Section: New Results

Numerical methods for cardiac electrophysiology

Participants : Muriel Boulakia, Jean-Frédéric Gerbeau, Damiano Lombardi.

In [58] we investigate the monodomain equation which describes the evolution of the cardiac electrical potential and which corresponds to a coupled system involving a reaction-diffusion equation and an ordinary differential equation. Lipschitz stability inequalities are shown for the identification of some parameters of the model from measurements on the cardiac potential and the ionic variable.

In [32] we studied the application of a Reduced-Order Modeling method (Approximated Lax Pairs) to the solution of the partial differential equations describing the polarisation of tissues. Due to the complexity of the scenarios involved and the presence of propagating waves, the performances of the standard methods proposed in the literature to provide a low computational cost solution are not always satisfactory. The ALP method consists of the construction of an adaptive time dependent basis that diagonalises, at each time, a Schrödinger-type operator. Its application to several 2D and 3D test-cases on the equations arising in electro-physiology was investigated, showing that the performances of the method in terms of speed-up and accuracy are promising.

In [62] we considered the simulation of full cycles of the electrical activity of the heart and the corresponding body surface potential. The model is based on a realistic torso and heart anatomy, including ventricles and atria. One of the specificities of our approach is to model the atria as a surface, which is the kind of data typically provided by medical imaging for thin volumes. The bidomain equations are considered in their usual formulation in the ventricles, and in a surface formulation on the atria. Two ionic models are used: the Courtemanche-Ramirez-Nattel model on the atria, and the " Minimal model for human Ventricular action potentials " (MV) by Bueno-Orovio, Cherry and Fenton in the ventricles. The heart is weakly coupled to the torso by a Robin boundary condition based on a resistor-capacitor transmission condition. Various ECGs are simulated in healthy and pathological conditions (left and right bundle branch blocks, Bachmann's bundle block, Wolff-Parkinson-White syndrome). To assess the numerical ECGs, we use several qualitative and quantitative criteria found in the medical literature. Our simulator can also be used to generate the signals measured by a vest of electrodes. This capability is illustrated at the end of the article.

In [24] we address the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the definition of a model of the electromechanical contraction which is registered on ECG data but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish in this work the convergence of a sequential estimator which combines for such coupled problems various state of the art sequential data assimilation methods in a unified consistent and efficient framework. Indeed we aggregate a Luenberger observer for the mechanical state and a Reduced Order Unscented Kalman Filter applied on the parameters to be identified and a POD projection of the electrical state. Then using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat compared with more classical strategies which only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the advantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart.