Perturbation theory as a new analytical approach to the MEG forward problem for realistic volume conductor modeling of the head

eBe
lic
Be
N
ted
al
e
l
ct
al
s
a
n
m
o
b
en
il
s.
JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 4 15 FEBRUARY 2001I. INTRODUCTION
Current source reconstructions from magnetoencephalo-
graphic1 ~MEG! measurements crucially depend on the accu-
racy of the forward solution, i.e., the calculation of the mag-
netic field due to a dipole placed in a volume conductor.
Exact analytical solutions both of the electric and magnetic
forward problem are only known for special volume
conductors2–10 with the sphere being the most prominent.3
These special volume conductors are in general an insuffi-
cient approximation of the inner boundary of the scull, the
most relevant part of the whole volume conductor, the head.
So far, solutions for complex geometries can only be
obtained by numerical methods solving the differential or the
integral form of the corresponding Maxwell equations by
means of the ‘‘finite element method’’ ~FEM!11 or the
‘‘boundary element method’’ ~BEM!,12–14 respectively.
While both methods are applicable for a large class of vol-
ume conductors they are very time consuming and require a
large amount of disk space. Furthermore, originally analyti-
cal operations like differentiation15–18 might lead to large
errors if applied on solutions given only numerically. While
in BEM this drawback can in principle be avoided it is in-
herent in FEM.
Here, we propose a new method based on the fact that
for MEG deviations of the realistic volume conductor from
the spherical approximation are small. The magnetic field
can therefore formally be expressed as a spherical solution
plus a correction. While the functional dependence of this
correction on the deviation from the sphere can, of course,
not be solved exactly, we will derive an equation to exactly
compute the low order Taylor expansion of this functional.
Remarkably, it will turn out that for a first order Taylor ex-
pansion the corresponding solution of the electric problem19
is not needed, and the solution for the magnetic field can be
expressed as a surprisingly simple integral with no unknown
variables left.
For an explicit evaluation of this integral we have to
express the source, the solution, and the deformation of the
sphere in finite series of spherical harmonics.19,20 The latter
implies that though there is no principal limit in the com-
plexity of the volume conductor there will be a practical one
depending on the specific computer implementation of the
algorithms. In this sense we make two assumptions on the
volume conductor: ~a! the deformation of the sphere is small
compared to its radius, and ~b! the deformation is smooth,
i.e., sufficiently fittable by the number of chosen surface pa-
rameters. For MEG, both conditions are matched, in particu-
lar for cortical sources where the upper hemisphere is the
relevant part of the volume conductor.
This article is organized as follows. In Sec. II A we re-
call the formal expressions for the magnetic field and the
parameterization of the volume conductor. The fundamentala!Electronic mail: nolte@CS.unm.eduPerturbative analytical solutions of th
for realistic volume conductors
Guido Noltea)
Neurophysics Group, Department of Neurology, Klinikum
Hindenburgdamm 30, D-12200 Berlin, Germany
Thomas Fieseler
Institute of Medicine, Research Center Ju¨lich, D-52425 Ju¨
Gabriel Curio
Neurophysics Group, Department of Neurology, Klinikum
Hindenburgdamm 30, D-12200 Berlin, Germany
~Received 23 March 2000; accepted for publication 8
The magnetic field induced by a current dipole situa
computed exactly. Here, we derive approximate an
magnetoencephalography the deviation of the volum
approximation is small. We present an explicit integra
Taylor expansion of the magnetic field with respe
solution of the electric problem of order n21. Especi
problem only the well-known electric solution for a
evaluation of this integral by a series of spherical h
computation of the external magnetic field which is a
smooth volume conductor deformations of realistic
magnetic field is exactly curl-free it is equally g
performance for a realistic magnitude of deformations
for a prolate spheroid. We found a relevant improvem
boundary element method for superficial sources wh
deep sources. © 2001 American Institute of Physic2360021-8979/2001/89(4)/2360/10/$18.00
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecmagnetic forward problem
njamin Franklin Freie Universita¨t Berlin,
h, Germany
njamin Franklin Freie Universita¨t Berlin,
ovember 2000!
in a realistic volume conductor cannot be
ytical solutions based on the fact that in
conductor ~i.e., the head! from a spherical
form which allows to calculate the nth order
to this deviation from the corresponding
ly, for a first order solution of the magnetic
pherical volume conductor is needed. The
rmonics results in a fast algorithm for the
excellent approximation of the true field for
agnitude. Since the approximation of the
od for all components. We estimate the
y comparing the results to the exact solution
t over corresponding solutions given by the
e the performance is in the same order for
@DOI: 10.1063/1.1337089#0 © 2001 American Institute of Physics
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

equation of this article is derived in Sec. II B where the radial
component of the magnetic field is expressed as a simple
integral. Based on this equation we present in Sec. II C the
algorithm to compute the magnetic fields in arbitrary direc-
tion. In Sec. III we present fundamental properties of the
forward calculation, namely convergence, the dependence of
the magnetic field on the spatial frequencies of the deformed
volume conductor, and computational cost. In Sec. IV we
make detailed comparisons of the perturbative, spherical, and
BEM solutions for a prolate spheroid, and we finally discuss
our results in Sec. V.
Let us make a final remark: we do not present calcula-
tions for realistic volume conductors found from magnetic
resonance imaging which is beyond the scope of this article.
Furthermore, since the accuracy of BEM solutions them-
selves crucially depends on dipole depth, direction, and the
component of the calculated magnetic field, there is no rig-
orous way to judge the two different solutions.
