Skip to main content

Forward Modeling and Tissue Conductivities

  • Reference work entry
  • First Online:
Book cover Magnetoencephalography

Abstract

The neuroelectromagnetic forward model describes the prediction of measurements from known sources. It includes models for the sources and the sensors as well as an electromagnetic description of the head as a volume conductor, which are discussed in this chapter. First we give a general overview on the forward problem and discuss various simplifications and assumptions that lead to different analytical and numerical methods. Next, we introduce important analytical models which assume simple geometries of the head. Then we describe numerical models accounting for realistic geometries. The most important numerical methods for head modeling are the boundary element method (BEM) and the finite element method (FEM). The boundary element method describes the head by a small number of compartments, each with a homogeneous isotropic conductivity. In contrast, the finite element method discretizes the 3D distribution of the anisotropic conductivity tensor with the help of small-volume elements. Subsequently, we discuss in some detail how electrical conductivity information is measured and how it is used in forward modeling. Finally, we briefly introduce the lead field concept.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 649.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 849.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Akhtari M, Bryant HC, Mamelak AN, Heller L, Shih JJ, Mandelkern M, Matlachov A, Ranken DM, Best ED, Sutherling WW (2000) Conductivities of three-layer human skull. Brain Topogr 13:29–42

    Article  CAS  PubMed  Google Scholar 

  • Akhtari M, Bryant HC, Marnelak AN, Flynn ER, Heller L, Shih JJ, Mandelkern M, Matlachov A, Ranken DM, Best ED, DiMauro MA, Lee RR, Sutherling WW (2002) Conductivities of three-layer live human skull. Brain Topogr 14:151–167

    Article  CAS  PubMed  Google Scholar 

  • Akhtari M, Salamon N, Duncan R, Fried I, Mathern GW (2006) Electrical conductivities of the freshly excised cerebral cortex in epilepsy surgery patients; correlation with pathology, seizure duration, and diffusion tensor imaging. Brain Topogr 18:281–290

    Article  CAS  PubMed  Google Scholar 

  • Akhtari M, Mandelkern M, Bui D, Salamon N, Vinters HV, Mathern GW (2010) Variable anisotropic brain electrical conductivities in epileptogenic foci. Brain Topogr 23:292–300

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Aydin U, Vorwerk J, Kupper 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 9:e93154

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Aydin U, Vorwerk J, Dumpelmann M, Kupper P, Kugel H, Heers M, Wellmer J, Kellinghaus C, Haueisen J, Rampp S, Stefan H, Wolters CH (2015) Combined EEG/MEG can outperform single modality EEG or MEG source reconstruction in presurgical epilepsy diagnosis. PLoS One 10:e0118753

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Bangera N, Schomer D, Dehghani N, Ulbert I, Cash S, Papavasiliou S, Eisenberg S, Dale A, Halgren E (2010) Experimental validation of the influence of white matter anisotropy on the intracranial EEG forward solution. J Comput Neurosci 29:371–387

    Article  PubMed  PubMed Central  Google Scholar 

  • Barnard ACL, Duck IM, Lynn MS (1967a) Application of electromagnetic theory to electrocardiology I. Derivation of integral equations. Biophys J 7:443–462

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Barnard ACL, Duck IM, Lynn MS, Timlake WP (1967b) Application of electromagnetic theory to electrocardiology II. Numerical solution of integral equations. Biophys J 7:463–491

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Basser PJ, Mattiello J, LeBihan D (1994) MR diffusion tensor spectroscopy and imaging. Biophys J 66:259–267

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Baumann S, Wozny D, Kelly S, Meno F (1997) The electrical conductivity of human cerebrospinal fluid at body temperature. IEEE Trans Biomed Eng 44:220–223

    Article  CAS  PubMed  Google Scholar 

  • Baysal U, Haueisen J (2004) Use of a priori information in estimating tissue resistivities – application to human data in vivo. Physiol Meas 25:737–748

    Article  PubMed  Google Scholar 

  • Boemmel F, Roeckelein R, Urankar L (1993) Boundary element solution of biomagnetic problems. IEEE Trans Magn 29:1395–1398

    Article  Google Scholar 

  • Brebbia C, Telles J, Wrobel L (1984) Boundary element techniques. Springer, Berlin

