The forward and inverse problems of electrocardiography

A Review of Fetal ECG Signal Processing; Issues and Promising Directions

The field of electrocardiography has been in existence for over a century, yet despite significant advances in adult clinical electrocardiography, signal processing techniques and fast digital processors, the analysis of fetal ECGs is still in its infancy. This is, partly due to a lack of availability of gold standard databases, partly due to the relatively low signal-to-noise ratio of the fetal ECG compared to the maternal ECG (caused by the various media between the fetal heart and the measuring electrodes, and the fact that the fetal heart is simply smaller), and in part, due to the less complete clinical knowledge concerning fetal cardiac function and development. In this paper we review a range of promising recording and signal processing techniques for fetal ECG analysis that have been developed over the last forty years, and discuss both their shortcomings and advantages. Before doing so, however, we review fetal cardiac development, and the etiology of the fetal ECG. A selection of relevant models for the fetal/maternal ECG mixture is also discussed. In light of current understanding of the fetal ECG, we then attempt to justify recommendations for promising future directions in signal processing, and database creation.

A computationally efficient method for determining the size and location of myocardial ischemia

The purpose of this paper is to introduce a new method for solving the inverse problem of locating ischemic regions in the heart. The electrical activity in the human heart is modeled by the bidomain equations, which can be expanded to compute the potentials on the body surface. The associated inverse problem is to use ECG recordings to gain information about ischemias. We propose an algorithm for doing this, combining the level set method with a simpler minimization problem. Instead of trying to determine the shape, as in the level set method, we simply make the approximation that the ischemia is spherical. The effects of ischemia on the electrical attributes of heart tissue are examined. The new method is tested with computer simulations on synthetic body surface potential maps (BSPMs) in a realistic geometry, using realistic values for the parameters. It is shown to be, in some respects, superior to the level set approach and may be a step toward a practical algorithm useful in medical diagnostics.

Towards a computational method for imaging the extracellular potassium concentration during regional ischemia

We investigate the possibility of using body surface potential maps to image the extracellular potassium concentration during regional ischemia. The problem is formulated as an inverse problem based on a linear approximation of the bidomain model, where we minimize the difference between the results of the model and observations of body surface potentials. The minimization problem is solved by a one-shot technique, where the original PDE system, an adjoint problem, and the relation describing the minimum, are solved simultaneously. This formulation of the problem requires the solution of a 5 x 5 system of linear partial differential equations. The performance of the model is investigated by performing tests based on synthetic data. We find that the model will in many cases detect the correct position and approximate size of the ischemic regions, while some cases are more difficult to locate. It is observed that a simple post-processing of the results produces images that are qualitatively very similar to the true solution.

Two hybrid regularization frameworks for solving the electrocardiography inverse problem

In this paper, two hybrid regularization frameworks, LSQR-Tik and Tik-LSQR, which integrate the properties of the direct regularization method (Tikhonov) and the iterative regularization method (LSQR), have been proposed and investigated for solving ECG inverse problems. The LSQR-Tik method is based on the Lanczos process, which yields a sequence of small bidiagonal systems to approximate the original ill-posed problem and then the Tikhonov regularization method is applied to stabilize the projected problem. The Tik-LSQR method is formulated as an iterative LSQR inverse, augmented with a Tikhonov-like prior information term. The performances of these two hybrid methods are evaluated using a realistic heart-torso model simulation protocol, in which the heart surface source method is employed to calculate the simulated epicardial potentials (EPs) from the action potentials (APs), and then the acquired EPs are used to calculate simulated body surface potentials (BSPs). The results show that the regularized solutions obtained by the LSQR-Tik method are approximate to those of the Tikhonov method, the computational cost of the LSQR-Tik method, however, is much less than that of the Tikhonov method. Moreover, the Tik-LSQR scheme can reconstruct the epcicardial potential distribution more accurately, specifically for the BSPs with large noisy cases. This investigation suggests that hybrid regularization methods may be more effective than separate regularization approaches for ECG inverse problems.

Truncated total least squares: a new regularization method for the solution of ECG inverse problems

The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation AX = B. In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart-lung-torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.

