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ML and MAP estimation of parameters for the Kalman filter and smoother applied to electrocardiographic imaging

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Abstract

In electrocardiographic imaging (ECGI), one solves the inverse problem of electrocardiography (ECG) to reconstruct equivalent cardiac sources based on the body surface potential measurements and a mathematical model of the torso. Due to attenuation and spatial smoothing within the torso, this inverse problem is ill-posed. Among many regularization approaches used in the ECG literature to overcome this ill-posedness, statistical techniques have received great attention because of their flexibility to represent the data, and ability to provide performance evaluation tools for quantification of uncertainties and errors in the model. However, despite their potential to accurately reconstruct the equivalent cardiac sources, one major challenge in these methods is how to best utilize the prior information available in terms of training data. In this paper, we address the question of how to define the prior probability distributions (pdf) of the sources and the error terms so that we can obtain more accurate and robust inverse solutions. We employ two methods, maximum likelihood (ML) and maximum a posteriori (MAP), for estimating the model parameters such as the prior pdfs, error pdfs, and the state-transition matrix, based on the same training data. These model parameters are then used for the state-space representation and estimation of the epicardial potentials, which constitute the equivalent cardiac sources in this study. The performances of ML- and MAP-based model parameter estimation methods are evaluated qualitatively and quantitatively at various noise levels and geometric disturbances using two different simulated datasets. Bayesian MAP estimation, which is also a well-known statistical inversion technique, and Tikhonov regularization, which can be formulated as a special and simplified version of Bayesian MAP estimation, have been included here for comparison with the Kalman filtering method. Our results show that the state-space approach outperforms Bayesian MAP estimation in all cases; ML yields accurate results when the test and training beats come from the same physiological model, but MAP is superior to ML, especially if the test and training beats are from different physiological models.

ML and MAP estimation of parameters for the Kalman filter and smoother applied to electrocardiographic imaging

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Acknowledgments

The authors would like to thank Dr. Robert S. MacLeod and his colleagues from University of Utah, Nora Eccles Harrison Cardiovascular Research and Training Institute for the data used in this study. The qualitative assessments in this work became possible by software, Map3d, which was supported by the National Institute of General Medical Sciences of the National Institutes of Health under grant number P41 GM103545-18. The authors also would like to thank Walther H. W. Schulze and his colleagues from Karlsruhe Institute of Technology (KIT) for the data used in this study, and the Consortium of ECGI for facilitating data sharing for all researchers.

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  1. Department of Electrical and Electronics Engineering, Middle East Technical University, Üniversiteler Mahallesi Dumlupınar Bulvarı No:1, 06800, Çankaya, Ankara, Turkey

    Taha Erenler & Yesim Serinagaoglu Dogrusoz

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  1. Taha Erenler
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  2. Yesim Serinagaoglu Dogrusoz
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Correspondence to Yesim Serinagaoglu Dogrusoz.

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Erenler, T., Serinagaoglu Dogrusoz, Y. ML and MAP estimation of parameters for the Kalman filter and smoother applied to electrocardiographic imaging. Med Biol Eng Comput 57, 2093–2113 (2019). https://doi.org/10.1007/s11517-019-02018-6

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