Improving electrocardiographic imaging solutions: A comprehensive study on regularization parameter selection in L-curve optimization in the Atria

Abstract

Background: In electrocardiographic imaging (ECGI), selecting an optimal regularization parameter (λ) is crucial for obtaining accurate inverse electrograms. The effects of signal and geometry uncertainties on the inverse problem regularization have not been thoroughly quantified, and there is no established methodology to identify when λ is sub-optimal due to these uncertainties. This study introduces a novel approach to λ selection using Tikhonov regularization and L-curve optimization, specifically addressing the impact of electrical noise in body surface potential map (BSPM) signals and geometrical inaccuracies in the cardiac mesh.

Methods: Nineteen atrial simulations (5 of regular rhythms and 14 of atrial fibrillation) ensuring variability in substrate complexity and activation patterns were used for computing the ECGI with added white Gaussian noise from 40 dB to -3dB. Cardiac mesh displacements (1-3 cm) were applied to simulate the uncertainty of atrial positioning and study its impact on the L-curve shape. The regularization parameter, the maximum curvature, and the most horizontal angle of the L-curve (β) were quantified. In addition, BSPM signals from real patients were used to validate our findings.

Results: The maximum curvature of the L-curve was found to be inversely related to signal-to-noise ratio and atrial positioning errors. In contrast, the β angle is directly related to electrical noise and remains unaffected by geometrical errors. Our proposed adjustment of λ, based on the β angle, provides a more reliable ECGI solution than traditional corner-based methods. Our findings have been validated with simulations and real patient data, demonstrating practical applicability.

Conclusion: Adjusting λ based on the amount of noise in the data (or on the β angle) allows finding optimal ECGI solutions than a λ purely found at the corner of the L-curve. It was observed that the relevant information in ECGI activation maps is preserved even under the presence of uncertainties when the regularization parameter is correctly selected. The proposed criteria for regularization parameter selection have the potential to enhance the accuracy and reliability of ECGI solutions.

Keywords: Electrical noise; Electrocardiographic imaging; Geometrical uncertainty; Inverse problem; L-curve.