An approach to inverse calculation of epicardial potentials from body surface maps

Electrocardiographic imaging in the atria

The inverse problem of electrocardiography or electrocardiographic imaging (ECGI) is a technique for reconstructing electrical information about cardiac surfaces from noninvasive or non-contact recordings. ECGI has been used to characterize atrial and ventricular arrhythmias. Although it is a technology with years of progress, its development to characterize atrial arrhythmias is challenging. Complications can arise when trying to describe the atrial mechanisms that lead to abnormal propagation patterns, premature or tachycardic beats, and reentrant arrhythmias. This review addresses the various ECGI methodologies, regularization methods, and post-processing techniques used in the atria, as well as the context in which they are used. The current advantages and limitations of ECGI in the fields of research and clinical diagnosis of atrial arrhythmias are outlined. In addition, areas where ECGI efforts should be concentrated to address the associated unsatisfied needs from the atrial perspective are discussed.

Tipping the scales of understanding: An engineering approach to design and implement whole-body cardiac electrophysiology experimental models

The study of cardiac electrophysiology is built on experimental models that span all scales, from ion channels to whole-body preparations. Novel discoveries made at each scale have contributed to our fundamental understanding of human cardiac electrophysiology, which informs clinicians as they detect, diagnose, and treat complex cardiac pathologies. This expert review describes an engineering approach to developing experimental models that is applicable across scales. The review also outlines how we applied the approach to create a set of multiscale whole-body experimental models of cardiac electrophysiology, models that are driving new insights into the response of the myocardium to acute ischemia. Specifically, we propose that researchers must address three critical requirements to develop an effective experimental model: 1) how the experimental model replicates and maintains human physiological conditions, 2) how the interventions possible with the experimental model capture human pathophysiology, and 3) what signals need to be measured, at which levels of resolution and fidelity, and what are the resulting requirements of the measurement system and the access to the organs of interest. We will discuss these requirements in the context of two examples of whole-body experimental models, a closed chest in situ model of cardiac ischemia and an isolated-heart, torso-tank preparation, both of which we have developed over decades and used to gather valuable insights from hundreds of experiments.

Efficient identification of scars using heterogeneous model hierarchies

AIMS

Detection and quantification of myocardial scars are helpful for diagnosis of heart diseases and for personalized simulation models. Scar tissue is generally characterized by a different conduction of excitation. We aim at estimating conductivity-related parameters from endocardial mapping data. Solving this inverse problem requires computationally expensive monodomain simulations on fine discretizations. We aim at accelerating the estimation by combining electrophysiology models of different complexity.

METHODS AND RESULTS

Distributed parameter estimation is performed by minimizing the misfit between simulated and measured electrical activity on the endocardial surface, subject to the monodomain model and regularization. We formulate this optimization problem, including the modelling of scar tissue and different regularizations, and design an efficient solver. We consider grid hierarchies and monodomain-eikonal model hierarchies in a recursive multilevel trust-region method. With numerical examples, efficiency and estimation quality, depending on the data, are investigated. The multilevel solver is significantly faster than a comparable single level solver. Endocardial mapping data of realistic density appears to be sufficient to provide quantitatively reasonable estimates of location, size, and shape of scars close to the endocardial surface.

CONCLUSION

In several situations, scar reconstruction based on eikonal and monodomain models differ significantly, suggesting the use of the more involved monodomain model for this purpose. Eikonal models can accelerate the computations considerably, enabling the use of complex electrophysiology models for estimating myocardial scars from endocardial mapping data.

A new approach to the intracardiac inverse problem using Laplacian distance kernel

Background

The inverse problem in electrophysiology consists of the accurate estimation of the intracardiac electrical sources from a reduced set of electrodes at short distances and from outside the heart. This estimation can provide an image with relevant knowledge on arrhythmia mechanisms for the clinical practice. Methods based on truncated singular value decomposition (TSVD) and regularized least squares require a matrix inversion, which limits their resolution due to the unavoidable low-pass filter effect of the Tikhonov regularization techniques.

