A smoothed boundary bidomain model for cardiac simulations in anatomically detailed geometries

Figures

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  Fig 1. Schematic diagram of the considered geometry. ϕi and Vm are defined in the heart domain H, whereas ϕo is defined both in H and in the external conductor T. ∂H indicates the boundary between the heart and torso domains. The phase field ψ introduces a smooth interface between the heart domain (where ψ = 1) and the external conductor (where ψ = 0).

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  Fig 2. Comparison between SBB solution (colored lines) and the reference solution (filled markers) in a 1D cable model. A) Membrane potential along the cable in three distinct time instants. B) Extracellular potential along the cable in three distinct time instants. C) Mean absolute error in the membrane potential at each time step. D) Mean absolute error in the extracellular potential at each time step. E) Extracellular potential along the cable in three different time instants corresponding to the arrival of the action potential at the right end of the cable. F) Membrane potential in the right end of the cable. The inset zooms on the upstroke phase.

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  Fig 3. Grid convergence tests of the SBB method for different values of ξ with and without operator splitting. A) Time-averaged mean absolute error in membrane potential and extracellular potential obtained with the SBB method without operator splitting for different values of ξ and dx. Missing elements indicate that the SBB algorithm without operator splitting did not achieve a solution. B) Time-averaged mean absolute error in membrane potential and extracellular potential obtained with the SBB method exploiting operator splitting for different values of ξ and dx.

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  Fig 4. Comparison between SBB and polar FD methods in a 2D isotropic annular geometry with cathodic extracellular stimulation. A) Contour plots of membrane potential (Vm = −40 mV) at different times for the SBB and polar FD methods. B) Comparison of extracellular potential field obtained with SBB and polar FD methods at two different time instants. Dashed black lines indicate the boundary between the cardiac tissue and the external conductor.

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  Fig 5. Comparison between SBB and polar FD methods in a 2D anisotropic annular geometry with cathodic extracellular stimulation. A) Contour plots of membrane potential (Vm = −40 mV) at different times for the SBB and polar FD methods. B) Comparison of extracellular potential field obtained with SBB and polar FD methods at two different time instants. Dashed black lines indicate the boundary between the cardiac tissue and the external conductor.

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  Fig 6. Comparison between SBB and polar FD methods in a 2D anisotropic annular geometry with anodic extracellular stimulation. A) Contour plots of membrane potential (Vm = −40 mV) at different times for the SBB and polar FD methods. B) Comparison of extracellular potential field (first line) and membrane potential (second line) obtained with SBB and polar FD methods at two different time instants. Dashed black lines indicate the boundary between the cardiac tissue and the external conductor. Membrane potential is zoomed at the stimulation region.

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  Fig 7. Grid convergence tests of the SBB method for different values of ξ with and without operator splitting in the 2D annular geometry. A) Time-averaged mean absolute error in membrane potential and extracellular potential obtained with the SBB method without operator splitting for different values of ξ and dx. Missing elements indicate that the SBB algorithm without operator splitting did not achieve a solution. B) Time-averaged mean absolute error in membrane potential and extracellular potential obtained with the SBB method exploiting operator splitting for different values of ξ and dx.

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  Fig 8. 3D anatomically detailed human cardiac simulation with a weak (top) and strong (bottom) defibrillation shock. A slice of the left ventricle (z = 2.25 cm) is shown on the right side. The weak shock induces membrane depolarization at cardiac surfaces, mainly on the epicardial site. Virtual electrodes were generated around the large valve openings. The strong shock induces larger depolarized regions with virtual electrodes located also inside the myocardium.