Atrial electrophysiology study using a surface-based model relying on asymptotic analysis

Atrial fibrillation is the most common type of cardiac arrhythmia. It is characterized by multiple waves of excitation coursing through myocardial tissue. Hence, there is an important medical need for modeling the electrical activity in the atria – e.g. for therapy planning assistance with radio-frequency ablation. The atria have very thin walls, mainly apparent as a thick surface in medical imaging. Our first objective is to derive a 2D electrophysiology model defined over the midsurface of the geometry, and valid for thin cardiac structures. The major challenge consists in taking into account the anisotropy resulting from the preferred conduction direction along the muscle fibers, which direction rapidly varies across the thickness. Of course, an important motivation also lies in computational effectiveness, which is expected to be far superior than with a 3D model. The electric wave propagating in the cardiac tissue can be represented by a nonlinear reaction-diffusion partial differential equation, coupled with an ordinary differential equation representing cellular activity. We consider here the bidomain model, see e.g. [6]. We make assumptions on the diffusion tensor in order to model the anisotropic directions. We formulate an asymptotic framework for an anisotropic diffusion model in the spirit of thin structural models – such as shells – in mechanics, see e.g. [1]. We thus identify a limit problem and we show the existence and uniqueness of a solution for this limit problem. We prove a preliminary weak convergence result prior to establishing the strong convergence. This allows us to propose a mathematically justified model defined over the midsurface, and designed for cardiac electrophysiology in thin structures. As a validation of our model, we compare the 3D and surface-based models, especially when the fibers are rapidly-varying across the thickness, and with non-planar geometries. A pathological case is proposed as further illustration featuring spiral waves, a captivating phenomenon in cardiac electrophysiology, often considered to be responsible for atrial and ventricular fibrillation. The procedure proposed in [2] is employed to obtain the spirals. The FELiScE finite element library developed at Inria is used for all simulations. The differences between the 3D and surface models are very limited and concentrated along the wave front. Furthermore, we obtain excellent results in terms of computational time (432 min. with 3D model versus 26 min. with surface model for 1500 ms simulation – i.e. 2 cardiac cycles – with 15,000 time steps). Then, we present physiological simulations with an anatomical surface mesh representing the mid-surface of the two atria. The conduction tracts between the atria have been defined, and the fibers directions at the endocardium and epicardium are prescribed based on [4]. The values of the conductivity parameters depend on the specific areas considered. The Bachman bundle, the Crista Terminalis, and the pectinate muscles are regions of established fast conduction. By contrast, the Fossa Ovalis and the isthmus of the right atrial floor are regions of known slow conduction. The Courtemanche-Ramirez-Nattel model – which has been specifically derived for human atrial cells – is used to model the reaction term representing the ionic current across the membrane. We compare the simulation results with other modeling studies [3,5], and excellent adequacy is obtained, with greatly improved effectiveness. References [1] D. Chapelle and K.J. Bathe. The Finite Element Analysis of Shells - Fundamentals. Springer, second edition, 2011. [2] S. Goktepe and E. Kuhl. Computational modeling of cardiac electrophysiology: A novel finite element approach. International Journal for Numerical Methods in Engineering, 79(2):156–178, 2009. [3] D.M. Harrild and S.H. Craig. A computer model of normal conduction in the human atria. Circulation Research, (87):e25–e36, 2000. [4] S.Y. Ho, R.H. Anderson, and D. Sanchez-Quintana. Atrial structure and fibers: morphologic bases of atrial conduction. Cardiovascular Research, (54):325–336, 2002. [5] M. Krueger, V. Schmidt, C. Tobon, F. Weber, C. Lorenz, D. Keller, H. Barschdorf, M. Burdumy, P. Neher, G. Plank, K. Rhode, G. Seemann, D. Sanchez-Quintana, J. Saiz, R. Razavi, and O. Dossel. Modeling atrial fiber orientation in patient-specific geometries: a semi-automatic rule-based approach. Functional Imaging and Modeling of the Heart, pages 223–232, 2011. [6] F.B. Sachse. Computational Cardiology: Modeling of Anatomy, Electrophysiology and Mechanics. Springer-Verlag, 2004.