Methods

We propose to use, for the first time, a Mercer’s kernel given by the Laplacian of the distance in the quasielectrostatic field equations, hence providing a Support Vector Regression (SVR) formulation by following the principles of the Dual Signal Model (DSM) principles for creating kernel algorithms.

Results

Simulations in one- and two-dimensional models show the performance of our Laplacian distance kernel technique versus several conventional methods. Firstly, the one-dimensional model is adjusted for yielding recorded electrograms, similar to the ones that are usually observed in electrophysiological studies, and suitable strategy is designed for the free-parameter search. Secondly, simulations both in one- and two-dimensional models show larger noise sensitivity in the estimated transfer matrix than in the observation measurements, and DSM−SVR is shown to be more robust to noisy transfer matrix than TSVD.

Conclusion

These results suggest that our proposed DSM−SVR with Laplacian distance kernel can be an efficient alternative to improve the resolution in current and emerging intracardiac imaging systems.

Introduction to noninvasive cardiac mapping

From the dawn of the twentieth century, the electrocardiogram (ECG) has revolutionized the way clinical cardiology has been practiced, and it has become the cornerstone of modern medicine today. Driven by clinical and research needs for a more precise understanding of cardiac electrophysiology beyond traditional ECG, inverse solution electrocardiography has been developed, tested, and validated. This article outlines the important progress from ECG development, through more extensive measurement of body surface potentials, and the fundamental leap to solving the inverse problem of electrocardiography, with a focus on mathematical methods and experimental validation.

Noninvasive reconstruction of cardiac electrical activity: update on current methods, applications and challenges

Electrical activity at the level of the heart muscle can be noninvasively reconstructed from body-surface electrocardiograms (ECGs) and patient-specific torso-heart geometry. This modality, coined electrocardiographic imaging, could fill the gap between the noninvasive (low-resolution) 12-lead ECG and invasive (high-resolution) electrophysiology studies. Much progress has been made to establish electrocardiographic imaging, and clinical studies appear with increasing frequency. However, many assumptions and model choices are involved in its execution, and only limited validation has been performed. In this article, we will discuss the technical details, clinical applications and current limitations of commonly used methods in electrocardiographic imaging. It is important for clinicians to realise the influence of certain assumptions and model choices for correct and careful interpretation of the results. This, in combination with more extensive validation, will allow for exploitation of the full potential of noninvasive electrocardiographic imaging as a powerful clinical tool to expedite diagnosis, guide therapy and improve risk stratification.

Noninvasive imaging of cardiac excitation: current status and future perspective

Noninvasive imaging of cardiac excitation using body surface potential mapping (BSPM) data and inverse procedures is an emerging technique that enables estimation of myocardial depolarization and repolarization. Despite numerous reports on the possible advantages of this imaging technique, it has not yet advanced into daily clinical practice. This is mainly due to the time consuming nature of data acquisition and the complexity of the mathematics underlying the used inverse procedures. However, the popularity of this field of research has increased and noninvasive imaging of cardiac electrophysiology is considered a promising tool to complement conventional invasive electrophysiological studies. Furthermore, the use of appropriately designed electrode vests and more advanced computers has greatly reduced the procedural time. This review provides descriptive overview of the research performed thus far and the possible future directions. The general challenges in routine application of BSPM and inverse procedures are discussed. In addition, individual properties of the biophysical models underlying the inverse procedures are illustrated.

