On the Geselowitz formula in biomagnetics

Authors:
George Dassios and Fotini Kariotou

Journal:
Quart. Appl. Math. **61** (2003), 387-400

MSC:
Primary 78A25

DOI:
https://doi.org/10.1090/qam/1976377

MathSciNet review:
MR1976377

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**L. Brand.*"Vector and Tensor Analysis"*, John Wiley & Sons, New York, 1947 MR**0021449****[2]**B. N. Cuffin and D. Cohen, ``Magnetic Fields of a Dipole in Special Volume Conductor Shapes", IEEE Trans. Biom. Eng.,**BME-24**, pp. 372-381, 1977**[3]**I. M. Gel'fand and G. E. Shilov,*"Generalized Functions"*, Vol. I-V, Academic Press, New York, 1964 MR**0166596****[4]**D. B. Geselowitz, ``On Bioelectric Potentials in an Inhomogeneous Volume Conductor", Biophys. J.,**7**, pp. 1-11, 1967**[5]**D. B. Geselowitz, ``On the Magnetic Field Generated Outside an Inhomogeneous Volume Conductor by Internal Current Sources", IEEE Trans. Magn.,**6**, pp. 346-347, 1970**[6]**F. Grynszpan and D. B. Geselowitz, ``Model Studies of the Magnetocardiogram", Biophys. J..**13**, pp. 911-925, 1973**[7]**M. S. Hämäläinen, R. Hari. R. Ilmoniemi, J. Knuutila, and O. Lounasmaa, ``Magnetoencephalography--Theory, Instrumentation, and Application to Noninvasive Studies of the Working Human Brain", Rev. Mod. Phys.,**65**, pp. 413-497, 1993**[8]**R. J. Ilmoniemi, M. S. Hämäläinen, and J. Knuutila, ``The Forward and Inverse Problems in the Spherical Model", pp. 278-282 in*Biomagnetics: Applications and Theory*, edited by Harold Weinberg, Gerhard Stroink, and Toiro Katila, Pergamon Press, New York, 1985**[9]**V. D. Kupradze,*"Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity*", North-Holland, Amsterdam, 1979 MR**530377****[10]**L. D. Landau and E. M. Lifshitz,*"Electrodynamics of Continuous Media"*, Pergamon Press, Oxford, 1960**[11]**J. Malmivuo and R. Plonsey,*"Bioelectromagnetism"*, Oxford, Univ. Press, New York, 1995**[12]**G. Nolte, T. Fieseler, and G. Curio, ``Perturbative Analytical Solutions of the Magnetic Forward Problem for Realistic Volume Conductors", J. Appl. Phys.,**89**, pp. 2360-2369, 2001**[13]**J. Sarvas, ``Basic Mathematical and Electromagnetic Concepts of the Biomagnetic Inverse Problem", Phys. Med. Biol.,**32**, pp. 11-22, 1987

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
78A25

Retrieve articles in all journals with MSC: 78A25

Additional Information

DOI:
https://doi.org/10.1090/qam/1976377

Article copyright:
© Copyright 2003
American Mathematical Society