The electrical activity of the heart is a complex phenomenon strictly related to its physiology, fiber structure and anatomy. At the cellular level the cell membrane separates both the intra- and extracellular environments consisting of a dilute aqueous solution of dissolved salts dissociated into ions. Differences in ion concentrations on opposite sides of the membrane lead to a voltage called the transmembrane potential, v_M, defined as the difference between the intra- and extracellular potentials, (u_I and u_E). The bioelectric activity of a cardiac cell is described by the time course of v_M, the so called action potential. At the tissue level the most complete mathematical model of cardiac electrophysi-ology is the Bidomain model, consisting of a degenerate reaction-diffusion system of a parabolic and an elliptic partial differential equation modelling v_m and u_E of the anisotropic cardiac tissue, coupled nonlinearly with a membrane model. The multiscale nature of the Bidomain models yields very high computational costs for its numerical resolution. The starting point for a spatial discretization is a geometrical representation that encompasses the required anatomical and structural details, and that is also suitable for computational studies. Detailed models were proposed based on structured grids with cubic Hermite interpolation functions, which enable a smooth representation of ventricular geometry with relatively few elements, see e.g. . In this study we used an alternative approach based on Isogeometric Analysis (IGA), a novel method for the discretization of partial differential equations introduced in. This method adopts the same spline or Non-Uniform Rational B-spline (NURBS) basis functions used to design domain geometries in CAD to construct both trial and test spaces in the discrete variational formulation of the differential problem, and provides a higher control on the regularity of the discrete space. The IGA discretization of the Bidomain model in space and semi-implicit (IMEX) finite differences in time lead to the resolution at each time step of a large and very ill-conditioned linear system. Since the iteration matrix is symmetric semidefinite, it is natural to use the preconditioned conjugate gradient method.
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