Assessment of regularization techniques for electrocardiographic imaging

A widely used approach to solving the inverse problem in electrocardiography involves computing potentials on the epicardium from measured electrocardiograms (ECGs) on the torso surface. The main challenge of solving this electrocardiographic imaging (ECGI) problem lies in its intrinsic ill-posedness. While many regularization techniques have been developed to control wild oscillations of the solution, the choice of proper regularization methods for obtaining clinically acceptable solutions is still a subject of ongoing research. However there has been little rigorous comparison across methods proposed by different groups. This study systematically compared various regularization techniques for solving the ECGI problem under a unified simulation framework, consisting of both 1) progressively more complex idealized source models (from single dipole to triplet of dipoles), and 2) an electrolytic human torso tank containing a live canine heart, with the cardiac source being modeled by potentials measured on a cylindrical cage placed around the heart. We tested 13 different regularization techniques to solve the inverse problem of recovering epicardial potentials, and found that non-quadratic methods (total variation algorithms) and first-order and second-order Tikhonov regularizations outperformed other methodologies and resulted in similar average reconstruction errors.

Application of L1-norm regularization to epicardial potential solution of the inverse electrocardiography problem

The electrocardiographic inverse problem of computing epicardial potentials from multi-electrode body-surface ECG measurements, is an ill-posed problem. Tikhonov regularization is commonly employed, which imposes penalty on the L2-norm of the potentials (zero-order) or their derivatives. Previous work has indicated superior results using L2-norm of the normal derivative of the solution (a first order regularization). However, L2-norm penalty function can cause considerable smoothing of the solution. Here, we use the L1-norm of the normal derivative of the potential as a penalty function. L1-norm solutions were compared to zero-order and first-order L2-norm Tikhonov solutions and to measured 'gold standards' in previous experiments with isolated canine hearts. Solutions with L1-norm penalty function (average relative error [RE] = 0.36) were more accurate than L2-norm (average RE = 0.62). In addition, the L1-norm method localized epicardial pacing sites with better accuracy (3.8 +/- 1.5 mm) compared to L2-norm (9.2 +/- 2.6 mm) during pacing in five pediatric patients with congenital heart disease. In a pediatric patient with Wolff-Parkinson-White syndrome, the L1-norm method also detected and localized two distinct areas of early activation around the mitral valve annulus, indicating the presence of two left-sided pathways which were not distinguished using L2 regularization.

Application of the method of fundamental solutions to potential-based inverse electrocardiography

Potential-based inverse electrocardiography is a method for the noninvasive computation of epicardial potentials from measured body surface electrocardiographic data. From the computed epicardial potentials, epicardial electrograms and isochrones (activation sequences), as well as repolarization patterns can be constructed. We term this noninvasive procedure Electrocardiographic Imaging (ECGI). The method of choice for computing epicardial potentials has been the Boundary Element Method (BEM) which requires meshing the heart and torso surfaces and optimizing the mesh, a very time-consuming operation that requires manual editing. Moreover, it can introduce mesh-related artifacts in the reconstructed epicardial images. Here we introduce the application of a meshless method, the Method of Fundamental Solutions (MFS) to ECGI. This new approach that does not require meshing is evaluated on data from animal experiments and human studies, and compared to BEM. Results demonstrate similar accuracy, with the following advantages: 1. Elimination of meshing and manual mesh optimization processes, thereby enhancing automation and speeding the ECGI procedure. 2. Elimination of mesh-induced artifacts. 3. Elimination of complex singular integrals that must be carefully computed in BEM. 4. Simpler implementation. These properties of MFS enhance the practical application of ECGI as a clinical diagnostic tool.

Meshless methods in potential inverse electrocardiography

Potential inverse electrocardiography (PIE) is a method for reconstruction of epicardial potentials from measured body surface electrocardiograms and heart-torso geometry. The method of choice for computing epicardial potentials has been either the boundary element method (BEM) or the finite element method (FEM). These methods require time-consuming meshing of the heart and torso surfaces or the volume between the two surfaces. Moreover, meshing can introduce artifacts in the reconstructed epicardial images if optimization is not carefully done. Here we introduce the application of a meshless method, the method of fundamental solutions (MFS) to PIE in the hope of overcoming such meshing-related problems. This study shows that MFS is a promising meshless alternative to BEM and FEM in PIE.

