A Weighting Matrix to Remove Depth Bias in the Linear Biomagnetic Inverse Problem with Application to Cardiology

Abstract

The determination of unknown electrical sources within the body from external magnetic field measurements is known as the biomagnetic inverse problem. For electrical sources modeled by multiple current dipoles of fixed location and orientation and variable strength, the inverse problem is linear, and it is often ill-posed [1]. The ill-posedness can be addressed by specifying that the inverse Solution be the minimum norm least squares estimate (MNLE) which exists and is unique [3]. The MNLE, however, has a major shortcoming in that it favors the sources that are closest to the magnetic field sensors [l][2]. This so called depth bias yields solutions that are physiologically implausible, especially for three dimensional source spaces. One method of altering the characteristics of the MNLE is to employ a weighting matrix in the formulation and to solve for the corresponding weighted minimum norm least squares estimate (WMNLE) [4]. The purpose of this paper is to present a specific weighting matrix that removes the bias to depth. Comparisons are made to both the unweighted case and the case where the weight is derived from column normalization of the forward matrix. Finally, the suggested weighting matrix is applied to magnetocardiographic data for a spherical multiple dipole model.