For electrical or magnetic measurements at a distance significantly larger than the extent of the source, the spatial fine structure of the field distribution is not detected, and the dominant contribution is the dipole associated with longitudinal intracellular currents. In MEG, the effects of ohmic currents through the head volume are minimal, and a reasonable analytical solution can be derived by treating the head as a homogeneous conducting sphere. The radius of the sphere is chosen to approximate the inner surface of the skull, based on individual anatomical images or external measures of head shape coupled with a knowledge of average anatomy. Unlike MEG, EEG signal topographies are strongly influenced by the conductivity properties of the head. Even simple models of elemental EEG signals must take into account the presence of multiple tissue layers of differing conductivity. Dipole response topographies for EEG can be computed in a volume model consisting of multiple spherical shells, with layers corresponding to brain, cerebral-spinal fluid, skull, and scalp, using a truncated series of Legendre polynomials. Many researchers in the field have argued that this class of model is more than adequate for source localization, given the uncertainties associated with the inverse problem. However, as inverse procedures have improved, a number of studies have underscored the value of more sophisticated forward models for source localization accuracy. Some of these approaches are summarized in Fig. 5.

A significant improvement in the accuracy of the forward calculation can be achieved by boundary element calculations incorporating the geometry of the major tissue classes within the head. Surface meshes are constructed to approximate conductivity boundaries. Because the conductivity of the skull is significantly lower than that of other tissues in the head, it is particularly important to capture the geometry of the skull near sensors. Most calculations of this sort are limited to simple topologies, e.g., nested compartments without intersections or penetrations. Conductivity values are typically taken from the literature and correspond to measurements originally made in cadaver tissue. Because there is no basis for further subdivision, conductivity is taken as homogeneous within a compartment. Although the boundary element method employs simple geometries, the calculations are relatively time-consuming, because the solution matrix typically contains terms for the interactions between every pair of nodes in the mesh. Numerical studies have demonstrated that it is possible to achieve accuracy approaching that of the boundary element methods at much lower computational cost by approximating the skull boundary with local spheres selected for each sensor.

3D methods provide an alternative strategy for highresolution forward calculations. In finite difference (FD) or finite element (FE) forward calculations, the head volume is divided into a collection of volume elements that form the computational mesh. Potentials are computed at nodes, typically associated with boundaries or vertices of the volume elements. These calculations assume a current source that generates the potential and account for the conductivity properties of the volume elements that strongly influence the spread of potential. FE methods typically employ an irregular mesh composed of tetrahedral elements. This allows the mesh to closely match conductivity boundaries within the medium that may give rise to sharp gradients in the potential distribution, providing greater accuracy with a smaller number of volume elements. However, most present methods for mesh generation and refinement require considerable human intervention. FD methods typically employ a regular (e.g., rectangular) mesh. Segmentation or voxel classification schemes can be applied to regular volumetric data such as MRI to produce a data volume that can be used for calculations without constructing specialized meshes.

3D methods are most useful if we have access to external information that can be used to define the geometry or electrical properties of the conductive medium. MRI provides the most accessible and flexible measure of tissue properties, although X-ray CT provides a better definition of the geometry and microstructure of the skull. New and evolving MRI techniques will eventually provide even more useful information for forward modeling. Current density MRI operates by applying external currents at the head surface and measuring the perturbation of the image by local volume currents. This allows an estimation of head conductivity on a voxel by voxel basis. Diffusion tensor MRI measures the magnitude and characterizes the direction of diffusion of water within tissue. Structures such as white matter tracts give rise to anisotropic physical properties that can be described by a tensor, e.g., the apparent diffusion coefficient of water and the measured conductivity are much higher along a fiber tract than transverse to it. MRI can be used to estimate the anisotropic conductivity of such tissue. FE and FD methods can accommodate anistropic conductivity (though again the FD application is simpler). By passing currents through pairs of surface electrodes and estimating the induced potentials at other electrodes in the array, electrical impedance tomography (EIT) allows an estimation of the bulk properties of major tissue

Figure 5 NEM forward modeling in realistic geometries. (a) Computational tools for interactive and semiautomatic segmentation of cortical anatomy allow extraction of computational geometries. Upper panel: Region-growing algorithms with adaptive criteria perform segmentation of white matter and identification of gray matter by dilation. Lower left: 3D rendering of the cortical surface identified by an automatic algorithm. Lower right: Rendering of the skull segmented by region-growing techniques from 3D MRI data. (b) Boundary element mesh based on simplified skull and scalp geometry derived from MRI volume imagery. (c) The regular computational mesh employed for finite difference computations in anisotropic media. The nodes for potential computation are at the corners of the spatial volume elements. (d) Finite difference calculation of potential distribution, using a detailed computational geometry derived from MRI. The current source is located within the temporal lobe, with a posterior to anterior orientation. The slices from the computed potential distribution show evidence of current leakage through the skull penetrations of the optic nerve.

Figure 5 NEM forward modeling in realistic geometries. (a) Computational tools for interactive and semiautomatic segmentation of cortical anatomy allow extraction of computational geometries. Upper panel: Region-growing algorithms with adaptive criteria perform segmentation of white matter and identification of gray matter by dilation. Lower left: 3D rendering of the cortical surface identified by an automatic algorithm. Lower right: Rendering of the skull segmented by region-growing techniques from 3D MRI data. (b) Boundary element mesh based on simplified skull and scalp geometry derived from MRI volume imagery. (c) The regular computational mesh employed for finite difference computations in anisotropic media. The nodes for potential computation are at the corners of the spatial volume elements. (d) Finite difference calculation of potential distribution, using a detailed computational geometry derived from MRI. The current source is located within the temporal lobe, with a posterior to anterior orientation. The slices from the computed potential distribution show evidence of current leakage through the skull penetrations of the optic nerve.

classes within the head and even a measure of 3D reconstruction. Tomographic reconstruction of head conductivity requires 3D computational techniques and is greatly facilitated by accurate geometrical information drawn from MRI.

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