On the Geselowitz formula in biomagnetics

Abstract

Abstract.A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density, is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.

The unique determination of neuronal currents in the brain via magnetoencephalography

“…Furthermore it has been correctly pointed out that this latter inversion is much simpler since the expression for ∂U/∂ |x| is simpler than the expression of U (see [5]). …”

“…For a recent rigorous derivation of this equation see [5]. Equation (1.4) relates J p inside the head with B outside the head.…”

“…and dS denotes the infinitesimal surface element on ∂Ω. For a recent rigorous derivation of this equation see [5].…”

“…where |x| denotes the length of the vector x, denotes the volume occupied by the head, ∂ is the boundary of , n denotes the unit outward vector normal to the surface ∂ and dS denotes the infinitesimal surface element on ∂ . For a recent rigorous derivation of this equation see [5]. Equation (1.4) relates J p inside the head with B outside the head.…”

Αντίστροφα Προβλήματα Στη Μαθηματική Θεωρία Της Ηλεκτρο-Μαγνητο-Εγκεφαλογραφίας

“…δημοσίευση[20] ισχύει επιπλέον και για χωρίο που περιέχουν ιδιομορφίες της μορφής (III-1.1).Τελικά, η έκφραση (III-1.21) μέσω των (III-1.29) και (III-1.30) καταλήγει στη θεμελιώδη ολοκληρωτική αναπαράσταση του Geselowitz για το μαγνητικό πεδίο…”