3.1 Complex models for the propagation of cardiac action potentials

The contraction of the heart is coordinated by a complex electrical activation process which relies on about a million ion channels, pumps, and exchangers of various kinds in the membrane of each cardiac cell. Their interaction results in a periodic change in transmembrane potential called an action potential. Action potentials in the cardiac muscle propagate rapidly from cell to cell, synchronizing the contraction of the entire muscle to achieve an efficient pump function. The spatio-temporal pattern of this propagation is related both to the function of the cellular membrane and to the structural organization of the cells into tissues. Cardiac arrythmias originate from malfunctions in this process. The field of cardiac electrophysiology studies the multiscale organization of the cardiac activation process from the subcellular scale up to the scale of the body. It relates the molecular processes in the cell membranes to the propagation process through the multiscale structure of the tissue and organ, to measurable signals in the heart and to the electrocardiogram, an electrical signal on the torso surface.

Several improvements of current models of the propagation of action potentials are being developed in the Carmen team, based on previous work 36 and on the data available at IHU Liryc:

• Enrichment of the current monodomain and bidomain models 36, 49 by accounting for structural heterogeneities of the tissue at cellular and intermediate scales. Here we focus on multiscale analysis techniques applied to the various high-resolution structural data available at IHU Liryc.
• Coupling of the tissues from the different cardiac compartments and conduction systems. Here, we develop models that couple 1D, 2D and 3D phenomena described by reaction- diffusion PDEs.

These models are essential to improve our understanding of cardiac electrical dysfunction. To this aim, we use high-performance computing techniques in order to numerically explore the complexity of these models.

We use these model codes for applied studies in two important areas of cardiac electrophysiology: atrial fibrillation 42 and sudden-cardiac-death (SCD) syndromes 7, 645. This work is performed in collaboration with several physiologists and clinicians both at IHU Liryc and abroad.

3.2 Simplified models and inverse problems

The medical and clinical exploration of the cardiac electric signals is based on accurate reconstruction of the patterns of propagation of the action potential. The correct detection of these complex patterns by non-invasive electrical imaging techniques has to be developed. This involves solving inverse problems that cannot be addressed with the more complex models. We want both to develop simple and fast models of the propagation of cardiac action potentials and improve the solutions to the reconstruction questions of cardiac electrical imaging techniques.

These questions concern the reconstruction of diverse information, such as cardiac activation maps or, more generally, the whole cardiac electrical activity, from high-density body surface electrocardiograms. It is a possibly powerful diagnosis technique, which success would be considered as a breakthrough. Although widely studied during the last decade, the reconstructed activation maps, for instance, are highly inacurate and have a poor clinical interest. It remains a challenge for the scientific community to understand how body surface signals can better inform on the fine details of arrhythmic mechanisms.

The most usual method consists in solving a Laplace equation on the volume delimited by the body surface and the epicardial surface, forwhich we contribute by:

• studying in depth the dependance of the inverse problem on inhomogeneities in the torso, conductivity values, the geometry, electrode positions, etc.,
• improving the solution to the inverse problem by using new regularization strategies, factorization of boundary value problems, and the theory of optimal control.

In addition, we have started to explore many alternative approaches including:

• using complete propagation models in the inverse problem, like the bidomain or monodomain equations, for instance in order to localize electrical sources,
• constructing data-based models using e.g. statistical learning techniques, which would accurately represent some families of well-identified pathologies, or allow to combine physics and biology-informed models and clinical data, and
• constructing simpler models of the propagation of the activation front, based on eikonal or level-set equations.

3.3 Numerical techniques

We want our numerical simulations to be efficient, accurate, and reliable with respect to the needs of the medical community. Based on previous work on solving the monodomain and bidomain equations 38, 37, 48, 28, we will focus on:

• High-order numerical techniques with respect to the variables with physiological meaning, like velocity, AP duration and restitution properties,
• Efficient, dedicated preconditioning techniques coupled with parallel computing.

Existing simulation tools used in our team rely, among others, on mixtures of explicit and implicit integration methods for ODEs, hybrid MPI-OpenMP parallellization, algebraic multigrid preconditioning, and Krylov solvers. New developments include high-order explicit integration methods and task-based dynamic parallellism.

