Inverse Problem Of Electrocardiography Research Articles

A comparison of higher-order generalized eigensystem techniques and tikhonov regularization for the inverse problem of electrocardiography*

In a recent series of papers we proposed a new class of methods, the generalized eigensystem (GES) methods, for solving the inverse problem of electrocardiography. In this paper, we compare zero, first, and second order regularized GES methods to zero, first, and second order Tikhonov methods. Both optimal results and results from parameter estimation techniques are compared in terms of relative error and accuracy of epicardial potential maps. Results from higher order regularization depend heavily on the exact form of the regularization operator, and operators generated by finite element techniques give the most accurate and consistent results. In the optimal parameter case, the GES techniques produce smaller average relative errors than the Tikhonov techniques. However, as the regularization order increases, the difference in average relative errors between the two techniques becomes less pronounced. We introduce the minimum distance to the origin (MDO) technique to choose the number of expansion modes for the GES techniques. This produces average relative errors similar to those obtained using the composite residual and smoothing operator (CRESO) with Tikhonov regularization. Second order regularization gives the smallest average relative errors but over-smoothes important epicardial features. In general, GES with MDO resolves the epicardial features better than Tikhonov with CRESO for the data set studied.

Application of high-order boundary elements to the electrocardiographic inverse problem

Eight-noded quadrilateral boundary elements are applied to the electrocardiographic inverse problem as an example for high-order boundary elements. It is shown that the choice of the shape functions used for approximation of the potentials has a remarkable influence on the solution obtained if the number of electrodes is smaller than the number of primary source points (under-determined equation system). Three different formulations are investigated considering a concentric spheres problem where an analytic solution is available: (a) the isoparametric formulation; (b) the quasi-first-order formulation; and (c) the pseudo-subparametric formulation as a new method. In a second step the pseudo-subparametric formulation (which provided the best results in the test problem) is applied to real word data. The transmembrane potential pattern of a 40 years old female suffering from severe heart failure and ventricular tachycardia after large anterior wall myocardial infarction is reconstructed for one time instant. Furthermore, an algorithm for the calculation of the transfer matrix is presented which avoids restrictions to the boundary element mesh caused by the placement of the electrodes.

Inverse electrocardiography by simultaneous imposition of multiple constraints.

We describe two new methods for the inverse problem of electrocardiography. Both employ regularization with multiple constraints, rather than the standard single-constraint regularization. In one method, multiple constraints on the spatial behavior of the solution are used simultaneously. In the other, spatial constraints are used simultaneously with constraints on the temporal behavior of the solution. The specific cases of two spatial constraints and one spatial and one temporal constraint are considered in detail. A new method, the L-Surface, is presented to guide the choice of the required pairs of regularization parameters. In the case when both spatial and temporal regularization are used simultaneously, there is an increased computational burden, and two methods are presented to compute solutions efficiently. The methods are verified by simulations using both dipole sources and measured canine epicardial data.

An admissible solution approach to inverse electrocardiography.

The goal of the inverse problem of electrocardiography is noninvasive discrimination and characterization of normal/abnormal cardiac electrical activity from measurements of body surface potentials. Smoothing and attenuation in the torso volume conductor cause the problem to be ill posed. Standard regularized solutions employ an a priori constraint to achieve reliability and may be biased by the constraint chosen as well as the regularization parameter used to weight it. In this paper, we describe an approach that reformulates this inverse problem as the search for a solution that is a member of an admissible solution set; admissibility is defined in terms of the available constraints. In principle, this approach can utilize as many constraints as may be available, unlike standard techniques which do not easily permit the use of multiple constraints. No regularization parameter is required; instead, we need to choose the nature and size of the constraint sets. Constraints described include several spatial constraints, weighted constraints, and temporal constraints. We describe a solution approach based on iterative convex optimization, and the algorithm--the ellipsoid algorithm--which we have used. Accuracy and feasibility of the method are illustrated with simulation results using dipole sources and measured epicardial potentials.

