Computing ischemic regions in the heart with the bidomain model--first steps towards validation

Hybrid Neural State-Space Modeling for Supervised and Unsupervised Electrocardiographic Imaging

State-space modeling (SSM) provides a general framework for many image reconstruction tasks. Error in a priori physiological knowledge of the imaging physics, can bring incorrectness to solutions. Modern deep-learning approaches show great promise but lack interpretability and rely on large amounts of labeled data. In this paper, we present a novel hybrid SSM framework for electrocardiographic imaging (ECGI) to leverage the advantage of state-space formulations in data-driven learning. We first leverage the physics-based forward operator to supervise the learning. We then introduce neural modeling of the transition function and the associated Bayesian filtering strategy. We applied the hybrid SSM framework to reconstruct electrical activity on the heart surface from body-surface potentials. In unsupervised settings of both in-silico and in-vivo data without cardiac electrical activity as the ground truth to supervise the learning, we demonstrated improved ECGI performances of the hybrid SSM framework trained from a small number of ECG observations in comparison to the fixed SSM. We further demonstrated that, when in-silico simulation data becomes available, mixed supervised and unsupervised training of the hybrid SSM achieved a further 40.6% and 45.6% improvements, respectively, in comparison to traditional ECGI baselines and supervised data-driven ECGI baselines for localizing the origin of ventricular activations in real data.

Noninvasive Reconstruction of Transmural Transmembrane Potential With Simultaneous Estimation of Prior Model Error

To reconstruct electrical activity in the heart from body-surface electrocardiograms (ECGs) is an ill-posed inverse problem. Electrophysiological models have been found effective in regularizing these inverse problems by incorporating a priori knowledge about how the electrical potential in the heart propagates over time. However, these models suffer from model errors arising from, for example, parameters associated with tissue properties and the earliest sites of excitation. We present a Bayesian approach to simultaneously estimate transmembrane potential (TMP) signals and prior model errors, exploiting sparsity of the error in the gradient domain in the form of a novel sparse prior based on variational lower bound of the generalized Gaussian distribution. In synthetic and real-data experiments, we demonstrate the improvement of accuracy in TMP reconstruction brought by simultaneous model error estimation. We further provide theoretical and empirical justifications for the change of performances in the presented method at the presence of different model errors.

Continuous analogue to iterative optimization for PDE-constrained inverse problems

The parameters of many physical processes are unknown and have to be inferred from experimental data. The corresponding parameter estimation problem is often solved using iterative methods such as steepest descent methods combined with trust regions. For a few problem classes also continuous analogues of iterative methods are available. In this work, we expand the application of continuous analogues to function spaces and consider PDE (partial differential equation)-constrained optimization problems. We derive a class of continuous analogues, here coupled ODE (ordinary differential equation)-PDE models, and prove their convergence to the optimum under mild assumptions. We establish sufficient bounds for local stability and convergence for the tuning parameter of this class of continuous analogues, the retraction parameter. To evaluate the continuous analogues, we study the parameter estimation for a model of gradient formation in biological tissues. We observe good convergence properties, indicating that the continuous analogues are an interesting alternative to state-of-the-art iterative optimization methods.

Activation recovery interval imaging of premature ventricular contraction

Dispersion of ventricular repolarization due to abnormal activation contributes to the susceptibility to cardiac arrhythmias. However, the global pattern of repolarization is difficult to assess clinically. Activation recovery interval (ARI) has been used to understand the properties of ventricular repolarization. In this study, we developed an ARI imaging technique to noninvasively reconstruct three-dimensional (3D) ARI maps in 10 premature ventricular contraction (PVC) patients and evaluated the results with the endocardial ARI maps recorded by a clinical navigation system (CARTO). From the analysis results of a total of 100 PVC beats in 10 patients, the average correlation coefficient is 0.86±0.05 and the average relative error is 0.06±0.03. The average localization error is 4.5±2.3 mm between the longest ARI sites in 3D ARI maps and those in CARTO endocardial ARI maps. The present results suggest that ARI imaging could serve as an alternative of evaluating global pattern of ventricular repolarization noninvasively and could assist in the future investigation of the relationship between global repolarization dispersion and the susceptibility to cardiac arrhythmias.

Parallelization and High-Performance Computing Enables Automated Statistical Inference of Multi-scale Models

Mechanistic understanding of multi-scale biological processes, such as cell proliferation in a changing biological tissue, is readily facilitated by computational models. While tools exist to construct and simulate multi-scale models, the statistical inference of the unknown model parameters remains an open problem. Here, we present and benchmark a parallel approximate Bayesian computation sequential Monte Carlo (pABC SMC) algorithm, tailored for high-performance computing clusters. pABC SMC is fully automated and returns reliable parameter estimates and confidence intervals. By running the pABC SMC algorithm for ∼10⁶ hr, we parameterize multi-scale models that accurately describe quantitative growth curves and histological data obtained in vivo from individual tumor spheroid growth in media droplets. The models capture the hybrid deterministic-stochastic behaviors of 10⁵-10⁶ of cells growing in a 3D dynamically changing nutrient environment. The pABC SMC algorithm reliably converges to a consistent set of parameters. Our study demonstrates a proof of principle for robust, data-driven modeling of multi-scale biological systems and the feasibility of multi-scale model parameterization through statistical inference.