    Book  Google Scholar 

  • Crile GW, Hosmer HR, Rowland AF (1922) The electrical conductivity of animal tissues under normal and pathological conditions. Am J Physiol 60:59–106

    Article  Google Scholar 

  • Cuffin BN, Cohen D (1977) Magnetic fields of a dipole in special volume conductor shapes. IEEE Trans Biomed Eng 24:372–381

    Article  CAS  PubMed  Google Scholar 

  • Dabek J, Kalogianni K, Rotgans E, van der Helm FCT, Kwakkel G, van Wegen EEH, Daffertshofer A, de Munck JC (2016) Determination of head conductivity frequency response in vivo with optimized EIT-EEG. Neuroimage 127:484–495

    Article  PubMed  Google Scholar 

  • Dannhauer M, Lanfer B, Wolters CH, Knösche TR (2011) Modeling of the human skull in EEG source analysis. Hum Brain Mapp 32:1383–1399

    Article  PubMed  Google Scholar 

  • de Munck JC (1988) The potential distribution in a llayered anisotropic spheroidal volume conductor. J Appl Phys 64:464–470

    Article  Google Scholar 

  • de Munck JC (1989) A mathematical and physical interpretation of the electromagnetic field of the brain. University of Amsterdam, Amsterdam

    Google Scholar 

  • Donchin E (1966) A multivariate approach to analysis of average evoked potentials. IEEE Trans Biomed Eng BM13:131–139

    Article  Google Scholar 

  • Drechsler F, Wolters CH, Dierkes T, Si H, Grasedyck L (2009) A full subtraction approach for finite element method based source analysis using constrained Delaunay tetrahedralisation. Neuroimage 46:1055–1065

    Article  CAS  PubMed  Google Scholar 

  • Fieseler T (1999) Analytic source and volume conductor models for biomagnetic fields. University Jena, Jena

    Google Scholar 

  • Fletcher D, Amir A, Jewett D, Fein G (1995) Improved method for computation of potentials in a realistic head shape model. IEEE Trans Biomed Eng 42:1094–1104

    Article  CAS  PubMed  Google Scholar 

  • Fuchs M, Wagner M, Wischmann HA, Kohler T, Theissen A, Drenckhahn R, Buchner H (1998) Improving source reconstructions by combining bioelectric and biomagnetic data. Electroencephalogr Clin Neurophysiol 107:93–111

    Article  CAS  PubMed  Google Scholar 

  • Gabriel C, Gabriel S, Corthout E (1996) The dielectric properties of biological tissues. 1. Literature survey. Phys Med Biol 41:2231–2249

    Article  CAS  PubMed  Google Scholar 

  • Gabriel C, Peyman A, Grant EH (2009) Electrical conductivity of tissue at frequencies below 1 MHz. Phys Med Biol 54:4863–4878

    Article  CAS  PubMed  Google Scholar 

  • Galeotti G (1902) The electric conductibility of animal tissues. Z Biol 43:289–340

    CAS  Google Scholar 

  • Geddes LA, Baker LE (1967) Specific resistance of biological material – a compendium of data for the biomedical engineer and physiologist. Med Biol Eng 5:271–293

    Article  CAS  PubMed  Google Scholar 

  • Gencer NG, Acar CE (2004) Sensitivity of EEG and MEG measurements to tissue conductivity. Phys Med Biol 49:701–717

    Article  PubMed  Google Scholar 

  • Geselowitz D (1967) On bioelectric potentials in an inhomogeneous volume conductor. Biophys J 7:1–11

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Geselowitz D (1970) On magnetic field generated outside an inhomogeneous volume conductor by internal current sources. IEEE Trans Magn 6:346–347

    Article  Google Scholar 

  • Giapalaki SN, Kariotou F (2006) The complete ellipsoidal shell-model in EEG imaging. Abstr Appl Anal 2006:1–18

    Article  Google Scholar 

  • Gonçalves S, de Munck JC, Verbunt JPA, Heethaar RM, da Silva FHL (2003) In vivo measurement of the brain and skull resistivities using an EIT-based method and the combined analysis of SEF/SEP data. IEEE Trans Biomed Eng 50:1124–1128

