Application of the Method of Fundamental Solutions to Potential-based Inverse Electrocardiography

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

Noninvasive Electrocardiographic Imaging (ECGI): Application of the Generalized Minimal Residual (GMRes) Method

Electrocardiographic imaging (ECGI) is a developing imaging modality for cardiac electrophysiology and arrhythmias. It reconstructs epicardial potentials, electrograms, and isochrones from electrocardiographic body-surface potentials noninvasively. Current ECGI methodology employs Tikhonov regularization, which imposes constraints on the reconstructed potentials or their derivatives. This approach can sometimes reduce spatial resolution by smoothing the solution. Accuracy depends on a priori knowledge of solution characteristics and determination of an optimal regularization parameter. These properties led us to implement an independent, iterative approach for ECGI—the generalized minimal residual (GMRes) method—which does not apply constraints. GMRes was applied to experimental data during activation/repolarization of normal and infarcted hearts. GMRes reconstructions were compared to Tikhonov reconstructions and to measured gold standards in isolated hearts. Overall, the accuracy of GMRes solutions was similar to Tikhonov regularization. However, in certain cases GMRes recovered localized potential features (e.g., multiple potential minima), which were lost in the Tikhonov solution. Simultaneous use of these two complementary methods in clinical ECGI will ensure reliability and maximal extraction of diagnostic information in the absence of a priori information about a patient's condition.© 2003 Biomedical Engineering Society. PAC2003: 8719Hh, 8757Gg     

Application of L1-Norm Regularization to Epicardial Potential Solution of the Inverse Electrocardiography Problem

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

Direct Inference of the Spectra of Pericardial Potentials Using the Boundary-Element Method

New methods, based on Tikhonov regularization, were developed to infer the magnitude and phase of pericardial potentials directly. These methods were tested in an adult-male torso model using measured human epicardial potentials. With 1% noise added to body-surface potentials, regularization with an optimal parameter at each frequency from 1 to 100 Hz gave an average relative error (RE) in inferred spectral magnitudes of 0.44. Regularization with the composite–residual–smoothing–operator (CRESO) parameter increased the RE slightly to 0.47. With 10% additive noise, 10 mm overestimation of heart radius, and a 10 mm error in heart position, the average CRESO parameter from 1 to 100 Hz gave an average RE of 0.71. Performance was frequency dependent. The smallest REs occurred at low frequencies. With 1% noise, optimal regularization gave average REs of 0.20, 0.40, and 0.53 in the 1–15, 15–46, and 46–100 Hz bands, respectively. Direct inference of spectral magnitudes was more accurate than Fourier transformation of inferred time-domain waveforms. Results suggest that when heart size and location are not known, minimum REs in spectral estimates are found using an overestimated heart size and a regularization parameter which is the average value over the frequency band of interest. © 1999 Biomedical Engineering Society. PAC99: 8719Hh, 8719Nn, 0260Lj     

Spatial Methods of Epicardial Activation Time Determination in Normal Hearts

The purpose of this study was to demonstrate errors in activation time maps created using the time derivative method on fractionated unipolar electrograms, to characterize the epicardial distribution of those fractionated electrograms, and to investigate spatial methods of activation time determination. Electrograms (EGs) were recorded using uniform grids of electrodes (1 or 2 mm spacing) on the epicardial surface of six normal canine hearts. Activation times were estimated using the time of the minimum time derivative, maximum spatial gradient, and zero Laplacian and compared with the time of arrival of the activation wave front as assessed from a time series of potential maps as the standard. When comparing activation times from the time derivative for the case of epicardial pacing, spatial gradient and Laplacian methods with the standard for EGs without fractionation, correlations were high (R²=0.98, 0.98, 0.97, respectively). Similar comparisons using results from only fractionated EGs (R²=0.85,0.97,0.95) showed a lower correlation between times from the time derivative method and the standard. The results suggest an advantage of spatial methods over the time derivative method only for the case of epicardial pacing where large numbers of fractionated electrograms are found. © 2003 Biomedical Engineering Society. PAC2003: 8719Hh, 8719Nn, 8780Tq     

A Novel Interpolation Method for Electric Potential Fields in the Heart during Excitation

