The electrocardiographic inverse problem.

Using the boundary element method in conjunction with Tikhonov zero-order regularization, we have computed epicardial potentials from body surface potential data in a realistic geometry heart-torso system. The inverse-reconstructed epicardial potentials were compared to the actual measured potentials throughout a normal cardiac cycle. Potential features (maxima, minima) were recovered with an accuracy better than 1 cm in their location. In this chapter, we use these data to illustrate and discuss computational issues related to the inverse-reconstruction procedure. These include the boundary element method, the choice of a regularization scheme to stabilize the inversion, and the effects of incorporating a priori information on the accuracy of the solution. In particular, emphasis is on the use of temporal information in the regularization procedure. The sensitivity of the solution to geometrical errors and to the spatial and temporal resolution of the data is discussed.

Regional regularization of the electrocardiographic inverse problem: a model study using spherical geometry.

This study examines the use of a new regularization scheme, called regional regularization, for solving the electrocardiographic inverse problem. Previous work has shown that different time frames in the cardiac cycle require varying degrees of regularization. This reflects differences in potential magnitudes, gradients, signal-to-noise ratio (SNR), and locations of electrical activity. One might expect, therefore, that a single regularization parameter and a uniform level of regularization may also be insufficient for a single potential map of a single time frame because in one map there are regions of high and low potentials and potential gradients. Regional regularization is a class of methods that subdivides a given potential map into functional "regions" based on the spatial characteristics of the potential ("spatial frequencies"). These individual regions are regularized separately and recombined into a complete map. This paper examines the hypothesis that such regionally regularized maps are more accurate than if all regions were taken together and solved with an averaged level of regularization. In a homogeneous concentric spheres model, Legendre polynomials are used to decompose a torso potential map into a set of submaps, each with a different degree of spatial variation. The original torso map is contaminated with data noise, or geometrical error or both, and regional regularization improves the epicardial potential reconstruction by up to 25% [relative error (RE)]. Regional regularization also improves the reconstructed location of peaks. A practical goal is to extend the application of this method to the realistic torso geometry, but because Legendre decomposition is limited to geometries with spherical symmetry, other methods of map decomposition must be found. Singular value decomposition (SVD) is used to decompose the maps into component parts. Its individual submaps also have different levels of spatial variation; moreover, it is generalizable to any vector, does not require spherical symmetry, and is extremely efficient numerically. Using SVD decomposition for regional regularization, significant improvement was achieved in the map quality in the presence of data noise.

The use of temporal information in the regularization of the inverse problem of electrocardiography.

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 Tikhonov 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 in the correct human anatomical position. The results show a marked improvement in the inverse solution when these temporal methods are used, and demonstrate that important physiological events (e.g., right ventricular breakthrough) not detected by the quasi-static approach, are reconstructed using these methods. In addition, the results indicate that as the time interval between sampled maps is reduced, the quality of the solutions that use this temporal regularization is greatly improved.

Noninvasive electrocardiographic imaging: reconstruction of epicardial potentials, electrograms, and isochrones and localization of single and multiple electrocardiac events.

BACKGROUND: The goal of noninvasive electrocardiographic imaging (ECGI) is to determine electric activity of the heart by reconstructing maps of epicardial potentials, excitation times (isochrones), and electrograms from data measured on the body surface. METHODS AND RESULTS: Local electrocardiac events were initiated by pacing a dog heart in a human torso-shaped tank. Body surface potential measurements (384 electrodes) were used to compute epicardial potentials noninvasively. The accuracy of reconstructed epicardial potentials was evaluated by direct comparison to measured ones (134 electrodes). Protocols included pacing from single sites and simultaneously from two sites with various intersite distances. Body surface potentials showed a single minimum for both single- and double-site pacing (intersite distances of 52, 35, and 17 mm). Noninvasively reconstructed epicardial electrograms, potentials, and isochrones closely approximated the measured ones. Single pacing sites were reconstructed to within < or = 10 mm of their measured positions. Dual sites were located accurately and resolved for the above intersite distances. Regions of sparse and crowded isochrones, indicating spatial nonuniformities of epicardial activation spread, were also reconstructed. CONCLUSIONS: The study demonstrates that ECGI can reconstruct epicardial potentials, electrograms, and isochrones over the entire epicardial surface during the cardiac cycle. It can provide detailed information on local activation of the heart noninvasively. Its uses could include localization of cardiac electric events (eg, ectopic foci), characterization of nonuniformities of conduction, characterization of repolarization properties (eg, dispersion), and mapping of dynamically changing arrhythmias (eg, polymorphic VT) on a beat-by-beat basis.

Electrocardiographic imaging: Noninvasive characterization of intramural myocardial activation from inverse-reconstructed epicardial potentials and electrograms.

BACKGROUND: A recent study demonstrated the ability of electrocardiographic imaging (ECGI) to reconstruct, noninvasively, epicardial potentials, electrograms, and activation sequences (isochrones) generated by epicardial activation. The current study expands the earlier work to the three-dimensional myocardium and investigates the ability of ECGI to characterize intramural myocardial activation noninvasively and to relate it to the underlying fiber structure of the myocardium. This objective is motivated by the fact that cardiac excitation and arrhythmogenesis involve the three-dimensional ventricular wall and its anisotropic structure. METHODS AND RESULTS: Intramural activation was initiated by pacing a dog heart in a human torso tank. Body surface potentials (384 electrodes) were used to compute epicardial potentials noninvasively. Accuracy of reconstructed epicardial potentials was evaluated by direct comparison to measured ones (134 electrodes). Protocols included pacing from five intramural depths. Epicardial potentials showed characteristic patterns (1) early in activation, central negative region with two flanking maxima aligned with the orientation of fibers at the depth of pacing; (2) counterclockwise rotation of positive potentials with time for epicardial pacing, clockwise rotation for subendocardial pacing, and dual rotation for midmyocardial pacing; and (3) central positive region for endocardial pacing. Noninvasively reconstructed potentials closely approximated these patterns. Reconstructed epicardial electrograms and epicardial breakthrough times closely resembled measured ones, demonstrating progressively later epicardial activation with deeper pacing. CONCLUSIONS: ECGI can noninvasively estimate the depth of intramyocardial electrophysiological events and provides information on the spread of excitation in the three-dimensional anisotropic myocardium on a beat-by-beat basis.