The magnetic lead field theorem in the quasi-static approximation and its use for magnetoencephalography forward calculation in realistic volume conductors.

INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 48 (2003) 3637–3652 PII: S0031-9155(03)66369-4
The magnetic lead field theorem in the
quasi-static approximation and its use for
magnetoencephalography forward calculation in
realistic volume conductors
Guido Nolte
Human Motor Control Section, NINDS, NIH, Bethesda, MD, USA
E-mail: nolteg@ninds.nih.gov
Received 21 July 2003
Published 24 October 2003
Online at stacks.iop.org/PMB/48/3637
Abstract
The equation for the magnetic lead field for a given magnetoencephalography
(MEG) channel is well known for arbitrary frequencies ω but is not directly
applicable to MEG in the quasi-static approximation. In this paper we derive an
equation for ω = 0 starting from the very definition of the lead field instead of
using Helmholtz’s reciprocity theorems. The results are (a) the transpose of the
conductivity times the lead field is divergence-free, and (b) the lead field differs
from the one in any other volume conductor by a gradient of a scalar function.
Consequently, for a piecewise homogeneous and isotropic volume conductor,
the lead field is always tangential at the outermost surface. Based on this
theoretical result, we formulated a simple and fast method for the MEG forward
calculation for one shell of arbitrary shape: we correct the corresponding lead
field for a spherical volume conductor by a superposition of basis functions,
gradients of harmonic functions constructed here from spherical harmonics,
with coefficients fitted to the boundary conditions. The algorithm was tested for
a prolate spheroid of realistic shape for which the analytical solution is known.
For high order in the expansion, we found the solutions to be essentially exact
and for reasonable accuracies much fewer multiplications are needed than in
typical implementations of the boundary element methods. The generalization
to more shells is straightforward.
1. Introduction
In magnetoencephalography (MEG), tiny magnetic fields are measured over a person’s head
and, if applicable, interpreted as arising from neural currents in the brain (Ha¨ma¨la¨inen et al
1993). Correctly interpreting the data in terms of neural generators (‘the inverse problem’)
0031-9155/03/223637+16$30.00 © 2003 IOP Publishing Ltd Printed in the UK 3637

3638 G Nolte
generally requires an accurate knowledge of the physical relationship between the two, i.e.
knowledge of the external magnetic field induced by a given source (‘the forward problem’).
The solution of the forward problem is inherently complicated by the fact that the magnetic
field induced by a single current element, a ‘current dipole’, depends on the characteristics
of the volume conductor, the head. A reasonable and popular, but in general insufficient,
assumption is that the head can be modelled by a sphere with homogeneous and isotropic
conductivity in the inside. In this case, the forward problem can be solved exactly in closed
form (Sarvas 1987). Although an exact solution also exists for spheroids (Cuffin and Cohen
1977), calculating the external field requires approximating rather complicated infinite sums.
With regard to the limited flexibility of these volume conductor models, the respective solutions
are probably only useful for test purposes.
In a more general approach, it is assumed that the volume conductor consists of piecewise
homogeneous and isotropic compartments such as grey matter, white matter, skull and scalp.
In this case the field equations, which are in general differential equations in the whole volume,
can be reduced to integral equations defined solely on the surfaces of the compartments. These
equations can be solved numerically by (various versions of) the boundary element method
(BEM) applicable both for MEG and electroencephalography (EEG) (Goncalves et al 2003,
Tissari and Rahola 2003, Fuchs et al 2002, Mosher et al 1999, de Munck 1992). Drawbacks
of BEM are that it is computationally quite costly, eventually requires a large amount of
memory and becomes singular in the vicinity of a surface separating two compartments.
One can also construct non-singular and fast methods both for EEG (Nolte and Curio 1999)
and MEG (Nolte et al 2001). There, the solutions are approximated by a first-order Taylor
expansion with respect to the deviation of the ‘true’ volume conductor, assumed to consist
of one shell, and its spherical approximation. A drawback of this method is that it requires
both sphere-like and relatively smooth surfaces. Furthermore, the method is already very
difficult in its present formulation and a generalization to a higher-order Taylor expansion and
to multiple shells is almost hopelessly complicated.
In order to solve the forward problem for arbitrary (and not piecewise homogeneous and
isotropic) conductivity distributions, it has been proposed that the field equations should be
solved as differential equations in the whole volume using the finite element method (FEM)
(Awada et al 1997, Haueisen et al 2000, Laarne et al 2000, Weinstein et al 2000). In this case,
conductivity becomes a space-dependent tensor which is, in principle, measurable by magnetic
resonance imaging (Tuch et al 2001). In FEM, the whole volume is divided into tetraheders
on which the local potential is approximated by a constant or low-order polynomials. As with
BEM, a problem is that this approximation becomes bad in the vicinity of a dipolar source
which induces a singular potential at its location.
In all of the above-mentioned techniques one attempts to ‘directly’ solve the field equations
of the forward problem. An alternative approach is to solve for each MEG or EEG channel
the ‘reciprocal problem’, i.e. the equation for the ‘lead field’ L(r), defined for EEG such that
V (r1) − V (r2) =
∫
d3r JI (r) ·L(r, r1, r2) (1)
where V (r1) and V (r2) are the potentials at electrode locations r1 and r2, respectively, induced
by an impressed current JI . If the lead field is known, the forward calculation for a current
dipole at position r with moment Q is reduced to a simple evaluation of the lead field
V (r1) − V (r2) = Q ·L(r, r1, r2). (2)
From Helmholtz’s reciprocity theorem, it can be shown that the lead field for EEG is given
by the electric field of a unit current inserted at r1 and extracted at r2 in the same volume

