scholarly journals Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model

2018 ◽  
Vol 28 (05) ◽  
pp. 979-1035 ◽  
Author(s):  
Annabelle Collin ◽  
Sébastien Imperiale
Keyword(s):  
Mathematical Analysis ◽  
Cardiac Electrophysiology ◽  
Convergence Theory ◽  
Bidomain Model ◽  
Periodic Homogenization ◽  
Homogenization Procedure ◽  
Uniform Estimates ◽  
Bidomain Equations ◽  
Homogenization Process

The aim of this paper is to provide a complete mathematical analysis of the periodic homogenization procedure that leads to the macroscopic bidomain model in cardiac electrophysiology. We consider space-dependent and tensorial electric conductivities as well as space-dependent physiological and phenomenological nonlinear ionic models. We provide the nondimensionalization of the bidomain equations and derive uniform estimates of the solutions. The homogenization procedure is done using 2-scale convergence theory which enables us to study the behavior of the nonlinear ionic models in the homogenization process.

scholarly journals Optimal monodomain approximations of the bidomain equations used in cardiac electrophysiology

2014 ◽  
Vol 24 (06) ◽  
pp. 1115-1140 ◽  
Author(s):  
Yves Coudière ◽  
Yves Bourgault ◽  
Myriam Rioux
Keyword(s):  
Boundary Conditions ◽  
Cardiac Electrophysiology ◽  
Propagation Speed ◽  
Neumann Boundary ◽  
Test Cases ◽  
Bidomain Model ◽  
Bidomain Equations ◽  
Monodomain Model ◽  
The Difference ◽  
Computationally Intensive

The bidomain model is the current most sophisticated model used in cardiac electrophysiology. The monodomain model is a simplification of the bidomain model that is less computationally intensive but only valid under equal conductivity ratio. We propose in this paper optimal monodomain approximations of the bidomain model. We first prove that the error between the bidomain and monodomain solutions is bounded by the error ‖B - A‖ between the bidomain and monodomain conductivity operators. Optimal monodomain approximations are defined by minimizing the distance ‖B - A‖, which reduces for solutions over all ℝd to minimize the Lp norm of the difference between the operator symbols. Similarly, comparing the symbols pointwise amounts to compare the propagation of planar waves in the bidomain and monodomain models. We prove that any monodomain model properly propagates at least d planar waves in ℝd. We next consider and solve the optimal problem in the L∞ and L2 norms, the former providing minimal propagation error uniformly over all directions. The quality of these optimal monodomain approximations is compared among themselves and with other published approximations using two sets of test cases. The first one uses periodic boundary conditions to mimic propagation in ℝd while the second is based on a square domain with common Neumann boundary conditions. For the first test cases, we show that the error on the propagation speed is highly correlated with the error on the symbols. The second test cases show that domain boundaries control propagation directions, with only partial impact from the conductivity operator used.

Numerical solution of the bidomain equations

2009 ◽  
Vol 367 (1895) ◽  
pp. 1931-1950 ◽  
Author(s):  
S. Linge ◽  
J. Sundnes ◽  
M. Hanslien ◽  
G.T. Lines ◽  
A. Tveito
Keyword(s):  
Mathematical Models ◽  
Cardiac Electrophysiology ◽  
Mathematical Reasoning ◽  
Full Potential ◽  
Processing Unit ◽  
Bidomain Model ◽  
Central Processing ◽  
Cellular Models ◽  
Bidomain Equations ◽  
The Past

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.

scholarly journals Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations

2022 ◽  
Author(s):  
Sunghan Kim ◽  
Ki-Ahm Lee
Keyword(s):  
Elliptic Equations ◽  
Linear Equations ◽  
Nonlinear Elliptic Equations ◽  
Periodic Homogenization ◽  
Fully Nonlinear Elliptic Equations ◽  
Fully Nonlinear ◽  
Approximation Techniques ◽  
Uniform Estimates ◽  
Nonlinear Elliptic ◽  
Linearized Operator

AbstractThis article is concerned with uniform $$C^{1,\alpha }$$ C 1 , α and $$C^{1,1}$$ C 1 , 1 estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.

scholarly journals Periodic homogenization and damage evolution in RVE composite material with inclusion

2021 ◽  
Vol 15 (58) ◽  
pp. 319-343
Author(s):  
Karim Benyahi ◽  
Youcef Bouafia ◽  
Mohand Said Kachi ◽  
Sarah Benakli ◽  
Amel Hamri
Keyword(s):  
Composite Materials ◽  
Homogenization Method ◽  
Periodic Homogenization ◽  
Homogenization Procedure ◽  
Element Analysis ◽  
Damage Models ◽  
Damage Process ◽  
The One ◽  
Calculation Code ◽  
Effective Mechanical Properties

