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
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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.
- Louis Brand, Vector and Tensor Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall Ltd., London, 1947. MR 0021449
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
- I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
D. B. Geselowitz, “On Bioelectric Potentials in an Inhomogeneous Volume Conductor", Biophys. J., 7, pp. 1–11, 1967
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
F. Grynszpan and D. B. Geselowitz, “Model Studies of the Magnetocardiogram", Biophys. J.. 13, pp. 911–925, 1973
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
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
- V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, Translated from the second Russian edition, North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1979. Edited by V. D. Kupradze. MR 530377
L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media", Pergamon Press, Oxford, 1960
J. Malmivuo and R. Plonsey, "Bioelectromagnetism", Oxford, Univ. Press, New York, 1995
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
J. Sarvas, “Basic Mathematical and Electromagnetic Concepts of the Biomagnetic Inverse Problem", Phys. Med. Biol., 32, pp. 11–22, 1987
L. Brand. "Vector and Tensor Analysis", John Wiley & Sons, New York, 1947
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
I. M. Gel’fand and G. E. Shilov, "Generalized Functions", Vol. I-V, Academic Press, New York, 1964
D. B. Geselowitz, “On Bioelectric Potentials in an Inhomogeneous Volume Conductor", Biophys. J., 7, pp. 1–11, 1967
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
F. Grynszpan and D. B. Geselowitz, “Model Studies of the Magnetocardiogram", Biophys. J.. 13, pp. 911–925, 1973
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
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
V. D. Kupradze, "Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity", North-Holland, Amsterdam, 1979
L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media", Pergamon Press, Oxford, 1960
J. Malmivuo and R. Plonsey, "Bioelectromagnetism", Oxford, Univ. Press, New York, 1995
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
J. Sarvas, “Basic Mathematical and Electromagnetic Concepts of the Biomagnetic Inverse Problem", Phys. Med. Biol., 32, pp. 11–22, 1987
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© Copyright 2003
American Mathematical Society
