The forward and inverse problems of electrocardiography

Unlocking the diagnostic potential of electrocardiograms through information transfer from cardiac magnetic resonance imaging

Cardiovascular diseases (CVD) can be diagnosed using various diagnostic modalities. The electrocardiogram (ECG) is a cost-effective and widely available diagnostic aid that provides functional information of the heart. However, its ability to classify and spatially localise CVD is limited. In contrast, cardiac magnetic resonance (CMR) imaging provides detailed structural information of the heart and thus enables evidence-based diagnosis of CVD, but long scan times and high costs limit its use in clinical routine. In this work, we present a deep learning strategy for cost-effective and comprehensive cardiac screening solely from ECG. Our approach combines multimodal contrastive learning with masked data modelling to transfer domain-specific information from CMR imaging to ECG representations. In extensive experiments using data from 40,044 UK Biobank subjects, we demonstrate the utility and generalisability of our method for subject-specific risk prediction of CVD and the prediction of cardiac phenotypes using only ECG data. Specifically, our novel multimodal pre-training paradigm improves performance by up to 12.19% for risk prediction and 27.59% for phenotype prediction. In a qualitative analysis, we demonstrate that our learned ECG representations incorporate information from CMR image regions of interest. Our entire pipeline is publicly available at https://github.com/oetu/MMCL-ECG-CMR.

Solving the Inverse Problem of Electrocardiography for Cardiac Digital Twins: A Survey

Cardiac digital twins (CDTs) are personalized virtual representations used to understand complex cardiac mechanisms. A critical component of CDT development is solving the ECG inverse problem, which enables the reconstruction of cardiac sources and the estimation of patient-specific electrophysiology (EP) parameters from surface ECG data. Despite challenges from complex cardiac anatomy, noisy ECG data, and the ill-posed nature of the inverse problem, recent advances in computational methods have greatly improved the accuracy and efficiency of ECG inverse inference, strengthening the fidelity of CDTs. This paper aims to provide a comprehensive review of the methods for solving ECG inverse problems, their validation strategies, their clinical applications, and their future perspectives. For the methodologies, we broadly classify state-of-the-art approaches into two categories: deterministic and probabilistic methods, including both conventional and deep learning-based techniques. Integrating physics laws with deep learning models holds promise, but challenges such as capturing dynamic electrophysiology accurately, accessing accurate domain knowledge, and quantifying prediction uncertainty persist. Integrating models into clinical workflows while ensuring interpretability and usability for healthcare professionals is essential. Overcoming these challenges will drive further research in CDTs.

A wavelet subband based LSTM model for 12-lead ECG synthesis from reduced lead set

Synthesis of a 12-lead electrocardiogram from a reduced lead set has previously been extensively studied in order to meet patient comfort, minimise complexity, and enable telemonitoring. Traditional methods relied solely on the inter-lead correlation between the standard twelve leads for learning the models. The 12-lead ECG possesses not only inter-lead correlation but also intra-lead correlation. Learning a model that can exploit this spatio-temporal information in the ECG could generate lead signals while preserving important diagnostic information. The proposed approach takes leverage of the enhanced inter-lead correlation of the ECG signal in the wavelet domain. Long-short-term memory (LSTM) networks, which have emerged as a powerful tool for sequential data mining, are a type of recurrent neural network architecture with an inherent capability to capture the spatiotemporal information of the heart signal. This work proposes the deep learning architecture that utilizes the discrete wavelet transform and the LSTM to reconstruct a generic 12-lead ECG from a reduced lead set. The experimental results are evaluated using different diagnostic measures and similarity metrics. The proposed framework is well founded, and accurate reconstruction is possible as it can capture clinically significant features and provides a robust solution against noise.

Forward problem of electrocardiography based on cardiac source vector orientations

To localize the unusual cardiac activities non-invasively, one has to build a prior forward model that relates the heart, torso, and detectors. This model has to be constructed to mathematically relate the geometrical and functional activities of the heart. Several methods are available to model the prior sources in the forward problem, which results in the lead field matrix generation. In the conventional technique, the lead field assumed the fixed prior sources, and the source vector orientations were presumed to be parallel to the detector plane with the unit strength in all directions. However, the anomalies cannot always be expected to occur in the same location and orientation, leading to misinterpretation and misdiagnosis. To overcome this, the work proposes a new forward model constructed using the VCG signals of the same subject. Furthermore, three transformation methods were used to extract VCG in constructing the time-varying lead fields to steer to the orientation of the source rather than just reconstructing its activities in the inverse problem. In addition, the unit VCG loop of the acute ischemia patient was extracted to observe the changes compared to the normal subject. The abnormality condition was achieved by delaying the depolarization time by 15ms. The results involving the unit vectors of VCG demonstrated the anisotropic nature of cardiac source orientations, providing information about the heart's electrical activity.

Non-invasive estimation of atrial fibrillation driver position using long-short term memory neural networks and body surface potentials

BACKGROUND AND OBJECTIVE

Atrial Fibrillation (AF) is a supraventricular tachyarrhythmia that can lead to thromboembolism, hearlt failure, ischemic stroke, and a decreased quality of life. Characterizing the locations where the mechanisms of AF are initialized and maintained is key to accomplishing an effective ablation of the targets, hence restoring sinus rhythm. Many methods have been investigated to locate such targets in a non-invasive way, such as Electrocardiographic Imaging, which enables an on-invasive and panoramic characterization of cardiac electrical activity using recording Body Surface Potentials (BSP) and a torso model of the patient. Nonetheless, this technique entails some major issues stemming from solving the inverse problem, which is known to be severely ill-posed. In this context, many machine learning and deep learning approaches aim to tackle the characterization and classification of AF targets to improve AF diagnosis and treatment.

METHODS

In this work, we propose a method to locate AF drivers as a supervised classification problem. We employed a hybrid form of the convolutional-recurrent network which enables feature extraction and sequential data modeling utilizing labeled realistic computerized AF models. Thus, we used 16 AF electrograms, 1 atrium, and 10 torso geometries to compute the forward problem. Previously, the AF models were labeled by assigning each sample of the signals a region from the atria from 0 (no driver) to 7, according to the spatial location of the AF driver. The resulting 160 BSP signals, which resemble a 64-lead vest recording, are preprocessed and then introduced into the network following a 4-fold cross-validation in batches of 50 samples.