Analytical validation of the BEM—application of the BEM to the electrocardiographic forward and inverse problem

The objective of this study is to analytically validate a boundary element (BE) formulation for the relationship between the transmembrane potential on the heart's surface and the potential on the body surface applying a concentric spherical test geometry. The relative difference (reldif) between the potential on the outer sphere of the test geometry computed analytically and numerically is determined by 3.59% for the coarse discretization (48 BEs) and by 0.46% in the case of the finer subdivision (192 BEs). In the inverse problem, the transmembrane potential on the inner sphere is estimated numerically from the electric potential on the outer sphere by using a minimum-norm least-square approach. The relative differences found are 20.2% when no measurement noise is added and 26.4% in the presence of 2% additional Gaussian noise. The BE formulation is also applied to real world data for solving the electrocardiographic inverse problem. A normal volunteer's inhomogeneous thorax (outer thorax surface, surfaces of the lungs, epicardial heart surface) is modelled by 424 BEs. The same inverse method is then applied in order to reconstruct the transmembrane potential on the epicardium from the measured body surface potential (BSP) data during normal ventricular depolarisation.

The forward and inverse problems of electrocardiography.

This article summarizes the theoretical underpinnings of both the forward and inverse problems of electrocardiography. Space limitations prohibit describing all of the research work done in these areas, and the author apologizes in advance for any omissions on this account or due to oversight. The article should enable one to gain a better qualitative and quantitative understanding of the heart's electrical activity.

Recent progress in inverse problems in electrocardiology.

The considerable progress achieved in the inverse problem of electrocardiography over the last decade has provided grounds for optimism about the possibility of approaching significant clinically relevant applications in the next decade. However, there are a number of basic questions that still remain. In addressing these questions, the authors feel it is important to seek solutions that emphasize physiological rather than mathematical significance. This approach leads to twin requirements for useful inverse solutions: accuracy, defined in a physiologically meaningful (and not just averaged and mathematical) sense, and reliability, not only to measurement noise but also to geometric modeling errors and other uncertainties that are inescapable in practical application. Studies using analytically tractable models may still be relevant, but it seems more important to find solutions to practical inverse problems, which will move the field toward wider acceptance and credibility.

An improved method for estimating epicardial potentials from the body surface

We present a new method for regularizing the illposed 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.

Performance of generalized eigensystem and truncated singular value decomposition methods for the inverse problem of electrocardiography

Singular Value Decomposition (SVD) and Generalized Eigensystem (GES) inverse techniques are compared for their ability to solve the inverse problem of electrocardiography. In the inverse problem of electrocardiography, electrical potential data for numerous locations on the body (torso) surface is used to infer the electrical potentials on the heart surface, while the governing equation and material properties are assumed known. This paper addresses two areas. First, the previously observed improved performance of GES compared to SVD is explained in terms of the unique nature of the GES vectors. Second, epicardial data from six in-vitro rabbit heart experiments are used to project body surface data for six different geometries, and inverse solutions are computed both with and without added noise. For concentric geometries, GES outperformed SVD in all instances. For eccentric heart/body geometries, GES outperformed SVD when the inverse errors themselves were small. In all cases, GES was less sensitive to added noise than SVD.

The inverse problem of electrocardiography: a solution in terms of single- and double-layer sources of the epicardial surface.

An approach to the inverse problem of electrocardiography that involves an estimation of the electric potentials (double-layer equivalent sources) on the heart's epicardial surface from the electrocardiographic potentials that are measurable on the body surface has received considerable attention. This report deals with a heretofore unexplored extension of this approach, one that yields, in addition to the electric potentials on the epicardial surface, the normal components of their gradients (single-layer equivalent sources). We show that this formulation has at least three advantages over the formulation in term of epicardial potentials alone: (1) single-layer equivalent sources, which reflect the flow of current across the epicardial surface, are well suited for the imaging of regional ischemia and infarction; (2) the transfer matrix linking the epicardial and body-surface potentials for this formulation is less ill conditioned than that for the formulation in terms of potentials alone; (3) the input vector for inverse calculations consists of spatially filtered (rather that directly measured and therefore noise) body-surface potentials. To establish the feasibility of this new formulation of the inverse problem and to compare it with the formulation in terms of potentials alone, we used a realistically shaped boundary-element model of human torso. By calculating singular values less ill conditioned. We then directly calculated epicardial and body-surface potentials for a single dipole located centrally and for three simultaneously active dipoles located eccentrically in the torso's heart region and used these results to test three methods that are prerequisites of a successful inverse solution: Tikhonov regularization, linearly constrained least squares, and an L-curve method. The feasibility of the new formulation was demonstrated by the fact that the method based on the linearly constrained least squares improved on overregularized Tikhonov solutions over a wide range of regularization parameters, and it yielded solutions that were more accurate than the best-possible Tikhonov solutions. Moreover, the L-curve solution procedure, which requires no a priori information about the solution, yielded slightly underregularized, but accurate, estimates for the optimal regularization parameter and the corresponding best-possible Tikhonov solution. Our results also showed that replacing--in the interest computational economy--quadrature formulas for the planar triangles with various approximate formulas for the nodes of the model reduces the accuracy of the inverse solution.

