The previous chapters described how to measure the magnetic fields generated by sources within the brain. The next step is to analyse the data with the purpose of estimating the region where the source of the activity seen in the signals would originate. Isofield and isopotential maps can be generated, depicting the magnetic field or potential distribution at a certain point in time, which can be analysed visually, but also numerically. From these distributions, one can calculate the current distribution that would yield these measurements.
The solution of inverse problems is based on the theory of electromagnetics, which is described by the Maxwell equations. Plonsey and Heppner (1967) showed that for the relatively slow bioelectric phenomena we are talking about, the time-dependency can be left out of these equations, and the quasi-static Maxwell equations can be applied. The permeability of the head is taken as that of free space, thus:
In these equations, 
is the electric
field, V is the electric potential, 
the
magnetic field, 
the current density, and 
.
the magnetic permeability of free space.
The head can be seen as a conducting region of space, in which the
conductivity changes with the various types of tissue. Helmholtz (1853)
already showed that the calculation of the current distribution within a volume
conductor from measurements of the fields at points outside of the volume
conductor does not yield a unique solution. Furthermore, since the inverse
problem is a non-lineair problem, it is necessary to Solve the inverse Solution
iteratively, i.e. estimate a current distribution, calculate the resulting field
and compare this with the measured field. The calculation of the magnetic
and electric fields from given sources is called the solution of the forward
problem.
MEG signals arise from impressed (non-ohmic) currents produced by
permeability changes in the neuronal membranes. These impressed currents
generate ohmic currents in the active cells, in the interstitial space adjacent
to the walls of active cells, in the non-active positive cells and at a larger
distance in the volume conductor. The impressed currents do not produce
MEG signals outside the volume conductor since the membrane is very thin, and
since the current elements tend to mutually cancel their magnetic fields (Swinney
and Wikswo, 1980). Therefore, It is convenient to split the total current
into two parts. First, the currents occurring in the source region,
denoted by the primary current density 
.
Second, the return
currents, or volume currents 
outside the
source region, which are ohmic and ran be written as
is the marcroscopic conductivity
within the volume conductor, in our case the head. The conductivity will
be a function of space coordinates and for an anisotropic tissue it will be
different for different directions and can mathematically be treated as a
tensor. We restrict ourselves to an isotropic conductivity.
Substitution of equation III.4 into III.3 leads to the Poisson equation for :
The solution of this Poisson equation reads:
This solution is known as the Biot and Savart law. r' is the
source location, and the point at which is
being calculated is called r . The integration is carried out over
the head (G), and all variables inside the integral depend on the location
inside the head.
Equation 6 can be simplified by assuming the head to be a piece-wise linear homogeneous isotropic medium, also called a simple medium, i.e. consisting of several regions, each with a conductivity which is constant. Such a model is called a compartment model of the head. By applying the Gauss theorem, we can arrive at the formula derived by Geselowitz (1970):
is equal to the first term on the
right-hand side of equation 6 and denotes the contribution of the primary
current, j denotes the number of the region over which the integration takes
place, although it is now an integration over the surface of the region and 
is the normal vector to the surface of the region. 
denotes the electric conductivity on the outside of the jth boundary, while 
denotes the electric conductivity on the inner side of that boundary. As
can be seen, the magnetic field depends on the potential at each surface.
Barnard et al. had already found an equation for the electric potential (1967):
with:
which is equal to the potential generated by the primary current density Jp. in an infinite homogeneous medium with unit conductivity.
The integral over the surface of a compartment can be discretised in order to
solve the forward problem numerically. Surface elements, for instance
triangles, are the basis of the boundary element method, which will be used in
chapter IV. Note that equations 7 and 8 are only valid for closed
homogeneous and isotropic compartments. This means that anisotropy of
tissues and breaches in the skull or skull-less regions cannot be taken into
account. If skull-less regions or anisotropy have to be taken into
account, the volume integral has to be discretised and solved numerically. The
most appropriate method used for this purpose is called the Finite Element
Method (Pruis et al., to be published; Miller and Henriquez, 1990). Its
use for modelling the surface of the head has been described by Yen et al.
