The exterior magnetic field for the multilayer ellipsoidal model of the brain

The magnetic induction field in the exterior of an ellipsoidally inhomogeneous, four-conducting-layer model of the human head is obtained analytically up to its quadrupole approximation. The interior ellipsoidal core represents the homogeneous brain while each one of the shells represents the cerebrospinal fluid, the skull and the scalp, all characterized by different conductivities. The inhomogeneities of these four domains, together with the anisotropy imposed by the use of the ellipsoidal geometry, provide the most realistic physical and geometrical model of the brain for which an analytic solution of the biomagnetic forward problem is possible. It is shown that in contrast to the spherical model, where shells of different conductivity are magnetically invisible, the magnetic induction field in ellipsoidal geometry is strongly dependent on the conductivity supports. The fact that spherical shells of different conductivity are invisible has enhanced the common belief that the biomagnetic forward solution does not depend on the conductivity profiles. As we demonstrate in the present work, this is not true. Hence, the proposed multilayered ellipsoidal model provides a qualitative improvement of the realistic interpretation of magnetoencephalography (MEG) measurements. We show that the presence of the shells of different conductivity can be incorporated in the form of the dipole vector for the homogeneous model. Numerical investigations show that the effects of shell inhomogeneities are almost as sound as the level of MEG measurements themselves. The degenerate cases, where either the differences of the conductivities within the shells disappear, or the ellipsoidal geometry is reduced to the spherical one, are also considered.