The Effects of Thoracic Inhomogeneities on the Relationship Between Epicardial and Torso Potentials

This study examines the effects of the lungs, spine, sternum, and the anisotropic skeletal muscle layer on the relationship between torso and epicardial potentials. Boundary integral equations representing potentials on the epicardial surface, the torso surface, and the internal conductivity interfaces were solved yielding a set of transfer coefficients valid for any source inside the epicardium and for any conductivity configuration outside the epicardial surface. These transfer coefficients relate potentials on the torso to potentials on the epicardial surface. Calculated torso potentials are generated via the transfer coefficients and measured epicardial potentials for comparison to measured torso potentials. This comparison indicates whether including the thoracic inhomogeneities improves attainable accuracy in calculations relating torso potentials to epicardial potentials.     

Three-Dimensional Simulation of the Ventricular Depolarization and Repolarization Processes and Body Surface Potentials: Nornal Heart and Bundle Branch Block

A three-dimensional computer model has been constructed to simulate the ventricular depolarization and repolarization processes in a human heart. The electrocardiogram (ECG), the vectorcardiogram(VCG), and the body surface potential map (BSPM) during the QRS-T period are obtained automatically under certain heart conditions such as bundle branch block and myocardial infarctions. The ventricles, together with bundle branches and the Purkinje fibers, are composed of approximately 50 000 cell units which are arranged in a cubic close-packed structure. A different action potential waveform was assigned to each unit. The heart model is mounted in a homogeneous human torso model. Electric dipoles, which are proportional to the spatial gradient of the action potential, are generated in all the cell units. These dipoles give rise to a potential distribution on the torso surface, which is calculated by means of the boundary element method. The resulting ECG's, VCG's, and BSPM's are within the expected range of clinical observations.     

Membrane polarization induced in the myocardium by defibrillationfields: an idealized 3-D finite element bidomain/monodomain torso model

This study develops a three-dimensional finite element torso model with bidomain myocardium to simulate the transmembrane potential (TMP) of the heart induced by defibrillation fields. The inhomogeneities of the torso are modeled as eccentric spherical volumes with both the curvature and the rotation features of cardiac fibers incorporated in the myocardial region. The numerical computation of the finite element bidomain myocardial model is validated by a semianalytic solution. The simulations show that rotation of fiber orientation through the depth of the myocardial wall changes the pattern of polarization and decreases the amount of cardiac tissue polarized compared to the idealized analytic model with no fiber rotation incorporated. The TMP induced by transthoracic and transvenous defibrillation fields are calculated and visualized. The TMP is quantified by a continuous measure of the percentage of myocardial mass above a potential gradient threshold. Using this measure, the root mean square differences in TMP distribution produced by reversing the electrode polarity for anterior-posterior and transvenous electrode configurations are 13.6 and 28.6%, respectively. These results support the claim that a bidomain model of the heart predicts a change of defibrillation threshold with reversed electrode polarity     

Forward and inverse problems of electrocardiography: modeling andrecovery of epicardial potentials in humans

To assess the accuracy of solutions to the inverse problem of electrocardiography in man, epicardial potentials computed from thoracic potential distributions were compared to potentials measured directly over the surface of the heart during arrhythmia surgery. Three-dimensional finite element models of the thorax with different mesh resolutions and conductivity inhomogeneities were constructed from serial computerized tomography scans of a patient. These torso models were used to compute transfer matrices relating the epicardial potentials to the thoracic potentials. Potential distributions over the torso and the ventricles were measured with 63 leads in the same patient whose anatomical data was used to construct the torso models. To solve the inverse problem, different methods based on Tykhonov regularization or regularization-truncation were applied. The recovered epicardial potential distributions closely resembled the epicardial potential distributions measured early during ventricular preexcitation, but not the more complex distributions measured later during the QRS complex. Several problems encountered as the validation process is applied in man are also discussed     

On the Possibility of Directly Relating the Pattern of Ventricular Surface Activation to the Pattern of Body Surface Potential Distribution

