Equivalent Dipole Layer Research Articles

Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models.

The body surface electrocardiogram (ECG) is a direct result of electrical activity generated by the myocardium. Using the body surface ECGs to reconstruct cardiac electrical activity is called the inverse problem of electrocardiography. The method to solve the inverse problem depends on the chosen cardiac source model to describe cardiac electrical activity. In this paper, we describe the theoretical basis of two inverse methods based on the most commonly used cardiac source models: the epicardial potential model and the equivalent dipole layer model. We discuss similarities and differences in applicability, strengths and weaknesses and sketch a road towards improved inverse solutions by targeted use, sequential application or a combination of the two methods.

Modeling ventricular repolarization gradients in normal cases using the equivalent dipole layer

Background Electrical activity underlying the T-wave is less well understood than the QRS-complex. This study investigated the relationship between normal T-wave morphology and the underlying ventricular repolarization gradients using the equivalent dipole layer (EDL).Methods Body-surface-potential-maps (BSPM, 67‑leads) were obtained in nine normal cases. Subject specific MRI-based anatomical heart/torso-models with electrode positions were created. The boundary element method was used to account for the volume conductor effects. To simulate the measured T-waves, the EDL was used to apply different ventricular repolarization gradients: a) transmural, b) interventricular c) apico-basal and d) all three gradients (a-c) combined. The combined gradient (d) was optimized using an inverse procedure (Levenberg-Marquardt). Correspondence between simulated and measured T-waves was assessed using correlation coefficient (CC) and relative difference (RD).Results Realistic T-waves were simulated if repolarization times of: (a) the epicardium were smaller than the endocardium; (b) the left ventricle were smaller than the right ventricle and (c) the apex increased towards the base. The apico-basal gradient resulted in the highest correspondence between measured and simulated T-waves (CC = 0.84(0.81–0.91);RD = 0.68(0.60–0.71)) compared to a transmural gradient (CC = 0.77(0.71–0.80);RD = 1.46(0.82–1.75)) and an interventricular gradient (CC = 0.71(0.67–0.80);RD = 0.85(0.75–0.87)). All three gradients combined further improved the correspondence between measured and simulated T-waves (CC = 0.83(0.82–0.89);RD = 0.60(0.51–0.63)), especially after optimization (CC = 0.96(0.94–0.98);RD = 0.27(0.22–0.34)).Conclusion The application of all repolarization gradients combined resulted in the largest agreement between simulated and measured T-waves, followed by the apico-basal repolarization gradient. With these findings, we will optimize our EDL-based inverse procedure to assess repolarization abnormalities.

Reduction to the pole near the equator with a Fourier-domain equivalent dipole layer

Reduction to the pole (RTP) transforms the phase of the observed total-field anomaly, making magnetic interpretation easier. However, RTP operations are notoriously unstable at low latitudes and produce singularities at the equator. The equivalent-source approach is an appropriate method to address the well-known difficulties of RTP at low latitudes. Although we routinely use equivalent-source RTP algorithms in the spatial domain, they consume massive computational resources. In contrast, the equivalent-source method in the Fourier domain is more efficient and also invalid for RTP because of the inevitable use of the RTP operator in the inversion. To overcome this problem, we introduce a simple approximation of the RTP operator instead and then develop a novel algorithm to calculate the magnetic moments of the equivalent dipole layer from the total-field anomaly near the equator. Because the calculations are mainly performed in the Fourier domain, this algorithm is more efficient than the conventional spatial-domain method. We also impose a nonnegative constraint on the dipole layer to improve the results and significantly reduce the undesired striation associated with the low-latitude RTP. We evaluate the effectiveness of our approach with synthetic data near the equator, and the results indicate that the algorithm can produce an RTP field that is an accurate representation of the real field at the pole, even when accounting for the existence of noise. The new approach is applied to field data collected near the geomagnetic equator in the South China Sea. In the central part of the anomaly, we obtain a robust spindle shape of the RTP field, which is identical to the shape of a basalt formation in that location. Recent 1:1,000,000 scale marine geologic mapping has confirmed this presence, proving that our solution is an applicable approach to the problem.