    Article  PubMed  Google Scholar 

  • 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 51:145–163

    Article  PubMed  Google Scholar 

  • Gutierrez D, Nehorai A, Muravchik CH (2004) Estimating brain conductivities and dipole source signals with EEG arrays. IEEE Trans Biomed Eng 51:2113–2122

    Article  PubMed  Google Scholar 

  • 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 50:3787–3806

    Article  PubMed  Google Scholar 

  • Hallez H, Vanrumste B, Grech R, Muscat J, De Clercq 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 4:46

    Article  PubMed  PubMed Central  Google Scholar 

  • Hämäläinen MS, Sarvas J (1989) Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data. IEEE Trans Biomed Eng 36:165–171

    Article  PubMed  Google Scholar 

  • 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

    Article  CAS  PubMed  Google Scholar 

  • Haueisen J, Hafner C, Nowak H, Brauer H (1996) Neuromagnetic field computation using the multiple multipole method. Int J Numer Modell 9:144–158

    Article  Google Scholar 

  • Haueisen J, Bottner A, Funke M, Brauer H, Nowak H (1997) The influence of boundary element discretization on the forward and inverse problem in electroencephalography and magnetoencephalography. Biomed Tech 42:240–248

    Article  CAS  Google Scholar 

  • Haueisen J, Tuch DS, Ramon C, Schimpf PH, Wedeen VJ, George JS, Belliveau JW (2002) The influence of brain tissue anisotropy on human EEG and MEG. Neuroimage 15:159–166

    Article  CAS  PubMed  Google Scholar 

  • Hoekema R, Wieneke GH, Leijten FSS, van Veelen CWM, van Rijen PC, Huiskamp GJM, Ansems J, van Huffelen AC (2003) Measurement of the conductivity of skull, temporarily removed during epilepsy surgery. Brain Topogr 16:29–38

    Article  CAS  PubMed  Google Scholar 

  • Huang MX, Mosher JC, Leahy RM (1999) A sensor-weighted overlapping-sphere head model and exhaustive head model comparison for MEG. Phys Med Biol 44:423–440

    Article  CAS  PubMed  Google Scholar 

  • Huang Y, Liu AA, Lafon B, Friedman D, Dayan M, Wang X, Bikson M, Doyle WK, Devinsky O, Parra LC (2017) Measurements and models of electric fields in the in vivo human brain during transcranial electric stimulation. Elife 6:e18834

    Google Scholar 

  • Ilmoniemi R (1985) Neuromagnetism: theory, techniques, and measurements. Helsinki University of Technology, Espoo

    Google Scholar 

  • Kariotou F (2004) Electroencephalography in ellipsoidal geometry. J Math Anal Appl 290:324–342

    Article  Google Scholar 

  • Kayser J, Tenke CE (2005) Trusting in or breaking with convention: towards a renaissance of principal components analysis in electrophysiology. Clin Neurophysiol 116:1747–1753

    Article  PubMed  Google Scholar 

  • Kybic J, Clerc M, Abboud T, Faugeras O, Keriven R, Papadopoulo T (2005) A common formalism for the integral formulations of the forward EEG problem. IEEE Trans Med Imaging 24:12–28

    Article  PubMed  Google Scholar 

  • Lai Y, van Drongelen W, Ding L, Hecox KE, Towle VL, Frim DM, He B (2005) Estimation of in vivo human brain-to-skull conductivity ratio from simultaneous extra- and intra-cranial electrical potential recordings. Clin Neurophysiol 116:456–465

    Article  CAS  PubMed  Google Scholar 

  • Lalancette M, Quraan M, Cheyne D (2011) Evaluation of multiple-sphere head models for MEG source localization. Phys Med Biol 56:5621–5635

    Article  CAS  PubMed  Google Scholar 

  • Lew S, Wolters CH, Anwander A, Makeig S, MacLeod RS (2009a) Improved EEG source analysis using low-resolution conductivity estimation in a four-compartment finite element head model. Hum Brain Mapp 30:2862–2878

    Article  PubMed  Google Scholar 

  • Lew S, Wolters CH, Dierkes T, Röer C, Macleod RS (2009b) 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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Lindenblatt G, Silny J (2001) A model of the electrical volume conductor in the region of the eye in the ELF range. Phys Med Biol 46:3051–3059

