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     

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.     

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     

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.     

On the theory of the electrocardiogram

The biophysical basis for understanding the electrocardiogram is set forth. Bioelectric sources arise from electrical activity in the heart at the cellular level. The relation of these sources, which can be formally represented as impressed currents, to potentials involves solution of the volume conductor problem. This solution is based on Green's theorem. Sources are related to the transmembrane action potential through a bidomain model of heart muscle. Microscopic and macroscopic aspects of the bidomain model are developed. Various transformations of the source are considered, including multipoles, multiple dipoles, and replacement of the volume distribution with distributions on the heart surface. Time integrals of the waveform are related to excitation time and action potential duration. The theoretical results form the basis of a computer model of the electrocardiogram that relates skin potentials to the spatial and temporal distribution of action potentials in the heart     

Theoretical Evaluation of the McFee and Frank Vectorcardiographic Lead Systems Using a Numerical Inhomogeneous Torso Model

This paper investigates the accuracy of the McFee and Frank vectorcardiographic (VCG) lead systems via numerical simulation. A 23 dipole heart model is used as the source. Unit dipoles are placed at each source dipole location and potentials are calculated at the appropriate vectorcardiographic electrode sites on the surface of a numerical inhomogeneous torso model comprising lungs, intraventricular blood masses, and an anisotropic-conductivity skeletal muscle layer. These potentials enable one to calculate the lead vectors relating a given source dipole to the voltage measured in a particular vectorcardiographic lead. Quantitative accuracy measures are utilized to compare lead vectors for the two vectorcardiographic systems. It was found that the McFee system is the superior if the blood masses are not included, but that this superiority is practically neutralized upon their inclusion. The McFee system is also the more accurate as regards uniformity of sensitivity in the X, Y, and Z directions.     

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.     

Studies of the Electrocardiogram Using Realistic Cardiac and Torso Models

Several aspects of the forward and inverse problems of electrocardiography are investigated through the use of digital computer models. Two forms of a fixed location, variable moment, 20-dipole cardiac model of QRS are developed from actual cardiac excitation data. One form uses time-varying orientation dipoles; the other uses fixed orientation dipoles. An electric multipole expansion (EME) cardiac model employing the dipole, quadrupole, and octupole terms is also developed and used as an equivalent forward and inverse cardiac model. Two realistically shaped torso models are used. The homogeneous torso has uniform conductivity; the inhomogeneous torso contains realistically shaped lung regions with reduced conductivity. It is found that when the EME model is used as an equivalent forward cardiac model, it can accurately represent the actual 20-dipole cardiac model in the homogeneous torso. Limb leads are accurately represented by the dipole terms alone while the precordial leads require the quadrupole and octupole terms. It is also found that while the lung regions have little effect on the ECG's produced by the models, these regions can have a significant effect on the inverse solutions for certain dipoles in the 20-dipole cardiac model. These lung regions appear to have a much smaller effect on the dipole terms in the EME model. Solutions of the inverse problem for the terms in the EME model indicate that when a limited number of measurements are used, the best results can be obtained by uniform distribution of the measurements over the torso.     

Electrocardiographic textbooks based on template hearts warped using ultrasonic images

Advances in technology make the application of sophisticated approaches to assessing electrical condition of the heart practical. Estimates of cardiac electrical features inferred from body-surface electrocardiographic (ECG) maps are now routinely found in a clinical setting, but errors in those inverse solutions are especially sensitive to the accuracy of heart model geometry and placement within the torso. The use of a template heart model allows for accurate generation of individualized heart models and also permits effective comparison of inferred electrical features among multiple subjects. A collection of features mapped onto a common template forms a textbook of anatomically specific ECG variability. Our template warping process to individualize heart models based on a template heart uses ultrasonic images of the heart from a conventional, phased-array system. We chose ultrasound because it is nonionizing, less expensive, and more convenient than MR or CT imaging. To find the orientation and position in the torso model of each image, we calibrated the ultrasound probe by imaging a custom phantom consisting of multiple N-fiducials and computing a transformation between ultrasound coordinates and measurements of the torso surface. The template heart was warped using a mapping of corresponding landmarks identified on both the template and the ultrasonic images. Accuracy of the method is limited by patient movement, tracking error, and image analysis. We tested our approach on one normal control and one obese diabetic patient using the mixed-boundary-value inverse method and compared results from both on the template heart. We believe that our novel textbook approach using anatomically specific heart and torso models will facilitate the identification of electrophysiological biomarkers of cardiac dysfunction. Because the necessary data can be acquired and analyzed within about 30?min, this framework has the potential for becoming a routine clinical procedure.     

The Eccentric Spheres Model as the Basis for a Study of the Role of Geometry and Inhomogeneities in Electrocardiography

To study the role of internal geometry and inhomogeneities in the "forward problem" of electrocardiography (ECG), a mathematical model was constructed which permitted manipulation of these variables. The model, which consists of two eccentric systems of concentric spheres, contains all the important torso compartments, namely the blood cavity, the myocardium, the pericardium, the lung region, the surface muscle layer, and the subcutaneous fat. An analytic solution is found in the form of a double series expansion in Legendre polynomials. The integrated effect of the inhomogeneities on the surface potential distribution is investigated. The model demonstrates the importance of interactions between the various torso components in determining the potential distribution at the surface.     