II. THEORY
A. Backgorund
Brain activity is quite generally described by a station-
ary, primary current JP(r). The primary current induces a
‘‘return’’ or ‘‘volume’’ current JV(r) which together with
JP(r) makes up the total current J(r)5JP(r)1JV(r). In
contrast to the active part JP(r), which can be arbitrary, the
volume current is assumed to be induced by an electric field
within a medium3
JV5s
~
r!E52s~r!„V~r!, ~1!
where V(r) is the electric potential at point r and s(r) is the
conductivity. Stationarity implies that the total current must
have vanishing divergence
„
@
JP
~
r!2s~r!„V~r!#50, ~2!
which completely determines J as a function JP(r).
The magnetic field can now formally be calculated from
J using Biot–Savart’s law as
B
~
r
8
!52
m0
4pE dVJ~r!3„
1
ur
8
2ru
. ~3!
In this article we assume that the conductivity s(r) is homo-
geneous and isotropic within a volume conductor C with
boundary ]C . This implies that the volume conductor con-
sists of one compartment which is known to be a sufficient
approximation for the magnetic forward calculation.21,22
However, generalization of the presented method to more
compartments is straightforward and will be briefly dis-
cussed later. For one compartment the magnetic field can be
expressed as a surface integral by means of Green’s theorem3
B
~
r
8
!5
m0s
4p E
]C
dS3
r2r
8
ur2r
8
u
3 V~r!1B
inf
~
r
8
!, ~4!
where Binf is the magnetic field resulting from the primary
J. Appl. Phys., Vol. 89, No. 4, 15 February 2001part of the current density alone and V is the electric poten-
tial on the surface of volume conductor.
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecIn order to find an approximate analytical solution we
have to analytically define the surface.19,20 We assume that a
function f exists such that the surface is given by the image
of a function G:
@
0,p
#
3
@
0,2p
#
→R3 with
G
~
Q ,F!5
F
r
~
Q ,F!sin Q cos F
r
~
Q ,F!sin Q sin F
r
~
Q ,F!cos Q
G
~5!
and
r
~
Q ,F!5R1 f ~Q ,F!. ~6!
R is the unperturbed, constant radius of the spherical ap-
proximation of the real volume conductor. For this param-
eterization of the realistic volume conductor the integral
measure dS can be explicitly expressed as19
dS5dQdFr sin Q
S
reR2
] f
]Q
e
Q
2
1
sin Q
] f
]F
e
F
D
. ~7!
Here we expand f in the basis of spherical harmonics up
to order P
f
~
Q ,F!5
(
p50
P
(
q52p
p
bpqY p ,q~Q ,F! ~8!
with bp2q5bpq
! since f is real. We refer to (bpq) as the
multipole coefficients of the realistic surface which are re-
garded as a given input which can be derived from a sub-
ject’s cranial magnetic resonance imaging. For phase and
normalization conventions of the spherical harmonics we re-
fer to Ref. 19.
If the volume conductor is sufficiently smooth we can
expect that we only need a few terms in Eq. ~8! to describe
the head shape with a high accuracy. It was found that an
expansion up to order P55 is sufficient to describe details
that are as complicated as the neck.20 Explicit calculations
are done here up to P56. Signals of brain activity which can
be measured with high signal-to-noise ratio ~and, hence,
making the consideration of a realistic volume conductor
worthwhile! basically originates from in the superficial hu-
man cortex close to the upper hemisphere, and we can expect
that our expansion provides a sufficient description of the
head for this case. In general, the splitting of a volume con-
ductor into a sphere and a deformation is not unique. The
most convenient choice depends on the specific volume con-
ductor and the region of interest — if one exists. The depen-
dence of the forward calculation on this very choice will be
demonstrated in Sec. IV.
B. Perturbative calculation of the magnetic field
In principle the perturbative calculation of B(r
8
) re-
quires the knowledge of the measure dS, the Green’s func-
tion and the potential V up to the same order as B. However,
the calculation can be tremendously simplified if one uses
the same trick as for the spherical case, i.e., we merely cal-
culate the radial component of B. From this component the
total magnetic field is easily found by means of integration in
radial direction ~yielding the magnetic scalar potential! and
2361Nolte, Fieseler, and Curiofinally taking the gradient. The crucial advantage is that the
radial part of the measure dS does not contribute to the radial
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

e
F
Q
!
Q
‘ k k
rive after partial integration at the well known result that the
magnetic field vanishes outside the volume conductor.
The crucial point is that there is no term of order O( f 0)
in Eq. ~14!, and hence, for a first order approximation we
may set V to be the potential on the surface of a spherical
0volume conductor (V→V ) and evaluate G on the spherical
surface (G→G0) resulting in
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecVmon
0
~
r,r0!5
1
s
(
k51
(
l52k
1
k
Y k ,l~Q ,F!Y k ,l* ~Q0 ,F0!
Nkl
2
r0
Rk11
.
~18!
The potential for a current dipole with moment J can beand
r
@
cos Q sin Q
8
cos
~
F
8
2F!2cos Q
8
sin Q
#
ur2r
8
u
3
5
1
r
8
]G
]Q
1
1
r
8
r2r
8
g
ur2r
8
u
3
] f
]Q
. ~13!
Inserting this into Eq. ~10! we finally arrive at the re-
markably simple result
Br
8
5
m0s
4pr
8
E
dQdFr
S
]G
]F
] f
]Q
2
]G
]Q
] f
]F
D
V1B
r
8
inf
. ~14!
