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Forward Modeling and Tissue Conductivities

  • Jens HaueisenEmail author
  • Thomas R. Knösche
Reference work entry

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.

Keywords

Volume conduction Field computation EEG/MEG modeling BEM FEM 

References

  1. 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 13:29–42PubMedCrossRefGoogle Scholar
  2. 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 14:151–167PubMedCrossRefGoogle Scholar
  3. 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 18:281–290PubMedCrossRefGoogle Scholar
  4. Akhtari M, Mandelkern M, Bui D, Salamon N, Vinters HV, Mathern GW (2010) Variable anisotropic brain electrical conductivities in epileptogenic foci. Brain Topogr 23:292–300PubMedPubMedCentralCrossRefGoogle Scholar
  5. 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:e93154PubMedPubMedCentralCrossRefGoogle Scholar
  6. 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:e0118753PubMedPubMedCentralCrossRefGoogle Scholar
  7. 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 29:371–387PubMedPubMedCentralCrossRefGoogle Scholar
  8. Barnard ACL, Duck IM, Lynn MS (1967a) Application of electromagnetic theory to electrocardiology I. Derivation of integral equations. Biophys J 7:443–462PubMedPubMedCentralCrossRefGoogle Scholar
  9. Barnard ACL, Duck IM, Lynn MS, Timlake WP (1967b) Application of electromagnetic theory to electrocardiology II. Numerical solution of integral equations. Biophys J 7:463–491PubMedPubMedCentralCrossRefGoogle Scholar
  10. Basser PJ, Mattiello J, LeBihan D (1994) MR diffusion tensor spectroscopy and imaging. Biophys J 66:259–267PubMedPubMedCentralCrossRefGoogle Scholar
  11. Baumann S, Wozny D, Kelly S, Meno F (1997) The electrical conductivity of human cerebrospinal fluid at body temperature. IEEE Trans Biomed Eng 44:220–223PubMedCrossRefGoogle Scholar
  12. Baysal U, Haueisen J (2004) Use of a priori information in estimating tissue resistivities – application to human data in vivo. Physiol Meas 25:737–748PubMedCrossRefGoogle Scholar
  13. Boemmel F, Roeckelein R, Urankar L (1993) Boundary element solution of biomagnetic problems. IEEE Trans Magn 29:1395–1398CrossRefGoogle Scholar
  14. Brebbia C, Telles J, Wrobel L (1984) Boundary element techniques. Springer, BerlinCrossRefGoogle Scholar
  15. Crile GW, Hosmer HR, Rowland AF (1922) The electrical conductivity of animal tissues under normal and pathological conditions. Am J Physiol 60:59–106CrossRefGoogle Scholar
  16. Cuffin BN, Cohen D (1977) Magnetic fields of a dipole in special volume conductor shapes. IEEE Trans Biomed Eng 24:372–381PubMedCrossRefGoogle Scholar
  17. 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 127:484–495PubMedCrossRefGoogle Scholar
  18. Dannhauer M, Lanfer B, Wolters CH, Knösche TR (2011) Modeling of the human skull in EEG source analysis. Hum Brain Mapp 32:1383–1399PubMedCrossRefGoogle Scholar
  19. de Munck JC (1988) The potential distribution in a llayered anisotropic spheroidal volume conductor. J Appl Phys 64:464–470CrossRefGoogle Scholar
  20. de Munck JC (1989) A mathematical and physical interpretation of the electromagnetic field of the brain. University of Amsterdam, AmsterdamGoogle Scholar
  21. Donchin E (1966) A multivariate approach to analysis of average evoked potentials. IEEE Trans Biomed Eng BM13:131–139CrossRefGoogle Scholar
  22. 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 46:1055–1065PubMedCrossRefPubMedCentralGoogle Scholar
  23. Fieseler T (1999) Analytic source and volume conductor models for biomagnetic fields. University Jena, JenaGoogle Scholar
  24. 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 42:1094–1104PubMedCrossRefPubMedCentralGoogle Scholar
  25. 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 107:93–111PubMedPubMedCentralCrossRefGoogle Scholar
  26. Gabriel C, Gabriel S, Corthout E (1996) The dielectric properties of biological tissues. 1. Literature survey. Phys Med Biol 41:2231–2249PubMedCrossRefPubMedCentralGoogle Scholar
  27. Gabriel C, Peyman A, Grant EH (2009) Electrical conductivity of tissue at frequencies below 1 MHz. Phys Med Biol 54:4863–4878PubMedCrossRefPubMedCentralGoogle Scholar
  28. Galeotti G (1902) The electric conductibility of animal tissues. Z Biol 43:289–340Google Scholar
  29. Geddes LA, Baker LE (1967) Specific resistance of biological material – a compendium of data for the biomedical engineer and physiologist. Med Biol Eng 5:271–293PubMedCrossRefPubMedCentralGoogle Scholar
