Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue

Figure 1.. Snapshots showing the rotation of a spiral pinned to the ischaemic heterogeneity (radius 0.6 cm, highlighted by the white dashed circle) in an isotropic medium (4 cm×4 cm). Without external stimulation the spiral is stably pinned to the obstacle and a rotation period (Tsp) of 380 ms is recorded. Positions of the spiral tip on the border of the ischaemic area are shown at 0 ms (a), 100 ms (b), 200 ms (c) and 340 ms (d) within one rotation period. (Online version in colour.)

Figure 2.. Isotropic medium: example of unpinning and repinning of the pinned spiral during overdrive pacing (Tp=0.9Tsp) with t1=360 ms, after the delivery of the third pulse. In (a), the application of the electric field induces areas of depolarization and hyperpolarization at the boundaries of the ischaemic zone. Since the excitation threshold is exceeded, a new wave (N) is generated. One of the ends of the newly initiated wave (A) collides with the original spiral and annihilates it. The other end (F) travels counterclockwise around the boundary of the obstacle (b), but its propagation is inhibited by the refractory tail of the original spiral wave (c) and (d). This causes the temporary detachment of the wave (G) (e), followed by reattachment, also called repinning (f). The snapshots are taken at 15 ms (a), 70 ms (b), 170 ms (c), 270 ms (d), 300 ms (e) and 385 ms (f) after the delivery of the pulse. (Online version in colour.)

Figure 3.. Isotropic medium: example of successful termination of the pinned spiral. The simulation is run in underdrive pacing (Tp=1.9Tsp) for t1=80 ms and snapshots are taken from the interval following the delivery of the third stimulus. In (a) a new wave N is induced at the obstacle by the pulse, but this time part of the boundary is still refractory and does not get excited by the new shock. Therefore, only one end of the wave (A) is able to propagate and to collide with the surrounding waves (in particular with the small active pinned wave F) (a). The dynamics of the system then undergoes a transient (c–e), showing wave annihilation and break-up. At the end, the transient results in a stable regime in which the electrical activity in the tissue has been completely terminated (f). The snapshots are taken at 25 ms (a), 75 ms (b), 120 ms (c), 185 ms (d), 350 ms (e) and 700 ms (f) after the delivery of the pulse. (Online version in colour.)

Figure 4.. Results obtained from simulations in the isotropic case. On the x-axis different time intervals between pulses (Tp) relative to the period of the spiral (Tsp) are shown, varying from overdrive to underdrive pacing. The y-axis reports several initial conditions at which the first pulse was delivered (t1), each one corresponding to a specific position of the spiral on the border of the heterogeneity within one rotation period (cf. figure 1). The outcome is expressed in terms of the number of pulses that have to be applied in order to terminate the pinned spiral (as shown in figure 3f). The maximum number of stimulations delivered during the pacing protocol was five, so the white area in the graph corresponds to sequences that were not successful within the five pulses. For this reason, no statement can be made about the possibility of terminating the spiral wave in that region. (Online version in colour.)

Figure 5.. Snapshots showing the rotation of a spiral pinned to the ischaemic heterogeneity (radius 0.6 cm, highlighted by the white dashed circle) in an anisotropic medium (4 cm × 4 cm). Without external stimulation the spiral is stably pinned to the obstacle and a rotation period (Tsp) of 350 ms is recorded. The positions of the spiral tip on the border of the ischaemic area are shown at 80 ms (a), 140 ms (b), 250 ms (c) and 300 ms (d) within one rotation period. (Online version in colour.)

Figure 6.. Anisotropic medium: example of successful termination of the pinned spiral after the second stimulus has been delivered. The overdrive pacing protocol was applied, with (Tp=0.9Tsp) and t1=300 ms. In (a) the applied far field pulse is strong enough to exceed the excitation threshold and to induce a new wave nucleating from the heterogeneity. Since part of the boundary is still refractory due to the tail of the original spiral, only a small portion of the new wave is able to propagate. Within this part, one end disconnects from the obstacle because its propagation is inhibited by the refractory tail of the spiral, as shown in (b). The other end forms a pinned wave and collides with the tip of the original spiral (c,d). Panel (e) shows how this collision leads to the annihilation of the two waves and hence to the termination of the pinned spiral, as shown in figure 6f. The snapshots are taken at 3 ms (a), 14 ms (b), 34 ms (c), 64 ms (d), 134 ms (e) and 180 ms (f) after the delivery of the pulse. (Online version in colour.)

Figure 7.. Results obtained from simulations in the anisotropic case during the transient regime. On the x-axis different time intervals between pulses (Tp) relative to the period of the spiral (Tsp) are shown, varying from overdrive to underdrive pacing. The y-axis reports several initial conditions at which the first pulse was delivered (t1), each one corresponding to a specific position of the spiral on the border of the heterogeneity within one rotation period; compare with figure 5. The outcome is expressed in terms of the number of pulses that have to be applied in order to terminate the pinned spiral (as shown in figure 6f). The maximum number of stimulations delivered during the pacing protocol was five, so the white area in the graph corresponds to sequences that were not successful within the five pulses. For this reason, no statement can be made about the possibility of terminating the spiral wave in that region. (Online version in colour.)

Figure 8.. Snapshots showing the rotation of a spiral pinned to the ischaemic heterogeneity (radius 0.6 cm, highlighted by the white dashed circle) in an anisotropic medium (4 cm×4 cm) during the asymptotic regime. Without external stimulation the spiral is stably pinned to the obstacle and a rotation period (Tsp) of 340 ms is recorded. Positions of the spiral tip on the border of the ischaemic area are shown at 40 ms (a), 110 ms (b), 220 ms (c) and 280 ms (d) within one rotation period. (Online version in colour.)

Figure 9.. Results obtained from simulations in the anisotropic case during an asymptotic regime. On the x-axis different time intervals between pulses (Tp) relative to the period of the spiral (Tsp) are shown, varying from overdrive to underdrive pacing. The y-axis reports several initial conditions at which the first pulse was delivered (t1), each one corresponding to a specific position of the spiral on the border of the heterogeneity within one rotation period (cf. figure 8). The maximum number of stimulations delivered during the pacing protocol was five, so the white area in the graph corresponds to sequences that were not successful within the five pulses. For this reason, no statement can be made about the possibility of terminating the spiral wave in that region. (Online version in colour.)