Solving the Electrocardiography Inverse Problem by Using an Optimal Algorithm Based on the Total Least Squares Theory   

The inverse problem of electrocardiography (ECG) is to estimate the activity of the heart from the measured body surface potentials (BSPs) using the numerical methods. Since it is ill-posed, it needs the dedicated numerical regularization process. The common regularization methods used in the ECG inverse problem, like Tikhonov regularization and Truncated Singular Decomposition (TSVD), are based on the linear least square, in which only the measurement noise in the right hand side BSP data is considered. In the paper, we attempt to investigate the ECG inverse problem with a full consideration of noises/errors appearing on both sides of the linear system equation Ax=b. Here, we solve the ECG inverse problem that reconstructs epicardial potentials (EPs) from BSPs by using an optimal algorithm based on the Total Least Squares (TLS) theory - Regularized TLS (RTLS), which has not been investigated in the bioelectromagnetic inverse problems. The algorithm is tested by using a realistic shaped heart-lung-torso model with inhomogeneous conductivities. The performance of the RTLS is compared with other conventional approaches like Tikhonov and TSVD. In the numerical experiments, the EPs are reconstructed with a combination of measurement noise and/or geometry errors. The simulation results demonstrate that, in most cases, RTLS performs better than Tikhonov and TSVD. With proper physiological constraints, RTLS is able to robustly reconstruct EPs from BSPs, and therefore is a promising, feasible alternative for the ECG inverse problem.

[1]  Y Rudy,et al.  The inverse problem in electrocardiography: solutions in terms of epicardial potentials. , 1988, Critical reviews in biomedical engineering.

[2]  R. Gulrajani.  The forward and inverse problems of electrocardiography. , 1998, IEEE engineering in medicine and biology magazine : the quarterly magazine of the Engineering in Medicine & Biology Society.

[3]  L. Xia,et al.  A comparison of different choices for the regularization parameter in inverse electrocardiography models , 2006, 2006 International Conference of the IEEE Engineering in Medicine and Biology Society.

[4]  Dana H. Brooks,et al.  A Method to Reconstruct Activation Wavefronts Without Isotropy Assumptions Using a Level Sets Approach , 2005, FIMH.

[5]  Gene H. Golub,et al.  Tikhonov Regularization and Total Least Squares , 1999, SIAM J. Matrix Anal. Appl..

[6]  Sabine Van Huffel,et al.  Total least squares problem - computational aspects and analysis , 1991, Frontiers in applied mathematics.

[7]  Gene H. Golub,et al.  An analysis of the total least squares problem , 1980, Milestones in Matrix Computation.

[8]  G. Golub,et al.  Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers , 2004 .

[9]  Y. Rudy,et al.  The Inverse Problem in Electrocardiography: A Model Study of the Effects of Geometry and Conductivity Parameters on the Reconstruction of Epicardial Potentials , 1986, IEEE Transactions on Biomedical Engineering.

[10]  Amir Beck,et al.  On the Solution of the Tikhonov Regularization of the Total Least Squares Problem , 2006, SIAM J. Optim..

[11]  Marc Teboulle,et al.  Finding a Global Optimal Solution for a Quadratically Constrained Fractional Quadratic Problem with Applications to the Regularized Total Least Squares , 2006, SIAM J. Matrix Anal. Appl..

[12]  R M Gulrajani,et al.  The inverse problem in electrocardiography: solutions in terms of equivalent sources. , 1988, Critical reviews in biomedical engineering.

[13]  Ling Xia,et al.  Forward and Inverse Solutions of Electrocardiography Problem Using an Adaptive BEM Method , 2007, FIMH.

[14]  S Crozier,et al.  Electrodynamic Heart Model Construction and ECG Simulation , 2006, Methods of Information in Medicine.

[15]  Y. Rudy,et al.  Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia , 2004, Nature Medicine.

[16]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[17]  Diana Maria Sima,et al.  Regularization Techniques in Model Fitting and Parameter Estimation (Regularisatietechnieken in modellering en parameterschatting) , 2006 .

[18]  Rosemary A. Renaut,et al.  Efficient Algorithms for Solution of Regularized Total Least Squares , 2005, SIAM J. Matrix Anal. Appl..

[19]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[20]  L. Xia,et al.  Combination of the LSQR method and a genetic algorithm for solving the electrocardiography inverse problem. , 2007, Physics in medicine and biology.