Comptes Rendus
no. 1-2
Radio science for Humanity / Radiosciences au service de l'humanité
Conforming discretizations of boundary element solutions to the electroencephalography forward problem
[Discrétisations conformes des solutions aux éléments de frontière du problème direct en électroencéphalographie]
Lyes Rahmouni1  ; Simon B. Adrian1, 2  ; Kristof Cools3  ; Francesco P. Andriulli1, 4
1 IMT Atlantique, Technopole Brest-Iroise, 29238 Brest, France
2 Technische Universität München, Arcisstr. 21, 80333 München, Germany
3 The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
4 Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
Comptes Rendus. Physique, Volume 19 (2018) no. 1-2, pp. 7-25.

Dans ce papier, nous présentons une nouvelle stratégie de discrétisation pour la formulation aux éléments de frontière du problème direct de l'électroencéphalographie (EEG). Les méthodes aux éléments frontières (BEM) sont largement utilisées en imagerie EEG à haute résolution dans divers scénarios, pour leur stabilité numérique et leur efficacité reconnues par rapport à d'autres techniques basées sur des équations différentielles.

Malheureusement, il est également reconnu dans la littérature que leur précision diminue particulièrement lorsque la source de courant est dipolaire et se situe près de la surface du cerveau. Ce défaut constitue une importante limitation, étant donné qu'au cours d'une session d'imagerie EEG à haute résolution, plusieurs solutions du problème direct EEG sont requises, pour lesquelles les sources de courant sont proches ou sur la surface de cerveau.

Ce travail présente d'abord une analyse des opérateurs intervenant dans le problème direct et leur discrétisation. Nous montrons que plusieurs discrétisations couramment utilisées ne conviennent que dans un cadre L2, nécessitant que le terme d'expansion soit une fonction de carré intégrable. Dès lors, ces techniques ne sont pas cohérentes avec les propriétés spectrales des opérateurs en termes d'espaces de Sobolev d'ordre fractionnaire.

Nous développons ensuite une nouvelle stratégie de discrétisation conforme aux espaces de Sobolev avec des fonctions d'expansion moins régulières, donnant lieu à une nouvelle formulation intégrale. Le solveur résultant présente des propriétés favorables par rapport aux méthodes existantes et réduit sensiblement le recours à un maillage adaptatif et autres stratégies actuellement requises pour améliorer la précision du calcul. Les résultats numériques présentés corroborent les développements théoriques et mettent en évidence l'impact positif de la nouvelle approche.

In this paper, we present a new discretization strategy for the boundary element formulation of the Electroencephalography (EEG) forward problem. Boundary integral formulations, classically solved with the Boundary Element Method (BEM), are widely used in high resolution EEG imaging because of their recognized advantages, in several real case scenarios, in terms of numerical stability and effectiveness when compared with other differential equation based techniques. Unfortunately, however, it is widely reported in literature that the accuracy of standard BEM schemes for the forward EEG problem is often limited, especially when the current source density is dipolar and its location approaches one of the brain boundary surfaces. This is a particularly limiting problem given that during an high-resolution EEG imaging procedure, several EEG forward problem solutions are required, for which the source currents are near or on top of a boundary surface.

This work will first present an analysis of standardly and classically discretized EEG forward problem operators, reporting on a theoretical issue of some of the formulations that have been used so far in the community. We report on the fact that several standardly used discretizations of these formulations are consistent only with an L2-framework, requiring the expansion term to be a square integrable function (i.e., in a Petrov–Galerkin scheme with expansion and testing functions). Instead, those techniques are not consistent when a more appropriate mapping in terms of fractional-order Sobolev spaces is considered. Such a mapping allows the expansion function term to be a less regular function, thus sensibly reducing the need for mesh refinements and low-precisions handling strategies that are currently required. These more favorable mappings, however, require a different and conforming discretization, which must be suitably adapted to them. In order to appropriately fulfill this requirement, we adopt a mixed discretization based on dual boundary elements residing on a suitably defined dual mesh. We devote also a particular attention to implementation-oriented details of our new technique that will allow the rapid incorporation of our finding in one's own EEG forward solution technology. We conclude by showing how the resulting forward EEG problems show favorable properties with respect to previously proposed schemes, and we show their applicability to real-case modeling scenarios obtained from Magnetic Resonance Imaging (MRI) data.

Publié le :
DOI : 10.1016/j.crhy.2018.02.002
Keywords: EEG, Inverse problem, Forward problem, Mixed discretizations, Indirect formulation
Mot clés : EEG, Problème inverse, Problème direct, Discrétisation mixte, Formulation indirecte
Lyes Rahmouni 1 ; Simon B. Adrian 1, 2 ; Kristof Cools 3 ; Francesco P. Andriulli 1, 4

1 IMT Atlantique, Technopole Brest-Iroise, 29238 Brest, France
2 Technische Universität München, Arcisstr. 21, 80333 München, Germany
3 The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
4 Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy 
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Lyes Rahmouni; Simon B. Adrian; Kristof Cools; Francesco P. Andriulli. Conforming discretizations of boundary element solutions to the electroencephalography forward problem. Comptes Rendus. Physique, Volume 19 (2018) no. 1-2, pp. 7-25. doi : 10.1016/j.crhy.2018.02.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2018.02.002/

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