The magnetic lead field theorem 3639
conductor (Malmivuo and Plonsey 1995). The advantage of the lead field approach is that it is,
opposite to the direct forward calculation, regular at the (variable) source locations and singular
only at the (fixed) electrode locations. This is especially useful for FEM (Laarne et al 2000,
Weinstein et al 2000) but has also been exploited for piecewise homogeneous and isotropic
volume conductors (Riera 1998, Fletcher 1995).
Similar to EEG, for MEG the lead field is defined by
n ·B(r′) =
∫
d3r JI (r) ·L(r, r′,n) (3)
where L is now the lead field of an MEG channel measuring the field at location r′ in direction
n. Again, using Helmholtz’s reciprocity theorem, one can show in an even more general form
that for sources oscillating at frequency ω
L(r, r′,n, ω) =
E(r, r′,n, ω)
iω
(4)
where E is the electric field induced by a unit magnetic dipole at location r′ in direction n at
frequency ω (Heller and van Hulsteyn 1992). E is in general a nonlinear function of ω which
vanishes at ω = 0. In spite of its generality, the above formulation of the lead field theorem
has a problem if one wants to use it for MEG, i.e. if one wants to calculate E to get L: we can
directly insert any frequencies exceptω = 0, i.e. the quasi-static approximation. Consequently,
its main merit is the solution of the frequency-dependent reciprocal problem as the primary
one, e.g., to calculate the electric field induced by transcranial magnetic stimulation (TMS)
(Heller and van Hulsteyn 1992).
If an analytic form for E(r, r′,n, ω) is known, as was done by Eaton (1992) for a multi-
shell spherical volume conductor and for non-pointlike coils, it is, with l’Hospital’s rule,
possible to calculate the lead field in the quasi-static approximation as the limit ω → 0 of the
right-hand side of (4). Although for realistic volume conductors it is conceivable to solve the
reciprocal field equations numerically for a sufficiently small ω, it is much simpler to solve a
lead field equation which is directly usable in the quasi-static approximation.
The first purpose of this paper is to provide such an equation, which, to our knowledge,
does not yet exist. Secondly, we will use this equation to propose a simple, fast and extremely
accurate algorithm for MEG forward calculation in realistic volume conductors. In contrast
to the methods proposed by Riera (1998) and Fletcher (1995), where the lead field equation
for EEG is reformulated as an integral equation on the boundaries of the compartments of
the volume conductor, we will solve the respective differential equations exactly in the inside
fulfilling the boundary conditions approximately. Conceptually, this is the usual approach for
analytic solutions but is, to our knowledge, new for numerical solutions for realistic volume
conductors.
The paper is organized as follows. In section 2.1, we derive a well-defined theorem for the
magnetic lead field for arbitrary conductivities in the quasi-static approximation. The boundary
conditions for piecewise homogeneous and isotropic volume conductors are presented
in section 2.2; in section 2.3, we show that the well-known lead field for a spherical volume
conductor is consistent with the theorem. Based on the theoretical results, we formulate
an algorithm for the magnetic forward calculation in section 3.1. This is made explicit
only for one shell, which is usually sufficient for MEG, but can easily be generalized to
multiple shells. The performance of this algorithm is demonstrated for a prolate spheroid in
section 3.2. Finally, we summarize and discuss our findings in section 4.

3640 G Nolte
2. Theory
2.1. The magnetic lead field theorem in the quasi-static approximation
Let us first recall the fundamental macroscopic equation for the physics of EEG and MEG
(Sarvas 1987). In the quasi-static approximation, it is assumed that the source, the impressed
current JI , is accompanied by a passive volume current JV = −σ∇V , where V is the electric
potential and σ is in the most general form an asymmetric and space-dependent rank two tensor.
From charge conservation it follows that the total current, J = JI + JV , is divergence-free
0 = ∇ · J = ∇ · JI − ∇ · σ∇V (5)
which, together with the boundary condition of vanishing fields at infinity, uniquely determines
V (and hence J) for given source JI . To obtain the magnetic field, one integrates Bio–Savart’s
law over the whole volume
B(r′) =
µ0
4π
∫
d3r J(r) × ∇ 1
|r − r′|
. (6)
To simplify reading the equations, we introduce the abbreviation
G(r, r′) = µ0
4π
∇
1
|r − r′|
(7)
and omit the arguments in the functions. Combining (6) with the definition of the magnetic
lead field (3) results in
∫
d3r L · JI =
∫
d3r(J × G) · n (8)
for any impressed current JI . The integrals are taken over the entire space and it is assumed
that the currents vanish at infinity. We first decompose the lead field into a divergence-free
part A and a curl-free part −∇U
L = A − ∇U. (9)
Using J = JI − σ∇V and rearranging the terms, (8) is equivalent to
∫
d3r JI · (A − G × n) =
∫
d3r(JI · ∇U − (G × n) · σ∇V ). (10)
With repeated partial integration and using (5), the first term on the right-hand side can be
rewritten as
∫
d3r JI · ∇U = −
∫
d3r(∇ · JI )U = −
∫
d3r(∇ · σ∇V )U =
∫
d3r(σ∇V ) · ∇U
=
∫
d3r ∇V · σT ∇U = −
∫
d3rV ∇ · (σ T ∇U). (11)
Similarly, the second term on the right-hand side can be transformed to
−
∫
d3r(G × n) · σ∇V =
∫
d3rV ∇ · σT G × n. (12)
Inserting (11) and (12) into (10) leads to
∫
d3r JI · (A − G × n) =
∫
d3rV ∇ · σT (G × n − ∇U). (13)
The last equation is fulfilled for any JI with the choice

