Experimental tests of the cortical imaging technique-applicationsto the response to median nerve stimulation and the localization ofepileptiform discharges

In a previous paper a method for simulating the electric potentials on the surface of the brain was introduced. This method consisted of the construction of a layer of radially oriented current dipoles in a conducting sphere that simulated the head so that the voltages generated by the layer would take the values measured on the surface of the medium (the scalp). The harmonic potential function for this layer was then evaluated in the interior of the medium in an attempt to approximate the potentials that would be generated by the actual neural sources but which could not be observed without recourse to invasive recording techniques. This method, the cortical imaging technique (CIT), has been previously tested by applying it to artificially generated data where the "cortical surface" potentials were known and could be compared with CIT-generated potentials. In this paper the method is tested by applying it to the scalp-recorded potentials evoked by right median nerve stimulation, where direct cortical recordings are available for comparison, and to the scalp-recorded epileptiform discharges from two patients where the spike foci were well defined. The effects of varying the "noise ratio," an input parameter in CIT which allows one to account for noise in scalp-recorded data, is discussed.     

The potential for Laplacian maps to solve the inverse problem ofelectrocardiography

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 results 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     

Determining Surface Potentials from Current Dipoles, with Application to Electrocardiography

This paper presents a method for determining the potentials over the surface of a three-dimensional volume due to internal current sources. The volume may be inhomogeneous and irregularly shaped. The method for determining the potentials uses N simultaneous equations which when solved produce the potentials at N different surface points. The N simultaneous equations are solved by an iterative technique on an IBM computer.     

Potential Field from an Active Nerve in an Inhomogeneous, Anisotropic Volume Conductor?The Inverse Problem

A previously described, forward solution for the problem of determining surface potentials on a long circular limb arising from electrical nerve activity within the limb is used to solve the inverse problem, namely, the recovery of source nerve potentials from limb surface potentials. The inverse problem is solved by means of a two-dimensional (2-D) digital filter which has the advantages of simplicity, speed, and ease of implementation compared to any other solution method.     

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.     

An Analysis of Transfer Coefficients Calculated Directly from Epicardial and Body Surface Potential Measurements in the Intact Dog

This study dealt with the question of how to estimate body surface potentials from epicardial potential distributions in intact dogs; in particular, the study considered the feasibility of obtaining transfer coefficients directly from sequences of epicardial and body surface measurements of ventricular excitation and repolarization potential distributions, rather than from measurements of the geometry of the volume conductor. The transfer coefficients were calculated from the measured potentials via the mathematical method of using a Bayes estimator. The merit of this approach was that it offered the possibility of accurately representing the characteristics of the volume conductor without directly measuring either the volume conductor's geometry or its inhomogeneities. The experimental protocol made use of measurements from two dogs. Data from the first dog were used to obtain two sets of transfer coefficients, one by the Bayes method as applied to measured sequences of epicardial and body surface potentials, and the other by a method based on the geometric position of each epicardial and body surface electrode. These two sets of transfer coefficients were found to be similar in pattern and value size. Additionally, results of forward simulations in the first dog, based on the measured epicardial potentials and each set of transfer coefficients, were compared to the measured body potentials. The results showed that the simulated potentials were closer to the measured potentials when the transfer coefficients obtained from the potentials were used, rather than when the geometric coefficients were used.     

Inversion of simulated evoked potentials to charge distribution inside the human brain using an algebraic reconstruction technique

An analytic method is presented to estimate the evolution of electrical charge distribution inside the human brain related to the evoked potentials observed on the head surface. A three-layer concentric spherical human head model is adopted to express the relation between the observed potentials on the head surface and the spatial charge distribution inside the brain. An integral equation associated with the three-layer concentric head model Green's function is employed. Assuming the electric potentials are measured on the head surface, the charge distributions inside the human brain are computed by solving an inverse problem. The Green's function integral equation is inverted by using an algebraic reconstruction technique widely employed in X-ray tomography imaging. The accuracy of the proposed technique is examined by employing computer simulations and by checking the self-consistency of the algorithm.     

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     

Three-dimensional filtering approach to brain potential mapping

The spatial distribution of electroencephalogram (EEG) features on the scalp surface, both in time or frequency, is of great importance in clinical applications and medical research. Traditionally, mathematical methods based on interpolation algorithms have been widely applied to obtain the EEG mappings. This paper presents an innovative approach to reconstructing the brain potential mappings from multichannel EEG's. The three-dimensional (3-D) filtering approach, differing from the numerical interpolating methods, considers the spatial distribution of brain potentials as a 3-D signal, which is processed and interpolated according to its spatial frequency characteristics. The performance of the 3-D filtering method evaluated on simulated brain potentials is shown to be comparable to the four-nearest-neighbors method. Moreover, the 3-D filtering method is superior to the spherical splines method in efficiency. Two main advantages of this method are: the prospect of developing real-time, animated EEG mappings utilizing powerful digital signal processors and its capability of processing and interpolating the brain potentials on the realistic irregular scalp surface.     

