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

Multi-Source Data and Knowledge Fusion via Deep Learning for Dynamical Systems: Applications to Spatiotemporal Cardiac Modeling

Advanced sensing and imaging provide unprecedented opportunities to collect data from diverse sources for increasing information visibility in spatiotemporal dynamical systems. Furthermore, the fundamental physics of the dynamical system is commonly elucidated through a set of partial differential equations (PDEs), which plays a critical role in delineating the manner in which the sensing signals can be modeled. Reliable predictive modeling of such spatiotemporal dynamical systems calls upon the effective fusion of fundamental physics knowledge and multi-source sensing data. This paper proposes a multi-source data and knowledge fusion framework via deep learning for dynamical systems with applications to spatiotemporal cardiac modeling. This framework not only achieves effective data fusion through capturing the physics-based information flow between different domains, but also incorporates the geometric information of a 3D system through a graph Laplacian for robust spatiotemporal predictive modeling. We implement the proposed framework to model cardiac electrodynamics under both healthy and diseased heart conditions. Numerical experiments demonstrate the superior performance of our framework compared with traditional approaches that lack the capability for effective data fusion or geometric information incorporation.

Frequency-Enhanced Geometric-Constrained Reconstruction for Localizing Myocardial Infarction in 12-Lead Electrocardiograms

Determining the location of myocardial infarction is crucial for clinical management and therapeutic stratagem. However, existing diagnostic tools either sacrifice ease of use or are limited by their spatial resolution. Addressing this, we aim to refine myocardial infarction localization via surface potential reconstruction of the ventricles in 12-lead electrocardiograms (ECG). A notable obstacle is the ill-posed nature of such reconstructions. To overcome this, we introduce the frequency-enhanced geometric-constrained iterative network (FGIN). FGIN begins by mining the latent features from ECG data across both time and frequency domains. Subsequently, it increases the data dimensionality of ECG and captures intricate features using convolutional layers. Finally, FGIN incorporates ventricular geometry as a constraint on surface potential distribution. It allocates variable weights to distinct edges. Experimental validation of FGIN confirms its efficacy over synthetic and clinical datasets. On the synthetic dataset, FGIN outperforms seven existing reconstruction methods, attaining the highest Pearson Correlation Coefficient of 0.8624, the lowest Root Mean Square Error of 0.1548, and the highest Structural Similarity Index Measure of 0.7988. On the clinical public dataset (2007 PhysioNet/Computers in Cardiology Challenge), FGIN achieves better localization results than other approaches, according to the clinical standard 17-segment model, achieving an average Segment Overlap of 87.2%. Clinical trials on 50 patients demonstrate FGIN's effectiveness, showing an average accuracy of 91.6% and an average Segment Overlap of 88.2%.

Impulse Data Model For Solving The Inverse Problem of Electrocardiography

OBJECTIVE

To develop, train and test neural networks for predicting heart surface potentials (HSPs) from body surface potentials (BSPs). The method re-frames traditional inverse problems of electrocardiograpy into regression problems, constraining the solution space by decomposing signals with multidimensional Gaussian impulse basis functions.

METHODS

Impulse HSPs were generated with single Gaussian basis functions at discrete heart surface locations and projected to corresponding BSPs using a volume conductor torso model. Both BSP (inputs) and HSP (outputs) were mapped to regular 2D surface meshes and used to train a neural network. Predictive capabilities of the network were tested with unseen synthetic and experimental data.

RESULTS

A dense full connected single hidden layer neural network was trained to map body surface impulses to heart surface Gaussian basis functions for reconstructing HSP. Synthetic pulses moving across the heart surface were predicted from the neural network with root mean squared error of 9.1 +/ 1.4%. Predicted signals were robust to noise up to 20 dB and errors due to displacement and rotation of the heart within the torso were bounded and predictable. A shift of the heart 40 mm toward the spine resulted in a 4% increase in signal feature localization error. The set of training impulse function data could be reduced and prediction error remained bounded. Recorded HSPs from in-vitro pig hearts were reliably decomposed using space-time Gaussian basis functions. Activation times calculated from predicted HSPs for left-ventricular pacing had a mean absolute error of 10.4 +/ 11.4 ms. Other pacing scenarios were analyzed with similar success.

CONCLUSION

Impulses from Gaussian basis functions are potentially an effective and robust way to train simple neural network data models for reconstructing HSPs from decomposed BSPs.

SIGNIFICANCE

The HSPs predicted by the neural network can be used to generate activation maps that non-invasively identify features of cardiac electrical dysfunction and can guide subsequent treatment options.

Machine-readable Yin-Yang imbalance: traditional Chinese medicine syndrome computer modeling based on three-dimensional noninvasive cardiac electrophysiology imaging

OBJECTIVES

The principal diagnostic methods of traditional Chinese medicine (TCM) are inspection, auscultation and olfaction, inquiry, and pulse-taking. Treatment by syndrome differentiation is likely to be subjective. This study was designed to provide a basic theory for TCM diagnosis and establish an objective means of evaluating the correctness of syndrome differentiation.

METHODS

We herein provide the basic theory of TCM syndrome computer modeling based on a noninvasive cardiac electrophysiology imaging technique. Noninvasive cardiac electrophysiology imaging records the heart's electrical activity from hundreds of electrodes on the patient's torso surface and therefore provides much more information than 12-lead electrocardiography. Through mathematical reconstruction algorithm calculations, the reconstructed heart model is a machine-readable description of the underlying mathematical physics model that reveals the detailed three-dimensional (3D) electrophysiological activity of the heart.