RESULTS

The results show a mean accuracy of 74.75% among the 4 folds, with a better performance in detecting sinus rhythm, and drivers near the left superior pulmonary vein (R1), and right superior pulmonary vein (R3) whose mean sensitivity bounds around 84%-87%. Significantly good results are obtained in mean sensitivity (87%) and specificity (83%) in R1.

CONCLUSIONS

Good results in R1 are highly convenient since AF drivers are commonly found in this area: the left atrial appendage, as suggested in some previous studies. These promising results indicate that using CNN-LSTM networks could lead to new strategies exploiting temporal correlations to address this challenge effectively.

Hearables: feasibility of recording cardiac rhythms from single in-ear locations

The ear is well positioned to accommodate both brain and vital signs monitoring, via so-called hearable devices. Consequently, ear-based electroencephalography has recently garnered great interest. However, despite the considerable potential of hearable based cardiac monitoring, the biophysics and characteristic cardiac rhythm of ear-based electrocardiography (ECG) are not yet well understood. To this end, we map the cardiac potential on the ear through volume conductor modelling and measurements on multiple subjects. In addition, in order to demonstrate real-world feasibility of in-ear ECG, measurements are conducted throughout a long-time simulated driving task. As a means of evaluation, the correspondence between the cardiac rhythms obtained via the ear-based and standard Lead I measurements, with respect to the shape and timing of the cardiac rhythm, is verified through three measures of similarity: the Pearson correlation, and measures of amplitude and timing deviations. A high correspondence between the cardiac rhythms obtained via the ear-based and Lead I measurements is rigorously confirmed through agreement between simulation and measurement, while the real-world feasibility was conclusively demonstrated through efficacious cardiac rhythm monitoring during prolonged driving. This work opens new avenues for seamless, hearable-based cardiac monitoring that extends beyond heart rate detection to offer cardiac rhythm examination in the community.

Basis and applicability of noninvasive inverse electrocardiography: a comparison between cardiac source models

The body surface electrocardiogram (ECG) is a direct result of electrical activity generated by the myocardium. Using the body surface ECGs to reconstruct cardiac electrical activity is called the inverse problem of electrocardiography. The method to solve the inverse problem depends on the chosen cardiac source model to describe cardiac electrical activity. In this paper, we describe the theoretical basis of two inverse methods based on the most commonly used cardiac source models: the epicardial potential model and the equivalent dipole layer model. We discuss similarities and differences in applicability, strengths and weaknesses and sketch a road towards improved inverse solutions by targeted use, sequential application or a combination of the two methods.

A two-step inverse solution for a single dipole cardiac source

Introduction: The inverse problem of electrocardiography noninvasively localizes the origin of undesired cardiac activity, such as a premature ventricular contraction (PVC), from potential recordings from multiple torso electrodes. However, the optimal number and placement of electrodes for an accurate solution of the inverse problem remain undetermined. This study presents a two-step inverse solution for a single dipole cardiac source, which investigates the significance of the torso electrodes on a patient-specific level. Furthermore, the impact of the significant electrodes on the accuracy of the inverse solution is studied. Methods: Body surface potential recordings from 128 electrodes of 13 patients with PVCs and their corresponding homogeneous and inhomogeneous torso models were used. The inverse problem using a single dipole was solved in two steps: First, using information from all electrodes, and second, using a subset of electrodes sorted in descending order according to their significance estimated by a greedy algorithm. The significance of electrodes was computed for three criteria derived from the singular values of the transfer matrix that correspond to the inversely estimated origin of the PVC computed in the first step. The localization error (LE) was computed as the Euclidean distance between the ground truth and the inversely estimated origin of the PVC. The LE obtained using the 32 and 64 most significant electrodes was compared to the LE obtained when all 128 electrodes were used for the inverse solution. Results: The average LE calculated for both torso models and using all 128 electrodes was 28.8 ± 11.9 mm. For the three tested criteria, the average LEs were 32.6 ± 19.9 mm, 29.6 ± 14.7 mm, and 28.8 ± 14.5 mm when 32 electrodes were used. When 64 electrodes were used, the average LEs were 30.1 ± 16.8 mm, 29.4 ± 12.0 mm, and 29.5 ± 12.6 mm. Conclusion: The study found inter-patient variability in the significance of torso electrodes and demonstrated that an accurate localization by the inverse solution with a single dipole could be achieved using a carefully selected reduced number of electrodes.

Comparison of dipole-based and potential-based ECGI methods for premature ventricular contraction beat localization with clinical data

Introduction: Localization of premature ventricular contraction (PVC) origin to guide the radiofrequency ablation (RFA) procedure is one of the prominent clinical goals of non-invasive electrocardiographic imaging. However, the results reported in the literature vary significantly depending on the source model and the level of complexity in the forward model. This study aims to compare the paced and spontaneous PVC localization performances of dipole-based and potential-based source models and corresponding inverse methods using the same clinical data and to evaluate the effects of torso inhomogeneities on these performances. Methods: The publicly available EP solution data from the EDGAR data repository (BSPs from a maximum of 240 electrodes) with known pacing locations and the Bratislava data (BSPs in 128 leads) with spontaneous PVCs from patients who underwent successful RFA procedures were used. Homogeneous and inhomogeneous torso models and corresponding forward problem solutions were used to relate sources on the closed epicardial and epicardial-endocardial surfaces. The localization error (LE) between the true and estimated pacing site/PVC origin was evaluated. Results: For paced data, the median LE values were 25.2 and 13.9 mm for the dipole-based and potential-based models, respectively. These median LE values were higher for the spontaneous PVC data: 30.2-33.0 mm for the dipole-based model and 28.9-39.2 mm for the potential-based model. The assumption of inhomogeneities in the torso model did not change the dipole-based solutions much, but using an inhomogeneous model improved the potential-based solutions on the epicardial-endocardial ventricular surface. Conclusion: For the specific task of localization of pacing site/PVC origin, the dipole-based source model is more stable and robust than the potential-based source model. The torso inhomogeneities affect the performances of PVC origin localization in each source model differently. Hence, care must be taken in generating patient-specific geometric and forward models depending on the source model representation used in electrocardiographic imaging (ECGI).