A Matlab library for solving quasi-static volume conduction problems using the boundary element method

The boundary element method (BEM) is commonly used in the modeling of bioelectromagnetic phenomena. The Matlab language is increasingly popular among students and researchers, but there is no free, easy-to-use Matlab library for boundary element computations. We present a hands-on, freely available Matlab BEM source code for solving bioelectromagnetic volume conduction problems and any (quasi-)static potential problems that obey the Laplace equation. The basic principle of the BEM is presented and discretization of the surface integral equation for electric potential is worked through in detail. Contents and design of the library are described, and results of example computations in spherical volume conductors are validated against analytical solutions. Three application examples are also presented. Further information, source code for application examples, and information on obtaining the library are available in the WWW-page of the library: (http://biomed.tkk.fi/BEM).

Convolutive blind source separation algorithms applied to the electrocardiogram of atrial fibrillation: study of performance

The analysis of the surface electrocardiogram (ECG) is the most extended noninvasive technique in medical diagnosis of atrial fibrillation (AF). In order to use the ECG as a tool for the analysis of AF, we need to separate the atrial activity (AA) from other cardioelectric signals. In this matter, statistical signal processing techniques, like blind source separation (BSS), are able to perform a multilead statistical analysis with the aim to obtain the AA. Linear BSS techniques can be divided in two groups depending on the mixing model: algorithms where instantaneous mixing of sources is assumed, and convolutive BSS (CBSS) algorithms. In this work, a comparison of performance between one relevant CBSS algorithm, namely Infomax, and one of the most effective independent component analysis (ICA) algorithms, namely FastICA, is developed. To carry out the study, pseudoreal AF ECGs have been synthesized by adding fibrillation activity to normal sinus rhythm. The algorithm performances are expressed by two indexes: the signal to interference ratio (SIRAA) and the cross-correlation (RAA) between the original and the estimated AA. Results empirically prove that the instantaneous mixing model is the one that obtains the best results in the AA extraction, given that the mean SIRAA obtained by the FastICA algorithm (37.6 +/- 17.0 dB) is higher than the main SIRAA obtained by Infomax (28.5 +/- 14.2 dB). Also the RAA obtained by FastICA (0.92 +/- 0.13) is higher than the one obtained by Infomax (0.78 +/- 0.16).

On the possibility for computing the transmembrane potential in the heart with a one shot method: an inverse problem

We analyze the possibility for using body surface potential maps (BSPMs), a priori information about the voltage distribution in the heart and the bidomain equations to compute the transmembrane potential throughout the myocardium. Our approach is defined in terms of an inverse problem for elliptic partial differential equations (PDEs). More precisely, we formulate it in terms of an output least squares framework in which a goal functional is minimized subject to suitable PDE constraints. The problem is highly unstable and, even under optimal recording conditions, it does not have a unique solution. We propose a methodology for stabilizing and enforcing uniqueness for this inverse problem. Moreover, a fully implicit method for solving the involved minimization problem is presented. In other words, we show how one may solve it in terms of a system consisting of three linear elliptic PDEs, i.e. we derive a so-called one shot method (also commonly referred to as an all-at-once method). Finally, our theoretical findings are illuminated by a series of numerical experiments. These examples indicate that, in the presence of regional ischemia, it might be possible to approximately recover the transmembrane potential during the resting and plateau phases of the heart cycle. This is probably due to the fact that rather accurate a priori information is available during these time intervals. The problem of computing the transmembrane potential at an arbitrary time instance during a heart beat is still an open problem.

An understanding for the abnormal spikes of the EEG simulation in a 2-d neural network

The work here presents an abnormal EEG simulation and an analysis for the abnormal spikes in the simulation by using the wavelet method. The simulation is derived from the electrophysiological model of an excitable neuron being in a disorder process. The spike wave and the multi-spike wave of the EEG morphology are reconstructed by step changes in the concentration of the intracellular calcium ions ([Ca]_i). In the further work, when the concentration of [Ca] _i is sufficiently large, the multi-spike wave can also be reconstructed and the spikes of the potentials are analyzed by the multi-layer wavelet method. The work will be helpful to understand how the EEG morphology is formed from the microcosmic viewpoint.