Note that if both the volume conductor and the source ~and,
hence, V) are axially symmetric ~independent of F) we ar-
G0
~
r
8
,r!5
1
ur2r
8
u
5
(
n50
(
m52n
4p
2n11
R
r
8
n11
3
Y n ,m* ~Q ,F!Y n ,m~Q8,F8!
Nnm
2 ~16!
with
E
dQdF sin QY n ,m* ~Q ,F!Y n ,m~Q8,F8!
5
4p
2n11
~
n1umu!!
~
n2umu!! 5:Nnm
2
~17!
we recall that for a unit charge monopole placed at r0 the
potential on a spherical surface reads19,23component of B, but the nonradial part is of order f. Thus, an
nth order calculation of
Br
8
[er
8
•B ~9!
requires only the Green’s function and V up to order n21.
Br
8
2B
r
8
inf
5
m0s
4p er8E dQdF sin QrS 2
] f
]Q
e
Q
2
1
sin Q
] f
]F
5
m0s
4p er8E dQdF sin Q rS
] f
]Q
e
F
2
1
sin Q
] f
]F
e
5
m0s
4p E dQdF sin Q rH
] f
]Q
F
rsinQ
8
sin
~
F
8
2F
ur2r
8
u
3
2
1
sin Q
] f
]F
F
r
~
cos Q sin Q
8
cos
~
F
8
2F!2cos
ur2r
8
u
3
With the abbreviations g5sin Q sin Q
8
cos(F
8
2F)
1cos Q
8
cos Q and
G5G
~
r,r
8
!5
1
ur2r
8
u
~11!
we express the terms in curly brackets as
r sin Q
8
sin
~
F
8
2F!
ur2r
8
u
3 5
1
r
8
sin Q
]G
]F
1
1
r
8
sin Q
r2r
8
g
ur2r
8
u
3
] f
]F
~12!
2362 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001Especially, to calculate B up to first order we only need to
know the electric potential for a spherically symmetric vol-
ume conductor.
For the nonspherical part of the radial component of B
we explicitly get
D
3
r
ur2r
8
u
3 V~Q ,F!
D
r
ur2r
8
u
3 V~Q ,F!
G
8
sin Q!
G
J
V
~
Q ,F!. ~10!
Br
8
5
m0sR
4pr
8
E
dQdF
S
]G0
]F
] f
]Q
2
]G0
]Q
] f
]F
D
V0
1O
~
f 2!1B
r
8
inf
. ~15!
C. Explicit computation of the magnetic field
To explicitly evaluate the magnetic field for a dipolar
source we express all functions in the integral kernel of Eq.
~15! in series of spherical harmonics. While f is already for-
mally given in Eq. ~8! and G0 is well known to be
‘ n n
Nolte, Fieseler, and Curiofound from Eq. ~18! upon differentiation with respect to the
origin.19,24 This leads to
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

Vdip
0
~
Q ,F!5
1
s
(
kl
aklY k ,l~Q ,F! ~19!
with
akl5J„0
1
k
r0
kY k ,l* ~Q0 ,F0!
Rk11Nkl
2 , ~20!
where „0 denotes the gradient with respect to r0 .
Insertion of G0, V0, and f into Eq. ~15! leads to a solu-
tion for the radial component of the magnetic field. Since B
is curl free ~in the quasistatic approximation! the complete
magnetic field can be found from first calculating the scalar
magnetic potential F and then taking its gradient
B5„F5„
E
‘
r
Br . ~21!
After expanding Br in a series of spherical harmonics
integration along the radial direction is straightforward end-
ing up with the final solution
B
~
r
8
!5
m0
4p (n51
N
(
m52n
n
(
p51
P
(
q52p
p
(
k51
N
(
l52k
k
„
8
3
Y n ,m
r
8
n11
Rn11bpqCnmpqklakl
~
2n11 !~n11 ! 1B
sph
~
r
8
! ~22!
with bpq and akl defined in Eq. ~8! and Eq. ~20!, respec-
tively, and
Cnmpqkl[E
0
p
dQ
E
0
2p
dF
1
Nnm
2
3
S
]Y n ,m*
]Q
]Y p ,q
]F
2
]Y n ,m*
]F
]Y p ,q
]Q
D
Y k ,l , ~23!
where Bsph denotes the solution for a spherical volume con-
ductor. Here, we have already limited the sums to finite N
and P as will be the case in a computer implementation. The
proper choices will be discussed in the next section. Note,
that the minus sign arising from integration of 1/r
8
n12 in Eq.
~21! has been put into the order of derivatives in Eq. ~23!.
At first sight it seems that evaluation of the sum in Eq.
~23! over six indices is extremely time consuming, making it
useless for practical applications. This is indeed not the case
if, as in BEM, the computation is split into an initialization
step which is independent of the source and a final step for
each source.
For the initialization note that the Cnmpqkl in Eq. ~23! are
fixed numbers which need to be computed only once and can
be stored ~up to given order!. Alternatively one may generate
formulas ~e.g., with Maple! which compute the nonvanishing
elements for given p ,q as a function of k and l. The latter is
possible because the Cnmpqkl are sparse: the nonvanishing
elements are constrained by
m5q1l ~24!
n5k12 j with 2upu11<2 j<upu21. ~25!
J. Appl. Phys., Vol. 89, No. 4, 15 February 2001The Cnmpqkl may be regarded as coupling constants between
different spherical harmonics of the source V0 and the solu-
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjection B. For p50 these couplings vanish; for p51 the cou-
plings correspond to a diagonal matrix, and for general p the
‘‘coupling matrix’’ has p side/main diagonals.
