A Domain Decomposition Approch in the Electrocardiography Inverse Problem

International audienceThe mostly used mathematical formulation of the inverse problem in electrocardiography is based on a least method using a transfer matrix that maps the electrical potential on the heart to the body surface potential (BSP). This mathematical model is ill based and a lot of works have been concentrating on the regularization term without thinking of reformulating the problem itself. We propose in this study to solve the inverse problem based on a domain decomposition technique on a fictitious mirror-like boundary conditions. We conduct BSP simulations to produce synthetic data and use it to evaluate the accuracy of the inverse problem solution

Evaluation of Fifteen Algorithms for the Resolution of the Electrocardiography Imaging Inverse Problem Using ex-vivo and in-silico Data

The electrocardiographic imaging inverse problem is ill-posed. Regularization has to be applied to stabilize the problem and solve for a realistic solution. Here, we assess different regularization methods for solving the inverse problem. In this study, we assess (i) zero order Tikhonov regularization (ZOT) in conjunction with the Method of Fundamental Solutions (MFS), (ii) ZOT regularization using the Finite Element Method (FEM), and (iii) the L1-Norm regularization of the current density on the heart surface combined with FEM. Moreover, we apply different approaches for computing the optimal regularization parameter, all based on the Generalized Singular Value Decomposition (GSVD). These methods include Generalized Cross Validation (GCV), Robust Generalized Cross Validation (RGCV), ADPC, U-Curve and Composite REsidual and Smoothing Operator (CRESO) methods. Both simulated and experimental data are used for this evaluation. Results show that the RGCV approach provides the best results to determine the optimal regularization parameter using both the FEM-ZOT and the FEM-L1-Norm. However for the MFS-ZOT, the GCV outperformed all the other regularization parameter choice methods in terms of relative error and correlation coefficient. Regarding the epicardial potential reconstruction, FEM-L1-Norm clearly outperforms the other methods using the simulated data but, using the experimental data, FEM based methods perform as well as MFS. Finally, the use of FEM-L1-Norm combined with RGCV provides robust results in the pacing site localization

A conduction velocity adapted eikonal model for electrophysiology problems with re-excitability evaluation

Computational models of heart electrophysiology achieved a great interest from the medical community since they represent a novel framework to study the mechanisms that underpin heart pathologies. The high demand of computational resources and the long computational times required to evaluate the model solution hamper the use of detailed computational models in clinical applications. In this paper, we propose a multi-front eikonal algorithm capable of adapting the conduction velocity (CV) to the activation frequency of the tissue substrate. We then couple the new eikonal model with a Mitchell-Schaeffer (MS) ionic model to determine the tissue electrical state. Compared to the standard eikonal model, this model introduces three novelties: first, the local value of the transmembrane potential and of the ionic variable are known from the solution of the ionic model; second, the action potential duration (AP D) and the diastolic interval (DI) are computed from the solution of the MS model and used to determine when a part of the tissue is re-excitable. Third, CV is locally adapted to the underpinning electrophysiological state through the analytical CV restitution expression and the computed local DI. We conduct series of simulations on a tissue slab and on 3D realistic heart geometry and compare results to the monodomain. Our results show that the new model is much more accurate than the standard eikonal model. This model enables the numerical simulation of the heart electrophysiology on a clinical time scale and thus constitutes a good model candidate for computer-guided cardiac therapy

On Simulating the Effect of Sodium Channel Block on Cardiac Electromechanics

International audienceObjective: The purpose of this paper is to investigate computationally the influence of sodium ion channel block on cardiac electro-mechanics. Methods: To do so, we implement a myofiber orientation dependent passive stress model (Holzapfel-Ogden) in the multiphysics solver Chaste to simulate an imaged physiological model of the human ventricles. A dosage of a sodium channel blocker was then applied and its inhibitory effects on the electrical propagation across ventricles modeled. We employ the Kirchhoff active stress model to generate electrically excited contractile behavior of myofibers. Results: Our predictions indicate that a delay in the electrical activation of ventricular tissue caused by the sodium channel block translates to a delay in the mechanical biomarkers that were investigated. Moreover, sodium channel block was found to increase left ventricular twist. Conclusion: A multiphysics computational framework from the cell level to the organ level was used to predict the effect of sodium channel blocking drugs on cardiac electromechanics. Significance: There is growing interest to better understand drug-induced cardiovascular complications and to predict undesirable side effects at as early a stage in the drug development process as possible

A coupled system of PDEs and ODEs arising in electrocardiograms modelling

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure

Towards the modelling of the Purkinje/ myocardium coupled problem: A well-posedness analysis

International audienceThe Purkinje network is the specialized conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, in an insulated heart framework, we model the free running Purkinje system, using the monodomain equation. The intra-myocardium part of the Purkinje fiber is coupled to the ventricular tissue using the bidomain equation. The coupling is performed through the extracellular potential. We discretize the problem in time using a semi-implicit scheme. Then, we write a variational formulation of the semi discrete problem in a non standard weighted Sobolev functional spaces. We prove the existence and uniqueness of the solution of the Purkinje/myocardium semi-discretized problem. We discretize in space by the finite element P 1 − Lagrange and conduct some numerical tests showing the anterograde and retrograde propagation of the electrical wave between the tissue and the Purkinje fibers

Space rescaling in the MFS method improves the ECGI reconstruction

International audienceThe method of fundamental solutions (MFS) has been extensively used for the electrocardiographic imaging (ECGI) inverse problem. One of its advantages is that it is a meshless method. We remarked that the using cm instead of mm as a space unit has a high impact on the reconstructed inverse solution. Our purpose is to refine this observation, by introducing a rescaling coefficient in space and study its effect on the MFS inverse solution. Results are provided using simulated test data prepared using a reaction-diffusion model. We then computed the ECGI inverse solution for rescaling coefficient values varying from 1 to 100, and computed the relative error (RE) and correlation coefficient (CC). This approach improved the RE and CC by at least 10% but can go up to 40% independently of the pacing site. We concluded that the optimal coefficient depends on the heterogeneity and anisotropy of the torso and does not depend on the stimulation site. This suggests that it is related to an optimal equivalent conductivity estimation in the torso domain

Ionic Parameters Estimation in Multi-Scale Cardiac Electrophysiology Modelling

In this work, we present an optimal control formulation for the bidomain model in order to estimate maximal conductance parameters in cardiac electrophysiology multiscale modelling. We consider a general Hodgkin-Huxley formalism to describe the ionic exchanges at the microscopic level. We treat the desired parameters as control variables in a cost function minimizing the gap between the measured and the computed transmembrane potentials. First, we establish the existence of an optimal control solution and we formally derive the optimality system. Second, we propose a strategy for solving the estimation problem for both single and multiple parameters cases. Our algorithm is based on a gradient descent method, where the gradient is obtained by solving an adjoint problem. Both the state and the adjoint problems are solved using the finite element method. Numerical simulations for single and multiple conductances estimations show the capability of this approach to identify the values of sodium, calcium and potassium ion channels conductances of the Luo Rudy phase I model

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