Comparing the bidomain and monodomain models in electro-cardiology through convergence analysis

The monodomain and bidomain models are widely used in electro cardiology to simulate spreading of excitation potential waves in the myocardium. The bidomain model is quite popular for its physiological foundation and relevance whereas the monodomain model simply is a heuristic approximation of the previous one, lacking this physiological foundation but providing computational facilities. The purpose of the present article is to numerically compare these two models using a method of (numerical) convergence analysis. This method enables to reach two different objectives of first importance in biomedical engineering. Firstly it provides the discrepancy between the models at the continuous level (and not between the discretised equations only) by getting rid of the discretisation errors. Secondly, it allows to estimate the discretisation error so providing necessary grid resolution in order to run accurate enough simulation. The comparison is held in terms of activation times, a quantity of major physiological importance. Two test cases are considered, both including enhanced cell membrane kinetic description, tissue anisotropy and realistic macroscopic tissue parameters. The first test case is based on an academic (a unit square) geometry. The second one involves a 2d cut of a segmented human heart. It has been built from 3D medical data of the two ventricles and incorporates anisotropy occurring from muscular fibre rotation around the ventricles. Two conclusions are drawn from this study. In terms of activation time relative error, the discrepancy between the two models is quite small: of order 1\% or even below. Moreover this error is smaller than the discretisation error resulting from commonly used mesh size in biomedical engineering