a percolation model of atrial fibrillation

The mechanisms of atrial fibrillation (AF) are poorly understood, resulting in disappointing success rates of ablative treatment. Different mechanisms defined largely by different atrial activation patterns have been proposed and, arguably, this dispute has slowed the progress of AF research. Recent clinical evidence suggests a unifying mechanism of local drivers based on sustained reentrant circuits in the complex atrial architecture. Here, we present a percolation inspired computational model showing spontaneous emergence of AF that strongly supports, and gives a theoretical explanation for, the clinically observed diversity of activation. We show that the difference in surface activation patterns is a direct consequence of the thickness of the discrete network of heart muscle cells through which electrical signals percolate to reach the imaged surface. The model naturally follows the clinical spectrum of AF spanning sinus rhythm, paroxysmal AF, and persistent AF as the decoupling of myocardial cells results in the lattice approaching the percolation threshold. This allows the model to make the prediction that, for paroxysmal AF, reentrant circuits emerge near the endocardium, but in persistent AF they emerge deeper in the bulk of the atrial wall. If experimentally verified, this may go towards explaining the lowering ablation success rate as AF becomes more persistent.

Figure 1

The emergence of AF in the inhomogeneous model with simultaneous endo- ($z=0$) and epicardial ($z=24$) imaging. Smoothing has been applied for clarity. Red, excited nodes; blue, resting (excitable) nodes; other, refractory (unexcitable) nodes. (a) Planar wavefronts propagating during sinus rhythm. The dotted box indicates a gap in the wavefront formed by conduction block. (b) Endo view: The arrow indicates an excitation reentering the gap in the wavefront. Epi view: Reentry is not observed. (c) Emergence of fibrillatory activity. (d) Maintenance of fibrillatory activity. Epi view: Activity emerges on the surface as a point source (breakthrough activity) located at the star. See video A in Supplemental Material [37].

Figure 2

A cross section of the model during AF showing a closeup of the $x-z$ plane (fixed $y$ coordinate). The cross section has been chosen to align with a reentrant circuit. Arrows indicate the movement of wavefronts. The top (bottom) surface of each panel corresponds to the epicardium (endocardium). A reentrant circuit can be seen near the endocardium with activity (a) moving around an isolated fiber, (b, c) reentering the isolated fiber, and (d) emitting fibrillatory waves from the left end of the isolated fiber. Fibrillatory activity spreads from the endocardium to the epicardium through the complex fiber structure of the model. This results in a wide variety of possible breakthrough patterns on the epicardial surface. When viewed from the endocardium, reentrant activity will be clearly visible since the isolated fiber lies along the endocardial surface. When viewed from the epicardial surface, the wavefronts propagating from the reentrant circuit may emerge as a single breakthrough point, or as multiple breakthrough points simultaneously. See video D in Supplemental Material [37]. Scale: $100\times 25$ nodes.

Figure 3

The average time spent in AF for the (a) homogeneous and (b) inhomogeneous models. Both models show a transition from sinus rhythm at large coupling (0% AF time) through paroxysmal AF (0% $<$ AF time $<$ 100%) to persistent AF (100% AF time). The gray parameter regions indicate coupling parameters below the percolation threshold, and are thus irrelevant to clinical AF. (a) Homogeneous coupling parameters, ${\nu }_{\parallel }$ and ${\nu }_{\perp }$. Blue, sinus rhythm; red, persistent AF. The white transition region corresponds to paroxysmal AF. Guidelines of positive (negative) gradient indicate constant $\mathrm{\Delta }\theta \left(\overline{\nu }\right)$. Risk curve as a function of the inhomogenous coupling parameter, $\overline{\nu }$, (red graph). There is no AF for $\overline{\nu }\gtrsim 0.4$. Decreasing $\overline{\nu }$ associated with decoupling nodes, there is a transition from sinus rhythm through paroxysmal AF to persistent AF. Inset: Histograms showing the distribution of reentrant circuits driving AF for paroxysmal AF (top) and persistent AF (bottom) as a function of depth $z$ from the endocardium. Drivers cluster in the subendocardial region ($z=0$) for paroxysmal AF but are uniformly distributed across the bulk for persistent AF.

Figure 4

(a) An activation map adapted with permission from Ref. [14] showing reentry activity from subendo imaging and a focal source on the subepi side with four marked reference points. (b) The equivalent for our model. Smoothing has been applied to images for clarity. Red, excited nodes; blue, resting nodes; other, refractory nodes.