One very important consequence is that for finite p the
convergence is controlled by r0 /r8, the ratio of the sensor
radius and the source radius, since, apart from constant fac-
tors, the radius of the unperturbed sphere R occurring in the
factors (R/r
8
)n in Eq. ~22! and (r0 /R)k contained in the
source coefficient akl cancels out. Now, in a practical appli-
cation r0 /r8 is always sufficiently smaller than 1: magnetic
sensors cannot be put directly on the head surface and corti-
cal current sources reside at least 15 mm below the skin-air
boundary; hence, convergence will be excellent as is shown
in the next section.
Coming back to the calculation of B the initialization
consists of two steps.
~1! For the surface coefficients bpq which parameterizes the
perturbation around the sphere with radius R calculate
C˜ nmkl[
Rn11
~
2n11!~n11! (pq
bpqCnmpqkl . ~26!
~2! For each sensor at position ri which measures B in di-
rection ni compute
Cnm
i
[
m0
4p ni„
Y n ,m
ri
n11 ~27!
and
Fkl
i
[
(
nm
Cnm
i C˜ nmkl . ~28!
The vectors F i are the input for the explicit calculation
of B for each source. We point out again that they have to be
calculated only once for each volume conductor and sensor
configuration. They correspond to the ‘‘lead field’’ of the ith
sensor mapping the surface potential in the basis of spherical
harmonics to the magnetic field.
Now, for the calculation of Bi , the magnetic field in the
ith sensor, we have to compute akl according to Eq. ~20! for
each source and finally arrive at
Bi5(
kl
Fkl
i
akl1Bi
sph
. ~29!
Remarks: For notational simplicity we formulated the above
algorithm using complex numbers. However, in practice one
can save computation time if one splits the terms into real
and imaginary parts and makes use of the fact that f, V0 and
G0 are all real.
In the electric case we ended up with sums over a single
index by making use of a coordinate transformation for each
source rotating it to the z axis.19 However, for large p this
gets extremely complicated, and cannot be recommended.
Furthermore, the convergence properties are far better for the
magnetic case than for the electric case. The computation of
the double sums is sufficiently fast, as shown in the next
section.
The calculation of the magnetic field requires the evalu-
2363Nolte, Fieseler, and Curioation of spherical harmonics and their derivatives. For the
former we use standard algorithms,25 and for the latter we
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

derived simple relations which reduce this problem to the
former ~see the Appendix!.
III. PROPERTIES OF THE FORWARD CALCULATION
A. Convergence
The final perturbative solution for the magnetic field in-
duced by a current dipole was given in Eq. ~22! in terms of a
sum of spherical harmonics. Practical applicability of this
formula depends on its convergence behavior. Similar to the
electric case the series converges exponentially with r0 /r8
where r0 is the radius of the source and r8 is the radius of the
measuring point. In contrast to the latter, magnetic sensors,
being inside a dewar filled with liquid helium, cannot be
placed directly on the skin. In practice this means that even
for very superficial sources r0 /r8 is rarely larger than 0.5,
and one can expect the series to converge very fast.
Here, some care has to be taken since the ‘‘matrix’’ Eq.
~23! contains off-diagonals extremely increasing the required
number of terms. In general, for a surface parameterization
with spherical harmonics up to order P corresponding to
(P11)2 parameters a ‘‘solution spherical harmonic’’ of or-
der n can couple ~at most! to a ‘‘source spherical harmonic’’
of order n6(P21). Hence, the required order of the expan-
sion of the solution may increase by P21 compared to the
naive expectation. At this point it is obvious that the sparse-
ness of the coupling matrices in Eq. ~26! is essential since
otherwise the relevant expansion parameters would have
been both r0 /R and R/r8 with R being the radius of the
unperturbed volume conductor.
In our computer implementation we go up to order P
56 corresponding to 49 parameters to describe the realistic
volume conductor. In this case calculating the sum in Eq.
~29! up to N520 is sufficient as one can see in Fig. 1, where
we plotted
e
~
N ![
H
( i@Bi~40!2Bi~N !#2
( iBi
2
~
40! J
1/2
~30!
FIG. 1. Convergence: The relative difference between a magnetic field cal-
culated up to order N of spherical harmonics and the same field calculated
up to order 40 for a radial dipole placed on the surface (z059 cm! of a
spheroid, covered by a MEG whole head system of radius r
8
. The series
converge exponentially as ;(z0 /r8)N.
2364 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001as a function of N, the order of spherical harmonics in Eq.
~22!. Bi(N) denotes the perturbative magnetic field in the ith
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecsensor calculated up to order N. As a sensor configuration we
have chosen a virtual whole head system measuring the ra-
dial component of the magnetic field equally distributed
around the whole volume conductor at radius r
8
. The source
was defined to be a radial dipole placed on the z axis at
height r059 cm. The volume conductor corresponds to the
prolate spheroid as used in the next section and the pertur-
bative correction was calculated with respect to a sphere of
radius R510 cm. As one can see, for a measuring system
with radius r
8
515 cm corresponding to a source eccentricity
of r0 /r850.6 the magnetic correction has converged for N
520 up to 0.1%. Indeed, the chosen sensor configuration
has ‘‘bad’’ convergence properties: for a planar system at
height z
8
512 cm the ‘‘error’’ e(N) is smaller than for the
corresponding whole head system. Moreover, for a radial
dipole individual terms in the series basically cancel: thus the
considered case is exceptionally difficult.
For more accurate descriptions of the volume conductor
corresponding to larger values of P a larger value of N is
eventually needed. However, choosing, e.g., N530 would
still result in very fast forward calculations. Furthermore, the
large spatial frequencies ~large p) of the surface deformation
have in general a small amplitude, and hence, for these co-
efficients a relatively less accurate forward calculation is suf-
ficient.