3.4 Cardiac electrophysiology at the microscopic scale

Traditional numerical models of whole-heart physiology are based on the approximation of a perfect muscle using homogenisation methods. However, due to aging and cardiomyopathies, the cellular structure of the tissue changes. These modifications can give rise to life-threatening arrhythmias. For our research on this subject and with cardiologists of the IHU Liryc, we aim to design and implement models that describe the strong heterogeneity of the tissue at the cellular level and to numerically explore the mechanisms of these diseases.

The literature on this type of model is still very limited 54. Existing models are two-dimensional  46 or limited to idealized geometries, and use a linear (purely resistive) behaviour of the gap-juction channels that connect the cells. We propose a three-dimensional approach using realistic cellular geometry (Fig. 1), nonlinear gap-junction behaviour, and a numerical approach that can scale to hundreds of cells while maintaining a sub-micrometer spatial resolution (10 to 100 times smaller than the size of a cardiomyocyte). Following preliminary work in this area by us  31, 30, 29 and by others 54 we proposed a European project with 10 partner institutes and a 5.8M€ budget to develop software that can simulate such models, with micrometer resolution, on the scale of millions of cells, using future exascale supercomputers (microcard.eu). This project runs from April 2021 to October 2024, and involves also the Inria teams CAMUS, STORM and CARDAMOM as well as the Inria-led MMG Consortium.

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3.5 Models and tools for ablative therapies

Today, the most effective way to treat arrhythmias is to ablate selected regions of the cardiac tissue. As the lesions have no particular electric property, this creates conduction blocks that stop the disorganized propagation of action potentials. The ablation procedure consists in placing a catheter in contact with the targeted site and deliver energy into the tissue. The energy can be overheating by radio-frequency current, electroporating electric pulses or temperature drop (cryotherapy). In practice, the choice of the ablation site is done by the clinician based on previous signal measurements and imagery, and is also guided during the procedure with real-time measurement of the electric signal.

Our team works on several subjects related to ablation techniques that may improve the success rate of the treatments:

• accurate computation of electric fields generated by catheters : complex catheter shapes, contact models, tissue heterogeneities,
• models of creation of the lesions, either through temperature rise (radio-frequency) or electroporation,
• localization tools to help clinicians target the optimal ablation sites, based on both data of previously ablated patients and synthetic data.

4.1 Scientific context: IHU Liryc

The University Hospital of Bordeaux (CHU de Bordeaux) is equipped with a specialized cardiology hospital, the Hôpital Cardiologique du Haut-Lévêque, where the group of Professor Michel Haïssaguerre has established itself as a global leader in the field of cardiac electrophysiology 44, 43, 33. Their discoveries in the area of atrial fibrillation and sudden cardiac death syndromes are widely acclaimed, and the group is a national and international referral center for treatment of cardiac arrhythmia. Thus the group also sees large numbers of patients with rare cardiac diseases.

In 2011 the group has won the competition for a 40 million euro Investissements d'Avenir grant for the establishment of IHU Liryc, an institute that combines clinical, experimental, and numerical research in the area of cardiac arrhythmia. The institute works in all areas of modern cardiac electrophysiology: atrial arrhythmias, sudden death due to ventricular fibrillation, heart failure related to ventricular dyssynchrony, and metabolic disorders. It is recognized worldwide as one of the most important centers in this area.

The Carmen team was founded as a part of IHU Liryc. We bring applied mathematics and scientific computing closer to experimental and clinical cardiac electrophysiology. In collaboration with experimental and clinical researchers at Liryc we work to enhance fundamental knowledge of the normal and abnormal cardiac electrical activity and of the patterns of the electrocardiogram, and we develop new simulation tools for training, biological, and clinical applications.

4.2 Basic experimental electrophysiology

Our modeling is carried out in coordination with the experimental teams from IHU Liryc. It helps to write new concepts concerning the multiscale organisation of the cardiac action potentials that will serve our understanding in many electrical pathologies. For example, we model the structural heterogeneities at the cellular scale  32 (the MICROCARD project), and at an intermediate scale between the cellular and tissue scales.

At the atrial level, we apply our models to understand the mechanisms of complex arrythmias and the relation with the heterogeneities at the insertion of the pulmonary veins. We will model the heterogeneities specific to the atria, like fibrosis or fatty infiltration   51, 42. These heterogeneities are thought to play a major role in the development of atrial fibrillation.

