Sensitivity Analysis of the Forward Electroencephalographic Problem Depending on Head Shape Variations

Abstract

A crucial aspect in clinical practice is the knowledge of whether Electroencephalographic (EEG) measurements can be assigned to the functioning of the brain or to geometrical deviations of the human cranium. The present work is focused on continuing to advance understanding on how sensitive the solution of the forward EEG problem is in regard to the geometry of the head. This has been achieved by developing a novel analytic algorithm by performing a perturbation analysis in the linear regime using a homogenous spherical model. Notably, the suggested procedure provides a criterion which recognizes whether surface deformations will have an impact on EEG recordings. The presented deformations represent two major cases: (1) acquired alterations of the surface inflicted by external forces and (2) deformations of the upper part of the human head where EEG signals are recorded. Our results illustrate that neglecting geometric variations present on the heads surface leads to errors in the recorded EEG measurements less than 2%. However, for severe instances of deformations combined with cortical brain activity in the vicinity of the distortion site, the errors rise to almost 25%. Therefore, the accurate description of the head shape plays an important role in understanding the forward EEG problem only in these cases.

Mathematical Foundation of Electroencephalography

“…[35] to what degree deformations present at the conductor's surface affect EEG measurements. Although the EEG data are evaluated in a view to a deformed conductor, the calculations are accomplished based on the spherical geometry, furnishing a fast analytic algorithm prone to almost minimum error.…”

On the geometrical perturbation of a three‐shell spherical model in electroencephalography

“…In the present study, extending the aforementioned model in Doschoris et al, 18 we propose a multilayer spherical conductor where all the interfaces between layers of different conductivities are considered to be locally perturbed. In this view, small deformations of the exterior head surface and/or inner interfaces are considered to stand for localized acquired alterations inflicted by external forces, such as a blow or a bump to the head, related to falls, vehicle accidents, sports, and violence, or to special features of the complicated structure of the individuals head that segmentation of the MRI data have failed to reproduce accurately.…”

“…The solution of the aforementioned boundary value problems is based upon a mathematical technique by which we introduce a Poincaré‐type expansion 18 for the electric potentials in each region: where the parameter is considered to be close to zero and functions for any are independent of this perturbation parameter. The latter, in combination with the fact that the left‐hand side of expansion () refers to the point of he perturbed case, whereas its right‐hand side depends on of the unperturbed case, constitute the key method of this project.…”

“…The method is especially effective in case of smooth surfaces, as the prolate spheroidal one that is used for validation. A similar approach that considers localized surface deformations of a spherical homogeneous head model is proposed in Doschoris et al, 18 where incorporating a perturbation parameter, which controls the depth/height of the deformation, the linear term of the perturbed electric potential is given in a closed form and provides a criterion for the validation of the significance of the deformations in the EEG forward results.…”

“…Following Doschoris et al, 18 the appropriate definition of the perturbed surface, along with the resulting perturbation of the involved operators and the introduction a Poincaré‐type expansion for the solution, leads to a system of boundary value problems associated to each other with continuity conditions and proper regularity behavior ensured by asymptotic conditions. Series expansion of each solution is obtained in compact form via the classical spherical harmonic eigenfunctions.…”