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 -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.
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 , 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.
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
@article{CRPHYS_2018__19_1-2_7_0,
author = {Lyes Rahmouni and Simon B. Adrian and Kristof Cools and Francesco P. Andriulli},
title = {Conforming discretizations of boundary element solutions to the electroencephalography forward problem},
journal = {Comptes Rendus. Physique},
pages = {7--25},
publisher = {Elsevier},
volume = {19},
number = {1-2},
year = {2018},
doi = {10.1016/j.crhy.2018.02.002},
language = {en},
}
TY - JOUR AU - Lyes Rahmouni AU - Simon B. Adrian AU - Kristof Cools AU - Francesco P. Andriulli TI - Conforming discretizations of boundary element solutions to the electroencephalography forward problem JO - Comptes Rendus. Physique PY - 2018 SP - 7 EP - 25 VL - 19 IS - 1-2 PB - Elsevier DO - 10.1016/j.crhy.2018.02.002 LA - en ID - CRPHYS_2018__19_1-2_7_0 ER -
%0 Journal Article %A Lyes Rahmouni %A Simon B. Adrian %A Kristof Cools %A Francesco P. Andriulli %T Conforming discretizations of boundary element solutions to the electroencephalography forward problem %J Comptes Rendus. Physique %D 2018 %P 7-25 %V 19 %N 1-2 %I Elsevier %R 10.1016/j.crhy.2018.02.002 %G en %F CRPHYS_2018__19_1-2_7_0
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/
[1] Simultaneous head tissue conductivity and EEG source location estimation, NeuroImage, Volume 124 (2016), pp. 168-180
[2] Measuring the cortical correlation structure of spontaneous oscillatory activity with EEG and MEG, NeuroImage (2016)
[3] Using patient-specific hemodynamic response function in epileptic spike analysis of human epilepsy: a study based on EEG–fNIRS, NeuroImage, Volume 126 (2016), pp. 239-255
[4] Determination of head conductivity frequency response in vivo with optimized EIT-EEG, NeuroImage (2015)
[5] Single-trial analysis of oddball event-related potentials in simultaneous EEG-fMRI, Hum. Brain Mapp., Volume 28 (2007) no. 7, pp. 602-613
[6] The New York head—a precise standardized volume conductor model for EEG source localization and TES targeting, NeuroImage (2015)
[7] Towards high-quality simultaneous EEG-fMRI at 7 T: detection and reduction of EEG artifacts due to head motion, NeuroImage, Volume 120 (2015), pp. 143-153
[8] et al. The role of blood vessels in high-resolution volume conductor head modeling of EEG, NeuroImage, Volume 128 (2016), pp. 193-208
[9] Systematic regularization of linear inverse solutions of the EEG source localization problem, NeuroImage, Volume 17 (2002) no. 1, pp. 287-301
[10] Source localization of ictal epileptic activity investigated by high resolution EEG and validated by sEEG, NeuroImage, Volume 51 (2010) no. 2, pp. 642-653
[11] Review on solving the inverse problem in EEG source analysis, J. NeuroEng. Rehabil., Volume 5 (2008) no. 1, p. 25
[12] Review of methods for solving the EEG inverse problem, Int. J. Bioelectromagn., Volume 1 (1999) no. 1, pp. 75-86
[13] Effects of forward model errors on EEG source localization, Brain Topogr., Volume 26 (2013) no. 3, pp. 378-396
[14] et al. Review on solving the forward problem in EEG source analysis, J. NeuroEng. Rehabil., Volume 4 (2007) no. 1, p. 46
[15] et al. Review on solving the forward problem in EEG source analysis, J. NeuroEng. Rehabil., Volume 4 (2007) no. 1, p. 46
[16] Electric dipole tracing in the brain by means of the boundary element method and its accuracy, IEEE Trans. Biomed. Eng., Volume 6 (1987), pp. 406-414
[17] Boundary element method volume conductor models for EEG source reconstruction, Clin. Neurophysiol., Volume 112 (2001) no. 8, pp. 1400-1407
[18] Estimating cortical potentials from scalp EEGs in a realistically shaped inhomogeneous head model by means of the boundary element method, IEEE Trans. Biomed. Eng., Volume 46 (1999) no. 10, pp. 1264-1268
[19] The neuronal sources of EEG: modeling of simultaneous scalp and intracerebral recordings in epilepsy, NeuroImage, Volume 42 (2008) no. 1, pp. 135-146
[20] An improved boundary element method for realistic volume-conductor modeling, IEEE Trans. Biomed. Eng., Volume 45 (1998) no. 8, pp. 980-997
[21] Fast realistic modeling in bioelectromagnetism using lead-field interpolation, Hum. Brain Mapp., Volume 14 (2001) no. 1, pp. 48-63
[22] On the numerical accuracy of the boundary element method (EEG application), IEEE Trans. Biomed. Eng., Volume 36 (1989) no. 10, pp. 1038-1049
[23] Individually shaped volume conductor models of the head in EEG source localisation, Med. Biol. Eng. Comput., Volume 33 (1995) no. 4, pp. 582-588
[24] Symmetric BEM formulation for the m/EEG forward problem, Information Processing in Medical Imaging, Springer, 2003, pp. 524-535
[25] A common formalism for the integral formulations of the forward EEG problem, IEEE Trans. Med. Imaging, Volume 24 (2005) no. 1, pp. 12-28
[26] Handbook of Mathematical Methods in Imaging, vol. 1, Springer Science & Business Media, 2011
[27] Numerical mathematics of the subtraction method for the modeling of a current dipole in EEG source reconstruction using finite element head models, SIAM J. Sci. Comput., Volume 30 (2007) no. 1, pp. 24-45
[28] Boundary Element Methods, Springer, 2011
