Genetic algorithm-based regularization parameter estimation for the inverse electrocardiography problem using multiple constraints

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

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.

Accuracy of electrocardiographic imaging using the method of fundamental solutions

Solving the inverse problem of electrocardiology via the Method of Fundamental Solutions has been proposed previously. The advantage of this approach is that it is a meshless method, so it is far easier to implement numerically than many other approaches. However, determining the heart surface potential distribution is still an ill-posed problem and thus requires some form of Tikhonov regularisation to obtain the required distributions. In this study, several methods for determining an "optimal" regularisation parameter are compared in the context of solving the inverse problem of electrocardiology via the Method of Fundamental Solutions. It is found that the Robust Generalised Cross-Validation method most often yields epicardial potential distributions with the least relative error when compared to the input distribution. The study also compares the inverse solutions obtained with the Method of Fundamental Solutions with those obtained in a previous study using the boundary element method. It is found that choosing the best solution methodology and regularisation parameter determination method depends on the particular scenario being considered.

Application of robust Generalised Cross-Validation to the inverse problem of electrocardiology

Robust Generalised Cross-Validation was proposed recently as a method for determining near optimal regularisation parameters in inverse problems. It was introduced to overcome a problem with the regular Generalised Cross-Validation method in which the function that is minimised to obtain the regularisation parameter often has a broad, flat minimum, resulting in a poor estimate for the parameter. The robust method defines a new function to be minimised which has a narrower minimum, but at the expense of introducing a new parameter called the robustness parameter. In this study, the Robust Generalised Cross-Validation method is applied to the inverse problem of electrocardiology. It is demonstrated that, for realistic situations, the robustness parameter can be set to zero. With this choice of robustness parameter, it is shown that the robust method is able to obtain estimates of the regularisation parameter in the inverse problem of electrocardiology that are comparable to, or better than, many of the standard methods that are applied to this inverse problem.