Department of Electrical and Electronics Engineering, Middle East Technical University, Üniversiteler Mahallesi Dumlupınar Bulvarı No:1, 06800, Çankaya, Ankara, Turkey.
Med Biol Eng Comput. 2019 Oct;57(10):2093-2113. doi: 10.1007/s11517-019-02018-6. Epub 2019 Jul 30.
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. Graphical Abstract ML and MAP estimation of parameters for the Kalman filter and smoother applied to electrocardiographic imaging.
在心电图成像（ECGI）中，人们根据体表电位测量值和体躯的数学模型来解决心电图（ECG）的逆问题，以重建等效心源。由于体躯内的衰减和空间平滑作用，这个逆问题是不适定的。在 ECG 文献中用于克服这种不适定性的许多正则化方法中，统计技术因其灵活性而受到极大关注，这种灵活性可以用来表示数据，并能为模型中的不确定性和误差的量化提供性能评估工具。然而，尽管这些方法有能力准确地重建等效心源，但它们面临的一个主要挑战是如何最好地利用现有训练数据中的先验信息。在本文中，我们探讨了如何定义源和误差项的先验概率分布（pdf），以使我们能够获得更准确和稳健的逆解。我们使用最大似然（ML）和最大后验（MAP）两种方法来估计模型参数，如先验 pdf、误差 pdf 和状态转移矩阵，这些模型参数都是基于相同的训练数据。然后，这些模型参数用于心外膜电位的状态空间表示和估计，这是本研究中的等效心源。我们使用两种不同的模拟数据集，在不同的噪声水平和几何干扰下，对 ML 和 MAP 基于模型参数估计方法的性能进行了定性和定量的评估。我们还包括了贝叶斯 MAP 估计（也是一种著名的统计反演技术）和 Tikhonov 正则化（可以表示为贝叶斯 MAP 估计的特殊和简化版本），以便与卡尔曼滤波方法进行比较。我们的结果表明，在所有情况下，状态空间方法都优于贝叶斯 MAP 估计；当测试和训练节拍来自相同的生理模型时，ML 会产生准确的结果，但 MAP 优于 ML，尤其是当测试和训练节拍来自不同的生理模型时。
