Perturbative analytical solutions of the magnetic forward problem for realistic volume conductors

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Abstract
The magnetic field induced by a current dipole situated in a realistic volume conductor cannot be computed exactly. Here, we derive approximate analytical solutions based on the fact that in magnetoencephalography the deviation of the volume conductor (i.e., the head) from a spherical approximation is small. We present an explicit integral form which allows to calculate the nth order Taylor expansion of the magnetic field with respect to this deviation from the corresponding solution of the electric problem of order n−1. Especially, for a first order solution of the magnetic problem only the well-known electric solution for a spherical volume conductor is needed. The evaluation of this integral by a series of spherical harmonics results in a fast algorithm for the computation of the external magnetic field which is an excellent approximation of the true field for smooth volume conductor deformations of realistic magnitude. Since the approximation of the magnetic field is exactly curl-free it is equally good for all components. We estimate the performance for a realistic magnitude of deformations by comparing the results to the exact solution for a prolate spheroid. We found a relevant improvement over corresponding solutions given by the boundary element method for superficial sources while the performance is in the same order for deep sources.