Wavefront-based models for inverse electrocardiography

We introduce two wavefront-based methods for the inverse problem of electrocardiography, which we term wavefront-based curve reconstruction (WBCR) and wavefront-based potential reconstruction (WBPR). In the WBCR approach, the epicardial activation wavefront is modeled as a curve evolving on the heart surface, with the evolution governed by factors derived phenomenologically from prior measured data. The body surface potential/wavefront relationship is modeled via an intermediate mapping of wavefront to epicardial potentials, again derived phenomenologically. In the WBPR approach, we iteratively construct an estimate of epicardial potentials from an estimated wavefront curve according to a simplified model and use it as an initial solution in a Tikhonov regularization scheme. Initial simulation results using measured canine epicardial data show considerable improvement in reconstructing activation wavefronts and epicardial potentials with respect to standard Tikhonov solutions. In particular the WBCR method accurately finds the anisotropic propagation early after epicardial pacing, and the WBPR method finds the wavefront (regions of sharp gradient of the potential) both accurately and with minimal smoothing.     

A Monte Carlo method for the Dirichlet problem of dielectric wedges

The Monte Carlo method considered can be used to numerically compute electrostatic potentials inside a closed surface where: (a) the potential on the surface is known, and (b) the dielectric constant inside the surface changes only on boundaries. A modification is proposed to previously used Monte Carlo methods to overcome problems presented by dielectric wedges. In addition it is shown how it can be easily determined whether or not a point is inside given domain. The connection between this topological problem and the Monte Carlo technique is explained     

Eccentric Dipole in a Spherical Medium: Generalized Expression for Surface Potentials

A new formulation is presented for surface potentials produced on a homogeneous spherical volume conductor by an eccentric current dipole contained therein. The formulation is free of interminancies for all relevant dipole locations and leads directly to the solution of surface potentials due to an eccentric quadripole.     

Method to reduce blur distortion from EEG's using a realistic headmodel

A mathematical procedure, called deblurring, was developed to reduce spatial blur distortion of scalp-recorded brain potentials due to transmission through the skull and other tissues. Deblurring estimates potentials at the superficial cerebral cortical surface from EEGs recorded at the scalp using a finite-element model of each subject's scalp, skull, and cortical surface constructed from their magnetic resonance images (MRIs). Simulations indicate that deblurring is numerically stable, and a comparison of deblurred data with a direct cortical recording from a neurosurgery patient suggests that the procedure is valid. Application of deblurring to somatosensory evoked potential data recorded at 124 scalp sites suggests that the method greatly improves spatial detail and merits further development     

The inverse problem utilizing the boundary element method for a nonstandard female torso

This paper proposes a new method of rapidly deriving the transfer matrix for the boundary element method (BEM) forward problem from a tailored female torso geometry in the clinical setting. The method allows rapid calculation of epicardial potentials (EP) from body surface potentials (BSP). The use of EPs in previous studies has been shown to improve the successful detection of the life-threatening cardiac condition--acute myocardial infarction. The MRI scanning of a cardiac patient in the clinical setting is not practical and other methods are required to accurately deduce torso geometries for calculation of the transfer matrix. The new method allows the noninvasive calculation of tailored torso geometries from a standard female torso and five measurements taken from the body surface of a patient. This scaling of the torso has been successfully validated by carrying out EP calculations on 40 scaled torsos and ten female subjects. It utilizes the BEM in the calculation of the transfer matrix as the BEM depends only upon the topology of the surfaces of the torso and the heart, the former can now be accurately deduced, leaving only the latter geometry as an unknown.     

Effect of Cardiac Motion on Solution of the Electrocardiography Inverse Problem

Previous studies of the ECG inverse problem often assumed that the heart was static during the cardiac cycle; consequently, a time-dependent geometrical error was thought to be unavoidably introduced. In this paper, cardiac motion is included in solutions to the electrocardiographic inverse problem. Cardiac dynamics are simulated based on a previously developed biventricular model that coupled the electrical and mechanical properties of the heart, and simulated the ventricular wall motion and deformation. In the forward computation, the heart surface source model method is employed to calculate the epicardial potentials from the action potentials, and then, the simulated epicardial potentials are used to calculate body surface potentials. With the inclusion of cardiac motion, the calculated body surface potentials are more reasonable than those in the case of static assumption. In the epicardial potential-based inverse studies, the Tikhonov regularization method is used to handle ill-posedness of the ECG inverse problem. The simulation results demonstrate that the solutions obtained from both the static ECG inverse problem and the dynamic ECG inverse problem approaches are approximately the same during the QRS complex period, due to the minimal deformation of the heart in this period. However, with the most obvious deformation occurring during the ST-T segment, the static assumption of heart always generates something akin to geometry noise in the ECG inverse problem causing the inverse solutions to have large errors. This study suggests that the inclusion of cardiac motion in solving the ECG inverse problem can lead to more accurate and acceptable inverse solutions.      