The magnetic lead field theorem 3641
A = G × n (14)
∇ · σT (G × n − ∇U) = 0. (15)
A = G×n is the lead field in an infinite homogeneous and isotropic volume conductor which
we will refer to as L
∞
. Combining the last two equations and using (9) leads to the final result
L = L
∞
− ∇U (16)
∇ · σT L = 0. (17)
Note that the conductivity of the ‘reciprocal volume conductor’, the one in which the lead
field equation is solved, is given by the transpose of the original one and is not identical to it.
This is indeed also true for EEG, which can be shown similarly by using (1) and writing
V (r1) − V (r2) =
∫
d3rV (r)(δ(r − r1) − δ(r − r2)). (18)
This can also be shown by slightly modifying the derivation given by Malmivuo and Plonsey
(1995) to allow for non-scalar conductivities. Here, we have presented the lead field equations
for most general conductivities leaving open the question whether or not the antisymmetric
part of the conductivity tensor is of any practical relevance.
Obviously, one can write the lead field equivalently as
L = LX − ∇U (19)
where LX is the lead field in an arbitrary volume conductor. This merely corresponds to a
redefinition of U with the same result for the full L. This is not surprising, since with Stoke’s
theorem this is equivalent to the trivial fact that the external magnetic field of a closed loop
impressed current is independent of the volume conductor. The lead field in a realistic volume
conductor can hence be written as the lead field for a sphere plus a typically small correction.
We will use this in section 3.
In practice, volume conductors are always finite. If we want to calculate the lead field
numerically in arbitrary but finite volume conductors, we need to impose a boundary condition.
Since σT L is divergence-free the normal component must be continuous across the boundary
and, furthermore, since σT L vanishes outside the volume conductor the normal component of
σT L must vanish at the boundary of the volume conductor
n(r) · σT (r)L(r) = 0 (20)
for all surface points r with normals n(r).
In the derivation of the lead field equations, we always integrated over the whole volume
with boundary conditions at infinity. This led to an ill-defined equation for the lead field
potential in regions of vanishing conductivity, i.e. outside the volume conductor, which may
be regarded as unsatisfactory. Although more tedious it is also possible to derive the same
equations and boundary conditions by integrating only over the volume conductor giving rise
to surface terms which need to be discussed separately. This will only be sketched here.
Repeatedly using Gauss’s theorem, the right-hand side of (10) can be rewritten as
∫
X
d3r(JI · ∇U − (G × n) · σ∇V ) =
∫
X
d3rV ∇ · σT (G × n − ∇U)
+
∫
∂X
dS · JU −
∫
∂X
dS · σT (G × n − ∇U)V (21)
where dS is the oriented infinitesimal element on the surface ∂X of the volume conductor X.
The requirement that the first term on the right-hand side of (21) vanishes leads to the equation
for U in the interior of X. The second term always vanishes because no current can flow into

3642 G Nolte
or out of X. The third term vanishes if we impose the boundary condition (20) and use (14)
which itself need not to be rederived since there was no partial integration involved.
2.2. Piecewise homogeneous and isotropic volume conductors
For MEG forward calculation, the head is most commonly divided into a set of compartments
such as white and grey matter, skull and scalp. It is usually assumed that the conductivity is
piecewise homogeneous and isotropic, i.e. the conductivity is scalar and constant within each
compartment. According to (19) U is the correction of the lead field potential of an arbitrary
volume conductor X. The following equations for U are formulated for LX = L∞ but are also
true if X is the spherical volume conductor having smooth and divergence-free lead fields.
Since L
∞
is divergence-free we obtain for the inside of each compartment
�U = 0 (22)
i.e. U is harmonic. Note that ∇ · L = 0 in the inside is, with Gauss’s theorem, equivalent
to the well-known fact that an impressed current with unit strength on a closed surface is
magnetically silent (Sarvas 1987).
At the boundary between two compartments we obtain the following boundary conditions.
First, since ∇U is curl-free, the tangential components of ∇U must be continuous. Second,
since σL is divergence-free, the normal component of σL must be continuous and hence,
since L
∞
is continuous,
σ
−
n · ∇U
−
(r) − σ+n · ∇U+(r) = (σ− − σ+)n ·L∞(r) (23)
where σ
−
(σ+) is the conductivity of the inside(outside) compartment, U− (U+) is the respective
lead field potential and r is a point on the boundary separating the two compartments with
normal n. Finally, in order to have a non-divergent lead field at the surface, U must be
continuous at the boundary. This last boundary condition is actually a consequence of the
first: since the tangential component of ∇U is continuous across the boundary U
−
(r) can
differ from U+(r) only by a global constant1 which is irrelevant for the lead field. Note that at
the outermost surface, we have σ+ = 0 and with (23), the lead field is always tangential at the
outermost surface.
2.3. The spherical volume conductor
As an example, let us briefly discuss the spherical volume conductor which can be solved
analytically. To show that the analytical solution agrees with the above theory we must show
that the lead field for a spherical volume conductor is tangential at the surface and that it differs
from L
∞
by a gradient of a scalar function which is harmonic inside the volume conductor.
The former is well known and can directly be read off from the formulation given by Sarvas
(1987). For the latter the formulation given by Nolte and Curio (1997) is more convenient.
It was shown there that the magnetic field at location r′ due to a dipole with moment Q at
location r in a sphere can be written as
B(r′, r,Q) = B
∞
(r′, r,Q) + µ0
4π
(Q · ∇) r × r
′
r2r ′2 − (r · r′)2
(
r · r′ − r2
|r′ − r|2
−
r · r′
r ′
)
(24)
where B
∞
is the corresponding field in an infinite homogeneous and isotropic volume
conductor. The convenient point is that the contribution of the volume currents to the external
1 To see this, write the difference of U at two locations on the boundary as a line integral of ∇U along the surface.