A simulation of the abnormal EEG morphology by the 3-d finite element method

It's usually called the calculation of potentials on the cortex as the forward problem in the nerve system. Nowadays, the viewpoints of the equivalent source being regarded as the electrical activity of an excitable neuron have been widely accepted. In this paper, a novel method is presented to simulate the abnormal EEG morphology such as the spikes with two phases in the 3-D solution space. The abnormal spikes often occur in the epileptic seizure. Here, the abnormal rhythm is regarded as dynamic activity of equivalent currents source and the head as a homogeneous sphere model as well as the solution space. The finite element method (FEM) is utilized with the help of the ANSYS7.0 software. It is concluded that the abnormal spikes are original from the abnormally discharge process in the brain and the interval of the spikes (IS) can be lengthened with the increasing in the concentration of the intracellular Ca²⁺([Ca²⁺]<inf>i</inf>). It can be helpful to understand the mechanism of the abnormal EEG morphology from the microcosmic viewpoints.

Simulation of intracardial potentials with anisotropic computer heart models

To date, most interests in simulation studies with whole heart models are in body surface or epicardial potentials in order to compare the results with measurements in experimental and clinical practices. The focus of this study is paid on intracardial potentials as they are comparable with measurements during the invasive Electrophysiological Study (EPS) and the catheter ablation, which are becoming a dominant means in the therapy of arrhythmias. In this paper, simulations are implemented based on anisotropic computer heart models and a simulation system previously developed. The heart model consists of about 50,000 discrete cell elements with 1.5 mm spatial resolution. An algorithm in computing intracardial potentials inside the four cavities of the heart is developed. The HRA (High Right Atrium), HIS (His Bundle), RVA (Right Ventricle Apex), and CS (Coronary Sinus) potentials are shown and compared between a normal heart model and a heart model with the Wolff-Parkinson-White (WPW) Syndrome. By employing the intracardial potential parameters and morphology, the position of the accessory pathway between the atria and ventricula in the WPW computer heart model can be qualitatively located. It is concluded that the simulated intracardial potentials based on an anisotropic whole heart model can well approximate the realistic intracardial potentials.

Body surface ECG signal shape dispersion

The spatial distribution of the shape of the electrocardiography (ECG) waves obtained by body surface potential mapping (BSPM) is studied, using a 64-channel high-resolution ECG system. The index associated to each lead is the shape difference between its ECG wave and a reference computed taking into account all the leads on the same column. The reference is either a selected real wave or a synthetic signal computed by integral shape averaging (ISA). Better results are obtained with the ISA signal using the distribution function method (DFM) for computing the shape difference. The spatial dispersion of ECG waves is showed to allow the separation of patients after myocardial infarction (MI) from healthy subjects. In addition, the reference signal position for each column is computed. The path linking these positions appears as an invariant, i.e., it is independent of the subject and the ECG wave.

Improved performance of bayesian solutions for inverse electrocardiography using multiple information sources

The usual goal in inverse electrocardiography (ECG) is to reconstruct cardiac electrical sources from body surface potentials and a mathematical model that relates the sources to the measurements. Due to attenuation and smoothing that occurs in the thorax, the inverse ECG problem is ill-posed and imposition of a priori constraints is needed to combat this ill-posedness. When the problem is posed in terms of reconstructing heart surface potentials, solutions have not yet achieved clinical utility; limitations include the limited availability of good a priori information about the solution and the lack of a "good" error metric. We describe an approach that combines body surface measurements and standard forward models with two additional information sources: statistical prior information about epicardial potential distributions and sparse simultaneous measurements of epicardial potentials made with multielectrode coronary venous catheters. We employ a Bayesian methodology which offers a general way to incorporate these information sources and additionally provides statistical performance analysis tools. In a simulation study, we first compare solutions using one or more of these information sources. Then, we study the effects of varying the number of sparse epicardial potential measurements on reconstruction accuracy. To evaluate accuracy, we used the Bayesian error covariance as well as traditional error metrics such as relative error. Our results show that including even sparsely sampled information from coronary venous catheters can substantially improve the reconstruction of epicardial potential distributions and that a Bayesian framework provides a feasible approach to using this information. Moreover, computing the Bayesian error standard deviations offers a means to indicate confidence in the results even in the absence of validation data.

