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.
In @cite_9 , the authors note that the usual regularization techniques have no physical ground. Instead, they propose to regularize the inverse solution with the monodomain equations coupled to the Fenton-Karma ionic model. The strategy proposed in the present paper has some similarities with this approach. We rely on a full electrophysiological model of the action potential coupled to the Poisson problem to estimate the solution of the heart electrical activation. This physical model is personalized on the fly with respect to its parameters in order to adapt it to a specific patient. Furthermore, we propose an additional step of modeling by considering the mechanical response to the electrophysiological activation, so we are able to also integrate mechanical measurements. Indeed, we believe that multimodal observations improve the identifiability of the complete model and therefore improve the quality of the electrical and mechanical state reconstruction. Another originality of our work is the use of a sequential data assimilation strategy that is adapted to a coupled electromechanical evolution model. Here we demonstrate how state-of-the-art gain filter on the electrophysiological model and on the mechanical model can be aggregated to propose a joint gain filter for the coupled problem.
@cite_9
2105969869 2112280747 2079961943 2106903293
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. 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 inverse problem of electrocardiography is solved in order to reconstruct electrical events within the heart from information measured noninvasively on the body surface. These electrical events can be deduced from measured epicardial potentials; therefore, a noninvasive method of recovering epicardial potentials from body surface data is useful in clinical and experimental work. The ill-posed nature of this problem necessitates the use of regularization in the solution procedure. Inversion using Tihonov zero-order regularization, a quasi-static method, had been employed previously and was able to reconstruct, with relatively good accuracy, important events in cardiac excitation (maxima, minima, etc.). Taking advantage of the fact that the process of cardiac excitation is continuous in time, one can incorporate information from the time progression of excitation in the regularization procedure using the Twomey technique. Methods of this type were tested on data obtained from a human-torso tank in which a beating canine heart was placed. The results show a marked improvement in the inverse solution when these temporal methods are used. >