Abstract
Accurate source localization of electroencephalographic (EEG) signals requires detailed information about the geometry and physical properties of head tissues. Indeed, these strongly influence the propagation of neural activity from the brain to the sensors. Finite difference methods (FDMs) are head modelling approaches relying on volumetric data information, which can be directly obtained using magnetic resonance (MR) imaging. The specific goal of this study is to develop a computationally efficient FDM solution that can flexibly integrate voxel-wise conductivity and anisotropy information. Given the high computational complexity of FDMs, we pay particular attention to attain a very low numerical error, as evaluated using exact analytical solutions for spherical volume conductor models. We then demonstrate the computational efficiency of our FDM numerical solver, by comparing it with alternative solutions. Finally, we apply the developed head modelling tool to high-resolution MR images from a real experimental subject, to demonstrate the potential added value of incorporating detailed voxel-wise conductivity and anisotropy information. Our results clearly show that the developed FDM can contribute to a more precise head modelling, and therefore to a more reliable use of EEG as a brain imaging tool.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akalin Acar Z, Acar CE, Makeig S (2016) Simultaneous head tissue conductivity and EEG source location estimation. NeuroImage. https://doi.org/10.1016/j.neuroimage.2015.08.032
Aydin Ü, Vorwerk J, Küpper P, Heers M, Kugel H, Galka A, Hamid L, Wellmer J, Kellinghaus C, Rampp S, Wolters CH (2014) Combining EEG and MEG for the reconstruction of epileptic activity using a calibrated realistic volume conductor model. PLoS ONE. https://doi.org/10.1371/journal.pone.0093154
Bashar R, Li Y, Wen P (2008) Influence of white matter inhomogeneous anisotropy on EEG forward computing. Australas Phys Eng Sci Med. https://doi.org/10.1007/BF03178586
Castaño-Candamil S, Höhne J, Martínez-Vargas JD, An XW, Castellanos-Domínguez G, Haufe S (2015) Solving the EEG inverse problem based on space-time-frequency structured sparsity constraints. NeuroImage. https://doi.org/10.1016/j.neuroimage.2015.05.052
Clark J, Plonsey R (1968) The extracellular potential field of the single active nerve fiber in a volume conductor. Biophys J. https://doi.org/10.1016/S0006-3495(68)86524-5
Cook MJ, Koles ZJ (2006) A high-resolution anisotropic finite-volume head model for EEG source analysis. Conf Proc IEEE Eng Med Biol Soc. https://doi.org/10.1109/IEMBS.2006.260314
Cuartas ME, Acosta MC, Castellanos DG (2015) iLU preconditioning of the anisotropic-finite-difference based solution for the EEG forward problem. In: IWINAC 2015. Springer, Cham, p 408–418. https://doi.org/10.1007/978-3-319-18914-7_43
DeMunck JC (1988) The potential distribution in a layered anisotropic spheroidal volume conductor. J Appl Phys. https://doi.org/10.1063/1.341983
Grech R, Cassar T, Muscat J, Camilleri KP, Fabri SG, Zervakis M, Xanthopoulos P, Sakkalis V, Vanrumste B (2008) Review on solving the inverse problem in EEG source analysis. J Neuroeng Rehabil. https://doi.org/10.1186/1743-0003-5-25
Güllmar D, Haueisen J, Reichenbach JR (2010) Influence of anisotropic electrical conductivity in white matter tissue on the EEG/MEG forward and inverse solution. A high-resolution whole head simulation study. Neuroimage. https://doi.org/10.1016/j.neuroimage.2010.02.014
Hallez H, Vanrumste B, Van Hese P, D’Asseler Y, Lemahieu I, Van de Walle R (2005) A finite difference method with reciprocity used to incorporate anisotropy in electroencephalogram dipole source localization. Phys Med Biol. https://doi.org/10.1088/0031-9155/50/16/009
