Evaluating the Performance of the Hybrid Boundary Element- Finite Element (BE-FE) Method to Solve Electroencephalography (EEG) Forward Problem Based on the Mesh Quality: A Simulation Study
Abstract
Purpose: The Boundary Element (BE) and Finite Element (FE) methods are widely used numerical techniques to solve the Electroencephalography (EEG) forward problem. However, the FE Method (FEM) has difficulty in simulating current dipoles due to singularity, and the BE method (BEM) cannot simulate inhomogeneous and anisotropic conductivity profiles. Recently, a hybrid BE-FE method has been proposed to benefit from the advantages of both BEM and FEM in solving the EEG forward problem. Generally, the type of mesh may significantly influence the results of numerical EEG forward solvers and should be carefully studied.
Materials and Methods: In this paper, the performance of the hybrid BE-FE method is compared with an approach of FEM (partial integration) using three types of meshes. The ground truth is the analytical EEG forward solutions obtained from inhomogeneous and isotropic/anisotropic four-layer spherical head models with dipoles of radial and tangential directions at four eccentricities.
Results: The minimum mean of Relative Difference Measure (RDM) obtained from Partial Integration (PI)-FEM is 0.0596 at 70% source eccentricity while by using the hybrid BE-FE method it is improved to 0.0251 at the same eccentricity. On the other hand, the maximum mean of MAG obtained from PI-FEM is 0.6216 at 50% source eccentricity while it is improved to 0.9734 at the same eccentricity.
Conclusion: The results show that the hybrid BE-FE method outperforms PI-FEM in solving the EEG forward problem using three types of meshes regarding RDM and MAG error criteria.
1- Zeynep Akalin Acar and Scott Makeig, "Effects of forward model errors on EEG source localization." Brain topography, Vol. 26 (No. 3), pp. 378-96, (2013).
2- Daniel Güllmar, Jens Haueisen, and Jürgen R Reichenbach, "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, Vol. 51 (No. 1), pp. 145-63, (2010).
3- JC De Munck, Carsten H Wolters, and Maureen Clerc, EEG and MEG: forward modeling. (Handbook of neural activity measurement). (2012), pp. 192-248.
4- Johannes Vorwerk, Christian Engwer, Sampsa Pursiainen, and Carsten H Wolters, "A mixed finite element method to solve the EEG forward problem." IEEE transactions on medical imaging, Vol. 36 (No. 4), pp. 930-41, (2016).
5- Matti Stenroos and J Sarvas, "Bioelectromagnetic forward problem: isolated source approach revis (it) ed." Physics in Medicine & Biology, Vol. 57 (No. 11), p. 3517, (2012).
6- Lyes Rahmouni, Rajendra Mitharwal, and Francesco P Andriulli, "Two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography." Journal of Computational Physics, Vol. 348pp. 732-43, (2017).
7- Leandro Beltrachini, "The analytical subtraction approach for solving the forward problem in EEG." Journal of neural engineering, Vol. 16 (No. 5), p. 056029, (2019).
8- Ibrahem Taha and Gregory Cook, "Brain sources estimation based on EEG and computer simulation technology (CST)." Biomedical Signal Processing and Control, Vol. 46pp. 145-56, (2018).
9- Hans Hallez et al., "Review on solving the forward problem in EEG source analysis." Journal of neuroengineering and rehabilitation, Vol. 4 (No. 1), p. 46, (2007).
10- Norio Takahashi, Yujie Zhang, Zhuoxiang Ren, and David Lautru, "Finite element modeling of current dipoles using direct and subtraction methods for EEG forward problem." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, (2014).
11- Matti Hämäläinen, Riitta Hari, Risto J Ilmoniemi, Jukka Knuutila, and Olli V Lounasmaa, "Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain." Reviews of modern Physics, Vol. 65 (No. 2), p. 413, (1993).
12- S Baillet, JC Mosher, and RM Leahy, "Electromagnetic brain mapping. Ieee Signal Proc Mag. 2001; 18 (6): 14–30." ed, (2001).
13- Jan C De Munck, Bob W Van Dijk, and HENK Spekreijse, "Mathematical dipoles are adequate to describe realistic generators of human brain activity." IEEE transactions on biomedical engineering, Vol. 35 (No. 11), pp. 960-66, (1988).
14- Kassem A Awada, David R Jackson, Jeffery T Williams, Donald R Wilton, Stephen B Baumann, and Andrew C Papanicolaou, "Computational aspects of finite element modeling in EEG source localization." IEEE transactions on biomedical engineering, Vol. 44 (No. 8), pp. 736-52, (1997).