Electrocardiographic imaging (ECGI), a novel diagnostic modality used for mapping of focal left ventricular tachycardia in a young athlete

We report the first clinical application of electrocardiographic imaging (ECGI), a new, noninvasive imaging modality for arrhythmias, in an athlete with focal ventricular tachycardia (VT) originating from a left ventricular (LV) diverticulum. A reconstructed map of the epicardial activation sequence during a single premature ventricular complex (PVC) of an identical QRS morphology to the clinical VT, generated from 224-electrode body surface ECGs and a chest CT (ECGI), localized the PVC to the site of the diverticulum. This correlated with subsequent maps obtained using standard techniques. We describe the first case that used ECGI to guide diagnosis and therapy of a clinical tachyarrhythmia.

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.

Atrial activity extraction for atrial fibrillation analysis using blind source separation

This contribution addresses the extraction of atrial activity (AA) from real electrocardiogram (ECG) recordings of atrial fibrillation (AF). We show the appropriateness of independent component analysis (ICA) to tackle this biomedical challenge when regarded as a blind source separation (BSS) problem. ICA is a statistical tool able to reconstruct the unobservable independent sources of bioelectric activity which generate, through instantaneous linear mixing, a measurable set of signals. The three key hypothesis that make ICA applicable in the present scenario are discussed and validated: 1) AA and ventricular activity (VA) are generated by sources of independent bioelectric activity; 2) AA and VA present non-Gaussian distributions; and 3) the generation of the surface ECG potentials from the cardioelectric sources can be regarded as a narrow-band linear propagation process. To empirically endorse these claims, an ICA algorithm is applied to recordings from seven patients with persistent AF. We demonstrate that the AA source can be identified using a kurtosis-based reordering of the separated signals followed by spectral analysis of the sub-Gaussian sources. In contrast to traditional methods, the proposed BSS-based approach is able to obtain a unified AA signal by exploiting the atrial information present in every ECG lead, which results in an increased robustness with respect to electrode selection and placement.

Estimation of Heart-Surface Potentials Using Regularized Multipole Sources

Direct inference of heart-surface potentials from body-surface potentials has been the goal of most recent work on electrocardiographic inverse solutions. We developed and tested indirect methods for inferring heart-surface potentials based on estimation of regularized multipole sources. Regularization was done using Tikhonov, constrained-least-squares, and multipole-truncation techniques. These multipole-equivalent methods (MEMs) were compared to the conventional mixed boundary-value method (BVM) in a realistic torso model with up to 20% noise added to body-surface potentials and +/-1 cm error in heart position and size. Optimal regularization was used for all inverse solutions. The relative error of inferred heart-surface potentials of the MEM was significantly less (p < 0.05) than that of the BVM using zeroth-order Tikhonov regularization in 10 of the 12 cases tested. These improvements occurred with a fourth-degree (24 coefficients) or smaller multipole moment. From these multipole coefficients, heart-surface potentials can be found at an unlimited number of heart-surface locations. Our indirect methods for estimating heart-surface potentials based on multipole inference appear to offer significant improvement over the conventional direct approach.

A Competitive Wavelet Network for Signal Clustering

This correspondence proposes a novel signal clustering method based on the unsupervised training of a wavelet network. The synaptic weights are parameterized by wavelet basis functions, which are adjusted by a competitive algorithm that makes use of the neighborhood concept proposed by Kohonen. The robustness of the wavelet network with respect to noise is illustrated in a simulated problem, in which dynamic systems are grouped on the basis of their step responses. An example involving clustering of electrocardiographic signals taken from the MIT-BIH database is also presented. In this case, the ability of the proposed network to perform clustering at successive resolution levels is illustrated. The possibility of interpreting the information encoded in the network at the end of training is also discussed.

