Electroencephalography in ellipsoidal geometry

Abstract

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.

Calculation of the magnetic field due to a bioelectric current dipole in an ellipsoid

“…The ellipsoidal system [17], [13], on the other hand, has coordinates (̺, µ, ν) with semifocal distances h 1 , h 2 and h 3 , defined as…”

“…This was first proposed by Kariotou [17] and Das-sios [8], who derived formulas for Φ up to order 2 in m n . In our most recent study [15], the electric field due to a dipole in an ellipsoid was computed in a generalized approach where the corresponding expansion can be carried out to arbitrary degree.…”

Untitled

“…The density of these dipole distributions is proportional to −σ α u α on S α and to σ α u α − σ b u b on S b . Therefore, our first task is to solve the following boundary value problem, for the electric potential ∆u(r) = 0, r ∈ V, (2.10) 12) where J(r) is given by (2.6). Continuity conditions demand that the fields u, u α , u b are connected through the surface conditions…”

“…But, as anatomy indicates, the actual geometry of the human brain is best approximated by a triaxial ellipsoid [16], a geometrical shape far more complicated than the sphere or even the spheroid. Intense efforts towards a complete analytic solution for EEG and MEG problems in ellipsoidal geometry led to results included in [3,4,11,12]. The present work aims in obtaining an analytic expression of the leading quadrupolic term for the exterior magnetic field in the…”

A finite-element reciprocity solution for EEG forward modeling with realistic individual head models

“…An analytical solution has been derived for obtaining the potential distribution on the surface of an ellipsoid given an arbitrarily located current dipole [15,16,17]. The analytical approach is used as the ground truth when testing new methods.…”