A new method for myocardial activation imaging.

Noninvasive images of the myocardial activation sequence are acquired, based on a new formulation of the inverse problem of electrocardiography in terms of the critical points of the ventricular surface activation map. It is shown that the method is stable with respect to substantial amounts of correlated noise common in the measurements and modeling of electrocardiography and that problems associated with conventional regularization techniques can be circumvented. Examples of application of the method to measured human data are presented. This first invasive validation of results compares well to previously published results obtained by using a standard approach. The method can provide additional constraints on, and thus improve, traditional methods aimed at solving the inverse problem of electrocardiography.

A new method for regularization parameter determination in the inverse problem of electrocardiography.

Computing the potentials on the heart's epicardial surface from the body surface potentials constitutes one form of the inverse problem of electrocardiography. An often-used approach to overcoming the ill-posed nature of the inverse problem and stabilizing the solution is via zero-order Tikhonov regularization, where the squared norms of both the surface potential residual and the solution are minimized, with a relative weight determined by a so-called regularization parameter. This paper looks at the composite residual and smoothing operator (CRESO) and L-curve methods currently used to determine a suitable value for this regularization parameter, t, and proposes a third method that works just as well and is much simpler to compute. This new zero-crossing method selects t such that the squared norm of the surface potential residual is equal to t times the squared norm of the solution. Its performance was compared with those of the other two methods, using three stimulation protocols of increasing complexity. The first of these protocols involved a concentric spheres model for the heart and torso and three current dipoles placed inside the inner sphere as the source distribution. The second replaced the spheres with realistic epicardial and torso geometries, while keeping the three-dipole source configuration. The final simulation kept the realistic epicardial and torso geometries, but used epicardial potential distributions corresponding to both normal and ectopic activation of the heart as the source model. Inverse solutions were computed in the presence of both geometry noise, involving assumed erroneous shifts in the heart position, and of Gaussian measurement noise added to the torso surface potentials. It was verified that in an idealistic situation, in which correlated geometry noise dominated the uncorrelated Gaussian measurement noise, only the CRESO approach arrived at a value for t. Both L-curve and zero-crossing approaches did not work. Once measurement noise dominated geometry noise, all three approaches resulted in comparable t values. It was also shown, however, that often under low measurement noise conditions none of the three resulted in an optimum solution.

Geometric modeling of the human torso using cubic hermite elements.

We discuss the advantages and problems associated with fitting geometric data of the human torso obtained from magnetic resonance imaging, with high-order (bicubic Hermite) surface elements. These elements preserve derivative (C1) continuity across element boundaries and permit smooth anatomically accurate surfaces to be obtained with relatively few elements. These elements are fitted to the data with a new nonlinear fitting procedure that minimizes the error in the fit while maintaining C1 continuity with nonlinear constraints. Nonlinear Sobelov smoothing is also incorporated into this fitting scheme. The structures fitted along with their corresponding root mean-squared error, number of elements used, and number of degrees of freedom (df) per variable are: epicardium (0.91 mm, 40 elements, 142 df), left lung (1.66 mm, 80 elements, 309 df), right lung (1.69 mm, 80 elements. 309 df), skeletal muscle surface (1.67 mm, 264 elements, 1,010 df), fat layer (1.79 mm, 264 elements, 1.010 df), and the skin layer (1.43 mm, 264 elements, 1,010 df). The fitted surfaces are assembled into a combined finite element/boundary element model of the torso in which the exterior surfaces of the heart and lungs are modeled with two-dimensional boundary elements and the layers of the skeletal muscle, fat, and skin are modeled with finite elements. The skeletal muscle and fat layers are modeled with bicubic Hermite linear elements and are obtained by joining the adjacent surface elements for each layer. Applications for the torso model include the forward and inverse problems of electrocardiography, defibrillation studies, radiation dosage studies, and heat transfer studies.