RESULTS

From part of the simulation results, the imaged 3D cardiac electrical source provides dynamic information regarding the heart's electrical activity at any given location within the 3D myocardium.

CONCLUSIONS

This noninvasive cardiac electrophysiology imaging method is suitable for translating TCM syndromes into a computable format of the underlying mathematical physics model to offer TCM diagnosis evidence-based standards for ensuring correct evaluation and rigorous, scientific data for demonstrating its efficacy.

Introduction to noninvasive cardiac mapping

From the dawn of the twentieth century, the electrocardiogram (ECG) has revolutionized the way clinical cardiology has been practiced, and it has become the cornerstone of modern medicine today. Driven by clinical and research needs for a more precise understanding of cardiac electrophysiology beyond traditional ECG, inverse solution electrocardiography has been developed, tested, and validated. This article outlines the important progress from ECG development, through more extensive measurement of body surface potentials, and the fundamental leap to solving the inverse problem of electrocardiography, with a focus on mathematical methods and experimental validation.

Noninvasive imaging of 3-dimensional myocardial infarction from the inverse solution of equivalent current density in pathological hearts

We propose a new approach to noninvasively image the 3-D myocardial infarction (MI) substrates based on equivalent current density (ECD) distribution that is estimated from the body surface potential maps (BSPMs) during S-T segment. The MI substrates were identified using a predefined threshold of ECD. Computer simulations were performed to assess the performance with respect to: 1) MI locations; 2) MI sizes; 3) measurement noise; 4) numbers of BSPM electrodes; and 5) volume conductor modeling errors. A total of 114 sites of transmural infarctions, 91 sites of epicardial infarctions, and 36 sites of endocardial infarctions were simulated. The simulation results show that: 1) Under 205 electrodes and 10-μV noise, the averaged accuracies of imaging transmural MI are 83.4% for sensitivity, 82.2% for specificity, 65.0% for Dice's coefficient, and 6.5 mm for distances between the centers of gravity (DCG). 2) For epicardial infarction, the averaged imaging accuracies are 81.6% for sensitivity, 75.8% for specificity, 45.3% for Dice's coefficient, and 7.5 mm for DCG; while for endocardial infarction, the imaging accuracies are 80.0% for sensitivity, 77.0% for specificity, 39.2% for Dice's coefficient, and 10.4 mm for DCG. 3) A reasonably good imaging performance was obtained under higher noise levels, fewer BSPM electrodes, and mild volume conductor modeling errors. The present results suggest that this method has the potential to aid in the clinical identification of the MI substrates.

Computing volume potentials for noninvasive imaging of cardiac excitation

BACKGROUND

In noninvasive imaging of cardiac excitation, the use of body surface potentials (BSP) rather than body volume potentials (BVP) has been favored due to enhanced computational efficiency and reduced modeling effort. Nowadays, increased computational power and the availability of open source software enable the calculation of BVP for clinical purposes. In order to illustrate the possible advantages of this approach, the explanatory power of BVP is investigated using a rectangular tank filled with an electrolytic conductor and a patient specific three dimensional model.

METHODS

MRI images of the tank and of a patient were obtained in three orthogonal directions using a turbo spin echo MRI sequence. MRI images were segmented in three dimensional using custom written software. Gmsh software was used for mesh generation. BVP were computed using a transfer matrix and FEniCS software.

RESULTS

The solution for 240,000 nodes, corresponding to a resolution of 5 mm throughout the thorax volume, was computed in 3 minutes. The tank experiment revealed that an increased electrode surface renders the position of the 4 V equipotential plane insensitive to mesh cell size and reduces simulated deviations. In the patient-specific model, the impact of assigning a different conductivity to lung tissue on the distribution of volume potentials could be visualized.

CONCLUSION

Generation of high quality volume meshes and computation of BVP with a resolution of 5 mm is feasible using generally available software and hardware. Estimation of BVP may lead to an improved understanding of the genesis of BSP and sources of local inaccuracies.

Noninvasive mapping of transmural potentials during activation in swine hearts from body surface electrocardiograms

The three-dimensional cardiac electrical imaging (3DCEI) technique was previously developed to estimate the initiation site(s) of cardiac activation and activation sequence from the noninvasively measured body surface potential maps (BSPMs). The aim of this study was to develop and evaluate the capability of 3DCEI in mapping the transmural distribution of extracellular potentials and localizing initiation sites of ventricular activation in an in vivo animal model. A control swine model (n = 10) was employed in this study. The heart-torso volume conductor model and the excitable heart model were constructed based on each animal's preoperative MR images and a priori known physiological knowledge. Body surface potential mapping and intracavitary noncontact mapping (NCM) were simultaneously conducted during acute ventricular pacing. The 3DCEI analysis was then applied on the recorded BSPMs. The estimated initiation sites were compared to the precise pacing sites; as a subset of the mapped transmural potentials by 3DCEI, the electrograms on the left ventricular endocardium were compared to the corresponding output of the NCM system. Over the 16 LV and 48 RV pacing studies, the averaged localization error was 6.1±2.3 mm, and the averaged correlation coefficient between the estimated endocardial electrograms by 3DCEI and from the NCM system was 0.62±0.09. The results demonstrate that the 3DCEI approach can well localize the sites of initiation of ectopic beats and can obtain physiologically reasonable transmural potentials in an in vivo setting during focal ectopic beats. This study suggests the feasibility of tomographic mapping of 3D ventricular electrograms from the body surface recordings.