Non-Contact Monitoring of ECG in the Home Environment-Selecting Optimal Electrode Configuration

Capacitive electrocardiography (cECG) is most often used in wearable or embedded measurement systems. The latter is considered in the paper. An optimal electrocardiographic lead, as an individual feature, was determined based on model studies. It was defined as the possibly highest value of the R-wave amplitude measured on the back of the examined person. The lead configuration was also analyzed in terms of minimizing its susceptibility to creating motion artifacts. It was found that the direction of the optimal lead coincides with the electrical axis of the heart. Moreover, the electrodes should be placed in the areas preserving the greatest voltage and at the same time characterized by the lowest gradient of the potential. Experimental studies were conducted using the developed measurement system on a group of 14 people. The ratio of the R-wave amplitude (as measured on the back and chest, using optimal leads) was less than 1 while the SNR reached at least 20 dB. These parameters allowed for high-quality QRS complex detection with a PPV of 97%. For the "worst" configurations of the leads, the signals measured were practically uninterpretable.

Generalization and Regularization for Inverse Cardiac Estimators

Electrocardiographic Imaging (ECGI) aims to estimate the intracardiac potentials noninvasively, hence allowing the clinicians to better visualize and understand many arrhythmia mechanisms. Most of the estimators of epicardial potentials use a signal model based on an estimated spatial transfer matrix together with Tikhonov regularization techniques, which works well specially in simulations, but it can give limited accuracy in some real data. Based on the quasielectrostatic potential superposition principle, we propose a simple signal model that supports the implementation of principled out-of-sample algorithms for several of the most widely used regularization criteria in ECGI problems, hence improving the generalization capabilities of several of the current estimation methods. Experiments on simple cases (cylindrical and Gaussian shapes scrutinizing fast and slow changes, respectively) and on real data (examples of torso tank measurements available from Utah University, and an animal torso and epicardium measurements available from Maastricht University, both in the EDGAR public repository) show that the superposition-based out-of-sample tuning of regularization parameters promotes stabilized estimation errors of the unknown source potentials, while slightly increasing the re-estimation error on the measured data, as natural in non-overfitted solutions. The superposition signal model can be used for designing adequate out-of-sample tuning of Tikhonov regularization techniques, and it can be taken into account when using other regularization techniques in current commercial systems and research toolboxes on ECGI.

Non-invasive Estimation of Atrial Fibrillation Driver Position With Convolutional Neural Networks and Body Surface Potentials

Atrial fibrillation (AF) is characterized by complex and irregular propagation patterns, and AF onset locations and drivers responsible for its perpetuation are the main targets for ablation procedures. ECG imaging (ECGI) has been demonstrated as a promising tool to identify AF drivers and guide ablation procedures, being able to reconstruct the electrophysiological activity on the heart surface by using a non-invasive recording of body surface potentials (BSP). However, the inverse problem of ECGI is ill-posed, and it requires accurate mathematical modeling of both atria and torso, mainly from CT or MR images. Several deep learning-based methods have been proposed to detect AF, but most of the AF-based studies do not include the estimation of ablation targets. In this study, we propose to model the location of AF drivers from BSP as a supervised classification problem using convolutional neural networks (CNN). Accuracy in the test set ranged between 0.75 (SNR = 5 dB) and 0.93 (SNR = 20 dB upward) when assuming time independence, but it worsened to 0.52 or lower when dividing AF models into blocks. Therefore, CNN could be a robust method that could help to non-invasively identify target regions for ablation in AF by using body surface potential mapping, avoiding the use of ECGI.

A Novel Data-Adaptive Regression Framework Based on Multivariate Adaptive Regression Splines for Electrocardiographic Imaging

OBJECTIVE

Noninvasive electrocardiographic imaging (ECGI) is a promising tool for revealing crucial cardiac electrical events with diagnostic potential. We propose a novel nonparametric regression framework based on multivariate adaptive regression splines (MARS) for ECGI.

METHODS

The inverse problem was solved by using the regression model trained with body surface potentials (BSP) and corresponding electrograms (EGM). Simulated data as well as experimental data from torso-tank experiments were used as to assess the performance of the proposed method. The robustness of the method to measurement noise and geometric errors were assessed in terms of electrogram reconstruction quality, activation time accuracy, and localization error metrics. The methods were compared with Tikhonov regularization and neural network (NN)-based methods. The resulting mapping functions between the BSPs and EGMs were also used to evaluate the most influential measurement leads.

RESULTS

MARS-based method outperformed Tikhonov regularization in terms of reconstruction accuracy and robustness to measurement noise. The effects of geometric errors were remedied to some extent by enriching the training set composition including model errors. The MARS-based method had a comparable performance with NN-based methods, which require the adjustment of many parameters.

CONCLUSION

MARS-based method successfully discovers the inverse mapping functions between the BSPs and EGMs yielding accurate reconstructions, and quantifies the contribution of each BSP lead.

SIGNIFICANCE

MARS-based method is adaptive, requires fewer parameter adjustments than NN-based methods, and robust to errors. Thus, it can be a feasible data-driven approach for accurately solving inverse imaging problems.

Electrocardiographic imaging for cardiac arrhythmias and resynchronization therapy

Use of the 12-lead electrocardiogram (ECG) is fundamental for the assessment of heart disease, including arrhythmias, but cannot always reveal the underlying mechanism or the location of the arrhythmia origin. Electrocardiographic imaging (ECGi) is a non-invasive multi-lead ECG-type imaging tool that enhances conventional 12-lead ECG. Although it is an established technology, its continuous development has been shown to assist in arrhythmic activation mapping and provide insights into the mechanism of cardiac resynchronization therapy (CRT). This review addresses the validity, reliability, and overall feasibility of ECGi for use in a diverse range of arrhythmias. A systematic search limited to full-text human studies published in peer-reviewed journals was performed through Medline via PubMed, using various combinations of three key concepts: ECGi, arrhythmia, and CRT. A total of 456 studies were screened through titles and abstracts. Ultimately, 42 studies were included for literature review. Evidence to date suggests that ECGi can be used to provide diagnostic insights regarding the mechanistic basis of arrhythmias and the location of arrhythmia origin. Furthermore, ECGi can yield valuable information to guide therapeutic decision-making, including during CRT. Several studies have used ECGi as a diagnostic tool for atrial and ventricular arrhythmias. More recently, studies have tested the value of this technique in predicting outcomes of CRT. As a non-invasive method for assessing cardiovascular disease, particularly arrhythmias, ECGi represents a significant advancement over standard procedures in contemporary cardiology. Its full potential has yet to be fully explored.