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 potential for Laplacian maps to solve the inverse problem of electrocardiography

This paper presents a method to solve the inverse problem of electrocardiography using the Laplacian of the body surface potentials. The method presented is studied first using trade-off curves from a concentric spheres model representing a heart-torso system. Then a more conventional study is undertaken where a limited number of current dipoles are placed within the inner sphere and noise is added to the resulting potentials and Laplacians on the surface of the outer sphere. The result indicate that measurements of the outer surface Laplacian can more accurately reconstruct epicardial potentials than measurements of the outer surface potentials. The reconstructions are more accurate in that extrema are placed very close to their correct positions and multiple extrema and high potential gradients are recovered. Identical conclusions are observed in the presence of noise and even when the Laplacians are subject to greater noise than the potentials.

Numerical method for electrocardiographic inverse problem. 2D model problem

Numerical method for electrocardiographic inverse problem. 2D model problem

Three-dimensional simulation of epicardial potentials using a microcomputer-based heart-torso model

Previous cardiac simulation studies have focused on simulating the activation isochrones and subsequently the body surface potentials. Epicardial potentials, which are important for clinical applications as well as for electrocardiography inverse problem studies, however, have usually been neglected. This paper presents a procedure of simulating epicardial potentials using a microcomputer-based heart-torso model with real geometry. The heart model developed earlier which was composed of more than 60,000 cell units was used in this study. To simulate the epicardial potentials, an epicardial surface model which enclosed the whole heart was constructed. The heart model, together with the epicardial surface model, are mounted in an inhomogeneous human torso model. Electric dipoles, which are proportional to the spatial gradient of the action potential, are generated in all cell units. These dipoles give rise to a potential distribution on the epicardial surface, which is calculated by means of the boundary element method. The simulated epicardial potential maps during a normal heart beat and in patients with left bundle branch block (LBBB) are in close agreement with those reported in the literature.

Computational issues arising in multidimensional elliptic inverse problems: the inverse problem of electrocardiography

We compare a recently proposed generalized eigensystem approach and a new modified generalized eigensystem approach to more widely used truncated singular value decomposition and zero‐order Tikhonov regularization for solving multidimensional elliptic inverse problems. As a test case, we use a finite element representation of a homogeneous eccentric spheres model of the inverse problem of electrocardiography. Special attention is paid to numerical issues of accuracy, convergence, and robustness. While the new generalized eigensystem methods are substantially more demanding computationally, they exhibit improved accuracy and convergence compared with widely used methods and offer substantially better robustness.

A generalized eigensystem approach to the inverse problem of electrocardiography.

We develop a new approach to the ill-conditioned inverse problem of electrocardiography which employs finite element techniques to generate a truncated eigenvector expansion to stabilize the inversion. Ordinary three-dimensional isoparametric finite elements are used to generate the conductivity matrix for the body. We introduce a related eigenproblem, for which a special two-dimensional isoparametric area matrix is used, and solve for the lowest eigenvalues and eigenvectors. The body surface potentials are expanded in terms of the eigenvectors, and a least squares fit to the measured body surface potentials is used to determine the coefficients of the expansion. This expansion is then used directly to determine the potentials on the surface of the heart. The number of measurement points on the surface of the body can be less than the number of finite element nodes on the body surface, and the number of modes employed in the expansion can be adjusted to reduce errors due to noise.

Inverse electrocardiographic transformations: dependence on the number of epicardial regions and body surface data points

The inverse problem of electrocardiography, the computation of epicardial potentials from body surface potentials, is influenced by the desired resolution on the epicardium, the number of recording points on the body surface, and the method of limiting the inversion process. To examine the role of these variables in the computation of the inverse transform, Tikhonov's zero-order regularization and singular value decomposition (SVD) have been used to invert the forward transfer matrix. The inverses have been compared in a data-independent manner using the resolution and the noise amplification as endpoints. Sets of 32, 50, 192, and 384 leads were chosen as sets of body surface data, and 26, 50, 74, and 98 regions were chosen to represent the epicardium. The resolution and noise were both improved by using a greater number of electrodes on the body surface. When 60% of the singular values are retained, the results show a trade-off between noise and resolution, with typical maximal epicardial noise levels of less than 0.5% of maximum epicardial potentials for 26 epicardial regions, 2.5% for 50 epicardial regions, 7.5% for 74 epicardial regions, and 50% for 98 epicardial regions. As the number of epicardial regions is increased, the regularization technique effectively fixes the noise amplification but markedly decreases the resolution, whereas SVD results in an increase in noise and a moderate decrease in resolution. Overall the regularization technique performs slightly better than SVD in the noise-resolution relationship. There is a region at the posterior of the heart that was poorly resolved regardless of the number of regions chosen. The variance of the resolution was such as to suggest the use of variable-size epicardial regions based on the resolution.