A system of three 320-element spheres was employed to represent the endocardial and epicardial surfaces of the left ventricle and the body surface. The two inner (heart) spheres were considered electrogenic, and each active subunit was given an onset time and a monophasic action potential; these subunits were treated as source dipoles for successive instants in time. The potential distribution at any instant resulting on the outer (torso) surface was calculated from adding together the corresponding proportionate effects of all active subunits, each treated as dipolar sources. This result was compared to multipolar reduction of simultaneous endocardial and epicardial action potential patterns which, when combined, gave a net multipolar generator content enabling outer pattern approximation. The identity between the patterns of torso surface potential, systematically calculated from multiple dipoles, and those produced from the multipolar reduction provided three insights: 1) the whole surface treatment of the multipolar method is faster, 2) both show an offset term related to the monophasic nature of the sources and similar to that found in live data, and 3) such a model may provide a vehicle for experimentally testing the contribution of intramural sources to body surface potential maps.     

Use of the finite element method to determine epicardial from body surface potentials under a realistic torso model

This paper presents a new method of solution for the inverse problem in electrocardiography using the finite element procedure. It is an application of the authors' earlier work which derived a solution method by means of an integral equation under a generalized configuration of geometry and conductivity of the torso. Based on prior geometry information, the human torso region is discretized into a series offinite elements and, then, electric fields are computed when a set of linearly independent functions chosen as a basis is imposed on the epicardial surface. The set of these forward solutions defines the forward transfer coefficients which relate epicardial to body surface potentials. By the use of the forward transfer coefficients, a constrained least-squares estimate of the epicardial potential distribution can be obtained from measured body surface potentials. The solution method is examined through numerical experiments carried out for a realistic model of the human torso. It is demonstrated that the rapid decrease in voltage far from the heart generator makes this inverse problem ill conditioned and, as a result, the accuracy of the inverse epicardial potentials calculated depends greatly upon both the signal-to-noise ratio and the number of lead points in measuring the body surface potentials.     

The effect of geometric and topologic differences in boundary element models on magnetocardiographic localization accuracy

This study was performed to evaluate the changes in magnetocardiographic (MCG) source localization results when the geometry and the topology of the volume conductor model were altered. Boundary element volume conductor models of three patients were first constructed. These so-called reference torso models were then manipulated to mimic various sources of error in the measurement and analysis procedures. Next, equivalent current dipole localizations were calculated from simulated and measured multichannel MCG data. The localizations obtained with the reference models were regarded as the "gold standard." The effect of each modification was investigated by calculating three-dimensional distances from the gold standard localizations to the locations obtained with the modified model. The results show that the effect of the lungs and the intra-ventricular blood masses is significant for deep source locations and, therefore, the torso model should preferably contain internal inhomogeneities. However, superficial sources could be localized within a few millimeters even with nonindividual, so called standard torso models. In addition, the torso model should extend long enough in the pelvic region, and the positions of the lungs and the ventricles inside the model should be known in order to obtain accurate localizations.     

The effect of torso impedance on epicardial and body surface potentials: a modeling study

Experimental results have been published that report marked changes in measured epicardial potentials when the conductivity of the material surrounding the heart is altered. These reports raise a question as to the validity of the traditional two step, equivalent cardiac source approach to modeling the forward problem of electrocardiology as the equivalent source calculation occurs in what is effectively an isolated cardiac region. In the physical situation the heart is surrounded by a torso that contains many different tissue types with different conductivities and is certainly not isolated. Here, a fully coupled model of the problem is employed where the electrical pathways are continuous from a cellular level through to the body surface. This model is used to investigate the effects that torso inhomogeneities have on epicardial and body surface potentials, including comparisons with a traditional two step approach. In particular, it is shown that adding lungs changes the epicardial potentials by 17%, which is consistent with the reported experimental results. In none of the tested situations did the equivalent source approach completely reproduce the fully coupled results, supporting the notion that a fully coupled approach is required to properly solve the forward problem of electrocardiology.     