  30. Gencer NG, Acar CE (2004) Sensitivity of EEG and MEG measurements to tissue conductivity. Phys Med Biol 49:701–717PubMedCrossRefGoogle Scholar
  31. Geselowitz D (1967) On bioelectric potentials in an inhomogeneous volume conductor. Biophys J 7:1–11PubMedPubMedCentralCrossRefGoogle Scholar
  32. Geselowitz D (1970) On magnetic field generated outside an inhomogeneous volume conductor by internal current sources. IEEE Trans Magn 6:346–347CrossRefGoogle Scholar
  33. Giapalaki SN, Kariotou F (2006) The complete ellipsoidal shell-model in EEG imaging. Abstr Appl Anal 2006:1–18CrossRefGoogle Scholar
  34. 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 50:1124–1128PubMedCrossRefGoogle Scholar
  35. 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 51:145–163PubMedCrossRefGoogle Scholar
  36. Gutierrez D, Nehorai A, Muravchik CH (2004) Estimating brain conductivities and dipole source signals with EEG arrays. IEEE Trans Biomed Eng 51:2113–2122PubMedCrossRefGoogle Scholar
  37. 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 50:3787–3806PubMedCrossRefGoogle Scholar
  38. 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 4:46PubMedPubMedCentralCrossRefGoogle Scholar
  39. Hämäläinen MS, Sarvas J (1989) Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data. IEEE Trans Biomed Eng 36:165–171PubMedCrossRefGoogle Scholar
  40. 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 23:728–739PubMedPubMedCentralCrossRefGoogle Scholar
  41. Haueisen J, Hafner C, Nowak H, Brauer H (1996) Neuromagnetic field computation using the multiple multipole method. Int J Numer Modell 9:144–158CrossRefGoogle Scholar
  42. 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 42:240–248CrossRefGoogle Scholar
  43. 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 15:159–166PubMedCrossRefGoogle Scholar
  44. 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 16:29–38PubMedCrossRefGoogle Scholar
  45. Huang MX, Mosher JC, Leahy RM (1999) A sensor-weighted overlapping-sphere head model and exhaustive head model comparison for MEG. Phys Med Biol 44:423–440PubMedCrossRefPubMedCentralGoogle Scholar
  46. 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:e18834Google Scholar
  47. Ilmoniemi R (1985) Neuromagnetism: theory, techniques, and measurements. Helsinki University of Technology, EspooGoogle Scholar
  48. Kariotou F (2004) Electroencephalography in ellipsoidal geometry. J Math Anal Appl 290:324–342CrossRefGoogle Scholar
  49. Kayser J, Tenke CE (2005) Trusting in or breaking with convention: towards a renaissance of principal components analysis in electrophysiology. Clin Neurophysiol 116:1747–1753PubMedCrossRefGoogle Scholar
  50. 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 24:12–28PubMedCrossRefGoogle Scholar
  51. 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 116:456–465PubMedCrossRefGoogle Scholar
  52. Lalancette M, Quraan M, Cheyne D (2011) Evaluation of multiple-sphere head models for MEG source localization. Phys Med Biol 56:5621–5635PubMedCrossRefGoogle Scholar
  53. 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 30:2862–2878PubMedPubMedCentralCrossRefGoogle Scholar
  54. 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 59:1970–1988PubMedPubMedCentralCrossRefGoogle Scholar
  55. 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 46:3051–3059PubMedCrossRefGoogle Scholar
  56. Logothetis NK, Kayser C, Oeltermann A (2007) In vivo measurement of cortical impedance spectrum in monkeys: implications for signal propagation. Neuron 55:809–823PubMedCrossRefGoogle Scholar
  57. 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. 6:103–118Google Scholar
  58. Mosher JC, Leahy RM, Lewis PS (1999) EEG and MEG: forward solutions for inverse methods. IEEE Trans Biomed Eng 46:245–259PubMedPubMedCentralCrossRefGoogle Scholar
  59. Murakami S, Okada Y (2006) Contributions of principal neocortical neurons to magnetoencephalography and electroencephalography signals. J Physiol Lond 575:925–936PubMedPubMedCentralCrossRefGoogle Scholar
  60. 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 48:3637–3652PubMedCrossRefGoogle Scholar
  61. 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 14:3034Google Scholar
  62. 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 72:742–753PubMedCrossRefPubMedCentralGoogle Scholar
  63. 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 47:1487–1492PubMedCrossRefPubMedCentralGoogle Scholar
