Lorraine G. Olson

A generalized eigensystem approach to the inverse problem of electrocardiography

The authors develop a new approach to the ill-conditioned inverse problem of electrocardiography which employs finite element techniques to generate a truncated eigenvector expansion to stabilize the inversion. Ordinary three-dimensional isoparametric finite elements are used to generate the conductivity matrix for the body. The authors introduce a related eigenproblem, for which a special two-dimensional isoparametric area matrix is used, and solve for the lowest eigenvalues and eigenvectors. The body surface potentials are expanded in terms, of the eigenvectors, and a least squares fit to the measured body surface potentials is used to determine the coefficients of the expansion. This expansion is then used directly to determine the potentials on the surface of the heart. The number of measurement points on the surface of the body can be less than the number of finite element nodes on the body surface, and the number of modes employed in the expansion can be adjusted to reduce errors due to noise.<<ETX>>

A SINGULAR FUNCTION BOUNDARY INTEGRAL METHOD FOR THE LAPLACE EQUATION

The authors present a new singular function boundary integral method for the numerical solution of problems with singularities which is based on approximation of the solution by the leading terms of the local asymptotic expansion. The essential boundary conditions are weakly enforced by means of appropriate Lagrange multipliers. The method is applied to a benchmark Laplace-equation problem, the Motz problem, giving extremely accurate estimates for the leading singular coefficients. The method converges exponentially with the number of singular functions and requires a low computational cost. Comparisons are made to the analytical solution and other numerical methods.

Generalized eigensystem techniques for the inverse problem of electrocardiography applied to a realistic heart-torso geometry

The authors have previously proposed two novel solutions to the inverse problem of electrocardiography, the generalized eigensystem technique (GES) and the modified generalized eigensystem technique (tGES), and have compared these techniques with other numerical techniques using both homogeneous and inhomogeneous eccentric spheres model problems. In those studies the authors found their generalized eigensystem approaches generally gave superior performance over both truncated singular value decomposition (SVD) and zero-order Tikhonov regularization (TIK). Here, the authors extend the comparison to the case of a realistic heart-torso geometry. With this model, the GES and tGES approaches again provide smaller relative errors between the true potentials and the numerically derived potentials than the other methods studied. In addition, the isopotential maps recovered using GES and tGES appear to be more accurate than the maps recovered using either SVD and TIK.

An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefftcients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.

Pre-surgical CT/FEA for craniofacial distraction : I. Methodology, development, and validation of the cranial finite element model

Recently, surgeons have begun to treat serious congenital craniofacial deformities including craniosynostoses with mechanical devices that gradually distract the skull. As a prospective means of treatment planning for such complex deformities, FE models derived from routine preoperative CT scans (CT/FEA) would provide ideal patient specific engineering analyses. The purpose of this study was to assess the dimensional and predictive accuracy of the CT/FEA process through the development of a 3D model of a dry human calvarium subjected to two-point distraction ex vivo. Comparative skull measurements revealed that CT/FEA construction error did not exceed 1% for transcranial dimensions, and the thickness error did not exceed 8.66% or 0.31 mm. CT/FEA strain predictions for the central region of the skull, between the distraction posts, were not statistically different from homologous gage values at P < 0.05. Peripherally, however, the strain fields were less well behaved and the FE predictions showed only general qualitative agreement with gage recordings.

Presurgical finite element analysis from routine computed tomography studies for craniofacial distraction: II. An engineering prediction model for gradual correction of asymmetric skull deformities.

&NA; Finite element analysis from routine computed tomography studies (CT/FEA) allows clinicians to predict the mechanical and anatomic consequences of specific distraction systems before human application. A realistic three‐dimensional CT/FEA engineering model of an actual plagiocephalic infant with unicoronal synostosis was developed using 4215 parabolic triangular shell elements and intracranial pressure conditions ranging from 10 to 20 mmHg. The completed finite element analysis model was used to predict the anatomic outcome of multiaxial distraction delivered by hypothetical patterns of rod and node distraction units. The predictions for the various patterns of distraction units were also compared quantitatively with respect to force, stress, strain, and intracranial volume. Best anatomic corrections were achieved with bilateral patterns of distraction units that simultaneously elongated the ipsilateral cranium and shortened the contralateral cranium. Greatest strain levels were experienced within the osteotomy callus, greatest stress levels at the appliance anchorage sites, and the greatest rod force at the ipsilateral lower coronal position. (Plast. Reconstr. Surg. 102: 1395, 1998.)

Computational issues arising in multidimensional elliptic inverse problems: the inverse problem of electrocardiography

We compare a recently proposed generalized eigensystem approach and a new modified generalized eigensystem approach to more widely used truncated singular value decomposition and zero‐order Tikhonov regularization for solving multidimensional elliptic inverse problems. As a test case, we use a finite element representation of a homogeneous eccentric spheres model of the inverse problem of electrocardiography. Special attention is paid to numerical issues of accuracy, convergence, and robustness. While the new generalized eigensystem methods are substantially more demanding computationally, they exhibit improved accuracy and convergence compared with widely used methods and offer substantially better robustness.

A comparison of higher-order generalized eigensystem techniques and tikhonov regularization for the inverse problem of electrocardiography *

In a recent series of papers we proposed a new class of methods, the generalized eigensystem (GES) methods, for solving the inverse problem of electrocardiography. In this paper, we compare zero, first, and second order regularized GES methods to zero, first, and second order Tikhonov methods. Both optimal results and results from parameter estimation techniques are compared in terms of relative error and accuracy of epicardial potential maps. Results from higher order regularization depend heavily on the exact form of the regularization operator, and operators generated by finite element techniques give the most accurate and consistent results. In the optimal parameter case, the GES techniques produce smaller average relative errors than the Tikhonov techniques. However, as the regularization order increases, the difference in average relative errors between the two techniques becomes less pronounced. We introduce the minimum distance to the origin (MDO) technique to choose the number of expansion modes ...

Eigenproblems from finite element analysis of fluid-structure interactions

We present two methods for calculating resonant frequencies and mode shapes for fluid-structure vibration problems. We model the fluid-structure interaction problem using the ?-U-P0 finite element technique. This eigenproblem is more complicated than for ordinary structural analyses, as the formulation produces a quadratic eigenproblem (similar to those for gyroscopic systems). We derive properties for this quadratic eigenproblem from an equivalent linear problem. From these properties we develop eigenvalue solution techniques based on the determinant search and subspace iteration procedures.

Performance of generalized eigensystem and truncated singular value decomposition methods for the inverse problem of electrocardiography

Singular Value Decomposition (SVD) and Generalized Eigensystem (GES) inverse techniques are compared for their ability to solve the inverse problem of electrocardiography. In the inverse problem of electrocardiography, electrical potential data for numerous locations on the body (torso) surface is used to infer the electrical potentials on the heart surface, while the governing equation and material properties are assumed known. This paper addresses two areas. First, the previously observed improved performance of GES compared to SVD is explained in terms of the unique nature of the GES vectors. Second, epicardial data from six in-vitro rabbit heart experiments are used to project body surface data for six different geometries, and inverse solutions are computed both with and without added noise. For concentric geometries, GES outperformed SVD in all instances. For eccentric heart/body geometries, GES outperformed SVD when the inverse errors themselves were small. In all cases, GES was less sensitive to a...