Source Localization Probability Maps for Uncertainty Quantification in Electrocardiographic Imaging

Abstract

This study aimed to develop a new probabilistic visualization analysis to study source localization uncertainty in electrocardiographic imaging (ECGI). Using Monte Carlo error propagation, we developed probability maps that illustrate uncertainty in source localization compared to the ground truth source location. We used these probability maps to quantify the impact of noise amplitude and iterative Krylov regularization on source localization. Artificial Gaussian white noise was added to the body surface potentials between (0.5% and 9% of their amplitudes) to simulate noisy observations. We solved the inverse problem to recover heart surface potentials using the conjugate gradient least squares (CGLS) and preconditioned CGLS (PCGLS) algorithms with the Laplacian over the heart surface as a right preconditioner. We forward propagated these inverse solutions, and performed 200 CGLS and PCGLS Monte Carlo inversions per noise level. For each sample, we recorded the top 1% of lowest potential locations, and normalized across all samples to form empirical probability maps for source localization. Increasing the noise amplitude increased both the uncertainty and inaccuracy for source localization, with PCGLS outperforming CGLS across all noise amplitudes. We conclude that the concept of a source localization probability map may be useful clinically in identifying origins of arrhythmia in cardiac tissue.Keywords-electrocardiographic imaging (ECGI), Krylov inversion, Monte Carlo sampling, uncertainty quantification IntroductionCardiac ablation is an interventional procedure in which a clinician internally locates and destroys regions of cardiac tissue responsible for spontaneous pathological heart beats or reentrant wave activity. However, this invasive therapy can last many hours, and abnormal heart rhythms, or arrhythmias, frequently re-occur following the procedure [1,2]. One noninvasive approach to reducing proceFrance, and Johnson are with the Scientific Computing and Imaging Institute, University of Utah. Email: {jfrance, crj}@sci.utah.edu. This project was supported by grants from the NIH National Institute of General Medical Sciences (P41 GM103545-18) from the National Institutes of Health. dure times and the risk of recurrence is known as electrocardiographic imaging (ECGI). ECGI uses a patient's CT and MRI images to approximate a computational geometry, and combines this with ECG recordings into a mathematical model that predicts the origin of an arrhythmia.This inverse problem of electrocardiography may one day become a standard in ablation procedures, allowing clinicians to more quickly and accurately locate pathological heart tissue, since the ECGI prediction for the source of an arrhythmia essentially tells a clinician ahead of time where to ablate [3]. ECGI involves solving Laplace's equation [4,5] to obtain the systemwhere A ∈ R m×n , h ∈ R n , and y ∈ R m . The transfer matrix A relates the heart surface voltages h to the true body surface voltages y * . However, the observed body surface ECG re...

Quantifying and Visualizing Uncertainty for Source Localization in Electrocardiographic Imaging

“…For probabilistic maps, similar to the early study by France and Johnson (2016), we located the top 3% of the lowest voltage values (with the lowest voltage denoting the source (Wang and Rudy 2006)), and averaged these locations over the 200 samples to form a probabilistic representation for source localization. In France and Johnson (2016), probabilistic maps were visualized with direct mapping of probability to opacity. In our probabilistic map visualizations, we used color maps to segment the regions of high probability (> 0.75), moderate probability (between 0.5 and 0.75/ between 0.25 and 0.5), and low probability (< 0.25).…”

“…Figures 3d-e and 3f-g illustrate the probability maps (France and Johnson 2016) and confidence maps, respectively. For the probability maps visualized in Figures 3d-e, the darker green areas indicate the regions of higher probability for source localization.…”

“…To perform Monte Carlo error analysis, we sampled a noisy solution , where σ represents the standard deviation of the noise, a value that was fixed to produce errors at the same percentage p as in the original Equation (1). We then used the CGLS and PCGLS routines to obtain individual inversion samples, just as we obtained h k in originally solving Equation (1) (Aster et al 2013; France and Johnson 2016). We obtained an ensemble of 200 Monte Carlo samples per simulation.…”

“…For probabilistic maps, similar to the early study by France and Johnson (2016), we located the top 3% of the lowest voltage values (with the lowest voltage denoting the source (Wang and Rudy 2006)), and averaged these locations over the 200 samples to form a probabilistic representation for source localization. In France and Johnson (2016), probabilistic maps were visualized with direct mapping of probability to opacity.…”

“…Building on preliminary work for source localization by France and Johnson (2016), we apply iterative Krylov methods with Monte Carlo error propagation to study the impact of ECG measurement errors on inverse solutions. We then propose a framework for deriving source localization confidence interval (CI) regions, and present applications of level-set and topology-based visualizations to visually analyze uncertainty in source localization.…”