  64. Pauly H, Schwan H (1964) The dielectric properties of the bovine eye lens. IEEE Trans Biomed Eng 11:103–109PubMedCrossRefPubMedCentralGoogle Scholar
  65. Plonsey R, Heppner DB (1967) Considerations of quasi-stationarity in electrophysiological systems. Bull Math Biophys 29:657–664PubMedCrossRefPubMedCentralGoogle Scholar
  66. Ranck JB (1963) Analysis of specific impedance of rabbit cerebral cortex. Exp Neurol 7:153–174PubMedCrossRefPubMedCentralGoogle Scholar
  67. Rush S, Driscoll DA (1968) Current distribution in the brain from surface electrodes. Anesth Analg Curr Res 47:717–723CrossRefGoogle Scholar
  68. Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys Med Biol 32:11–22PubMedCrossRefPubMedCentralGoogle Scholar
  69. Schimpf PH (2007) Application of quasi-static magnetic reciprocity to finite element models of the MEG lead-field. IEEE Trans Biomed Eng 54:2082–2088PubMedCrossRefPubMedCentralGoogle Scholar
  70. Schimpf PH, Ramon C, Haueisen J (2002) Dipole models for the EEG and MEG. IEEE Trans Biomed Eng 49:409–418PubMedCrossRefGoogle Scholar
  71. 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 33:571–586PubMedCrossRefGoogle Scholar
  72. Seo JK, Woo EJ (2011) Magnetic resonance electrical impedance tomography (MREIT). SIAM Rev 53:40–68CrossRefGoogle Scholar
  73. Seo JK, Woo EJ (2014) Electrical tissue property imaging at low frequency using MREIT. IEEE Trans Biomed Eng 61:1390–1399PubMedCrossRefGoogle Scholar
  74. 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 55:2124–2133PubMedCrossRefGoogle Scholar
  75. Stenroos M, Hauk O (2013) Minimum-norm cortical source estimation in layered head models is robust against skull conductivity error. Neuroimage 81:265–272PubMedPubMedCentralCrossRefGoogle Scholar
  76. Stenroos M, Nenonen J (2012) On the accuracy of collocation and Galerkin BEM in the EEG/MEG forward problem. Int J Bioelectromagn 14:29–33Google Scholar
  77. Stenroos M, Sarvas J (2012) Bioelectromagnetic forward problem: isolated source approach revis(it)ed. Phys Med Biol 57:3517–3535PubMedCrossRefGoogle Scholar
  78. 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 88:256–263CrossRefGoogle Scholar
  79. Stenroos M, Hunold A, Haueisen J (2014) Comparison of three-shell and simplified volume conductor models in magnetoencephalography. Neuroimage 94:337–348PubMedCrossRefGoogle Scholar
  80. 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 55:2286–2292PubMedCrossRefGoogle Scholar
  81. 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 72:209–222CrossRefGoogle Scholar
  82. 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 98:11697–11701PubMedCrossRefGoogle Scholar
  83. 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 34:21–26PubMedCrossRefGoogle Scholar
  84. 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, InnsbruckGoogle Scholar
  85. 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 32:766–770PubMedCrossRefGoogle Scholar
  86. 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 55:2481–2486PubMedPubMedCentralCrossRefGoogle Scholar
  87. 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 2010:397272–397272PubMedPubMedCentralCrossRefGoogle Scholar
  88. Witwer JG, Trezek GJ, Jewett DL (1972) Effect of media inhomogeneities upon intracranial electrical fields. IEEE Trans Biomed Eng BM19:352–362CrossRefGoogle Scholar
  89. 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 20:1099–1116CrossRefGoogle Scholar
  90. 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 1300:189–192CrossRefGoogle Scholar
  91. Zhang YC, Zhu SA, He B (2004) A second-order finite element algorithm for solving the three-dimensional EEG forward problem. Phys Med Biol 49:2975–2987PubMedCrossRefPubMedCentralGoogle Scholar
  92. Zhang YC, van Drongelen W, He B (2006) Estimation of in vivo brain-to-skull conductivity ratio in humans. Appl Phys Lett 89:223903PubMedPubMedCentralCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Biomedical Engineering and InformaticsTechnische Universität IlmenauIlmenauGermany
  2. 2.Max Planck Institute for Human Cognitive and Brain SciencesLeipzigGermany

Section editors and affiliations

  • Seppo P. Ahlfors
    • 1
    • 2
  1. 1.Department of Radiology, MGH/HST Athinoula A. Martinos Center for Biomedical ImagingMassachusetts General HospitalCharlestown, MAUSA
  2. 2.Harvard-MIT Division of Health Sciences and TechnologyCambridgeUSA

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