ED Mathématiques et Informatique
Finite volume methods on unstructured meshes for the microscopic bidomain model in cardiac electrophysiology
by Zeina CHEHADE (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 10h00 - Ada Lovelace 200, avenue de la Vieille Tour 33405 Talence CEDEX
in front of the jury composed of
- Yves COUDIÈRE - Professeur des universités - Université de Bordeaux - Directeur de these
- Mazen SAAD - Professeur des universités - École Centrale de Nantes - Rapporteur
- Stella KRELL - Maîtresse de conférences - Université Nice Côte d'Azur - Rapporteur
- Héloïse BEAUGENDRE - Professeure des universités - Université de Bordeaux - Examinateur
- Luc MIEUSSENS - Professeur des universités - Université de Bordeaux - Examinateur
- Ayman MOURAD - Professeur - Université Libanaise - Examinateur
- Flore NABET - Maîtresse de conférences - École polytechnique - Examinateur
- Charles PIERRE - Ingénieur de recherche - Université de Pau et des Pays de l'Adour - Examinateur
This manuscript is dedicated to the development and numerical analysis of the cell-by-cell bidomain model in cardiac electrophysiology at the microscopic scale, using Finite Volume Methods (FVM) on unstructured meshes. This model, also known as the Extracellular-Membrane-Intracellular (EMI) model, consists of a system of reaction-diffusion equations describing the evolution of electric potential within each subdomain, coupled with non-standard and time-dependent transmission conditions. The model dynamics are defined exclusively at cardiomyocyte interfaces, where the electrical potential undergoes jumps, and the current flux is continuous, making FVM a suitable approach for solving this model. However, EMI model introduces various mathematical and numerical challenges due to its unusual formulation as a degenerate parabolic system, strong heterogeneity, severe stiffness from nonlinear ionic dynamics, high computational cost, and complex geometry requiring advanced mesh generation. Finite Element (FEM), Finite Difference (FDM), and Boundary Element (BEM) Methods have been used for this model, but FV approaches have not yet been explored. This work represents the first step toward developing a robust and efficient FV scheme for the EMI model, ensuring both numerical accuracy and efficiency in terms of convergence order, linear system symmetry, and reduced stencils. The advantage of FVM lies in its ability to generate two formulations: a large sparse system, similar to FEM, and a more reduced system defined only on the subdomain interfaces, similar to BEM. The latter significantly reduces the number of degrees of freedom and thus the computational cost. First, a rigorous mathematical analysis of the EMI model with mixed non-homogeneous Dirichlet and Neumann boundary conditions was conducted, using a broad set of functional analysis tools. Second, a comprehensive review of FVM and the numerical approaches used in cardiology was carried out to highlight the originality of this work. Due to the isotropic diffusion coefficient, we developed a cell-centered Two-Point Flux Approximation (TPFA) scheme for the EMI model, along with Forward-Backward Euler and SBDF2 methods, as well as Rush-Larsen time integrations. A major challenge was the construction of an admissible mesh for the TPFA scheme, for which numerical tools were developed to maintain the orthogonality condition in the used geometries. Convergence analysis was performed by establishing a priori error estimates with various regularity assumptions on the data, and numerical experiments validated the theoretical convergence rates for both sparse and dense versions in a two-dimensional framework. Once the precision of the schemes was evaluated, the EMI model was further studied by investigating the effects of discretization and model parameters on electrical wave propagation, conduction velocity, and activation time. Due to the constraints of complex geometries and three-dimensional cases, such as realistic cardiomyocytes, we developed and studied a hybrid SUSHI scheme for the EMI model. This scheme approximates the solution by using values on both cell-centered and all internal faces, offering advantages for EMI simulations in complex scenarios, increasing computational cost while maintaining flux consistency on heterogeneous interfaces. However, for cost-efficiency, a symmetric scheme close to a cell-centered scheme seems more suitable. The "SUSHI-NP" (non-parametric) scheme, which retains face unknowns only at the heterogeneous interfaces, represents a promising solution for reducing computational costs while preserving consistency and precision.