Mathematical methods for magnetoencephalography forward and inverse modeling with enhanced spatial resolutions

Abstract

Next-generation magnetoencephalography (MEG) sensors, optically-pumped magnetometers (OPMs), can be placed closer to the head and are expected to measure signals with improved spatial resolutions. This introduces many challenges and opportunities. For one, presently-used methods for lower frequency signals, especially approximations, may now require improvements in order to adequately resolve the new signals' higher frequency components. Moreover, underlying errors may become more pronounced, since they are also measured with higher resolutions. The Nyquist sampling theorem indicates that a denser sensor array is required for OPM systems; however, this may not always be practical, hence inquiries into areas such as sensor selection are can also be opened. In this dissertation, three mathematical considerations that may be found in each of the three contexts described above are assessed. First, we consider cubature approximations of the flux signal. Novel (near)-analytical methods of flux evaluations are developed, and the errors due to cubature approximations are evaluated. Second, we consider the effects of inaccurate boundary element head models on the signal and source localizations at varying sensor array distances. An approximate signal-to-noise ratio bound is established such that any sufficiently noiseless signal above bound will experience a noticeable effect from head model inaccuracies. Third, we propose using the QR pivoting algorithm to determine a sparse number of optimal sensor locations, deviating from the usual strive towards denser sensor arrays. As part of establishing the theory that leads up to these three specific considerations, we also derive novel quasi-static magnetic equations that are the general forms of current equations used in MEG. From these equations, the translational non-invariances of open current segments can be observed, which leads to many implications in forward and inverse models especially when it comes to source interpretations.