At the ventricular level, we focus on (1) modeling the complex coupling between the Purkinje network and the ventricles, which is supposed to play a major role in sudden cardiac death, and (2) modeling the heteogeneities related to the complex organization and disorganization of the myocytes and fibroblasts, which is important in the study of infarct scars for instance.

4.3 Clinical electrophysiology

Treatment of cardiac arrhythmia is possible by pharmacological means, by implantation of pacemakers and defibrillators, and by curative ablation of diseased tissue by local heating, freezing or electroporation. In particular the ablative therapies create challenges that can be addressed by numerical means. Cardiologists would like to know, preferably by noninvasive means, where an arrhythmia originates and by what mechanism it is sustained.

We address this issue in the first place using inverse models, which attempt to estimate the cardiac activity from a (high-density) electrocardiogram. A new project aims at performing this estimation on-site in the catheterization laboratory and presenting the results, together with the cardiac anatomy, on the screen that the cardiologist uses to monitor the catheter positions 47.

4.4 Application in Deep Brain Stimulation

Since 2017, we have been working with neurosurgeons from the Bordeaux University Hospital (Pr Cuny and Dr. Engelhardt) on improving the planning technique for deep brain surgery (DBS) for Parkinson's and Essential tremor diseases. DBS is the last resort to treat the symptoms of Parkinson's disease after the drug Levodopa. The surgery consists in placing electrodes in very specific regions of the patient's brain. These regions are unfortunately not visible on the 1.5 Tesla MRI, the most widely available MRI machines in hospitals. The most effective solution to date is to introduce 5 micro-electrodes (MER) to record the activity of neurons in the patient's brain and to prospect by moving the electrodes in order to find the best location. However, this approach renders the surgery very cumbersome because the patient must be awake during the exploration phase. In addition, this phase takes at least 3 hours and mobilizes a neurologist with his staff. The total duration of the operation is between 7 and 8 hours. Many elderly patients do not tolerate this surgery. We have proposed an approach that avoids the prospecting phase and performs surgery under general anesthesia. The idea is to learn on pairs of clinical landmarks and the position of active electrodes in order to predict the optimal position of the DBS from a pre-operative image. This approach simplifies and standardizes surgery planning. We tested several approaches doi:10.3389/fneur.2021.620360. We continue to seek approaches to fully automate the targeting process. We carried out a proof of concept by learning on the clinical database of the Bordeaux University Hospital. The clinical validation of our approach is in progress through a clinical trial which includes patients from the University Hospitals of Bordeaux and Lyon. Pr Cuny has submitted a phase 3 national clinical research hospital project (PHRCN) including 11 CHUs in France which has been accepted by the General Directorate for Care Offers (DGOS). The aim is to compare our new approach to the ones used in the other centers. Inria Bordeaux is a partner in this project and we maintain the OptimDBS software and solve any technical problem related to the compatibility of the MRIs exported by our software and the surgical robots in the different centers.

6.1 2021 Facts and Features

6.2 New software

6.3 New platforms

CEMPACK

CEMPACK is a new collection of software that was previously archived in different places. It includes the high-performance simulation code Propag and a suite of software for the creation of geometric models, preparing inputs for Propag, and analysing its outputs. In 2017 the code was collected in an archive on Inria's GitLab platform. The main components of CEMPACK are the following.

• Propag-5.1

  Applied modeling studies performed by the Carmen team in collaboration with IHU Liryc and foreign partners 752, 42, 41, 34 rely on high-performance computations on the national supercomputers Irene, Occigen, and Turing. The Propag-5 code is optimized for these systems. It is the result of a decades-long development first at the Université de Montréal in Canada, then at Maastricht University in the Netherlands, and finally at the Institute of Computational Science of the Università della Svizzera italiana in Lugano, Switzerland. Since 2016 most of the development on Propag has been done by M. Potse at the Carmen team 53. The code scales excellently to large core counts and, as it is controlled completely with command-line flags and configuration files, it can be used by non-programmers. It also features:

  • a plugin system for membrane models,
  • a completely parallel workflow, including the initial anatomy input and mesh partitioning, which allows it to work with meshes of more than ${10}^9$ nodes,
  • a flexible output scheme allowing hundreds of different state variables and transient variables to be output to file, when desired, using any spatial and temporal subsampling,
  • a configurable, LUSTRE-aware parallel output system in which groups of processes write HDF5/netCDF files, and
  • CWEB documentation of the entire code base.