Solving the inverse problem of electrocardiography using a Duncan and Horn formulation of the Kalman filter

Numeric regularization methods most often used to solve the ill-posed inverse problem of electrocardiography are spatial and ignore the temporal nature of the problem. In this paper, a Kalman filter reformulation incorporated temporal information to regularize the inverse problem, and was applied to reconstruct left ventricular endocardial electrograms based on cavitary electrograms measured by a noncontact, multielectrode probe. These results were validated against in situ electrograms measured with an integrated, multielectrode basket-catheter. A three-dimensional, probe-endocardium model was determined from multiplane fluoroscopic images. The boundary element method was applied to solve the boundary value problem and determine a linear relationship between endocardial and probe potentials. The Duncan and Horn formulation of the Kalman filter was employed and was compared to the commonly used zero- and first-order Tikhonov spatial regularization as well as the Twomey temporal regularization method. Endocardial electrograms were reconstructed during both sinus and paced rhythms. The Paige and Saunders solution of the Duncan and Horn formulation reconstructed endocardial electrograms at an amplitude relative error of 13% (potential amplitude) which was superior to solutions obtained with zero-order Tikhonov (relative error, 31%), first-order Tikhonov (relative error, 19%), and Twomey regularization (relative error, 44%). Likewise, activation time error in the inverse solution using the Duncan and Horn formulation (2.9 ms) was smaller than that of zero-order Tikhonov (4.8 ms), first-order Tikhonov (5.4 ms), and Twomey regularization (5.8 ms). Therefore, temporal regularization based on the Duncan and Horn formulation of the Kalman filter improves the solution of the inverse problem of electrocardiography.

Modeling Total Heart Function

Computational models of the electrical and mechanical function of the heart are reviewed. These models attempt to explain the integrated function of the heart in terms of ventricular anatomy, the structure and material properties of myocardial tissue, the membrane ion channels, and calcium handling and myofilament mechanics of cardiac myocytes. The models have established the computational framework for linking the structure and function of cardiac cells and tissue to the integrated behavior of the intact heart, but many more aspects of physiological function, including metabolic and signal transduction pathways, need to be included before significant progress can be made in understanding many disease processes.

Effects of experimental and modeling errors on electrocardiographic inverse formulations

The inverse problem of electrocardiology aims to reconstruct the electrical activity occurring within the heart using information obtained noninvasively on the body surface. Potentials obtained on the torso surface can be used as input for the inverse problem and an electrical image of the heart obtained. There are a number of different inverse algorithms currently used to produce electrical images of the heart. The relative performances of these inverse algorithms at this stage is largely unknown. Although there have been many simulation studies investigating the accuracy of each of these algorithms, to date, there has been no comprehensive study which compares a wide variety of inverse methods. By performing a detailed simulation study, we compare the performances of epicardial potential [Tikhonov, Truncated singular value decomposition (TSVD), and Greensite] and myocardial activation-based (critical point) inverse simulations along with different methods of choosing the appropriate level of regularization (optimal, L-curve, composite residual and smoothing operator, zero-crossing) to apply to each of these inverse methods. We also examine the effects of a variety of signal error, material property error, geometric error and a combination of these errors on each of the electrocardiographic inverse algorithms. Results from the simulation study show that the activation-based method is able to produce solutions which are more accurate and stable than potential-based methods especially in the presence of correlated errors such as geometric uncertainty. In general, the Greensite-Tikhonov method produced the most realistic potential-based solutions while the zero-crossing and L-curve were the preferred method for determining the regularization parameter. The presence of signal or material property error has little effect on the inverse solutions when compared with the large errors which resulted from the presence of any geometric error. In the presence of combined Gaussian and correlated errors representing conditions which may be encountered in an experimental or clinical environment, there was less variability between potential-based solutions produced by each of the inverse algorithms.

Geometrical aspects of the interindividual variability of multilead ECG recordings

The electrocardiogram (ECG) as measured from healthy subjects shows a considerable interindividual variability. This variability is caused by geometrical as well as by physiological factors. In this study, the relative contribution of the geometrical factors is estimated. In addition a method aimed at correcting for these factors is described. First, a measure (RV) for quantifying the overall variability is presented, and for healthy individuals its value is estimated as 0.52. Next, based on a simulation study using the individual (heart-lung-torso) geometry of 25 subjects, the variability caused by geometrical factors is estimated as 0.40, indicating that in healthy subjects the RV for healthy individuals resulting from electrophysiology is of the order of 0.33. In an evaluation of the correction procedure, applied to realistic, simulated body surface potentials, it is shown that RV caused by geometrical factors can be reduced from 0.40 to 0.06. When applying the correction procedure to measured ECG data no reduction of the RV value could be demonstrated. These results indicate that the involved procedure of the inverse computation of a cardiac equivalent source, at the present time, is of insufficient quality to cash in on the substantial reduction of RV values from 0.52 down to 0.33 that might be obtainable.

