Numerical Simulations of Cardiac Dynamics. What Can We Learn from Simple and Complex Models

Modeling the electrical activity of the heart, and the complex signaling patterns that underlay dangerous arrhythmias such as tachycardia and fibrillation, requires a quantitative model of action potential propagation. At present, there exist detailed ioni

than with the MBR-model, we used it to study the stability of spiral wave as function of tissue size and periodic boundary conditions by simulating a cylindrical ventricle with out thickness. For cylinders with a perimeter larger than the spiral’s wave length, we found no difference in dynamics when compared to a spiral rotating on a square sheet of tissue with zero flux boundary conditions. Nevertheless, when the perimeter is compared to the wavelength, then the spiral wave tip becomes perturbed by the incoming waves generated by itself. The collision of incoming waves, with a higher frequency that the one of rotation, produces a drift on the spiral wave. It has been shown [16] that the drift of a spiral is proportional to the high frequency pacing. Therefore the drift is proportional only to the ratio between the cylinder’s perimeter and the length of the wave-tip-trajectory (spiral core) and independent of the shape. Different tip trajectories can be induced in the BR-model as well as in the 3V-model by varying the sodium conductance [11-12]. We have check that in all cases drift is present when the perimeter is comparable to the core of the spiral.

For cylindrical perimeters smaller than the tip trajectory, it is not possible to maintain a spiral wave, however when a spiral is in the hypermeander regime, there is a window of perimeter sizes between the drift and termination, for which stable-drifting spirals will break. The breakup is produce because the hypermeander wave repolarizes unevenly some regions along the cylinder, therefore incoming waves produced by itself can block the wave tip front and form new spiral waves. This mechanism for breakup can occur in the ventricles when a spiral wave is created close to the apex where the ventricular perimeter decreases. We have checked that the MBR-model also produce the same effect of breakup on a tissue with equal size to the one used with the 3V-model (a perimeter of about 4cm) . 2.3.Three-dimensional slabs of cardiac tissue

We have also studied the stability of spiral waves in three-dimensions considering a slab of cardiac tissue and including the natural rotational anisotropy of the fibers [11-12,17]. By using the 3V-model fitted to the MBR-model we found that stable scroll waves (spiral waves in 3D) become unstable and break into multiple as function of tissue thickness and rotation anisotropy. This is in agreement with various experiments of thinning ventricular tissue similar to Garrey’s [3] where it is shown that VT degenerates into VF if the tissue is thick however VT remains stable if the tissue is thin.

We found that for thick slabs of tissue with little rotational anisotropy the scroll wave vortex (the equivalent of the spiral wave tip, which becomes a line in 3D) is stable to perturbations and behaves similarly as in a two-dimensional sheet. However, as the rotational anisotropy is increased, the vortex elongates and curves because of a highly localized twist induced by the fiber rotation. This elongated vortices collide with the tissue boundaries and produce a wave breakup which then leads to multiple waves characteristic of VF. To illustrate the nature of the twist that produces the filament instability we show in Fig. 3a the contour of a spiral wave rotating in the epi-cardium and superimposed the one from the endo-cardium. The spiral wave is from the 3V-model fitted to the MBR-model with sodium conductance changed from 4 to 2.47 (see I_fi) [12] so the spiral tip, follows a circular core (in this case distorted ellipses because of the tissue anisotropy with a 1:0.33 ratio). The figure shows how the major axes of the ellipses are rotated by an angle which is roughly equal to the total fiber rotation angle. It also shows how the anisotropic velocity induces a phase difference between the spiral in the epi- and endo-cardium. Therfore when the spiral wave turns around the highly curved part of the distorted ellipse (pivot turn) a twist is produced because the wave front on the epi-leads in time the pivot turn on the endo-cardium. Notice how in the figure the spiral in the epi- already finish the turn while the one in the endo-cardium has not started yet. The amount of twist in the vortex is then produce every pivot turn (i.e. twice per period) and the amount of twist depends not only on the fiber rotation but also on the curvature of the tip trajectory. The higher the twist induced the larger the elongation of the vortex line and the easiest to produce breakup. The 3V-model fitted to the MBR-model produces very highly curved tip trajectories and is the reason for which breakup can occur given an specific thickness and fiber rotation rate

(see Fig. 3b).

Fig. 3 (A) The rotational anisotropy is 120/mm in counter-clock direction from epi- to endo-cardium and the thickness of the tissue is 2.2 mm. Thick line represents the contour of the spiral and tip trajectory on the epi-cardium and the dash line on the endo-cardium.

(B) Summary of simulation results using the 3V-model fitted to MBR, the LR-I model with speedup calcium, and from an experimental APD restitution (see [11]). The gray area represents mammalian hearts. The lines separates the boundaries between stable VT (bellow the

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