  The code has been stable and reliable for several years. It can be considered the workhorse for our HPC work until CEPS takes over.

• Gepetto
  The Gepetto suite, named after a famous model maker, transforms a surface mesh of the heart into a set of (semi-)structured meshes for use by the Propag software or others. It creates the different fiber orientations in the model, including the transmurally rotating ventricular fibers and the various bundle structures in the atria (figure 2), and creates layers with possibly different electrophysiological properties across the wall. A practically important function is that it automatically builds the matching heart and torso meshes that Propag uses to simulate potentials in the torso (at a resolution of 1 mm) after projecting simulation results from the heart model (at 0.1 to 0.2 mm) on the coarser torso mesh 50. Like Propag, the Gepetto software results from a long-term development that started in Montreal, Canada, around 2002. The code for atrial fiber structure was developed by our team.
• Blender plugins
  Blender is a free software package for the production of 3-D models, renderings, and animations, comparable to commercial software such as Cinema4D. CEMPACK includes a set of plugins for Blender that facilitate the production of anatomical models and the visualization of measured and simulated data. It uses the MMG remeshing library, which is developed by the CARDAMOM team at Inria Bordeaux.

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7.1 Partial differential equations analysis

• Three Scale Unfolding Homogenization Method Applied to Cardiac Bidomain Model 12, 11. In this work, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electrophysiology. 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.
• Optimal control for nonlocal reaction-diffusion system describing calcium dynamics in cardiac cell 13. In this work, we introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction-diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well-posedness result of the adjoint problem using Faedo-Galerkin approximation, a priori estimates, and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the L2-norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme.

7.2 Numerical analysis and development of numerical methods

• Development of an Eulerian method to numerically solve incompressible bifluid problems with high density ratios with the specificity of a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure 15.
• A positive cell vertex Godunov scheme for a Beeler-Reuter based model of cardiac electrical activity. 14. The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this work, 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.

7.3 Modeling

• We proposed a model of cardiac mitochondria using a simplification approach in order to calibrate the model parameters to experimental data 21. The model describes the major features of mitochondrial activity willingly omitting detailed descriptions of these features in order to reduce its complexity (Fig. 3). In the end, our model consists of 6 state variables and 32 parameters, which is much less than the current models proposed by biologists. Preliminary sensitivity analysis was done on our model using Sobol indices, so as to run the calibration with identifiable parameters only. Our calibrated model was able to reproduce robustly respiration data.

  

  A cardiac mitochondron and the flux considered in the model; graph of the respiration rates (model vs data)

  This model is now being extended to include the effects of Reactive Oxygen Species (ROS) and the mitochondrial permeability transition pore (mPTP). Further data analysis and likelihood function tuning are required to calibrate this extended model to fluorescence data of individual mitochondria, using bayesian techniques. Ph.D. thesis of Bachar Tarraf. Interdisciplinary work between Liryc teams (modeling and metabolism) and other laboratories (ISM Bordeaux and LASS Toulouse), in the framework of the ANR Mitocard project.

• We modeled microstructural heterogeneities in the cardiac tissue by mean of a homogenized model. It enables to include, into macroscopic equivalent conductivity coefficients, quantitative information about the microstructure available from high resolution imaging modalities, such as fibrosis or diverse types of inclusions (follow up of Andjela Davidovic Ph.D thesis). The complete procedure to derive the macroscopic coefficients from MR images, first published online in 2020, was finally published in a special issue of the International Journal on Numerical Methods in Biomedical Engineering 40.
• Spiral Waves Generation Using an Eikonal-Reaction Cardiac Electrophysiology Model 27 In this work, we used the conduction velocity adapted eikonal model 35 in order to simulate reentry spiral waves. The new model that we present here combines the CV-adapted eikonal model with a specific parametrized stimulation current obtained from the diffusion term of the monodomain model. The monodomain model is computed once off-line and the generated diffusion current has been approximated, parametrized and used to stimulate the MS ode system. The preliminary results showed that with this model we have been able to simulate spiral waves with the same pattern as the monodomain model. To the best of our knowledge, this is the first work able to generate spiral waves using the eikonal model. Differently from the monodomain model, [8], the EK-CV-DIFF formulation captures the correct value of the conduction velocity even using a coarse resolution. This eikonal model is easy to parametrize and thus could be a good candidate to conduct personalized simulations in order to simulate the electrical wave of a given patient. Our future work will be to parametrize this model in order to provide a new electrocardiography imaging tool especially in atrial fibrillation conditions