Localization of the site of origin of reentrant arrhythmia from body surface potential maps: a model study

We have developed a model-based imaging approach to estimate the site of origin of reentrant arrhythmia from body surface potential maps (BSPMs), with the aid of a cardiac arrhythmia model. The reentry was successfully simulated and maintained in the cardiac model, and the simulated ECG waveforms over the body surface corresponding to a maintained reentry have evident characteristics of ventricular tachycardia. The performance of the inverse imaging approach was evaluated by computer simulations. The present simulation results show that an averaged localization error of about 1.5 mm, when 5% Gaussian white noise was added to the BSPMs, was detected. The effects of the heart-torso geometry uncertainty on the localization were also initially assessed and the simulation results suggest that no significant influence was observed when 10% torso geometry uncertainty or 10 mm heart position shifting was considered. The present simulation study suggests the feasibility of localizing the site of origin of reentrant arrhythmia from non-invasive BSPMs, with the aid of a cardiac arrhythmia model.

Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model

We have developed a new approach for imaging cardiac transmembrane potentials (TMPs) within the three-dimensional (3-D) myocardium by means of an anisotropic heart model. The cardiac TMP distribution is estimated from body surface electrocardiograms by minimizing objective functions of the "measured" body surface potential maps (BSPMs) and the heart-model-generated BSPMs. Computer simulation studies have been conducted to evaluate the present 3-D TMP imaging approach using pacing protocols. Simulations of single-site pacing at 24 sites throughout the ventricles, as well as dual-site pacing at 12 pairs of sites in the vicinity of atrio-ventricular ring were performed. The present simulation results show that the correlation coefficient (CC) and relative error (RE) between the "true" and inversely estimated TMP distributions were 0.9915 +/- 0.0041 and 0.1266 +/- 0.0326, for single-site pacing, and 0.9889 +/- 0.0034 and 0.1473 +/- 0.0237 for dual-site pacing, respectively, when 10 microV Gaussian white noise (GWN) was added to the BSPMs. The effects of heart and torso geometry uncertainty were also evaluated by shifting the heart position by 10 mm and altering the torso size by 10%. The CC between the "true" and inversely estimated TMP distributions was above 0.97 when these geometry uncertainties were considered. The present simulation results demonstrate the feasibility of noninvasive estimation of TMP distribution throughout the ventricles from body surface electrocardiographic measurements, and suggest that the present method may become a useful alternative in noninvasive imaging of distributed cardiac electrophysiological processes within the 3-D myocardium.

Noninvasive localization of the site of origin of paced cardiac activation in human by means of a 3-D heart model

A recently developed heart-model-based localization approach is experimentally evaluated in noninvasively localizing the site of origin of cardiac activation in a patient with a pacemaker. The heart-torso model of the patient was constructed from the contrast ultrafast computed tomography images. The site of initial paced activation in the patient was quantitatively localized and compared with the tip position of the pacemaker lead. The localization error of the inverse estimation was found to be 5.2 mm with respect to the true lead tip position. The promising result of this pilot experimental study suggests the feasibility of localizing the site of origin of cardiac activation in an experimental setting. The heart-model-based localization approach may become an alternative tool in localizing the site of origin of cardiac activation.

Noninvasive three-dimensional activation time imaging of ventricular excitation by means of a heart-excitation model

We propose a new method for imaging activation time within three-dimensional (3D) myocardium by means of a heart-excitation model. The activation time is estimated from body surface electrocardiograms by minimizing multiple objective functions of the measured body surface potential maps (BSPMs) and the heart-model-generated BSPMs. Computer simulation studies have been conducted to evaluate the proposed 3D myocardial activation time imaging approach. Single-site pacing at 24 sites throughout the ventricles, as well as dual-site pacing at 12 pairs of sites in the vicinity of atrioventricular ring, was performed. The present simulation results show that the average correlation coefficient (CC) and relative error (RE) for single-site pacing were 0.9992+/-0.0008/0.9989+/-0.0008 and 0.05+/-0.02/0.07+/-0.03, respectively, when 5 microV/10 microV Gaussian white noise (GWN) was added to the body surface potentials. The average CC and RE for dual-site pacing were 0.9975+/-0.0037 and 0.08+/-0.04, respectively, when 10 microV GWN was added to the body surface potentials. The present simulation results suggest the feasibility of noninvasive estimation of activation time throughout the ventricles from body surface potential measurement, and suggest that the proposed method may become an important alternative in imaging cardiac electrical activity noninvasively.

