Session S34.6

Finite Element Refinements for Inverse Electrocardiography: Hybrid-Shaped Elements, High-Order Element Truncation and Variational Gradient Operator

DF Wang*, RM Kirby, CR Johnson

University of Utah
Salt Lake City, UT, USA

The inverse electrocardiographic (ECG) problem is ill-posed and hence requires regularization. By means of the finite element method, we present a study of the regularizing effects of discretization and the optimal numerical approximation for the inverse problem in practical situations. We show that conventional refinement strategies effective for the forward problem may become inappropriate for the inverse problem because of the latter’s ill-posed nature. We propose refinement strategies that specifically consider the ill-posedness of the continuum inverse problem and attempt to alleviate the ill-conditioning of its discretized version. The strategies include hybrid-shaped finite elements and truncation from high-order finite elements.
In addition, we present two criteria for numerically evaluating the quality of discretization: 1)inspecting the singular values of the transfer matrix combined with Fourier analysis and 2)solving the discretized problem in a generalized Tikhonov framework employing multiple constraints simultaneously. Solving inverse problems inevitably requires the constraining of the gradient of the solution, but calculating the gradient operator over unstructured meshes (especially point-based meshes) is difficult and often performed ad hoc. By adopting the variational principles underlying the finite element method we formulated an implicit gradient operator as part of the generalized Tikhonov regularization scheme. The parameter weighing the gradient constraint has units of length and is responsive to different refinement strategies, revealing the regularization by discretization. Our study was conducted based on a realistic 3-D torso model, and preliminary results indicated our strategies might provide guidelines for 3-D mesh generation from segmented images in practical biomedical simulations.

(Abstract Control Number: 112)