Yesim Serinagaoglu

Use of Activation Time Based Kalman Filtering in Inverse Problem of Electrocardiography

The goal of this study is to solve inverse problem of electrocardiography (ECG) in terms of epicardial potentials using body surface (torso) potential measurements. The problem is ill-posed and regularization must be applied. Kalman filter is one of the regularization approaches, which includes both spatial and temporal correlations of epicardial potentials. However, in order to use the Kalman filter, one needs the state transition matrix (STM) that models the time evolution of the epicardial potentials. In inverse ECG literature, STM is either chosen as identity matrix or calculated from true epicardial potentials. The latter approach gives better results, however 1) It yields a matrix equation with a large size. 2) Not realistic. In this study we address the 1st shortcoming. Usually epicardial potential in one lead only depends on a limited number of leads; STM entries are close to zero for the remaining leads. In this study, we used simulated torso potentials, and constructed STM from true epicardial potentials. We used three different approaches to reduce the dimension of the problem: epicardial potential at one lead is assumed to be related to 1) Only the leads in its neighborhood, 2) The leads that are activated at around the same time (close activation times), (3) Both the leads with close activation times and its first order neighbors. The STM estimation problem is redefined to calculate only the limited number of related entries; the remaining STM entries are set to zero, hence reducing the problem size. The calculated STM is then used in the Kalman filter to estimate the epicardial potential distribution and later in the Kalman smoother to further reduce errors.

Spatio-Temporal Solutions in Inverse Electrocardiography

Conventional electrocardiography (ECG) is an essential tool for investigating cardiac disorders such as arrhythmias or myocardial infarction. It involves interpretation of potentials recorded at the body surface that occur due to the electrical activity of the heart. Estimation of epicardial potentials from these recorded signals is known as inverse problem of ECG. It is difficult to solve this problem for effective cardiac imaging due to the ill-posed nature and high dimensionality of the problem. There are many solution approaches in order to cope with these difficulties. We used Tikhonov regularization and Bayesian Maximum A Posteriori (MAP) estimation methods in this study. The traditional approach is to solve the problem at each time instant separately (column sequential approach). This is the fastest and easiest approach; however it does not include temporal correlations of the epicardial potentials. Greensite (2002) proposed certain specific assumptions about structures inherent in the problem formulation that allow the use of spatial and temporal constraints simultaneously. In this study, we applied this framework to solve the spatiotemporal inverse ECG problem. We first applied a temporal whitening filter to the original problem. The new set of equations, which became temporally decorrelated, was solved using the column sequential approach. The desired solution was obtained by transforming the result back into the original domain. In the spatiotemporal Bayesian MAP approach, the covariance matrix also changes after the application of the whitening transformation. We also derived an expression for the new covariance matrix in terms of the transformation matrix and the original covariance matrix of the epicardial potentials.

Electromechanical heart tissue model using cellular automaton

In this study, electrical and mechanical properties of the heart tissue are modeled for normal heart beat. Contraction of the tissue via electrical activation has also been considered in terms of time synchronization. “Cellular automaton” method is used for modeling the 2 dimensional heart tissue and electromechanical interactions. Using this method, both the normal heart beats electrical activation and the arrhythmia excitation could be taken on, without using complex differential equations. To consider the anisotropy of the heart tissue, fiber orientations have also been added to the model.

Estimation of state transition matrix in the Kalman filter based inverse ECG solution with the help of training sets

At this study the main motivation is to solve inverse problem of ECG with Kalman filter. In order to obtain feasible solutions determination of the state transition matrix (STM) correctly is vital. In literature the STM is usually found by using the test data itself which is not a realistic scenario. The major goal of this study is to determine STM without using test data. For that purpose a two stage method is suggested. At the first step the probability density function (pdf) is calculated using training sets and then this pdf is used to find Bayes-MAP solution which uses only spatial information. At the second step, the Bayes-MAP solution is used to find STM and later on, that STM is used in Kalman filter to obtain final results. It is seen that the results obtained with this method are better then normal Bayes-MAP results and the errors are within acceptable limits. So it is concluded that the usage of Bayes-MAP solutions in STM determination is a serious alternative for STM estimation.