Numerical aspects of spatio-temporal current density reconstruction from EEG-/MEG-data

The determination of the sources of electric activity inside the brain from electric and magnetic measurements on the surface of the head is known to be an ill-posed problem. In this paper, a new algorithm which takes temporal a priori information modeled by the smooth activation model into account is described and compared with existing algorithms such as Tikhonov-Phillips.

A theoretical analysis of acute ischemia and infarction using ECG reconstruction on a 2-D model of myocardium

We developed a two-dimensional ventricular tissue model in order to probe the determinants of electrocardiographic (ECG) morphology during acute and chronic ischemia. Hyperkalemia was simulated by step changes in [K+]out, while acidosis was induced by reducing Na+ and Ca2+ conductances. Hypoxia was introduced by its effect on potassium activity. During the initial moments of ischemia, ECG changes were characterized by increases in QRS amplitude and ST segment shortening, followed in the advanced phase by ST baseline elevation, T conformation changes, widening of the QRS and significant decreases in QRS amplitude in spite of an enlarged Q. During each phase, potential proarrhythmic mechanisms were investigated. The presence of unexcitable regions of simulated myocardial infarction led to polymorphic ECG. We also observed a nonuniform deflection of the ST segment from beat to beat. We used similar protocols to explore the responses of infarcted myocardium after impairment resolving. We found that despite irreversible uncoupling of the necrotic region, the restored normal ionic concentrations produced an isopotential ST segment and monomorphic ECG complexes, while an enlarged Q wave was still visible. In summary, these numerical experiments indicate the possibility to track in the ECG pathologic changes following the altered electrophysiology of the ischemic heart.

Spatial regularization of the electrocardiographic inverse problem and its application to endocardial mapping

Numeric regularization methods for solving the inverse problem of electrocardiography in realistic volume conductor models have been mostly limited to uniform regularization in the spatial domain. A method of spatial regularization (SR) was developed and tested in canine, where each spatial spectral component of the volume conductor model was considered separately, and a SR operator was selected based on explicit a posteriori criterion at each time instant through the heartbeat. The inverse problem was solved in the left ventricle by reconstructing endocardial surface electrograms based on cavitary electrograms measured with the use of a noncontact, multielectrode probe. The results were validated based on electrograms measured in situ at the same endocardial locations using an integrated, multielectrode basket-catheter. A probe-endocardium three-dimensional model was determined from multiplane fluoroscopic images. The boundary element method was applied to solve the boundary value problem and derive the relationship between endocardial and probe potentials. Endocardial electrograms were reconstructed during both normal and paced rhythms using SR as well as standard, uniform, zeroth-order Tikhonov (ZOT) regularization. Compared to endocardial electrograms measured by the basket, electrograms reconstructed using SR [relative error (RE) = 0.32, correlation coefficient (CC) = 0.97, activation error = 3.3 ms] were superior to electrograms reconstructed using ZOT regularization (RE = 0.59, CC = 0.79, activation error = 4.9 ms). Therefore, regularization based on spatial spectral components of the model improves the solution of the inverse problem of electrocardiography compared to uniform regularization.