Incorporating structural abnormalities in equivalent dipole layer based ECG simulations.

Introduction: Electrical activity of the myocardium is recorded with the 12-lead ECG. ECG simulations can improve our understanding of the relation between abnormal ventricular activation in diseased myocardium and body surface potentials (BSP). However, in equivalent dipole layer (EDL)-based ECG simulations, the presence of diseased myocardium breaks the equivalence of the dipole layer. To simulate diseased myocardium, patches with altered electrophysiological characteristics were incorporated within the model. The relation between diseased myocardium and corresponding BSP was investigated in a simulation study. Methods: Activation sequences in normal and diseased myocardium were simulated and corresponding 64-lead BSP were computed in four models with distinct patch locations. QRS-complexes were compared using correlation coefficient (CC). The effect of different types of patch activation was assessed. Of one patient, simulated electrograms were compared to electrograms recorded during invasive electro-anatomical mapping. Results: Hundred-fifty-three abnormal activation sequences were simulated. Median QRS-CC of delayed versus dyssynchronous were significantly different (1.00 vs. 0.97, p < 0.001). Depending on the location of the patch, BSP leads were affected differently. Within diseased regions, fragmentation, low bipolar voltages and late potentials were observed in both recorded and simulated electrograms. Discussion: A novel method to simulate cardiomyopathy in EDL-based ECG simulations was established and evaluated. The new patch-based approach created a realistic relation between ECG waveforms and underlying activation sequences. Findings in the simulated cases were in agreement with clinical observations. With this method, our understanding of disease progression in cardiomyopathies may be further improved and used in advanced inverse ECG procedures.

The Effect of Segmentation Variability in Forward ECG Simulation.

Segmentation of patient-specific anatomical models is one of the first steps in Electrocardiographic imaging (ECGI). However, the effect of segmentation variability on ECGI remains unexplored. In this study, we assess the effect of heart segmentation variability on ECG simulation. We generated a statistical shape model from segmentations of the same patient and generated 262 cardiac geometries to run in an ECG forward computation of body surface potentials (BSPs) using an equivalent dipole layer cardiac source model and 5 ventricular stimulation protocols. Variability between simulated BSPs for all models and protocols was assessed using Pearson's correlation coefficient (CC). Compared to the BSPs of the mean cardiac shape model, the lowest variability (average CC = 0.98 ± 0.03) was found for apical pacing whereas the highest variability (average CC = 0.90 ± 0.23) was found for right ventricular free wall pacing. Furthermore, low amplitude BSPs show a larger variation in QRS morphology compared to high amplitude signals. The results indicate that the uncertainty in cardiac shape has a significant impact on ECGI.

Spectral consistency of satellite and airborne data: Application of an equivalent dipole layer for combining satellite and aeromagnetic data sets

We present a novel approach for the combination of aeromagnetic and satellite data applying an equivalent dipole layer and spherical harmonic (SH) expansion of the dipoles. The method involves two steps: (1) inversion for the magnetic parameters of the equivalent dipole layer and (2) conversion of the magnetic parameters into SH coefficients. With this approach, SH analysis can be used for a regional study area, and, for example, the long-wavelength range of aeromagnetic data can be replaced with satellite data. We test our approach on the third, fourth, and fifth editions of the Magnetic Anomaly Map of Australia, which were releveled using independent aeromagnetic data sets for long-wavelength correction. The results indicate that the magnetic compilations leveled to long-haul control lines have a good agreement with the LCS-1 satellite model in the range of SH degree 40 to 110 (corresponding to half-wavelengths between 180 to 500 km), whereas the third edition of the Magnetic Anomaly Map of Australia demonstrates poor control over long- wavelengths in this spectral range. Our analysis suggests that even the carefully processed fifth edition would benefit from the replacement of long-wavelength data with satellite data.

