Representation of cardiac electrical activity by a moving dipole for normal and ectopic beats in the intact dog.

The inverse problem in electrocardiography involves determining, noninvasively, epicardial potentials and activation isochrones from measured body surface potentials. Although much theoretical work and some preliminary clinical work has been done in this area. One major obstacle to bringing this technique into the clinical arena involves determining, noninvasively, body surface and epicardial surface geometry. Clinical procedures coupling body surface potential mapping (BSPM) with computer tomography (CT)-yielded body surface and epicardial geometries, as well as BSPM electrode position, in six healthy volunteers, accurately and reproducibly. As part of these procedures, CT scans localized electrode position on the body surface of a 224 electrode BSPM vest. >

Localization of the initial site of cardiac ectopic activity has direct clinical benefits for treating focal cardiac arrhythmias. The aim of the present study is to experimentally evaluate the performance of the equivalent moving dipole technique on noninvasively localizing the origin of the cardiac ectopic activity from the recorded body surface potential mapping (BSPM) data in a well-controlled experimental setting. The cardiac ectopic activities were induced in four well-controlled intact pigs by either single-site pacing or dual-site pacing within the ventricles. In each pacing study, the initiation sites of cardiac ectopic activity were localized by estimating the locations of a single moving dipole (SMD) or two moving dipoles (TMDs) from the measured BSPM data and compared with the precise pacing sites (PSs). For the single-site pacing, the averaged SMD localization error was 18.6 ± 3.8 mm over 16 sites, while the averaged distance between the TMD locations and the two corresponding PSs was slightly larger (24.9 ± 6.2 mm over five pairs of sites), both occurring at the onset of the QRS complex (10-25 ms following the pacing spike). The obtained SMD trajectories originated near the stimulus site and then traversed across the heart during the ventricular depolarization. The present experimental results show that the initial location of the moving dipole can provide the approximate site of origin of a cardiac ectopic activity in vivo, and that the migration of the dipole can portray the passage of an ectopic beat across the heart.

Body surface Laplacian mapping of cardiac electrical activity

The solution of the inverse problem in electrocardiography, if it can be obtained, is clinically significant since information on cardiac excitation can be obtained non‐invasively from the surface electrocardiogram. This paper describes a solution for the three‐dimensional inverse problem in electrocardiography and examines the possibility of clinical application based on a numerical example for the human torso model. In the inverse problem in electrocardiography the potential transfer matrix relating the epicardial potential distribution to the body surface potential distribution is determined, which enables estimation of the former from the latter. Owing to the smoothing effect in the human thorax, however, the potential transfer matrix is modified such that the calculated epicardial potential distribution becomes oscillatory. In our study, this problem is remedied by using the conditional least‐squares filter and applying Tikhonov's regularization. The relation between the number of electrodes, measurement accuracy of the body surface electrocardiograph and the error in estimation of the epicardial potential distribution is discussed. Based on this discussion, it is concluded that estimation of the epicardial potential distribution with practicable accuracy requires at least 180 electrodes and three‐place measurement accuracy for the body surface electrocardiogram.