Figure 5

(a) The global success rate of full depth focal ablation if the first reentrant circuit location is destroyed [single focal ablation (SFA), circles], and if the first ten locations are destroyed [multiple focal ablations (MFA), crosses] as a function of the coupling parameter, $\overline{\nu }$. The dashed line indicates the coupling value below which all simulated tissues harbour reentrant circuits. (b) The local success rate of a single focal ablation destroying reentrant circuits at the ablation site as a function of the ablation depth, $z$, for tissues with high coupling, $\overline{\nu }=0.33$ (plus), and low coupling, $\overline{\nu }=0.27$ (up triangle). The dashed (dot-dashed) lines correspond to the cumulative sum of the reentrant circuit depth probabilities for $\overline{\nu }=0.33$ ($\overline{\nu }=0.27$).

Figure 6

Simulated ablation in a tissue with multiple drivers demonstrating how the ability to terminate fibrillatory activity is closely related to driver depth. AF emerges (a) in the bottom left corner from a reentrant circuit near the epicardium, and (b) at the top of the tissue from a separate reentrant circuit near the endocardium. (c) Two shallow ablations (solid red in bottom left and top center) are applied to the endocardial surface at the locations of the reentrant circuits penetrating part of the way into the tissue. (d) The circuit near the epicardium has not been destroyed by the ablation since the ablation has not penetrated far enough into the tissue. The circuit near the endocardium has been destroyed successfully. However, since an active circuit remains in the tissue, AF has not been terminated. (e) and (f) show the same ablation process as demonstrated in (c) and (d); however, the ablation is now transmural, penetrating the full distance through the tissue. This is indicated by the solid red regions on the subepi view in (e) and (f). The ablations have successfully destroyed both reentrant circuits. Hence, fibrillatory waves dissipate as shown in (f) and sinus rhythm will be restored. Note, transmural ablation is not always possible in clinical practice. Ablation lesions typically struggle to penetrate more than $2\mathrm{mm}$ [41] into the atrial wall, which can be up to $7\mathrm{mm}$ thick in places [14]. Scale: $200\times 200$ nodes (full tissue).

Figure 7

(a) The excitation wave propagates across neighboring nodes. Nodes are connected longitudinally with frequency ${\nu }_{\parallel }$ and transversely (along $y$ and $z$ directions) with frequency ${\nu }_{\perp }$. At time $t$, the red bordered node is resting. At time $t+1$, the node activates in response to the stimulus coming from its neighbor to the left. At time $t+2$, the red bordered node becomes refractory and the excitation wave activates the two nodes in the resting state to which it is coupled. The node to the left of the red bordered node cannot be excited as it is in a refractory state. (b) Temporal dynamics of a node. Each node sequentially passes across three distinct states: resting (black), excited (white), and refractory (gray scale).

Figure 8

Progression of the excitation wave through a simple reentrant circuit situated on the surface of the 3D topology used to mimic the heart muscle fiber network. The red bordered node is susceptible to conduction block. (a, b) The planar excitation wave reaches the node that is susceptible to conduction block. The red bordered node does not excite and blocks the propagation in the lower fiber while the planar propagation of the wavefront proceeds in the upper fiber. (b, c) The activation wavefront reaches the penultimate node in the upper fiber that connects the two adjacent fibers. The excitation advances in multiple directions, indicated by the green arrows. In particular, the wavefront reenters the fiber where the original conduction was blocked. (c, d) If the path length indicated in the lower fiber is longer than $\tau /2$, the node placed to the left of the red bordered node will have reached its resting state when the backward propagation arrives. Therefore, the region indicated in light blue will act as a reentrant circuit until the red bordered node blocks the excitation again. The likelihood of creating a reentrant circuit is determined by the coupling frequencies ${\nu }_{\parallel }$ and ${\nu }_{\perp }$ and the fraction $\delta$ of nodes susceptible to conduction block.

Figure 9

Three-dimensional visualization of an AF driver located in the bulk of the atrial tissue. The black path represents the connectivity structure between the nodes. In this particular example we observe that longitudinal connections are more frequent than transverse connections ($y$ and $z$ directions), representative of reentrant circuits that can be observed when ${\nu }_{\parallel }$ is substantially larger than ${\nu }_{\perp }$.