Ex vivo Validation of Noninvasive Epicardial and Endocardial Repolarization Mapping.

Background: The detection and localization of electrophysiological substrates currently involve invasive cardiac mapping. Electrocardiographic imaging (ECGI) using the equivalent dipole layer (EDL) method allows the noninvasive estimation of endocardial and epicardial activation and repolarization times (AT and RT), but the RT validation is limited to in silico studies. We aimed to assess the temporal and spatial accuracy of the EDL method in reconstructing the RTs from the surface ECG under physiological circumstances and situations with artificially induced increased repolarization heterogeneity.Methods: In four Langendorff-perfused pig hearts, we simultaneously recorded unipolar electrograms from plunge needles and pseudo-ECGs from a volume-conducting container equipped with 61 electrodes. The RTs were computed from the ECGs during atrial and ventricular pacing and compared with those measured from the local unipolar electrograms. Regional RT prolongation (cooling) or shortening (pinacidil) was achieved by selective perfusion of the left anterior descending artery (LAD) region.Results: The differences between the computed and measured RTs were 19.0 ± 17.8 and 18.6 ± 13.7 ms for atrial and ventricular paced beats, respectively. The region of artificially delayed or shortened repolarization was correctly identified, with minimum/maximum RT roughly in the center of the region in three hearts. In one heart, the reconstructed region was shifted by ~2.5 cm. The total absolute difference between the measured and calculated RTs for all analyzed patterns in selectively perfused hearts (n = 5) was 39.6 ± 27.1 ms.Conclusion: The noninvasive ECG repolarization imaging using the EDL method of atrial and ventricular paced beats allows adequate quantitative reconstruction of regions of altered repolarization.

Comparing Non-invasive Inverse Electrocardiography With Invasive Endocardial and Epicardial Electroanatomical Mapping During Sinus Rhythm.

This study presents a novel non-invasive equivalent dipole layer (EDL) based inverse electrocardiography (iECG) technique which estimates both endocardial and epicardial ventricular activation sequences. We aimed to quantitatively compare our iECG approach with invasive electro-anatomical mapping (EAM) during sinus rhythm with the objective of enabling functional substrate imaging and sudden cardiac death risk stratification in patients with cardiomyopathy. Thirteen patients (77% males, 48 ± 20 years old) referred for endocardial and epicardial EAM underwent 67-electrode body surface potential mapping and CT imaging. The EDL-based iECG approach was improved by mimicking the effects of the His-Purkinje system on ventricular activation. EAM local activation timing (LAT) maps were compared with iECG-LAT maps using absolute differences and Pearson’s correlation coefficient, reported as mean ± standard deviation [95% confidence interval]. The correlation coefficient between iECG-LAT maps and EAM was 0.54 ± 0.19 [0.49–0.59] for epicardial activation, 0.50 ± 0.27 [0.41–0.58] for right ventricular endocardial activation and 0.44 ± 0.29 [0.32–0.56] for left ventricular endocardial activation. The absolute difference in timing between iECG maps and EAM was 17.4 ± 7.2 ms for epicardial maps, 19.5 ± 7.7 ms for right ventricular endocardial maps, 27.9 ± 8.7 ms for left ventricular endocardial maps. The absolute distance between right ventricular endocardial breakthrough sites was 30 ± 16 mm and 31 ± 17 mm for the left ventricle. The absolute distance for latest epicardial activation was median 12.8 [IQR: 2.9–29.3] mm. This first in-human quantitative comparison of iECG and invasive LAT-maps on both the endocardial and epicardial surface during sinus rhythm showed improved agreement, although with considerable absolute difference and moderate correlation coefficient. Non-invasive iECG requires further refinements to facilitate clinical implementation and risk stratification.