Body Surface Potential Mapping Based on Cylindrical Regression

A new statistical method is proposed for evaluating the electric potentials of the human heart measured on the body surface (body surface potential mapping). The method is based on the representation of the measured values in terms of a cylindrical regression model, where the regression is set up horizontally by means of a trigonometric polynomial and vertically by means of an ordinary one. The special experimental equipment required for the optimal exploitation of the model is briefly described. The estimated potential distribution can be graphically displayed in three dimensions; the polynomial degree can be chosen in such a way as to resolve all statistical detail desired. The statistical optimality of the new approximation method is established. The method requires relatively little computational effort and, thus, is suitable for implementation on small computers.     

The depolarization sequence of the human heart surface computedfrom measured body surface potentials

A method for computing the activation sequence at the ventricular surface from body surface potentials, has been adapted to handle measured data. By using measured anatomical data together with a 64-channel ECG (electrocardiogram) recording of the same subject for three subjects, it is shown that the model is able to determine activation sequences on the heart surface which closely resemble similar data obtained through invasive measurement as reported in literature     

Assessing the effect of uncertainty in intracavitary electrodeposition on endocardial potential estimates

The aim of this simulation study is to determine the effect of uncertainty in intracavitary probe electrode position on the accuracy of estimated endocardial potentials. Intracavitary probe position uncertainty is simulated by randomly moving an idealized probe surface about the center of an idealized left ventricular endocardial surface. These random deviations represent possible probe locations that are incorporated as correlated noise. An optimum inverse transfer coefficient matrix, relating intracavitary potentials to endocardial potentials, is computed and subsequently used to calculate the best linear estimate of the true endocardial potentials. For uncorrelated endocardial potentials and probe position uncertainty within 1.5 mm of the coordinates of the exact probe electrode locations, a root-mean-square (rms) error of 34.0% is obtained. Increasing probe position uncertainties to 3.0 and 6.0 mm results in rms errors of 60.8 and 88.3%, respectively. For endocardial potentials that are 90% dipolar, the rms errors for probe position uncertainties of 1.5, 3.0, and 6.0 mm are 11.3, 19.6, and 28.5%, respectively. These simulation results imply that position uncertainty of a multielectrode, intracavitary probe can be a major source of error in estimating endocardial potentials from intracavitary potentials.     

Numerical tests of a method for simulating electrical potentials onthe cortical surface

A mathematical imaging method for simulating cortical surface potentials was introduced at recent neurosciences meetings [1a], [1b], [2] and was applied to elucidate the neural origins of evoked responses in normal volunteers and certain patient populations. This method consists of the solution of an inward harmonic continuation problem and its effect is to simulate data that has not been attenuated and smeared by the skull. This cortical imaging technique (CIT) is validated by applying it to artificially derived data. Pairs of dipolar sources with different depths and separations are introduced into a spherical conducting medium simulating the head. Scalp potential maps are constructed by interpolating the simulated data between 28 "scalp" electrode positions. Noise is added to the data to approximate the variability in measured potentials that would be observed in practice. CIT is used in each case to construct potential maps on layers concentric to and within the layer representing the scalp. In several instances when the dipole pair is deep and closely spaced, the sources cannot be separated by the scalp topographical maps but are easily separated by the "cortical" topographical maps. CIT is also applied to scalp-recorded potentials evoked by bilateral median nerve stimulation and pattern-reversal visual stimulation.     

Improved surface laplacian estimates of cortical potential using realistic models of head geometry

Surface Laplacian of scalp EEG can be used to estimate the potential distribution on the cortical surface as an alternative to invasive approaches. However, the accuracy of surface Laplacian estimation depends critically on the geometric shape of the head model. This paper presents a new method for computing the surface Laplacian of scalp potential directly on realistic scalp surfaces in the form of a triangular mesh reconstructed from MRI scans. Unlike previous methods, this algorithm does not resort to any surface fitting proxy and can improve the surface Laplacian estimation of cortical potential patterns by as much as 34% on realistically shaped head models. Simulations and experimental data are presented to demonstrate the advantage of the proposed method over the conventional spherical approximation and the utility of a more accurate surface Laplacian method for estimating cortical potentials from scalp electrodes.