Heart Surface Potentials Research Articles

Influence of the Tikhonov Regularization Parameter on the Accuracy of the Inverse Problem in Electrocardiography.

The electrocardiogram (ECG) is the standard method in clinical practice to non-invasively analyze the electrical activity of the heart, from electrodes placed on the body's surface. The ECG can provide a cardiologist with relevant information to assess the condition of the heart and the possible presence of cardiac pathology. Nonetheless, the global view of the heart's electrical activity given by the ECG cannot provide fully detailed and localized information about abnormal electrical propagation patterns and corresponding substrates on the surface of the heart. Electrocardiographic imaging, also known as the inverse problem in electrocardiography, tries to overcome these limitations by non-invasively reconstructing the heart surface potentials, starting from the corresponding body surface potentials, and the geometry of the torso and the heart. This problem is ill-posed, and regularization techniques are needed to achieve a stable and accurate solution. The standard approach is to use zero-order Tikhonov regularization and the L-curve approach to choose the optimal value for the regularization parameter. However, different methods have been proposed for computing the optimal value of the regularization parameter. Moreover, regardless of the estimation method used, this may still lead to over-regularization or under-regularization. In order to gain a better understanding of the effects of the choice of regularization parameter value, in this study, we first focused on the regularization parameter itself, and investigated its influence on the accuracy of the reconstruction of heart surface potentials, by assessing the reconstruction accuracy with high-precision simultaneous heart and torso recordings from four dogs. For this, we analyzed a sufficiently large range of parameter values. Secondly, we evaluated the performance of five different methods for the estimation of the regularization parameter, also in view of the results of the first analysis. Thirdly, we investigated the effect of using a fixed value of the regularization parameter across all reconstructed beats. Accuracy was measured in terms of the quality of reconstruction of the heart surface potentials and estimation of the activation and recovery times, when compared with ground truth recordings from the experimental dog data. Results show that values of the regularization parameter in the range (0.01-0.03) provide the best accuracy, and that the three best-performing estimation methods (L-Curve, Zero-Crossing, and CRESO) give values in this range. Moreover, a fixed value of the regularization parameter could achieve very similar performance to the beat-specific parameter values calculated by the different estimation methods. These findings are relevant as they suggest that regularization parameter estimation methods may provide the accurate reconstruction of heart surface potentials only for specific ranges of regularization parameter values, and that using a fixed value of the regularization parameter may represent a valid alternative, especially when computational efficiency or consistency across time is required.

Localization of magnetocardiographic sources for myocardial infarction cases using deterministic and Bayesian approaches

In this paper, the inverse problems of cardiac sources using analytical and probabilistic methods are solved and discussed. The standard Tikhonov regularization technique is solved initially to estimate the under-determined heart surface potentials from Magnetocardiographic (MCG) signals. The results of the deterministic method subjected to noise in the measurements are discussed and compared with the probabilistic models. Hierarchical Bayesian modeling with fixed Gaussian prior is employed to quantify the uncertainties in source reconstructions. A novel application of Variational Bayesian inference approach has been presented to estimate the heart sources. The reconstruction results of Variational Bayesian model with non-stationary priors are compared with solutions of simplistic Bayesian approach; and the performances are evaluated using Root Mean Square Error (RMSE) and correlation co-efficient metrics. The Bayesian solutions in the study are also extended to localize the MCG sources for two types of Myocardial infarction cases.

Impulse Data Models for the Inverse Problem of Electrocardiography

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 electrocardiography into regression problems, constraining the solution space by decomposing signals with multidimensional Gaussian impulse basis functions. 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. 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. 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. 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.

Which bidomain conductivity is the most important for modelling heart and torso surface potentials during ischaemia?

Mathematical simulations using the bidomain model, which represents cardiac tissue as consisting of an intracellular and an extracellular space, are a key approach that can be used to improve understanding of heart conditions such as ischaemia. However, key inputs to these models, such as the bidomain conductivity values, are not known with any certainty. Since efforts are underway to measure these values, it would be useful to be able to quantify the effect on model outputs of uncertainty in these inputs, and also to determine, if possible, which are the most important values to focus on in experimental studies. Our previous work has systematically studied the sensitivity of heart surface potentials to the bidomain conductivity values, and this was performed using a half-ellipsoidal model of the left ventricle. This study uses a bi-ventricular heart in a torso model and this time looks at the sensitivity of the torso surface potentials, as well as the heart surface potentials, to various conductivity values (blood, torso and the six bidomain conductivities). We found that both epicardial and torso potentials are the most sensitive to the intracellular longitudinal (along the cardiac fibres) conductivity (gil) with more minor sensitivity to the torso conductivity, and that changes in gil have a significant effect on the surface potential distributions on both the torso and the heart.

