(Color online). (a) Example of the phase field along a one-dimensional section from the slice of rabbit ventricles shown in (b). The solid line indicates the original values of 1 or 0 assigned to and the open circles represent the final steady-state values of after solving Eq. (3) using . (c) Different solutions for using , 0.05, and corresponding to diamonds, circles and stars, respectively. Note that for the choice of in Eq. (3) , matches , as shown by the long dashed curve when using and the short dashed curve when using . The width of the interface is approximately .
(Color online). Fiber orientation information for the anatomical model of rabbit ventricles of Ref. 20 . (a) and (b) Slices (a) and (b) from the apex showing the projections of the three-dimensional fibers in the plane. (c) The three components of the fiber orientation vectors along the line in (b) for points inside the original domain (white area, length between 0.57 and 0.9 cm) and points in the enlarged domain (gray areas) whose fiber values originally were undefined. Continuity is present in all components. See text for details of the iterative process used to determine values for previously undefined fiber orientations.
Representative action potentials as measured in uniform cables from (a) the Nygren et al. human atrial model (Ref. 38 ), (b) the Fox et al. canine ventricular model (Ref. 39 ), and (c) the phenomenological ionic model in Ref. 12 with parameters to reproduce canine epicardial cells. Note that the Fox et al. model has a faster upstroke than the phenomenological model (values of are 270 and , respectively). (d) The upstrokes corresponding to the models shown in (a) (long dashes), (b) (solid), and (c) (dashes). Models with different values of are used to show its effect on the accuracy of the solutions obtained using the phase-field method (see Fig. 5 ).
Membrane potential distribution along a -long cable at different times using the phase-field method with , , and (symbols) and using a standard zero-flux finite-difference code with the same discretization. The initial condition is a brief excitation at the center of the cable at . This produces a symmetric excitation that propagates to the edges. (a) Voltage distribution during depolarization (all voltage values are ). Initial time is , final time is , and the voltage distribution is plotted every . (b) Voltage distribution during repolarization (all voltage values are ). Initial time is , final time is , and the voltage profile is plotted every . (c) Comparison of action potentials at the boundary using finite differences (solid) and the phase-field method (dotted), with the two upstrokes highlighted in the inset. The phenomenological model is used.
(a) Relative error in the maximum upstroke velocity as a function of the phase field width for one-dimensional cable simulations as shown in Fig. 4 for the Fox et al. (short dashes), Nygren et al. (long dashes), and phenomenological (solid) models. Note that for the values of used throughout this paper, the relative error is less than 10%. (b) Cumulative error in action potential for the same cases. The cumulative error is obtained by computing the absolute error in voltage over the time course of one action potential and then computing the ratio of the area under that curve to the area under the curve of the action potential calculated with the finite difference code. (c) Relative error in action potential duration for the same cases. In all cases, and . Slight increases in error can be observed for the smallest values of because for these values the interface is very steep and is not adequately resolved by the fixed used.
(Color online). Propagation of a point stimulus applied off-center in a quarter-annulus. (a)–(d) Electrical excitation (orange) propagating into quiescent tissue (black) at times , 20, 35, and , respectively. (e) Comparison of wave front contours at intervals for the solution using the phase-field method in Cartesian coordinates (red, symbols) and the reference solution using polar coordinates (black, solid). Grid spacings are and or 0.45°, with .
(Color online). Propagation of a point stimulus applied (a) to the lower right corner of a quarter-annulus, (b) to the lower right corner of a quarter-annulus with a hole, and (c) near the right edge of an anisotropic square domain with a hole. Wave front contours are shown at intervals for the solution using the phase-field method (red, symbols) and for the reference solution using finite differences (black, solid). Reference solutions are obtained using polar coordinates in (a) and (b) and using standard finite differences in (c). Note that the contours are normal to all the boundaries for both solutions in (a) and (b). The ratio of diffusion constants parallel and perpendicular to the fibers in (c) is and the fiber angle is . Grid spacings are and or 0.45° with for (a) and (b), while for (c) the grid spacing is with and .
Maximum relative error in propagation velocity in the domain shown in Fig. 7(c) as a function of the , the width of the phase field. For values of below , the error is less than 10%. The inclusion of anisotropy increases the error by 2%–4% compared to an isotropic simulation for nearly all values of tested.
(Color). Example simulations using the phase-field method in complex cardiac geometries. (a) Single scroll wave in the rabbit ventricular model. The left and right images show posterior and anterior views, respectively. (b) Slabs of the rabbit ventricles during scroll wave propagation (posterior view). The slabs are perpendicular to the apex-base axis and proceed toward the apex. (c) Propagation of an electrical wave in the canine ventricles after a stimulus along a simulated Purkinje network. The left image shows an anterior view of the ventricles with a small portion cut out to allow the endocardium to be seen. The cut-out view on the right shows the anterior endocardium. (d) Two views of a spiral wave in the anatomical model of the human atria. The Nygren et al. model of human atrial cells is used. Electrical potential is color-coded with red corresponding to strongly depolarized tissue and blue corresponding to repolarized tissue. In all cases, grid spacing is , and the phase-field control parameter is . Time steps are for (a) and (b), for (c), and for (d).
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