The direct MEG problem in the presence of an ellipsoidal shell inhomogeneity

Authors:
George Dassios and Fotini Kariotou

Journal:
Quart. Appl. Math. **63** (2005), 601-618

MSC (2000):
Primary 78M99, 35QXX

DOI:
https://doi.org/10.1090/S0033-569X-05-00971-2

Published electronically:
July 26, 2005

MathSciNet review:
2187922

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The forward problem of Magnetoencephalography for an ellipsoidal inhomogeneous shell-model of the brain is considered. The inhomogeneity enters through a confocal ellipsoidal shell exhibiting different conductivity than the one of the brain tissue. It is shown that, as far as the leading quadrupolic moment of the exterior magnetic field is concerned, the complicated expression associated with the field itself is the same as in the homogeneous case, while the effect of the shell is focused on the form of the generalized dipole moment. In contrast to the spherical case, where no shell inhomogeneities are ``readable'' outside the skull, the ellipsoidal shells establish their existence on the exterior magnetic induction field in a way that depends not only on the geometry but also on the conductivity of the shell. The degenerated spherical results are fully recovered.

**1.**B.N. Cuffin and D. Cohen, ``Magnetic Fields of a Dipole in Special Volume Conductor Shapes'',*IEEE Trans. Biomedical Eng.*,**BME-24**, pp. 372-381, 1997**2.**George Dassios and Fotini Kariotou,*On the Geselowitz formula in biomagnetics*, Quart. Appl. Math.**61**(2003), no. 2, 387–400. MR**1976377**, https://doi.org/10.1090/qam/1976377**3.**George Dassios and Fotini Kariotou,*Magnetoencephalography in ellipsoidal geometry*, J. Math. Phys.**44**(2003), no. 1, 220–241. MR**1946700**, https://doi.org/10.1063/1.1522135**4.**George Dassios and Fotini Kariotou,*On the exterior magnetic field and silent sources in magnetoencephalography*, Abstr. Appl. Anal.**4**(2004), 307–314. MR**2064143**, https://doi.org/10.1155/S1085337504306032**5.**A. S. Fokas, Y. Kurylev, and V. Marinakis,*The unique determination of neuronal currents in the brain via magnetoencephalography*, Inverse Problems**20**(2004), no. 4, 1067–1082. MR**2087980**, https://doi.org/10.1088/0266-5611/20/4/005**6.**J.C. de Munck, ``The Potential Distribution in a Layered Anisotropic Spheroidal Volume Conductor'',*J. Appl. Phys.*,**64**, pp. 464-470, 1988**7.**D. B. Geselowitz, ``On Bioelectric Potentials in an Inhomogeneous Volume Conductor'',*Biophys. J.*,**7**, pp. 1-11, 1967**8.**D. B. Geselowitz, ``On the Magnetic Field Generated Outside an Inhomogeneous Volume Conductor by Internal Current Sources'',*IEEE Trans. Magn.*,**MAG-6**, pp. 346-347, 1970**9.**E. W. Hobson,*The theory of spherical and ellipsoidal harmonics*, Chelsea Publishing Company, New York, 1955. MR**0064922****10.**R. J. Ilmoniemi, M. S. Hämäläinen and J. Knuutila, ``The Forward and Inverse Problems in the Spherical model'', pp. 278-282, in*Biomagnetism: Applications and Theory*, edited by Harold Weinberg, Gerhard Stroink, and Toivo Katila, Pergamon Press, New York, 1985**11.**Kamvyssas, G. and Kariotou, F., ``Confocal Ellipsoidal Boundaries in EEG Modeling'', Bulletin of the Greek Mathematical Society (in press)**12.**Fotini Kariotou,*Electroencephalography in ellipsoidal geometry*, J. Math. Anal. Appl.**290**(2004), no. 1, 324–342. MR**2032245**, https://doi.org/10.1016/j.jmaa.2003.09.066**13.**J. Malmivuo and R. Plonsey, ``*Bioelectromagnetism*'', Oxford University Press, New York, 1995**14.**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**15.**J. Sarvas, ``Basic Mathematical and Electromagnetic Concepts of the Biomagnetic Inverse Problem'',*Phys. Med. Biol.*,**32**, pp. 11-22, 1987**16.**W.S. Snyder, M.R. Ford, G.G. Warner and H.L. Fisher, Jr., ``Estimates of Absorbed Fractions for Monoenergetic Photon Sources Uniformly Distributed in Various Organs of a Heterogeneous Phantom'',*Journal of Nuclear Medicine*, Supplement Number 3, August 1969, Volume 10, Pamphlet No. 5, Revised 1978

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
78M99,
35QXX

Retrieve articles in all journals with MSC (2000): 78M99, 35QXX

Additional Information

**George Dassios**

Affiliation:
Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, and ICEHT/FORTH

**Fotini Kariotou**

Affiliation:
Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, and Hellenic Open University

DOI:
https://doi.org/10.1090/S0033-569X-05-00971-2

Received by editor(s):
August 13, 2004

Published electronically:
July 26, 2005

Article copyright:
© Copyright 2005
Brown University