The behavior of the heart is governed by electrical currents generated in the myocardium, and therefore, the study of the cardiac electrical activity is essential for the diagnosis of cardiac diseases. Electrical currents in the myocardium generate an electric field that propagates through the conductive tissues of the body to reach the torso surface and, consequently, recording the surface potential distribution provides indirect information of the myocardial behavior. The Body Surface Potential Mapping (BSPM) technique allows the noninvasive multichannel recording on the surface of the torso, providing a more complete view of the electrical activity by observing events undetectable by conventional techniques. The mathematical relationship between the myocardial electrical activity and fields recorded on the torso surface can be found by solving either the forward or the inverse problem of electrocardiography. The forward problem of electrocardiography entails the calculation of the torso potentials from the electrical activity of the heart and the 3D body model, while the inverse problem resolution allows the noninvasive reconstruction of the electrical activity of the heart from surface potentials. The inverse problem is of great importance in clinical applications since it allows estimating the electrical activity of the myocardium with only noninvasive recordings. However, inverse problem resolution is still a big challenge in electrocardiography since it is ill-posed, very unstable and has multiple solutions. In this thesis different algorithms and strategies based on the inverse problem resolution were developed and applied in the noninvasive diagnosis of ventricular and atrial arrhythmias and evaluated with mathematical cellular models and clinical data bases. The thesis focuses on the inverse problem resolution for the noninvasive reconstruction of the myocardial electrical activity for different diseases and propagation patterns, implementing a novel system for complex propagation patterns. The results obtained and propagation patterns were evaluated and classified with the corresponding optimal resolution strategy that minimizes the error and increases the stability of the system, proving its advantages and disadvantages depending on the different diseases and their activation pattern. A novel iterative method was implemented for the inverse problem dipolar resolution optimized for representing simple propagation patterns, achieving a high stability and robustness against noise by constraining the solution to a limited number of dipoles. However, propagation patterns not representable by few dipoles need to be computed with the inverse problem in terms of epicardial solutions which provide a more detailed estimation of the myocardial activity. Inverse problem resolution in the voltage and phase domains showed a good accuracy for simple and organized propagation patterns. This method allowed the noninvasive diagnosis of the Brugada syndrome or the location of ectopic focus in atrial arrhythmias by performing a parametric analysis of the electrograms morphology or the activation map reconstruction. However, mathematical and patient results presented in this thesis proved that, for complex propagation patterns like atrial fibrillation (AF), inverse solutions in the voltage and phase domains are over-smoothed and over-optimistic, simplifying the complex AF activity, leading to non-physiological results that do not match the complex intracardiac electrograms recorded in AF patients. In this thesis we proposed a novel technique for the noninvasive identification and location of high dominant frequency AF sources, based on the assumption that in many cases atrial drivers present the highest activation rate with an intermittent propagation to the rest of the tissue that activates at a slower rate. Although, voltage and phase inverse solutions for AF complex propagation patterns were over smoothed and inaccurate, the noninvasive estimation of frequency maps was significantly more accurate, allowing the identification of the AF frequency gradient and location of high frequency sources. This technique may help in planning ablation procedures, avoiding unnecessary interseptal punctures for right-to-left frequency gradients cases and facilitating the targeting of AF drivers, reducing risk and time of the clinical procedure.

The ability to relate data collected on the body surface to the electrical activity on the epicardial surface is of great importance in noninvasive detection and localization of many diseases of the heart. The computation of the potential distribution on the body surface from the potential distribution on the surface of the hearty (epicardial potentials) constitutes the forward problem in electrocardiography. Conversely, the problem of the reconstruction of epicardial potentials from body surface potentials is called the inverse problem in electrocardiography. Modeling and analysis related to these two problems are presented. >

The objective of this study is to evaluate the spatial resolution of body surface Laplacian maps (BSLMs) in localizing ventricular electrical activity by means of computer simulation. A 3-D computer heart-torso model was used to simulate cardiac electrical activity and the body surface maps. A two-site pacing protocol was used to generate two simultaneously activated myocardial events on the anterior epicardial wall and the anterior endocardial wall. Following the pacing, the BSLMs and the body surface potential maps (BSPMs) were calculated and compared with the known activation pattern. As a result, the BSLMs showed superior resolution than the BSPMs for localized initial ventricular activity. In summary, the present study suggests that body surface Laplacian mapping may provide a useful methodology for the clinical diagnosis of cardiac electrical abnormalities.

In the inverse problem in electrocardiography, the information concerning the tissue is known, and the electrical phenomenon in the heart is to be estimated using the measured body surface potential. In the already‐estimated solution of the inverse problem in electrocardiography, the forward solution is determined first, and then the backward solution is derived using the forward solution. In this procedure, an ill‐posed problem may be generated, resulting in an unstable solution, necessitating smoothing. This paper proposes a new method of solution where the backward solution is calculated directly from the surface potential. Due to the ill‐posed condition inherent in the inverse problem in electrocardiography, the potential distribution derived inside the body is oscillatory. The local spatial smoothing is applied to suppress the oscillation. The tissue information is given from Computerized Tomography through an image‐input device. Then a nonlinear network is constructed. The earlier solution method for the inverse problem in electro‐cardiography can only be applied to the linear electrical medium. The proposed method can also be applied to the nonlinear system. This paper describes the solution method for the inverse problem in electrocar‐diography taking the nonlinear conductance into consideration. Examples solved by applying this method are shown. In this method, the deficit of information by excessive smoothing can be avoided. Especially, in the precardial part, the epicardial potential close to the true distribution can be estimated. The proposed method does not require the forward solution. The computation time can be reduced. By employing a successive solution method the memory capacity can be reduced.

