Abstract
The finite element method (FEM) is a numerical method that is often used for solving electroencephalography (EEG) forward problems involving realistic head models. In this study, FEM solutions obtained using three different mesh structures, namely coarse, densely refined, and adaptively refined meshes, are compared. The simulation results showed that the accuracy of FEM solutions could be significantly enhanced by adding a small number of elements around regions with large estimated errors. Moreover, it was demonstrated that the adaptively refined regions were always near the current dipole sources, suggesting that selectively generating additional elements around the cortical surface might be a new promising strategy for more efficient FEM-based EEG forward analysis.
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References
Akalin Acar Z, Makeig S (2013) Effects of forward model errors on EEG source localization. Brain Topogr 26:378–396. https://doi.org/10.1007/s10548-012-0274-6
Awada KA, Jackson DR, Williams JT et al (1997) Computational aspects of finite element modeling in EEG source localization. IEEE Trans Biomed Eng 44:736–752
Aydin Ü, Vorwerk J, Küpper P et al (2014) Combining EEG and MEG for the reconstruction of epileptic activity using a calibrated realistic volume conductor model. PLoS ONE 9:e93154. https://doi.org/10.1371/journal.pone.0093154
Aydin Ü, Rampp S, Wollbrink A et al (2017) Zoomed MRI guided by combined EEG/MEG source analysis: a multimodal approach for optimizing presurgical epilepsy work-up and its application in a multi-focal epilepsy patient case study. Brain Topogr 30:417–433. https://doi.org/10.1007/s10548-017-0568-9
Baillet S, Mosher JC, Leahy RM (2001) Electromagnetic brain mapping. IEEE Signal Process Mag 18:14–30. https://doi.org/10.1109/79.962275
Buchner H, Knoll G, Fuchs M et al (1997) Inverse localization of electric dipole current sources in finite element models of the human head. Electroencephalogr Clin Neurophysiol 102:267–278. https://doi.org/10.1016/S0013-4694(96)95698-9
Bugeda G (2002) A comparison between new adaptive remeshing strategies based on point wise stress error estimation and energy norm error estimation. Commun Numer Methods Eng 18:469–482. https://doi.org/10.1002/cnm.505
Cheng DK (1989) Field and wave electromagnetics, 2nd edn. Addison Wesley, New York
Cho J-H, Vorwerk J, Wolters CH, Knösche TR (2015) Influence of the head model on EEG and MEG source connectivity analyses. NeuroImage 110:60–77. https://doi.org/10.1016/j.neuroimage.2015.01.043
Choi J-H (2013) Source reconstruction algorithm considering intrinsic characteristics of neuroelectromagnetic source. Dissertation, Seoul National University
Engwer C, Vorwerk J, Ludewig J, Wolters CH (2017) A discontinuous Galerkin method to solve the EEG forward problem using the subtraction approach. SIAM J Sci Comput 39:B138–B164. https://doi.org/10.1137/15M1048392
Fang Q, Boas DA (2009) Tetrahedral mesh generation from volumetric binary and grayscale images. In: IEEE international symposium on Biomedical imaging: from nano to macro, 2009. ISBI’09, pp. 1142–1145. IEEE, 2009. https://doi.org/10.1109/ISBI.2009.5193259
Fuchs M, Drenckhahn R, Wischmann H-A, Wagner M (1998) An improved boundary element method for realistic volume-conductor modeling. IEEE Trans Biomed Eng 45:980–997. https://doi.org/10.1109/10.704867
Grätsch T, Bathe K-J (2005) A posteriori error estimation techniques in practical finite element analysis. Comput Struct 83:235–265. https://doi.org/10.1016/j.compstruc.2004.08.011
Hahn S-Y, Calmels C, Meunier G, Coulomb JL (1988) A posteriori error estimate for adaptive finite element mesh generation. IEEE Trans Magn 24:315–317. https://doi.org/10.1109/20.43920