Constructing a 3-D mesh model for electrical cardiac activity simulation

The 3-D ventricle model in this study was reconstructed from a series of MRI torso cross-section data. We used a 3-D voxel array to represent the ventricle. As in cardiac simulations proposed by previous studies, the activation sequence and body surface ECG were simulated in this model. But to reduce the amount of elements in the model, so that the amount of parameters in the model can be handled numerically, we propose another approach to simulate cardiac activity. A mesh model was constructed on the closed surface formed by epicardiac and endocardiac surfaces of the ventricle. We propose a method to simulate the activation sequence on the epicardiac and endocardiac surfaces of the mesh model. As with the uniform double layer theorem, body surface ECG can be estimated in terms of epicardiac and endocardiac surface current source. Consequently, we can also generate ECG waveforms corresponding to this mesh simulation. Both the depolarization sequence and ECG simulated by the mesh model resemble those generated by the 3-D voxel model. However, the mesh model greatly simplified the process of ECG simulation. Both the simulation of depolarization and ECG estimation were expressed in terms of clear and simple mathematical representations. Consequently, we can analytically investigate the effects of the mesh model's parameters on the cardiac activation sequence and ECG. It could be a useful tool to numerically study the relation of ECG waveforms and electrical activity of the heart.

Reconstruction of 3-D geometry using 2-D profiles and a geometric prior model

A method has been developed to reconstruct three-dimensional (3-D) surfaces from two-dimensional (2-D) projection data. It is used to produce individualized boundary element models, consisting of thorax and lung surfaces, for electro- and magnetocardiographic inverse problems. Two orthogonal projections are utilized. A geometrical prior model, built using segmented magnetic resonance images, is deformed according to profiles segmented from projection images. In our method, virtual X-ray images of the prior model are first constructed by simulating real X-ray imaging. The 2-D profiles of the model are segmented from the projections and elastically matched with the profiles segmented from patient data. The displacement vectors produced by the elastic 2-D matching are back projected onto the 3-D surface of the prior model. Finally, the model is deformed, using the back-projected vectors. Two different deformation methods are proposed. The accuracy of the method is validated by a simulation. The average reconstruction error of a thorax and lungs was 1.22 voxels, corresponding to about 5 mm.

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.

Improved estimation of pericardial potentials from body-surface maps using individualized torso models

Clinical applicability of inferred pericardial potentials is limited because accuracy is significantly affected by noise in surface electrocardiograms (ECGs), errors in electrode location on the torso model, and errors in the geometry and inhomogeneities of the torso model itself. To quantify effects of electrode location and geometric errors in torso-surface models, we measured locations of 190 electrodes used in body-surface mapping of 11 adults, along with over 2,000 sites on each torso surface. Measurements were made to within 2 mm with an Immersion Personal Digitizer. To quantify effects of errors in pericardial-surface models we also estimated heart position, size, and orientation in each subject from ultrasonic images registered to the body-surface coordinates. Known pericardial potentials were taken from epicardial measurements made during QRS with a 90-electrode sock in an adult male undergoing cardiac surgery. Body-surface ECGs were calculated for each individual from the pericardial maps, using standard boundary-element methods. Accuracy of zero-order-Tikhonov inverse solutions was tested in 91-node pericardial and 1,026-node torso models, individualized for each subject. With 10 microv rms noise added to surface potentials, the optimal regularization constant at each instant in QRS gave a relative error of 0.44 +/- 0.03; it was 0.47 +/- 0.03 using the composite residual and smoothing operator (CRESO) technique. When calculated body-surface potentials from the first 10 subjects were placed at corresponding electrode positions on the torso of the eleventh subject, whose heart size and orientation was the mean of the other 10 subjects, relative error increased to 0.87 +/- 0.06 for optimal regularization. CRESO failed in the fixed torso model. Results demonstrate that a fixed model does not provide useful estimates of pericardial potentials, and that individualized models enhance the performance of techniques for the estimation of regularization parameters.

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.

An improved method for estimating epicardial potentials from the body surface

We present a new method for regularizing the illposed problem of computing epicardial potentials from body surface potentials. The method simultaneously regularizes the equations associated with all time points, and relies on a new theorem which states that a solution based on optimal regularization of each integral equation associated with each principal component of the data will be more accurate than a solution based on optimal regularization of each integral equation associated with each time point. The theorem is illustrated with simulations mimicking the complexity of the inverse electrocardiography problem. As must be expected from a method which imposes no additional a priori constraints, the new approach addresses uncorrelated noise only, and in the presence of dominating correlated noise it is only successful in producing a "cleaner" version of a necessarily compromised solution. Nevertheless, in principle, the new method is always preferred to the standard approach, since it (without penalty) eliminates pure noise that would otherwise be present in the solution estimate.

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.

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.

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.