Computer simulation of epicardial potentials using a heart-torsomodel with realistic geometry

Previous cardiac simulation studies have focused on simulating the activation isochrones and subsequently the body surface potentials. Epicardial potentials, which are important for clinical application as well as for electrocardiographic inverse problem studies, however, have usually been neglected. This paper describes a procedure of simulating epicardial potentials using a microcomputer-based heart-torso model with realistic geometry. The authors' heart model developed earlier is composed of approximately 65000 cell units which are arranged in a cubic close-packed structure. An action potential waveform with variable in duration is assigned to each unit. The heart model, together with the epicardial surface model constructed recently, 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 the 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 a preexcited beat to mimic Wolff-Parkinson-White (WPW) syndrome are in close agreement with those reported in the literature     

Influence of interface charge inhomogeneities on the measurement of surface state densities in SiSiO2 interfaces by means of the MOS a.c. conductance technique

The relationship between interface charge and surface potential of a MOS capacitor is examined when interface charge inhomogeneities are present. For practical values of the interface charge variance, the relation between interface charge and surface potential is found to be quite linear. High surface state densities and high impurity concentrations tend to damp the potential fluctuations and to increase the linearity. The magnitude of the potential deviation for a given charge deviation increases from flat band to weak inversion and decreases again in strong inversion, due to screening, but the linearity is found to be best in weak inversion.The original Nicollian-Goetzberger analysis of the MOS a.c. conductance technique uses a Gaussian potential distribution and an equivalent circuit consisting of an array of parallel surface state branches connected to a single oxide capacitance. We compare this model with a patchwork model, using a Gaussian interface charge density distribution and an equivalent circuit with distributed oxide capacitance. It is found that in depletion, for practical charge densities, the patchwork model interpretation of conductance peaks does not lead to a very different result than the random charge distribution model interpretation. Both models agree very well on surface state density and variance of the interface charge distribution, but a large discrepancy on the capture cross section of the surface states is possible.     

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.     

Statistically Constrained Inverse Electrocardiography

This paper examines the feasibility of utilizing statistical constraints on the inverse potential model to determine the potential distribution over a 4 cm sphere surrounding the heart from perturbed torso potentials. These perturbed torso potentials reflect instrumentation, quadrature, electrode placement, and heart position uncertainties. This work is an extension of the authors' previous work which concluded that it is not feasible to determine this same potential distribution using unconstrained solutions. However, the results of the present work indicate that with the use of approximate signal and noise covariance matrices, it is possible to achieve estimates of this potential distribution with an average sum squared error of twenty-five percent. Further, the estimation of the signal and noise covariance matrices can be accomplished with a knowledge of heart geometry, torso geometry, The approximate measurement error, and a rough estimate of the time an average section of myocardium is depolarized, but without an a priori specification of the activation sequence.     

Inherent potential inhomogeneity on the semiconductor surface for equilibrium impurity distribution

The inherent inhomogeneity of potential on the doped-semiconductor surface during the formation of equilibrium diffusion distribution for an electroactive impurity in the space-charge layers is discussed. The characteristic random-potential values are determined in the case of nondegenerate surface electron gas. The dependence of these inhomogeneities on the surface and bulk parameters is shown.     

Native disorder potential at the surface of a heavily doped semiconductor

The dependence of native potential inhomogeneities on spatial dispersion of the dielectric response of the two-dimensional electron gas at the surface of a heavily doped semiconductor is discussed. The amplitude and scale of the disorder potential in the case of a strongly degenerate surface electron gas are determined. It is shown that the inhomogeneities considered depend on the surface and bulk parameters.     

An assessment of variable thickness and fiber orientation of theskeletal 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)     

Inverse Recovery of Two Moving Dipoles from Simulated Surface Potential Distributions on a Realistic Human Torso Model