An enhanced method for accessory pathway localization in case of Wolff-Parkinson-White syndrome

This paper presents an analysis of the Arruda accessory pathway localization method for patients suffering from Wolff-Parkinson-White syndrome, with modifications to increase the overall accuracy. The Arruda method was tested on a total of 79 cases, and 91.1% localization performance was reached. After a deeper analysis of each decision point of the Arruda localization method, we considered that the lead aVF was not as relevant as other leads (I, II, III, V1) used. The branch of the decision tree, which evaluates the left ventricle positions, was entirely replaced using different decision criteria based on the same biological parameters. The modified algorithm significantly improves the localization accuracy in the left ventricle, reaching 94.9%. An accurate localization performance of non-invasive methods is relevant because it can enlighten the necessary invasive interventions, and it also reduces the discomfort caused to the patient.

A patient specific electro-mechanical model of the heart

This paper presents a patient specific deformable heart model that involves the known electrical and mechanical properties of the cardiac cells and tissue. The whole heart model comprises ten Tusscher's ventricular and Nygren's atrial cell models, the anatomical and electrophysiological model descriptions of the atria (introduced by Harrild et al.) and ventricle (given by Winslow et al.), and the mechanical model of the periodical cardiac contraction and resting phenomena proposed by Moireau et al. During the propagation of the depolarization wave, the kinetic, compositional and rotational anisotropy is handled by the tissue, organ and torso model. The applied patient specific parameters were determined by an evolutionary computation method. An intensive parameter reduction was performed using the abstract formulation of the searching space. This patient specific parameter representation enables the adjustment of deformable model parameters in real-time. The validation process was performed using simultaneously measured ECG and ultrasound image records that were compared with simulated signals and shapes using an abstract, parameterized evaluation criterion.

Noninvasive estimation of global activation sequence using the extended Kalman filter

A new algorithm for 3-D imaging of the activation sequence from noninvasive body surface potentials is proposed. After formulating the nonlinear relationship between the 3-D activation sequence and the body surface recordings during activation, the extended Kalman filter (EKF) is utilized to estimate the activation sequence in a recursive way. The state vector containing the activation sequence is optimized during iteration by updating the error variance/covariance matrix. A new regularization scheme is incorporated into the "predict" procedure of EKF to tackle the ill-posedness of the inverse problem. The EKF-based algorithm shows good performance in simulation under single-site pacing. Between the estimated activation sequences and true values, the average correlation coefficient (CC) is 0.95, and the relative error (RE) is 0.13. The average localization error (LE) when localizing the pacing site is 3.0 mm. Good results are also obtained under dual-site pacing (CC = 0.93, RE = 0.16, and LE = 4.3 mm). Furthermore, the algorithm shows robustness to noise. The present promising results demonstrate that the proposed EKF-based inverse approach can noninvasively estimate the 3-D activation sequence with good accuracy and the new algorithm shows good features due to the application of EKF.

Mathematical modeling of electrocardiograms: a numerical study

This paper deals with the numerical simulation of electrocardiograms (ECG). Our aim is to devise a mathematical model, based on partial differential equations, which is able to provide realistic 12-lead ECGs. The main ingredients of this model are classical: the bidomain equations coupled to a phenomenological ionic model in the heart, and a generalized Laplace equation in the torso. The obtention of realistic ECGs relies on other important features--including heart-torso transmission conditions, anisotropy, cell heterogeneity and His bundle modeling--that are discussed in detail. The numerical implementation is based on state-of-the-art numerical methods: domain decomposition techniques and second order semi-implicit time marching schemes, offering a good compromise between accuracy, stability and efficiency. The numerical ECGs obtained with this approach show correct amplitudes, shapes and polarities, in all the 12 standard leads. The relevance of every modeling choice is carefully discussed and the numerical ECG sensitivity to the model parameters investigated.

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

We consider distributed parameter identification problems for the FitzHugh-Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton-Krylov type that combines Newton's method for numerical optimization with Krylov subspace solvers for the resulting Karush-Kuhn-Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions.

The next generation EKG--in vivo demonstration of noninvasive electrocardiographic imaging during normal sinus rhythm

Noninvasive, in vivo, reconstruction of epicardial electrical activity is needed to help better study, understand, and treat electrical rhythm abnormalities. Here, a new method for noninvasive electrocardiographic imaging is used to reconstruct epicardial potentials in vivo during normal sinus rhythm. This method used measured body surface potentials (BSPMs) and the relative geometry between the body surface and epicardial surface from computed tomography (CT) to reconstruct in vivo epicardial potentials during normal sinus rhythm. The reconstructed epicardial potentials correlated qualitatively with those expected for various aspects of normal sinus rhythm (NSR). This study shows that noninvasively reconstructed epicardial potentials could provide useful information on the electrical activity of the heart during normal activation and repolarization sequences not otherwise available.