(1991) and Bertrand et al. (1991).
Often the forward (and hence the inverse) problem is solved using a model
consisting of concentric spheres. Each sphere is a homogeneous compartment
with a constant conductivity, which is usually also taken to be isotropic, or
where anisotropy is allowed in the restricted sense that the conductivity in the
radial direction differs from that in the tangential direction.
The most frequently used model of the source is the equivalent current dipole. A current dipole P is defined by:
P is finite. a is a vector pointing from a
current sink to a current source, |a| is the distance between the
sink and the source and I is the total current coming out of the source, which
is equal to the total current going into the sink.
Okada and Nicholson (1988) studied the neural basis of magnetic evoked fields of
the brain with an isolated turtle cerebellum. From their measurements it
followed that the field is dipolar at a distance of the order of the dimensions
of the active region, the dipole is oriented along the longitudinal axis of the
active cells (i.e. perpendicular to the cerebellar surface) and that a
synchronous activity in a 1 mm3 of nerve tissue will lead to a magnetic field of
0. 1 pT at a source-to-field distance of 2 cm.
In this thesis we shall restrict ourselves to an equivalent current dipole to represent the primary current distribution within a source region. The dipole term is the lowest-order term in the multipole expansion of V and of B . At a distance which is much larger than the dimension of the source region, the higher-order terms can be neglected. The magnetic field due to the dipole is, according to the law of Biot and Savart:
This expression can be entered in equation III.7 to obtain the magnetic field. The similar expression for V0(r), necessary for equation III.8 reads:
The medium is assumed to be infinite and simple. Two different approaches can be used to analyse measured data of the magnetic field and the electric potential. First, at each point in time, one or more dipoles can be assumed to be active. Therefore, in principle, each time instant would yield a different solution. This is called a moving dipole solution. Second, one may assume that over a certain period of time the dipole or dipoles remain in the same position. In this case, more information is used to determine their positions. The strength and orientation of the dipoles may vary over time. This is the fixed dipole solution.
As noted earlier, any measurable response from the brain is the result of many neurons firing simultaneously. The neurons are organised in layers in the cortex. The activated area has a finite extent, and the active patch of cortex may be more appropriately described by a dipole layer. If this dipole layer is assumed to be homogeneous, i.e. the dipole density r is constant and the dipoles are oriented perpendicular to the layer, the magnetic field and electric potential in a homogeneous isotropic conductor of infinite extent are given by (Peters, 1981):
where 
is the dipole density in the
direction of the local normal, dS is a surface-element, šS is the
rim of the layer, dl is a line element of this rim and d
is the solid angle at which the dipole layer is seen from the observation
point. Consequently, any homogeneous dipole layer with the same rim gives
rise to the same magnetic field and electric potential distribution.
Simulation studies for layers with a circular rim show that at distances larger
than the dimensions of the rim the field and potential can be described in good
approximation by the field and potential distribution generated by a dipole in
the centre of the layer. The condition that the measurement is taken at a
sufficient distance from the extended source region to represent it by a dipole
is normally satisfied since the measurements are performed outside the head, at
a distance of two to three centimeters from the brain in the case of magnetic
measurements and on the scalp at a distance of at least I cm in the case of
electric measurements. Note that, if the dipole layer representing the
cortex is not planar, the location found for the equivalent dipole will not be
located in that dipole layer (De Munck, 1989). This is of consequence in
interpreting and evaluating the location of dipoles in MRI-scans.
Instead of describing sources which represent a restricted area of space,
which can be modeled by current dipoles, it is also possible to model the brain
activity as distributed sources. An approach for calculating such sources
is the Current Density method (loannides et al., 1990), in which an algorithm
searches for a current distribution in a restricted region of space.
Kullmann et al. (1989) used this method to search for sources inside a plane in
the head, and then he repeated the procedure for a different plane, effectively
scanning the head for sources. The algorithm as used by Kullmann et al. or
Wang et al. (I 992) tries to find a solution in a two-dimensional region.