The magnetic lead field theorem 3643
field is written in terms of a gradient with respect to dipole (and not sensor) location. From
the definition of the lead field and using (16)
n ·B(r′, r,Q) = Q ·L(r, r′,n) = Q ·L
∞
(r, r′,n) − Q · ∇U(r, r′,n) (25)
we can directly read off U to be
U(r, r′,n) = −
µ0
4π
n · r × r′
r2r ′2 − (r · r′)2
(
r · r′ − r2
|r′ − r|2
−
r · r′
r ′
)
. (26)
Let us now write r in a shifted and rotated coordinate system (r = r′+R) such that the direction
r′ corresponds to the z-axis and the direction of n corresponds to the x-axis. (Note that due to
the term n · r × r′ in (26) we can assume without loss of generality that n is orthogonal to r′.)
In the respective spherical coordinates, U assumes a very simple form
U(R, r′,n) =
µ0
8π
(R + r ′) sin � cot(�/2)
Rr ′
. (27)
Writing the Laplace operator in spherical coordinates
� =
1
R2
∂
∂R
(
R2
∂
∂R
)
+
1
R2 sin �
∂
∂�
(
sin �
∂
∂�
)
+
1
R2 sin2 �
∂2
∂�2
(28)
it is now straightforward to show that �U = 0 wherever U is non-singular. The singularities
of U occur at R = 0, i.e. at the sensor location, and at � = 0, i.e. on the straight line from
the sensor to infinity. Note that U is not singular at � = π and hence the singularities are
constrained to a half-axis lying completely outside the volume conductor.
3. Application to forward calculation
3.1. Algorithm for a single shell volume conductor
3.1.1. General algorithm. We now want to use the theorem for the magnetic lead field to
derive an algorithm for the standard magnetic forward calculation for the one-shell volume
conductor which is usually sufficient for MEG (Ha¨ma¨la¨inen et al 1993). Omitting in the
notation the dependences on channel location and orientation, one can write the full lead field
L(r) for one specific channel as
L(r) = Lsph(r) − ∇U(r) (29)
where Lsph(r) is the lead field for a spherical volume conductor and U(r) is a harmonic function
chosen such that the full lead field is tangential at the surface of the volume conductor. Here,
we expand U as
U(r) =
M
∑
m=1
amUm(r) (30)
where (Um) is an appropriate set of fixed harmonic functions. These functions are constructed
from spherical harmonics as explained below, but other choices are also possible and,
depending on the volume conductor, eventually better. E.g., spherical harmonics will in
general lead to inaccurate solutions even for an infinite expansion if the volume conductor
is not starlike2. Finding the lead field is now reduced to finding the coefficients am for each
channel.
2 Starlike means that a point in the interior exists such that the linear connection of this point to an arbitrary surface
point is completely inside the volume conductor.