Reconstruction of 12-Lead Electrocardiogram from a Three-Lead Patch-Type Device Using a LSTM Network

Reconstructing a standard 12-lead electrocardiogram (ECG) from signals received from electrodes packed into a patch-type device is a challenging task in the field of medical instrumentation. All attempts to obtain a clinically valid 12-lead ECG using a patch-type device were not satisfactory. In this study, we designed the hardware for a three-lead patch-type ECG device and employed a long short-term memory (LSTM) network that can overcome the limitations of the linear regression algorithm used for ECG reconstruction. The LSTM network can overcome the issue of reduced horizontal components of the vector in the electric signal obtained from the patch-type device attached to the anterior chest. The reconstructed 12-lead ECG that uses the LSTM network was tested against a standard 12-lead ECG in 30 healthy subjects and ECGs of 30 patients with pathologic findings. The average correlation coefficient of the LSTM network was found to be 0.95. The ability of the reconstructed ECG to detect pathologic abnormalities was identical to that of the standard ECG. In conclusion, the reconstruction of a standard 12-lead ECG using a three-lead patch-type device is feasible, and such an ECG is an equivalent alternative to a standard 12-lead ECG.

Spatial-dependent regularization to solve the inverse problem in electromyometrial imaging

Recently, electromyometrial imaging (EMMI) was developed to non-invasively image uterine contractions in three dimensions. EMMI collects body surface electromyography (EMG) measurements and uses patient-specific body-uterus geometry generated from magnetic resonance images to reconstruct uterine electrical activity. Currently, EMMI uses the zero-order Tikhonov method with mean composite residual and smoothing operator (CRESO) to stabilize the underlying ill-posed inverse computation. However, this method is empirical and implements a global regularization parameter over all uterine sites, which is sub-optimal for EMMI given the severe eccentricity of body-uterus geometry. To address this limitation, we developed a spatial-dependent (SP) regularization method that considers both body-uterus eccentricity and EMG noise. We used electrical signals simulated with spherical and realistic geometry models to compare the reconstruction accuracy of the SP method to those of the CRESO and the L-Curve methods. The SP method reconstructed electrograms and potential maps more accurately than the other methods, especially in cases of high eccentricity and noise contamination. Thus, the SP method should facilitate clinical use of EMMI and can be used to improve the accuracy of other electrical imaging modalities, such as Electrocardiographic Imaging. Graphical abstract The spatial-dependent regularization (SP) technique was designed to improve the accuracy of Electromyometrial Imaging (EMMI). The top panel shows the eccentricity of body-uterus geometry and four representative body surface electrograms. The bottom panel shows boxplots of correlation coefficients and relative errors for the electrograms reconstructed with SP and two conventional methods, the L-Curve and mean CRESO methods.

Noninvasive high-resolution electromyometrial imaging of uterine contractions in a translational sheep model

In current clinical practice, uterine contractions are monitored via a tocodynamometer or an intrauterine pressure catheter, both of which provide crude information about contractions. Although electrohysterography/electromyography can measure uterine electrical activity, this method lacks spatial specificity and thus cannot accurately measure the exact location of electrical initiation and location-specific propagation patterns of uterine contractions. To comprehensively evaluate three-dimensional uterine electrical activation patterns, we describe here the development of electromyometrial imaging (EMMI) to display the three-dimensional uterine contractions at high spatial and temporal resolution. EMMI combines detailed body surface electrical recording with body-uterus geometry derived from magnetic resonance images. We used a sheep model to show that EMMI can reconstruct uterine electrical activation patterns from electrodes placed on the abdomen. These patterns closely match those measured with electrodes placed directly on the uterine surface. In addition, modeling experiments showed that EMMI reconstructions are minimally affected by noise and geometrical deformation. Last, we show that EMMI can be used to noninvasively measure uterine contractions in sheep in the same setup as would be used in humans. Our results indicate that EMMI can noninvasively, safely, accurately, robustly, and feasibly image three-dimensional uterine electrical activation during contractions in sheep and suggest that similar results might be obtained in clinical setting.

Early Signs of Critical Slowing Down in Heart Surface Electrograms of Ventricular Fibrillation Victims

Ventricular fibrillation (VF) is a dangerous type of cardiac arrhythmia which, without intervention, almost always results in sudden death. Implantable automatic defibrillators are among the most successful devices to prevent sudden death by automatically applying a shock to the heart when fibrillation occurs. However, the electric shock is very painful and could lead to dangerous situations when a patient is, for example, driving or biking. An early warning signal for VF could reduce the risk in such situations or, in the future, reduce the need for defibrillation altogether. Here, we test for the presence of critical slowing down (CSD), which has proven to be an early warning indicator for critical transitions in a range of different systems. CSD is characterized by a buildup of autocorrelation; we therefore study the residuals of heart surface electrocardiograms (ECGs) of patients that suffered VF to investigate if we can measure positive trends in autocorrelation. We consider several methods to extract these residuals from the original signals. For three out of four VF victims, we find a significant amount of positive autocorrelation trends in the residuals, which might be explained by CSD. We show that these positive trends may not be measurable from the original body surface ECGs, but only from certain areas around the heart surface. We argue that additional experimental studies involving heart surface ECG data of subjects that did not suffer VF are required to quantify the prediction accuracy of the promising results we get from the data of VF victims.

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.