7.4 Modeling and inverse problems

• Cardiac Activation Maps Reconstruction: A Comparative Study Between Data-Driven and Physics-Based Methods 20. Proof of concept of AI based methods for the development of new ECGi algorithms: second part of Amel Karoui Ph.D. thesis. The PhD was defended in 2021.
• Validation on experimental data of the patchwork method, a ECGI method selecting among several numerical method the one minimizing the residuals on the torso 23. (Ph.D thesis of Oumayma Bouhamama, co-supervised with Laura Bear from Signal Processing team).
• Resolution of the inverse problem of Electrical Impedance Tomography with an immersed boundary method on a cartesian grid in 2 dimensions (Ph.D thesis of Niami Nasr, co-supervised by J. Dardé, IMT Toulouse).
• Development and validation on academic test-cases of an eikonal model to solve the inverse problem of the propagation of the front on the heart surface from body-surface measurements, in collaboration with J. Fehrenbach (IMT, Toulouse).
• We developed a new machine-learning approach for the localization of tissue that is provoking premature ventricular contractions (PVC) 22. The approach is original in several ways. Firstly, as it is meant to be used during an intervention, it has to be very fast – eventually close to real-time. Secondly, it was trained on simulated data only, but then showed good performance for real patients. This was possible due to the use of highly realistic patient-based models to produce the training data. Finally, it uses only the 8 ECG electrodes that are needed to record a standard electrocardiogram, in contrast to traditional inverse models which use hundreds of electrodes. This limited lead set could be used because the problem of PVC localisation is intrinsically very constrained. The use of a limited lead set is very important for clinical applicability.
• A volume source method for solving ECGI inverse problem. 26 In this paper, we present a volumic model to solve the electrocardiography inverse problem that takes into account structural informations obtained by imaging techniques. The direct problem maps a volumic current in the cardiac muscle (ventricles) to the body surface electrical measures. The model is based on coupling bidomain heart model with torso conduction. The corresponding inverse problem is solved with the Tikhonov regularization. Simulated public database EDGAR is used for the evaluation of this method and we compared them to standard method of fundamental solutions (MFS). The sensitivity to noise is also assessed. The correlation coefficients and the relative error of activation times were computed. Results show that the volume method performs better than MFS for any noise level, and reconstruct in addition endocardial information.

7.5 High performance computing

• For the MICROCARD project, we developed a prototype software that creates artificial geometric models of cardiac muscle tissue at the cellular scale. We need such models for our simulations, because current imaging data at this scale is limited to a handful of cells, while we require thousands to, ultimately, billions of cells (each represented by roughly a thousand tetrahedral elements). Our software defines a level-set function on a tailored mesh. The MMG software is then used to discretize the level set and improve the mesh quality. With the current prototype it is feasible to produce networks of a few hundred cells, and using replication techniques we can make these meshes as large as can be stored: our largest test so far required 16TB storage, while a full human heart would require in the order of 2PB. These results will be presented at the ECCOMAS meeting in June 2022.

  

  A view of a set of cardiac cell surface meshes.

• With R. Turpault and K. Khadra, collaborators from IMB (Institut de Mathématiques de Bordeaux), we continue the development of a very high order finite volume method for the monodomain equation, following the method published in 39. During 2021, we developped ands started to test the MPI parallel version that will be used ot extend our numerical experiments to large three-dimensional models. The numerical scheme also serves as a sandbox for the method to be developped in the MICROCARD project. This foreseen method will have to account for the Laplace equation in the whole computational domain, like the previous one, and in addition with jumps on all the cell membranes, which are spread among the whole computational domain.