15- Marion Darbas and Stephanie Lohrengel, "Review on mathematical modelling of electroencephalography (EEG)." Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 121 (No. 1), pp. 3-39, (2019).
16- Helmut Buchner et al., "Inverse localization of electric dipole current sources in finite element models of the human head." Electroencephalography and clinical Neurophysiology, Vol. 102 (No. 4), pp. 267-78, (1997).
17- Y Yan, PL Nunez, and RT Hart, "Finite-element model of the human head: scalp potentials due to dipole sources." Medical and Biological Engineering and Computing, Vol. 29 (No. 5), pp. 475-81, (1991).
18- Maxime Monin, Lyes Rahmouni, Adrien Merlini, and Francesco P Andriulli, "A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography." IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology, (2020).
19- Chany Lee and Chang-Hwan Im, "New Strategy for Finite Element Mesh Generation for Accurate Solutions of Electroencephalography Forward Problems." Brain topography, Vol. 32 (No. 3), pp. 354-62, (2019).
20- Sergey N Makarov, Matti Hämäläinen, Yoshio Okada, Gregory M Noetscher, Jyrki Ahveninen, and Aapo Nummenmaa, "Boundary element fast multipole method for enhanced modeling of neurophysiological recordings." IEEE transactions on biomedical engineering, Vol. 68 (No. 1), pp. 308-18, (2020).
21- Sergey N Makarov, Gregory M Noetscher, Tommi Raij, and Aapo Nummenmaa, "A quasi-static boundary element approach with fast multipole acceleration for high-resolution bioelectromagnetic models." IEEE transactions on biomedical engineering, Vol. 65 (No. 12), pp. 2675-83, (2018).
22- Sergey N Makarov, Matti Hämäläinen, Yoshio Okada, Gregory M Noetscher, Jyrki Ahveninen, and Aapo Nummenmaa, "Solving High-Resolution Forward Problems for Extra-and Intracranial Neurophysiological Recordings Using Boundary Element Fast Multipole Method." bioRxiv, p. 567933, (2019).
23- Chris P Bradley, Glen M Harris, and Andrew J Pullan, "The computational performance of a high-order coupled FEM/BEM procedure in electropotential problems." IEEE transactions on biomedical engineering, Vol. 48 (No. 11), pp. 1238-50, (2001).
24- J Sikora, SR Arridge, RH Bayford, and L Horesh, "The application of hybrid BEM/FEM methods to solve electrical impedance tomography's forward problem for the human head." in XII ICEBI & V EIT Conference., Gdansk. ed, (2004).
25- Subhadra Srinivasan, Hamid R Ghadyani, Brian W Pogue, and Keith D Paulsen, "A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging." Biomedical optics express, Vol. 1 (No. 2), pp. 398-413, (2010).
26- Emmanuel Olivi, Maureen Clerc, and Théodore Papadopoulo, "Domain decomposition for coupling finite and boundary element methods in EEG." in 17th International Conference on Biomagnetism Advances in Biomagnetism–Biomag2010, (2010): Springer, pp. 120-23.
27- P Ghaderi Daneshmand and R Jafari, "A 3D hybrid BE–FE solution to the forward problem of electrical impedance tomography." Engineering Analysis with Boundary Elements, Vol. 37 (No. 4), pp. 757-64, (2013).
28- Nasireh Dayarian and Ali Khadem, "A hybrid boundary element-finite element method to solve the EEG forward problem." bioRxiv, (2021).
29- Johannes Vorwerk, Ümit Aydin, Carsten H Wolters, and Christopher R Butson, "Influence of head tissue conductivity uncertainties on EEG dipole reconstruction." Frontiers in neuroscience, Vol. 13p. 531, (2019).
30- Moritz Dannhauer, Benjamin Lanfer, Carsten H Wolters, and Thomas R Knösche, "Modeling of the human skull in EEG source analysis." Human brain mapping, Vol. 32 (No. 9), pp. 1383-99, (2011).
31- S Gedney, "The finite element method in electromagnetics [Book Review]." IEEE Antennas and Propagation Magazine, Vol. 36 (No. 3), pp. 75-76, (1994).
32- Paul H Schimpf, Ceon Ramon, and Jens Haueisen, "Dipole models for the EEG and MEG." IEEE transactions on biomedical engineering, Vol. 49 (No. 5), pp. 409-18, (2002).
33- Whye-Teong Ang, A beginner's course in boundary element methods. Universal-Publishers, (2007).
34- Jan WH Meijs, Onno W Weier, Maria J Peters, and ADRIAAN Van Oosterom, "On the numerical accuracy of the boundary element method (EEG application)." IEEE transactions on biomedical engineering, Vol. 36 (No. 10), pp. 1038-49, (1989).
