# Volume Conduction

Carsten Wolters and Jan C de Munck (2007), Scholarpedia, 2(3):1738. | doi:10.4249/scholarpedia.1738 | revision #137330 [link to/cite this article] |

**Volume conduction**, a term used in **bioelectromagnetism**, can be defined as the transmission of electric or magnetic fields from an electric primary current source through biological tissue towards measurement sensors. In the considered low-frequency band (frequencies below 1000 Hz), the capacitive component of tissue impedance, the inductive effect and the electromagnetic propagation effect can be neglected, so that the transmission can be modeled with the quasi-static Maxwell equations (Plo67; Sar87; Mun91). The volume conductor is represented through the conductivity distribution of the different tissues through which the fields are transmitted.

## Contents |

## Relation to forward and inverse problem

Volume conductor models are a basis for source analysis in Electrocardiography (ECG), Magnetocardiography (MCG), Electroencephalography (EEG) and Magnetoencephalography (MEG). The activity that is measured in ECG/MCG (EEG/MEG) is the result of movements of ions, the so-called *impressed currents*, within activated regions in the human heart (in the cortex sheet of the human brain). The *current dipole* is generally used as the ''atomic structure'' of the impressed current (Sar87; Mun88a). The current dipole causes ohmic *return currents* to flow through the surrounding medium. The ECG (EEG) measures the potential differences of the return currents at the torso (scalp) surface, whereas the MCG (MEG) measures the magnetic flux of both impressed and return currents in a distance of a few centimeters from the torso (head) surface. The reconstruction of the impressed primary current distribution from measured surface fields is called the *inverse problem*. Its solution requires the simulation of the field distribution for a current dipole in the corresponding volume conductor using the quasi-static Maxwell equations, the so-called *forward problem*. All inverse methods (see, e.g., the article about source analysis) such as, e.g., dipole fit and scan methods, current density reconstructions and beamformers are based on the solutions of the forward problem and are thus influenced by the degree of realism in the modeling of the volume conductor.

## Volume conduction effects in EEG and MEG

In order to understand the relationship between EEG and MEG and the underlying primary source configuration, the electrical conduction properties of the human head (the volume conductor) have to be modeled. The progress made in developing forward modeling techniques led to a variety of source analysis applications. It is obvious that a completely realistic volume conductor model currently cannot be accomplished in routine source analysis, but it is important to specify those characteristics of the system which play a dominant role.

### Knowledge about tissue conductivities

Inverse source analysis in the brain is sensitive to the conductivities of the head tissues, which vary across individuals and within the same individual due to variations in age, disease state, and environmental factors. First attempts to measure the conductivities of biological tissues were made *in vitro*, often using samples taken from animals (Ged67). Brain white matter was measured to have a direction dependent (*anisotropic*) conductivity with a ratio of about 1:9 normal:parallel to the fibers ((Nic65), see further references in (Gue06)). The conductivity of human cerebrospinal fluid was measured in (Bau97) and that of skull in (Hoe03). The human skull consists of a soft bone layer (spongiosa) enclosed by two hard bone layers (compacta). The spongiosa has a much higher measured conductivity than the compacta (Akh02), i.e., the skull has an inhomogeneous conductivity. Recently, methods were proposed to determine *in vivo* conductivities of head tissues by using an Electrical Impedance Tomography (EIT) based approach (Gon03a; Gon03b) or by estimating them from measured EEG data (Gut04; Lew09), EEG and simultaneous intracranial data (Lai05) or combined EEG and MEG data (Fuc98b; Gon03a;Hua07;Wol10). With regard to the intracranial tissues such as brain grey and white matter, an indirect determination of the anisotropic conductivity through Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) was proposed by (Bas94; Tuc01).

### Numerical approaches for volume conductor modeling

#### Quasi-analytical solutions

Different numerical approaches for the forward problem have been used. For the EEG, the first model described the head by a single homogeneously conducting compartment (Fra52). Then the three (Ges67) and four (Hos78) concentric sphere models were developed, wherein the layers represent the skin, (cerebrospinal fluid), skull and brain. In each of these layers, the conductivity was assumed to be homogeneous and isotropic. In (Mun88b; Mun93), a quasi-analytical solution was presented of a volume conductor model consisting of an arbitrary number of concentric/confocal anisotropic layers of different conductivities. For the MEG, it appears that the magnetic field outside the head is completely independent of the conductivity profile, provided that the conductor is spherically symmetric (Mun91). For this model an analytical formula has been derived (Sar87). Besides the fact that those models are still frequently used in source analysis routine, they also serve as validation tools for more realistic numeric modeling. Finally, we mention a promising approach to solve very general volume conductor models based on a perturbation analysis of quasi-analytical solutions (Nol99; Nol01).