Cardiac resynchronization therapy (CRT) has proven efficacious in the treatment of patients with heart failure and dyssynchronous activation. Currently, we select suitable CRT candidates based on the QRS complex duration (QRSd) and morphology with left bundle branch block being the optimal substrate for resynchronization. To improve CRT response rates, recommendations emphasize attention to electrical parameters both before implant and after it. Therefore, we decided to study activation times before and after CRT on the body surface potential maps (BSPM) and to compare thus obtained results with data from electroanatomical mapping using the CARTO system. Total of 21 CRT recipients with symptomatic heart failure (NYHA II-IV), sinus rhythm, and QRSd >/=150 ms and 7 healthy controls were studied. The maximum QRSd and the longest and shortest activation times (ATmax and ATmin) were set in the BSPM maps and their locations on the chest were compared with CARTO derived time interval and site of the latest (LATmax) and earliest (LATmin) ventricular activation. In CRT patients, all these parameters were measured during both spontaneous rhythm and biventricular pacing (BVP) and compared with the findings during the spontaneous sinus rhythm in the healthy controls. QRSd was 169.7+/-12.1 ms during spontaneous rhythm in the CRT group and 104.3+/-10.2 ms after CRT (p<0.01). In the control group the QRSd was significantly shorter: 95.1+/-5.6 ms (p<0.01). There was a good correlation between LATmin(CARTO) and ATmin(BSPM). Both LATmin and ATmin were shorter in the control group (LATmin(CARTO) 24.8+/-7.1 ms and ATmin(BSPM) 29.6+/-11.3 ms, NS) than in CRT group (LATmin(CARTO) was 48.1+/-6.8 ms and ATmin(BSPM) 51.6+/-10.1 ms, NS). BVP produced shortening compared to the spontaneous rhythm of CRT recipients (LATmin(CARTO) 31.6+/-5.3 ms and ATmin(BSPM) 35.2+/-12.6 ms; p<0.01 spontaneous rhythm versus BVP). ATmax exhibited greater differences between both methods with higher values in BSPM: in the control group LATmax(CARTO) was 72.0+/-4.1 ms and ATmax (BSPM) 92.5+/-9.4 ms (p<0.01), in the CRT candidates LATmax(CARTO) reached only 106.1+/-6.8 ms whereas ATmax(BSPM) 146.0+/-12.1 ms (p<0.05), and BVP paced rhythm in CRT group produced improvement with LATmax(CARTO) 92.2+/-7.1 ms and ATmax(BSPM) 130.9+/-11.0 ms (p<0.01 before and during BVP). With regard to the propagation of ATmin and ATmax on the body surface, earliest activation projected most often frontally in all 3 groups, whereas projection of ATmax on the body surface was more variable. Our results suggest that compared to invasive electroanatomical mapping BSPM reflects well time of the earliest activation, however provides longer time-intervals for sites of late activation. Projection of both early and late activated regions of the heart on the body surface is more variable than expected, very likely due to changed LV geometry and interposed tissues between the heart and superficial ECG electrode.

The inverse problem in electrocardiography is ill-conditioned, and small noise included in the measured potentials causes large errors in the solution. Since the inverse problem is mostly described as a linear problem, the entire problem has often been treated in terms of a transfer matrix. The degree of linear independence among the vectors in the transfer matrix, which is directly related to the stability of the solution, is well represented by the singular values of the transfer matrix. By means of the singular value decomposition of the transfer matrix, the stability of solution to the inverse problem has been discussed when the potential data contain noise or the transfer matrix includes some error. We have derived expressions of maximum possible error magnification and a root-mean-square error magnification and, in terms of these parameters, found that only 4 equivalent cardiac dipoles or only 15 independent epicardial potentials can be estimated from body surface potentials when they are measured with an accuracy as high as 99 percent.