B. Dependence on spatial frequencies
For the algorithm to compute the external magnetic field
the order of spherical harmonics to parametrize the realistic
surface is limited by an in principle arbitrary though finite
number P. The larger P is chosen the higher is the compu-
tational cost. However, for a square integrable deformation f
~any continuous deformation is square integrable! the se-
quence of surface coefficients, if written in the basis of nor-
malized spherical harmonics, converges to zero. Moreover,
for smooth deformations this convergence will be rapid.
Apart from this general considerations the question
arises of how large is the impact of individual frequency
components on the external field. In fact, if the surface po-
tential is relatively smooth and the sensor configuration is
not too close to the surface, contributions from higher order
frequency surface deformations will basically cancel out, i.e.,
the mapping of the surface deformations to the magnetic
field effectively acts as a spatial low pass filter.
To show this we have again used a spherical whole head
system, as described in the last subsection, with radius r
8
. To
the spherical volume conductor of radius R510 cm we
added ‘‘pure’’ normalized multipolar deformations
bpq5dp0p
dq0q
Npq ~31!
resulting in a correction to the magnetic field Bp0q0
i in the ith
sensor. Now for each order p0 we calculated the mean
g
~
p0![
1
2p011 (q052p0
p0
F
(
i
~
Bp0q0
i
!
2
G
1/2
. ~32!
As a source we have chosen a radial dipole placed on the
Nolte, Fieseler, and Curioz-axis at height z0 . In Fig. 2 we show g(p0) for three values
of r
8
and for various dipole depths. We see a clear exponen-
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

for deep sources high frequency oscillations of the volume C
˜ has ;P2 off-diagonals the total cost is proportional toconductor can always be neglected.
C. Computational cost
The computation of the magnetic field consists of three
steps: ~a! the parameterization of a given surface in a series
of spherical harmonics, ~b! the initialization for given surface
parameters and sensor configuration, and ~c! the actual cal-
culation for each source. Computation times for each step are
given for a HP 9000 ~B180L! with 180 MHz.
To describe a surface by spherical harmonics we assume
that it is given as a set of Ns points. Here, we first fit a sphere
to these points which is a simple nonlinear fit with 4 param-
eters: the cost is proportional to Ns . Then we keep these
parameters fixed and fit the deviation up to order P with
(P11)2 parameters, which is a linear fit: the cost is propor-
tional to (P11)4Ns . Since the Ns points of the surface are
in general not equally distributed as a function of surface
angles we do not make use of the orthogonality of the spheri-
cal harmonics. For P56 and Ns51600, as was used for the
prolate spheroid in the next section, this whole fit takes about
1.2 s. The splitting of the nonlinear sphere fit and the linear
P2N2Nc with Nc being the number of channels. For Nc
550, P56, and N520 this takes about 1 s resulting in a
total cost of 2 s for the initialization.
Finally, in addition to the spherical solution we have to
construct akl according to Eq. ~20! and calculate the ‘‘scalar
product’’ with Fkl
i for each sensor to obtain the magnetic
field @see Eq. ~29!#. For Nc550 channels a single forward
calculation takes about 14 ms.
Both the initialization and the final computation of the
magnetic field involve the evaluation of spherical harmonics
and their derivatives. We would like to note that, with the
help of the rules in the Appendix, the respective computa-
tional cost can be neglected.
The required disk space is considerably low because
relatively small sparse matrices are involved. The coupling
matrix C˜ in Eq. ~26! has only ;P2N2 nonvanishing ele-
ments and is constructed from ;P3N2 nonvanishing fixed
numbers Cnmpqkl . Additionally, the ‘‘lead fields’’ Fkl
i re-
quire the storage of ;NcN2 numbers which is in general
negligible.
IV. COMPARISON WITH THE PROLATE SPHEROIDtial decay of the norms of the magnetic fields. This exponen-
tial decay is the larger the farer the system is from the vol-
ume conductor. For more superficial sources this decay is
less pronounced eventually resulting in an essentially con-
stant g(p0) for sources placed on the surface of the volume
conductor (z05R510 cm!.
In practice the distance from the dewar to the inner
boundary of the skull is always large whereas the source can
in general be quite close to this boundary. To conclude on
this issue, the assumption of smoothness of the volume con-
ductor is only necessary if superficial sources are considered:
J. Appl. Phys., Vol. 89, No. 4, 15 February 2001deviation fit speeds up ~and simplifies! the calculation. How-
ever, we want to emphasize that this rather follows from the
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecnature of perturbation theory: the unperturbed, spherical ap-
proximation should be as accurate as possible in order to
obtain a small correction.
For the initialization we first construct the matrix C˜ ac-
cording to Eq. ~26!. For each p ,q and going up to order N
both in the expansion of the source and the solution requires
the calculation of ;pN2 matrix elements, and hence for (P
11)2 different values of p ,q the total cost increases as
P3N2. For N520 and P56 this takes about 1 s. Now, for
each sensor C˜ has to be applied on Cnm
i
@see Eq. ~28!#. Since
FIG. 2. The average contribution of different orders of
the surface parameterization to the external magnetic
field for whole head systems of radius r
8
and for vari-
ous dipole locations (z0).
2365Nolte, Fieseler, and CurioAs an illustrative example we will calculate the pertur-
bative solution for the prolate spheroid, which roughly cor-
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

responds to the form of a realistic head volume conductor.