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The forward calculation is divided into two steps: in the first step, we estimate the
coefficients am independent of the source; and in the second, we actually calculate the magnetic
field for a given source by evaluating the lead field. For the first step, let us assume that the
volume conductor is given by a set of locations rn and normals nn for n = 1, . . . , N . Generally,
having fewer ‘basis functions’ Um than points on the surface (N > M), the boundary condition
nn ·L(rn) = 0 for all n cannot be fulfilled exactly for any choice of coefficients (am). Instead,
these coefficients are now defined to be the ones which minimize the error in the boundary
condition
∑
n
(
nn ·Lsph(rn) −
∑
m
amnn · ∇Um(rn)
)2
. (31)
Let us denote by a the vector with coefficients am, by C the matrix with coefficients
Cnm = nn · ∇Um(rn) and by b the vector with coefficients bn = nn ·Lsph(rn). The minimizing
coefficients are then given by
a = (CT C)−1CT b (32)
which completes the estimate of the lead field. For a given volume conductor and channel
location and orientation, a needs to be calculated only once which can be regarded as an
initialization.
In the second step the actual fields are calculated for a given dipole. If a is known, the
magnetic field in the specified channel due to a dipole at location r with dipole moment Q is
now simply given by
n ·B(r′) = Q ·Lsph(r) −
∑
m
amQ · ∇Um(r). (33)
3.1.2. Technical details. The choice of basis functions Um is arbitrary as long as they are
harmonic. For the examples below we chose Um as the real and imaginary parts of
rpYpq(θ, φ)/Npq (34)
where Ypq are spherical harmonics with indices p = 1, . . . , P and q = 0, . . . , p (q =
1, . . . , p) for the real (imaginary) parts and
Npq = s
pp
(
(p + q)!
(p − q)!(2p + 1)
)1/2
(35)
where s is a reasonable scale of the system set to the average radius of the surface points.
Using spherical harmonics up to order P gives a total of M = (P + 1)2 − 1 basis functions.
Note that p = 0 cannot be included because the gradient vanishes and does not contribute to
the lead field. The idea of the normalization is that with this choice, CT C is approximately
(exactly, for an infinite number of equally distributed surface points) proportional to a unit
matrix for a spherical volume conductor of radius s. However, the normalization is in principle
arbitrary and we did not observe any dependence on the choice of s. A stable algorithm to
calculate spherical harmonics is well known (Press et al 1992). We computed the gradients
with recursion relations (Nolte et al 2001) but they can also be calculated numerically.
In principle, it is possible to choose a different set of basis functions for each channel.
Note, however, that both for the initialization and calculation of the magnetic field for each
dipole, the most time-consuming parts ((CT C)−1CT and Q · ∇Um(r), respectively) depend
on the chosen basis functions and not on the channel location or orientation. Hence, choosing
the same basis functions for all channels will significantly reduce the computational cost.

The magnetic lead field theorem 3645
Another possible choice is Um(r) = 1/|r − rm| with appropriate expansion points rm
placed outside the volume conductor3, which is certainly easier to programme than spherical
harmonics. This corresponds to virtual charges placed outside the volume conductor. However,
the optimal choice of rm is unclear and, although we have not tried, the orthogonality of
spherical harmonics on a spherical surface probably leads to a better conditioned CT C. On
the other hand, it can make very much sense to combine the truncated expansion in spherical
harmonics with virtual charges placed, e.g., only in the vicinity of local deformations of the
volume conductor which are difficult to properly include with (globally important) spherical
harmonics.
Although channel-dependent basis functions are more costly they can occasionally make
sense. For example, constructing the basis functions from Ypq(θ, φ)/rp+1 where the centre
of the coordinates is now the location of the channel automatically takes into account that the
lead field is less smooth in the vicinity of the channel. This can be useful if a channel is very
close to the volume conductor.
In the example presented in more detail in the next section, we used 3154 surface points
(with normals) and 169 MEG channels. Here, the initialization takes on a PC with a 2.5 GHz
processor 1.5 s for P = 10, 12 s for P = 20 and 120 s for P = 40. The forward calculation for
all channels for a single dipole takes 150 µs for P = 0, 550 µs for P = 10, 2.3 ms for P = 20
and 9.5 ms for P = 40. Finally, we note that for this relatively large number of MEG channels
the cost for calculating the spherical harmonics and their gradients is negligible in both steps
of the algorithm.
3.2. Testing the algorithm for a prolate spheroid
We have formulated the algorithm for calculating the lead field in a single-shell realistic
volume conductor as the analytic solution for a spherical volume conductor plus a numerical
correction implying that the algorithm is trivially exact for a sphere. In order to test it in a
non-trivial way, we compare our solutions to the exact solutions for a prolate spheroid (Fieseler
2000). For the ‘exact’ solution, the Cuffin/Cohen expansion was used up to order 60. The
relative error could be estimated to be always less than 10−4. The relative error is smaller for
deeper sources, and for sources more than 1 cm away from the surface, the relative error is
smaller than 10−6 (Nolte et al 2001).
The volume conductor, sensor system and analysed dipole locations are illustrated in
figure 1. The prolate spheroid has a long axis of 12 cm oriented in x-direction and two
short 9 cm axes. We analyse dipoles placed on the z-axis at various depths. The magnetic
field is calculated in z-direction at 169 sensor locations covering a 12 cm by 12 cm plane at
z = 12 cm. Sensor orientation reflects typical MEG-devices measuring the field in close-to-
radial direction.
We consider two types of error measures for assessing the error in the forward calculation.
The relative error of the ‘field’ B, the vector of field values in all channels, is defined as
�rel(B,Btrue) ≡
|B − Btrue|
|Btrue|
(36)
where Btrue is the true field. Since the relative error can be misleading when the true field
is vanishing we also present results with an absolute error measure defined by the maximum
norm
�abs(B,Btrue) ≡ max
k
|B(k) − Btrue(k)| (37)
3 For multiple shells: outside the respective compartment.