In mapping the electrical activity of the heart, interpolation of electric potentials plays two important roles. First, it permits the estimation of potentials in regions that could not be sampled or where signal quality was poor, and second, it supports the construction of isopotential lines and surfaces for visualization. The difficulty in developing robust interpolation techniques for cardiac applications lies in the abrupt change in potential in the vicinity of the activation wave front. Despite the resulting nonlinearities in spatial potential distributions, simple linear interpolation methods are the current standard and the resulting errors due to aliasing can be large if electrode spacing does not lie on the order of 0.5–2 mm—the thickness of the activation wave front. We have developed a novel interpolation method that is based on two observations specific to the spread of excitation in the heart: (1) that propagation velocity changes smoothly within a region large enough to contain several measurement electrodes and (2) that electrogram morphology varies very little in the neighborhood of each sample point except for a time shift in the potential wave forms. The resulting interpolation scheme breaks the interpolation of one highly nonlinear variable—extracellular potential—into two separate interpolations of variables with much less drastic spatial variation—activation time and electrogram morphology. We have applied this method to potentials originally recorded at 1.5 mm spacing and then subsampled at a range of densities for testing of the interpolation. The results based both on reconstruction of isopotential contour maps and statistical comparison showed significant improvement of this novel approach over standard linear techniques. The applications of the new method include improved determination of electrophysiological parameters such as spatial gradients of potential and the path of cardiac activation and recovery, estimation of electrograms at desired locations, and visualization of electric potential distributions. © 1998 Biomedical Engineering Society. PAC98: 8790+y, 0260Ed, 8710+e     

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

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

The electrocardiographic inverse problem.

Using the boundary element method in conjunction with Tikhonov zero-order regularization, we have computed epicardial potentials from body surface potential data in a realistic geometry heart-torso system. The inverse-reconstructed epicardial potentials were compared to the actual measured potentials throughout a normal cardiac cycle. Potential features (maxima, minima) were recovered with an accuracy better than 1 cm in their location. In this chapter, we use these data to illustrate and discuss computational issues related to the inverse-reconstruction procedure. These include the boundary element method, the choice of a regularization scheme to stabilize the inversion, and the effects of incorporating a priori information on the accuracy of the solution. In particular, emphasis is on the use of temporal information in the regularization procedure. The sensitivity of the solution to geometrical errors and to the spatial and temporal resolution of the data is discussed.     

Three-dimensional panoramic imaging of cardiac arrhythmias in rabbit heart

Cardiac fluorescent optical imaging provides the unique opportunity to investigate the dynamics of propagating electrical waves during ventricular arrhythmias and the termination of arrhythmias by strong electric shocks. Panoramic imaging systems using charge-coupled device (CCD) cameras as the photodetector have been developed to overcome the inability to monitor electrical activity from the entire cardiac surface. Photodiode arrays (PDAs) are known to have higher temporal resolution and signal quality, but lower spatial resolution compared to CCD cameras. We construct a panoramic imaging system with three PDAs and image Langendorff perfused rabbit hearts (n=18) during normal sinus rhythm, epicardial pacing, and arrhythmias. The recorded spatiotemporal dynamics of electrical activity is texture mapped onto a reconstructed 3-D geometrical heart model specific to each heart studied. The PDA-based system provides sufficient spatial resolution (1.72 mm without interpolation) for the study of wavefront propagation in the rabbit heart. The reconstructed 3-D electrical activity provides us with a powerful tool to investigate the fundamental mechanisms of arrhythmia maintenance and termination.     

Adaptive sampling of intracellular and extracellular cardiac potentials with the fan method

Uniform sampling rates of 5000 samples s⁻¹ and 15000 samples s⁻¹ or more are required to accurately represent intracellular and extracellular cardiac electrograms. These high rates are necessary because of short intervals where rapid deflections occur; thus uniform sampling results in far too many samples from slow phases when voltages change little between consecutive samples. Adaptive sampling with the fan algorithm selects samples with an irregular temporal spacing that specifies each waveform with the minimum number of samples required for a given maximum error or tolerance ε. This paper describes the performance of the fan as a function of tolerance on cardiac action potentials and electrograms simulated with the Beeler-Reuter model or measured directly from the heart, including examples showing signal quality, average sampling rate and intervals between samples. With low tolerances of ɛ^α = 100 μV (intracellular) or ɛ^α = 25 μV (extracellular), the average fan sampling rates were about 8000 samples s⁻¹ for measured waveforms, and about 200 samples s⁻¹ for simulated ones. With higher tolerances of ɛ^β = 1 mV (intracellular) or ɛ^β = 60 μV (extracellular) average fan sampling rates near 200 samples s⁻¹ were found in all cases. The results suggest that good quality measurements of action potentials and electrograms can be obtained with remarkably low average sampling rates.     