Sensibility analysis of the Arruda localization method

This paper presents an analysis of the Arruda accessory pathway localization method (for patients suffering from Wolff-Parkinson-White syndrome) with suggestions to increase the overall performance. The Arruda method was tested on a total of 121 patients, and a 90% localization performance was reached. This was considered almost as performing result as the highest published (90%) by L. Boersma in 2002. After a deeper analysis of each decision point of Arruda localization method we considered that the lead AVF is not as relevant as other used leads (I, II, III, VI). The overall performance (90%) was slightly lower then the correct decision rate (91.67%) at the weakest decision element (AVF+) of the method. The vectorial space constructed from the most used leads (II, VI, AVF) is not orthogonal which can be a reason for weaker rate in case of AVF.

Noninvasive three-dimensional electrocardiographic imaging of ventricular activation sequence

Imaging the myocardial activation sequence is critical for improved diagnosis and treatment of life-threatening cardiac arrhythmias. It is desirable to reveal the underlying cardiac electrical activity throughout the three-dimensional (3-D) myocardium (rather than just the endocardial or epicardial surface) from noninvasive body surface potential measurements. A new 3-D electrocardiographic imaging technique (3-DEIT) based on the boundary element method (BEM) and multiobjective nonlinear optimization has been applied to reconstruct the cardiac activation sequences from body surface potential maps. Ultrafast computerized tomography scanning was performed for subsequent construction of the torso and heart models. Experimental studies were then conducted, during left and right ventricular pacing, in which noninvasive assessment of ventricular activation sequence by means of 3-DEIT was performed simultaneously with 3-D intracardiac mapping (up to 200 intramural sites) using specially designed plunge-needle electrodes in closed-chest rabbits. Estimated activation sequences from 3-DEIT were in good agreement with those constructed from simultaneously recorded intracardiac electrograms in the same animals. Averaged over 100 paced beats (from a total of 10 pacing sites), total activation times were comparable (53.3 +/- 8.1 vs. 49.8 +/- 5.2 ms), the localization error of site of initiation of activation was 5.73 +/- 1.77 mm, and the relative error between the estimated and measured activation sequences was 0.32 +/- 0.06. The present experimental results demonstrate that the 3-D paced ventricular activation sequence can be reconstructed by using noninvasive multisite body surface electrocardiographic measurements and imaging of heart-torso geometry. This new 3-D electrocardiographic imaging modality has the potential to guide catheter-based ablative interventions for the treatment of life-threatening cardiac arrhythmias.

Finite element modelling and simulations in cardiovascular mechanics and cardiology: A bibliography 1993–2004

The paper gives a bibliographical review of the finite element modelling and simulations in cardiovascular mechanics and cardiology from the theoretical as well as practical points of views. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1993 and 2004. At the end of this paper, more than 890 references are given dealing with subjects as: Cardiovascular soft tissue modelling; material properties; mechanisms of cardiovascular components; blood flow; artificial components; cardiac diseases examination; surgery; and other topics.

Localization of the site of origin of reentrant arrhythmia from body surface potential maps: a model study

We have developed a model-based imaging approach to estimate the site of origin of reentrant arrhythmia from body surface potential maps (BSPMs), with the aid of a cardiac arrhythmia model. The reentry was successfully simulated and maintained in the cardiac model, and the simulated ECG waveforms over the body surface corresponding to a maintained reentry have evident characteristics of ventricular tachycardia. The performance of the inverse imaging approach was evaluated by computer simulations. The present simulation results show that an averaged localization error of about 1.5 mm, when 5% Gaussian white noise was added to the BSPMs, was detected. The effects of the heart-torso geometry uncertainty on the localization were also initially assessed and the simulation results suggest that no significant influence was observed when 10% torso geometry uncertainty or 10 mm heart position shifting was considered. The present simulation study suggests the feasibility of localizing the site of origin of reentrant arrhythmia from non-invasive BSPMs, with the aid of a cardiac arrhythmia model.

Detection of the fingerprint of the electrophysiological abnormalities that increase vulnerability to life-threatening ventricular arrhythmias

Reduction of sudden death requires accurate identification of patients at risk for ventricular tachycardia (VT) and effective therapies. The Multicenter Unsustained Tachycardia Trial and Multicenter Automatic Defibrillator Implantation Trials demonstrate that the implantable cardioverter defibrillator impacts favorably on the incidence of VT in patients with myocardial infarction, underscoring the need to detect the electrophysiologic abnormalities required for the development of VT. Methods used for this purpose include: Holter monitoring, ejection fraction, signal-averaged ECG, heart rate variability, T-wave alternans, baroreflex sensitivity, and programmed stimulation. Performance of each method alone has demonstrated high-negative but low-positive predictive values. Recent studies confirm that their use in combination augments performance.A second approach for improving performance has been to reexamine how well each method detects the electrophysiological derangements that lead to VT. Our recent work has focused on the signal-averaged ECG. Judging from transmural maps of ventricular activation during VT and sinus rhythm obtained from patients, late potentials fail to detect completely signals from myocardium responsible for VT. To obviate this limitation we developed an approach based on inferred epicardial potentials in the frequency domain from 190-surface ECGs using individualized heart-torso models. Torso geometry and electrode positions are measured with a 3-armed digitizer. The location of cardiac structures is determined using echocardiography. The pericardial surface is approximated by a sphere that encloses the heart. Epicardial potentials are inferred using the boundary element method with zero-order Tikhonov regularization and the Composite Residual Smoothing Operator over the QRS complex. Studies are underway to determine if analysis of bioelectrical signals enveloping arrhythmogenic tissue improves identification of patients vulnerable to VT.

