Machine learning methods for locating re-entrant drivers from electrograms in a model of atrial fibrillation

Figures

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  Figure 1.. The formation of a simple re-entrant circuit at the cellular level. An excited cell is shown in white and resting (excitable) cells are black. Cells shown in grey are refractory (unexcitable) for the duration of the refractory period. All cells are coupled to their nearest neighbours longitudinally—this reflects the strong coupling of cells along muscle fibres. Transverse couplings exist with probability ν—reducing ν reflects the decoupling between adjacent muscle fibres caused by fibrosis. An excitation is initiated along the left wall of the heart tissue and propagates left to right. When the excitation reaches a dysfunctional cell which fails to fire (marked by a red cross), (a) the excitation in fibre B is blocked but excitations continue in fibre A above. When a coupling between the excited and blocked strands is reached, the excited cell can send a signal propagating backwards from right to left down the blocked strand (b). If the path length of the re-excitation is sufficiently long, the re-entrant excitation can excite tissue behind the main wavefront. This signal can then move to the adjacent strands forming a continuously re-excited circuit. The simple re-entrant…

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  Figure 2.. The formation of fibrillatory re-entrant circuits in the Christensen–Manani–Peters model for a refractory period of τ=50. Excited cells are shown in white, resting (excitable) cells are black and cells which are refractory (unexcitable) are shown in grey. (a) A planar wave initiated by the pacemaker cells at the left boundary of the cylindrical heart tissue propagates from left to right along muscle fibres. (b) A dysfunctional cell fails to fire, blocking the propagation of the signal along that fibre. Excited cells in the adjacent muscle fibre can re-excite the blocked fibre at the next vertical coupling (figure 1). (c) If the path length of the re-excitation is sufficiently long, the re-entrant excitation can escape the unexcitable region. (d) This escaped excitation initiates a continuously activated circuit from which chaotic waves propagate—this circuit is the driver of AF. (e) Fibrillatory waves spread across the whole tissue and prevent the formation of new planar wavefronts at the pacemaker cells on the left boundary. (f) If a dysfunctional cell in the re-entrant circuit fails to fire, the circuit is stopped and fibrillatory behaviour dissipates.

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  Figure 3.. Electrograms simulated on perfect isotropic tissue (ν=1.0) in the CMP model using equation (A.1). Voltage is measured in arbitrary units. Time is measured in steps of 3 ms such that at sinus rhythm the pacemaker cells activate every 660 ms (220 time steps). (a) An electrogram recorded during regular sinus rhythm. The large depolarizations at approximately t=100 and t=320 correspond to the planar wavefront crossing the electrogram recording probe. The small fluctuations at approximately t=200 and t=420 correspond to the activation of a new planar wavefront at the left boundary by the pacemaker cells and the dissipation of electrical activity at the open boundary on the right after the wave propagates through the tissue—this is a small discretization effect in the CMP model. (b) Electrograms recorded during the rapid pacing of the heart where electrical wavefronts originate from a single point in the tissue. The dashed and solid lines show electrograms recorded at the centre of the electrical activity and 30 cells displaced from the centre, respectively. There are clear differences between electrograms recorded at different locations relative to the centre of electrical activity.…

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  Figure 4.. Electrical wavefronts propagate in all directions from an AF driver (shown as the star in the bottom left). A 3×3 electrode grid is used to record 9 simultaneous electrograms (large boxes numbered 1–9) across a region of 7×7 cells in the CMP model (small squares shown in grey). By spacing out the electrodes, the approaching electrical wavefront will cross each of the electrodes in a slightly different order and in a slightly different direction, affecting the feature behaviour seen at each electrode. Hence, taking the gradient of features across the electrode grid can give detailed information about the wavefront flow across the grid, enhancing our knowledge of the driver’s position. The local zig-zagging of the electrical wavefront due to the discrete structure of the CMP models can occasionally result in errors when calculating the direction of global wavefront propagation. However, by taking the mean value of the gradient of features for a fixed unit vector across different parts of the electrode grid these errors are minimized. Any remaining discrepancies are accounted for using the constraint mechanisms described in §(a) and appendix D. The intermediate electrode spacing…

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  Figure 5.. The vector feature map (VFM) for the dominant frequency of the electrogram’s Fourier transform on an arbitrary scale, shown by the greyscale. The image is generated from tissues with a single ectopic cell beating every 60 time steps placed at a random location in the tissue with transverse coupling fraction ν=0.2. The image shows strong separation between regions of unidirectional wavefront propagation (light grey) and bidirectional wavefront propagation (dark grey). There is also a strong indicator of the driver’s centre, black region at (0,0). Notice that the image is symmetric across the X- and Y-axes and that different feature maps highlight different wavefront dynamics. There is significant differentiation between bulk regions but little differentiation in small, localized regions. Combining multiple vector feature maps can improve local differentiation. More details can be found in appendix B.