#### Boundary element method

In order to better take into account the *realistic shape* of the scalp and skull surfaces, *Boundary Element* (BE) head models have been developed, being adequate for piecewise homogeneous isotropic compartments. Numerical accuracy of the BE approaches could be improved through the isolated problem approach (Mei89; Hae89), the use of linear basis functions with analytically integrated elements (Mun92), quadratic elements (Frij00), local mesh refinement techniques (Yve95; Zan95), virtual mesh refinement (Fuc98a), Galerkin approach (Mosh99; Lynn68) or
a symmetric BEM that uses a combination of a single- and a double-layer potential approach (Kyb05).

#### 3D discretization methods

Besides the *Finite Difference* (FD) (Sal97; Moh03; Moh04; Hal05) and the *Finite Volume* (FV) methods (Moh04; Coo06), *Finite Element* (FE) volume conductor modeling is able to treat both realistic geometries and inhomogeneous and anisotropic material parameters (Yan91; Ber91; Awa97; Buc97; Bro98; Mar98; Oll99; Wei00; Sch02; Wol02; Gen04; Wol04; Wol06; Wol07a;Dan10). When compared to FD and FV modeling, the FE approach is often considered to be more flexible with regard to the representation of complicated geometries and of higher accuracy, especially when using higher-order basis functions. One impediment to using 3D methods has been the high computational cost of carrying out the simulations. The use of recently developed advances in efficient solver techniques for EEG and MEG 3D forward modeling (Wol02; Moh03; Moh04; Lew09b), transfer matrix approaches for both EEG and MEG (Gen04; Wol04; Dre09) and reciprocity approaches for EEG and MEG (Wei00; Moh04; Hal05; Sch07; Val09) dramatically reduce the complexity of the computations, so that the main disadvantage of 3D modeling no longer exists. Sensitivity studies have been carried out in realistic 3D models for the influence of the CSF (Ram04;Wol06;Wen08;Rul09), skull inhomogeneity (Oll99;Ram04;Sad07; Dan10) and realistic brain anisotropy (Hau02;Wol06;Rul09;Gue10) on EEG and MEG. Those studies support the hypothesis that modeling CSF, skull inhomogeneity and brain anisotropy is important for accurate EEG and MEG source reconstruction. It is furthermore widely accepted that skull conductivity inhomogeneities such as skull sutures (Poh97) or skull fontanels and other skull holes have a non-negligible effect on especially the EEG source analysis. Local conductivity changes around the primary source as caused by brain lesions (Bro98;Rul09), local anisotropies (Wol05) or skull-holes from trepanation (Bro98) have a non-negligible effect on both EEG and MEG.

## Validation and results in realistic head models

For the following FE computations, the software SimBio was used (https://www.mrt.uni-jena.de/simbio).

#### Error measures

Before a numerical code for the solution of the forward problem in more realistic head volume conductors can be used in source analysis practice, numerical errors have to be evaluated. Therefore, the numerical solutions can be compared to quasi-analytical solutions, which only exist in simplified volume conductors such as, e.g., the multi-layer sphere model described above. A statistical metric, the Relative Difference Measure (RDM), is often used to express the topographic error between the quasi-analytically and the numerically computed surface potential distributions for dipolar current sources in the inner sphere ('brain') compartment. The RDM is defined as (Mar98; Sch02; Wol07a) \[\tag{1} RDM=\sqrt{\sum_i\left(\frac{\Phi^i_{Num}}{||\Phi_{Num}||_2}-\frac{\Phi^i_{Ana}}{||\Phi_{Ana}||_2}\right)^2} \]

where \(\Phi^i_{Num}\) is the potential (magnetic field) determined by the numerical approximation at sensor \(i\ ,\) \(\Phi^i_{Ana}\) is the corresponding quasi-analytical solution and \(||\cdot||_2\) expresses the L2-norm of a vector. To test the accuracy of a numerical method, one often assumes a standard electrode/magnetometer geometry and computes the RDM as a function of dipole eccentricity, which is defined as the distance of the dipole location to the midpoint of the sphere model divided by the radius of the inner sphere.

