Modeling head tissues
Predicting the electromagnetic fields produced by an elementary source model at a given sensor array requires another modeling step, which concerns a large part of the MEG/EEG literature. Indeed, MEG/EEG ‘head modeling’ studies the influence of the head geometry and electromagnetic properties of head tissues on the magnetic fields and electrical potentials measured outside the head.
Given a model of neural currents, the physics of MEG/EEG are ruled by the theory of electrodynamics (Feynman, 1964), which reduces in MEG to Maxwell’s equations, and to Ohm’s law in EEG, under quasistatic assumptions. These latter consider that the propagation delay of the electromagnetic waves from brain sources to the MEG/EEG sensors is negligible. The reason is the relative proximity of MEG/EEG sensors to the brain with respect to the expected frequency range of neural sources (up to 1KHz) (Hämäläinen et al., 1993). This is a very important, simplifying assumption, which has immediate consequences on the computational aspects of MEG/EEG head modeling.
Indeed, the equations of electro and magnetostatics determine that there exist analytical, closed-form solutions to MEG/EEG head modeling when the head geometry is considered as spherical. Hence, the simplest, and consequently by far most popular model of head geometry in MEG/EEG consists of concentric spherical layers: with one sphere per major category of head tissue (scalp, skull, cerebrospinal fluid and brain).
The spherical head geometry has further attractive properties for MEG in particular. Quite remarkably indeed, spherical MEG head models are insensitive to the number of shells and their respective conductivity: a source within a single homogeneous sphere generates the same MEG fields as when located inside a multilayered set of concentric spheres with different conductivities. The reason for this is that conductivity only influences the distribution of secondary, volume currents that circulate within the head volume and which are impressed by the original primary neural currents. The analytic formulation of Maxwell’s equations in the spherical geometry shows that these secondary currents do not generate any magnetic field outside the volume conductor (Sarvas, 1987). Therefore in MEG, only the location of the center of the spherical head geometry matters. The respective conductivity and radius of the spherical layers have no influence on the measured MEG fields. This is not the case in EEG, where both the location, radii and respective conductivity of each spherical shell influence the surface electrical potentials.
This relative sensitivity to tissue conductivity values is a general, important difference between EEG and MEG.
A spherical head model can be optimally adjusted to the head geometry, or restricted to regions of interest e.g., parieto-occipital regions for visual studies. Geometrical registration to MRI anatomical data improves the adjustment of the best-fitting sphere geometry to an individual head.
Another remarkable consequence of the spherical symmetry is that radially oriented brain currents produce no magnetic field outside a spherically symmetric volume conductor. For this reason, MEG signals from currents generated within the gyral crest or sulcal depth are attenuated, with respect to those generated by currents flowing perpendicularly to the sulcal walls. This is another important contrast between MEG and EEG’s respective sensitivity to source orientation (Hillebrand & Barnes, 2002).
Finally, the amplitude of magnetic fields decreases faster than electrical potentials’ with the distance from the generators to the sensors. Hence it has been argued that MEG is less sensitive to mesial and subcortical brain structures than EEG. Experimental and modeling efforts have shown however that MEG can detect neural activity from deeper brain regions (Tesche, 1996, Attal et al.., 2009).
Though spherical head models are convenient, they are poor approximations of the human head shape, which has some influence on the accuracy of MEG/EEG source estimation (Fuchs, Drenckhahn, Wischmann, & Wagner, 1998). More realistic head geometries have been investigated and all require solving Maxwell’s equations using numerical methods. Boundary Element (BEM) and Finite Element (FEM) methods are generic numerical approaches to the resolution of continuous equations over discrete space. In MEG/EEG, geometric tessellations of the different envelopes forming the head tissues need to be extracted from the individual MRI volume data to yield a realistic approximation of their geometry.
Three approaches to MEG/EEG head modeling: (a) Spherical approximation of the geometry of head tissues, with analytical solution to Maxwell’s and Ohm’s equations; (b) Tessellated surface envelopes of head tissues obtained from the segmentation of MRI data; (c) An alternative to (b) using volume meshes – here built from tetrahedra. In both (b) and (c) Maxwell’s and Ohm’s equations need to be solved using numerical methods: BEM and FEM, respectively.
In BEM, the conductivity of tissues is supposed to be homogeneous and isotropic within each envelope. Therefore, each tissue envelope is delimited using surface boundaries defined over a triangulation of each of the segmented envelopes obtained from MRI.
FEM assumes that tissue conductivity may be anisotropic (such as the skull bone and the white matter), therefore the primary geometric element needs to be an elementary volume, such as a tetrahedron (Marin, Guerin, Baillet, Garnero, & Meunier, 1998).
The main obstacle to a routine usage of BEM, and more pregnantly of FEM, is the surface or volume tessellation phase. Because the head geometry is intricate and not always well-defined from conventional MRI due to signal drop-outs and artifacts, automatic segmentation tools sometimes fail to identify some important tissue structures. The skull bone for instance, is invisible on conventional T1-weighted MRI. Some image processing techniques however can estimate the shape of the skull envelope from high-quality T1-weighted MRI data (Dogdas, Shattuck, & Leahy, 2005). However, the skull bone is a highly anisotropic structure, which is difficult to model from MRI data. Recent progress using MRI diffusion-tensor imaging (DTI) helps reveal the orientation of major white fiber bundles, which is also a major source of conductivity anisotropy (Haueisen et al.., 2002).
Computation times for BEM and FEM remain extremely long (several hours on a conventional workstation), and are detrimental to rapid access to source localization following data acquisition. Both algorithmic (Huang, Mosher, & Leahy, 1999, Kybic, Clerc, Faugeras, Keriven, & Papadopoulo, 2005) and pragmatic (Ermer, Mosher, Baillet, & Leah, 2001, Darvas, Ermer, Mosher, & Leahy, 2006) solutions to this problem have however been proposed to make realistic head models more operational. They are available in some academic software packages.
Finally, let us close this section with an important caveat: Realistic head modeling is bound to the correct estimation of tissues conductivity values. Though solutions for impedance tomography using MRI (Tuch, Wedeen, Dale, George, & Belliveau, 2001) and EEG (Goncalves et al.., 2003) have been suggested, they remain to be matured before entering the daily practice of MEG/EEG. So far, conductivity values from ex-vivo studies are conventionally integrated in most spherical and realistic head models (Geddes & Baker, 1967).