An Inverse Source Problem for Maxwell's Equations in Magnetoencephalography

Consider an inverse source problem for Maxwell's equations which arises in determining locations of epileptic foci in the living human brain. The human brain is modeled as a heterogeneous medium, where the electric permittivity, magnetic permeability, and conductivity may all be functions. A current dipole is used to model the epilepsy. The inverse source problem in this context is to determine the current dipole from boundary measurements of the fields. In this paper, a new computational method is introduced for solving the inverse problem. The method is based on a low-frequency asymptotic analysis of Maxwell's equations. Our method is constructive and has good convergence properties. A crucial step of the method is to construct special test functions. Uniqueness and stability results for the inverse problem are also established. The method and results are expected to find applications particularly in magnetoencephalography.

[1]  Identification of cracks by boundary measurements at low frequencies , 2000 .

[2]  J. Sarvas.  Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. , 1987, Physics in medicine and biology.

[3]  Sailing He,et al.  Identification of dipole sources in a bounded domain for Maxwell's equations , 1998 .

[4]  I. Gelfand,et al.  Inversion method for magnetoencephalography , 1996 .

[5]  Sailing He,et al.  Frequency series expansion of an explicit solution for a dipole inside a conducting sphere at low frequency. , 1998, IEEE transactions on bio-medical engineering.

[6]  David Colton,et al.  The uniqueness of a solution to an inverse scattering problem for electromagnetic waves , 1992 .

[7]  Victor Isakov,et al.  Inverse Source Problems , 1990 .

[8]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[9]  David Isaacson,et al.  A linearized inverse boundary value problem for Maxwell's equations , 1992 .

[10]  H. Ammari,et al.  Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume , 2003 .

[11]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[12]  C. Tai,et al.  Dyadic green functions in electromagnetic theory , 1994 .

[13]  Gunther Uhlmann,et al.  An inverse boundary value problem for Maxwell's equations , 1992 .

[14]  David Colton,et al.  Time harmonic electromagnetic waves in an inhomogeneous medium , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  H. Ammari,et al.  Time-harmonic electromagnetic fields in thin chiral curved layers , 1998 .

[16]  R. Ilmoniemi,et al.  Magnetoencephalography-theory, instrumentation, and applications to noninvasive studies of the working human brain , 1993 .

[17]  Vijay K. Varadan,et al.  Time-Harmonic Electromagnetic Fields in Chiral Media , 1989 .

[18]  Petri Ola,et al.  An inverse boundary value problem in electrodynamics , 1993 .

[19]  J. Nédelec,et al.  Propagation d'ondes électromagnétiques à basses fréquences , 1998 .