#### Validation studies

To validate the FE code, the anisotropic multi-layer sphere model described above is used. Extensive validation was performed, e.g., in (Ber91; Mar98; Sch02; Wol07a; Dre09; Lew09b). In Figure 2 the RDM, computed at 134 regularly placed electrodes on the outer surface of a three compartment isotropic (anisotropic) sphere model with outer surface radii of 90 mm ('skin'), 80 mm ('skull') and 70 mm ('brain') and conductivities of 0.33 S/m, 0.0042 S/m (0.0042 S/m:0.042 radial:tangential) and 0.33 S/m, resp., is shown for sources with different eccentricities. The figure shows the 'impact of the skull anisotropy', i.e., the topography error between quasi-analytical forward computations in models with isotropic ('Anaiso') and 1:10 anisotropic 'skull' ('Anaaniso') and compares it to the numerical accuracy of FE subtraction approach (see mathematics below) computations in a tetrahedral model (FEtet156: 156K nodes, linear basis functions) and of a geometry-adapted (node-shifted: *ns*) hexahedral model (FEcube398ns: 398K nodes, trilinear basis functions). It can be observed that, for sufficiently eccentric sources, the effect of 1:10 'skull' anisotropy is much larger than the numerical inaccuracies of the computed FEM solution. The FEM solution accuracy can be further increased by increasing the integration order or reducing the element edge length especially in elements close to the source (e.g., Dre09; Lew09b) or increasing the degree of the basis function at the expense of a higher computational complexity.

#### Results in realistic head models

##### Influence of tissue conductivity inhomogeneity

As a first example we study the influence of head tissue conductivity inhomogeneity. A pre-surgical T1-MRI (256 sagittal, 120 coronal and 256 axial slices, voxel-size of 0.86x1.6x0.86mm) and, after skull trepanation, a post-surgical CT (512 sagittal, 635 coronal and 68 axial slices, voxel-size of 0.49x0.49x2.65mm) were measured from a patient with medically intractable epilepsy (we thank G. Worrell from The Mayo Clinic, Rochester, USA, for kindly providing these datasets). A voxel-based affine registration using mutual information was used to register the T1-MRI onto the CT image. The registered dataset was then segmented into four tissue classes: skin, skull with the trepanation hole, cerebrospinal fluid (*CSF*) and brain. Skin, skull and the CSF in the surgery opening were extracted from the CT while the remaining part of the CSF-compartment and the brain were segmented out of the T1-MRI. In order to study the influence of a skull hole on the forward problem, a second four-tissue-model was created where the trepanation hole was manually closed. Both segmented datasets were tessellated into tetrahedral FE models of about 140K nodes and 850K elements. Figure 3 and Figure 4 show sagittal slices of the conductivity tensor ellipsoids in the barycenters of the tetrahedra elements when modeling the tissues as isotropic. The tensors were normalized and colored by trace. The highest trace values could be found in the CSF compartment (red) and the lowest in the skull compartment (dark blue).

In order to show the impact of the skull trepanation hole on the forward problem, electric potential computations were performed in both FE models for dipole sources with strengths of 100 nAm as shown by the yellow cones in Figure 3 and Figure 4. The distributions of the electric potentials at the surfaces of both head models are shown in Figure 5 and Figure 6. As it is well visible, neglecting the modelling of the skull opening results in a much blurred potential distribution, corroborating the importance of patient-specific volume conductor modeling. It was shown in (Wol06) that skull anisotropy has a smearing effect on the forward EEG computations and doesn't affect the MEG. However, newer investigations (Dan10; Sad07) show, that the concept of skull anisotropy seems outdated and should be replaced by the much more important concept of skull inhomogeneity, which is due to the three-layeredness of the skull in the very low conducting skull compacta and the much higher conducting skull spongiosa.