This work deals with the coupling between a periodic homogenization procedure and a damage process occurring in a RVE of inclusion composite materials. We mainly seek on the one hand to determine the effective mechanical properties according to the different volume fractions and forms of inclusions for a composite with inclusions at the macroscopic level, and on the other hand to explore the rupture mechanisms that can take place at the microstructure level. To do this; the first step is to propose a periodic homogenization procedure to predict the homogenized mechanical characteristics of an inclusion composite. This homogenization procedure is applied to the theory based on finite element analysis by the Abaqus calculation code. The inclusions are modeled by a random object modeler, and the periodic homogenization method is implemented by python scripts. It is then a matter of introducing the damage into the problem of homogenization, that is to say; once the homogenized characteristics are assessed in the absence of the damage initiated by microcracks and micro cavitations, it is then possible to introduce damage models by a subroutine (Umat) in the Abaqus calculation code. The verifications carried out focused on RVE of composite materials with inclusions.

scholarly journals Parametrization of activation based cardiac electrophysiology models using bidomain model simulations

2016 ◽  
Vol 2 (1) ◽  
pp. 611-615
Author(s):  
Birgit Stender
Keyword(s):  
Conduction Velocity ◽  
Cardiac Electrophysiology ◽  
Approximate Solutions ◽  
Cell Types ◽  
Bidomain Model ◽  
Computationally Efficient ◽  
Model Equations ◽  
Cardiac Excitation ◽  
Different Cell Types ◽  
Excitation Propagation

AbstractEikonal models are useful to compute approximate solutions of cardiac excitation propagation in a computationally efficient way. In this work the underlying conduction velocities for different cell types were computed solving the classical bidomain model equations for planar wavefront propagation. It was further investigated how changes in the conductivity tensors within the bidomain model analytically correspond to changes in the conduction velocity. The error in the presence of local front curvature for the derived eikonal model parametrization were analyzed. The conduction velocity simulated based on the bidomain model was overestimated by a maximum of 10%.

scholarly journals An evaluation of some assumptions underpinning the bidomain equations of electrophysiology

2019 ◽  
Vol 37 (2) ◽  
pp. 262-302
Author(s):  
Jonathan P Whiteley
Keyword(s):  
Cardiac Electrophysiology ◽  
Cardiac Tissue ◽  
Tissue Level ◽  
Almost Periodic ◽  
Governing Equations ◽  
Fibre Direction ◽  
Common Assumption ◽  
Bidomain Equations ◽  
Extracellular Conductivity ◽  
Electrical Equilibrium

Abstract Tissue level cardiac electrophysiology is usually modelled by the bidomain equations, or the monodomain simplification of the bidomain equations. One assumption made when deriving the bidomain equations is that both the intracellular and extracellular spaces are in electrical equilibrium. This assumption neglects the disturbance of this equilibrium in thin regions close to the cell membrane known as Debye layers. We first demonstrate that the governing equations at the cell, or microscale, level may be adapted to take account of these Debye layers with little additional complexity, provided the permittivity within the Debye layers satisfies certain conditions that are believed to be satisfied for biological cells. We then homogenize the microscale equations using a technique developed for an almost periodic microstructure. Cardiac tissue is usually modelled as sheets of cardiac fibres stacked on top of one another. A common assumption is that an orthogonal coordinate system can be defined at each point of cardiac tissue, where the first axis is in the fibre direction, the second axis is orthogonal to the first axis but lies in the sheet of cardiac fibres and the third axis is orthogonal to the cardiac sheet. It is assumed further that both the intracellular and extracellular conductivity tensors are diagonal with respect to these axes and that the diagonal entries of these tensors are constant across the whole tissue. Using the homogenization technique we find that this assumption is usually valid for cardiac tissue, but highlight situations where the assumption may not be valid.

Homogenization Procedure Effect on Microscopical Performance of Concrete Containing Supplementary Cementitious Materials

2020 ◽  
Vol 995 ◽  
pp. 168-173
Author(s):  
Zdeněk Prošek ◽  
Vladimír Hrbek ◽  
Petr Bílý ◽  
Lukáš Vráblík
Keyword(s):  
High Performance ◽  
Chemical Resistance ◽  
High Performance Concrete ◽  
Cementitious Materials ◽  
Supplementary Cementitious Materials ◽  
Macro Level ◽  
Homogenization Procedure ◽  
Homogenization Process ◽  
Concrete Mixtures ◽  
Major Factors

The advantages of supplementary cementitious materials (SCM) use in concrete, such as reduced cement consumption and overall material improvement (durability, chemical resistance, etc.), are widely known. Our research focuses on two major factors connected to high performance concrete (HPC) containing SCM, the content of selected SCM and the homogenization process used for concrete mixture. Both of these aspects of this research were addressed by the authors from macro-level of the material. In this contribution, the focus is directed on microscopical performance of concrete mixtures with highest macro-mechanical features with respect to both homogenization procedure and SCM containment.

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