Cardiac Activation Maps Reconstruction: A Comparative Study Between Data-Driven and Physics-Based Methods.

One of the essential diagnostic tools of cardiac arrhythmia is activation mapping. Noninvasive current mapping procedures include electrocardiographic imaging. It allows reconstructing heart surface potentials from measured body surface potentials. Then, activation maps are generated using the heart surface potentials. Recently, a study suggests to deploy artificial neural networks to estimate activation maps directly from body surface potential measurements. Here we carry out a comparative study between the data-driven approach DirectMap and noninvasive classic technique based on reconstructed heart surface potentials using both Finite element method combined with L1-norm regularization (FEM-L1) and the spatial adaptation of Time-delay neural networks (SATDNN-AT). In this work, we assess the performance of the three approaches using a synthetic single paced-rhythm dataset generated on the atria surface. The results show that data-driven approach DirectMap quantitatively outperforms the two other methods. In fact, we observe an absolute activation time error and a correlation coefficient, respectively, equal to 7.20 ms, 93.2% using DirectMap, 14.60 ms, 76.2% using FEM-L1 and 13.58 ms, 79.6% using SATDNN-AT. In addition, results show that data-driven approaches (DirectMap and SATDNN-AT) are strongly robust against additive gaussian noise compared to FEM-L1.

Improving Localization of Cardiac Geometry Using ECGI.

Electrocardiographic imaging (ECGI) requires a model of the torso, and inaccuracy in the position of the heart is a known source of error. We previously presented a method to localize the heart when body and heart surface potentials are known. The goal of this study is to extend this approach to only use body surface potentials. We used an iterative coordinate descent optimization to estimate the positions of the heart for several consecutive heartbeats relying on the assumption that the epicardial potential sequence is the same in each beat. The method was tested with data synthesized using measurements from a isolated-heart, torso-tank preparation. Improvement was evaluated in terms of both heart localization and ECGI accuracy. The geometric correction resulted in cardiac geometries closely matching ground truth geometry. ECGI accuracy increased dramatically by all metrics using the corrected geometry. Future studies will employ more realistic animal models and then human subjects. Success could impact clinical ECGI by reducing errors from respiratory movement and perhaps decrease imaging requirements, reducing both cost and logistical difficulty of ECGI, widening clinical applicability.

Spatiotemporal regularization for inverse ECG modeling

Advanced sensing such as the wearable sensor network provides an unprecedented opportunity to capture a wealth of information pertinent to space-time electrical activity of the heart, and facilitate the inverse electrocardiographic (ECG) modeling with the readily available data of body surface potential mapping. However, it is often challenging to derive heart-surface potentials from body-surface measurements, which is called the “inverse ECG problem.” Traditional regression is not concerned about spatiotemporal dynamic variables in complex geometries, and tends to be limited in the ability to handle high-dimensional spatiotemporal data for solving the inverse ECG problem. This paper presents a comparison study of regularization methods in the performance to achieve robust solutions of the inverse ECG problem. We first introduce the forward and inverse ECG problems. Second, we propose two spatiotemporal regularization (STRE) models to increase the robustness of inverse ECG modeling. Finally, case studies are conducted on the two-sphere geometry, as well as a real-world torso-heart geometry to evaluate the performance of different regularization methods. Experimental results show that STRE models effectively tackle the ill-conditioned inverse ECG problem and yield 56.3% and 67.3% performance improvement compared to the traditional Tikhonov regularization in the two-sphere and the torso-heart geometries, respectively. The spatiotemporal regularization methodology is shown to have strong potential to achieve robust solutions for high-dimensional predictive modeling in the inverse ECG problem.

Tracking the Position of the Heart From Body Surface Potential Maps and Electrograms.