ECG Adapted Fastest Route Algorithm to Localize the Ectopic Excitation Origin in CRT Patients

Although model-based solution strategies for the ECGI were reported to deliver promising clinical results, they strongly rely on some a priori assumptions, which do not hold true for many pathological cases. The fastest route algorithm (FRA) is a well-established method for noninvasive imaging of ectopic activities. It generates test activation sequences on the heart and compares the corresponding test body surface potential maps (BSPMs) to the measured ones. The test excitation propagation patterns are constructed under the assumption of a global conduction velocity in the heart, which is violated in the cardiac resynchronization (CRT) patients suffering from conduction disturbances. In the present work, we propose to apply dynamic time warping (DTW) to the test and measured ECGs before measuring their similarity. The warping step is a non-linear pattern matching that compensates for local delays in the temporal sequences, thus accounting for the inhomogeneous excitation propagation, while aligning them in an optimal way with respect to a distance function. To evaluate benefits of the temporal warping for FRA-based BSPMs, we considered three scenarios. In the first setting, a simplified simulation example was constructed to illustrate the temporal warping and display the resulting distance map. Then, we applied the proposed method to eight BSPMs produced by realistic ectopic activation sequences and compared its performance to FRA. Finally, we assessed localization accuracy of both techniques in ten CRT patients. For each patient, we noninvasively imaged two paced ECGs: from left and right ventricular implanted leads. In all scenarios, FRA-DTW outperformed FRA in terms of LEs. For the clinical cases, the median (25-75% range) distance errors were reduced from 16 (8-23)mm to 5 (2-10)mm for all pacings, from 15 (11-25)mm to 8 (3-13)mm in the left, and from 19 (6-23)mm to 4 (2-8)mm in the right ventricle, respectively. The obtained results suggest the ability of temporal ECG warping to compensate for an inhomogeneous conduction profile, while retaining computational efficiency intrinsic to FRA.

A review of bioelectrodes for clinical electrophysiologists

The theory of bioelectrodes describes the rules governing the passage of electrical charge between electrodes and electrolytes. In this review, we explain the basis of bioelectrodes with focus on clinical electrophysiology. The central concept is the double-layer capacitance that forms in the interface between the electrode and tissue. This phenomenon controls charge transfer between electrodes and tissues and contributes to detrimental effects such as electrode polarization and motion artifacts. Many methods critical to the practice of electrophysiology, including fractally coated pacemaker leads, biphasic stimuli, signal filtering, and the use of nonpolarizable electrodes, are devised to mitigate these problems. Our goal is to provide a robust and intuitive background on these topics for practicing electrophysiologists to help them better understand how catheters and leads work and to assist them with optimizing and troubleshooting electrophysiology systems.

ECG Imaging to Detect the Site of Ventricular Ischemia Using Torso Electrodes: A Computational Study

Electrocardiography provides some information useful for ischemic diagnosis. However, more recently there has been substantial growth in the area of ECG imaging, which by solving the inverse problem of electrocardiography aims to produce high-resolution mapping of the electrical and magnetic dynamics of the heart. Most inverse studies use the full resolution of the body surface potential (BSP) to reconstruct the epicardial potentials, however using a limited number of torso electrodes to interpolate the BSP is more clinically relevant and has an important effect on the reconstruction which must be quantified. A circular ischemic lesion on the right ventricle lateral wall 27 mm in radius is reconstructed using three Tikhonov methods along with 6 different electrode configurations ranging from 32 leads to 1,024 leads. The 2nd order Tikhonov solution performed the most accurately (~80% lesion identified) followed by the 1st (~50% lesion identified) and then the 0 order Tikhonov solution performed the worst with a maximum of ~30% lesion identified regardless of how many leads were used. With an increasing number of leads the solution produces less error, and the error becomes more localised around the lesion for all three regularisation methods. In noisy conditions, the relative performance gap of the 1st and 2nd order Tikhonov solutions was reduced, and determining an accurate regularisation parameter became relatively more difficult. Lesions located on the left ventricle walls were also able to be identified but comparatively to the right ventricle lateral wall performed marginally worse with lesions located on the interventricular septum being able to be indicated by the reconstructions but not successfully identified against the error. The quality of reconstruction was found to decrease as the lesion radius decreased, with a lesion radius of <20 mm becoming difficult to correctly identify against the error even when using >512 torso electrodes.

Solving the Inverse Problem of Electrocardiography on the Endocardium Using a Single Layer Source

The inverse problem of electrocardiography consists in reconstructing cardiac electrical activity from given body surface electrocardiographic measurements. Despite tremendous progress in the field over the last decades, the solution of this problem in terms of electrical potentials on both epi- and the endocardial heart surfaces with acceptable accuracy remains challenging. This paper presents a novel numerical approach aimed at improving the solution quality on the endocardium. Our method exploits the solution representation in the form of electrical single layer densities on the myocardial surface. We demonstrate that this representation brings twofold benefits: first, the inverse problem can be solved for the physiologically meaningful single layer densities. Secondly, a conventional transfer matrix for electrical potentials can be split into two parts, one of which turned out to posess regularizing properties leading to improved endocardial reconstructions. The method was tested in-silico for ventricular pacings utilizing realistic CT-based heart and torso geometries. The proposed approach provided more accurate solution on the ventricular endocardium compared to the conventional potential-based solutions with Tikhonov regularization of the 0th, 1st, and 2nd orders. Furthermore, we show a uniform spatio-temporal behavior of the single layer densities over the heart surface, which could be conveniently employed in the regularization procedure.

Effect of Segmentation Variation on ECG Imaging

ECG imaging (ECGI) is the process of calculating electrical cardiac activity from body surface recordings from the geometry and conductivity of the torso volume. A key first step to create geometric models for ECGI and a possible source of considerable variability is to segment the surface of the heart. We hypothesize that this variation in cardiac segmentation will produce variation in the computed ventricular surface potentials from ECGI. To evaluate this hypothesis, we leveraged the resources of the Consortium for ECG Imaging (CEI) to carry out a comparison of ECGI results from the same body surface potentials and multiple ventricular segmentations. We found that using the different segmentations produced variability in the computed ventricular surface potentials. Not surprisingly, locations of greater variance in the computed potential correlated to locations of greater variance in the segmentations, for example near the pulmonary artery and basal anterior left ventricular wall. Our results indicate that ECGI may be more sensitive to segmentation errors on the anterior epicardial surface than on other areas of the heart.