In silico validation of electrocardiographic imaging to reconstruct the endocardial and epicardial repolarization pattern using the equivalent dipole layer source model

The solution of the inverse problem of electrocardiology allows the reconstruction of the spatial distribution of the electrical activity of the heart from the body surface electrocardiogram (electrocardiographic imaging, ECGI). ECGI using the equivalent dipole layer (EDL) model has shown to be accurate for cardiac activation times. However, validation of this method to determine repolarization times is lacking. In the present study, we determined the accuracy of the EDL model in reconstructing cardiac repolarization times, and assessed the robustness of the method under less ideal conditions (addition of noise and errors in tissue conductivity). A monodomain model was used to determine the transmembrane potentials in three different excitation-repolarization patterns (sinus beat and ventricular ectopic beats) as the gold standard. These were used to calculate the body surface ECGs using a finite element model. The resulting body surface electrograms (ECGs) were used as input for the EDL-based inverse reconstruction of repolarization times. The reconstructed repolarization times correlated well (COR > 0.85) with the gold standard, with almost no decrease in correlation after adding errors in tissue conductivity of the model or noise to the body surface ECG. Therefore, ECGI using the EDL model allows adequate reconstruction of cardiac repolarization times.Graphical abstractValidation of electrocardiographic imaging for repolarization using forward calculated body surface ECGs from simulated activation-repolarization sequences.

Assessment of the equivalent dipole layer source model in the reconstruction of cardiac activation times on the basis of BSPMs produced by an anisotropic model of the heart

Promising results have been reported in noninvasive estimation of cardiac activation times (AT) using the equivalent dipole layer (EDL) source model in combination with the boundary element method (BEM). However, the assumption of equal anisotropy ratios in the heart that underlies the EDL model does not reflect reality. In the present study, we quantify the errors of the nonlinear AT imaging based on the EDL approximation. Nine different excitation patterns (sinus rhythm and eight ectopic beats) were simulated with the monodomain model. Based on the bidomain theory, the body surface potential maps (BSPMs) were calculated for a realistic finite element volume conductor with an anisotropic heart model. For the forward calculations, three cases of bidomain conductivity tensors in the heart were considered: isotropic, equal, and unequal anisotropy ratios in the intra- and extracellular spaces. In all inverse reconstructions, the EDL model with BEM was employed: AT were estimated by solving the nonlinear optimization problem with the initial guess provided by the fastest route algorithm. Expectedly, the case of unequal anisotropy ratios resulted in larger localization errors for almost all considered activation patterns. For the sinus rhythm, all sites of early activation were correctly estimated with an optimal regularization parameter being used. For the ectopic beats, all but one foci were correctly classified to have either endo- or epicardial origin with an average localization error of 20.4 mm for unequal anisotropy ratio. The obtained results confirm validation studies and suggest that cardiac anisotropy might be neglected in clinical applications of the considered EDL-based inverse procedure.

Three‐dimensional cortical dipole imaging of brain electrical activity considering spherical and median planes

AbstractEquivalent cortical dipole layer imaging is proposed for high‐resolution electroencephalogram (EEG) visualization. Equivalent dipole layer distribution parallel to the brain surface has been used in previous head models. In the present study, the additional dipole distribution on the median plane is introduced to identify the three‐dimensional (3D) position and orientation of dipole sources. The parametric Wiener filter (PWF) using statistical signal and noise information was applied to solve an inverse problem of cortical dipole imaging. Simulations were performed in an inhomogeneous, concentric, three‐sphere volume conductor head model under various signal conditions. Cortical dipole imaging provided localized brain electrical activity with better spatial resolution compared to the scalp potential map. These simulation results suggested that the depth and orientation of individual dipole sources could be observed by mapping on both spherical and median planes. We applied the proposed imaging technique to human experimental data of the visual evoked potential (VEP) and obtained reasonable results that coincide with physiological knowledge. © 2010 Institute of Electrical Engineers of Japan. Published by John Wiley &amp; Sons, Inc.

