Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 612528, 14 pages

http://dx.doi.org/10.1155/2015/612528

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

^{1}Division of Applied Mathematics, Department of Chemical Engineering, University of Patras, 26504 Patras, Greece^{2}Institute of Chemical Engineering Sciences, Stadiou Street, P.O. Box 1414, 26504 Platani, Patras, Greece

Received 2 October 2014; Revised 22 December 2014; Accepted 31 December 2014

Academic Editor: Kalyana C. Veluvolu

Copyright © 2015 Michael Doschoris et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Reconstruction of cerebral activity via Electroencephalographic (EEG) recordings is an established tool and is of great medical significance. However, its reliability strongly depends on accurate algorithms which can efficiently handle the inverse solution, which, on the other hand, essentially depends on the preciseness of the forward problem. The geometry of the brain-head model holds a decisive role allowing the installation of analytic algorithms in a very limited number of cases [1–4] where else for realistic models computer simulations have to be introduced. In any event, an important question arises: how strong does the presence of cranial deformations influence the forward problem and therefore the accurate reconstruction of the source? A precise answer is of significant importance in clinical applications where, as an example, the precise mapping of neuronal activity is a prerequisite for neurosurgical preoperative planning [5, 6].

As of today, computer simulations studies [7–13] have provided strong evidence on the dependence of geometrical properties of the head model used on source localization. They all agree that neglecting such variations would affect the accuracy of the forward problem and therefore the reconstruction of the source as well.

Nonetheless, in order to gain a deeper comprehension of the problem a rigorous mathematical analysis is essential providing a vital step towards the recognition of the underlying phenomena as well as identifying the limitations of the developed algorithms. The forward EEG problem has been extensively scrutinized for over sixty years since Wilson and Bayley [14] attempted to quantify the interplay between neuronal activity and the potentials they generate at the scalp. Apparently, for realistic head shapes the problem can only be solved numerically. Notwithstanding, the development of analytical algorithms in the event where the surface deviates from geometries which permit the installation of closed form solutions, such as the sphere or the ellipsoid, can be attempted in the framework of perturbation methods [15, 16]. A first step towards understanding the contribution of surface deformations on the forward problem has been taken by Nolte and Curio [17]. The aforementioned authors, by realizing a perturbation technique different than the one displayed in the sequel, exploit the fact that if the shape of the head slightly varies from the sphere, one can accurately express the surface potential as a linear combination of spherical harmonics using only a small number of coefficients. Their analytical solution, derived utilizing Geselowitz’s integral formula [18], does not involve a perturbation* parameter* in the usual sense.

The present paper pursues a similar perspective, that is, approximating the human brain by a homogeneous spherical conductor. Next, with the aid of explicit mathematical relations the surface of the model is distorted. First, we investigate the effect of surface deformations by localized impact on the forward EEG problem. In general, head injuries show a high incidence rate followed by devastating neurological outcome [19]. On the other hand, anthropometric analysis has documented a difference in the form of the human head depending on gender and ethnicity [20, 21]. These differences on the forward EEG problem are examined in the sequel. To the authors’ knowledge no data exists regarding the extent of closed head injuries (CHI) or in that manner geometric variations of the head in general. Under these circumstances, the deformations under considerations are considered small. Compared to the model by Nolte and Curio, the explicit solution presented incorporates a perturbation parameter which allows, independently of the chosen surface, an in-deep examination of the effect of the extent of surface deformations on the scalp potential. As a result, the first-order correction provides a criterion in order to validate the significance of head shape deformations on EEG recordings.

The paper is structured as follows. Section 2 presents the mathematical background leading to closed form solutions for the forward EEG problem in spherical coordinates. The necessary adjustments which have to be made in order to obtain explicit solutions for a deformed spherical head are laid down in Section 3. In both sections, effort has been made to keep the mathematical display to a minimum, featuring only key relations. The interested reader will find the required details in order to obtain these expressions in the appendices. Finally, Section 4 is devoted to the numerical implementation and interpretation of the derived results whereas the Discussion summarizes the presented method and findings.

#### 2. Mathematical Formulation of the Forward Electroencephalographic Problem for a Spherical Conductor

Presuppose a spherical homogeneous conductor with radius and conductivity occupying a finite domain confined by a smooth boundary serving as an approximation for the brain. Identify by the exterior domain to where the conductivity is zero.

Activation of a localized region in the brain triggers a primary neuronal current generating an electric field as well as a magnetic induction field , respectively. In the case where the neuronal current is represented by a single equivalent dipole at the point with moment , then , denoting the Dirac measure.

Plonsey and Heppner [22] demonstrated that the electromagnetic activity of the brain is governed by the quasistatic theory of Maxwell’s equations, namely, where the magnetic permeability is assumed to be constant everywhere in .

Equation (1) allows the introduction of an electric potential such that

Moreover, by taking the divergence of (2), we immediately conclude that the* interior* electric potential solves the following Neumann boundary value problem in :
where the operators and act on the point of observation .

Once the above problem is solved, knowledge of the solution leads to the* exterior* electric potential satisfying the Dirichlet problem
According to the right-hand side of (5) it is evident that each nontrivial solution of the EEG problem is generated by the source activity, which we rewrite in the form
and therefore it suffices to analyze the action of the directional derivative . Further, it is eminent that the action of this specific directional derivative on the field of a monopole generates the field of a dipole [23]. Hence, the general problem can be solved by considering a monopole source while the corresponding solution for a dipole source is easily evaluated by acting with the directional derivative correlated to the pair . For this purpose we introduce the potential associated with a unit monopole source located at as
Under the above assumption, boundary-value problems (5), (6), and (7) simplify as follows:
as well as
Employing analytic techniques (see [3, 4] for details), it is not hard to show that the solution regarding (5) and (6) is
whereas the solution concerning (7) is
where
On the other hand, the corresponding surface values are easily evaluated to be

#### 3. Contribution of Surface Deformations on the Forward EEG Problem

The present section provides the analysis leading to closed form solutions for the forward EEG problem implicating deformations of the heads surface by explicit relations. To this end, replace the conductor considered in Section 2 by a locally deformed spherical conductor, depicted in Figure 1, occupying a finite domain confined by a smooth boundary , sharing the same physical characteristics as in the unperturbed case. The interior boundary value problem (BVP) (10), (11) together with the exterior BVP (12)–(14) described in Section 2 can now not be solved analytically in or in , respectively.