ECG imaging of ventricular tachycardia: evaluation against simultaneous non-contact mapping and CMR-derived grey zone

ECG imaging is an emerging technology for the reconstruction of cardiac electric activity from non-invasively measured body surface potential maps. In this case report, we present the first evaluation of transmurally imaged activation times against endocardially reconstructed isochrones for a case of sustained monomorphic ventricular tachycardia (VT). Computer models of the thorax and whole heart were produced from MR images. A recently published approach was applied to facilitate electrode localization in the catheter laboratory, which allows for the acquisition of body surface potential maps while performing non-contact mapping for the reconstruction of local activation times. ECG imaging was then realized using Tikhonov regularization with spatio-temporal smoothing as proposed by Huiskamp and Greensite and further with the spline-based approach by Erem et al. Activation times were computed from transmurally reconstructed transmembrane voltages. The results showed good qualitative agreement between the non-invasively and invasively reconstructed activation times. Also, low amplitudes in the imaged transmembrane voltages were found to correlate with volumes of scar and grey zone in delayed gadolinium enhancement cardiac MR. The study underlines the ability of ECG imaging to produce activation times of ventricular electric activity-and to represent effects of scar tissue in the imaged transmembrane voltages.

Examining the Impact of Prior Models in Transmural Electrophysiological Imaging: A Hierarchical Multiple-Model Bayesian Approach

Noninvasive cardiac electrophysiological (EP) imaging aims to mathematically reconstruct the spatiotemporal dynamics of cardiac sources from body-surface electrocardiographic (ECG) data. This ill-posed problem is often regularized by a fixed constraining model. However, a fixed-model approach enforces the source distribution to follow a pre-assumed structure that does not always match the varying spatiotemporal distribution of actual sources. To understand the model-data relation and examine the impact of prior models, we present a multiple-model approach for volumetric cardiac EP imaging where multiple prior models are included and automatically picked by the available ECG data. Multiple models are incorporated as an Lp-norm prior for sources, where p is an unknown hyperparameter with a prior uniform distribution. To examine how different combinations of models may be favored by different measurement data, the posterior distribution of cardiac sources and hyperparameter p is calculated using a Markov Chain Monte Carlo (MCMC) technique. The importance of multiple-model prior was assessed in two sets of synthetic and real-data experiments, compared to fixed-model priors (using Laplace and Gaussian priors). The results showed that the posterior combination of models (the posterior distribution of p) as determined by the ECG data differed substantially when reconstructing sources with different sizes and structures. While the use of fixed models is best suited in situations where the prior assumption fits the actual source structures, the use of an automatically adaptive set of models may have the ability to better address model-data mismatch and to provide consistent performance in reconstructing sources with different properties.

Sensitivity of Noninvasive Cardiac Electrophysiological Imaging to Variations in Personalized Anatomical Modeling

OBJECTIVE

Noninvasive cardiac electrophysiological (EP) imaging techniques rely on anatomically-detailed heart-torso models derived from high-quality tomographic images of individual subjects. However, anatomical modeling involves variations that lead to unresolved uncertainties in the outcome of EP imaging, bringing questions to the robustness of these methods in clinical practice. In this study, we design a systematic statistical approach to assess the sensitivity of EP imaging methods to the variations in personalized anatomical modeling.

METHODS

We first quantify the variations in personalized anatomical models by a novel application of statistical shape modeling. Given the statistical distribution of the variation in personalized anatomical models, we then employ unscented transform to determine the sensitivity of EP imaging outputs to the variation in input personalized anatomical modeling.

RESULTS

We test the feasibility of our proposed approach using two of the existing EP imaging methods: epicardial-based electrocardiographic imaging and transmural electrophysiological imaging. Both phantom and real-data experiments show that variations in personalized anatomical models have negligible impact on the outcome of EP imaging.

CONCLUSION

This study verifies the robustness of EP imaging methods to the errors in personalized anatomical modeling and suggests the possibility to simplify the process of anatomical modeling in future clinical practice.

SIGNIFICANCE

This study proposes a systematic statistical approach to quantify anatomical modeling variations and assess their impact on EP imaging, which can be extended to find a balance between the quality of personalized anatomical models and the accuracy of EP imaging that may improve the clinical feasibility of EP imaging.