The accurate generation of forward models is an important element in general research in electrocardiography, and in particular for the techniques for ElectroCardioGraphic Imaging (ECGI). Recent research efforts have been devoted to the reliable and fast generation of forward models. However, these model can suffer from several sources of inaccuracy, which in turn can lead to considerable error in both the forward simulation of body surface potentials and even more so for ECGI solutions. In particular, the accurate localization of the heart within the torso is sensitive to movements due to respiration and changes in position of the subject, a problem that cannot be resolved with better imaging and segmentation alone. Here, we propose an algorithm to localize the position of the heart using electrocardiographic recordings on both the heart and torso surface over a sequence of cardiac cycles. We leverage the dependency of electrocardiographic forward models on the underlying geometry to parameterize the forward model with respect to the position (translation) and orientation of the heart, and then estimate these parameters from heart and body surface potentials in a numerical inverse problem. We show that this approach is capable of localizing the position of the heart in synthetic experiments and that it reduces the modeling error in the forward models and resulting inverse solutions in canine experiments. Our results show a consistent decrease in error of both simulated body surface potentials and inverse reconstructed heart surface potentials after re-localizing the heart based on our estimated geometric correction. These results suggest that this method is capable of improving electrocardiographic models used in research settings and suggest the basis for the extension of the model presented here to its application in a purely inverse setting, where the heart potentials are unknown.

A machine learning approach to reconstruction of heart surface potentials from body surface potentials.

Invasive cardiac catheterisation is a precursor to ablation therapy for ventricular tachycardia. Invasive cardiac diagnostics are fraught with risks. Decades of research has been conducted on the inverse problem of electrocardiography, which can be used to reconstruct Heart Surface Potentials (HSPs) from Body Surface Potentials (BSPs), for non-invasive cardiac diagnostics. State of the art solutions to the inverse problem are unsatisfactory, since the inverse problem is known to be ill-posed. In this paper we propose a novel approach to reconstructing HSPs from BSPs using a Time-Delay Artificial Neural Network (TDANN). We first design the TDANN architecture, and then develop an iterative search space algorithm to find the parameters of the TDANN, which results in the best overall HSP prediction. We use recorded BSPs and HSPs from individuals suffering from serious cardiac conditions to validate our TDANN. The results are encouraging, in that the predicted and recorded HSPs have an average correlation coefficient of 0.7 under diseased conditions.

Localization of ventricular activation origin using patient-specific geometry: Preliminary results.

Catheter ablation of ventricular tachycardia (VT) may include induction of VT and localization of VT-exit site. Our aim was to assess localization performance of a novel statistical pace-mapping method and compare it with performance of an electrocardiographic inverse solution. Seven patients undergoing ablation of VT (4 with epicardial, 3 with endocardial exit) aided by electroanatomic mapping underwent intraprocedural 120-lead body-surface potential mapping (BSPM). Two approaches to localization of activation origin were tested: (1) A statistical method, based on multiple linear regression (MLR), which required only the conventional 12-lead ECG for a sufficient number of pacing sites with known origin together with patient-specific geometry of the endocardial/epicardial surface obtained by electroanatomic mapping; and (2) a classical deterministic inverse solution for recovering heart-surface potentials, which required BSPM and patient-specific geometry of the heart and torso obtained via computed tomography (CT). For the MLR method, at least 10-15 pacing sites with known coordinates, together with their corresponding 12-lead ECGs, were required to derive reliable patient-specific regression equations, which then enabled accurate localization of ventricular activation with unknown origin. For 4 patients who underwent epicardial mapping, the median of localization error for the MLR was significantly lower than that for the inverse solution (10.6vs. 27.3mm, P=0.034); a similar result held for 3 patients who underwent endocardial mapping (7.7vs. 17.1mm, P=0.017). The pooled localization error for all epicardial and endocardial sites was also significantly smaller for the MLR compared with the inverse solution (P=0.005). The novel pace-mapping approach to localizing the origin of ventricular activation offers an easily implementable supplement and/or alternative to the preprocedure inverse solution; its simplicity makes it suitable for real-time applications during clinical catheter-ablation procedures.

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

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

Physiology-based regularization of the electrocardiographic inverse problem

The inverse problem of electrocardiography aims at noninvasively reconstructing electrical activity of the heart from recorded body-surface electrocardiograms. A crucial step is regularization, which deals with ill-posedness of the problem by imposing constraints on the possible solutions. We developed a regularization method that includes electrophysiological input. Body-surface potentials are recorded and a computed tomography scan is performed to obtain the torso–heart geometry. Propagating waveforms originating from several positions at the heart are simulated and used to generate a set of basis vectors representing spatial distributions of potentials on the heart surface. The real heart-surface potentials are then reconstructed from the recorded body-surface potentials by finding a sparse representation in terms of this basis. This method, which we named ‘physiology-based regularization’ (PBR), was compared to traditional Tikhonov regularization and validated using in vivo recordings in dogs. PBR recovered details of heart-surface electrograms that were lost with traditional regularization, attained higher correlation coefficients and led to improved estimation of recovery times. The best results were obtained by including approximate knowledge about the beat origin in the PBR basis.