##### Influence of tissue conductivity anisotropy

In a second example we study the influence of head tissue conductivity anisotropy. Therefore, a five compartment (skin, skull, CSF, brain grey and white matter) realistically shaped head model was segmented out of a bimodal T1-/PD-MRI dataset and a tetrahedral FE model (147K nodes and 892K elements) was generated as shown in Figure 7. The human skull shows a conductivity with higher resistance in radial than in tangential direction. We determined the radial direction from a strongly smoothed triangular mesh, which was shrunken from the outer skull onto the outer spongiosa surface using a discrete deformable surface model. We started from the commonly used isotropic conductivity value and used the volume constraint to model the skulls conductivity anisotropy, as described in detail in (Wol06). Measured DT-MRI data were used as the basis for the determination of white matter conductivity anisotropy. Following the proposition of (Bas94), we assumed that the conductivity tensors share the eigenvectors with the measured diffusion tensors. (Shi99) measured diffusion anisotropy in 12 regions of interest in human white and gray matter and showed that in commissural, projection, and also association white matter, the shape of the diffusion ellipsoids is strongly prolate ("cigar-shaped"). Therefore, we assumed prolate rotationally symmetric tensor ellipsoids for the white matter compartment. We started from the commonly used isotropic conductivity value (Ged67) and used the volume constraint to model white matter conductivity anisotropy, as described in detail in (Wol06). Figure 8 shows an axial slice of the conductivity tensor ellipsoids in the barycenter of the tetrahedra elements from Figure 7 when modeling skull and white matter as anisotropic. The tensors were again normalized and colored by trace. Note the mainly top-bottom fiber directions of the pyramidal tracts and the mainly left-right orientation over the corpus callosum. In order to show the impact of white matter conductivity anisotropy on the forward problem, a line integral convolution technique was used to visualize the volume currents computed for a thalamic dipole source using either an isotropic or a 1:10 anisotropic white matter compartment as shown in the animated Figure 1. As concluded in (Wol06), anisotropic white matter conductivity causes return currents to flow in directions parallel to the white matter fiber tracts. The deeper a source lies and the more it is surrounded by anisotropic tissue, the larger the influence of this anisotropy on the resulting EEG and MEG.

## Mathematical Background

In the quasistatic approximation of Maxwell's equations, the distribution of electric potentials \(\Phi \) in the head domain \( \Omega \) of conductivity \( \sigma \ ,\) resulting from a primary current \( j^p \) is governed by the Poisson equation \[\tag{2} \nabla \cdot \left(\sigma\nabla\Phi\right) = \nabla \cdot j^p \ \ in\ \Omega \]

with homogeneous Neumann boundary conditions on the head surface \[ \langle\sigma\nabla\Phi, n\rangle = 0\ \ on\ \Gamma=\partial\Omega \] with \(n \) the unit surface normal, and a reference electrode with given potential, i.e., \(\Phi(x_{ref}) = 0 \) (Sar87). The primary currents are generally modeled by a mathematical dipole at position \(x_0 \) with the moment \( M \) (Sar87; Mun88a): \[\tag{3} j^p\left(x\right) := M \delta\!\left(x-x_0\right). \]

Let \(F\) be the surface enclosed by an MEG magnetometer flux transformer \(\Upsilon=\partial F\ .\) The magnetic flux \(\Psi\) through \(\Upsilon\) can be determined as a surface integral over the magnetic induction for the coil
area \(F\ ,\) or, using Stokes theorem, as
\[
\Psi = \int_{F} B\cdot df=\oint_{\Upsilon} A(x)\cdot dx=\oint_{\Upsilon}\frac{\mu}{4\pi}\int\limits_{\Omega}\frac{j^p(y)}{|x-y|}dy\cdot dx+\oint_{\Upsilon}\frac{\mu}{4\pi}\int_{\Omega}\frac{-\sigma(y)\nabla\Phi(y)}{|x-y|}dy\cdot dx =: \Psi_p + \Psi_{sec}
\]
The first part of this magnetic flux is called the *primary magnetic flux* which is in the following denoted with \(\Psi_p\) and the second is the so-called *secondary magnetic flux* \(\Psi_{sec}\ .\) \(\Psi_p\) is only dependent on the source model and can in general be computed by simply evaluating an analytical formula (Sar87). If we define
\[\tag{4}
C(y) = \oint_{\Upsilon}\frac{1}{|x-y|} dx,
\]

and if the potential distribution \(\Phi\) is given, the final equation for \(\Psi_{sec}\) emerges from the secondary (return) currents and can be given by \[\tag{5} \Psi_{sec} = -\frac{\mu}{4\pi}\int_{\Omega} (\sigma(y)\nabla\Phi(y),C(y)) dy \]

### Quasi-analytical solutions

When the geometry of the conductivity distribution \(\sigma_(x)\) has a spherical symmetry, a quasi analytical solution of (2) can be found by expressing it in spherical coordinates (Mor53). The angular part of this equation has Legendre polynomials as solution, whereas for the radial part one has \[\tag{6} \frac{\partial}{\partial r}\left(r^2\epsilon(r)\frac{\partial}{\partial r}R_n(r_0,r)\right)-n(n+1)\eta(r)R_n(r_0,r)=\delta(r_0-r). \]