Statistical modeling of the geometric error in cardiac electrical imaging

Kalman filter approach provides a natural way to include the spatio-temporal prior information in cardiac electrical imaging. This study focuses on the performance of Kalman filter approach with geometric errors present in inverse Electrocardiography (ECG) problem. The geometric errors considered here are the wrong determination of the hearts size and location. In addition to Kalman filtering, we also compare the performances of Tikhonov regularization and Bayesian MAP estimation when geometric errors are present. After presenting the effects of geometric errors on the solutions, a possible model to reduce the effects of the geometric errors in the inverse ECG problem for Bayes-MAP and Kalman solution is studied. To this purpose, a method that is suggested to overcome modeling errors in inverse problem solutions by Heino et. al. is modified and its effectiveness for inverse ECG problem is shown. Here the main idea is to assume geometric errors as additive noise and adding them to the covariance matrices used in the algorithms [1]. To the best of our knowledge, this is the first study in which it has been applied to the inverse problem of ECG.

Implementation of the minimum relative (cross) entropy method for the solution of inverse ECG problem

Minimum relative (cross) entropy method can be used to solve linear inverse electrocardiography (ECG) problem. Inverse ECG problem has a form d=Gm where m is a vector of unknown model parameters (epicardial potentials), d is a vector of measurements (torso potentials) and G is the forward transfer matrix. MRE method treats the elements of m as random variables and obtains the potential distribution and the solution as the expected value of the posterior distribution. The prior information about lower and upper bounds of m and a prior expected value of m are needed to obtain the epicardial potential distribution. In this study, MRE method is tested with various lower and upper bounds and expected values. Test results are compared with the true potentials, with each other and with Tikhonov regularization method results. These results show that similar solutions to Tikhonov are obtained with robust prior information and better results are obtained with better prior information. In addition, prior expected value of m is more effective than the lower and upper bounds of m.

Graphical user interface for inverse problem of electrocardiography

In this study several inverse problem of electrocardiography (ECG) solution algorithms are combined to be accessed with a single graphical user interface. This interface is designed to be used for both research and educational purposes. Although this interface is mainly designed for the inverse problem of ECG it could be benefited to solve other inverse problems as well. This interface provides a number of error functions as well as a direct link to Map3D program which provides a 3 dimensional display of the results on the heart surface. It is certain that the quantitive and visual results displayed in this interface ease the comparison of different inverse ECG algorithms significantly.

The effects of geometric errors on inverse ECG solutions using Kalman filter and Bayesian MAP estimation

Geometric errors in inverse ECG are usually the errors occur in the mathematical model used for solution due to wrong interpretation of hearts position and size, conductivities of organs in the model and electrode positions. In this study the effects of geometric errors in inverse ECG problem for Kalman filter and Bayes-MAP methods are studied. Furthermore the method suggested by Kaipio et. al., which assumes that these geometric errors are additive noise and independent of the epicardial potentials, is implemented. With this method, the effects of geometric errors on Kalman filter and Bayes-MAP solutions are reduced at the cost of smoothing the wavefront.

Imaging the electrical activity of the heart using a Kalman filter based approach: Comparison of results using different STM's

Kalman filter based solutions have been of particular interest in inverse problem of Electrocardiography (ECG) in recent years. One of the major problems with this approach however is the determination of the state transition matrix (STM) that relates the epicardial potentials at the current time instant to the potentials at the previous time instant. In this work, we use the solutions of Tikhonov regularization and Bayes-MAP algorithm to construct a STM, and we use these STMs in Kalman filtering. Our results indicate that the Kalman filter that uses a STM obtained from Bayes-MAP solutions produces accurate solutions.

Comparison of Bayesian MAP Estimation and Kalman Filter Methods in the Solution of Spatio-Temporal Inverse ECG Problem

In this study, spatial only, and spatio-temporal Bayesian Maximum a Posteriori (MAP) methods and an another spatio-temporal method, the Kalman filter approach, are used to solve the inverse electrocardiography (ECG) problem. Training sets are used to obtain the required a priori information for all methods. Two different approaches are employed to calculate the state transition matrix (STM), which maps the epicardial potentials in two consecutive time instants in the Kalman filter method. The first one uses the training set itself to iteratively estimate the STM, and the second one uses the candidate solution obtained using the spatial only Bayesian MAP estimate. The results are quantitatively compared using the correlation coefficient, the relative difference measurement star, the computation time measures, and qualitatively compared using spatial and temporal displays of epicardial potentials.