Imaging and visualization of 3-D cardiac electric activity

Noninvasive imaging of cardiac electric activity is of importance for better understanding the underlying mechanisms and for aiding clinical diagnosis and intervention of cardiac abnormalities. We propose to image the three-dimensional (3-D) cardiac bioelectric source distribution from body-surface electrocardiograms. Cardiac electrical sources were modeled by a current dipole distribution throughout the entire myocardium, and estimated by using the Laplacian weighted minimum norm (LWMN) algorithm from body-surface potentials. The estimated inverse solution of the current distribution was further improved by using a recursive weighting strategy for localized sources, such as origins of cardiac arrhythmias. Computer simulations were conducted to test the feasibility of the proposed approach by using a 3-D ventricle model embedded in a realistically shaped torso model. The boundary element method was used to solve the forward problem from assumed cardiac sources to the body-surface potentials. Two testing dipoles were placed in the left and right ventricles, simulating the early activation associated with ventricular arrhythmias. The LWMN inverse solution showed an equivalent source distribution over the entity of both ventricles, with spread areas of activity overlying the positions of the testing dipoles. The sharpened inverse image provides well-localized focal sources near the testing dipole positions. In summary, the present computer simulation suggests that the proposed 3-D cardiac current source imaging and localization approach appears to be a promising candidate for localizing and imaging sites of origins of cardiac activation.

Localization of the site of origin of cardiac activation by means of a heart-model-based electrocardiographic imaging approach

We have developed a new approach to solve the inverse problem of electrocardiography in terms of heart model parameters. The inverse solution of the electrocardiogram (ECG) inverse problem is defined, in the present study, as the parameters of the heart model, which are closely related to the physiological and pathophysiological status of the heart, and is estimated by using an optimization system of heart model parameters, instead of solving the matrix equation relating the body surface ECGs and equivalent cardiac sources. An artificial neural network based preliminary diagnosis system has been developed to limit the searching space of the optimization algorithm and to initialize the model parameters in the computer heart model. The optimal heart model parameters were obtained by minimizing the objective functions, as functions of the observed and model-generated body surface ECGs. We have tested the feasibility of the newly developed technique in localizing the site of origin of cardiac activation using a pace mapping protocol. The present computer simulation results show that, the present approach for localization of the site of origin of ventricular activation achieved an averaged localization error of about 3 mm [for 5-muV Gaussian white noise (GWN)] and 4 mm (for 10-muV GWN), with standard deviation of the localization errors of being about 1.5 mm. The present simulation study suggests that this newly developed approach provides a robust inverse solution, circumventing the difficulties of the ECG inverse problem, and may become an important alternative to other ECG inverse solutions.

A computer simulation study of cortical imaging from scalp potentials

In this paper, computer simulation studies were conducted to test the feasibility of imaging brain electrical activity from the scalp electroencephalograms. The inhomogeneous three-concentric-sphere head model was used to represent the head volume conductor. Closed spherical dipole layers, consisting of several thousand uniformly distributed dipoles, were used to reconstruct the cortical potential maps corresponding to neuronal sources located inside the brain. Simulation results indicate that the present procedure can image both cortical and deep sources, and for the cortical sources, a spatial resolution as high as 1.2 cm can be achieved.

