Electrical wave propagation in an anisotropic model of the left ventricle based on analytical description of cardiac architecture

Abstract

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher-Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.

Figure 1. A radial section of the endocardial (solid red line) and epicardial (dashed blue line) surfaces of the LV model, from .

Figure 2. A spiral surface.

The lines on the surface have equations ρ = const and ϕ = const. Color corresponds to height (z coordinate).

Figure 3. A spiral surface viewed from the top (left panel) and side (right panel).

Two myofibers are displayed in red and black.

Figure 4. Arrival times, in ms, of the waves after point stimulation at the apex for various values of anisotropy and fiber rotation.

The values of anisotropy are shown at the top of the figure and the values of the fiber rotation are shown in the left column. For details, see Table 1. Arrival times are color-coded in ms.

Figure 5. Arrival times, in ms, of the waves after point stimulation at the epicardial surface for a large anisotropy ratio D ₁∶D ₂ = 1∶0.111.

The notation is the same as in Fig. 4.

Figure 6. Arrival times, in ms, of the waves after point stimulation at the endocardial surface for a large anisotropy ratio D ₁∶D ₂ = 1∶0.111.

The notation is the same as in Fig. 4.

Figure 7. Arrival times, in ms, of the waves after point stimulation at the epicardial surface for an intermediate anisotropy ratio D ₁∶D ₂ = 1∶0.25.

The notation is the same as in Fig. 4.

Figure 8. Arrival times, in ms, of the waves after point stimulation at the endocardial surface for an intermediate anisotropy ratio D ₁∶D ₂ = 1∶0.25.

The notation is the same as in Fig. 4.

Figure 9. Arrival times, in ms, as a function of the distance from the stimulation point for the apical (A), epicardial (B), and endocardial (C) stimulation.

The distance on the horizontal axis is measured in ms as the arrival time of the wave in the isotropic model (see text for more details). The red lines represent numerical experiments for total rotation angle α = 174°; the black lines represent for α = 16°; and the blue lines represent isotropy. The solid lines correspond to the case D ₁∶D ₂ = 1∶0.111, while the dashed lines correspond to the case D ₁∶D ₂ = 1∶0.25. The vertical segments display minimal and maximal arrival times in each group of nodes. The average, min, and max arrival times are displayed on the leftmost panels for D ₁∶D ₂ = 1∶0.111 and in the middle column for D ₁∶D ₂ = 1∶0.25. The right column compares the average arrival times.

Figure 10. Scroll wave rotation period T, ms, as a function of total fiber rotation angle α, deg.

Figure 11. Potential, mV, on the LV surface during scroll wave rotation (left) and tip trajectory for D ₁∶D ₂ = 1∶0.111 (red line) and for D ₁∶D ₂ = 1∶0.25 (black line) (right).

The results are shown for model 2 (γ ₀ = 0.2, γ ₁ = 0.7, see text and Table 1 for details).

Figure 12. Velocity of scroll wave filament drift for the simulation of 8 s.

Average filament velocity V_c, mm/s (A). Velocity components multiplied by 1000, per second: latitudinal component v_ψ (B) and longitudinal component v_ϕ (C).

Figure 13. Scroll wave filaments in the LV model.

The anisotropy ratio is D ₁∶D ₂ = 1∶0.111. Panels A, B, C, D: models 1, 2, 3, 4 (see Table 1), fiber rotation angle in the LV wall increases from panel A to panel D. The epicarduim (semitransparent colored surface; color denotes height from the red base to the purple apex), the endocardium (opaque white meshy surface), and filaments (black lines and dots).