Herein is \(\epsilon(r)\) the radial conductivity profile and \(\eta(r)\) the tangential conductivity profile of the conductor. Equation (6) can be solved by assuming that \(\epsilon(r)\) and \(\eta(r)\) are constant in each shell (Mun93). When \(R_n(r_0,r)\) is solved, the quasi-analytic solution of the multi sphere model for a current dipole (3) can be presented as \[\tag{7} \Phi^{dip}(x_0,x)=\frac{1}{4\pi}<M,\nabla_0\Phi^{mon}(x_0,x)>\quad with\quad \Phi^{mon}(x_0,x)=\sum_{n=0}^{\infty}(2n+1)R_n(r_0,r)P_n(\frac{<x_0,x>}{r_0 r}) \]

where \(P_n()\) is the Legendre polynomial of order n. Note that the series expansion was first derived for a monopolar source in order to obtain maximum symmetry and simplicity in the geometry of the problem. The dipole solution is obtained by computing the derivative with regard to the source position. In practice, the infinite series of (7) has to be approximated with a finite number of terms. It appears that the closer \(r\) and \(r_0\ ,\) the more terms are needed. The number of terms can be reduced by adding and subtracting the same function, once in analytically closed form and once as series expansion. By choosing the appropriate multiplication factor, a series of differences appears with a faster convergence behaviour than the original series. Depending on the ratio of \(r\) and \(r_0\) the number of required terms ranges between 10 and 300. The function that is added and subtracted can be interpreted as the potential due to a current dipole in an infinite medium with constant conductivity. Details are presented in (Mun93).

### Boundary element approach

The boundary element method to solve equation (2) is based on the assumption that the conductor consists of compartments wherein the conductivity is constant and isotropic. For each compartment not containing the dipole source the potential is harmonic. At the interface between two compartments it is assumed that the potential and the normal component of the current are continuous. For such a compartment model, the following integral equation can be derived (Bar67) \[\tag{8} \left(\sigma_j^+\Omega_j^+(x)+\sigma_j^-\Omega_j^-(x)\right)\Phi(x)+\sum_{k=1}^{K} \left(\sigma_k^+-\sigma_k^-\right)\oint\!\!\oint_{S_k}\left(\Phi(x')\nabla\frac{1}{|x-x'|}\right)\cdot dS'=4\pi\Phi^{\infty}(x;M,x_0), \]

where \(K\) is the number of surfaces separating two compartments and \(\Phi^{\infty}\) is the potential due to a dipole in an infinite and homogeneous medium with unit conductivity (see Equations (13) and (14)). Once this equation is solved, i.e. an approximation of \(\Phi\) on each of the interfaces \(S_k\) has been found, the magnetic field can be computed by means of the following integral \[\tag{9} B(x)=B^{\infty}(x;M,x_0)+\mu_0\sum_{k=1}^{K}\left(\sigma_k^+-\sigma_k^-\right) \oint\!\!\oint_{S_k}\left(\Phi(x')\nabla\frac{1}{|x-x'|}\right)\times dS'. \]

Here \(B^{\infty}()\) is the magnetic field due to a current dipole in an infinite medium of constant conductivity. This field is independent of the conductivity.

In the simplest view on the BEM, the solution of (8) consists of two steps. First, the unknown potential \(\Phi(x)\) is expanded in a finite series of base functions \(\varphi_n(x)\ ,\) that are often derived from a triangular grid (see (Figure 9)): \[\tag{10} \Phi(x)\approx\sum\limits_{n=1}^N \varphi_n(x) u_n \]

Several choices for \(\varphi_n(x)\) can be made. One can assume that each \(\varphi_n(x)\) takes on a constant value on each triangle or, alternatively, one can use the same triangular grid and assume that \(\varphi_n(x)\) varies linearly from one triangle to the next. In the former case, \(\Phi(x)\) is approximated by a piecewise constant function and the number of unknowns is equal to the number of triangles.

In the second case, the approximation of \(\Phi(x)\) is by a piecewise linear function, and the number of unknowns equals the number of nodes, which is only about half the number of triangles. The second step of the BEM consists of substituting (10) in (8). This results in an infinite number of equations (for all \(x\)), and a finite number \(N\) of unknowns. To deal with this situation, one can chose for the collocation or the Galerkin approach. In the former case, \(N\) equations are obtained by letting x run over the triangle centres (piecewise constant case) or the nodes (piecewise linear case). A linear equation system of \(N\) equations with \(N\) unknowns of which the matrix elements can be computed analytically (Mun92; Oos83) then results: \[\tag{11} \sum_{n=1}^{N}A_{nm} u_n=u_n^{\infty} \]