3646 G Nolte
–15 –10 –5 0 5 10 15
–10
–5
0
5
10
15
x [cm]
z
[cm
]
measuring plane
prolate spheroid
source
Figure 1. Dipolar sources are placed on the z-axis in a spheroidal volume conductor with a long
(short) axis of 12 cm (9 cm). The z-component of the magnetic field is measured on a plane at
z = 12 cm.
where B(k) and Btrue(k) denote the field values in the kth channel. The absolute error is more
useful in practice as it indicates whether the difference of the approximated and true field is
measurable. On the other hand, it also depends on the, for the assessment of the algorithm
irrelevant, dipole moment, which is here always taken to be 20 nAm.
We recall that the forward calculations consist of two steps. In the first step, which is
independent of the source, we estimate the coefficients of the expansion of the lead field up
to order P in spherical harmonics corresponding to (P + 1)2 − 1 basis functions. For each
sensor, the coefficients are fitted to 3154 points and normals on the spheroid. Preferably, the
points are distributed more or less equally over the surface. However, in general, it is also
conceivable that higher densities at more complicated regions of the surface will be used. In
the second step, the basis functions are evaluated up to order P ′ with P ′ � P . We usually
assume P ′ = P . However, as we will show below, for not too superficial dipoles only few
terms are needed in the final sum for accurate forward solutions.
The relative errors for dipoles in all three directions as a function of dipole location along
the z-axis are seen in figure 2. For dipoles in the x- and y-directions, already for P = 10
the relative error is up to 1000 times smaller than the one for the spherical solution (P = 0).
Even for the most superficial sources considered here, namely z = ±8.9 cm, the numerical
solution is still about 10 times better than the spherical one. We observe a sudden decrease
in accuracy in the centre of the volume conductor for dipoles in the x-direction. This is due
to the fact that the true field vanishes for this source as a consequence of axial symmetry.
For the relative error, z = 0 was excluded; the closest points analysed were z = ±0.1 cm.
Another consequence is that the relative error for the spherical approximation at z = 0 (to
be understood as the limit z → 0) is not equal to one which is necessarily the case if we

The magnetic lead field theorem 3647
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
R
el
at
iv
e
er
ro
r 0
10
20
40
Dipole in x-direction
Dipole in z-direction
Dipole in y-direction
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
Dipole location z[cm]
0
10
20
40
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
Dipole location z[cm]
R
el
at
iv
e
er
ro
r
0
10
20
40
Figure 2. Relative error of the magnetic field as a function of dipole location on the z-axis for
various dipole directions and orders P = 0, 10, 20 and 40 in the expansion in spherical harmonics.
z = −9 cm corresponds to the bottom, z = 0 cm to the centre and z = 9 cm to the top of the
volume conductor. Note that P = 0 corresponds to the spherical approximation of the volume
conductor.
approximate a non-vanishing quantity by zero (here, the field due to a dipole in the centre of
the sphere).
If we increase the order up to P = 40, we typically obtain a relative error of about
10−7, which can be considered as exact, with a slight increase at the boundary of the volume
conductor. The typical increase of the error close to the boundary arises because for larger
eccentricities higher order harmonics become increasingly important. However, this is not
a singularity at the surface as in BEM. The algorithm, indeed, allows the sources to be put
directly on the surface4. Once the coefficients of the expansion are fitted (independent of
the source location), the algorithm does not ‘know’ anything about the surface. Apparently,
the error mainly depends on the eccentricity leading to relative symmetric results although in
absolute terms the lead fields are very different in the lower and upper part of the spheroid.
The only case where the numerical solution is worse than the spherical one is for very
superficial dipoles in the z-direction calculated up to order P = 10. This is, indeed, a
coincidence since at those surface locations the surface normal of the spheroid coincides with
the surface normal of the sphere: both the true solution and the spherical approximation vanish
for a dipole in the z-direction at z = ±9 cm. In this case, the absolute error as shown in
figure 3 is more meaningful. We observe that the absolute error for the numerical solutions is
relatively independent of the dipole direction. We found the largest error for a dipole in the
4 For our comparison, the problem is that we cannot calculate the exact solution on the surface since there the series
does not converge.

3648 G Nolte
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
104
Ab
so
lu
te
e
rro
r [
fT
]
0
10
20
40
Dipole in x-direction
Dipole in z-direction
Dipole in y-direction
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
104
Dipole location z[cm]
0
10
20
40
–9 –6 –3 0 3 6 9
10–8
10–6
10–4
10–2
100
102
Dipole location z[cm]
Ab
so
lu
te
e
rro
r [
fT
] 0
10
20
40
Figure 3. Absolute error of the magnetic field for a dipole with amplitude Q = 20 nAm as a
function of the dipole location on the z-axis for various dipole directions and orders P = 0, 10, 20
and 40 in the expansion in spherical harmonics (full lines). The dotted lines show the maximum
norm of the exact solutions themselves. Note that for a dipole in z-direction this dotted line is
invisible since it is identical to the error for P = 0.
x-direction at z = 8.9 cm. For a dipole moment of 20 nAm, the absolute error for the spherical
approximation is 43 fT, which is certainly measurable, but for this source the maximum norm
of the true field is with 625 fT also unrealistically large. For order P = 20, this error is reduced
to 0.8 fT. For deeper sources, the absolute error is typically below 1 aT. Although one can
reduce the error even further, we think that P = 20 is sufficient for any practical application.
To achieve the same accuracy, deeper sources require the summation of fewer terms in
the expansion than do more superficial sources. We can exploit this by initializing the forward
solution with a relatively high order, say P = 40, and for given dipole location, summing only
up to order P ′ � P . For example, for sources precisely in the centre of the coordinate system,
P ′ = 1 is absolutely sufficient since higher orders do not contribute to the lead field. Since
the ‘system matrix’ CT C is in general not diagonal, the estimate of a specific coefficient am
depends on which other coefficients we are estimating at the same time. Hence, the actual
calculation in the second step up to order P ′ depends on the order P of the first step. The
results of this method are illustrated in figure 4 for ‘deep’ sources at z = 5 cm (which is not
really deep) and for superficial sources at z = 8.9 cm. As a general result we observe that
the convergence is slightly but significantly better for the choice P = 40 than for P = P ′.
We found that for non-superficial sources P ′ = 5 is practically sufficient. This corresponds to
as few as 35 additional multiplications per channel compared to the spherical approximation.
Note that an analogue method in BEM would be highly non-trivial since one cannot just ignore
a subset of triangles.