The effects of inhomogeneities and anisotropies on electrocardiographic fields: a 3-D finite-element study

The aim of this study was to quantify the effects of selected inhomogeneities and anisotropies on computed electric potential fields associated with the electrocardiographic forward problem. The model construction was based on the Utah Torso model and included geometry for major anatomical structures such as subcutaneous fat, skeletal muscle, and lungs, as well as for epicardial fatpads, major arteries and veins, and the sternum, ribs, spine, and clavicles. Measured epicardial potentials served as the electrical source for solutions to the electrocardiographic forward problems computed using the finite element method (FEM). The geometry of the torso model for each simulation was constant, but different combinations of conductivities were assigned to individual organs or tissues. Comparisons of different conductivity combinations followed one of two basic schemes: 1) a homogeneous torso served as the reference against which we compared simulations with a single organ or tissue and assigned its nominal conductivity, or 2) a fully inhomogeneous torso served as the reference and we removed the effect of individual organs or tissues by assigning it the homogeneous conductivity value. When single inhomogeneities were added to an otherwise homogeneous isotropic model, anisotropic skeletal muscle (at a 15:1 anisotropy ratio) and the right and left lung had larger average effects (12.8, 12.7, and 12.1% relative error (RE), respectively) than the other inhomogeneities tested. Our results for removing single inhomogeneities show that the subcutaneous fat, the anisotropic skeletal muscle (with the degree of anisotropy equal to 7:1), and the lungs have larger average impacts on the body surface potential distributions than other elements of the model (with values of 14.9, 12.6, and 11.7% RE, respectively). The results also show that the size of the effect depended strongly on the distribution of epicardial potentials. The results of this study suggest that accurate representation of tissue inhomogeneity has a significant effect on the accuracy of the forward solution, with regions near the torso surface playing a larger role, in general, than those near the heart.

A bioelectric inverse imaging technique based on surface Laplacians

A new approach is proposed to solve bioelectric inverse problems by employing the surface Laplacian of the bioelectrical potential. A theoretical investigation was conducted to test the feasibility of epicardial inverse imaging of cardiac electrical activity. A two-sphere homogeneous volume conductor model, where the inner sphere represents the epicardium and the outer sphere the body surface, was used. Radial and tangential current dipoles were used to approximate localized wavefronts propagating from the endocardium to the epicardium, and ectopic myocardial activities. The epicardial potential distribution was reconstructed from the body surface Laplacians with the aid of the Tikhonov zero-order regularization technique, which then was compared with the results obtained from the body surface potentials using the same regularization scheme. The two inverse solutions were compared qualitatively via visual inspection of the reconstructed epicardial potential maps, and quantitatively by examining relative errors and correlation coefficients between the "true" and the reconstructed epicardial potentials. Both qualitative and quantitative results indicate that the surface Laplacians play a positive role in improving the ill-posed nature of the bioelectric inverse problem, which would enhance our capability of reconstructing important epicardial events such as extrema in the epicardial potential distribution. The present theoretical study suggests that the Laplacian-based inverse imaging technique may have important applications to epicardial inverse imaging and other bioelectric inverse imaging.

An equivalent body surface charge model representing three-dimensional bioelectrical activity

A new surface-source model has been developed to account for the bioelectrical potential on the body surface. A single-layer surface-charge model on the body surface has been developed to equivalently represent bioelectrical sources inside the body. The boundary conditions on the body surface are discussed in relation to the surface-charge in a half-space conductive medium. The equivalent body surface-charge is shown to be proportional to the normal component of the electric field on the body surface just outside the body. The spatial resolution of the equivalent surface-charge distribution appears intermediate between those of the body surface potential distribution and the body surface Laplacian distribution. An analytic relationship between the equivalent surface-charge and the surface Laplacian of the potential was found for a half-space conductive medium. The effects of finite spatial sampling and noise on the reconstruction of the equivalent surface-charge were evaluated by computer simulations. It was found through computer simulations that the reconstruction of the equivalent body surface-charge from the body surface Laplacian distribution is very stable against noise and finite spatial sampling. The present results suggest that the equivalent body surface-charge model may provide an additional insight to our understanding of bioelectric phenomena.