Noninvasive electroanatomic mapping of human ventricular arrhythmias with electrocardiographic imaging

The rapid heartbeat of ventricular tachycardia (VT) can lead to sudden cardiac death and is a major health issue worldwide. Efforts to identify patients at risk, determine mechanisms of VT, and effectively prevent and treat VT through a mechanism-based approach would all be facilitated by continuous, noninvasive imaging of the arrhythmia over the entire heart. Here, we present noninvasive real-time images of human ventricular arrhythmias using electrocardiographic imaging (ECGI). Our results reveal diverse activation patterns, mechanisms, and sites of initiation of human VT. The spatial resolution of ECGI is superior to that of the routinely used 12-lead electrocardiogram, which provides only global information, and ECGI has distinct advantages over the currently used method of mapping with invasive catheter-applied electrodes. The spatial resolution of this method and its ability to image electrical activation sequences over the entire ventricular surfaces in a single heartbeat allowed us to determine VT initiation sites and continuation pathways, as well as VT relationships to ventricular substrates, including anatomical scars and abnormal electrophysiological substrate. Thus, ECGI can map the VT activation sequence and identify the location and depth of VT origin in individual patients, allowing personalized treatment of patients with ventricular arrhythmias.     

A Field-Compatible Method for Interpolating Biopotentials

Mapping of bioelectric potentials over a given surface (e.g., the torso surface, the scalp) often requires interpolation of potentials into regions of missing data. Existing interpolation methods introduce significant errors when interpolating into large regions of high potential gradients, due mostly to their incompatibility with the properties of the three-dimensional (3D) potential field. In this paper, an interpolation method, inverse-forward (IF) interpolation, was developed to be consistent with Laplace's equation that governs the 3D field in the volume conductor bounded by the mapped surface. This method is evaluated in an experimental heart–torso preparation in the context of electrocardiographic body surface potential mapping. Results demonstrate that IF interpolation is able to recreate major potential features such as a potential minimum and high potential gradients within a large region of missing data. Other commonly used interpolation methods failed to reconstruct major potential features or preserve high potential gradients. An example of IF interpolation with patient data is provided to illustrate its applicability in the actual clinical setting. Application of IF interpolation in the context of noninvasive reconstruction of epicardial potentials (the inverse problem) is also examined. © 1998 Biomedical Engineering Society. PAC98: 8710+e, 0260Ed     

Quantitative Characterization of Epicardial Wave Fronts During Regional Ischemia and Elevated Extracellular Potassium Ion Concentration

This study applied zero-delay wave number spectral estimation as a means of quantifying the changes in activation and recovery sequences of propagating plane waves on the epicardial surface of in situ porcine hearts during regional hyperkalemia and ischemia. Unipolar electrograms (104) were recorded from the left ventricular surface of nine hearts using a plaque electrode array with 1 mm spatial sampling intervals. The objectives were (1) to define a set of parameters capable of quantifying the spatial and temporal changes in measured extracellular potentials associated with localized ischemia prior to the onset of conduction block; (2) to elevate regional levels of extracellular potassium ion concentration and quantify potential changes due to this known physiologic manipulation; and (3) to use quantitative parameters to make statistical comparisons in order to distinguish wave fronts during normal, ischemic and hyperkalemic conditions. Results showed that the parameters of wave number and average temporal frequency and the associated power, as determined from the wave number spectrum, provided statistically significant (p < 0.05) quantification of changes in wave front features during normal and ischemic or hyperkalemic conditions. The results were consistent with results obtained from conventional time–space domain methods like isochronal mapping and electrograms, with the advantage of a quantitative result enabling simple comparisons and trend analysis for large numbers of heart beats. © 1998 Biomedical Engineering Society. PAC98: 8759Wc, 8722Fy, 8780+s     

犬急性心肌梗塞后心外膜电图碎裂电位和室性心律失常的关系

10只急性心肌梗塞后2～6天的犬中,4只在窦性心律时缺血区心外膜电图上出现迟发的碎裂电活动。碎裂电位的特点为:同一部位的碎裂电位可随时间延长而有所变化,可出现在梗塞区或缺血边缘区。可以引起室性早搏或隐匿性地激动全心室。有碎裂电位的犬心经心脏程序电刺激均可诱发出室性心律失常;并且心肌梗塞面积显著大于无碎裂电位的其它犬心。提示局部心外膜电图中的碎裂电位是心室内存在潜在折返环路的重要标志。     

Wavelet-promoted sparsity for non-invasive reconstruction of electrical activity of the heart

We investigated a novel sparsity-based regularization method in the wavelet domain of the inverse problem of electrocardiography that aims at preserving the spatiotemporal characteristics of heart-surface potentials. In three normal, anesthetized dogs, electrodes were implanted around the epicardium and body-surface electrodes were attached to the torso. Potential recordings were obtained simultaneously on the body surface and on the epicardium. A CT scan was used to digitize a homogeneous geometry which consisted of the body-surface electrodes and the epicardial surface. A novel multitask elastic-net-based method was introduced to regularize the ill-posed inverse problem. The method simultaneously pursues a sparse wavelet representation in time-frequency and exploits correlations in space. Performance was assessed in terms of quality of reconstructed epicardial potentials, estimated activation and recovery time, and estimated locations of pacing, and compared with performance of Tikhonov zeroth-order regularization. Results in the wavelet domain obtained higher sparsity than those in the time domain. Epicardial potentials were non-invasively reconstructed with higher accuracy than with Tikhonov zeroth-order regularization (p <?0.05), and recovery times were improved (p <?0.05). No significant improvement was found in terms of activation times and localization of origin of pacing. Next to improved estimation of recovery isochrones, which is important when assessing substrate for cardiac arrhythmias, this novel technique opens potentially powerful opportunities for clinical application, by allowing to choose wavelet bases that are optimized for specific clinical questions.