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     

An Inverse Electrocardiographic Solution with an ON-OFF Model

The electrical activity of a dog's heart is modeled as originating in ten regions, called zones, each of which is either completely active or completely inactive at any one time. Using this model, the electrical activation sequence of the dog's heart during QRS is computed from potentials measured on the dog's torso, and from his geometry. The sequence computed is compared with measurements directly from the dog's heart. Additionally, the sequence is recomputed with perturbations in the location of the heart within the torso and with noise added to the torso potential data to see how greatly these changes affect the solution. The results support the possibility of using an ON-OFF model to determine the electrical activity of the heart.     

Use of Time Integrals of the ECG to Solve the Inverse Problem

With the use of a bisyncytial model of the heart, it is shown that time integrals of QRS and QSR-T are related to the amplitude (A), area (?d activation time (?) the cellular action potential (AP) on the closed surface surrounding the ventricles. For the normal heart, solution of the inverse problem would give ?and?on the heart surface and, by interpolation, in the myocardium, allowing reconstruction of the AP. In the case of ischemia and infarction, ?and A?would be available which, while not defining the AP, might provide valuable information. Necrosis introduces an unknown perturbation.     

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.     

Electromagnetic Fields and Power Deposition in Body-of-Revolution Models of Man

Internal electromagnetic (EM) fields and power absorption in a homogeneous Iossy dielectric body of revolution are evaluated using the surface integral equation method. The method yields moment method solutions for the induced current densities on the body surface. The interior fields to the body are then evafuated via the reciprocity theorem and the measurement matrix concept. The bulk body power deposition is obtained by the integration of the surface Poynting vector. The method applies for a wide range of dielectric parameters (with epsilon/sub r/ from 1.1 to 10/sup 2/ and sigma from 0 to 10/sup 3/ mhos/m) in the resonance region. Numerical results for EM fields and power deposition in a body-of-revolution model of a human torso with height of 1.78 m are evaluated for frequencies of 30, 80, and 300 MHz. It is found that the strongest power deposition in the torso model occurs for fields polarized along the longest dimension and for frequencies near the first resonance (i.e., 80 MHz) of the torso body. Hot spots are also observed in the neck region of the torso body.     

The Effects of Torso Geometry on Magnetocardiographic Lead Fields

A study of the effects of the torso on the sensitivity of magnetocardiograms (MCG's) is performed; comparisons with the effects on electrocardiograms (ECG's) are made. The effects are determined by comparing MCG or ECG lead fields (LF's) in a realistically shaped computer model of the human torso with the LF's that the same MCG or ECG measurements have when made on a semi-infinite volume conductor. Detailed data concerning the effects of the external torso boundary and lung regions on MCG and ECG LF's at various points in the ventricular region are presented. It is found that the boundary has effects on MCG LF's that are equal to or greater than the effects on ECG LF's. The effects of the lung regions on MCG LF's are found to be smaller than the effects of the boundary and to be equal to or greater than the effects of the lung regions on ECG LF's. On the basis of these results, it is concluded that MCG's have no advantage over ECG's in being able to simply and'accurately determine the nature and location of sources in the heart. In addition, the MCG LF's are compared with magnetic heart vector (MHV) LF's, i.e., the LF's that measure only the magnetic dipolar components of the sources in the heart. The MCG LF's are found to differ significantly from the MHV LF's; this indicates that accurate measurements of the MIHV in a real torso would be difficult to make.     

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     

The Inverse Problem in Electrocardiography: A Model Study of the Effects of Geometry and Conductivity Parameters on the Reconstruction of Epicardial Potentials

An idealized, analytic model using spherical harmonics was developed to analyze the effects of variations in torso geometry and volume conductivity parameters on the recovery of epicardial potentials from torso potentials. The model was also used to analyze the effects of these variations on individual terms in the orthogonal series expansion. The ability to reconstruct separate, local electrical events on the epicardium was examined under the following simulated situations: 1) all conductivity and geometry parameters were known accurately, 2) the conductivity of individual torso tissue layers was varied, 3) the torso-air boundary was eliminated (the "infinite medium" assumption), 4) the heart position was not accurately known, and 5) the heart size was not accurately known. Variation in conductivity and geometry parameters was found to exert a quantitative and qualitative effect on the amplitude, resolution, and position of the reconstructed epicardial maxima and minima. Significant differences were found in the ability of the inverse procedure to recover epicardial potentials resulting from posterior as opposed to anterior myocardial sources. Important conclusions regarding the narrow allowance for error in heart size and position, and the relative contributions of the torso tissue layer conductivities can provide guidelines for inverse reconstruction of epicardial potentials with a realistic model utilizing the true geometry.     

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.     

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