Hallez H, Vanrumste B, Grech R et al (2007) Review on solving the forward problem in EEG source analysis. J Neuroeng Rehabil 4:46. https://doi.org/10.1186/1743-0003-4-46
Hämäläinen MS, Hari R, Ilmoniemi RJ et al (1993) Magnetoencephalography—theory, instrumentation, and applications to noninvasivee studies of the working human brain. Rev Mod Phys 65:413
Haueisen J, Ramon C (1997) Influence of tissue resistivities on neuromagnetic fields and electric potentials studied with a finite element model of the head. IEEE Trans Biomed Eng 44:9
Haueisen J, Ramon C, Czapski P, Eiselt M (1995) On the influence of volume currents and extended sources on neuromagnetic fields: a simulation study. Ann Biomed Eng 23:728–739. https://doi.org/10.1007/BF02584472
Haueisen J, Tuch DS, Ramon C et al (2002) The influence of brain tissue anisotropy on human EEG and MEG. NeuroImage 15:159–166. https://doi.org/10.1006/nimg.2001.0962
Jin J (2014) The finite element method in electromagnetics, 3rd ed. Wiley, New Jersey
Kim H-S, Hong S-P, Choi K et al (1991) A three dimensional adaptive finite element method for magnetostatic problems. IEEE Trans Magn 27:4081–4084. https://doi.org/10.1109/20.104998
Lee WH, Kim T-S (2012) Methods for high-resolution anisotropic finite element modeling of the human head: automatic MR white matter anisotropy-adaptive mesh generation. Med Eng Phys 34:85–98. https://doi.org/10.1016/j.medengphy.2011.07.002
Lee WH, Kim T-S, Cho MH et al (2006) Methods and evaluations of MRI content-adaptive finite element mesh generation for bioelectromagnetic problems. Phys Med Biol 51:6173–6186. https://doi.org/10.1088/0031-9155/51/23/016
Lee WH, Liu Z, Mueller BA et al (2009) Influence of white matter anisotropic conductivity on EEG source localization: comparison to fMRI in human primary visual cortex. Clin Neurophysiol 120:2071–2081. https://doi.org/10.1016/j.clinph.2009.09.007
Lew S, Wolters CH, Dierkes T et al (2009) Accuracy and run-time comparison for different potential approaches and iterative solvers in finite element method based EEG source analysis. Appl Numer Math 59:1970–1988. https://doi.org/10.1016/j.apnum.2009.02.006
Liu J, Zhu S, Zhang Y, He B (2008) Finite Element Modeling of a Realistic Head Based on Medical Images. Int J Bioelectromagn 10:149–164
Logan DL, Veitch E, Carson C et al (2007) A first course in the finite element method fourth edition
Nemtsas P, Birot G, Pittau F et al (2017) Source localization of ictal epileptic activity based on high-density scalp EEG data. Epilepsia 58:1027–1036. https://doi.org/10.1111/epi.13749
Nusing A, Wolters CH, Brinck H, Engwer C (2016) The unfitted discontinuous Galerkin method for solving the EEG forward problem. IEEE Trans Biomed Eng 63:2564–2575. https://doi.org/10.1109/TBME.2016.2590740
Pellegrino G, Hedrich T, Chowdhury R et al (2016) Source localization of the seizure onset zone from ictal EEG/MEG data. Hum Brain Mapp 37:2528–2546. https://doi.org/10.1002/hbm.23191
Piastra MC, Nüßing A, Vorwerk J et al (2018) The discontinuous Galerkin finite element method for solving the MEG and the combined MEG/EEG forward problem. Front Neurosci. https://doi.org/10.3389/fnins.2018.00030
Pursiainen S (2012) Raviart–Thomas-type sources adapted to applied EEG and MEG: implementation and results. Inverse Prob 28:065013. https://doi.org/10.1088/0266-5611/28/6/065013
Pursiainen S, Sorrentino A, Campi C, Piana M (2011) Forward simulation and inverse dipole localization with the lowest order Raviart–Thomas elements for electroencephalography. Inverse Prob 27:045003. https://doi.org/10.1088/0266-5611/27/4/045003
