Effects of modeling errors on least squares error solutions to the inverse problem of electrocardiology

Abstract

Modeling errors produced by differences between an actual electrical source in the heart and a model of the source have effects on least squares error (LSE) solutions for the model. These effects are analyzed using linear algebra theory and some modeling errors representative of those for actual heart measurements. It is found that increasing the number of dipoles in a multiple-dipole (MD) model increases the sensitivity of the solutions to small modeling error changes while increasing the number of terms in a multipole expansion (ME) model is likely to improve the accuracy of the solution for the dipolar terms. The results show that a MD model with more than two free-orientation dipoles will not provide accurate, useful, or reliable information for the representative modeling errors. Accurate solutions for the dipolar terms in a ME model are obtained for a model containing dipole, quadrupole, and octupole terms for these modeling errors. A new type of “average” LSE (AVL) solution for MD models with fixed-orientation dipoles is developed and compared with LSE and nonnegative LSE (NNL) solutions; the AVL solutions are found to be least sensitive to small modeling error changes but to provide misleading information for certain modeling errors. Finally, it is found that once a certain density of measurements is reached, the effects of modeling errors are nearly independent of the measurements used and, hence, no improvement in the solutions can be obtained by increasing the number of measurements.