Mathématiques en biologie et médecine
(Système cardiovasculaire et poumons)
Lundi, 17 juin
Salle Du Manège
On the Performance of Anisotropic Mesh Adaptation for Simulating Complex Cardiac Dynamics
Belhamadia, Youssef
Université de l'Alberta, Campus St-Jean

The simulation of cardiac electrophysiological waves are known to require extremely fine meshes, limiting the applicability of current numerical models to simplified geometries and ionic models. Mesh adaptation methods are extremely helpful to improve the accuracy of finite element simulations and can reduce the number of degrees of freedom required for a given accuracy. However, they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes especially for complex cardiac dynamics. The purpose of this work is to explore the efficiency of a three-dimensional anisotropic adaptive finite element method for simulating the spatio-temporal chaos in cardiac tissue. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. In addition to the  monodomain, the bidomain model will be also employed and examples of ventricular fibrillation in cardiac tissue will be illustrated. Qualitative results will be presented showing that the proposed methodology reduces significantly the number of elements leading to huge saving in memory as well as in computational time.

Optimal monodomain approximations of the bidomain model used in cardiac electrophysiology
Bourgault, Yves (1), Yves Coudière (2), Charles Pierre (3) and Myriam Rioux (2,4)
(1) University of Ottawa, (2) INRIA, Bordeaux, (3) Université de Pau, (4) Université Laval

As we will illustrate, during forward propagation the bidomain and monodomain models predict strikingly similar depolarization/repolarization isochrons and electrocardiograms. We provide some mathematical foundations as to why the two models give similar results, by estimating the distance between the mono- and bidomain conductance operators. Our formalism provides new ways of approximating the bidomain operator using a properly set monodomain model. We then compare the two models through simple test cases.

A bilayer surface model of the human atria
Coudière, Yves
Université de Bordeaux

Atrial fibrillation is the most commonly encountered clinical arrhythmia. Despite recent advances in treatment by catheter ablation, its origin is still incompletely understood and it may be difficult to treat. Computer modelling offers an attractive complement to experiment. Simulations of fibrillation, however, are computationally demanding since the phenomenon requires long periods of observation. Because the atria are thin walled structures, they are often modelled as surfaces. However, this may not always be appropriate because abrupt changes of fibre direction are encountered through the atrial wall, that have been observed to affect the electrical phenomenon.

I will present a bilayer model obtained by asymptotic analysis that remains computationally simple while retaining the main 3D features of the atrial wall.  Afterwards, I will describe the semi-automatic procedure used to construct actual bilayer atrial models from medical images. Finally, I will show some numerical illustrations, such as a complex atrial arrhythmia developing over several heart beats, proving the medical relevance of the model.

The Current-Lifted Monodomain Model for Numerical Computations of the Heart Pacing
Rioux, Myriam
Université Laval

The response of the heart to an external electrical current stimulus remains a very active subject of research, e.g. for investigating defibrillation and pacing mechanisms. The bidomain model has always been recognized to be the only model suitable for these specific modelling applications. In this work, we describe a monodomain model with a source term that is derived from a lifting principle applied to the resolution of the electrostatic balance equation in the parabolic-elliptic formulation of the bidomain equations. The final model is a monodomain equation with lifted current, hence the appellation current-lifted monodomain (CLM). It allows any contribution of the current to be solved separately, making possible to account for unequal anisotropy ratios, the excitation via the intracellular, the extracellular, and the extracardiac media.  As this extra part of the calculation is performed once before all time iterations, the CLM performs as well as the standard monodomain model, i.e. about ten times faster than the bidomain model. For proving the use of the CLM model, virtual electrode polarizations (VEP) are simulated. VEP represents a whole class of phenomena strongly related to unequal anisotropy ratios, where the extracellular cardiac stimulation by an unipolar electrode induces characteristic regions of membrane polarization other than those in the vicinity of the electrode. Finally, the important advancement of this work is the replacement of the bidomain model for long-duration simulations with current stimulation, e.g., investigations of fibrillation mechanisms where current stimulations turn out to be an effective treatment