    Article  CAS  PubMed  Google Scholar 

  • Logothetis NK, Kayser C, Oeltermann A (2007) In vivo measurement of cortical impedance spectrum in monkeys: implications for signal propagation. Neuron 55:809–823

    Article  CAS  PubMed  Google Scholar 

  • Lütkenhöner B, Pantev C, Hoke M (1990) Comparison between different methods to approximate an area of the human head by a sphere. In: Grandori F, Hoke M, Romani GL (eds) Advances in audiology, Karger, Basel. 6:103–118

    Google Scholar 

  • Mosher JC, Leahy RM, Lewis PS (1999) EEG and MEG: forward solutions for inverse methods. IEEE Trans Biomed Eng 46:245–259

    Article  CAS  PubMed  Google Scholar 

  • Murakami S, Okada Y (2006) Contributions of principal neocortical neurons to magnetoencephalography and electroencephalography signals. J Physiol Lond 575:925–936

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Nolte G (2003) The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors. Phys Med Biol 48:3637–3652

    Article  PubMed  Google Scholar 

  • Oh S, Lee S, Cho M, Kim T, Kim I (2006) Electrical conductivity estimation from diffusion tensor and T2: a silk yarn phantom study. Proc Int Soc Magn Reson Med 14:3034

    Google Scholar 

  • Okada Y (1994) Origin of the apparent tissue conductivity in the molecular and granular layers of the in vitro turtle cerebellum and the interpretation of current source-density analysis. J Neurophysiol 72:742–753

    Article  CAS  PubMed  Google Scholar 

  • Oostendorp TF, Delbeke J, Stegeman DF (2000) The conductivity of the human skull: results of in vivo and in vitro measurements. IEEE Trans Biomed Eng 47:1487–1492

    Article  CAS  PubMed  Google Scholar 

  • Pauly H, Schwan H (1964) The dielectric properties of the bovine eye lens. IEEE Trans Biomed Eng 11:103–109

    Article  CAS  PubMed  Google Scholar 

  • Plonsey R, Heppner DB (1967) Considerations of quasi-stationarity in electrophysiological systems. Bull Math Biophys 29:657–664

    Article  CAS  PubMed  Google Scholar 

  • Ranck JB (1963) Analysis of specific impedance of rabbit cerebral cortex. Exp Neurol 7:153–174

    Article  PubMed  Google Scholar 

  • Rush S, Driscoll DA (1968) Current distribution in the brain from surface electrodes. Anesth Analg Curr Res 47:717–723

    Article  CAS  Google Scholar 

  • Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys Med Biol 32:11–22

    Article  CAS  PubMed  Google Scholar 

  • Schimpf PH (2007) Application of quasi-static magnetic reciprocity to finite element models of the MEG lead-field. IEEE Trans Biomed Eng 54:2082–2088

    Article  PubMed  Google Scholar 

  • Schimpf PH, Ramon C, Haueisen J (2002) Dipole models for the EEG and MEG. IEEE Trans Biomed Eng 49:409–418

    Article  PubMed  Google Scholar 

  • Sengül G, Baysal U (2012) An extended Kalman filtering approach for the estimation of human head tissue conductivities by using EEG data: a simulation study. Physiol Meas 33:571–586

    Article  PubMed  Google Scholar 

  • Seo JK, Woo EJ (2011) Magnetic resonance electrical impedance tomography (MREIT). SIAM Rev 53:40–68

    Article  Google Scholar 

  • Seo JK, Woo EJ (2014) Electrical tissue property imaging at low frequency using MREIT. IEEE Trans Biomed Eng 61:1390–1399

    Article  PubMed  Google Scholar 

  • Stenroos M, Haueisen J (2008) Boundary element computations in the forward and inverse problems of electrocardiography: comparison of collocation and Galerkin weightings. IEEE Trans Biomed Eng 55:2124–2133

    Article  PubMed  Google Scholar 

  • Stenroos M, Hauk O (2013) Minimum-norm cortical source estimation in layered head models is robust against skull conductivity error. Neuroimage 81:265–272