Cortical Dipole Imaging of Movement-related Potentials by Means of Parametric Inverse Filters Incorporating with Signal and Noise Covariance

The objective of this study is to explore suitable spatial filters for inverse estimation of cortical equivalent dipole layer imaging from the scalp electroencephalogram. We utilize cortical dipole source imaging to locate the possible generators of scalp-measured movement-related potentials (MRPs) in human. The effects of incorporating signal and noise covariance into inverse procedures were examined by computer simulations and experimental study. The parametric projection filter (PPF) and parametric Weiner filter (PWF) were applied to an inhomogeneous three-sphere head model under various noise conditions. The present simulation results suggest that the PWF incorporating signal information provides better cortical dipole layer imaging results than the PPF and Tikhonov regularization under the condition of moderate and high correlation between signal and noise distributions. On the other hand, the PPF has better performance than other inverse filters under the condition of low correlation between signal and noise distributions. The proposed methods were applied to self-paced MRPs in order to identify the anatomic substrate locations of neural generators. The dipole layer distributions estimated by means of PPF are well-localized as compared with blurred scalp potential maps and dipole layer distribution estimated by Tikhonov regularization. The proposed methods demonstrated that the contralateral premotor cortex was preponderantly activated in relation to movement performance. In cortical dipole source imaging, the PWF has better performance especially when the correlation between the signal and noise is high. The proposed inverse method was applicable to human experiments of MRPs if the signal and noise covariances were obtained.

Reconstruction of electrostatic force microscopy images

An efficient algorithm to restore the actual surface potential image from Kelvin probe force microscopy measurements of semiconductors is presented. The three-dimensional potential of the tip-sample system is calculated using an integral equation-based boundary element method combined with modeling the semiconductor by an equivalent dipole-layer and image-charge model. The derived point spread function of the measuring tip is then used to restore the actual surface potential from the measured image, using noise filtration and deconvolution algorithms. The model is then used to restore high-resolution Kelvin probe microscopy images of semiconductor surfaces.

Artefact elimination in spatiotemporal cortical dipole layer imaging with parametric projection filter

We explore suitable spatiotemporal filters for inverse estimation of an equivalent cortical dipole layer (DL) distribution from the scalp electroencephalogram (EEG) for imaging of brain electric sources. We have previously developed the parametric projection filter (PPF)-based cortical dipole layer imaging technique, which allows estimating cortical dipole layer inverse solutions in the presence of noise covariance. We have expanded the PPF to the time-varying filter in order to handle the spatiotemporally varying nature of brain electrical activity. The present simulation results indicate that the estimation error is reduced substantially by taking the spatiotemporal properties of the noise into consideration, such as eye-blink artefacts, and the proposed time-variant PPF method provides enhanced performance in rejecting time-varying noise.

Spatio-temporal cortical source imaging of brain electrical activity by means of time-varying parametric projection filter.

In the present study, we explore suitable spatio-temporal filters for inverse estimation of an equivalent dipole-layer distribution from the scalp electroencephalogram (EEG) for imaging of brain electric sources. We propose a time-varying parametric projection filter (tPPF) for the spatio-temporal EEG analysis. The performance of this tPPF algorithm was evaluated by computer simulation studies. An inhomogeneous three-concentric-spheres model was used in the present simulation study to represent the head volume conductor. An equivalent dipole layer was used to represent equivalently brain electric sources and estimated from the scalp potentials. The tPPF filter was tested to remove time-varying noise such as instantaneous artifacts caused by eyes-blink. The present simulation results indicate that the proposed time-variant tPPF method provides enhanced performance in rejecting time-varying noise, as compared with the time-invariant parametric projection filter.

Equivalent physical models and formulation of equivalent source layer in high-resolution EEG imaging

In high-resolution EEG imaging, both equivalent dipole layer (EDL) and equivalent charge layer (ECL) assumed to be located just above the cortical surface have been proposed as high-resolution imaging modalities or as intermediate steps to estimate the epicortical potential. Presented here are the equivalent physical models of these two equivalent source layers (ESL) which show that the strength of EDL is proportional to the surface potential of the layer when the outside of the layer is filled with an insulator, and that the strength of ECL is the normal current of the layer when the outside is filled with a perfect conductor. Based on these equivalent physical models, closed solutions of ECL and EDL corresponding to a dipole enclosed by a spherical layer are given. These results provide the theoretical basis of ESL applications in high-resolution EEG mapping.