35- A Zhi Zhang, "fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres Phys." Med. Biol, (1995).
36- Sven Wagner et al., "Using reciprocity for relating the simulation of transcranial current stimulation to the EEG forward problem." NeuroImage, Vol. 140pp. 163-73, (2016).
37- Qianqian Fang and David A Boas, "Tetrahedral mesh generation from volumetric binary and grayscale images." in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, (2009): Ieee, pp. 1142-45.
2- Daniel Güllmar, Jens Haueisen, and Jürgen R Reichenbach, "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, Vol. 51 (No. 1), pp. 145-63, (2010).
3- JC De Munck, Carsten H Wolters, and Maureen Clerc, EEG and MEG: forward modeling. (Handbook of neural activity measurement). (2012), pp. 192-248.
4- Johannes Vorwerk, Christian Engwer, Sampsa Pursiainen, and Carsten H Wolters, "A mixed finite element method to solve the EEG forward problem." IEEE transactions on medical imaging, Vol. 36 (No. 4), pp. 930-41, (2016).
5- Matti Stenroos and J Sarvas, "Bioelectromagnetic forward problem: isolated source approach revis (it) ed." Physics in Medicine & Biology, Vol. 57 (No. 11), p. 3517, (2012).
6- Lyes Rahmouni, Rajendra Mitharwal, and Francesco P Andriulli, "Two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography." Journal of Computational Physics, Vol. 348pp. 732-43, (2017).
7- Leandro Beltrachini, "The analytical subtraction approach for solving the forward problem in EEG." Journal of neural engineering, Vol. 16 (No. 5), p. 056029, (2019).
8- Ibrahem Taha and Gregory Cook, "Brain sources estimation based on EEG and computer simulation technology (CST)." Biomedical Signal Processing and Control, Vol. 46pp. 145-56, (2018).
9- Hans Hallez et al., "Review on solving the forward problem in EEG source analysis." Journal of neuroengineering and rehabilitation, Vol. 4 (No. 1), p. 46, (2007).
10- Norio Takahashi, Yujie Zhang, Zhuoxiang Ren, and David Lautru, "Finite element modeling of current dipoles using direct and subtraction methods for EEG forward problem." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, (2014).
11- Matti Hämäläinen, Riitta Hari, Risto J Ilmoniemi, Jukka Knuutila, and Olli V Lounasmaa, "Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain." Reviews of modern Physics, Vol. 65 (No. 2), p. 413, (1993).
12- S Baillet, JC Mosher, and RM Leahy, "Electromagnetic brain mapping. Ieee Signal Proc Mag. 2001; 18 (6): 14–30." ed, (2001).
13- Jan C De Munck, Bob W Van Dijk, and HENK Spekreijse, "Mathematical dipoles are adequate to describe realistic generators of human brain activity." IEEE transactions on biomedical engineering, Vol. 35 (No. 11), pp. 960-66, (1988).
14- Kassem A Awada, David R Jackson, Jeffery T Williams, Donald R Wilton, Stephen B Baumann, and Andrew C Papanicolaou, "Computational aspects of finite element modeling in EEG source localization." IEEE transactions on biomedical engineering, Vol. 44 (No. 8), pp. 736-52, (1997).
15- Marion Darbas and Stephanie Lohrengel, "Review on mathematical modelling of electroencephalography (EEG)." Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 121 (No. 1), pp. 3-39, (2019).
16- Helmut Buchner et al., "Inverse localization of electric dipole current sources in finite element models of the human head." Electroencephalography and clinical Neurophysiology, Vol. 102 (No. 4), pp. 267-78, (1997).
17- Y Yan, PL Nunez, and RT Hart, "Finite-element model of the human head: scalp potentials due to dipole sources." Medical and Biological Engineering and Computing, Vol. 29 (No. 5), pp. 475-81, (1991).
18- Maxime Monin, Lyes Rahmouni, Adrien Merlini, and Francesco P Andriulli, "A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography." IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology, (2020).
19- Chany Lee and Chang-Hwan Im, "New Strategy for Finite Element Mesh Generation for Accurate Solutions of Electroencephalography Forward Problems." Brain topography, Vol. 32 (No. 3), pp. 354-62, (2019).
20- Sergey N Makarov, Matti Hämäläinen, Yoshio Okada, Gregory M Noetscher, Jyrki Ahveninen, and Aapo Nummenmaa, "Boundary element fast multipole method for enhanced modeling of neurophysiological recordings." IEEE transactions on biomedical engineering, Vol. 68 (No. 1), pp. 308-18, (2020).