Reducing Error in ECG Forward Simulations With Improved Source Sampling

A continuing challenge in validating electrocardiographic imaging (ECGI) is the persistent error in the associated forward problem observed in experimental studies. One possible cause of this error is insufficient representation of the cardiac sources; cardiac source measurements often sample only the ventricular epicardium, ignoring the endocardium and the atria. We hypothesize that measurements that completely cover the pericardial surface are required for accurate forward solutions. In this study, we used simulated and measured cardiac potentials to test the effect of different levels of spatial source sampling on the forward simulation. Not surprisingly, increasing the source sampling over the atria reduced the average error of the forward simulations, but some sampling strategies were more effective than others. Uniform and random distributions of samples across the atrial surface were the most efficient strategies in terms of lowest error with the fewest sampling locations, whereas “single direction” strategies, i.e., adding to the atrioventricular (AV) plane or atrial roof only, were the least efficient. Complete sampling of the atria is needed to eliminate errors from missing cardiac sources, but while high density sampling that covers the entire atria yields the best results, adding as few as 11 electrodes on the atria can significantly reduce these errors. Future validation studies of the ECG forward simulations should use a cardiac source sampling that takes these considerations into account, which will, in turn, improve validation and understanding of ECGI.

Quantum theory of mass potentials

Probabilistic formalism of quantum mechanics is used to quantitatively link the global scale mass potential with the underlying electrical activity of excitable cells. Previous approaches implemented methods of classical physics to reconstruct the mass potential in terms of explicit physical models of participating cells and the volume conductor. However, the multiplicity of cellular processes with extremely intricate mixtures of deterministic and random factors prevents the creation of consistent biophysical parameter sets. To avoid the uncertainty inherent in physical attributes of cell ensembles, we undertake here a radical departure from deterministic equations of classical physics, instead applying the probabilistic reasoning of quantum mechanics. Crucial steps include: (1) the relocation of the elementary bioelectric sources from a cellular to a molecular level; (2) the creation of microscale particle models in terms of a non-homogenous birth-and-death process. To link the microscale processes with macroscale potentials, time-frequency analysis was applied for estimation of the empirical characteristic functions for component waveforms of electroencephalogram (EEG), eye-blink electromyogram (EMG), and electrocardiogram (ECG). We describe universal models for the amplitude spectra and phase functions of functional components of mass potentials. The corresponding time domain relationships disclose the dynamics of mass potential components as limit distribution functions produced by specific microscale transients. The probabilistic laws governing the microscale machinery, founded on an empirical basis, are presented. Computer simulations of particle populations with time dependent transition probabilities reveal that hidden deterministic chaos underlies development of the components of mass potentials. We label this kind of behaviour “transient deterministic chaos”.

A new approach to the intracardiac inverse problem using Laplacian distance kernel

Background

The inverse problem in electrophysiology consists of the accurate estimation of the intracardiac electrical sources from a reduced set of electrodes at short distances and from outside the heart. This estimation can provide an image with relevant knowledge on arrhythmia mechanisms for the clinical practice. Methods based on truncated singular value decomposition (TSVD) and regularized least squares require a matrix inversion, which limits their resolution due to the unavoidable low-pass filter effect of the Tikhonov regularization techniques.

Methods

We propose to use, for the first time, a Mercer’s kernel given by the Laplacian of the distance in the quasielectrostatic field equations, hence providing a Support Vector Regression (SVR) formulation by following the principles of the Dual Signal Model (DSM) principles for creating kernel algorithms.

Results

Simulations in one- and two-dimensional models show the performance of our Laplacian distance kernel technique versus several conventional methods. Firstly, the one-dimensional model is adjusted for yielding recorded electrograms, similar to the ones that are usually observed in electrophysiological studies, and suitable strategy is designed for the free-parameter search. Secondly, simulations both in one- and two-dimensional models show larger noise sensitivity in the estimated transfer matrix than in the observation measurements, and DSM−SVR is shown to be more robust to noisy transfer matrix than TSVD.

Conclusion

These results suggest that our proposed DSM−SVR with Laplacian distance kernel can be an efficient alternative to improve the resolution in current and emerging intracardiac imaging systems.

Reconstruction of 12-lead ECG Using a Single-patch Device

Objectives: The aim of this study is to develop an optimal electrode system in the form of a small and wearable single-patch ECG monitoring device that allows for the faithful reconstruction of the standard 12-lead ECG.

Methods: The optimized universal electrode positions on the chest and the personalized transformation matrix were determined using linear regression as well as artificial neural networks (ANNs). A total of 24 combinations of 4 neighboring electrodes on 35 channels were evaluated on 19 subjects. Moreover, we analyzed combinations of three electrodes within the four-electrode combination with the best performance.

Results: The mean correlation coefficients were all higher than 0.95 in the case of the ANN method for the combinations of four neighboring electrodes. The reconstructions obtained using the three and four sensing electrodes showed no significant differences. The reconstructed 12-lead ECG obtained using the ANN method is better than that using the MLR method. Therefore, three sensing electrodes and one ground electrode (forming a square) placed below the clavicle on the left were determined to be suitable for ensuring good reconstruction performance.

Conclusions: Since the interelectrode distance was determined to be 5 cm, the suggested approach can be implemented in a single-patch device, which should allow for the continuous monitoring of the standard 12-lead ECG without requiring limb contact, both in daily life and in clinical practice.

Determination of the excitation origin in the ventricles from the ECG using support vector regression

A common treatment of focal ventricular tachycardia is the catheter ablation of triggering sites. They have to be found manually by the physician during an intervention in a catheter lab. Thus, a method for determining the position of the focus automatically is desired. The inverse problem of electrocardiography addresses this problem by reconstructing the source of the ectopic beats using the surface ECG. This problem is ill-posed and therefore needs specific methods for solving it. We propose a machine learning approach for localisation of the ectopic foci in the heart to assist cardiologists with their therapy planning.We simulated 600 120-lead ECGs with different known excitation origins in the heart using a cellular automaton followed by a forward calculation. Features from the ECGs were used as input for a support vector regression (SVR). We assumed a functional relation between features from the ECG and the excitation origin. To benchmark SVR, we also used the well-known Tikhonov 0th order regularisation to reconstruct the transmembrane potentials in the heart and detect the location of the ectopic foci. Parameters for SVR and regularisation were chosen using a grid search minimising the error between estimated and true excitation origin. Compared to the Tikhonov regularisation method, SVR achieved a smaller deviation between estimated and real excitation origin evaluated with 6-fold cross validation. Future work could investigate on the behaviour on data from simulations with other torso and electrophysiological models, the influence of other methods for feature extraction and finally the evaluation with clinical data.