Figure 10

Surface activation patterns in the model (a) before and (b) after processing. (a) The raw node states of a single layer of nodes in the model. White indicates active nodes, black nodes are resting (excitable), and gray nodes are refractory. This raw image only shows the node states in a single layer at a fixed $z$ coordinate. (b) To aid data interpretation and match the visualization methods applied in Ref. [14] we process the raw image using Gaussian smoothing. A 3D Gaussian kernel is used to convolve the raw node states. The Gaussian is centered on the imaged surface with standard deviations ${\sigma }_x={\sigma }_y=1$ and ${\sigma }_z=5$. The color scheme for the activation maps after processing has been chosen to match that used in Ref. [14]. Active wavefronts are shown in red, resting nodes are in blue, and refractory nodes are represented by all other colors. This filtered image improves the coherence of wavefronts in the model and aids data interpretation. Scale: $100\times 75$ nodes.

Figure 11

Examples of breakthrough patterns in the model. Close-up view of activity for five different realizations (tissues). Tissues (a) and (b) show one primary breakthrough point. Tissues (c) and (d) show the activity emerging across a wider area almost simultaneously. Tissue (e) shows two clearly divided breakthrough points. Globally, all five tissues have breakthrough activity that will generate waves that appear to originate from a focal point. These activation patterns are typically observed on the epicardium during paroxysmal AF, but may, in rare cases, also be seen on the endocardium. Quantitatively differentiating between breakthrough types is difficult. See videos A, B, C, and E. Scale: $100\times 75$ nodes.

Figure 12

Examples of partial and full reentry in the model. (a) An example of partial surface reentry where a full rotation is not observed in the surface activity. This is because the reentrant circuit stretches to a depth of approximately four nodes from the endocardial surface. (b) Complete surface reentry, where no gaps are observed in the reentrant circuit. Reentrant circuits are typically observed on the endocardium during paroxysmal AF, but may also be found on the epicardium in rare cases. See video D. Scale: $90\times 70$ nodes.

Figure 13

An example of surface activation patterns where reentrant activity is not observed on either atrial surface. Close-up view of the activation pattern in the case of a reentrant circuit with depth $z=16, \overline{\nu }=0.31$. (a) The tissue in sinus rhythm with regularly propagating wavefronts. A signature of fibrillatory activity characterizing AF is not observed. (b) Initial fibrillatory activity emerges on both epicardium and endocardium as breakthrough points. Note that this reentrant circuit is eight layers closer to the epicardium than the endocardium, as can be seen by the delay between activation patterns. (c) Fibrillatory activity spreads across the tissue. (d) AF is sustained. Reentrant circuits are not observed on either surface. Scale: $100\times 100$ nodes.

Figure 14

An example of reentrant activity observed on the endocardial surface. Close-up view of activation patterns in the case of a reentrant circuit near the endocardial surface with depth $z=1, \overline{\nu }=0.33$. (a) The model in sinus rhythm. A faint reentrant excitation is observed at the center of the subendo view. (b) Fibrillatory activity emerges in the model. An unexcited tract on the right side of the active region in the subendo view indicates the isolated fiber for the reentrant circuit. Activity spreads on the subepi view resembling a focal source. (c) The wavefront reenters the unexcited tract in the subendo view. The activity breaks through onto the epicardial surface at a single stable breakthrough point. (d) Fibrillatory activity is sustained. Scale: $100\times 100$ nodes.

Figure 15

An example of reentrant activity observed on the epicardial surface. Close-up view of the activation pattern in the case of a reentry circuit near the epicardium with depth $z=22, \overline{\nu }=0.32$. Circuits forming near the epicardium are much rarer than those forming near the endocardium in paroxysmal AF. (a) The model in sinus rhythm. A faint reentrant excitation is observed above the center of the subepi view. (b) Fibrillatory activity emerges in the model. An excitation is just starting to enter the unexcited tract on the right side of the active region in the subepi. Fibrillatory activity has broken through to the endocardial surface at multiple points simultaneously. (c, d) Fibrillatory activity is driven and sustained. A reentrant circuit is clearly observed from subepi imaging. Activity appears to originate from a focal point in subendo imaging. Scale: $110\times 100$ nodes.