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Graphical Abstract The inverse problem of electrocardiography is to reconstruct heart-surface potentials from recorded bodysurface electrocardiograms (ECGs) and a torso-heart geometry. However, it is ill-posed and solving it requires additional constraints for regularization. We introduce a regularization method that simultaneously pursues a sparse wavelet representation in time-frequency and exploits correlations in space. Our approach reconstructs epicardial (heart-surface) potentials with higher accuracy than common methods. It also improves the reconstruction of recovery isochrones, which is important when assessing substrate for cardiac arrhythmias. This novel technique opens potentially powerful opportunities for clinical application, by allowing to choose wavelet bases that are optimized for specific clinical questions.

     

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.     

Novel Technique for Cardiac Electromechanical Mapping with Magnetic Resonance Imaging Tagging and an Epicardial Electrode Sock

Near-simultaneous measurements of electrical and mechanical activation over the entire ventricular surface are now possible using magnetic resonance imaging tagging and a multielectrode epicardial sock. This new electromechanical mapping technique is demonstrated in the ventricularly paced canine heart. A 128–electrode epicardial sock and pacing electrodes were placed on the hearts of four anesthetized dogs. In the magnetic resonance scanner, tagged cine images (8–15 ms/frame) and sock electrode recordings (1000 Hz) were acquired under right-ventricular pacing and temporally referenced to the pacing stimulus. Electrical recordings were obtained during intermittent breaks in image acquisition, so that both data sets represented the same physiologic state. Since the electrodes were not visible in the images, electrode recordings and cine images were spatially registered with Gd-DTPA markers attached to the sock. Circumferential strain was calculated at locations corresponding to electrodes. For each electrode location, electrical and mechanical activation times were calculated and relationships between the two activation patterns were demonstrated. This method holds promise for improving understanding of the relationships between the patterns of electrical activation and contraction in the heart. © 2003 Biomedical Engineering Society. PAC2003: 8761Pk, 8719Rr, 8719Hh     

Electrophysiological determinants of hypokalaemia‐induced arrhythmogenicity in the guinea‐pig heart

Aim: Hypokalaemia is an independent risk factor contributing to arrhythmic death in cardiac patients. In the present study, we explored the mechanisms of hypokalaemia‐induced tachyarrhythmias by measuring ventricular refractoriness, spatial repolarization gradients, and ventricular conduction time in isolated, perfused guinea‐pig heart preparations. Methods: Epicardial and endocardial monophasic action potentials from distinct left ventricular (LV) and right ventricular (RV) recording sites were monitored simultaneously with volume‐conducted electrocardiogram (ECG) during steady‐state pacing and following a premature extrastimulus application at progressively reducing coupling stimulation intervals in normokalaemic and hypokalaemic conditions. Results: Hypokalaemic perfusion (2.5 mm K⁺ for 30 min) markedly increased the inducibility of tachyarrhythmias by programmed ventricular stimulation and rapid pacing, prolonged ventricular repolarization and shortened LV epicardial and endocardial effective refractory periods, thereby increasing the critical interval for LV re‐excitation. Hypokalaemia increased the RV‐to‐LV transepicardial repolarization gradients but had no effect on transmural dispersion of APD₉₀ and refractoriness across the LV wall. As determined by local activation time recordings, the LV‐to‐RV transepicardial conduction and the LV transmural (epicardial‐to‐endocardial) conduction were slowed in hypokalaemic heart preparations. This change was attributed to depressed diastolic excitability as evidenced by increased ventricular pacing thresholds. Conclusion: These findings suggest that hypokalaemia‐induced arrhythmogenicity is attributed to shortened LV refractoriness, increased critical intervals for LV re‐excitation, amplified RV‐to‐LV transepicardial repolarization gradients and slowed ventricular conduction in the guinea‐pig heart.     

心外膜临时起搏在心内直视手术后的应用

心外膜起搏电极更多情况下作为心内膜起搏电极的补充,主要在无法应用心内膜起搏电极或经静脉途径心内膜起搏电极植入失败以及心脏直视术中,成为治疗心脏手术后房室传导阻滞,心动过缓及暂时性节律紊乱的主要措施。本文概述了临时心外膜起搏器的各种适应证,及何时考虑过渡到永久性起搏,起搏器的种类的选择,起搏器导线移除等问题。