Identification of weakly coupled multiphysics problems. Application to the inverse problem of electrocardiography

This work addresses the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the defini-tion 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 ag-gregate 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 ad-vantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart

Doctor of Philosophy

dissertationInverse Electrocardiography (ECG) aims to noninvasively estimate the electrophysiological activity of the heart from the voltages measured at the body surface, with promising clinical applications in diagnosis and therapy. The main challenge of this emerging technique lies in its mathematical foundation: an inverse source problem governed by partial differential equations (PDEs) which is severely ill-conditioned. Essential to the success of inverse ECG are computational methods that reliably achieve accurate inverse solutions while harnessing the ever-growing complexity and realism of the bioelectric simulation. This dissertation focuses on the formulation, optimization, and solution of the inverse ECG problem based on finite element methods, consisting of two research thrusts. The first thrust explores the optimal finite element discretization specifically oriented towards the inverse ECG problem. In contrast, most existing discretization strategies are designed for forward problems and may become inappropriate for the corresponding inverse problems. Based on a Fourier analysis of how discretization relates to ill-conditioning, this work proposes refinement strategies that optimize approximation accuracy o f the inverse ECG problem while mitigating its ill-conditioning. To fulfill these strategies, two refinement techniques are developed: one uses hybrid-shaped finite elements whereas the other adapts high-order finite elements. The second research thrust involves a new methodology for inverse ECG solutions called PDE-constrained optimization, an optimization framework that flexibly allows convex objectives and various physically-based constraints. This work features three contributions: (1) fulfilling optimization in the continuous space, (2) formulating rigorous finite element solutions, and (3) fulfilling subsequent numerical optimization by a primal-dual interiorpoint method tailored to the given optimization problem's specific algebraic structure. The efficacy o f this new method is shown by its application to localization o f cardiac ischemic disease, in which the method, under realistic settings, achieves promising solutions to a previously intractable inverse ECG problem involving the bidomain heart model. In summary, this dissertation advances the computational research of inverse ECG, making it evolve toward an image-based, patient-specific modality for biomedical research

Electrodynamic Model of the Heart to Detect Necrotic Areas in a Human Heart

To diagnose the conditions and diseases of the cardiovascular system is the main task of electrocardiology. The problem of the cardiovascular system diagnostics is caused by a complex multi-level mechanism of its functioning, and only experienced specialists are able to establish a correct diagnosis. Since the working heart is inaccessible to direct observations in real life, diagnostics of diseases is based on noninvasive methods such as electrocardiography. By assumption, weak "bursts" (micropotentials) of electrocardiographic signals in different areas are the precursors of dangerous arrhythmias. The amplitude of these signals on the body surface is insignificant and tends to be commensurate with the noise level of the measuring system. Advances in electrocardiography make it possible to generate a high resolution ECG signal and to detect the heart micropotentials. The method of modeling helps to understand causes of micropotentials in the ECG signal by selecting the model parameters. The model of the heart should allow generating a signal close to the high resolution ECG signal. The research aims to find a numerical model that allows solving the inverse problem of the heart tissue characteristics recovery using a high resolution ECG signal and CT data on the heart geometry. The proposed computer model and highly sensitive methods for the ECG measurement are the part of the hardware-software complex to detect dangerous precursors of cardiac arrhythmias

A Bayesian approach for uncertainty quantification in elliptic Cauchy problem

International audienceWe study the Cauchy problem in the framework of static linear elasticity and its resolution via the Steklov-Poincaré approach. In the linear Gaussian framework, the straightforward application of Bayes theory leads to formulas allowing to deduce the uncertainty on the identified field from the noise level. We use a truncated Ritz decomposition of the Steklov-Poincaré operator, which reduces the number of degrees of freedom and significantly lowers the computational cost