The magnetic lead field theorem 3649
0 5 10 15 20
10–6
10–4
10–2
100
102
Ab
so
lu
te
e
rro
r [
fT
]
Ab
so
lu
te
e
rro
r [
fT
]
Ab
so
lu
te
e
rro
r [
fT
]
Ab
so
lu
te
e
rro
r [
fT
]
P=P′
P=40
0 5 10 15 20
10–2
10–1
100
101
102
P=P′
P=40
Superficial dipole in x-direction
0 5 10 15 20
10–6
10–4
10–2
100
102
Order P′
P=P′
P=40
0 5 10 15 20
10–2
10–1
100
101
102
Order P′
P=P′
P=40
Superficial dipole in z-direction
Deep dipole in x-direction
Deep dipole in z-direction
Figure 4. Absolute error for dipole with amplitude Q = 20 nAm as a function of order P ′ in the
actual calculation of the fields, i.e. the second step in the algorithm. The lead field coefficients
were calculated in the first step up to fixed order P = 40 (dotted) or up to variable order P = P ′
(full). The deep dipoles were placed at z = 5 cm and the superficial ones at z = 8.9 cm.
4. Summary and discussion
In this paper, we derived a theorem for the magnetic lead field in the quasi-static approximation
stating that: (a) the transpose of the conductivity times the lead field is divergence-free, and (b)
the lead field differs from the one in an infinite homogeneous and isotropic volume conductor
by a gradient of a scalar function, the ‘lead field potential’. As a direct consequence, the
difference of any two lead fields for different volume conductors and for the same channel can
be written as a gradient of a scalar function U. For a piecewise homogeneous and isotropic
conductivity, it follows that the lead field potential is harmonic in the inside of the compartments
and fulfils specified conditions at the boundaries. Based on this theorem, we have proposed a
two-step algorithm for the forward calculation for a single-shell volume conductor correcting
the lead field of a spherical volume conductor: in the initialization step, which is independent
of the source, expand U for each channel in a basis of harmonic functions (here constructed
from spherical harmonics) and fit the coefficients to the boundary condition at a set of surface
points and normals. For the actual calculation for a given dipole, evaluate the lead field at the
source location. Let us summarize the essential properties of the algorithm.
(i) Accuracy. Testing the algorithm for a prolate spheroid, we found the relative errors were
as low as 10−7 for most dipole locations and even for the most eccentric dipoles were
lower than 10−5 for order P = 40 in the spherical harmonics corresponding to 1680 basis
functions. We emphasize that with this algorithm it is possible to put the source directly
on the surface of the volume conductor. The error is larger for eccentric sources because
more terms are needed in the expansion but the expansion itself is regular. We do not

3650 G Nolte
think that it is possible to achieve similar accuracies with BEM with reasonable effort.
Of course, such an accuracy is never needed in practice. The point we want to make is
that an essentially exact method for a relatively simple problem gives confidence that the
accuracy is also reasonable for more difficult cases.
(ii) Speed. There is always a tradeoff between speed and accuracy. Both initializing and
actually calculating the field up to a very high order probably is not much faster than
BEM. However, to achieve typical BEM accuracies, an expansion up to order 10 in
spherical harmonics corresponding to 120 basis functions is sufficient. In this case,
one has to construct and invert an 120 × 120 ‘system matrix’. A corresponding BEM
matrix with 120 surface nodes or triangles would be clearly insufficient. Furthermore, it is
possible to initialize up to high order and then to use this high order only for very eccentric
dipoles. In this case, we maintain high accuracies for all dipoles and an extremely fast
(in the order of the analytic solution of a sphere) forward solution for most dipole
locations.
(iii) Memory requirement. The number of surface points is typically much larger than the
necessary number of basis functions M. For the proposed algorithm, one needs to store
two matrices of size M2 and MNc, respectively, where Nc is the number of channels. For
any practically needed accuracies (M < 500 should be more than enough) that memory
requirement is negligible.
(iv) Simplicity. The algorithm contains one non-trivial step, namely, the evaluation of
spherical harmonics and their gradients. However, calculation of the spherical harmonics
themselves is standard and although we prefer to use perhaps non-standard recursion
relations for the gradients, it is also possible to calculate them numerically. Other
technicalities typically occurring in BEM such as a careful treatment of the autosolid
angle (Goncalves et al 2003), the analytic integration over triangles (de Munck 1992) and
in fact the whole triangularization of a surface, do not occur here.
It is worthwhile to discuss the nature of our approximation and the possible limitations
of the proposed method. We can always regard an approximate lead field in a ‘true’ volume
conductor as an exact lead field in an approximate or ‘effective’ volume conductor. Unless
we have not chosen the approximate lead field to be an exact solution for some, e.g.,
spherical, volume conductor, this effective volume conductor is different for each channel.
Choosing different effective volume conductors for different channels can be useful for
substantially improving the forward calculation using only the spherical volume conductor
model (Huang et al 1999).
Let us discuss what these effective volume conductors qualitatively look like for the
approach presented here. A finite order expansion of a lead field (correction) obviously means
a smoothing of the lead field which may contain substantial high spatial frequencies because of
two different reasons: (a) the sensor is very close to the surface, and (b) the surface itself is not
smooth. If the sensors are very close to a smooth surface then the true lead field is (within the
volume conductor) highly focal and contains substantial high frequency components. A low
order expansion would then be a bad approximation and the corresponding effective volume
conductor would be substantially deformed with assumingly large deformations in the vicinity
of the sensors implying that the effective volume conductors are different for each channel
and contain higher spatial frequencies than the original one.
On the other hand, in practice the sensors are not close to the surface and we can assume
that the order of expansion is sufficiently large to include the higher spatial frequencies arising
from the sensors. In this case, the finite order expansion corresponds effectively to a smoothing
of the volume conductor which will also be essentially independent of the sensors. Hence,
for not too close sensors the proposed method gives essentially exact solutions in approximate