We strongly emphasize again that the perturbation theory
can be applied to arbitrary volume conductors as long as they
are sufficiently smooth and as long as the deviation from a
spherical fit is not too large. However, the spheroid is the
only nonspherical volume conductor for which an exact so-
lution exists, thus making it possible to evaluate the pro-
posed approximate solution. Note, that the halfspace can be
regarded as a special case of the spherical volume
conductor.3 Moreover, one cannot perturb around the half-
space within the proposed framework because the eccentric-
ity of any source is 1.
The prolate spheroid is an egg-shaped surface defined by
the image of the function
G
~
Q ,F!5
S
2j cos Q
A
j
2
2c2sin QsinF
A
j
2
2c2sin Q cos F
D
, ~33!
where j and c are fixed numbers. Here, we rotated the vol-
ume conductor as compared to the standard definition by 90°
around the y axis in order to match the convention that the z
coordinate corresponds to vertical direction.
Here we choose j512 cm and c5A65 cm correspond-
ing to l59 cm (12 cm! for the short ~long! half axis of the
spheroid. These values roughly correspond to the typical dis-
tance ear to ear and front to back for a realistic case. This
spheroid rather fits the skin than the inner skull of a typical
head. However, relative errors do not depend on equal scale
transformations of source, volume conductor, and sensor
configuration. Moreover, the present choice is rather pessi-
mistic in the sense that superficial sources have larger eccen-
tricities than in typical real cases.
To use perturbation theory, the spheroid is approximated
by a sphere plus a correction. This description is not unique.
A reasonable choice is to use a fit both for the sphere and for
the correction. However, it can be better to choose the sphere
such that it approximates the realistic volume conductor in a
region of interest: especially, for realistic cases, the sphere
should rather fit the upper hemisphere if one is interested in
cortical sources. Here, we discuss only two out of an infinite
number of possibilities: ~a! the sphere is chosen as a fit re-
sulting in a radius of R510 cm and ~b! the sphere is defined
to be the largest inner sphere having a radius of R59 cm in
order to be accurate on ‘‘top’’ of the volume conductor.
For both spheres the correction according to Eqs. ~6! and
~8! is defined as a least squares fit to Ns51600 surface
points. The spheroid and the two spheres are shown in Fig. 3.
The spheres plus corrections up to order P56 fit the spher-
oid with an accuracy of 99.99%: they are practically indis-
tinguishable from the spheroid and are, hence, omitted in the
figure. As measuring device we choose in this section a pla-
nar array of magnetometers at height z512 cm. For definite-
ness, dipole moments are set to 20 nAm throughout this sec-
tion.
To calculate the solutions of the prolate spheroid we use
our arbitrary-precision implementation26 of the Cuffin/Cohen
10
2366 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001series expansion. The first 60 terms of the expansion were
considered for the calculations. To make sure that 60 terms
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecare sufficient for a satisfying convergence, we compared the
results to the results obtained with a considerable higher
number of terms, namely 100. The relative differences of the
solutions for 60 and 100 terms were less than 1024 for all
sources used in this article, which is sufficient to consider the
solutions as ‘‘exact’’ for the comparisons to the different
approximating solutions. The cutoff error is highest for the
sources closest to the border of the volume conductor (z0
5689 mm! and decreases rapidly with increasing distance
from the border, e.g., for uz0u<80 mm the relative cutoff
error with 60 terms is only about 1026.
Let us first show two examples. In Fig. 4 we show the z
components of the magnetic fields corresponding to a dipole
placed on the z axis at height z054.5 cm pointing into x
direction. The perturbative solution was calculated with re-
spect to the fitted sphere (R510 cm!. The spherical solution
~upper, middle! deviates from the exact ~upper, left! by 14%
while the perturbative solution ~upper, right! is 20 times
more accurate having an error of only 0.7%. In the lower
panel we show the respective difference fields. The perturba-
tive correction ~lower, middle! is in very good agreement to
the ideal correction ~lower, left!, the exact field minus the
spherical solution: the difference of the latter ~lower, right! is
about 20 times smaller than the ideal correction.
An interesting second example can be seen in Fig. 5.
Here the dipole is located at z0521 cm and points into y
direction — the most difficult case for a nonsuperficial di-
pole. The perturbative solution was calculated with the inner
sphere. While the spherical approximation breaks down
completely, the perturbative solution correctly represents the
complicated structure of the exact solution.
In the following we want to discuss in more detail the
dependence of the accuracy of the forward calculations on
the source parameters. We first restrict ourselves to the
spherical and the perturbative solutions; the performance of
FIG. 3. The prolate spheroid and two spherical approximations. The ap-
proximation of the spheroid by spherical harmonics up to order P56 is
practically exact and not included in the figure. The dipole location and
orientation are varied in the analysis — the plot shows a typical example.
Nolte, Fieseler, and CurioBEM, which was also studied for comparison, will be dis-
cussed separately.
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

FIG. 5. Same as Fig. 4 for a dipole
located at (x0 ,y0 ,z0)5(0,0,21) cm
pointing in y direction.In Fig. 6 this accuracy, defined as
e
a
5
F
( i~Bexact
i
2Bapprox
i
!
2
( i~Bexact
i
!
2 G
1/2
, ~34!
where a5x ,y ,z denotes the measured component of the
magnetic field, is shown as a function of dipole depth z0 .