A comparison of measured and calculated intracavitary potentials for electrical stimuli in the exposed dog heart

The objective of this paper is to test the feasibility of using a multielectrode, intracavitary probe to solve a forward problem in which measured intracavitary potentials are compared to those calculated from subendocardial potentials and left ventricular (LV) cavity geometry. Intracavitary potentials and subendocardial potentials are measured simultaneously during electrical pacing stimuli from the LV apex, LV anterior base, LV posterior base, and right ventricular (RV) outflow tract of three exposed dog hearts. The LV cavity geometry is measured from postmortem magnetic resonance microscopy images of fixed hearts. Boundary integrals are approximated using a boundary element method and solved for intracavitary potentials. Correlation coefficients for LV apical pacing episodes are 0.989 +/- 0.002 while those for nonapical pacing episodes are 0.873 +/- 0.092. These results indicate that for electrical pacing from the apex, intracavitary stimulus potentials can be calculated with a high degree of accuracy. For nonapical pacing locations, the accuracy decreases since the calculations are more sensitive to errors in measuring probe position and LV cavity geometry near the septum. These results show that accurate geometric measurements of the intracavitary probe position and subendocardial surface are the primary concerns in solving future forward and inverse problems using an intracavitary probe.

An assessment of variable thickness and fiber orientation of the skeletal muscle layer on electrocardiographic calculations

This paper assesses the effectiveness of including variable thickness and fiber orientation characteristics of the skeletal muscle layer in calculations relating epicardial and torso potentials. A realistic model of a canine torso which includes extensive detail about skeletal muscle layer thickness and fiber orientation is compared with two other uniformly anisotropic models: one of constant thickness and the other of variable thickness. First, transfer coefficients are calculated from the model data. Then torso potentials for each model are calculated from the transfer coefficients and measured epicardial potentials. The comparison of calculated and observed torso potentials indicates that a simple model consisting of a uniformly anisotropic skeletal muscle layer of 1.0-1.5 cm constant thickness significantly improves the model. However, if photographic slices of the canine torso are used to introduce more detailed data about the variation in skeletal muscle thickness and fiber orientation into the model, the agreement and between calculated and measured torso potentials decreased, although a finite element mesh of over 5000 nodes was used to describe the skeletal muscle in the more detailed model. One source of error increase was considered to be due to numerical discretization and could be reduced with a much finer mesh or by utilizing higher order polynomials to represent the potential distribution within each finite element. However, the results presented in this paper show that high precision computation (64-bit word length) on the mainframe IBM 3081 with an attached FPS-164 gives a slow rate of improvement with reduced discretization intervals and that utilizing higher order polynomials within each finite element gives an even slower rate of improvement.(ABSTRACT TRUNCATED AT 250 WORDS)

Comparison and testing of least-squares time domain inverse solutions in electrocardiography

The use of several mathematical methods for estimating epicardial ECG potentials from arrays of body surface potentials has been reported in the literature; most of these methods are based on least-squares reconstruction principles and operate in the time-space domain. In this paper we introduce a general Bayesian maximum a posteriori (MAP) framework for time domain inverse solutions in the presence of noise. The two most popular previously applied least-squares methods, constrained (regularized) least-squares and low-rank approximation through the singular value decomposition, are placed in this framework, each of them requiring the a priori knowledge of a 'regularization parameter', which defines the degree of smoothing to be applied to the inversion. Results of simulations using these two methods are presented; they compare the ability of each method to reconstruct epicardial potentials. We used the geometric configuration of the torso and internal organs of an individual subject as reconstructed from CT scans. The accuracy of each method at each epicardial location was tested as a function of measurement noise, the size and shape of the subarray of torso sensors, and the regularization parameter. We paid particular attention to an assessment of the potential of these methods for clinical use by testing the effect of using compact, small-size subarrays of torso potentials while maintaining a high degree of resolution on the epicardium.