We tested a procedure to recover two moving dipole (TMD) parameters from bidipolar potential distributions generated over the surface of a numerical human torso model, using 120 surface sampling points. The surface distributions were computed for either a finite homogeneous torso (T1), a finite torso with lungs (T2), or a finite torso with lungs and blood masses (T3). Inverse calculations were carried out to initially recover the multipole series components (15, 24, or 35 terms) using either a finite homogeneous torso (I1) or a finite torso with lungs (I2), and a least-squares difference procedure. Next, the TMD parameters were obtained by fitting these multipole series components (15 or 24 terms) to the multipole series components estimated from the Brody shift equations, using the Levenberg-Marquardt iterative algorithm and a series of initial estimates for the TMD solution. A simulation run involved 253 different input dipole-pair combinations and 20 initial estimates. The correct TMD solution almost always coincided with that yielding the minimum residue, which was also the solution obtained by the largest fraction of initial estimates. The lowest rms position error of 2.7 mm was obtained with a T1-I1 torso combination, and with 35 recovered multipole series components and 24 Brody shift equations. Larger errors were obtained using a lower number of recovered multipole series components and Brody shift equations, or pairs of parallel or antiparallel dipoles, or dipoles of unequal amplitude.     

Effect of torso boundaries on electric potential and magnetic fieldof a dipole

A numerical model of a human torso was used to study and compare the effect of outer torso, lung, and intracavitary blood mass boundaries on the body surface distribution of electric potential and normal component of magnetic field due to a single current dipole placed at various locations in the heart. Results are presented in the form of isopotential and isofield maps and are also compared to the maps of a dipole in a semi-infinite homogenous model in the context of single dipole inverse solutions. The inclusion of the boundaries has a large effect on the magnitudes of the maps and modest effects on their topology. The electric and magnetic maps show similar responses to the boundaries for X (leftward) and Y (upward) directed dipoles. The electric maps of Z (back-to-front) dipoles are comparatively unaffected by the boundaries, unlike the magnetic maps of Z dipoles, to which the outer boundary makes a substantial contribution. The results indicate electric and magnetic maps have complementary sensitivities for certain dipole components in the presence of realistic boundaries     

Extracting isovolumes from three-dimensional torso geometry usingPROLOG

Three-dimensional (3D) finite element torso models are widely used to simulate defibrillation field quantities, such as potential, gradient and current density. These quantities are computed at spatial nodes that comprise the torso model. These spatial nodes typically number between 10⁵ and 10⁶, which makes the comprehension of torso defibrillation simulation output difficult. Therefore, the objective of this study is to rapidly prototype software to extract a subset of the geometric model of the torso for visualization in which the nodal information associated with the geometry of the model meets a specified threshold value (e.g., minimum gradient). The data extraction software is implemented in PROLOG, which is used to correlate the coordinate, structural and nodal data of the torso model. A PROLOG-based environment has been developed and is used to rapidly design and test new methods for sorting, collecting and optimizing data extractions from defibrillation simulations in a human torso model for subsequent visualization     

The effects of errors in assumed conductivities and geometry onnumerical solutions to the inverse problem of electrocardiography

Here, the authors used a previously proposed model problem to examine the effects of inhomogeneities on four techniques for numerically solving the inverse problem of electrocardiography. The layered inhomogeneous eccentric spheres system contains three regions representing the lungs, muscle, and subcutaneous fat, and is numerically modeled using finite elements. The authors simulated both anterior and posterior spherical cap activation fronts. They examined inverse solutions based on zero order Tikhonov regularization, truncated singular value decomposition, their new generalized eigensystem approach, and a modification of the generalized eigensystem approach. The effects on the inverse solutions of geometrical errors, errors in the assumed conductivities, and homogeneous torso assumptions were examined     

The use of the spatial covariance in computing pericardialpotentials

This paper investigates the incorporation of the spatial covariance of the pericardial potentials, assumed known a priori as a regularization function, when computing the pericardial potential distribution from observed body surface potentials. The resulting inverse solutions are compared with those using as a regularization function: 1) the norm of the solution, 2) the norm of the surface Laplacian of the solution, as well as with those based on using the truncated singular value decomposition. The study uses a realistic source model to simulate potentials throughout the QRS-interval. This source is placed in an anatomically accurate inhomogeneous volume conductor model of the torso. The use of a single value of the regularization parameter is shown to be feasible: for data incorporating 2% noise, the use of the spatial covariance is demonstrated to result in a relative error over the entire QRS interval as low as 10%. Major errors are demonstrated to result if the effect of the inhomogeneity of the lungs is ignored. The spatial covariance based inverse is shown to be more robust with respect to the perturbations (noise; inhomogeneity) than the other estimators included in this study.