Impact of electrode orientation, myocardial wall thickness, and myofiber direction on intracardiac electrograms: numerical modeling and analytical solutions

Figures

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  FIGURE 1. Setup of the simulation. The wave propagation is in all simulations in the X-direction. Here, L is the wall thickness, ψ is the angle between the myofiber direction and the direction of wave propagation, α is the elevation angle of the catheter, and β is the azimuthal angle. The interelectrode distance on the catheter is d, and h is the distance of the catheter tip to the myocardial wall. The highlighted (yellow) tissue is the stimulated tissue and, thus, the onset of the wave.

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  FIGURE 2. Electrical potential of a traveling wave within the cardiac wall, as a function of the lateral position (or time) x = vt and electrode distance from the myocardium z. (A) 3D view. (B) Top view of the same profile. It should be noted that there is no singularity of the potential, but a discontinuity (branch cut) at the wave front within the cardiac wall. (C) Same view as (B) but observed with simulated data.

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  FIGURE 3. Trigonometric and inverse trigonometric functions used in the analytical calculations.

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  FIGURE 4. Analytical solution for the potential caused by a finite traveling dipole sheet in the X-direction. (A) Spatial potential profile at different distances h to the myocardium, for subsequent time steps. In the direction of front propagation (positive X), the potential is elevated. (B) Regarded as a function of time at a fixed recording position, there is first an upward deflection in the unipolar signal. In both cases, we used ϕ max = 30 mV, ϕ min = −86 mV, g i,xx = 0.1527 S/m, g~=0.8341 S/m, L = 5 mm, and v = 1 mm/ms.

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  FIGURE 5. Interpretation of the arctan(x) and solid angles in the myocardium.

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  FIGURE 6. (A) Exact solution of the bipolar signal as a local potential difference. Here, ϕ max = 30 mV, ϕ min = −86 mV, g i,xx = 0.1527 S/m, g~=0.8341 S/m, L = 5 mm, and v = 1 mm/ms. The green solid line denotes V bip,⊥ in (A) and V bip,‖ in (B). (C) Three main measuring directions of the bipolar extracellular potential V bipolar.

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  FIGURE 7. Comparison between the analytical numerical solution to the EGM near a slab with wave propagation along the myofiber direction. (A) Unipolar signal (gray), showing a far-field artefact in the simulation result, occurring since the wave front hits the end of the myocardial slab. (B) Bipolar signal measured parallel to the myocardial surface in the direction of wave propagation. (C) Bipolar signal measured perpendicular to the myocardial wall. The slab thickness was L = 5 mm. Furthermore, ϕ max = 30 mV, ϕ min = −86 mV, g i,xx = 0.1527 S/m, g~=0.8341 S/m, v = 0.42 mm/ms, and d = 2 mm. Wave speed v is the only free parameter in the theory and was measured in the simulation. The RMSE (Eq. 9) between the simulated signal and the anisotropic theoretical prediction was computed between 0 and 250 ms.

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  FIGURE 8. Solid angle contributions caused by mirror sources. (A) Isotropic three-layered medium. (B) Anisotropic three-layered medium, after re-scaling of the Z-coordinate. If hL≪1 , then the first term of the series offers a good approximation.

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  FIGURE 9. Effect of myofiber direction on the EGM amplitude: homogenized (see Eqs 23, 33), re-scaled anisotropic (one term) (see Eq. 32), and full series solution (see Eq. 31). Solutions shown here have h = 1 mm, L = 5 mm, and d = 2 mm. In every plot, the RMSE (Eq. 9) was calculated between 0 and 300 ms between the simulated signal and the anisotropic series prediction.

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  FIGURE 10. Comparison between two analytical bipolar signals as a function of time: exact difference between two unipolar signals (Eq. 35) (solid color) and the simulated signal (solid gray line). Here, we used ϕ max = 30 mV, ϕ min = −86 mV, g i,xx = 0.1527 S/m, g~=0.8341 S/m, L = 5 mm, v = 0.42 mm/ms, and d = 2 mm. The RMSE (Eq. 9) between the two signals is shown on every plot, calculated between 0 and 150 ms.

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  FIGURE 11. Effect of (A) the distance of the electrode to endocardium h and (B) interelectrode distance d on bipolar electrogram signals in the isotropic homogeneous medium. Here again, the two signals were given as follows: exact difference between two unipolar signals (Eq. 35) (solid colored) and the simulated signal (solid gray) (only for h = 1 mm and d = 2 mm). The comparison was carried out both for the parallel and normal electrical field component. Here, we used x = 0 mm, ϕ max = 30 mV, ϕ min = −86 mV, g i,xx = 0.1527 S/m, g~=0.8341 S/m, L = 5 mm, and v = 0.42 mm/ms. For parameters for which there is simulated data, the RMSE (Eq. 9) is given on the plot, calculated between 0 and 150 ms.

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  FIGURE 12. Results of feature extraction on numerically simulated electrograms for different catheter orientations (α, β ∈ [0°, 30°, 60°, 90°]). The results are shown for simulation parameters L = 5 mm, d = 2 mm, and h = 1 mm. Extracted features are the local maximum (orange), minimum (purple), and EGM width (green).