The accuracy was calculated for the spherical approximationDownloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecand for the perturbative solutions with respect to a fitted
sphere of radius R510 cm ~‘‘pert. a!’’ and to an inner sphere
of radius R59 cm ~‘‘pert. b!.’’ Apart from the central dipole
pointing in z direction the inner sphere works better than the
fitted sphere. Especially, if a ‘‘radial’’ dipole approaches the
surface at z059 cm the fitted sphere may result in large
relative errors. The reason for this is that the exact solutionFIG. 4. Calcutated fields for a dipole
located at (x0 ,y0 ,z0)5(0,0,4.5) cm
pointing in x direction. Upper panel:
exact solution ~left!, spherical approxi-
mation ~middle!, perturbative approxi-
mation ~right!. Lower panel: ideal cor-
rection to the sphere ~left!, calculated
correction ~middle!, difference ~ideal
calculated! of the corrections ~right!.
2367J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Nolte, Fieseler, and Curiot to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

radial dipoles approaching the surface, the improvement due
to perturbative corrections can be much more dramatic for
intermediate dipoles eventually giving rise to an error de-
crease by a factor 100.
Finally, we want to compare the results for the analytical
approximation with the corresponding numerical solution
given by BEM using the program ‘‘Curry’’ ~Philips!. For the
latter we used a surface parameterization consisting of 2610
triangles with a typical side length of about 8 mm. In fact,
we used a given liquor triangularization and transformed the
triangles and normals to parametrize the spheroid.
As can be seen from Fig. 6 the relative accuracy of BEM
depends very much on the location and the direction of the
dipole. Generally, BEM becomes poor in the vicinity of the
surface of the volume conductor: for the z component of the
magnetic field this increase of the error starts at about 1 cm
distance (’ triangle length! from the surface and becomes
unacceptable at half the triangle length in agreement with
other findings.12
V. CONCLUSION
We presented the theory to compute an analytical ap-
proximation of the external magnetic field due to a source
inside a realistic volume conductor. We assumed that the
volume conductor can be described by a sphere plus a small
correction. It was shown that a first order Taylor expansion
of the magnetic field with respect to this correction can be
given without referring to the corresponding electric solu-
tion. The central result of the proposed theory, given in Eq.
~14!, is a remarkably simple integral for the calculation of
the radial component of the magnetic field, from which all
other components follow resulting in an exactly curl-free ap-
proximation.
An explicit computer implementation of the theory can
only be done for a finite parameterization of the realistic
volume conductor. Then, perturbation theory leads to a cou-
pling of input, the surface potential calculated in the spheri-converges to zero in this limit: if the perturbative solution is
calculated with a sphere of correct radius ~at the considered
location! it has this same property and hence the error stays
finite. However, also for other dipole directions the inner
sphere is more appropriate for superficial sources as was the
purpose of this very choice.
In the center of the volume conductor the relative error
eventually increases as can be seen for a dipole pointing in y
direction. The basic reason is that tangential dipoles get more
radial when approaching the center and, hence, the exact
solution decreases. Remarkably, this increase cannot be seen
for the dipole in x direction: the exact and the perturbative
solution converge to zero due to axial symmetry resulting in
a finite error. We note, that for this case the point at z050
was excluded in the figure since for the accuracy one divides
zero by zero.
Quite generally, perturbation theory improves the spheri-
cal approximation by about a factor 10–20 with exceptional
cases in both directions: while one should be careful with
2368 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001BEM works satisfactory for nonsuperficial sources apart
from the case of a central dipole pointing in x direction.
Downloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecHowever, this extreme breakdown of performance is basi-
cally caused by the vanishing of the true solution. Perturba-
tion theory based on the inner sphere works always better
than BEM while the corresponding solution for the fitted
sphere is eventually worse in regions of good performance of
BEM.
The breakdown of BEM in the vicinity of the surface is
more pronounced if a nonvertical field component is studied.
From the lower panel of Fig. 6 it can be seen that analytical
solutions, being exactly curl-free, behave similar for all field
components. In contrast, a large increase of the error of BEM
is observed at the boundaries: the contribution of the volume
current arises from relatively few, basically vertical second-
ary currents ~i.e., triangles! which induce a basically nonver-
tical magnetic field. If the x component of the magnetic field
is studied then also the ‘‘bad’’ perturbative solution ~fitted
sphere! has always smaller errors than BEM for sources
closer than 3 cm to the surface, and even the spherical solu-
tion is better than BEM for sources closer than 1 cm.
FIG. 6. Upper panel: Error of the forward calculation
for various approximations of the forward calculation
as a function of dipole height for the z component of the
magnetic field. Lower panel: Ratio of the corresponding
error of the x component and the z component. The
terms pert. a and pert. b refer to the perturbative expan-
sions around a sphere of radius R510 cm and R59
cm, respectively.
Nolte, Fieseler, and Curiocal approximation, and output, the magnetic field at some
specific location, consisting of a sparse and diagonally domi-
t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.

nant matrix. The latter property ensures that the final solu-
tion, which is written as a series of spherical harmonics,
converges as ;(r0 /r8)n where r0 (r8) denotes the radial
coordinate of the source ~sensor!. In practice, even for super-
Fn ,m~r![rnY n ,m~Q ,F! ~A1!
then from standard recursion relations it follows that
] ]
2369J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Nolte, Fieseler, and Curioficial sources r0 /r8 is substantially smaller than 1 resulting
in very fast algorithms for both the initialization and the
actual forward calculation.