Hallez H, Vanrumste B, Grech R, Muscat J, De W, Vergult A, D’Asseler Y, Camilleri KP, Fabri SG, Van Huffel S, Lemahieu I (2007) Review on solving the forward problem in EEG source analysis. J NeuroEng Rehabil. https://doi.org/10.1186/1743-0003-4-46
Hallez H, Staelens S, Lemahieu I (2009) Dipole estimation errors due to not incorporating anisotropic conductivities in realistic head models for EEG source analysis. Phys Med Biol. https://doi.org/10.1088/0031-9155/54/20/004
Haueisen J, Ramon C, Eiselt M, Brauer H, Nowak H (1997) Influence of tissue resistivities on neuromagnetic fields and electric potentials studied with a finite element model of the head. IEEE Trans Bio-med Eng. https://doi.org/10.1109/10.605429
Herrendorf G, Steinhoff BJ, Kolle R, Baudewig J, Waberski TD, Buchner H, Paulus W (2000) Dipole-source analysis in a realistic head model in patients with focal epilepsy. Epilepsia. https://doi.org/10.1111/j.1528-1157.2000.tb01508.x
Irimia A, Bradshaw LA (2005) Ellipsoidal electrogastrographic forward modelling. Phys Med Biol. https://doi.org/10.1088/0031-9155/50/18/012
Irimia A, Goh SYM, Torgerson CM, Chambers MC, Kikinis R, Horn JD Van (2013a) Clinical neurophysiology forward and inverse electroencephalographic modeling in health and in acute traumatic brain injury. Clin Neurophysiol. https://doi.org/10.1016/j.clinph.2013.04.336
Irimia A, Goh S-YM, Torgerson CM, Stein NR, Chambers MC, Vespa PM, Van Horn JD (2013b) Electroencephalographic inverse localization of brain activity in acute traumatic brain injury as a guide to surgery, monitoring and treatment. Clin Neurol Neurosurg. https://doi.org/10.1016/j.clineuro.2013.08.003
Le Bihan D, Johansen-Berg H (2012) Diffusion MRI at 25: exploring brain tissue structure and function. NeuroImage. https://doi.org/10.1016/j.neuroimage.2011.11.006
Liu Q, Farahibozorg S, Porcaro C, Wenderoth N, Mantini D (2017) Detecting large-scale networks in the human brain using high-density electroencephalography. Hum Brain Mapp. https://doi.org/10.1002/hbm.23688
Meijs JW, Weier OW, Peters MJ, van Oosterom A (1989) On the numerical accuracy of the boundary element method. IEEE Trans Biomed Eng. https://doi.org/10.1109/10.40805
Michel CM, Murray MM (2012) Towards the utilization of EEG as a brain imaging tool. Neuroimage. https://doi.org/10.1016/j.neuroimage.2011.12.039
Michel E, Hernandez D, Lee SY (2016) Electrical conductivity and permittivity maps of brain tissues derived from water content based on T1 -weighted acquisition. Magn Reson Med. https://doi.org/10.1002/mrm.26193
Oostenveld R, Fries P, Maris E, Schoffelen JM (2011) FieldTrip: Open source software for advanced analysis of MEG, EEG, and invasive electrophysiological data. Comput Intell Neurosci. https://doi.org/10.1155/2011/156869
Panizo M, Castellanos A, Rivas J (1977) Finite-difference operators in inhomogeneous anisotropic media. J Appl Phys. https://doi.org/10.1063/1.323779
Saleheen HI, Ng KT (1997) New finite difference formulations for general inhomogeneous anisotropic bioelectric problems. IEEE Trans Biomed Eng. https://doi.org/10.1109/10.623049
Saleheen HI, Ng KT (1998) A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues. IEEE Trans Biomed Eng. https://doi.org/10.1109/10.650347
Salmelin R, Baillet S (2009) Electromagnetic brain imaging. IEEE Signal Process Mag. https://doi.org/10.1002/hbm.20795
Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys Med Biol. https://doi.org/10.1088/0031-9155/32/1/004
