A Kalman filter-based approach to reduce the effects of geometric errors and the measurement noise in the inverse ECG problem

In this article, we aimed to reduce the effects of geometric errors and measurement noise on the inverse problem of Electrocardiography (ECG) solutions. We used the Kalman filter to solve the inverse problem in terms of epicardial potential distributions. The geometric errors were introduced into the problem via wrong determination of the size and location of the heart in simulations. An error model, which is called the enhanced error model (EEM), was modified to be used in inverse problem of ECG to compensate for the geometric errors. In this model, the geometric errors are modeled as additive Gaussian noise and their noise variance is added to the measurement noise variance. The Kalman filter method includes a process noise component, whose variance should also be estimated along with the measurement noise. To estimate these two noise variances, two different algorithms were used: (1) an algorithm based on residuals, (2) expectation maximization algorithm. The results showed that it is important to use the correct noise variances to obtain accurate results. The geometric errors, if ignored in the inverse solution procedure, yielded incorrect epicardial potential distributions. However, even with a noise model as simple as the EEM, the solutions could be significantly improved

Lanczos bidiagonalization-based inverse solution methods applied to electrical imaging of the heart by using reduced lead-sets: A simulation study

In inverse problem of electrocardiography (ECG), electrical activity of the heart is estimated from body surface potential measurements. This electrical activity provides useful information about the state of the heart, thus it may help clinicians diagnose and treat heart diseases before they cause serious health problems. For practical application of the method, having fewer number of electrodes for data acquisition is an advantage. Additionally, inverse problem of ECG is ill-posed due to attenuation and smoothing within the body. Therefore, the solution of ECG inverse problem has to be regularized. In this study, we constrain ourselves to two Lanczos-bidiagonalization-based inverse solution methods, namely, Lanczos least-squares QR (L-LSQR) factorization and Lanczos truncated total least-squares (L-TTLS). Tikhonov regularization is also implemented as a base for comparison for these methods. We use body surface measurements simulated using epicardial potentials measured from the surface of canine hearts. In these experiments, the hearts are stimulated from the ventricles at various sites, mimicking ectopic beats. Torso potentials are obtained from these epicardial measurements by multiplying them with the forward transfer matrix and adding Gaussian distributed noise. We solve the inverse problem using different number of leads on the body surface (771, 192, 64, and 32 leads), and assess the performances of these regularization methods for the reduced lead-sets. These reduced lead-sets are selected from the primary 771-lead configuration by using two main approaches. The first approach is manually selecting appropriate leads, and the second one uses the inverse problem approach to select leads sequentially. The results show that the L-TTLS method is more successful in reconstructing epicardial potentials than the L-LSQR method. The L-TTLS method is faster than the Tikhonov regularization, since it benefits from bidiagonal form of the forward matrix. Reducing the number of electrodes to 64 has a small effect on the solutions, but with 32 leads, inverse solutions get less precise, and the difference between the results of Tikhonov regularization and L-TTLS method becomes less significant

The effect of interpolating low amplitude leads on the inverse reconstruction of cardiac electrical activity

Electrocardiographic imaging is an imaging modality that has been introduced recently to help in visualizing the electrical activity of the heart and consequently guide the ablation therapy for ventricular arrhythmias. One of the main challenges of this modality is that the electrocardiographic signals recorded at the torso surface are contaminated with noise from different sources. Low amplitude leads are more affected by noise due to their low peak-to-peak amplitude. In this paper, we have studied 6 datasets from two torso tank experiments (Bordeaux and Utah experiments) to investigate the impact of removing or interpolating these low amplitude leads on the inverse reconstruction of cardiac electrical activity. Body surface potential maps used were calculated by using the full set of recorded leads, removing 1, 6, 11, 16, or 21 low amplitude leads, or interpolating 1, 6, 11, 16, or 21 low amplitude leads using one of the three interpolation methods – Laplacian interpolation, hybrid interpolation, or the inverse-forward interpolation. The epicardial potential maps and activation time maps were computed from these body surface potential maps and compared with those recorded directly from the heart surface in the torso tank experiments. There was no significant change in the potential maps and activation time maps after the removal of up to 11 low amplitude leads. Laplacian interpolation and hybrid interpolation improved the inverse reconstruction in some datasets and worsened it in the rest. The inverse forward interpolation of low amplitude leads improved it in two out of 6 datasets and at least remained the same in the other datasets. It was noticed that after doing the inverse-forward interpolation, the selected lambda value was closer to the optimum lambda value that gives the inverse solution best correlated with the recorded one

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

In inverse electrocardiography, the goal is to estimate cardiac electrical sources from potential measurements on the body surface. It is by nature an ill-posed problem, and regularization must be employed to obtain reliable solutions. This paper employs the multiple constraint solution approach proposed in Brooks et al. (IEEE Trans Biomed Eng 46(1):3-18, 1999) and extends its practical applicability to include more than two constraints by finding appropriate values for the multiple regularization parameters. Here, we propose the use of real-valued genetic algorithms for the estimation of multiple regularization parameters. Theoretically, it is possible to include as many constraints as necessary and find the corresponding regularization parameters using this approach. We have shown the feasibility of our method using two and three constraints. The results indicate that GA could be a good approach for the estimation of multiple regularization parameters