Variational Bayesian electrophysiological imaging of myocardial infarction

The presence, size, and distribution of ischemic tissue bear significant prognostic and therapeutic implication for ventricular arrhythmias. While many approaches to 3D infarct detection have been developed via electrophysiological (EP) imaging from noninvasive electrocardiographic data, this ill-posed inverse problem remains challenging especially for septal infarcts that are hidden from body-surface data. We propose a variational Bayesian framework for EP imaging of 3D infarct using a total-variation prior. The posterior distribution of intramural action potential and all regularization parameters are estimated from body-surface data by minimizing the Kullback-Leibler divergence. Because of the uncertainty introduced in prior models, we hypothesize that the solution uncertainty plays as important a role as the point estimate in interpreting the reconstruction. This is verified in a set of phantom and real-data experiments, where regions of low confidence help to eliminate false-positives and to accurately identify infarcts of various locations (including septum) and distributions. Owing to the ability of total-variation prior in extracting the boundary between smooth regions, the presented method also has the potential to outline infarct border that is the most critical region responsible for ventricular arrhvthmias.

Computational generation of the Purkinje network driven by clinical measurements: the case of pathological propagations

To properly describe the electrical activity of the left ventricle, it is necessary to model the Purkinje fibers, responsible for the fast and coordinate ventricular activation, and their interaction with the muscular propagation. The aim of this work is to propose a methodology for the generation of a patient-specific Purkinje network driven by clinical measurements of the activation times related to pathological propagations. In this case, one needs to consider a strongly coupled problem between the network and the muscle, where the feedback from the latter to the former cannot be neglected as in a normal propagation. We apply the proposed strategy to data acquired on three subjects, one of them suffering from muscular conduction problems owing to a scar and the other two with a muscular pre-excitation syndrome (Wolff-Parkinson-White). To assess the accuracy of the proposed method, we compare the results obtained by using the patient-specific Purkinje network generated by our strategy with the ones obtained by using a non-patient-specific network. The results show that the mean absolute errors in the activation time is reduced for all the cases, highlighting the importance of including a patient-specific Purkinje network in computational models.

Noninvasive transmural electrophysiological imaging based on minimization of total-variation functional

While tomographic imaging of cardiac structure and kinetics has improved substantially, electrophysiological mapping of the heart is still restricted to the surface with little or no depth information beneath. The progress in reconstructing 3-D action potential from surface voltage data has been hindered by the intrinsic ill-posedness of the problem and the lack of a unique solution in the absence of prior assumptions. In this work, we propose a novel adaption of the total-variation (TV) prior to exploit the unique spatial property of transmural action potential of being piecewise smooth with a steep boundary (gradient) separating depolarized and repolarized regions. We present a variational TV-prior instead of a common discrete TV-prior for improved robustness to mesh resolution, and solve the TV-minimization by a sequence of weighted, first-order L2-norm minimization. In a large set of phantom experiments, the proposed method is shown to outperform existing quadratic methods in preserving the steep gradient of action potential along the border of infarcts, as well as in capturing the disruption to the normal path of electrical wavefronts. Real-data experiments also further demonstrate the potential of the proposed method in revealing the location and shape of infarcts when quadratic methods fail to do so.

Comparison of minimum-norm estimation and beamforming in electrocardiography with acute ischemia

In the electrocardiographic (ECG) inverse problem, the electrical activity of the heart is estimated from measured electrocardiogram. A model of thorax conductivities and a model of the cardiac generator is required for the ECG inverse problem. Limitations and errors in methods, models, and data will lead to errors in the estimates. However, in experimental applications, the use of limited or erroneous models is often inevitable due to necessary model simplifications and the difficulty of obtaining accurate 3D anatomical imaging data. In this work, we focus on two methods for solving the inverse problem of ECG in the case of acute ischemia: minimum-norm (MN) estimation and linearly constrained minimum-variance beamforming. We study how these methods perform with different sizes of ischemia and with erroneous conductivity models. The results indicate that the beamformer can localize small ischemia given an accurate model, but it cannot be used for estimating the size of ischemia. The MN estimator is tolerant to geometry errors and excels in estimating the size of ischemia, although the beamformer performs better with accurate model and small ischemia.

Using transmural regularization and dynamic modeling for noninvasive cardiac potential imaging of endocardial pacing with imprecise thoracic geometry

Cardiac electrical imaging from body surface potential measurements is increasingly being seen as a technology with the potential for use in the clinic, for example for pre-procedure planning or during-treatment guidance for ventricular arrhythmia ablation procedures. However several important impediments to widespread adoption of this technology remain to be effectively overcome. Here we address two of these impediments: the difficulty of reconstructing electric potentials on the inner (endocardial) as well as outer (epicardial) surfaces of the ventricles, and the need for full anatomical imaging of the subject's thorax to build an accurate subject-specific geometry. We introduce two new features in our reconstruction algorithm: a nonlinear low-order dynamic parameterization derived from the measured body surface signals, and a technique to jointly regularize both surfaces. With these methodological innovations in combination, it is possible to reconstruct endocardial activation from clinically acquired measurements with an imprecise thorax geometry. In particular we test the method using body surface potentials acquired from three subjects during clinical procedures where the subjects' hearts were paced on their endocardia using a catheter device. Our geometric models were constructed using a set of CT scans limited in axial extent to the immediate region near the heart. The catheter system provides a reference location to which we compare our results. We compare our estimates of pacing site localization, in terms of both accuracy and stability, to those reported in a recent clinical publication , where a full set of CT scans were available and only epicardial potentials were reconstructed.