Computational techniques for solving the bidomain equations in three dimensions

Vigmond, Edward J.; Aguel, Felipe; Trayanova, Natalia A.

doi:10.1109/tbme.2002.804597

Cited by 204 publications

(157 citation statements)

References 18 publications

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“…The monodomain equation for impulse propagation, ƒ⅐G si ƒV m ϭ ␤ sv (C m dV m ͞dt ϩ I ion ), was solved by using a Galerkin finite element method and a linear triangular discretization with internodal distance of 100 m (24). For validation of the interpolation method, a meandering spiral wave was simulated on a 20-mm circular substrate by using the Karma action potential model (25).…”

Section: Methodsmentioning

confidence: 99%

Multiarm spirals in a two-dimensional cardiac substrate

Bursac

Aguel

Tung

2004

Proc. Natl. Acad. Sci. U.S.A.

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A variety of chemical and biological nonlinear excitable media, including heart tissue, can support stable, self-organized waves of activity in a form of rotating single-arm spirals. In particular, heart tissue can support stationary and meandering spirals of electrical excitation, which have been shown to underlie different forms of cardiac arrhythmias. In contrast to single-arm spirals, stable multiarm spirals (multiple spiral waves that rotate in the same direction around a common organizing center) have not been demonstrated and studied yet in living excitable tissues. Here, we show that persistent multiarm spirals of electrical activity can be induced in monolayer cultures of neonatal rat heart cells by a short, rapid train of electrical point stimuli applied during single-arm-spiral activity. Stable formation is accomplished only in monolayers that show a relatively broad and steep dependence of impulse wavelength and propagation velocity on rate of excitation. The resulting multiarm spirals emit waves of electrical activity at rates faster than for single-arm spirals and exhibit two distinct behaviors, namely ''arm-switching'' and ''tip-switching.'' The phenomenon of rate acceleration due to an increase in the number of spiral arms possibly may underlie the acceleration of functional reentrant tachycardias paced by a clinician or an antitachycardia device.R otating single-arm spirals can exist in different chemical(1-4) and biological systems (5-8). If a spiral is stationary or weakly meandering, it represents a persistent source of highly periodic activity with a period of rotation that is determined by the properties of excitation, recovery, and diffusion in the medium. In the heart, stable single-arm spirals can underlie periodic activity such as monomorphic tachycardia, whereas unstable spirals that continuously form and break up are shown to underlie aperiodic and lethal heart activity, namely fibrillation (9). So far, stable rotating structures with a higher degree of organization, such as multiarmed spirals, have been demonstrated experimentally in chemically (2) or light-controlled (10) Belousov-Zhabotinsky reactions, in starving social amoebae Dictyostelium discoideum (11), and in FitzHugh-Nagumo-type computer models of homogenous weakly excitable media (12)(13)(14). The modes of induction of these multiarmed spirals [i.e., use of an unexcitable central obstacle that is subsequently removed (2, 10) and spontaneous formation from dense multiple wavebreaks (12) or from preset spatial distributions of excited and recovered medium (13, 14)] are not applicable to cardiac tissue. Moreover, healthy cardiac tissue exhibits normal excitability, in contrast to two-variable FitzHugh-Nagumo models, which require weak excitability to support stable multiarm spirals (12-15). In a normally excitable medium, the existence of a stable multiarm spiral as an entity distinct from negligibly interacting adjacent single spirals was questioned by Winfree (16,17), who considered that in this case distinct rates of rotat...

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Section: Methodsmentioning

confidence: 99%

Multiarm spirals in a two-dimensional cardiac substrate

Bursac

Aguel

Tung

2004

Proc. Natl. Acad. Sci. U.S.A.

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“…This process of code platform development in parallel by multiple groups has led to a wide range of numerical techniques and solution strategies [13]. Furthermore, each software code is designed for a specific purpose, with customized input and output file formats, specific mesh structures and numerical schemes for the governing equations.…”

Section: Introductionmentioning

confidence: 99%

Verification of cardiac tissue electrophysiology simulators using an N -version benchmark

Niederer

Kerfoot

Benson

et al. 2011

Phil. Trans. R. Soc. A.

237

308

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One contribution of 11 to a Theme Issue 'Towards the virtual physiological human: mathematical and computational case studies'. Ongoing developments in cardiac modelling have resulted, in particular, in the development of advanced and increasingly complex computational frameworks for simulating cardiac tissue electrophysiology. The goal of these simulations is often to represent the detailed physiology and pathologies of the heart using codes that exploit the computational potential of high-performance computing architectures. These developments have rapidly progressed the simulation capacity of cardiac virtual physiological human style models; however, they have also made it increasingly challenging to verify that a given code provides a faithful representation of the purported governing equations and corresponding solution techniques. This study provides the first cardiac tissue electrophysiology simulation benchmark to allow these codes to be verified. The benchmark was successfully evaluated on 11 simulation platforms to generate a consensus gold-standard converged solution. The benchmark definition in combination with the gold-standard solution can now be used to verify new simulation codes and numerical methods in the future.

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“…Previous works have considered several variants of these splitting and/or uncoupling techniques. In particular, the uncoupled approach has been considered in [1,22,35,36,45,47,[51][52][53][54]. The splitting of reaction and diffusion terms has been considered in [35,53,54], while a three steps Strang splitting, with reaction -diffusion -reaction steps, has been considered in [22,47,51].…”

Section: Time Discretizationmentioning

confidence: 99%

“…Many different preconditioners have been proposed in order to devise efficient iterative solvers for the linear systems deriving from both the splitting techniques: Symmetric Successive Over Relaxation [32], block diagonal or triangular [6,17,33,34,52], optimized Schwarz [18], multigrid [1,[33][34][35]46,54], multilevel Schwarz [29,30,40,42].…”

Section: Introductionmentioning

confidence: 99%

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model

2013

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Abstract. The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.Mathematics Subject Classification. 65N55, 65M55, 65F10, 92C30.

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Computational techniques for solving the bidomain equations in three dimensions

Cited by 204 publications

References 18 publications

Multiarm spirals in a two-dimensional cardiac substrate

Multiarm spirals in a two-dimensional cardiac substrate

Verification of cardiac tissue electrophysiology simulators using an N -version benchmark

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model

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