Mathematical and numerical methods for reaction-diffusion models in electrocardiology

Colli-Franzone, P.; Pavarino, Luca F.; Scacchi, Simone

doi:10.1007/978-88-470-1935-5_5

Cited by 12 publications

(25 citation statements)

References 136 publications

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“…For a model that contains K different ion channels for different ionic species (the most important ones being K + , Na + , and Ca 2+ ), a system of ordinary differential equations expressing the conservation of charges reads [19]:…”

Section: Single-cell Electrophysiology Modelsmentioning

confidence: 99%

“…Finally, by combining (16) and (19) in a coupled problem, we obtain the full spatial approximation of the monodomain problem (7) by means of NURBS-based IGA, which at the discrete level reads:…”

Section: Spatial Approximation: Iga For Surface Pdesmentioning

confidence: 99%

“…A further assumption of equal anisotropic conductivities in the intra-and extracellular compartments leads to a simplified formulation in terms of the transmembrane potential that is defined as the difference of the two potentials, called the monodomain equation, requiring only the solution of a single reaction-diffusion equation. For details on the derivation of the equations and further bibliography on their mathematical approximation, see [19,20].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

Patelli

Dedè

Lassila

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ≥ 1 and globally C k -continuous, with k = 0 or p − 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom.We show that B-spline basis functions of degree p ≥ 1, which are C p−1 -continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that, for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting.

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Section: Single-cell Electrophysiology Modelsmentioning

confidence: 99%

Section: Spatial Approximation: Iga For Surface Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

Patelli

Dedè

Lassila

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…[7,27,31] for a derivation of the Bidomain model from homogenization of cellular models. We recall that the cardiac tissue consists of an arrangement of fibers that rotate counterclockwise from epi-to endocardium, and that have a laminar organization modeled as a set of muscle sheets running radially from epi-to endocardium.…”

Section: The Anisotropic Bidomain Modelmentioning

confidence: 99%

“…A good compromise between stability and efficiency is obtained using linear implicit methods, studied e. g. in [9,10,24], which require at each time step the solution of 2-4 linear systems, or semi-implicit methods, studied e.g. in [6,7,14,43,55]. For a detailed comparative study on the stability and accuracy of several Bidomain time discretizations (implicit, semi-implicit, explicit), we refer to the recent work [14].…”

Section: Time Discretizationmentioning

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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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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Mathematical and numerical methods for reaction-diffusion models in electrocardiology

Cited by 12 publications

References 136 publications

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

Isogeometric approximation of cardiac electrophysiology models on surfaces: An accuracy study with application to the human left atrium

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

Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model

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