Pursiainen S, Vorwerk J, Wolters CH (2016) Electroencephalography (EEG) forward modeling via H (div) finite element sources with focal interpolation. Phys Med Biol 61:8502–8520. https://doi.org/10.1088/0031-9155/61/24/8502
Rahmouni L, Mitharwal R, Andriulli FP (2016) A mixed discretized surface-volume integral equation for solving EEG forward problems with inhomogeneous and anisotropic head models. In: 2016 IEEE 13th international symposium on biomedical imaging (ISBI). IEEE, pp 763–766
Raizer A, Meunier G, Coulomb J-L (1989) An approach for automatic adaptive mesh refinement in finite element computation of magnetic fields. IEEE Trans Magn 25:2965–2967. https://doi.org/10.1109/20.34339
Rivière B, Wheeler MF, Girault V (2001) A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J Numer Anal 39:902–931. https://doi.org/10.1137/S003614290037174X
Schimpf PH, Haynor DR, Kim Y (1996) Object-free adaptive meshing in highly heterogeneous 3-D domains. Int J Biomed Comput 40:209–225. https://doi.org/10.1016/0020-7101(95)01146-3
Schimpf P, Haueisen J, Ramon C, Nowak H (1998) Realistic computer modelling of electric and magnetic Fields of human head and torso. Parallel Comput 24:1433–1460
Schimpf PH, Ramon C, Haueisen J (2002) Dipole models for the EEG and MEG. IEEE Trans Biomed Eng 49:409–418. https://doi.org/10.1109/10.995679
Shahid S, Wen P (2010) Analytic and numeric evaluation of EEG forward problem using spherical volume conductor models. In: IEEE/ICME International conference on complex medical engineering. IEEE, pp 28–33
Si H (2015) TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans Math Softw 41:1–36. https://doi.org/10.1145/2629697
Ubertini F (2004) Patch recovery based on complementary energy. Int J Numer Methods Eng 59:1501–1538. https://doi.org/10.1002/nme.924
Vorwerk J, Clerc M, Burger M, Wolters CH (2012) Comparison of boundary element and finite element approaches to the EEG forward problem. Biomed Eng Biomed Tech. https://doi.org/10.1515/bmt-2012-4152
Vorwerk J, Cho J-H, Rampp S et al (2014) A guideline for head volume conductor modeling in EEG and MEG. NeuroImage 100:590–607. 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 36:930–941. https://doi.org/10.1109/TMI.2016.2624634
Wolters CH, Grasedyck L, Hackbusch W (2004) Efficient computation of lead field bases and influence matrix for the FEM-based EEG and MEG inverse problem. Inverse Prob 20:1099–1116. https://doi.org/10.1088/0266-5611/20/4/007
Wolters CH, Köstler H, Möller C et al (2007) Numerical approaches for dipole modeling in finite element method based source analysis. Intern Congr Ser 1300:189–192
Yan Y, Nunez PL, Hart RT (1991) Finite-element model of the human head: scalp potentials due to dipole sources. Med Biol Eng Comput 29:475–481. https://doi.org/10.1007/BF02442317
Ziegler E, Chellappa SL, Gaggioni G et al (2014) A finite-element reciprocity solution for EEG forward modeling with realistic individual head models. NeuroImage 103:542–551. https://doi.org/10.1016/j.neuroimage.2014.08.056
Acknowledgements
This work was supported in part by Institute for Information & communications Technology Promotion (IITP) Grant funded by the Korea government (MSIT) (2017-0-00432) and in part by the Brain Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2015M3C7A1031969).
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Lee, C., Im, CH. New Strategy for Finite Element Mesh Generation for Accurate Solutions of Electroencephalography Forward Problems. Brain Topogr 32, 354–362 (2019). https://doi.org/10.1007/s10548-018-0669-0
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DOI: https://doi.org/10.1007/s10548-018-0669-0