Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology

Bendahmane, Mostafa; Mroue, Fatima; Saad, Mazen; Talhouk, Raafat

doi:10.1016/j.nonrwa.2019.05.006

Cited by 11 publications

(26 citation statements)

References 31 publications

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“…Then, the existence of the weak solution is given in the following theorem with the proof can be found in [11] where the mesoscopic domain is ignored.…”

Section: Definition 2 (Weak Formulation)mentioning

confidence: 99%

“…The model that describes the electrical activity of the heart, is called by "Bidomain model". The authors in [11] established the well-posedness of this microscopic bidomain model under different conditions and proved the existence and uniqueness of their solutions (see also [28,45,13] for other approaches).…”

Section: Introductionmentioning

confidence: 99%

“…Amar et al [6] studied a hierarchy of electrical conduction problems in biological tissues via two-scale convergence. Recently, the authors in [11] proved the existence and uniqueness of solution of the microscopic bidomain model based on Faedo-Galerkin method. Further, they used the unfolding homogenization method at two scales to show that the solution of the microscopic biodmain model converges to the solution of the macroscopic one.…”

Section: Introductionmentioning

confidence: 99%

“…Further, to pass to the limit in nonlinear terms, we use the technique involving the unfolding operator introduced in [26] and a Kolmogorov-Riesz compactness's result. Second, we will follow the standard unfolding method on the extracellular one (similar derivation may be found in [11]). An important motivation for our investigations is their application to unfolding homogenization method proposed to the effective properties of the cardiac tissue at micro-meso-macro scales.…”

Section: Introductionmentioning

confidence: 99%

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Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

et al. 2021

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In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra-and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness's result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. 2020 Mathematics Subject Classification. 65N55 and 35A01 and 35B27 and 35K57 and 65M..

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“…Then, the existence of the weak solution is given in the following theorem with the proof can be found in [11] where the mesoscopic domain is ignored.…”

Section: Definition 2 (Weak Formulation)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

et al. 2021

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“…In this model, the anisotropic cardiac tissue is represented by averaging electric properties over a length scale greater than that of a single cell. We refer to [4] for a derivation of the bidomain equations, to [5, 6] for detailed reviews and to [7] for a rigorous derivation of the bidomain equations from a microscopic model of cardiac electrophysiology. Assuming an additional condition on the anisotropy, a simpler version is obtained and is called the monodomain model.…”

Section: Introductionmentioning

confidence: 99%

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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On uniqueness theorems for the inverse problem of electrocardiography in the Sobolev spaces

Kalinin¹,

Shlapunov

Ushenin³

2022

Z Angew Math Mech

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We consider a mathematical model related to reconstruction of cardiac electrical activity from ECG measurements on the body surface. An application of recent developments in solving boundary value problems for elliptic and parabolic equations in Sobolev type spaces allows us to obtain uniqueness theorems for the model. The obtained results can be used as a sound basis for creating numerical methods for non-invasive mapping of the heart.

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Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology

Cited by 11 publications

References 31 publications

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

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

On uniqueness theorems for the inverse problem of electrocardiography in the Sobolev spaces

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