1. C. Pierre, F. Plouraboué   Numerical analysis of a new mixed-formulation for eigenvalue convection-diffusion problems.
   SIAM Applied Maths, Vol. 70(3), 2009.
   PDF file.       Journal link
   
   
2. Coudiere Y., Pierre C., Rousseau O., Turpault R.   A 2D/3D Discrete Duality Finite Volume Scheme. Application to ECG simulation.
   International Fourmal on Finite Volumes, Vol. 6(1), 2009.
   PDF file.       Journal Link
3. Y. Bourgault, Y. Coudiere and C. Pierre.  Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology.
   Nonlinear Analysis: Real World Applications. Vol 10(1), pp. 458-482, 2009.
   PDF file.       Journal Link
   
   
4. F. Plouraboué, C. Pierre.  Stationary convection-diffusion between two co-axial cylinders.
   Int. J. of Heat and Mass Transfer. Vol 50(23-24), pp. 4901-4907, 2007.
   PDF file.       Journal Link
   
   
5. Y. Coudière, C. Pierre.   Stability and convergence of a finite volume method for two systems of reaction-diffusion equations in electro-cardiology.
   Nonlinear Analysis: Real World Applications. Vol. 7(4), pp. 916-935, 2006.
   PDF file.       Journal Link
   
   
6. C. Pierre, F. Plouraboué and M. Quintard.   Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective.
   SIAM J. Appl. Math. Vol. 66(1): pp. 122-152, 2005.
   PDF file.       Journal Link

1. Pierre C., Rousseau O., Bourgault, Y.   Segmented medical images based simulations of Cardiac electrical activity and electrocardiogram: a model comparison.
   PDF file       HAL preprint
   
   

1. Coudière Y., Pierre C., Turpault R.  A 2D/3D Finite Volume Method used to solve the bidomain equations of electrocardiology.
   ALGORITHMY 2009. 18th Conference on Scientific Computing: pp. 1-10. Slovakia. March 15 ~ 20, 2009.
   PDF file
   
   
2. Y. Coudiere, C. Pierre, O. Rousseau and R. Turpault.   2D/3D Discrete Duality Finite Volume (DDFV) scheme for anisotropic- heterogeneous elliptic equations, application to the electrocardiogram simulation. .
   Int. symposium on Finite Volumes for Complex Applications V: pp. 313-320. Aussois, France, june 8-13, 2008.
   PDF file
   
   
3. Y. Coudiere and C. Pierre.   Stability and Convergence of a Finite Volume Method for a Reaction Diffusion System in Electro-Cardiology.
   In proc., Int. symposium on Finite Volumes for Complex Applications IV: pp. 163-172, Marrakech, july 2005.
   PDF file

Modelling and Simulating the electrical Activity of the Heart
embedded in the Torso, Numerical Analysis and Finite Volumes Methods.

Université de Nantes, 2005.

Applied Mathematics PhD thesis, Université de Nantes, 2005.
Supervision: Yves Coudière and Francois Jauberteau.
Reviewers: P. Colli-Franzone (Pavia), R. Herbin (Marseille).
       PDF File     Thèses en ligne

Abstract.  This PhD thesis deals with the two following purposes: modelling and simulation in the field of bio mathematics on the first hand, numerical analysis and scientific computing on the other hand.
  Within the field of bio-mathematics, attention is paid to the modelling of complex biological systems at the whole organ scale. Namely the description of the electrical field generated by the heart activity. This activity is multi scale.

The micro-scale phenomena are nowadays well understood and quantitative models are available, describing ionic exchanges on the cell membranes.

The bidomain model presents a macro-scale homogenised version of cell membrane models, taking the anisotropic structure of the cardiac tissues into account.

This heart model has been coupled with a torso electrical model, thus providing an ECG (electrocardiogram) simulator.    Finite volumes methods have been developed to solve the model.
Firstly for a simplified version of the bidomain model, classical finite volumes schemes have been studied: stability and the convergence is proven theoretically and tested numerically.
The complete model (heart/torso coupled system) exhibits conceptual and technical difficulties (tissues anisotropy, coupling conditions, distorted and unstructured meshes). To overcome these difficulties, a new class of finite volumes schemes has been developed in 3D: DDFV methods (Discrete Duality Finite Volumes). This method has been implemented and used both to simulate the action potential propagation in the heart and the ECG.

Key words. Bio mathematics ~ Finite volumes methods ~ Electrocardiology ~ Simulation and numerical modelling ~ Stability and convergence of numerical methods ~ Reaction diffusion equations