Section: New Results

Construction of numerical models

• Implementation of an accurate bilayer model of human atria, including realistic geometry and qualitative fibre direction [19] , [21] , [16]

  We introduce a bilayer model of the human atria. We set a specific mathematical model based on two surface monodomain problems coupled by a coupling term. We recalled convergence results of the bilayer model towards a 3D model for thin tissues, we formalized an optimization method to set the coupling coefficient and we present two different asymptotically equivalent numerical implementation of the model. We then present a geometrically and electrophysiologically accurate model of the atrial heterogeneities, including two layers of fibre directions and ionic function heterogeneities based on histological and modelling works. We assess the physiological relevance of the model during a sinus wave and we check the occurrence of three-dimensional electrical behaviour such as slight electrical dissociation. This bilayer model is able to take into account transmural heterogeneities only accessible since then with full 3D models, while keeping the low computational load associated with surface models. It is then a light and relevant tool for long-lasting simulations designed to investigate atrial arrhythmia.

• Personalization method of the bilayer model to registrate the geometry to a patient dependant geometry [16]

  If the generic atrial bilayer model developed in [21] allows to conduct general experiments, greater customization of the model is necessary to carry out more specific studies on a given patient. We present a methodology to obtain a patient-dependant model containing the geometry of the patient, a generic fibrous organization and an image of the patient's fibrosis obtained by late-enhancement MRI. This is a common work with the clinical team of the CHU du Haut-Lévêque (H. Cochet and P. Jaïs) and the Asclepios Inria team.

  The methodology is based on a registration method developed by Durrleman et al. [34] that allows to register surfaces : the generic model is registrated towards a patient-specific geometry (work by M. Sermseant and R. Cabrera-Lozoya, Asclepios Team). The fibre organisation is transported by the same linear local transformations. The late-enhancement is projected on the model to take into account the complex patient-specific fibrotic repartition. A similar methodology was presented by McDowell et al. [37] . However, the authors took as a starting point a three-dimensional geometry and a different methodology to registrate the geometry. The work presented here is therefore innovative.

• Faster solvers for cardiac electro-physiology problems [27]

  There are many applications in cardiac electro-physiology where computational time is the main requirement to fulfill, even by sacrificing accuracy. Some techniques were investigated in this direction, in order to obtain a break-even point between accuracy and speed. The complete problem involves solving some ODEs on each mesh node and inverting large sparse matrices, often ill-conditionned.

  We first designed a method based on the Proper Orthogonal Decomposition (POD) technique: we project the linear system onto a well-chosen orthogonal basis of smaller dimension while still solving the ODES. We tried the method on both the bidomain and monodomain equations, and extended the tests on an HPC machine, in order to observe scalability performances. There is no improvment for the monodomain equations because its linear systems are well-conditionned. For the bidomain equations, the CPU time decreases by a factor of 10 between the full and reduced models, and better scalability performances.

  We secondly developped an eikonal model, in view of serious games applications for the Medic Activ project. The Dijkstra algorithm is used to solve the eikonal equation and the transmembrane potential is determined by the solution of a Mitchell-Schaeffer model on each mesh node. Some modifications where introduced to take into account re-excitability and allox re-entrant waves. Compared by the algorithm proposed by [38] , the transmembrane potential comes from the solution of an underlying model, not through an approximation. This represent an innovation, to our knowledge not present in literature.