[29] Improving the mfie's accuracy by using a mixed discretization, IEEE Antennas and Propagation Society International Symposium, 2009, APSURSI'09, IEEE, 2009, pp. 1-4
[30] Accurate and conforming mixed discretization of the MFIE, IEEE Antennas Wirel. Propag. Lett., Volume 10 (2011), pp. 528-531
[31] Improving the accuracy of the second-kind Fredholm integral equations by using the Buffa–Christiansen functions, IEEE Trans. Antennas Propag., Volume 59 (2011) no. 4, pp. 1299-1310
[32] Conforming boundary element methods in acoustics, Eng. Anal. Bound. Elem., Volume 50 (2015), pp. 447-458
[33] Mixed discretization formulations for the direct EEG problem, EuCAP ( April 2014 ), pp. 3183-3185
[34] A comparison of different numerical methods for solving the forward problem in EEG and MEG, Physiol. Meas., Volume 14 (1993) no. 4A (A1)
[35] The need for correct realistic geometry in the inverse EEG problem, IEEE Trans. Biomed. Eng., Volume 46 (1999) no. 11, pp. 1281-1287
[36] Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements, 2008
[37] Mathematical dipoles are adequate to describe realistic generators of human brain activity, IEEE Trans. Biomed. Eng., Volume 35 (1988) no. 11, pp. 960-966
[38] Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol., Volume 32 (1987) no. 1, p. 11
[39] Dipole models for the EEG and MEG, IEEE Trans. Biomed. Eng., Volume 49 (2002) no. 5, pp. 409-418
[40] Bioelectromagnetic forward problem: isolated source approach revis(it)ed, Phys. Med. Biol., Volume 57 (2012) no. 11, p. 3517
[41] MNE software for processing MEG and EEG data, NeuroImage, Volume 86 (2014), pp. 446-460
[42] Head model and electrical source imaging: a study of 38 epileptic patients, NeuroImage: Clinical, Volume 5 (2014), pp. 77-83
[43] The construction of some efficient preconditioners in the boundary element method, Adv. Comput. Math., Volume 9 (1998) no. 1–2, pp. 191-216
[44] Integral Equations: Theory and Numerical Treatment, vol. 120, Birkhäuser, 2012
[45] Static and dynamic potential integrals for linearly varying source distributions in two-and three-dimensional problems, IEEE Trans. Antennas Propag., Volume 35 (1987) no. 6, pp. 662-669
[46] On the numerical integration of the linear shape functions times the 3-d Green's function or its gradient on a plane triangle, IEEE Trans. Antennas Propag., Volume 41 (1993) no. 10, pp. 1448-1455
[47] A linear discretization of the volume conductor boundary integral equation using analytically integrated elements, IEEE Trans. Biomed. Eng., Volume 39 (1992) no. 9, pp. 986-990
[48] A dual finite element complex on the barycentric refinement, Math. Comput., Volume 76 (2007) no. 260, pp. 1743-1769
[49] Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG, Phys. Med. Biol., Volume 47 (2002) no. 6, p. 961
[50] Deflated decomposition of solutions of nearly singular systems, SIAM J. Numer. Anal., Volume 21 (1984) no. 4, pp. 738-754
[51] High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int. J. Numer. Methods Eng., Volume 21 (1985) no. 6, pp. 1129-1148
[52] Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data, IEEE Trans. Biomed. Eng., Volume 36 (1989) no. 2, pp. 165-171
[53] Use of the isolated problem approach for multi-compartment BEM models of electro-magnetic source imaging, Phys. Med. Biol., Volume 50 (2005) no. 13, p. 3007
[54] The potential distribution in a layered anisotropic spheroidal volume conductor, J. Appl. Phys., Volume 64 (1988) no. 2, pp. 464-470
[55] A fast method to compute the potential in the multisphere model, IEEE Trans. Biomed. Eng., Volume 40 (1993) no. 11, pp. 1166-1174
[56] A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres, Phys. Med. Biol., Volume 40 (1995) no. 3, p. 335
[57] The conductivity of the human skull: results of in vivo and in vitro measurements, IEEE Trans. Biomed. Eng., Volume 47 (2000) no. 11, pp. 1487-1492
[58] Fieldtrip: open source software for advanced analysis of MEG, EEG, and invasive electrophysiological data, Comput. Intell. Neurosci., Volume 2011 (2010)
[59] Fast robust automated brain extraction, Hum. Brain Mapp., Volume 17 (2002) no. 3, pp. 143-155
[60] et al. Automatically parcellating the human cerebral cortex, Cereb. Cortex, Volume 14 (2004) no. 1, pp. 11-22
[61] Brainvisa: a complete software platform for neuroimaging, Python in Neuroscience Workshop, Euroscipy, Paris, 2011
[62] Brainsuite: an automated cortical surface identification tool, Med. Image Anal., Volume 6 (2002) no. 2, pp. 129-142
[63] et al. 3d slicer as an image computing platform for the quantitative imaging network, Magn. Reson. Imaging, Volume 30 (2012) no. 9, pp. 1323-1341
[64] Brainvoyager—past, present, future, NeuroImage, Volume 62 (2012) no. 2, pp. 748-756
[65] et al. In vivo measurement of the brain and skull resistivities using an EIT-based method and realistic models for the head, IEEE Trans. Biomed. Eng., Volume 50 (2003) no. 6, pp. 754-767
[66] Estimation of in vivo brain-to-skull conductivity ratio in humans, Appl. Phys. Lett., Volume 89 (2006) no. 22
[67] In vivo conductivity estimation with symmetric boundary elements, Int. J. Bioelectromagn., Volume 7 (2005), pp. 307-310
[68] An Introduction to Sobolev Spaces and Interpolation, Springer, 2007
[69] Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, vol. 144, Springer, 2001
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