Probability-based current dipole localization from biomagnetic fields

Focal biomagnetic sources are described as pointlike current dipoles. The dipole parameters, position, and moment coordinates are commonly determined from biomagnetic data using iterative nonlinear optimization algorithms such as the Levenberg-Marquardt algorithm. However, even for single-dipole sources, mislocalizations can occur due to side minima of the cost function or due to a wrong choice of the start vector. This can be shown by introducing a cost function where the independent variables are only the position coordinates instead of position and moment coordinates. This dimensional reduction-which is also possible for multiple dipole sources-is achieved by calculating the cost function at each position with the position and data-dependent, optimum dipole moments. The authors call these dipoles with-in a least squares sense-optimum moments, locally optimal dipoles. The visualization of such a single-dipole cost function and of the iteration steps of the Levenberg-Marquardt algorithm show why mislocalizations cannot be avoided. Therefore, the authors propose an alternative noniterative localization algorithm for single-dipole sources without this drawback. It uses localization probabilities calculated by means of the locally optimal dipoles. Besides the determination of the dipole parameters, the proposed algorithm furnishes a reliable error for each localization. Its effectiveness is shown with simulated and real patient data.