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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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. > The authors present a new method for regularizing the ill-posed problem of computing epicardial potentials from body surface potentials. The method simultaneously regularizes the equations associated with all time points, and relies on a new theorem which states that a solution based on optimal regularization of each integral equation associated with each principal component of the data will be more accurate than a solution based on optimal regularization of each integral equation associated with each time point. The theorem is illustrated with simulations mimicking the complexity of the inverse electrocardiography problem. As must be expected from a method which imposes no additional a priori constraints, the new approach addresses uncorrelated noise only, and in the presence of dominating correlated noise it is only successful in producing a "cleaner" version of a necessarily compromised solution. Nevertheless, in principle, the new method is always preferred to the standard approach, since it (without penalty) eliminates pure noise that would otherwise be present in the solution estimate. Background—The last decade witnessed an explosion of information regarding the genetic, molecular, and mechanistic basis of heart disease. Translating this information into clinical practice requires the development of novel functional imaging modalities for diagnosis, localization, and guided intervention. A noninvasive modality for imaging cardiac arrhythmias is not yet available. Present electrocardiographic methods cannot precisely localize a ventricular tachycardia (VT) or its key reentrant circuit components. Recently, we developed a noninvasive electrocardiographic imaging modality (ECGI) that can reconstruct epicardial electrophysiological information from body surface potentials. Here, we extend its application to image reentrant arrhythmias. Methods and Results—Epicardial potentials were recorded during VT with a 490 electrode sock during an open chest procedure in 2 dogs with 4-day-old myocardial infarctions. Body surface potentials were generated from these epicardial potentials in a human tor... 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. 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. # Geometric Modelling # Cell Modelling # Tissue Modelling # Whole-Heart Modelling # Organ in the Body -- The Forward Problem of Electrocardiology # The Inverse Problem of Electrocardiology # Modelling Other Cardiac Processes Although it has been known throughout this century that a complex sequence of electrical events is produced on the body surface by the electrophysiological properties of the heart, the question of how well these body surface events can be explained mathematically on the basis of experimental measurements of cardiac geometry and electrical activity remains unanswered. Recent advances in experimental capabilities have made possible the near simultaneous measurement of both cardiac epicardial and corresponding body surface potential distributions from in vivo animal preparations using chronically implanted electrodes to keep the volume conductor intact. This report provides a method for finding transfer coefficients that relate the epicardial and body surface potential distributions to each other. The method is based on knowing the geometric location of each electrode, and on having enough electrodes to establish the geometric shape and the potential distribution of closed epicardial and body surfaces. However, the method does not require that either the heart or body surfaces have any special shape, such as that of a sphere, or that any electrical quantities, such as voltage gradients, be known in addition to the potentials. The use of potential distributions to represent heart electrical activity is advantageous since such distributions can be directly measured experimentally, without a transformation to any other form, such as multiple current-generating dipoles, being required. This report includes a statement of the underlying integral equations, the procedure. for finding the equations' coefficients from geometry measurements, some considerations for computer algorithms, and an example. This paper reviews and updates the single moving dipole (SMD) and two moving dipole (TMD) inverse electrocardiographic and electroencephalographic solutions. These inverse solutions are particularly appropriate when the electrical activity of the heart or brain may be represented by one or two well-localized foci. They attempt to match the measured body surface or evoked scalp potentials to potentials generated on the surface of a model of the intervening volume conductor by one or two moving current dipoles, respectively. The two alternative methods of solution are discussed initially. The first is a direct least-squares error match of measured and model-generated surface potentials, the second an indirect solution based on the least-squares error match of the potentials due to equivalent multipole series representations of the real and model sources, respectively. Next, brief reviews of moving dipole inverse solutions in the EEG and ECG fields are presented. Simulation studies, as well as experimental and clinical studies in animals and humans, are described. The Discussion section summarizes the optimum solution approaches that should be used in clinical EEG and ECG studies in man. It also cautions against the temptation to translate the numerical adequacy of inverse SMD and TMD solutions into physiological validity, without independent knowledge as to the nature of the underlying sources. 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. 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.