Rush-Larsen time-stepping methods of high order for stiff problems in cardiac electrophysiology

To address the issues of stability and accuracy for reaction-diffusion equations, the development of high order and stable time-stepping methods is necessary. This is particularly true in the context of cardiac electrophysiology, where reaction-diffusion equations are coupled with stiff ODE systems. Many research have been led in that way in the past 15 years concerning implicit-explicit methods and exponential integrators. In 2009, Perego and Veneziani proposed an innovative time-stepping method of order 2. In this paper we present the extension of this method to the orders 3 and 4 and introduce the Rush-Larsen schemes of order k (shortly denoted RL\_k). The RL\_k schemes are explicit multistep exponential integrators. They display a simple general formulation and an easy implementation. The RL\_k schemes are shown to be stable under perturbation and convergent of order k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL\_k method is numerically studied as applied to the membrane equation in cardiac electrophysiology. The RL k schemes are shown to be stable under perturbation and convergent oforder k. Their Dahlquist stability analysis is performed. They have a very large stability domain provided that the stabilizer associated with the method captures well enough the stiff modes of the problem. The RL k method is numerically studied as applied to the membrane equation in cardiac electrophysiology

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

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A domain decomposition strategy for a very high-order finite volumes scheme applied to cardiac electrophysiology

International audienceIn this paper, a domain decomposition technique for a very high-order finite volumes scheme is proposed. The objective is to obtain an efficient way to perform numerical simulations in cardiac electrophysiology. The aim is to extend a very high-order numerical scheme previously designed, where large stencils are used for polynomial reconstructions. Therefore, a particular attention has to be paid to maintain the scalability in parallel. Here, we propose to constrain the stencils inside the subdomains or their first layer of neighbors. The method is shown to remain accurate and to scale perfectly up to the level where there are not enough cells in the subdomains. Hence, these high-order schemes are proved to be efficient tools to perform realistic simulations in cardiac electrophysiology

The Discrete Duality Finite Volume Method for Convection Diffusion Problems

In this paper we extend the Discrete Duality Finite Volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cell-based gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to how convergence of the approximation and a global first-order convergence rate is derived. The theoretical results are confirmed bysome numerical experiments

A CeVeFE DDFV scheme for discontinuous anisotropic permeability tensors

International audienceIn this work we derive a formulation for discontinuous diffusion tensor for the Discrete Duality Finite Volume (DDFV) framework that is exact for affine solutions. In fact, DDFV methods can naturally handle anisotropic or non-linear problems on general distorded meshes. Nonetheless, a special treatment is required when the diffusion tensor is discontinuous across an internal interfaces shared by two control volumes of the mesh. In such a case, two different gradients are considered in the two subdiamonds centered at that interface and the flux conservation is imposed through an auxiliary variable at the interface

Image-Based Modeling of the Heterogeneity of Propagation of the Cardiac Action Potential. Example of Rat Heart High Resolution MRI

International audienceIn this paper we present a modified bidomain model, derived with homogenization technique from assumption of existence of diffusive inclusions in the cardiac tissue. The diffusive inclusions represent regions without electrically active myocytes, e.g. fat, fibrosis etc. We present the application of this model to a rat heart. Starting from high resolution (HR) MRI, geometry is built and meshed using image processing techniques. We perform a study on the effects of tissue heterogeneities induced with diffusive inclusions on the velocity and shape of the depolarization wavefront. We study several test cases with different geometries for diffusive inclusions, and we find that the velocity might be affected by 5% and up to 37% in some cases. Additionally, the shape of the wavefront is affected

Virtual electrode polarization and current activation with monodomain equations

The bidomain model is nowadays one of the most accurate mathematical descriptions of the electrical activity in the heart. From now on, it was believed to be the only model for accurate simulations of cardiac muscle stimulation in a clinically relevant manner, i.e. through extracellular electrodes. In this paper, we develop a computationally efficient and accurate approximation of the bidomain model that allows for extracellular stimulation and accounts for unequal anisotropy ratios between intra-and extra-cellular media: the current-lifted monodomain model. We prove its use in the isolated heart by reproducing four types of extracellular activation that exhibit virtual electrode polarization. The simplicity of the code implementation and the fact that the computational cost is equal to the standard monodomain model henceforth give an excellent alternative to the bidomain model for expensive simulations like arrhythmia, fibrillation, and their respective treatments, e.g. signal guided catheter ablation and defibrillation

Considering New Regularization Parameter-Choice Techniques for the Tikhonov Method to Improve the Accuracy of Electrocardiographic Imaging