Computational assessment of drug‐induced effects on the electrocardiogram: from ion channel to body surface potentials

Background and PurposeUnderstanding drug effects on the heart is key to safety pharmacology assessment and anti-arrhythmic therapy development. Here our goal is to demonstrate the ability of computational models to simulate the effect of drug action on the electrical activity of the heart, at the level of the ion-channel, cell, heart and ECG body surface potential.Experimental ApproachWe use the state-of-the-art mathematical models governing the electrical activity of the heart. A drug model is introduced using an ion channel conductance block for the hERG and fast sodium channels, depending on the IC50 value and the drug dose. We simulate the ECG measurements at the body surface and compare biomarkers under different drug actions.Key ResultsIntroducing a 50% hERG-channel current block results in 8% prolongation of the APD90 and 6% QT interval prolongation, hERG block does not affect the QRS interval. Introducing 50% fast sodium current block prolongs the QRS and the QT intervals by 12% and 5% respectively, and delays activation times, whereas APD90 is not affected.Conclusions and ImplicationsBoth potassium and sodium blocks prolong the QT interval, but the underlying mechanism is different: for potassium it is due to APD prolongation; while for sodium it is due to a reduction of electrical wave velocity. This study shows the applicability of in silico models for the investigation of drug effects on the heart, from the ion channel to the ECG-based biomarkers.

Comparison of epicardial potential maps derived from the 12-lead electrocardiograms with scintigraphic images during controlled myocardial ischemia

Our aim was to cross-validate electrocardiographic (ECG) and scintigraphic imaging of acute myocardial ischemia. The former method was based on inverse calculation of heart-surface potentials from the body-surface ECGs, and the latter, on a single photon emission computed tomography (SPECT). A boundary-element torso model with 352 body-surface and 202 heart-surface nodes was used to perform the ECG inverse solution. Potentials at 352 body-surface nodes were calculated from those acquired at 12-lead ECG measurement sites using regression coefficients developed from a design set (n = 892) of body-surface potential mapping (BSPM) data. The test set (n = 18) consisted of BSPM data from patients who underwent a balloon-inflation angioplasty of either the left anterior descending coronary artery (LAD) (n = 7), left circumflex coronary artery (LCx) (n = 2), or the right coronary artery (RCA) (n = 9). Body-surface potential mapping distributions at J point for 352 nodes were estimated from the 12-lead ECG, and an agreement with those estimated from 120 leads was assessed by a correlation coefficient (CC) (in percent). These estimates yielded very similar BSPM distributions, with a CC of 91.0% ± 8.1% (mean ± SD) for the entire test set and 94.1% ± 1.4%, 96.7% ± 0.8%, and 87.4% ± 10.3% for LAD, LCx, and RCA subgroups, respectively. Corresponding heart-surface potential distributions obtained by inverse solution correlated with a lower CC of 69.3% ± 18.0% overall and 73.7% ± 10.8%, 84.7% ± 1.1%, and 62.6% ± 21.8%, respectively, for subgroups. Bull's-eye displays of heart-surface potentials calculated from estimated BSPM distributions had an area of positive potentials that qualitatively corresponded, in general, with the underperfused territory suggested by SPECT images. For the LAD and LCx groups, all 9 ECG-derived bull's-eye images indicated the expected territory; for the RCA group, 6 of 9 ECG-derived images were as expected; 2 of 3 misclassified cases had very small ECG changes in response to coronary-artery occlusion, and their SPECT images showed indiscernible patterns. In conclusion, our findings demonstrate that noninvasive ECG imaging based on just the 12-lead ECG might provide useful estimates of the regions of myocardial ischemia that agree with those provided by scintigraphic techniques.