The solution of the system (11) of equations can be performed by standard means, provided one adds a constraint like \(\sum_{n=1}^{N}w_n u_n=0\ ,\) to remove the singularity of (11). With the Galerkin approach (e.g. Mosh99) the left and right hand side of (8) are multiplied with the base functions \(\varphi_n(x)\ ,\) and integrated over the surfaces. For each \(n\) this yields an equation with \(N\) unknowns. This system of equations can also be solved by standard means. The disadvantage of the Galerkin approach is that the computation of the matrix elements is more time consuming than in the collocation approach, but the great advantage is that it yields more accurate results, in particular for eccentric sources. Many more variants of the BEM have been described in the literature. To comply with the large conductivity jump of the skull, the isolated problem approach can be used (Mei89; Hae89; Fuc98a). In this approach, first the potential is computed on a model which is isolated at the brain, and next this potential is propagated through the skull to the skin. A more profound way to improve accuracy is to solve (11) iteratively. The disadvantage being of course that these iterations need to be performed for each right hand side anew. However, by introducing tiny modifications in \(A_{nm}\) using the panel clustering method (Hac89), the multiplication of \(A_{nm}\) with a \(N\)-vector is sped up enormously. In this way much finer triangulations become feasible, despite the full system matrix. The panel clustering method has given rise to many new variants of the BEM. Finally, it is important to mention the symmetric BEM approach of (Kyb05), where a combination of a single- and a double-layer potential approach is used, leading to very high accuracies for three isotropic compartment BEM EEG and MEG forward modeling.

### Finite element approach

Besides direct potential approaches, where the mathematical dipole (3) is modeled through monopolar sources at neighboring FE mesh nodes (Yan91; Buc97; Moh04; Wei00; Sch02; Wol07a; Lew09b), a subtraction method can be used to subtract the dipole singularity out of the Poisson equation (Ber91; Awa97; Bro98; Mar98; Sch02; Wol07a; Dre09). For the FE subtraction method, the total potential \(\Phi\) is split into two parts, \[\tag{12} \Phi=\Phi^{\infty}+\Phi^{corr}, \]

where the singularity potential \(\Phi^{\infty}\) is defined as the solution for a dipole in an unbounded homogeneous conductor with the conductivity \(\sigma^{\infty}\) (the tensor-valued conductivity at the source position). The solution of Poisson's equation for the singularity potential \[\tag{13} \Delta\Phi^{\infty}=\frac{\nabla \cdot j^p}{\sigma^{\infty}} \]

can be formed analytically by use of (3) (Sar87): \[\tag{14} \Phi^{\infty}(x) = \frac{1}{4\pi\sigma^{\infty}}\frac{\langle M,(x-x_0)\rangle}{\left|x-x_0\right|^3}. \]

In case of \(\sigma^{\infty}\) being anisotropic, we find (Wol07c): \[\tag{15} \Phi^{\infty}(x) = \frac{1}{4 \pi \sqrt{\det \sigma^{\infty}}} \frac{\langle M, (\sigma^{\infty})^{-1} (x - x_0)\rangle}{\langle(\sigma^{\infty})^{-1} (x - x_0), x - x_0 \rangle^{3/2}}. \]

Subtracting (13) from (2) yields a Poisson equation for the correction potential \[\tag{16} -\nabla \cdot \left(\sigma\nabla\Phi^{corr}\right) = \nabla \cdot \left((\sigma-\sigma^{\infty})\nabla\Phi^{\infty}\right)\ \ in\ \Omega, \]

with inhomogeneous Neumann boundary conditions at the surface: \[\tag{17} \langle \sigma \nabla \Phi^{corr},n\rangle = -\langle \sigma \nabla\Phi^{\infty}, n\rangle\ \ on\ \Gamma. \]

The advantage of (16) is that the right-hand side is free of any source singularity, because in a subdomain around the dipole, the conductivity \(\sigma-\sigma^{\infty}\) is zero. In (Wol07c), a proof is given for existence and uniqueness of a weak solution in the function space of zero-mean potential functions and convergence properties of the FE method are stated for the numerical solution to the correction potential. For the numerical approximation of the correction potential, the FE method can be used with, e.g., piecewise linear basis functions \(\varphi_i\) at nodes \(\xi_i\ ,\) i.e., \(\varphi_i(x)=1\) for \(x=\xi_i\) and \(\varphi_j(x)=0\) for all \(j\ne i\ .\) When projecting both the singularity and the correction potential into the FE space, i.e., \(\Phi^{\infty}(x) \approx \Phi^{\infty}_h(x) = \sum\limits_{i=1}^N \varphi_i(x) u^{\infty}_i\) with \(u^{\infty}_i = \Phi^{\infty}(\xi_i) \) and \(\Phi^{corr}(x) \approx \Phi_h^{corr}(x)=\sum_{j=1}^{N}\varphi_j(x) u^{corr}_j \ ,\) and applying variational and FE techniques to (16),(17), the linear equation system \[\tag{18} K \underline{u}^{corr} = \underline{J}^{corr}, \qquad \underline{J}^{corr}=- K^{corr} \underline{u}^{\infty} -S \underline{u}^{\infty} \]