Figure 16

The distribution of reentrant circuit depth $z$ from the endocardium in the homogeneous model. Fiber angle $\mathrm{\Delta }\theta$ is constant across the tissue. AF progresses from paroxysmal AF (top row) to persistent AF (bottom row). The left column is for tissues with strong longitudinal coupling and weak transverse coupling. The central column is for tissues with equal longitudinal and transverse coupling. The right column is for tissues with weak longitudinal coupling and strong transverse coupling. Depth $z=0$ ($z=24$) corresponds to the endocardium (epicardium). In paroxysmal AF, reentrant circuits are predominantly found near the atrial surfaces. When visualized from the atrial surface, these circuits will typically appear as reentrant or rotational activity. In persistent AF, reentrant circuits are uniformly distributed in the bulk of the atrial wall, away from the epicardium and endocardium. These circuits are more likely to result in breakthrough point (focal) activation patterns when visualized from the atrial surfaces. In the homogeneous model, the endocardium and epicardium are equivalent due to the symmetry.

Figure 17

The distribution of reentrant circuit depth $z$ from the endocardium in the inhomogeneous model. AF progresses from paroxysmal AF ($\overline{\nu }=0.32$) to persistent AF ($\overline{\nu }=0.27$). Fiber angle varies linearly across the tissue from $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={24}^{\circ }$ on the endocardium to $\mathrm{\Delta }{\theta }_{\mathrm{epi}}={42}^{\circ }$ on the epicardium. Depth $z=0$ ($z=24$) corresponds to the endocardium (epicardium). In the inhomogeneous model, the symmetry between the epicardium and endocardium is broken with the stronger longitudinal coupling on the endocardium than the epicardium. For paroxysmal AF, reentrant circuits are predominantly found near the atrial surfaces, and significantly more circuits cluster near the endocardium than the epicardium. This is because fibers have much stronger longitudinal coupling on the endocardial surface than on the epicardial surface. When visualized from the endocardium, these circuits will typically appear as reentrant or rotational activity. In persistent AF, reentrant circuits are uniformly distributed in the bulk of the atrial wall, away from the epicardium and endocardium. These circuits are more likely to result in breakthrough point (focal) activation patterns when visualized from the endocardium. As AF has progressed from paroxysmal to persistent, the average depth of circuits from the endocardium at $z=0$ has significantly increased—this hinders the effective use of catheter ablation for those circuits deep in the atrial bulk.

Figure 18

The termination of AF by destroying the reentrant circuit using ablation in the inhomogeneous model. The images show the activation patterns for a close-up of the endocardial surface. (a) A faint reentrant excitation is visible just above the center of the image. This corresponds to the reentrant circuit which is driving AF. (b) Nodes surrounding the reentrant circuit are destroyed (solid red region). The reentry circuit is not sustained since the reentrant excitation collides with the ablation lesion. The fibrillatory waves dissipate. (c) The model has returned to sinus rhythm. The ablation lesion disrupts the sinus wavefront, but does not induce fibrillation. See video F. Scale: $100\times 110$ nodes.

Figure 19

Inhomogeneous coupling parameter risk curves (red graph) for various choices of fiber orientation and tissue thickness. Inset: Histograms showing the distribution of reentrant circuits driving AF for paroxysmal AF (top) and persistent AF (bottom) as a function of depth $z$ from the endocardium. Left column: Tissue thickness of ten nodes ($1.0\mathrm{mm}$). Right column: Tissue thickness of 50 nodes ($5.0\mathrm{mm}$). (a, b) Strong longitudinal coupling on endocardium, $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={10}^{\circ }$, and uniform coupling on the epicardium, $\mathrm{\Delta }{\theta }_{\mathrm{epi}}={45}^{\circ }$. (c, d) Moderate longitudinal coupling on endocardium, $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={24}^{\circ }$, and nearly uniform coupling on epicardium, $\mathrm{\Delta }{\theta }_{\mathrm{epi}}={42}^{\circ }$. Parameters correspond to those used in Fig. 3 of the main paper. (e, f) Very mild longitudinal coupling on endocardium, $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={40}^{\circ }$, and very mild transverse coupling on epicardium, $\mathrm{\Delta }{\theta }_{\mathrm{epi}}={60}^{\circ }$. (g, h) Uniform coupling on endocardium, $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={45}^{\circ }$, and strong longitudinal coupling on epicardium, $\mathrm{\Delta }{\theta }_{\mathrm{endo}}={10}^{\circ }$. (g) and (h) are identical to (a) and (b) with the endocardium and epicardium swapped.