The use of calculated epicardial potentials improves significantly the sensitivity of a diagnostic algorithm in the detection of acute myocardial infarction

Inverse electrocardiography can calculate epicardial potentials (EP) from body surface potentials (BSP) taking into account a thoracic volume conductor model (TVCM). Previous studies have shown that a tailored TVCM is superior to a general TVCM in calculating EP. However, construction of a tailored TVCM for a patient in an acute clinical setting is impractical. In this study we used a general TVCM in our EP calculations to determine whether this improves detection of acute myocardial infarction (AMI) using a diagnostic algorithm. BSP were derived from the 80-lead body surface map (BSM). Consecutive patients (n=379) with ischemic type chest pain were recruited. The BSM and a 12-lead electrocardiogram (ECG) were recorded at initial presentation and creatine kinase (CK) and/or CK-MB were measured initially, 12 and 24 hours postsymptom onset. A physician interpreted the 12-lead electrocardiogram and documented ST elevation if present. AMI was defined by the World Health Organization (WHO) criteria. The diagnostic algorithm result for each patient using BSP and calculated EP were documented. AMI occurred in 171 patients. The diagnostic algorithm using BSP identified 106 of these as ST elevation AMI (STEMI) (sensitivity 62%, specificity 80%). The same algorithm using EP identified 133 as STEMI (sensitivity 78%, specificity 80%). Calculated EP improved the algorithm's diagnostic sensitivity by a factor of 1.25 (P<.001) with no significant difference in specificity. Calculated EP using a general TVCM significantly improves the sensitivity of a diagnostic algorithm based on BSP in detection of AMI with no significant loss in specificity.

Noninvasive imaging of cardiac transmembrane potentials within three-dimensional myocardium by means of a realistic geometry anisotropic heart model

We have developed a new approach for imaging cardiac transmembrane potentials (TMPs) within the three-dimensional (3-D) myocardium by means of an anisotropic heart model. The cardiac TMP distribution is estimated from body surface electrocardiograms by minimizing objective functions of the "measured" body surface potential maps (BSPMs) and the heart-model-generated BSPMs. Computer simulation studies have been conducted to evaluate the present 3-D TMP imaging approach using pacing protocols. Simulations of single-site pacing at 24 sites throughout the ventricles, as well as dual-site pacing at 12 pairs of sites in the vicinity of atrio-ventricular ring were performed. The present simulation results show that the correlation coefficient (CC) and relative error (RE) between the "true" and inversely estimated TMP distributions were 0.9915 +/- 0.0041 and 0.1266 +/- 0.0326, for single-site pacing, and 0.9889 +/- 0.0034 and 0.1473 +/- 0.0237 for dual-site pacing, respectively, when 10 microV Gaussian white noise (GWN) was added to the BSPMs. The effects of heart and torso geometry uncertainty were also evaluated by shifting the heart position by 10 mm and altering the torso size by 10%. The CC between the "true" and inversely estimated TMP distributions was above 0.97 when these geometry uncertainties were considered. The present simulation results demonstrate the feasibility of noninvasive estimation of TMP distribution throughout the ventricles from body surface electrocardiographic measurements, and suggest that the present method may become a useful alternative in noninvasive imaging of distributed cardiac electrophysiological processes within the 3-D myocardium.

Noninvasive localization of the site of origin of paced cardiac activation in human by means of a 3-D heart model

A recently developed heart-model-based localization approach is experimentally evaluated in noninvasively localizing the site of origin of cardiac activation in a patient with a pacemaker. The heart-torso model of the patient was constructed from the contrast ultrafast computed tomography images. The site of initial paced activation in the patient was quantitatively localized and compared with the tip position of the pacemaker lead. The localization error of the inverse estimation was found to be 5.2 mm with respect to the true lead tip position. The promising result of this pilot experimental study suggests the feasibility of localizing the site of origin of cardiac activation in an experimental setting. The heart-model-based localization approach may become an alternative tool in localizing the site of origin of cardiac activation.

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.

Computational framework for simulating the mechanisms and ECG of re-entrant ventricular fibrillation

Ventricular arrhythmias remain an important cause of morbidity and mortality in the Western world. Although the underlying mechanisms of these arrhythmias can be studied experimentally, these investigations are in general limited to mapping electrical activity on the heart surface. Computational models of action potential propagation offer a potentially powerful way to study electrical activation and arrhythmias, but current models are not easy to link to the clinical environment. In this paper, we describe a framework for computing action potential propagation in which the geometry, electrophysiology and regional properties of ventricular myocardium can be specified so that, for example, different models for cardiac cellular electrophysiology can be used. We have computed action potential propagation during both normal beats and re-entry in an anatomically accurate model of ventricular geometry. By computing the resultant electric current flow in the torso we have also generated simulated ECG signals that result from specific activation patterns in the ventricular model. Models can be powerful tools for explaining observations, and this approach is able to provide a direct link between the different configurations of re-entry observed in computational and experimental studies, and the ECG signals observed in patients.