with large but sparse matrices
\[
K^{[i,j]} = \int_\Omega \langle\sigma\nabla\varphi_i,\nabla\varphi_j\rangle, \qquad \left(K^{corr}\right)^{[i,j]} = \int_\Omega \langle\left(\sigma-\sigma^{\infty}\right)\nabla\varphi_i,\nabla\varphi_j\rangle, \qquad S^{[i,j]} = \int_\Gamma \langle\sigma^{\infty}\nabla\varphi_j,n\rangle\varphi_i
\]
can be derived with \(\underline{u}^{\infty}=(u^{\infty}_1,\ldots,u^{\infty}_N)\) being the coefficient vector for \(\Phi^{\infty}_h\) (Wol07c). The equation system is then solved for the coefficient vector \(\underline{u}^{corr}=(u^{corr}_1,\ldots,u^{corr}_N)\) and the total potential is computed using (12). In a small subdomain around the dipole position, the linear approximation of the singularity potential \(\Phi^{\infty}\) through \(\Phi^{\infty}_h\) is quite rough, but \(\sigma-\sigma^{\infty}\) is zero so that, under the condition that the source is not too close to a next conductivity jump, (16) and (17) are appropriately modeled with the presented linear FE approach (Wol07c). An algebraic multigrid preconditioned conjugate gradient (AMG-CG) method can then be used as an efficient solver for equation system (18) with symmetric positive definite stiffness matrix (Wol02;Lew09b). The above approach was named *projected subtraction approach* in (Dre09). However, when additionally modeling the CSF, the sources in the grey matter compartment are as close as a millimeter from the next conductivity jump (the one from grey matter to CSF). It was shown in (Dre09) that in this case, the so-called *full subtraction approach*, which more accurately integrates the right-hand side in (16), has important advantages with regard to numerical accuracy compared to the presented projected subtraction approach (Dre09), however, at the expense of higher computation time.

The FE method also allows to model extended sources that might be more realistic in practical applications than the traditional mathematical dipole (see, e.g., (Tan05;Pur07)).

For the magnetic forward problem, the flux transformers of the MEG device have to be modeled (see Equation (4)). Such a coil can be modelled by means of a thin, closed conductor loop, using isoparametric quadratic row elements (Lan07). When approximating the potential by means of its Galerkin projection, i.e., \(\Phi(x) \approx \Phi_h(x)=\sum_{i=1}^{N}\varphi_i(x) u_i \ ,\) Equation (5) can be written in matrix form \(S\; \underline{u} =: \underline{\Psi}_{sec}\) with \(S\) the so-called *secondary flux matrix* which maps the potential onto the secondary flux vector \(\underline{\Psi}_{sec}\ .\) The secondary flux matrix has the entries
\[\tag{GLS1:label exists!}
S^{[j,i]} = -\frac{\mu}{4\pi}\int_{\Omega} (\sigma(y)\nabla\varphi_i(y),C_j(y))dy \quad \forall 1\le i\le N
\]

where \(C_j(y)\) denotes the function (4) for the \(j\)-th MEG magnetometer \(\Upsilon_j\ .\) For the computation of the matrix entries of \(S\ ,\) a FE ansatz for the integrand and Gauss integration can be used.

A *transfer matrix approach* (see (Wei00) for EEG and (Gen04;Wol04;Dre09) for EEG and MEG) should be used in order to keep the computational complexity of the FE based EEG and MEG inverse problem in reasonable limits, where a large number of forward computations have to be carried out. The transfer matrix approach limits the total number of FE linear equation systems to be solved for any inverse method to the number of measurement sensors \(s\ .\) After computing the \(s\) vectors of the transfer matrix, each forward problem can be solved by a single multiplication of the FE right hand side with the matrix, resulting in a much reduced computational effort for FE forward approaches (Wol04;Dre09;Lew09b).

