Two-Step Tikhonov method for solving ECGI inverse problem

Background: Electrocardiographic Imaging (ECGI) is a non-invasive technology that reconstructs cardiac electrical activity from body-surface electrocardiograms. Its implementation has two steps: forward model construction and inverse problem solving. The existing methods for solving inverse problems typically assume that the forward problem has been accurately solved, ignoring the associated uncertainties. Objective: This study aims to integrate forward numerical methods with inverse regularization algorithms to enhance ECGI inverse solution accuracy. Methods: The Two-Step Tikhonov (2S-Tik) method combines the forward numerical and inverse regularization methods, employing the inverse-derived regularization parameter to refine the transfer matrix. First, the optimal regularization parameter λ_opt is identified via the L-curve method in the 0th-order Tikhonov algorithm. Then, λ_opt is fixed, and the optimal A_opt is sought within the search space of the transfer matrix A. Finally, the selected λ_opt and A_opt are combined for solution. Results: For sinus rhythm, the spatial (sCC) and temporal (tCC) correlation coefficients of cardiac surface reconstructed electrical potentials using the 2S-Tik method are approximately 0.72 ± 0.21 and 0.90 ± 0.19, respectively, improving sCC and tCC by 10% and 6% over the traditional Tikhonov algorithm (0.65 ± 0.26 and 0.85 ± 0.21). Moreover, when Gaussian noise is added to body-surface potentials, the 2S-Tik method exhibits robust and stable performance. Conclusion: Compared with the traditional Tikhonov algorithm, the 2S-Tik method significantly improves the accuracy of ECGI inverse solution and has great potential in clinical ECGI applications.