21- Sergey N Makarov, Gregory M Noetscher, Tommi Raij, and Aapo Nummenmaa, "A quasi-static boundary element approach with fast multipole acceleration for high-resolution bioelectromagnetic models." IEEE transactions on biomedical engineering, Vol. 65 (No. 12), pp. 2675-83, (2018).
22- Sergey N Makarov, Matti Hämäläinen, Yoshio Okada, Gregory M Noetscher, Jyrki Ahveninen, and Aapo Nummenmaa, "Solving High-Resolution Forward Problems for Extra-and Intracranial Neurophysiological Recordings Using Boundary Element Fast Multipole Method." bioRxiv, p. 567933, (2019).
23- Chris P Bradley, Glen M Harris, and Andrew J Pullan, "The computational performance of a high-order coupled FEM/BEM procedure in electropotential problems." IEEE transactions on biomedical engineering, Vol. 48 (No. 11), pp. 1238-50, (2001).
24- J Sikora, SR Arridge, RH Bayford, and L Horesh, "The application of hybrid BEM/FEM methods to solve electrical impedance tomography's forward problem for the human head." in XII ICEBI & V EIT Conference., Gdansk. ed, (2004).
25- Subhadra Srinivasan, Hamid R Ghadyani, Brian W Pogue, and Keith D Paulsen, "A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging." Biomedical optics express, Vol. 1 (No. 2), pp. 398-413, (2010).
26- Emmanuel Olivi, Maureen Clerc, and Théodore Papadopoulo, "Domain decomposition for coupling finite and boundary element methods in EEG." in 17th International Conference on Biomagnetism Advances in Biomagnetism–Biomag2010, (2010): Springer, pp. 120-23.
27- P Ghaderi Daneshmand and R Jafari, "A 3D hybrid BE–FE solution to the forward problem of electrical impedance tomography." Engineering Analysis with Boundary Elements, Vol. 37 (No. 4), pp. 757-64, (2013).
28- Nasireh Dayarian and Ali Khadem, "A hybrid boundary element-finite element method to solve the EEG forward problem." bioRxiv, (2021).
29- Johannes Vorwerk, Ümit Aydin, Carsten H Wolters, and Christopher R Butson, "Influence of head tissue conductivity uncertainties on EEG dipole reconstruction." Frontiers in neuroscience, Vol. 13p. 531, (2019).
30- Moritz Dannhauer, Benjamin Lanfer, Carsten H Wolters, and Thomas R Knösche, "Modeling of the human skull in EEG source analysis." Human brain mapping, Vol. 32 (No. 9), pp. 1383-99, (2011).
31- S Gedney, "The finite element method in electromagnetics [Book Review]." IEEE Antennas and Propagation Magazine, Vol. 36 (No. 3), pp. 75-76, (1994).
32- Paul H Schimpf, Ceon Ramon, and Jens Haueisen, "Dipole models for the EEG and MEG." IEEE transactions on biomedical engineering, Vol. 49 (No. 5), pp. 409-18, (2002).
33- Whye-Teong Ang, A beginner's course in boundary element methods. Universal-Publishers, (2007).
34- Jan WH Meijs, Onno W Weier, Maria J Peters, and ADRIAAN Van Oosterom, "On the numerical accuracy of the boundary element method (EEG application)." IEEE transactions on biomedical engineering, Vol. 36 (No. 10), pp. 1038-49, (1989).
35- A Zhi Zhang, "fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres Phys." Med. Biol, (1995).
36- Sven Wagner et al., "Using reciprocity for relating the simulation of transcranial current stimulation to the EEG forward problem." NeuroImage, Vol. 140pp. 163-73, (2016).
37- Qianqian Fang and David A Boas, "Tetrahedral mesh generation from volumetric binary and grayscale images." in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, (2009): Ieee, pp. 1142-45.
| Files | ||
| Issue | Vol 10 No 2 (2023) | |
| Section | Original Article(s) | |
| DOI | https://doi.org/10.18502/fbt.v10i2.12219 | |
| Keywords | ||
| Electroencephalography Forward Problem Boundary Element Method Finite Element Method Hybrid Boundary Element–Finite Element Method Spherical Head Model | ||
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How to Cite
1.
Dayarian N, Khadem A. Evaluating the Performance of the Hybrid Boundary Element- Finite Element (BE-FE) Method to Solve Electroencephalography (EEG) Forward Problem Based on the Mesh Quality: A Simulation Study. Frontiers Biomed Technol. 2023;10(2):150-160.