Boundary element method-based cortical potential imaging of somatosensory evoked potentials using subjects' magnetic resonance images

A boundary element method-based cortical potential imaging technique has been developed to directly link the scalp potentials with the cortical potentials with the aid of magnetic resonance images of the subjects. First, computer simulations were conducted to evaluate the new approach in a concentric three-sphere inhomogeneous head model. Second, the corresponding cortical potentials were estimated from the patients' preoperative scalp somatosensory evoked potentials (SEPs) based on the boundary element models constructed from subjects' magnetic resonance images and compared to the postoperative direct cortical potential recordings in the same patients. Simulation results demonstrated that the cortical potentials can be estimated from the scalp potentials using different scalp electrode configurations and are robust against measurement noise. The cortical imaging analysis of the preoperative scalp SEPs recorded from patients using the present approach showed high consistency in spatial pattern with the postoperative direct cortical potential recordings. Quantitative comparison between the estimated and the directly recorded subdural grid potentials resulted in reasonably high correlation coefficients in cases studied. Amplitude difference between the estimated and the recorded potentials was also observed as indexed by the relative error, and the possible underlying reasons are discussed. The present numerical and experimental results validate the boundary element method-based cortical potential imaging approach and demonstrate the feasibility of the new approach in noninvasive high-resolution imaging of brain electric activities from scalp potential measurement and magnetic resonance images.

Noninvasive characterisation of multiple ventricular events using electrocardiographic imaging

Distributions of epicardial potentials, calculated from body surface electrocardiograms (ECGs), were investigated to determine if they could enable detection of multiple sites of ventricular activity. An anatomical model of the human ventricular myocardium was used to simulate activation sequences initiated at nine different ventricular pairs of sites. From these sequences, body surface ECGs were simulated at 352 sites on the torso surface and then used to reconstruct epicardial potentials at 202 sites. The criterion for detection of dual ventricular events was the presence of two distinct primary potential minima in the reconstructed epicardial potentials. The shortest distance between the two events in the right ventricle that resulted in the reconstruction of epicardial potential patterns, featuring two minima, was 27 mm; the distance between the two events in the left ventricle was 23 mm. When Gaussian white noise in the simulated body surface potentials was increased from 3 microV to 15 microV and 50 microV, dual events became more difficult to distinguish. Findings indicate that calculated epicardial potentials provide useful visual information about the presence of multiple ventricular events that is not apparent in features of body surface ECGs, and could be particularly helpful in optimising mapping procedures during difficult or unsuccessful radiofrequency ablations of accessory pathways.

Localization of the site of origin of cardiac activation by means of a heart-model-based electrocardiographic imaging approach

We have developed a new approach to solve the inverse problem of electrocardiography in terms of heart model parameters. The inverse solution of the electrocardiogram (ECG) inverse problem is defined, in the present study, as the parameters of the heart model, which are closely related to the physiological and pathophysiological status of the heart, and is estimated by using an optimization system of heart model parameters, instead of solving the matrix equation relating the body surface ECGs and equivalent cardiac sources. An artificial neural network based preliminary diagnosis system has been developed to limit the searching space of the optimization algorithm and to initialize the model parameters in the computer heart model. The optimal heart model parameters were obtained by minimizing the objective functions, as functions of the observed and model-generated body surface ECGs. We have tested the feasibility of the newly developed technique in localizing the site of origin of cardiac activation using a pace mapping protocol. The present computer simulation results show that, the present approach for localization of the site of origin of ventricular activation achieved an averaged localization error of about 3 mm [for 5-muV Gaussian white noise (GWN)] and 4 mm (for 10-muV GWN), with standard deviation of the localization errors of being about 1.5 mm. The present simulation study suggests that this newly developed approach provides a robust inverse solution, circumventing the difficulties of the ECG inverse problem, and may become an important alternative to other ECG inverse solutions.

A new computational approach for cortical imaging

Estimation of current or potential distribution on the cortex is used to obtain information about neural sources from the scalp recorded electroencephalogram. If the active sources in the brain are superficial, the estimated field distribution on the cortex also yields information about the active source configuration. In these cases, these methods can be used as source localization methods. In this study, we concentrate on finite-element-based cortex potential estimation. Usually these methods require surface interpolation of the recorded voltages at the electrodes onto the entire scalp surface. We propose a new computational approach which does not require the use of surface interpolation but does it implicitly and uses only the recorded data at the electrodes. We refer to this method as the systematic approach (SA). We compare the SA with the surface interpolation approach (IA) and show that the SA is able to produce somewhat better accuracy than the IA. However, the main asset is that the sensitivity of the cortical potential maps to the regularization parameter is significantly lower than with the IA.

The use of the spatial covariance in computing pericardial potentials

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

Inverse electrocardiography by simultaneous imposition of multiple constraints

We describe two new methods for the inverse problem of electrocardiography. Both employ regularization with multiple constraints, rather than the standard single-constraint regularization. In one method, multiple constraints on the spatial behavior of the solution are used simultaneously. In the other, spatial constraints are used simultaneously with constraints on the temporal behavior of the solution. The specific cases of two spatial constraints and one spatial and one temporal constraint are considered in detail. A new method, the L-Surface, is presented to guide the choice of the required pairs of regularization parameters. In the case when both spatial and temporal regularization are used simultaneously, there is an increased computational burden, and two methods are presented to compute solutions efficiently. The methods are verified by simulations using both dipole sources and measured canine epicardial data.