The transfer matrix for epicardial potential in a piece-wise homogeneous thorax model: the boundary element formulation

In epicardial potential imaging, the epicardial potential is reconstructed computationally from the measured body surface potential. The transfer function that relates the heart and body surface potentials is commonly constructed with some point-collocation-weighted boundary element technique, assuming an electrically homogeneous volume conductor. This assumption causes modeling errors. In this study, the system of surface integral equations that describes the relationship between the heart and body surface potentials is thoroughly derived in a piece-wise homogeneous volume conductor. The equations are discretized with the method of weighted residuals, enabling the use of Galerkin weighting in the numerical solution of the equations. The construction of the transfer matrix is described in detail for constant and linear collocation and Galerkin methods, and the resulting forward transfer matrices are validated via simple numerical simulations. The linear Galerkin method is found to generate the smallest errors. The presented method increases the accuracy of the forward-computed body surface potential and thus prepares the way for more accurate inverse reconstructions of epicardial potential.

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.

Computing and visualizing electric potentials and current pathways in the thorax

The long-term goal of electrocardiography is to relate electric potentials on the body surface with activities in the heart. Many previously reported studies have focused on direct links between heart and body surface potentials. The goals of this study were first to validate computational methods of determining volume potentials and currents with high-resolution experimental measurements and then to use interactive visualization of thoracic currents to understand features of the electrocardiographic fields from measured cardiac sources. We developed both simulation and experimental studies based on a realistic shaped torso phantom containing an isolated, perfused dog heart. Interventions included atrial pacing, single pacing and simultaneously pacing at multiple locations on the ventricles. Simulated torso volume potentials closely matched measured potentials in the torso-tank preparation (mean correlation coefficients of 0.95). Simulation further provided a means of estimating the current field in the torso from the computed torso volume potentials and the local geometric and conductive properties of the medium. Applying these techniques to the torso electric fields under a variety of pacing conditions, we have further demonstrated that thoracic current can provide many insights into the relationship between heart surface potential and body surface potentials. Specifically, we have shown that geometric factors including cardiac source configuration and location play an important role in determining to what extent electric activity in the heart is directly visible on the body surface electrocardiogram. The computation and visualization toolkit we developed in this study to explore current fields associated with cardiac events may provide new insights into electrocardiology.

Adaptive local regularization methods for the inverse ECG problem.

One of the fundamental problems in theoretical electrocardiography can be characterized by an inverse problem. We present new methods for achieving better estimates of heart surface potential distributions in terms of torso potentials through an inverse procedure. First, we outline an automatic adaptive refinement algorithm that minimizes the spatial discretization error in the transfer matrix, increasing the accuracy of the inverse solution. Second, we introduce a new local regularization procedure, which works by partitioning the global transfer matrix into sub-matrices, allowing for varying amounts of smoothing. Each submatrix represents a region within the underlying geometric model in which regularization can be specifically 'tuned' using an a priori scheme based on the L-curve method. This local regularization method can provide a substantial increase in accuracy compared to global regularization schemes. Within this context of local regularization, we show that a generalized version of the singular value decomposition (GSVD) can further improve the accuracy of ECG inverse solutions compared to standard SVD and Tikhonov approaches. We conclude with specific examples of these techniques using geometric models of the human thorax derived from MRI data.

Computation of Heart Surface Potentials Using the Surface Source Model

The bidomain model of the heart leads to the result that the volume density of cardiac current source moment is proportional to the gradient of the macroscopic transmembrane action potential distribution. If the anisotropy ratios of the inner and outer domains (syncytia) of the myocardium are equal, then the volume distribution of cardiac sources can be replaced by an appropriate double layer on the heart surface. The double layer source distribution (heart surface source model) provides a basis for calculating heart surface potentials from cardiac sources. The heart surface model was used to calculate epicardial potentials for the normal heart as well as for a case of ischemia and of infarction. The model was also used to determine the effect of insulating the heart surface. Insulating the heart surface caused an almost fourfold increase in peak-to-peak amplitude of simulated electrograms, with little change in waveshape. Simulated electrograms showed good agreement with recorded electrograms reported in the literature. The heart surface source model appears to provide a basis for relating heart surface potentials to the distribution of cellular action potential.

The surface source model of the electrocardiogram

The bi-syncytial model of the heart leads to the result that the volume density of cardiac current source moment is proportional to the gradient of the macroscopic transmembrane action potential distribution. If the anisotropy ratios of inner and outer domains (syncytia) of the myocardium are equal, then the volume distribution of cardiac sources can be replaced by an appropriate double layer on the heart surface. For an intramyocardial electrode there is an additional term proportional to the transmembrane potential. The double layer source distribution (heart surface source model) provides a basis for calculating heart surface potentials from cardiac sources.