The magnetic lead field theorem 3651
(smoothed) volume conductors. Since in our example the volume conductor was already
smooth, the solutions were essentially exact. This also implies that the practical limitation
of the proposed method is given by the fluctuations of the volume conductor surface. We
emphasize, however, that this is not a principle limitation as we can go, at least in principle, to
arbitrary high orders in the expansion, or consider a different basis for the harmonic functions.
We also recall that with the variants of the method suggested in section 3.1.2 it should be
possible to handle local deformations also with a low order expansion.
We have presented here an algorithm for one shell but the generalization to N shells is
conceptually straightforward: expand the lead field potential for each compartment separately
in harmonic functions5 and fit the set of coefficients to the boundary conditions at all surfaces.
Note that the actual forward calculation is essentially not slower since one needs to evaluate
the lead field only in that compartment which contains the source. However, the generalization
to N shells still contains one conceptual difficulty: the boundary conditions involve both U
and its gradient. Hence, the relative weighting of the two depends on the units. On the other
hand, in a hypothetical expansion up to infinite order, both types of boundary conditions are
both necessary and sufficient, and the solution does not depend on the units. Therefore, we
assume that for a finite order expansion, the final result is not very sensitive to the chosen
units. Whether this assumption really holds is the subject for a future investigation.
Acknowledgment
I wish to thank Thomas Fieseler for providing the exact solutions for the prolate spheroid.
References
Awada K A, Jackson D R, Williams J T, Wilton D R, Baumann S B and Papanicolaou A C 1997 Computational
aspects of finite element modeling in EEG source localization IEEE Trans. Biomed. Eng. 44 736–52
Cuffin B N and Cohen D 1977 Magnetic fields of a dipole in special volume conductor shapes IEEE Trans. Biomed.
Eng. 24 372–81
Eaton H 1992 Electric field induced in a spherical volume conductor from arbitrary coild: application to magnetic
stimulation and MEG Med. Biol. Eng. Comput. 30 433–40
Fieseler T 2002 Analytic Source and Volume Conductor Models for Biomagnetic Fields (Aachen, Germany: Shaker)
Fletcher D J 1995 Improved method for computation of potentials in a realistic head shape model IEEE Trans. Biomed.
Eng. 42 1094–3
Fuchs M, Kastner J, Wagner M, Hawes S and Ebersole J S 2002 A standardized boundary element method volume
conductor model Clin. Neurophysiol. 113 702–12
Goncalves S I, de Munck J C, Verbunt J P, Bijma F, Heethaar R M and Lopes da Silva F 2003 In vivo measurement
of the brain and skull resistivities using an EIT-based method and realistic models for the head IEEE Trans.
Biomed. Eng. 50 754–67
Ha¨ma¨la¨inen M S, Hari R, Ilmoniemi R J, Knuutila J and Lounasma O V 1993 Magnetoencephalography—theory,
instrumentation, and application to noninvasive studies of the human brain Rev. Mod. Phys. 65 413–97
Haueisen J, Ramon C, Brauer H and Nowak H 2000 The influence of local tissue conductivity changes on the
magnetoencephalogram and the electroencephalogram Biomed. Tech. 45 211–4
Heller L and van Hulsteyn D B 1992 Brain stimulation using electromagnetic sources: theoretical aspects Biophys.
J. 63 129–38
Huang M X, Mosher J C and Leahy R M 1999 A sensor-weighted overlapping-sphere head model and exhaustive
head model comparison for MEG Phys. Med. Biol. 44 423–40
Laarne P, Hyttinen J, Dodel S, Malmivuo J and Eskola H 2000 Accuracy of two dipolar inverse algorithms applying
reciprocity for forward calculation Comput. Biomed. Res. 33 172–85
Malmivuo J and Plonsey R 1995 Bioelectromagnetism—Principles and Applications of Bioelectric and Biomagnetic
Fields (New York: Oxford University Press)
5 Note that apart from the innermost compartment one also needs the functions Ypq(�, �)/rp+1.