Value of epicardial potential maps in localizing pre-excitation sites for radiofrequency ablation. A simulation study

Using computer simulations, we systematically investigated the limitations of an inverse solution that employs the potential distribution on the epicardial surface as an equivalent source model in localizing pre-excitation sites in Wolff-Parkinson-White syndrome. A model of the human ventricular myocardium that features an anatomically accurate geometry, an intramural rotating anisotropy and a computational implementation of the excitation process based on electrotonic interactions among cells, was used to simulate body surface potential maps (BSPMs) for 35 pre-excitation sites positioned along the atrioventricular ring. Two individualized torso models were used to account for variations in torso boundaries. Epicardial potential maps (EPMs) were computed using the L-curve inverse solution. The measure for accuracy of the localization was the distance between a position of the minimum in the inverse EPMs and the actual site of pre-excitation in the ventricular model. When the volume conductor properties and lead positions of the torso were precisely known and the measurement noise was added to the simulated BSPMs, the minimum in the inverse EPMs was at 12 ms after the onset on average within 0.65 +/- 0.26 cm of the pre-excitation site. When the standard torso model was used to localize the sites of onset of the pre-excitation sequence initiated in individualized male and female torso models, the mean distance between the minimum and the pre-excitation site was 0.67 +/- 0.31 cm for the male torso and 0.82 +/- 0.53 cm for the female torso. The findings of our study indicate that a location of the minimum in EPMs computed using the inverse solution can offer non-invasive means for pre-interventional planning of the ablative treatment.

Noninvasive electrocardiographic imaging: reconstruction of epicardial potentials, electrograms, and isochrones and localization of single and multiple electrocardiac events

BACKGROUND

The goal of noninvasive electrocardiographic imaging (ECGI) is to determine electric activity of the heart by reconstructing maps of epicardial potentials, excitation times (isochrones), and electrograms from data measured on the body surface.

METHODS AND RESULTS

Local electrocardiac events were initiated by pacing a dog heart in a human torso-shaped tank. Body surface potential measurements (384 electrodes) were used to compute epicardial potentials noninvasively. The accuracy of reconstructed epicardial potentials was evaluated by direct comparison to measured ones (134 electrodes). Protocols included pacing from single sites and simultaneously from two sites with various intersite distances. Body surface potentials showed a single minimum for both single- and double-site pacing (intersite distances of 52, 35, and 17 mm). Noninvasively reconstructed epicardial electrograms, potentials, and isochrones closely approximated the measured ones. Single pacing sites were reconstructed to within < or = 10 mm of their measured positions. Dual sites were located accurately and resolved for the above intersite distances. Regions of sparse and crowded isochrones, indicating spatial nonuniformities of epicardial activation spread, were also reconstructed.

CONCLUSIONS

The study demonstrates that ECGI can reconstruct epicardial potentials, electrograms, and isochrones over the entire epicardial surface during the cardiac cycle. It can provide detailed information on local activation of the heart noninvasively. Its uses could include localization of cardiac electric events (eg, ectopic foci), characterization of nonuniformities of conduction, characterization of repolarization properties (eg, dispersion), and mapping of dynamically changing arrhythmias (eg, polymorphic VT) on a beat-by-beat basis.

A bioelectric inverse imaging technique based on surface Laplacians

A new approach is proposed to solve bioelectric inverse problems by employing the surface Laplacian of the bioelectrical potential. A theoretical investigation was conducted to test the feasibility of epicardial inverse imaging of cardiac electrical activity. A two-sphere homogeneous volume conductor model, where the inner sphere represents the epicardium and the outer sphere the body surface, was used. Radial and tangential current dipoles were used to approximate localized wavefronts propagating from the endocardium to the epicardium, and ectopic myocardial activities. The epicardial potential distribution was reconstructed from the body surface Laplacians with the aid of the Tikhonov zero-order regularization technique, which then was compared with the results obtained from the body surface potentials using the same regularization scheme. The two inverse solutions were compared qualitatively via visual inspection of the reconstructed epicardial potential maps, and quantitatively by examining relative errors and correlation coefficients between the "true" and the reconstructed epicardial potentials. Both qualitative and quantitative results indicate that the surface Laplacians play a positive role in improving the ill-posed nature of the bioelectric inverse problem, which would enhance our capability of reconstructing important epicardial events such as extrema in the epicardial potential distribution. The present theoretical study suggests that the Laplacian-based inverse imaging technique may have important applications to epicardial inverse imaging and other bioelectric inverse imaging.

Generalized eigensystem techniques for the inverse problem of electrocardiography applied to a realistic heart-torso geometry

We have previously proposed two novel solutions to the inverse problem of electrocardiography, the generalized eigensystem technique (GES) and the modified generalized eigensystem technique (tGES), and have compared these techniques with other numerical techniques using both homogeneous and inhomogeneous eccentric spheres model problems. In those studies we found our generalized eigensystem approaches generally gave superior performance over both truncated singular value decomposition (SVD) and zero-order Tikhonov regularization (TIK). In this paper we extend the comparison to the case of a realistic heart-torso geometry. With this model, the GES and tGES approaches again provide smaller relative errors between the true potentials and the numerically derived potentials than the other methods studied. In addition, the isopotential maps recovered using GES and tGES appear to be more accurate than the maps recovered using either SVD and TIK.

A new method for myocardial activation imaging

Noninvasive images of the myocardial activation sequence are acquired, based on a new formulation of the inverse problem of electrocardiography in terms of the critical points of the ventricular surface activation map. It is shown that the method is stable with respect to substantial amounts of correlated noise common in the measurements and modeling of electrocardiography and that problems associated with conventional regularization techniques can be circumvented. Examples of application of the method to measured human data are presented. This first invasive validation of results compares well to previously published results obtained by using a standard approach. The method can provide additional constraints on, and thus improve, traditional methods aimed at solving the inverse problem of electrocardiography.

