Stable time integration suppresses unphysical oscillations in the bidomain model

Ziaratgahi, Saeed Torabi; Marsh, Megan E.; Sundnes, Joakim; Spiteri, Raymond J.

doi:10.3389/fphy.2014.00040

Cited by 9 publications

(4 citation statements)

References 22 publications

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“…Indeed, such numerical artifacts may lead to errant physiological conclusions as in the case of the extinction of a spiral wave illustrated in [11]. Such oscillations in the evolution of the wavefront have been also observed in [12] (for electromechanics) with different numerical integration schemes for the ionic currents and in [13] (for the bidomain model) with the Crank–Nicolson method. The main purpose of this paper is to study a positive CVFE scheme, eliminating the generation of oscillations, for the monodomain model coupled to a physiologically based ionic model: the Beeler–Reuter model.…”

Section: Introductionmentioning

confidence: 82%

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

Bendahmane

Mroue

Saad

2020

Numerical Methods Partial

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The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.

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Section: Introductionmentioning

confidence: 82%

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

Bendahmane

Mroue

Saad

2020

Numerical Methods Partial

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“…For the SM split we refrain using higher order schemes, as our experimental studies suggest that the presented schemes preserve physiological features to sufficient precision for practical use. Note that some higher order schemes can introduce additional numerical oscillations 67 …”

Section: Discussionmentioning

confidence: 99%

“…Note that some higher order schemes can introduce additional numerical oscillations. 67 It should be noted that the proposed scheme was not combined with other advanced techniques to improve solver efficiency. These techniques include lookup tables and partial evaluation 68 or partitioned exponential schemes as the one developed by Rush and Larsen.…”

Section: Limitationsmentioning

confidence: 99%

A simple and efficient adaptive time stepping technique for low‐order operator splitting schemes applied to cardiac electrophysiology

Ogiermann

Perotti

Balzani

2023

Numer Methods Biomed Eng

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We present a simple, yet efficient adaptive time stepping scheme for cardiac electrophysiology (EP) simulations based on standard operator splitting techniques. The general idea is to exploit the relation between the splitting error and the reaction's magnitude—found in a previous one‐dimensional analytical study by Spiteri and Ziaratgahi—to construct the new time step controller for three‐dimensional problems. Accordingly, we propose to control the time step length of the operator splitting scheme as a function of the reaction magnitude, in addition to the common approach of adapting the reaction time step. This conforms with observations in numerical experiments supporting the need for a significantly smaller time step length during depolarization than during repolarization. The proposed scheme is compared with classical proportional–integral–differential controllers using state‐of‐the‐art error estimators, which are also presented in details as they have not been previously applied in the context of cardiac EP with operator splitting techniques. Benchmarks show that choosing the time step as a sigmoidal function of the reaction magnitude is highly efficient and full cardiac cycles can be computed with precision even in a realistic biventricular setup. The proposed scheme outperforms common adaptive time stepping techniques, while depending on fewer tuning parameters.

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“…The oscillations are a manifestation of the Gibbs phenomenon (Gottlieb and Shu, 1997 ). Methods to eliminate these oscillations include mass lumping and modified ODE integration schemes (Torabi Ziaratgahi et al, 2014 ). It has been suggested that these methods may not improve the underlying numerical stability of the solutions (Gresho and Lee, 1981 ).…”

Section: Discussionmentioning

confidence: 99%

High-order finite element methods for cardiac monodomain simulations

et al. 2015

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Computational modeling of tissue-scale cardiac electrophysiology requires numerically converged solutions to avoid spurious artifacts. The steep gradients inherent to cardiac action potential propagation necessitate fine spatial scales and therefore a substantial computational burden. The use of high-order interpolation methods has previously been proposed for these simulations due to their theoretical convergence advantage. In this study, we compare the convergence behavior of linear Lagrange, cubic Hermite, and the newly proposed cubic Hermite-style serendipity interpolation methods for finite element simulations of the cardiac monodomain equation. The high-order methods reach converged solutions with fewer degrees of freedom and longer element edge lengths than traditional linear elements. Additionally, we propose a dimensionless number, the cell Thiele modulus, as a more useful metric for determining solution convergence than element size alone. Finally, we use the cell Thiele modulus to examine convergence criteria for obtaining clinically useful activation patterns for applications such as patient-specific modeling where the total activation time is known a priori.

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Stable time integration suppresses unphysical oscillations in the bidomain model

Cited by 9 publications

References 22 publications

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

A simple and efficient adaptive time stepping technique for low‐order operator splitting schemes applied to cardiac electrophysiology

High-order finite element methods for cardiac monodomain simulations

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