8.1 International initiatives

8.2 International research visitors

8.3 European initiatives

8.4 National initiatives

8.5 Regional initiatives

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 Major publications

• 1 articleB.Boris Andreianov, M.Mostafa Bendahmane, K. H.Kenneth H. Karlsen and C.Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model.Networks and Heterogeneous Media622011, 195-240
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• 2 articleA.Adnane Azzouzi, Y.Yves Coudière, R.Rodolphe Turpault and N.Nejib Zemzemi. A mathematical model of Purkinje-Muscle Junctions.Mathematical Biosciences and Engineering842011, 915-930
• 3 articleY.Yves Bourgault, Y.Yves Coudière and C.Charles Pierre. Existence And Uniqueness Of The Solution For The Bidomain Model Used In Cardiac Electrophysiology.Nonlinear Anal. Real World Appl.1012009, 458-482URL: http://hal.archives-ouvertes.fr/hal-00101458/fr
• 4 articleY.Yves Coudière, C.Charles Pierre, O.Olivier Rousseau and R.Rodolphe Turpault. A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation.International Journal on Finite Volumes612009, URL: http://hal.archives-ouvertes.fr/hal-00328251/fr
• 5 articleY.Yves Coudière and C.Charles Pierre. Stability And Convergence Of A Finite Volume Method For Two Systems Of Reaction-Diffusion Equations In Electro-Cardiology.Nonlinear Anal. Real World Appl.742006, 916--935URL: http://hal.archives-ouvertes.fr/hal-00016816/fr
• 6 articleP. W.Peter W. Macfarlane, C.Charles Antzelevitch, M.Michel Haïssaguerre, H. V.Heikki V. Huikuri, M.Mark Potse, R.Raphael Rosso, F.Frederic Sacher, J. T.Jani T. Tikkanen, H.Hein Wellens and G.-X.Gan-Xin Yan. The Early Repolarization Pattern; A Consensus Paper.Journal of the American College of Cardiology66resulting from the Symposium on J Wave Patterns and a J Wave Syndrome, Glasgow, August 2013. Defines terminology for the J point: Jo, Jp, Jt. Provisionally accepted 19 May 2015.2015, 470-477URL: http://dx.doi.org/10.1016/j.jacc.2015.05.033
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• 7 articleV. M.Veronique M F Meijborg, M.Mark Potse, C. E.Chantal E Conrath, C. N.Charly N W Belterman, J. M.Jacques M T de Bakker and R.Ruben Coronel. Reduced Sodium Current in the Lateral Ventricular Wall Induces Inferolateral J-Waves.Front Physiol7365August 2016
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• 8 articleC.Charles Pierre. Preconditioning the bidomain model with almost linear complexity.Journal of Computational Physics2311January 2012, 82--97URL: http://www.sciencedirect.com/science/article/pii/S0021999111005122
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10.2 Publications of the year