Regularization Techniques for ECG Imaging during Atrial Fibrillation: A Computational Study

The inverse problem of electrocardiography is usually analyzed during stationary rhythms. However, the performance of the regularization methods under fibrillatory conditions has not been fully studied. In this work, we assessed different regularization techniques during atrial fibrillation (AF) for estimating four target parameters, namely, epicardial potentials, dominant frequency (DF), phase maps, and singularity point (SP) location. We use a realistic mathematical model of atria and torso anatomy with three different electrical activity patterns (i.e., sinus rhythm, simple AF, and complex AF). Body surface potentials (BSP) were simulated using Boundary Element Method and corrupted with white Gaussian noise of different powers. Noisy BSPs were used to obtain the epicardial potentials on the atrial surface, using 14 different regularization techniques. DF, phase maps, and SP location were computed from estimated epicardial potentials. Inverse solutions were evaluated using a set of performance metrics adapted to each clinical target. For the case of SP location, an assessment methodology based on the spatial mass function of the SP location, and four spatial error metrics was proposed. The role of the regularization parameter for Tikhonov-based methods, and the effect of noise level and imperfections in the knowledge of the transfer matrix were also addressed. Results showed that the Bayes maximum-a-posteriori method clearly outperforms the rest of the techniques but requires a priori information about the epicardial potentials. Among the purely non-invasive techniques, Tikhonov-based methods performed as well as more complex techniques in realistic fibrillatory conditions, with a slight gain between 0.02 and 0.2 in terms of the correlation coefficient. Also, the use of a constant regularization parameter may be advisable since the performance was similar to that obtained with a variable parameter (indeed there was no difference for the zero-order Tikhonov method in complex fibrillatory conditions). Regarding the different targets, DF and SP location estimation were more robust with respect to pattern complexity and noise, and most algorithms provided a reasonable estimation of these parameters, even when the epicardial potentials estimation was inaccurate. Finally, the proposed evaluation procedure and metrics represent a suitable framework for techniques benchmarking and provide useful insights for the clinical practice.

A priori model independent inverse potential mapping: the impact of electrode positioning

INTRODUCTION

In inverse potential mapping, local epicardial potentials are computed from recorded body surface potentials (BSP). When BSP are recorded with only a limited number of electrodes, in general biophysical a priori models are applied to facilitate the inverse computation. This study investigated the possibility of deriving epicardial potential information using only 62 torso electrodes in the absence of an a priori model.

METHODS

Computer simulations were used to determine the optimal in vivo positioning of 62 torso electrodes. Subsequently, three different electrode configurations, i.e., surrounding the thorax, concentrated precordial (30 mm inter-electrode distance) and super-concentrated precordial (20 mm inter-electrode distance) were used to record BSP from three healthy volunteers. Magnetic resonance imaging (MRI) was performed to register the electrode positions with respect to the anatomy of the patient. Epicardial potentials were inversely computed from the recorded BSP. In order to determine the reconstruction quality, the super-concentrated electrode configuration was applied in four patients with an implanted MRI-conditional pacemaker system. The distance between the position of the ventricular lead tip on MRI and the inversely reconstructed pacing site was determined.

RESULTS

The epicardial potential distribution reconstructed using the super-concentrated electrode configuration demonstrated the highest correlation (R = 0.98; p < 0.01) with the original epicardial source model. A mean localization error of 5.3 mm was found in the pacemaker patients.

CONCLUSION

This study demonstrated the feasibility of deriving detailed anterior epicardial potential information using only 62 torso electrodes without the use of an a priori model.

A theoretical analysis of the electrogastrogram (EGG)

In this study, a boundary element model was developed to investigate the relationship between the gastric electrical activity, also known as slow waves, and the electrogastrogram (EGG). A dipole was calculated to represent the equivalent net activity of gastric slow waves. The dipole was then placed in an anatomically-realistic torso model to simulate EGG. The torso model was constructed from a laser-scanned geometry of an adult male torso phantom with 190 electrode sites equally distributed around the torso so that simulated EGG could be directly compared between the physical model and the mathematical model. The results were analyzed using the Fast Fourier Transforms (FFT), spatial distribution of EGG potential and a resultant EGG based on a 3-lead configuration. The FFT results showed both the dipole and EGG contained identical dominant frequency component of 3 cycles per minute (cpm), with this result matching known physiological phenomenon. The -3 dB point of the EGG was 110 mm from the region directly above the dipole source. Finally, the results indicated that electrode coupling could theoretically be used in a similar fashion to ECG coupling to gain greater understanding of how EGG correlate to gastric slow waves.

Noninvasive identification of two lesions with local repolarization changes using two dipoles in inverse solution simulation study

BACKGROUND

The method for inverse localization and identification of two distinct simultaneous lesions with changed repolarization in the ventricular myocardium (two-vessel disease) is proposed and its robustness to errors in input data is tested in this simulation study.

METHOD

The inverse solution was obtained from the difference between STT integral body surface potential map computed with repolarization changes and the STT integral map from normal activation. In a numerical model of ventricles 48 cases of two simultaneous lesions and 48 cases of a single lesion were modeled. The effect of the lesions was taken to be represented by two dipoles. The input data were disturbed by three types of added noise. Twenty three characteristics of every obtained inverse solution were defined and four of them were used as the features in discriminant analysis task distinguishing the correct inverse solutions identifying two lesions.

RESULTS

The mean localization error for identified two lesions was 1.1±0.7cm. The sensitivity and specificity of quadratic discriminant analysis with cross-validation and feature selection was higher than 90%.

CONCLUSIONS

The combination of the inverse solution with two dipoles and discriminant analysis allows the identification of two simultaneous lesions without a priori information about the number of lesions.