High-resolution EEG mapping: an equivalent charge-layer approach

Brain electrical signal is one of the windows to understanding neural activities. Various high-resolution imaging techniques have been developed to reveal the electrical activities underneath the cortical surface from scalp electroencephalographic recordings, such as scalp Laplacian, cortical surface potential, equivalent charge layer (ECL) and equivalent dipole layer (EDL). In this work, we develop forward density formulae for the ECL and the EDL of neural electric sources in a 4-concentric-sphere head model, and compare ECL with EDL in theory, simulation and real evoked data tests. The results confirm that the ECL map may be of higher spatial resolution than the EDL map.

Very fast numerical analysis of benchmark models of eddy-current testing for steam generator tube

In eddy-current testing (ECT), a thin-opening crack is frequently encountered. In this paper, such a crack is modeled by using an ideal crack, in which the eddy-current field perturbation due to the crack is described as the effect produced by an equivalent current dipole layer located on its surface. The dipole density can be found by solving an electric field integral equation. By following a moment method scheme, numerical predictions are made for steam generator tube ECT benchmark models Step 2 and Step 3, where a conductor plate and a tube wall, respectively, are detected. The probe impedance changes are evaluated and are compared with experiments, and good agreements are found. By using rapid algorithms to compute the dyadic Green's function, the total required CPU time is only several seconds for each problem.

High-resolution EEG: on the cortical equivalent dipole layer imaging

Background: Brain electrical activity is a spatio-temporally distributed process. Cortical imaging techniques have been developed to reconstruct cortical activity from the scalp electroencephalographic or magnetoencephalographic measurements. Several cortical imaging approaches, such as the epicortical potentials and a dipole layer accounting for the cortical activity, have been used to represent brain electrical activity. Methods: A closed cortical dipole layer source model is used to equivalently represent brain electrical activity. The relationship between the primary brain electrical sources and the cortical equivalent dipole layer is derived from the theory of electromagnetics. Computer simulation studies were conducted using a 3-concentric-sphere head model to validate the proposed theory. The cortical equivalent dipole layer imaging approach was tested in both computer simulation and human visual evoked potential (VEP) experiments. Results: The strength of the cortical equivalent dipole layer is shown to be proportional to the electrical potential over the same surface generated by primary electrical sources, had the outer medium been replaced by air. The proposed theory was validated by computer simulation in a discrete system. Simulation and VEP experimental studies suggest the feasibility of applying the cortical equivalent dipole layer imaging approach for brain imaging. Conclusions: The cortical equivalent dipole layer model can equivalently represent the primary brain electrical sources throughout the entire brain surrounded by the dipole layer. The strength of the cortical equivalent dipole layer due to primary sources can be directly calculated according to the theory developed in the present study.

Equivalent dipole source imaging of brain electric activity by means of parametric projection filter.

In the present study, spatial filters for inverse estimation of an equivalent dipole layer from the scalp-recorded potentials have been explored for their suitability in achieving high-resolution electroencephalogram (EEG) imaging. The performance of the parametric projection filter (PPF), which we propose to use for high-resolution EEG imaging, has been evaluated by computer simulations in the presence of a priori information on noise. An inhomogeneous three-concentric-sphere head model was used in the present simulation study to represent the head volume conductor. An equivalent dipole layer was used to model brain electric sources and estimated from the scalp potentials. Various noise conditions were simulated and the parametric projection filter was compared with standard regularization procedures such as the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TKNV). The present simulation results suggest that the proposed method performs better than that of commonly used inverse regularization techniques, such as the general inverse using the TSVD and the TKNV, when the correlation between the original source distribution and the noise distribution is low, and performs similarly when the correlation is high. A method for determining the optimum regularization parameter, which can be applied to parametric inverse techniques, has also been developed.