Schimpf PH, Ramon C, Haueisen J (2002) Dipole models for the EEG and MEG. IEEE Trans Biomed Eng. https://doi.org/10.1109/10.995679
Stenroos M, Sarvas J (2012) Bioelectromagnetic forward problem: Isolated source approach revisited. Phys Med Biol. https://doi.org/10.1088/0031-9155/57/11/3517
Tadel F, Baillet S, Mosher JC, Pantazis D, Leahy RM (2011) Brainstorm: a user-friendly application for MEG/EEG analysis. Comput Intell Neurosci. https://doi.org/10.1155/2011/879716
Truong DQ, Magerowski G, Blackburn GL, Bikson M, Alonso AM (2013) Computational modeling of transcranial direct current stimulation (tDCS) in obesity: Impact of head fat and dose guidelines. NeuroImage. https://doi.org/10.1016/j.nicl.2013.05.011
Turovets SI, Poolman P, Salman A, Malony AD, Tucker DM (2008) Conductivity analysis for high-resolution EEG. In: 2008 International conference on biomedical engineering and informatics, IEEE. https://doi.org/10.1109/BMEI.2008.358
Turovets S, Volkov V, Zherdetsky A, Prakonina A, Malony AD (2014) A 3D finite-difference BiCG iterative solver with the Fourier-Jacobi preconditioner for the anisotropic EIT/EEG forward problem. Comput Math Methods Med. https://doi.org/10.1155/2014/426902
Vanrumste B, Van Hoey G, Van de Walle R, D’Have MR, Lemahieu IA, Boon PA (2001) The validation of the finite difference method and reciprocity for solving the inverse problem in EEG dipole source analysis. Brain Topogr. https://doi.org/10.1023/A:1012909511833
Vatta F, Meneghini F, Esposito F, Mininel S, Di Salle F (2010) Realistic and spherical head modeling for EEG forward problem solution: a comparative cortex-based analysis. Comput Intell Neurosci. https://doi.org/10.1155/2010/972060
Vorwerk JO (2018) The FieldTrip-SimBio pipeline for EEG forward solutions. Biomed Eng. https://doi.org/10.1186/s12938-018-0463-y
Vorwerk J, Cho J-H, Rampp S, Hamer H, Knosche TR, Wolters CH (2014) A guideline for head volume conductor modeling in EEG and MEG. NeuroImage. https://doi.org/10.1016/j.neuroimage.2014.06.040
Vorwerk J, Engwer C, Pursiainen S, Wolters CH (2017) A mixed finite element method to solve the EEG forward problem. IEEE Trans Med Imaging. https://doi.org/10.1109/TMI.2016.2624634
Wolters CH, Anwander A, Tricoche X, Weinstein D, Koch MA, Macleod RS (2006) Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: a simulation and visualization study using high-resolution finite element modeling. NeuroImage. https://doi.org/10.1016/j.neuroimage.2005.10.014
Ziegler E, Chellappa SL, Gaggioni G, Ly JQ, Vandewalle G, André E, Geuzaine C, Phillips C (2014) A finite-element reciprocity solution for EEG forward modeling with realistic individual head models. NeuroImage. https://doi.org/10.1016/j.neuroimage.2014.08.056
Funding
The work was supported the Colombian Program for Researcher Training (Grant 2012-528), the KU Leuven Special Research Fund (Grant C16/15/070), the Research Foundation Flanders (FWO) (Grants G0F76.16N, G0936.16N, and EOS.30446199).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there are no potential conflicts of interest.
Ethical Approval
All procedures performed in human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.
Informed Consent
Informed consent was obtained from all individual participants included in the study.
Additional information
Handling Editor: Jens Haueisen.
Rights and permissions
About this article
Cite this article
Cuartas Morales, E., Acosta-Medina, C.D., Castellanos-Dominguez, G. et al. A Finite-Difference Solution for the EEG Forward Problem in Inhomogeneous Anisotropic Media. Brain Topogr 32, 229–239 (2019). https://doi.org/10.1007/s10548-018-0683-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10548-018-0683-2