## Mesh generation

### Boundary element meshing techniques

Triangular meshes for the BEM can be derived from an MR scan using image segmentation techniques (Wag95). In this way very accurate brain segmentations can be obtained. Since these methods are never completely automatic they can be very time-consuming in practice. The question is also whether very accurate geometrical models are really needed, considering the numerical and modelling inaccuracies that are present anyhow in BEM models. Simpler and less accurate meshes can be obtained from MR images by fitting shape functions on a manually delineated brain, skull and skin (Ent01). By constraining the base functions only very course delineation suffice to derive BEM models for MEG.

### Finite element meshing techniques

In FE modeling, a 3D mesh has to be generated to represent the geometric and electric properties of the volume conductor.

FE mesh generation if often considered to be extremely difficult in practice, but it can in contrast be very easy when using a hexahedral approach. Voxels from a segmented MR volume can be used directly as hexahedral elements, possibly reducing resolution by prior subsampling of the volume. In order to increase conformance to the real geometry and to mitigate the staircase effects of a voxel mesh, a technique to shift nodes on material interfaces can be applied in order to obtain smoother and more accurate boundaries (Wol07a). Nodes on a two-material interface are moved into the direction of the centroid of the set of incident voxels with *minority material*, i.e., the material occurring three times or less in the 8 surrounding voxels. If the centroid of these minority voxels relative to a node is \((x,y,z)\ ,\) it is shifted by \((\Delta x,\Delta y,\Delta z)=(ns*x,ns*y,ns*z)\) with the user-defined node-shift factor \(ns \in [0,0.5)\ .\) The choice \(ns\in[0,0.5)\) ensures that interior angles at element vertices remain convex and the Jacobian determinant remains positive. An isoparametric FE approach is then needed to process the geometry-adapted hexahedral elements (Wol07a). In Figure 10, the isopotential distribution was computed using the projected subtraction approach in a three compartment realistically shaped geometry-adapted (\(ns =0.49\)) hexahedral FE head model with a 1:10 anisotropic skull compartment. For geometry-adapted hexahedral mesh generation, the software (VGRID) was used. Another interesting hexahedral approach was presented in (Val10), where the tissue interfaces were represented with level sets, and the finite element space was locally modified to better approximate the discontinuities of the solution.

A more sophisticated way of FE meshing uses surface-based tetrahedral tessellations of the segmented tissue compartments. The procedure is more difficult in practice, because a qualitatively good tetrahedral mesh needs non-intersecting surfaces from the tissue compartments (just like in BEM modeling), which are often not easy to generate. The software TetGen uses a *constrained Delaunay tetrahedralization* (CDT) approach (Si08;Dre09). The meshing procedure starts with the preparation of a suitable boundary discretization of the model. Surface meshes of different layers are not allowed to intersect each other. The CDT approach is then used to construct a tetrahedralization conforming to the surface meshes. It first builds a Delaunay tetrahedralization of the vertices of the surface meshes. It then uses a local degeneracy removal algorithm combining vertex perturbation and vertex insertion to construct a new set of vertices which includes the input set of vertices. In the last step, a fast facet recovery algorithm is used to construct the CDT (Si08;Dre09).
This approach is combined with two further constraints to the size and shape of the tetrahedra. The first constraint can be used to restrict the volume of the generated tetrahedra in a certain compartment. The second constraint is important for the generation of quality tetrahedra.
If \(R\) denotes the radius of the unique circumsphere of a tetrahedron and \(L\) its shortest edge length, the so-called *radius-edge ratio* of the tetrahedron can be defined as\(Q = \frac{R}{L}\ .\) The radius-edge ratio can detect almost all badly-shaped tetrahedra except one type of tetrahedra, so-called \(slivers\ .\) A sliver is a very flat tetrahedron which has no small edges, but can have arbitrarily large dihedral angles (close to \(\pi\)). For this reason, an additional mesh smoothing and
optimization step should be used to remove the slivers and improve the overall mesh quality (Si08;Dre09).
As shown in (Dre09), a qualitatively good CDT mesh in combination with the full subtraction approach can lead to very high numerical accuracies. The tetrahedral FE model from Figure 7 was created with a meshing approach described in detail in (Wag98) using the software CURRY.

## Acknowledgement

This project has been supported by the German research foundation (DFG) WO1425/1-1, JU445/5-1.

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**Internal references**

- Valentino Braitenberg (2007) Brain. Scholarpedia, 2(11):2918.
- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
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## External Links

## See also

Source analysis, Electroencephalography (EEG), Magnetoencephalography (MEG), Cortex