    Article  PubMed  Google Scholar 

  • Stenroos M, Nenonen J (2012) On the accuracy of collocation and Galerkin BEM in the EEG/MEG forward problem. Int J Bioelectromagn 14:29–33

    Google Scholar 

  • Stenroos M, Sarvas J (2012) Bioelectromagnetic forward problem: isolated source approach revis(it)ed. Phys Med Biol 57:3517–3535

    Article  CAS  PubMed  Google Scholar 

  • Stenroos M, Mantynen V, Nenonen J (2007) A Matlab library for solving quasi-static volume conduction problems using the boundary element method. Comput Methods Prog Biomed 88:256–263

    Article  CAS  Google Scholar 

  • Stenroos M, Hunold A, Haueisen J (2014) Comparison of three-shell and simplified volume conductor models in magnetoencephalography. Neuroimage 94:337–348

    Article  PubMed  Google Scholar 

  • Tang C, You F, Cheng G, Gao D, Fu F, Yang G, Dong X (2008) Correlation between structure and resistivity variations of the live human skull. IEEE Trans Biomed Eng 55:2286–2292

    Article  PubMed  Google Scholar 

  • Tissari S, Rahola J (2003) Error analysis of a Galerkin method to solve the forward problem in MEG using the boundary element method. Comput Methods Prog Biomed 72:209–222

    Article  Google Scholar 

  • Tuch DS, Wedeen VJ, Dale AM, George JS, Belliveau JW (2001) Conductivity tensor mapping of the human brain using diffusion tensor MRI. Proc Natl Acad Sci USA 98:11697–11701

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • van den Broek SP, Zhou H, Peters MJ (1996) Computation of neuromagnetic fields using finite-element method and Biot-Savart law. Med Biol Eng Comput 34:21–26

    Article  PubMed  Google Scholar 

  • van Uitert R, Weinstein D, Johnson C, Zhukov L (2001) Finite element EEG and MEG simulations for realistic head models: quadratic vs. linear approximations. In: International conference on non-invasive functional source imaging, Innsbruck

    Google Scholar 

  • Vesanen PT, Nieminen JO, Zevenhoven KCJ, Hsu YC, Ilmoniemi RJ (2014) Current-density imaging using ultra-low-field MRI with zero-field encoding. Magn Reson Imaging 32:766–770

    Article  PubMed  Google Scholar 

  • Wang K, Li J, Zhu S, Mueller B, Lim K, Liu Z, He B (2008) A new method to derive white matter conductivity from diffusion tensor MRI. IEEE Trans Biomed Eng 55:2481–2486

    Article  PubMed  PubMed Central  Google Scholar 

  • Wendel K, Vaisanen J, Seemann G, Hyttinen J, Malmivuo J (2010) The influence of age and skull conductivity on surface and subdermal bipolar EEG leads. Comput Intell Neurosci 2010:397272–397272

    Article  PubMed  PubMed Central  Google Scholar 

  • Witwer JG, Trezek GJ, Jewett DL (1972) Effect of media inhomogeneities upon intracranial electrical fields. IEEE Trans Biomed Eng BM19:352–362

    Article  Google Scholar 

  • 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 Probl 20:1099–1116

    Article  Google Scholar 

  • Wolters C, Köstler H, Möller C, Härdtlein J, Anwander A (2007) Numerical approaches for dipole modeling in finite element method based source analysis. Int Congr Ser 1300:189–192

    Article  Google Scholar 

  • Zhang YC, Zhu SA, He B (2004) A second-order finite element algorithm for solving the three-dimensional EEG forward problem. Phys Med Biol 49:2975–2987

    Article  CAS  PubMed  Google Scholar 

  • Zhang YC, van Drongelen W, He B (2006) Estimation of in vivo brain-to-skull conductivity ratio in humans. Appl Phys Lett 89:223903

    Article  PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jens Haueisen .

Editor information

Editors and Affiliations

Section Editor information

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Haueisen, J., Knösche, T.R. (2019). Forward Modeling and Tissue Conductivities. In: Supek, S., Aine, C. (eds) Magnetoencephalography. Springer, Cham. https://doi.org/10.1007/978-3-030-00087-5_4

Download citation

Publish with us

Policies and ethics