Regional regularization of the electrocardiographic inverse problem: a model study using spherical geometry

This study examines the use of a new regularization scheme, called regional regularization, for solving the electrocardiographic inverse problem. Previous work has shown that different time frames in the cardiac cycle require varying degrees of regularization. This reflects differences in potential magnitudes, gradients, signal-to-noise ratio (SNR), and locations of electrical activity. One might expect, therefore, that a single regularization parameter and a uniform level of regularization may also be insufficient for a single potential map of a single time frame because in one map there are regions of high and low potentials and potential gradients. Regional regularization is a class of methods that subdivides a given potential map into functional "regions" based on the spatial characteristics of the potential ("spatial frequencies"). These individual regions are regularized separately and recombined into a complete map. This paper examines the hypothesis that such regionally regularized maps are more accurate than if all regions were taken together and solved with an averaged level of regularization. In a homogeneous concentric spheres model, Legendre polynomials are used to decompose a torso potential map into a set of submaps, each with a different degree of spatial variation. The original torso map is contaminated with data noise, or geometrical error or both, and regional regularization improves the epicardial potential reconstruction by up to 25% [relative error (RE)]. Regional regularization also improves the reconstructed location of peaks. A practical goal is to extend the application of this method to the realistic torso geometry, but because Legendre decomposition is limited to geometries with spherical symmetry, other methods of map decomposition must be found. Singular value decomposition (SVD) is used to decompose the maps into component parts. Its individual submaps also have different levels of spatial variation; moreover, it is generalizable to any vector, does not require spherical symmetry, and is extremely efficient numerically. Using SVD decomposition for regional regularization, significant improvement was achieved in the map quality in the presence of data noise.

A new method for regularization parameter determination in the inverse problem of electrocardiography

Computing the potentials on the heart's epicardial surface from the body surface potentials constitutes one form of the inverse problem of electrocardiography. An often-used approach to overcoming the ill-posed nature of the inverse problem and stabilizing the solution is via zero-order Tikhonov regularization, where the squared norms of both the surface potential residual and the solution are minimized, with a relative weight determined by a so-called regularization parameter. This paper looks at the composite residual and smoothing operator (CRESO) and L-curve methods currently used to determine a suitable value for this regularization parameter, t, and proposes a third method that works just as well and is much simpler to compute. This new zero-crossing method selects t such that the squared norm of the surface potential residual is equal to t times the squared norm of the solution. Its performance was compared with those of the other two methods, using three stimulation protocols of increasing complexity. The first of these protocols involved a concentric spheres model for the heart and torso and three current dipoles placed inside the inner sphere as the source distribution. The second replaced the spheres with realistic epicardial and torso geometries, while keeping the three-dipole source configuration. The final simulation kept the realistic epicardial and torso geometries, but used epicardial potential distributions corresponding to both normal and ectopic activation of the heart as the source model. Inverse solutions were computed in the presence of both geometry noise, involving assumed erroneous shifts in the heart position, and of Gaussian measurement noise added to the torso surface potentials. It was verified that in an idealistic situation, in which correlated geometry noise dominated the uncorrelated Gaussian measurement noise, only the CRESO approach arrived at a value for t. Both L-curve and zero-crossing approaches did not work. Once measurement noise dominated geometry noise, all three approaches resulted in comparable t values. It was also shown, however, that often under low measurement noise conditions none of the three resulted in an optimum solution.

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

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

Clinical applications of BSM

Body surface mapping (BSM) has now become a feasible clinical technique, providing useful information applicable to the diagnosis of cardiac arrhythmias and their treatment by surgical and endocardial catheter ablation. In WPW patients, validation of preexcitation patterns has been obtained by computer simulation and by direct epicardial mapping at surgery. BSM pacemapping has subsequently been developed to be used during radiofrequency catheter ablation. This method has been evaluated prospectively and its predictive accuracy assessed. The recognition of two distinct BSM patterns in idiopathic ventricular tachycardia, has led to the application of successful pacemapping for radiofrequency catheter ablation. The use of a realistic tri-dimensional heart-torso computer model has shown that specific sites of endocardial stimulation are related to distinct thoracic map patterns.

Impedance tomography: computational analysis based on finite element models of a cylinder and a human thorax

A direct image reconstruction method of electrical impedance tomography (EIT) is evaluated using three-dimensional (3-D) finite element models of cylindrical and torso-shaped volume conductors. The cylindrical model is used to examine the effect of electrode configurations and the sensitivity to off-plane objects and to noise in the measured data. It is also used to validate the modeling procedures by comparison with experimental data acquired from a similar cylindrical tank filled with saline. Simulation results show only minor differences in performance between the various electrode configurations. In the second part, a realistic human thorax model constructed from CT images is used to evaluate monitoring of pulmonary edema by EIT. The conductivity, volume, and vertical position of an abnormal region in the lungs are varied to simulate the progress of edema. Dynamic EIT images are reconstructed from data computed for the inhomogeneous thorax (heart and lungs) as the reference set and a realistic amount of noise is added to reproduce the conditions in which the technique would be used in practice. Simulation results show that a 10 ml edema region with a conductivity equal to that of blood can be detected at a 40 dB signal-to-noise ratio (SNR). Detection of a smaller volume, in the order of 2 ml, should be possible by improving either the instrumentation to achieve 60 dB SNR or the performance of reconstruction algorithms.