10.3 Cited publications

• 28 articleB.Boris Andreianov, M.Mostafa Bendahmane, K. H.Kenneth H. Karlsen and C.Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model.Networks and Heterogeneous Media622011, 195-240
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• 29 inproceedingsP.-E.Pierre-Elliott Bécue, F.Florian Caro, M.Mostafa Bendahmane and M.Mark Potse. Modélisation et simulation de l'électrophysiologie cardiaque à l'échelle microscopique.43e Congrès National d'Analyse Numérique (CANUM)oral presentationSMAIObernai, Alsace, FranceMay 2016, URL: http://smai.emath.fr/canum2016/resumesPDF/peb/Abstract.pdf
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• 30 miscP.-E.Pierre-Elliott Bécue, F.Florian Caro, M.Mark Potse and Y.Yves Coudière. Theoretical and Numerical Study of Cardiac Electrophysiology Problems at the Microscopic Scale..PosterJuly 2016
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• 31 inproceedingsP.-E.Pierre-Elliott Bécue, M.Mark Potse and Y.Yves Coudière. A Three-Dimensional Computational Model of Action Potential Propagation Through a Network of Individual Cells.Computing in Cardiology 2017Rennes, FranceSeptember 2017, 1-4
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• 32 inproceedingsP.-E.Pierre-Elliott Bécue, M.Mark Potse and Y.Yves Coudière. Microscopic Simulation of the Cardiac Electrophysiology: A Study of the Influence of Different Gap Junctions Models.Computing in CardiologyMaastricht, NetherlandsSeptember 2018
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• 33 articleB.Benjamin Berte, F.Frederic Sacher, S.Saagar Mahida, S.Seigo Yamashita, H. S.Han S. Lim, A.Arnaud Denis, N.Nicolas Derval, M.Mélèze Hocini, M.Michel Haïssaguerre, H.Hubert Cochet and P.Pierre Jaïs. Impact of Septal Radiofrequency Ventricular Tachycardia Ablation; Insights From Magnetic Resonance Imaging.Circulation1302014, 716-718
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• 34 inproceedingsJ.Judit Chamorro Servent, L.Laura Bear, J.Josselin Duchateau, M.Mark Potse, R.Rémi Dubois and Y.Yves Coudière. Do we need to enforce the homogeneous Neuman condition on the Torso for solving the inverse electrocardiographic problem by using the method of fundamental solution ?Computing in Cardiology 201643Computing in Cardiology 2016Vancouver, CanadaSeptember 2016, 425-428
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• 35 articleC.Cesare Corrado and N.Nejib Zemzemi. A conduction velocity adapted eikonal model for electrophysiology problems with re-excitability evaluation.Medical Image Analysis43January 2018, 186-197
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• 36 articleY.Yves Coudière, Y.Yves Bourgault and M.Myriam Rioux. Optimal monodomain approximations of the bidomain equations used in cardiac electrophysiology.Mathematical Models and Methods in Applied Sciences246February 2014, 1115-1140
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• 37 articleY.Yves Coudière, C.Charles Pierre, O.Olivier Rousseau and R.Rodolphe Turpault. A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation.International Journal on Finite Volumes612009, URL: http://hal.archives-ouvertes.fr/hal-00328251/fr
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• 38 articleY.Yves Coudière and C.Charles Pierre. Stability And Convergence Of A Finite Volume Method For Two Systems Of Reaction-Diffusion Equations In Electro-Cardiology.Nonlinear Anal. Real World Appl.742006, 916--935URL: http://hal.archives-ouvertes.fr/hal-00016816/fr
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• 39 articleY.Yves Coudière and R.Rodolphe Turpault. A domain decomposition strategy for a very high-order finite volumes scheme applied to cardiac electrophysiology.Journal of Computational Science372019, 101025URL: http://www.sciencedirect.com/science/article/pii/S1877750319303862
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• 40 articleA.An\djela Davidović, Y.Yves Coudière and Y.Yves Bourgault. Modelling the action potential propagation in a heart with structural heterogeneities: From high-resolution MRI to numerical simulations.International Journal for Numerical Methods in Biomedical EngineeringFebruary 2020, e3322
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• 41 articleJ.Josselin Duchateau, M.Mark Potse and R.Remi Dubois. Spatially Coherent Activation Maps for Electrocardiographic Imaging.IEEE Transactions on Biomedical Engineering64May 2017, 1149-1156
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• 42 inproceedingsA.Ali Gharaviri, M.Mark Potse, S.Sander Verheule, R.Rolf Krause, A.Angelo Auricchio and U.Ulrich Schotten. Epicardial Fibrosis Explains Increased Transmural Conduction in a Computer Model of Atrial Fibrillation .Computing in CardiologyVancouver, CanadaSeptember 2016
