This work addresses 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.
The different strategies can be distinguished by the cardiac electrical source models they rely on. One of the first approaches was to estimate equivalent electrical dipoles @cite_45 @cite_13 . Another popular approach is to estimate the heart surface potential, usually called epicardial potential (even though pericardial potential would be more appropriate as noted in @cite_8 ). The potential @math within the torso @math is assumed to be solution of the Poisson problem: where @math denotes the boundary of the heart and @math is the electrical conductivity of the torso. The inverse problem then consists in estimating @math on @math (see e.g. @cite_1 @cite_22 @cite_41 ). This problem being notoriously ill-posed, various regularizations have been proposed: Tikhonov @cite_20 , the use of temporal information @cite_62 @cite_18 , truncated Singular Value Decomposition or truncated Total Least Square @cite_21 .
@cite_18 @cite_62 @cite_22 @cite_8 @cite_41 @cite_21 @cite_1 @cite_45 @cite_13 @cite_20
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We compare two source formulations for the electrocardiographic forward problem in consideration of their implications for regularizing the ill-posed inverse problem. The established epicardial potential source model is compared with a bidomain-theory-based transmembrane potential source formulation. The epicardial source approach is extended to the whole heart surface including the endocardial surfaces. We introduce the concept of the numerical null and signal space to draw attention to the problems associated with the nonuniqueness of the inverse solution and show that reconstruction of null-space components is an important issue for physiologically meaningful inverse solutions. Both formulations were tested with simulated data generated with an anisotropic heart model and with clinically measured data of two patients. A linear and a recently proposed quasi-linear inverse algorithm were applied for reconstructions of the epicardial and transmembrane potential, respectively. A direct comparison of both formulations was performed in terms of computed activation times. We found the transmembrane potential-based formulation is a more promising source formulation as stronger regularization by incorporation of biophysical a priori information is permitted. Abstract An analytic, eccentric-spheres model was used to test the efficacy of different regularization techniques based on the Tikhonov family of regularizers. The model, although simple, retains the relative size and position of the heart within the body and may incorporate all the inhomogeneities of the human torso. The boundary-element method was used to construct a transfer matrix relating the body surface potentials to the epicardial potentials, for the homogeneous form of the model. Different regularization techniques were compared in the presence of surface potential noise and in the presence of errors in estimating the conductivities, the heart size and the heart position. Results indicate that the relative error in the inverse-recovered epicardial potential with regularization does not rise proportionally to the noise level. The relative error (RE) with a 5 Gaussian noise level is 0.17; with 20 it is 0.29. Additionally, the regularized inverse procedure is shown to restore smoothness and accuracy to the inverse-recovered epicardial potentials in the presence of errors in estimating the heart position and heart size, which, using an unregularized inversion, would lead to large-amplitude oscillations in the solution. The ability of a numerical procedure to detect and to localize two experimentally induced, epicardial dipolar generators was tested in 24 isolated, perfused rabbit heart preparations, suspended in an electrolyte-filled spherical tank. Electrocardiograms were recorded from 32 electrodes on the surface of the test chamber before and after placement of each of two epicardial burns. The second lesion was located either 180 degrees, 90 degrees, or 45 degrees from the first. Signals were processed by iterative routines that computed the location of one or two independent dipoles that best reconstruced the observed surface potentials. The computed single dipole acounting for 99.68 of root mean sequare (RMS) surface potential recorded after the first burn was located 0.26 + - 0.10 cm from the centroid of the lesion. Potentials recorded after the second lesions were fit with two dipoles that accounted for 99.36 + - 1.51 of RMS surface potentials and that were located 0.42 + - 0.26 cm and 0.57 + - 0.49 cm from the centers of the corresponding burn. Seventy-one percent of computed dipoles were located within the visible perimeter of the burn. Thus, two simultaneously active dipolar sources can be detected and accurately localized by rigorous study of the generated electrical field. We consider the inverse electrocardiographic problem of computing epicardial potentials from a body-surface potential map. We study how to improve numerical approximation of the inverse problem when the finite-element method is used. Being ill-posed, the inverse problem requires different discretization strategies from its corresponding forward problem. We propose refinement guidelines that specifically address the ill-posedness of the problem. The resulting guidelines necessitate the use of hybrid finite elements composed of tetrahedra and prism elements. Also, in order to maintain consistent numerical quality when the inverse problem is discretized into different scales, we propose a new family of regularizers using the variational principle underlying finite-element methods. These variational-formed regularizers serve as an alternative to the traditional Tikhonov regularizers, but preserves the L2 norm and thereby achieves consistent regularization in multiscale simulations. The variational formulation also enables a simple construction of the discrete gradient operator over irregular meshes, which is difficult to define in traditional discretization schemes. We validated our hybrid element technique and the variational regularizers by simulations on a realistic 3-D torso heart model with empirical heart data. Results show that discretization based on our proposed strategies mitigates the ill-conditioning and improves the inverse solution, and that the variational formulation may benefit a broader range of potential-based bioelectric problems.