Current dipole localization with an ideal magnetometer system

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Abstract
The goal of the study was to explore the most fundamental aspects of a magnetoencephalography (MEG)-based dipole source analysis. For that purpose, a MEG measurement with an ideal magnetometer system (providing the radial component of the magnetic field as a continuous function) is considered. The analytical formulas derived for the variances and covariances of the parameter estimation errors, validated by means of Monte Carlo simulations, allow quantitative predictions in terms of dipole depth, radius and span of the magnetometer system, signal-to-noise (SNR) ratio, and other parameters. A negative correlation exists between radial coordinate and longitudinal component of the moment (perpendicular to radial direction, same plane as actual dipole moment and center of sphere), whereas the other parameters are independent. The standard deviations of the five dipole parameters show fundamental differences with respect to their asymptotic behavior for deep dipoles: If the root mean square (rms) value of the magnetic field is kept constant (moment with depth-dependent amplitude), the error for the transverse coordinate (perpendicular to radial and longitudinal coordinate) is proportional to the distance R between dipole and center of sphere, the errors for the other dipole coordinates, and the relative error for the transverse component of the dipole moment are constant, and the relative error for the longitudinal component of the dipole moment follows a 1/R law.