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• 43 articleM.Michel Haïssaguerre, N.N. Derval, F.F. Sacher, L.L. Jesel, I.I. Deisenhofer, L.L. de Roy, J. L.J. L. Pasquié, A.A. Nogami, D.D. Babuty, S.S. Yli-Mayry, C.C. De Chillou, P.P. Scanu, P.P. Mabo, S.S. Matsuo, V.V. Probst, S.S. Le Scouarnec, P.P. Defaye, J.J. Schlaepfer, T.T. Rostock, D.D. Lacroix, D.D. Lamaison, T.T. Lavergne, Y.Y. Aizawa, A.A. Englund, F.F. Anselme, M.M. O'Neill, M.M. Hocini, K. T.K. T. Lim, S.S. Knecht, G. D.G. D. Veenhuyzen, P.P. Bordachar, M.M. Chauvin, P.P. Jaïs, G.G. Coureau, G.G. Chene, G. J.G. J. Klein and J.J. Clémenty. Sudden cardiac arrest associated with early repolarization.N. Engl. J. Med.3582008, 2016--2023
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• 44 articleM.M. Haïssaguerre, P.P. Jaïs, D. C.D. C. Shah, S.S. Garrigue, A.A. Takahashi, T.T. Lavergne, M.M. Hocini, J. T.J. T. Peng, R.R. Roudaut and J.J Clémenty. Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins.N. Engl. J. Med.3391998, 659-666
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• 45 articleM. G.Mark G. Hoogendijk, M.Mark Potse, A. C.André C. Linnenbank, A. O.Arie O. Verkerk, H. M.Hester M. den Ruijter, S. C.Shirley C. M. van Amersfoorth, E. C.Eva C. Klaver, L.Leander Beekman, C. R.Connie R. Bezzina, P. G.Pieter G. Postema, H. L.Hanno L. Tan, A. G.Annette G. Reimer, A. C.Allard C. van der Wal, A. D.Arend D. J. ten Harkel, M.Michiel Dalinghaus, A.Alain Vinet, A. A.Arthur A. M. Wilde, J. M.Jacques M. T. de Bakker and R.Ruben Coronel. Mechanism of Right Precordial ST-Segment Elevation in Structural Heart Disease: Excitation Failure by Current-to-Load Mismatch.Heart Rhythm72010, 238-248URL: http://dx.doi.org/10.1016/j.hrthm.2009.10.007
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• 46 articleM. L.Marjorie Letitia Hubbard and C. S.Craig S. Henriquez. A microstructural model of reentry arising from focal breakthrough at sites of source-load mismatch in a central region of slow conduction.Am. J. Physiol. Heart Circ. Physiol.3062014, H1341-1352
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• 47 inproceedingsM.Michal Kania, Y.Yves Coudière, H.Hubert Cochet, M.Michel Haissaguerre, P.Pierre Ja\"is and M.Mark Potse. A new ECG-based method to guide catheter ablation of ventricular tachycardia.iMAging and eLectrical TechnologiesUppsala, SwedenApril 2018
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• 48 articleC.Charles Pierre. Preconditioning the bidomain model with almost linear complexity.Journal of Computational Physics2311January 2012, 82--97URL: http://www.sciencedirect.com/science/article/pii/S0021999111005122
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• 49 articleM.Mark Potse, B.Bruno Dubé, J.Jacques Richer, A.Alain Vinet and R. M.Ramesh M. Gulrajani. A Comparison of monodomain and bidomain -diffusion models for action potential propagation in the human heart.IEEE Transactions on Biomedical Engineering53122006, 2425-2435URL: http://dx.doi.org/10.1109/TBME.2006.880875
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• 50 articleM.Mark Potse, B.Bruno Dubé and A.Alain Vinet. Cardiac Anisotropy in Boundary-Element Models for the Electrocardiogram.Medical and Biological Engineering and Computing472009, 719--729URL: http://dx.doi.org/10.1007/s11517-009-0472-x
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• 51 inproceedingsM.Mark Potse, A.Ali Gharaviri, S.Simone Pezzuto, A.Angelo Auricchio, R.Rolf Krause, S.Sander Verheule and U.Ulrich Schotten. Anatomically-induced Fibrillation in a 3D model of the Human Atria.Computing in CardiologyMaastricht, NetherlandsSeptember 2018
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• 52 miscM.Mark Potse, V. M.Veronique M F Meijborg, C. N.Charly N W Belterman, J. M.Jacques M T de Bakker, C. E.Chantal E Conrath and R.Ruben Coronel. Regional conduction slowing can explain inferolateral J waves and their attenuation by sodium channel blockers.PosterSeptember 2016
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• 53 articleM.Mark Potse. Scalable and Accurate ECG Simulation for Reaction-Diffusion Models of the Human Heart.Frontiers in Physiology9April 2018, 370
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• 54 articleA.Aslak Tveito, K. H.Karoline H. J\aeger, M.Miroslav Kuchta, K.-A.Kent-Andre Mardal and M. E.Marie E. Rognes. A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue.Front. Phys.5This is very very similar to what we were doing with PEB... Looks like we're scooped at least for the approach, but we do have a few abstracts ([becue:cinc17], [becue16a], MMCE meeting in Ottawa November 2017). Note it's in Frontiers in (Biomedical) Physics, not Physiology.2017, 48
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