We presented a detailed analysis of the performance of
the proposed approximation by comparing it to the analytical
solution for a ~prolate! spheroid, the only nontrivial volume
conductor where an exact solution is known. With the long
axis being 33% larger than the two short axes the assumed
deformation of the sphere is rather an overestimate of real-
istic deformations. Quite generally we found that the pertur-
bative solution improves the spherical approximation by a
factor 10–20 with exceptions in both directions. Especially
we found no problems with very superficial tangential
sources; radial sources eventually show up a diverging rela-
tive error because the exact solution vanishes for sources
placed on the surface giving rise to an intrinsically singular
performance measure.
For comparison we also calculated the corresponding
BEM solutions. We found BEM to be satisfactory for deep
sources with an error being in the same order as the pertur-
bative approach. Details, however, depend on the dipole di-
rection and the location and on the specific realization of the
perturbative approximation. BEM eventually breaks down
for superficial sources. While the corresponding increase of
the error occurs only for sources very close to the surface
(’5 mm! if an essentially radial ~here: vertical! component
of the magnetic field is studied, the tangential ~here: horizon-
tal! field components show a breakdown of the performance
much earlier (’2 cm!. This is in sharp contrast to analytical
methods which lead to exactly curl-free solutions.
We demonstrated the performance of a first order pertur-
bation theory for a single surface parametrized by 49 param-
eters. Generalizations can be done with respect to all aspects:
especially in conjunction with the corresponding electric
solution19 the theoretical basis is given, and the computa-
tional cost, being so far extremely low, can be expected to be
within acceptable limits.
ACKNOWLEDGMENTS
Supported by DFG Ma 1782/3. The authors would like
to extend our appreciations to Samual J. Williamson and Jan
C. de Munck for helpful discussions.
APPENDIX
Here, we present some very useful relations to calculate
derivatives of spherical harmonics, which are not known so
far. Phase and normalization conventions are as in Ref. 19.
Let us defineDownloaded 31 Jan 2001 to 64.106.43.203. Redistribution subjecS
]x
1i
]y DFn ,m52Fn21,m11 ;m>0, ~A2!
S
]
]x
2i
]
]y DFn ,m5~n1m21 !~n1m !Fn21,m21 ;m.0,
~A3!
]
]z
Fn ,m5~n1m !Fn21,m ;m>0. ~A4!
The corresponding rules for m,0 in Eq. ~A2! follow from
complex conjugation of Eq. ~A3! and similarly for the other
rules. Calculation of „Fnm corresponds to simple combina-
tions of Eqs. ~A2!–~A4!. Here, we have to calculate the gra-
dients of rnY n ,m and Y n ,m /rn11. The latter can be calculated
from Eqs. ~A2!–~A4! by writing Y n ,m /rn115Fn ,m /r2n11.
1 M. S. Ha¨ma¨la¨inen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lou-
nasma, Rev. Mod. Phys. 65, 413 ~1993!.
2 E. Frank, J. Appl. Phys. 23, 1225 ~1952!.
3 J. Sarvas, Phys. Med. Biol. 32, 11 ~1987!.
4 J. C. de Munck and M. J. Peters, IEEE Trans. Biomed. Eng. 40, 1166
~1993!.
5 D. A. Brody, F. H. Terry, and R. E. Ideker, IEEE Trans. Biomed. Eng. 20,
141 ~1973!.
6 P. M. Berry, Ann. N.Y. Acad. Sci. 65, 1126 ~1956!.
7 J. C. de Munck, J. Appl. Phys. 64, 464 ~1988!.
8 P. Lambin and J. Troquet, J. Appl. Phys. 54, 4174 ~1983!.
9 A. S. Ferguson and G. Stroink, J. Appl. Phys. 76, 7671 ~1994!.
10 B. N. Cuffin and D. Cohen, IEEE Trans. Biomed. Eng. 24, 372 ~1977!.
11 M. Rosenfeld, R. Tanami, and S. Abboud, IEEE Trans. Biomed. Eng. 43,
679 ~1996!.
12 J. C. de Munck, IEEE Trans. Biomed. Eng. 39, 986 ~1992!.
13 A. S. Ferguson, X. Zhang, and G. Stroink, IEEE Trans. Biomed. Eng. 41,
445 ~1994!.
14 M. Fuchs, R. Drenckhahn, H.-A. Wischmann, and M. Wagner, IEEE
Trans. Biomed. Eng. 45, 980 ~1998!.
15 G. Nolte and G. Curio, Biophys. J. 73, 1253 ~1997!.
16 T. F. Oostendorp and A. van Oosterom, IEEE Trans. Biomed. Eng. 43,
394 ~1996!.
17 Y. Wang, IEEE Trans. Biomed. Eng. 45, 131 ~1998!.
18 G. Nolte and G. Curio, IEEE Trans. Biomed. Eng. 46, 400 ~1999!.
19 G. Nolte and G. Curio, J. Appl. Phys. 86, 2800 ~1999!.
20 C. Purcell, T. Mashiko, K. Okada, and K. Ueno, IEEE Trans. Biomed.
Eng. 38, 303 ~1991!.
21 M. X. Huang, J. C. Mosher, and R. M. Leahy, Phys. Med. Biol. 44, 423
~1999!.
22 M. S. Ha¨ma¨la¨inen and J. Sarvas, IEEE Trans. Biomed. Eng. 36, 165
~1989!.
23 Z. Zhang and D. L. Jewett, Electroencephalogr. Clin. Neurophysiol. 88, 1
~1993!.
24 A. S. Ferguson and D. Durand, J. Appl. Phys. 71, 3107 ~1992!.
25 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
Numerical Recipes in C ~Cambridge University Press, Cambridge, 1992!.
26 T. Fieseler, Analytic Source and Volume Conductor Models for Biomag-
netic Fields ~Shaker, Aachen, Germany, 2000!.t to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html.