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  Figure 6.. A flow chart describing the basic AF driver location algorithm. When starting the algorithm, the simplified CMP model is run with one or two drivers placed at random locations—the algorithm could easily be extended to more drivers if desired as outlined in figure 7. A random position is chosen to record the first set of electrograms. Search regions are constrained as described in appendix D. Predictions from the X and Y regressors are forced to lie within search constraints. ‘Prediction error’ initiates if the constraint region becomes too small or predictions loop between repeated coordinates.

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  Figure 7.. A diagram highlighting the key mechanism behind the extension of a single driver location algorithm to an arbitrary number of drivers. (a) A single driver tissue where the driver has been placed at the circular boundary to indicate wavefronts propagating from above and below (small stars). Wavefront collisions are expected (dotted line) and observed (wavy line) at the centre of the tissue. (b/c) The grey regions indicate the search regions for a second driver inferred by observing the absence of wavefront collisions at the expected position. (d) A second driver is found (big star). This two-driver system can be conceptually split into two single driver tissues, (e/f). These can be searched using the same recursive strategy starting at (a). Note that, in the current work, the mechanism for finding multiple drivers is simplified to only predict the position of expected collisions in one dimension; however, this could easily be extended to the full geometry in future work.

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  Figure 8.. An example of an electrogram showing a short period of fibrillation driven by a re-entrant circuit. The electrogram has been recorded on the point of re-entry. More details can be found in [15].

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  Figure 9.. Consider tissues A & B, where the drivers are located at coordinate (150,150) and (50,50), and electrograms are recorded at (170,170) and (70,70), respectively. Despite the electrograms being at different coordinates in the heart tissue, the relative vector from the driver to the electrogram is the same in both tissues, r=(20,20). Hence, both electrograms typically record very similar wavefront behaviour. Note that not every tissue has electrograms recorded at every vector relative to the driver. The periodic boundary conditions of the CMP model in the Y (transverse) direction mean that the relative vector from driver to recording position is always taken as the shortest distance between the two points—this is expressed by the individual tissues being wrapped. This method used to generate vector feature maps. Thousands of instances (tissues) of the CMP model are generated with a single re-entrant circuit randomly positioned in each tissue—two such tissues, A & B, are shown in this image. At a fixed vector away from the driver, r, we expect largely similar wavefront dynamics, and hence largely consistent electrogram features. These vectors can be mapped onto a single image by…

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  Figure 10.. A schematic showing the different regions on a vector feature map associated with different wavefront dynamics. The oval region at the centre corresponds to electrograms recorded near the AF driver—wavefronts in this region spread in all directions. The white strips either side of the centre indicate convex wavefronts which propagate along the X-axis centred on the driver. The black strips correspond to positions near the circular boundary where wavefronts propagating in the +Y and −Y directions from the driver collide. Finally, the grey regions represent the ‘bulk’, usually crossed by wavefronts that are approximately planar. The dark grey regions correspond to tissue which statistically only observes unidirectional flow in X, i.e. all electrograms in the dark grey region on the left only observe wavefronts propagating from the right. The light grey regions represent positions relative to the ectopic beat which can statistically observe bidirectional flow in X, i.e. positions in the light grey regions could see wavefronts approaching from either +X or −X over various tissue instances. The transition between these regions is not always sharp and varies over tissue instances. Note…

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  Figure 11.. Vector feature maps for the mean vertical gradient across the electrode grid for the electrogram sampling start index. The above image has been created using the regular single rotor procedure illustrated in figure 9. The image below follows the same procedure but has been created from tissues with a second rotor at a fixed vector 30 cells above the initial rotor (note the image is generated from a smaller dataset which accounts for a small increase in noise compared to the above image). Rotor positions are indicated by white crosses. Wavefronts propagate away from the dashed lines and collide at the dotted lines, marking the boundary of a given rotor’s sphere of influence. The feature behaviour within a rotor’s sphere of influence (the regions up to the nearest dotted lines) in a double rotor system are almost identical to the dynamics observed in a single rotor system. The dotted line at the centre shows a sharp transition from the first rotor dynamics centred at (0,0) to the second rotor dynamics centred at (0,30). Note that the shift of the collision line from Y =±100 in the single rotor tissue to Y =−85 in the double rotor tissue can be exploited when searching for multiple…

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  Figure 12.. Simple qualitative decision tree with a maximum depth of two. The responses arewhether or not the probe is on or off the AF driver. Three features (X1,X2 and X3) with their corresponding thresholds (s1,s2 and s3) are used to create the response regions.