The electrocardiographic imaging (ECGI) inverse problem highly relies on adding constraints, a process called regularization, as the problem is ill-posed. When there are no prior information provided about the unknown epicardial potentials, the Tikhonov regularization method seems to be the most commonly used technique. In the Tikhonov approach the weight of the constraints is determined by the regularization parameter. However, the regularization parameter is problem and data dependent, meaning that different numerical models or different clinical data may require different regularization parameters. Then, we need to have as many regularization parameter-choice methods as techniques to validate them. In this work, we addressed this issue by showing that the Discrete Picard Condition (DPC) can guide a good regularization parameter choice for the two-norm Tikhonov method. We also studied the feasibility of two techniques: The U-curve method (not yet used in the cardiac field) and a novel automatic method, called ADPC due its basis on the DPC. Both techniques were tested with simulated and experimental data when using the method of fundamental solutions as a numerical model. Their efficacy was compared with the efficacy of two widely used techniques in the literature, the L-curve and the CRESO methods. These solutions showed the feasibility of the new techniques in the cardiac setting, an improvement of the morphology of the reconstructed epicardial potentials, and in most of the cases of their amplitude

Impact of the Endocardium in a Parameter Optimization to Solve the Inverse Problem of Electrocardiography

Electrocardiographic imaging aims at reconstructing cardiac electrical events from electrical signals measured on the body surface. The most common approach relies on the inverse solution of the Laplace equation in the torso to reconstruct epicardial potential maps from body surface potential maps. Here we apply a method based on a parameter identification problem to reconstruct both activation and repolarization times. From an ansatz of action potential, based on the Mitchell-Schaeffer ionic model, we compute body surface potential signals. The inverse problem is reduced to the identification of the parameters of the Mitchell-Schaeffer model. We investigate whether solving the inverse problem with the endocardium improves the results or not. We solved the parameter identification problem on two different meshes: one with only the epicardium, and one with both the epicardium and the endocardium. We compared the results on both the heart (activation and repolarization times) and the torso. The comparison was done on validation data of sinus rhythm and ventricular pacing. We found similar results with both meshes in 6 cases out of 7: the presence of the endocardium slightly improved the activation times. This was the most visible on a sinus beat, leading to the conclusion that inclusion of the endocardium would be useful in situations where endo-epicardial gradients in activation or repolarization times play an important role

C.E.P.S. : an efficient tool for cardiac electrophysiology simulations

International audienceNumerical models become a new and important tool to understand the mechanisms of cardiac arrythmias, delivering more and more accurate in-silico experiments. Beyond the development of mathematical models or numerical algorithms, a software tool must be developed to support this research. C.E.P.S. (Cardiac ElectroPhysiology Simulator) is a software tool under development by Inria Carmen team. Its purpose is to provide researchers from the modelling group, and collaborators, with a common environment to develop efficiently new models and numerical methods for cardiac electrophysiology. CEPS is designed to run on massively parallel architectures, and to make use of state-of-the-art and well known computing libraries to achieve realistic and complex heart simulations. Our short-term goals include solving monodomain and bidomain equations on 3D domain representing major structures of the heart (ventricles, atria and Purkinje fibers). CEPS supports the coupling surface/volume elements, surface/cable elements and volume/cable elements in order to include the complete structure of the heart. It is also designed to simulate electrocardiograms following heart/torso coupling. We also aim to automatically incorporate ionic models from CellML or JSIM databases. The structure of the code allows to easily include new PDE/ODE systems, to account for progresses in modelling, but also elements or numerical methods of arbitrary order of accuracy, for research on more efficient numerical solvers.Les modèles numériques sont un outil nouveau et important pour la compréhension des mécanismes des arythmies cardiaques, fournissant des expériences in-silico de plus en plus précises. Au-delà du développement de modèles mathématiques et d'algorithmes numériques, un logiciel doit être développé pour soutenir cette recherche. C.E.P.S. (Cardiac ElectroPhysiology Simulator ) est un outil logiciel en cours de développement par l'équipe-projet Inria Carmen. Son but est de fournir aux chercheurs du groupe de modélisation et ses collaborateurs avec un environnement commun pour développer de nouveaux modèles et méthodes numériques efficaces pour l'électrophysiologie cardiaque. CEPS est conçu pour fonctionner sur les architectures massivement parallèles, et utilise des bibliothèques de calcul bien connues pour réaliser des simulations cardiaques réalistes et complexes. Nos objectifs à court terme comprennent la résolution des équations monodomaine et bidomaine sur le domaine 3D représentant les grandes structures du cœur (ventricules , oreillettes et réseau de conduction cardique). CEPS prend en charge les éléments de couplage surface / volume, surface / câble et volume / câble afin d'y inclure la structure complète du cœur. CEPS est également conçu pour simuler les électrocardiogrammes suivants le couplage coeur / torse . Nous visons également à incorporer automatiquement des modèles ioniques des bases de données de CellML ou JSIM. La structure du code permet d'inclure facilement de nouvelles EDP / systèmes d'EDO