Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem

TLDR

The review covers fundamental mathematical and physical concepts of the biomagnetic inverse problem, including the forward problem for homogeneous and inhomogeneous media. The authors present Gesselowitz’s formulae and a surface integral equation for piecewise homogeneous conductors, discuss special cases such as spherical symmetry and layered media, analyze non‑uniqueness and electric‑potential effects, and propose practical inverse solutions via weighted least‑squares with confidence limits and minimum‑norm estimation.

Abstract

Basic mathematical and physical concepts of the biomagnetic inverse problem are reviewed with some new approaches. The forward problem is discussed for both homogeneous and inhomogeneous media. Gesselowitz' formulae and a surface integral equation are presented to handle a piecewise homogeneous conductor. The special cases of a spherically symmetric conductor and a horizontally layered medium are discussed in detail. The non-uniqueness of the solution of the magnetic inverse problem is discussed and the difficulty caused by the contribution of the electric potential to the magnetic field outside the conductor is studied. As practical methods of solving the inverse problem, a weighted least-squares search with confidence limits and the method of minimum norm estimate are discussed.