The spatial resolution (SR) of the body surface Laplacian map (BSLM) was assessed using a three-dimensional, realistically shaped, heart-torso model. The BSLMs were estimated from the body surface potential maps (BSPMs) generated by pacing different sites of the ventricle of a three-dimensional computer heart model using a novel three-dimensional spline Laplacian algorithm. Pacing was performed at a total of 88 myocardial units in five regions of the AV ring (anterior, left wall, posterior, right wall, and septum) and three regions adjacent to the AV ring in the middle anterior and posterior of the ventricles. The SR of the BSPMs and BSLMs were investigated by means of the correlation coefficient (CC) of maps. When 5 microV and 10 microV Gaussian white noises were added into the simulated BSPMs, the SR, at 36 ms after the onset of pacing, was about 5.0 +/- 1.2 mm and 5.4 +/- 1.3 mm for the BSPMs, and 3.3 +/- 0.8 mm and 4.0 +/- 0.9 mm for the BSLMs, respectively. The results of the present simulation study suggest that the BSLM has higher SR and may provide a more accurate means than the BSPMs for differentiating between the accessory pathways or the sites of other ectopic cardiac beats along the AV ring and in its neighboring regions.

Introduction: Focal atrial tachycardia is commonly treated by radio frequency ablation with an acceptable long-term success. Although the location of ectopic foci tends to appear in specific hot-spots, they can be located virtually in any atrial region. Multi-electrode surface ECG systems allow acquiring dense body surface potential maps (BSPM) for non-invasive therapy planning of cardiac arrhythmia. However, the activation of the atria could be affected by fibrosis and therefore biomarkers based on BSPM need to take these effects into account. We aim to analyze the effect of fibrosis on a BSPM derived index, and its potential application to predict the location of ectopic foci in the atria.Methodology: We have developed a 3D atrial model that includes 5 distributions of patchy fibrosis in the left atrium at 5 different stages. Each stage corresponds to a different amount of fibrosis that ranges from 2 to 40%. The 25 resulting 3D models were used for simulation of Focal Atrial Tachycardia (FAT), triggered from 19 different locations described in clinical studies. BSPM were obtained for all simulations, and the body surface potential integral maps (BSPiM) were calculated to describe atrial activations. A machine learning (ML) pipeline using a supervised learning model and support vector machine was developed to learn the BSPM patterns of each of the 475 activation sequences and relate them to the origin of the FAT source.Results: Activation maps for stages with more than 15% of fibrosis were greatly affected, producing conduction blocks and delays in propagation. BSPiMs did not always cluster into non-overlapped groups since BSPiMs were highly altered by the conduction blocks. From stage 3 (15% fibrosis) the BSPiMs showed differences for ectopic beats placed around the area of the pulmonary veins. Classification results were mostly above 84% for all the configurations studied when a large enough number of electrodes were used to map the torso. However, the presence of fibrosis increases the area of the ectopic focus location and therefore decreases the utility for the electrophysiologist.Conclusions: The results indicate that the proposed ML pipeline is a promising methodology for non-invasive ectopic foci localization from BSPM signal even when fibrosis is present.

The distributions of transmembrane voltage (TMV) within the cardiac tissue are linearly connected with the patient's body surface potential maps (BSPMs) at every time instant. The matrix describing the relation between the respective distributions is referred to as the transfer matrix. This matrix can be employed to carry out forward calculations in order to find the BSPM for any given distribution of TMV inside the heart. Its inverse can be used to reconstruct the cardiac activity non-invasively, which can be an important diagnostic tool in the clinical practice. The computation of this matrix using the finite element method can be quite time-consuming. In this work, a method is proposed allowing to speed up this process by computing an approximate transfer matrix instead of the precise one. The method is tested on three realistic anatomical models of real-world patients. It is shown that the computation time can be reduced by 50% without loss of accuracy.

We present numerical results related to the direct and the inverse problems in electrocardiography. The electrical activity of the heart is described by the bidomain equations. The electrocardiograms (ECGs) recorded in different points on the body surface are obtained by coupling the bidomain equation to a Laplace equation in the torso. The simulated ECGs are quite satisfactory. As regards the inverse problem, our goal is to estimate the parameters of the bidomain‐torso model. Here we present some preliminary results of a parameter estimation for the torso model.

The inverse problem in electrocardiography on a spherical model