Computing volume potentials for noninvasive imaging of cardiac excitation

BACKGROUND

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

METHODS

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

RESULTS

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

CONCLUSION

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

Data-driven estimation of cardiac electrical diffusivity from 12-lead ECG signals

Diagnosis and treatment of dilated cardiomyopathy (DCM) is challenging due to a large variety of causes and disease stages. Computational models of cardiac electrophysiology (EP) can be used to improve the assessment and prognosis of DCM, plan therapies and predict their outcome, but require personalization. In this work, we present a data-driven approach to estimate the electrical diffusivity parameter of an EP model from standard 12-lead electrocardiograms (ECG). An efficient forward model based on a mono-domain, phenomenological Lattice-Boltzmann model of cardiac EP, and a boundary element-based mapping of potentials to the body surface is employed. The electrical diffusivity of myocardium, left ventricle and right ventricle endocardium is then estimated using polynomial regression which takes as input the QRS duration and electrical axis. After validating the forward model, we computed 9500 EP simulations on 19 different DCM patients in just under three seconds each to learn the regression model. Using this database, we quantify the intrinsic uncertainty of electrical diffusion for given ECG features and show in a leave-one-patient-out cross-validation that the regression method is able to predict myocardium diffusion within the uncertainty range. Finally, our approach is tested on the 19 cases using their clinical ECG. 84% of them could be personalized using our method, yielding mean prediction errors of 18.7ms for the QRS duration and 6.5° for the electrical axis, both values being within clinical acceptability. By providing an estimate of diffusion parameters from readily available clinical data, our data-driven approach could therefore constitute a first calibration step toward a more complete personalization of cardiac EP.

Noninvasive imaging of cardiac excitation: current status and future perspective

Noninvasive imaging of cardiac excitation using body surface potential mapping (BSPM) data and inverse procedures is an emerging technique that enables estimation of myocardial depolarization and repolarization. Despite numerous reports on the possible advantages of this imaging technique, it has not yet advanced into daily clinical practice. This is mainly due to the time consuming nature of data acquisition and the complexity of the mathematics underlying the used inverse procedures. However, the popularity of this field of research has increased and noninvasive imaging of cardiac electrophysiology is considered a promising tool to complement conventional invasive electrophysiological studies. Furthermore, the use of appropriately designed electrode vests and more advanced computers has greatly reduced the procedural time. This review provides descriptive overview of the research performed thus far and the possible future directions. The general challenges in routine application of BSPM and inverse procedures are discussed. In addition, individual properties of the biophysical models underlying the inverse procedures are illustrated.

MANIFOLD LEARNING FOR ANALYSIS OF LOW-ORDER NONLINEAR DYNAMICS IN HIGH-DIMENSIONAL ELECTROCARDIOGRAPHIC SIGNALS

The dynamical structure of electrical recordings from the heart or torso surface is a valuable source of information about cardiac physiological behavior. In this paper, we use an existing data-driven technique for manifold identification to reveal electrophysiologically significant changes in the underlying dynamical structure of these signals. Our results suggest that this analysis tool characterizes and differentiates important parameters of cardiac bioelectric activity through their dynamic behavior, suggesting the potential to serve as an effective dynamic constraint in the context of inverse solutions.

Cardiac position sensitivity study in the electrocardiographic forward problem using stochastic collocation and boundary element methods

The electrocardiogram (ECG) is ubiquitously employed as a diagnostic and monitoring tool for patients experiencing cardiac distress and/or disease. It is widely known that changes in heart position resulting from, for example, posture of the patient (sitting, standing, lying) and respiration significantly affect the body-surface potentials; however, few studies have quantitatively and systematically evaluated the effects of heart displacement on the ECG. The goal of this study was to evaluate the impact of positional changes of the heart on the ECG in the specific clinical setting of myocardial ischemia. To carry out the necessary comprehensive sensitivity analysis, we applied a relatively novel and highly efficient statistical approach, the generalized polynomial chaos-stochastic collocation method, to a boundary element formulation of the electrocardiographic forward problem, and we drove these simulations with measured epicardial potentials from whole-heart experiments. Results of the analysis identified regions on the body-surface where the potentials were especially sensitive to realistic heart motion. The standard deviation (STD) of ST-segment voltage changes caused by the apex of a normal heart, swinging forward and backward or side-to-side was approximately 0.2 mV. Variations were even larger, 0.3 mV, for a heart exhibiting elevated ischemic potentials. These variations could be large enough to mask or to mimic signs of ischemia in the ECG. Our results suggest possible modifications to ECG protocols that could reduce the diagnostic error related to postural changes in patients possibly suffering from myocardial ischemia.

Application of L1-norm regularization to epicardial potential reconstruction based on gradient projection

The epicardial potential (EP)-targeted inverse problem of electrocardiography (ECG) has been widely investigated as it is demonstrated that EPs reflect underlying myocardial activity. It is a well-known ill-posed problem as small noises in input data may yield a highly unstable solution. Traditionally, L2-norm regularization methods have been proposed to solve this ill-posed problem. But the L2-norm penalty function inherently leads to considerable smoothing of the solution, which reduces the accuracy of distinguishing abnormalities and locating diseased regions. Directly using the L1-norm penalty function, however, may greatly increase computational complexity due to its non-differentiability. We propose an L1-norm regularization method in order to reduce the computational complexity and make rapid convergence possible. Variable splitting is employed to make the L1-norm penalty function differentiable based on the observation that both positive and negative potentials exist on the epicardial surface. Then, the inverse problem of ECG is further formulated as a bound-constrained quadratic problem, which can be efficiently solved by gradient projection in an iterative manner. Extensive experiments conducted on both synthetic data and real data demonstrate that the proposed method can handle both measurement noise and geometry noise and obtain more accurate results than previous L2- and L1-norm regularization methods, especially when the noises are large.

A Kalman filter-based approach to reduce the effects of geometric errors and the measurement noise in the inverse ECG problem

In this article, we aimed to reduce the effects of geometric errors and measurement noise on the inverse problem of Electrocardiography (ECG) solutions. We used the Kalman filter to solve the inverse problem in terms of epicardial potential distributions. The geometric errors were introduced into the problem via wrong determination of the size and location of the heart in simulations. An error model, which is called the enhanced error model (EEM), was modified to be used in inverse problem of ECG to compensate for the geometric errors. In this model, the geometric errors are modeled as additive Gaussian noise and their noise variance is added to the measurement noise variance. The Kalman filter method includes a process noise component, whose variance should also be estimated along with the measurement noise. To estimate these two noise variances, two different algorithms were used: (1) an algorithm based on residuals, (2) expectation maximization algorithm. The results showed that it is important to use the correct noise variances to obtain accurate results. The geometric errors, if ignored in the inverse solution procedure, yielded incorrect epicardial potential distributions. However, even with a noise model as simple as the EEM, the solutions could be significantly improved.

Equivalent moving dipole localization of cardiac ectopic activity in a swine model during pacing

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.