This surface should be known a priori. A logical choice for the
two-dimensional surface is the cortex, because in many experiments this is the
location where the sources are expected to be. Although the cortex can be
extracted successfully from MRI scans with our software, it may not be trivial
to reconstruct the complex shape of the cortex in three dimensions into a single
two dimensional surface.
The use of a sphere model for the head greatly simplifies equation III.7. The outer product with the normal vector clearly states that the integral in equation III.7 gives no contribution to the radial part of the magnetic field outside the sphere. Since the divergence of B has to be zero (equation III.2), there is no contribution to the magnetic field outside of the sphere at all from this part of equation III.7 (Peters and Wieringa, to be published). Therefore, the magnetic field produced by a current in a spherically symmetric volume conductor is the same as that of a current in an infinite medium: B0. Consequently, the magnetic field does not depend on the conductivities or radii of the concentric spheres. The expression for the potential is also simplified by using a spherical volume conductor, but no closed analytical expression for V at the outer surface can be found if the number of compartments is more than one.
The expression for the electric potential at the outer surface of a set of
concentric spheres is given by the general solution of Laplace's equation 
V=0
and the particular solution of the dipole is: V = Vpart + Vgen. Taking the
boundary conditions into account.
where b is the distance between the dipole P and
the centre of the sphere, the
conductivity of the region where the dipole is situated and Pnm
are the associated Legendre functions. The boundary conditions for this
problem are:
1) V is continuous at r = Rs
2) The normal component of j is continuous at r = Rs, therefore
3) The potential for r goes to zero should be finite.
Without loss of generality we can choose a coordinate system such that the
dipole is at the Z-axis. As an example, we calculate the solution for a
radial source Pz within a homogeneous sphere. In this case the
potential distribution possesses axial symmetry and therefore is independent of
the azimuthal angle 
. In this case the
general solution of Laplace's equation reads:
where Pn. = Pn0 are the Legendre polynomials.
The particular solution expressed in spherical harmonic functions reads:
The total solution is the sum of the general and the particular solution (equations 17 and 19). Taking boundary condition 3 into account it becomes:
From boundary condition3 it follows that Bn=0 for all n.
Applying boundary condition 2 yields:
Combining equation 20 and 21 and taking r=Rs. results in the solution:
We can do the same for a dipole oriented in the X-direction. The solution reads:
The solution for a dipole in the Y-direction is analoguous to the one in the X-direction. The problem ii only rotated over 90 degrees. The total solution for the potential at the surface of a homogeneous sphere due to a dipole inside the sphere on the Z-axis, with dipole moment p = (px, py, pz), is:
A solution for an arbitrary number of spheres was derived by Zhang and Jewett
(I 993). In the solution derived by De Munck (I 989) even anisotropic
conductivity, in the restricted sense that the conductivity in the radial
direction could differ from that in the tangential directions, was
allowed. If concentric spheres are used to represent the head, it is
important that the spheres are fit locally to the inside of the skull in the
region of interest nearest to the (expected) location of the source, since the
volume currents are confined to the conductor.
The solution of the inverse problem can be found by an iterative
procedure. An initial estimate of the parameters for the chosen source
model is made, and the resulting magnetic and electric field are calculated,
using a head model. The calculated fields are compared with the measured
ones, and the source parameters are adjusted accordingly. We use the
Marquardt algorithm (Marquardt, 1963) for this procedure. The iterations
continue until the difference between the measured and calculated field is
sufficiently small. If no satisfactory solution can be found, or if the
solution of the problem using the magnetic and the electric data differ
significantly, it indicates that one or more of our assumptions could be
wrong. Perhaps the model of the volume conductor is not adequate for this
situation, or the number of sources which are simultaneously active differ from
the number which was anticipated.
The generated magnetic fields are not uniform across the individual coils of the
gradiometer. Consequently, the inverse procedure has to take into account
the spatial separation, size and orientation of the gradiometer coils. The
location and inclination of the axis of the gradiometer has to be known with
respect to the head or with respect to the sphere describing the head.
(c) MEG, EEG and the integration with Magnetic Resonance Images, H.J. Wieringa, 1993
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