Magnetic Susceptibility Tomography
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5062 IEEE TRANSACTIONS ON MAGNETICS,
VOL.
30, NO. 6 ,
NOVEMBER
1994
Magnetic Susceptibility Tomography for Three
Dimensional Imaging of Diamagnetic and
Paramagnetic Objects
NCstor
G .
Septi lveda, Ian
M .
T h o m a s , a n d J o h n
P .
Wiksw o, J r .
Abstract-A tomographic technique for reconstructing .the
three-dimensional distribution of magnetic susceptibility in an
object is described.
A SQUID
magnetometer may be used to
measure the perturbations imposed by the object on an applied
magnetic field and these data contain information about the
susceptibility distribution. To assess the technique, a model ob-
ject was defined, simulated magnetic field data were generated,
and a matrix inversion was carried out with singular value de-
composition to yield a least-squares solution for the suscepti-
bility distribution. Various relative geometries of the three in-
teracting physical systems (the applied field, the object and the
measurement space) were used and the algorithm’s perform-
ance was investigated for each of the cases in which one of the
systems was moved while keeping the other two fixed. With
either strategy involving relative motion between the object and
the measurement space, accurate, convergent solutions were
obtained, but the algorithm failed when only the direction of
the uniform applied field was varied.
A
suitable nonuniform
applied field may make the algorithm robust. Applications for
a tomographic imaging susceptometer in biomedical imaging,
nondestructive evaluation, and geophysics are envisioned.
I.
INTRODUCTION
HE term tomography, originally defined as the mea-
T
urement of plane sections of a three-dimensional ob-
ject, has recently acquired a broader meaning and
is
now
used for any method of imaging the interior of an object
from measurements made entirely outside it. Most to-
mographic techniques have been developed during the last
twenty years for biomedical diagnostic applications
[
11
where the capability to “see ” inside a living body using
noninvasive external measurements has obvious advan-
tages. The techniques a re computationally demanding a nd
their rapid progress from research prototypes to standard
clinical procedures owes much to advances in computer
capabilities over that period.
The two broad classes of computer assisted tomography
(CAT)
[2]
are transmission imaging and emission imag-
ing.
In
transmission imaging [3], a source of radiation
(usually X-rays) and a suitable detector are placed on op-
posite sides of the body. Analysis of multiple measure-
Manuscript received June 21, 1993; revised May 17, 1994. This work
was carried out under contract with D uPont.
The authors are with the Electromagnetics Laboratory, Department of
Physics and Astronomy, Vanderbilt University, Box 1807, Station B,
Nashville, TN 37235.
IEEE Log Number 9403996.
ments taken along different ray-paths in a single plane al-
lows reconstruction of the absorption distribution and,
hence, the internal structure. Emission imaging differs in
that the source of radiation is a radionuclide-labeled sub-
stance inside the body. The images obtained depend on
the distr ibution of this substance which, in turn, d epends
on the morphology and physiological condition of organs
and tissues.
In
positron emission tomography (PET)
[4],
a position emitted by the radionuclide is detected indi-
rectly via the two oppositely-directed,
51 1
keV gamma
rays that are created when the positron ann ihilates with an
electron. Single-photon emission computed tomography
(SPECT)
[5]
is more flexible than PE T becau se it uses the
larger set of radioisotopes whose nuclei decay, emitting
gamma rays as individual photons. Tomo graphic mapping
may also be performed with nuclear magnetic resonance
(NMR ) imaging
[6],
and this technique is most commo nly
used to image the distribution of hydrogen atoms in the
body. Firstly, a strong magnetic field is applied, aligning
the nuclear dipole moments. Under these circumstances,
a pulse of radio waves of the correct resonant frequency
(which depends on the magnetic field experienced by a
nucleus) can modify the direction of the nuclear moment
which then rapidly returns to its previous state, radiating
a radio wave in all directions. By adding a field gradient
to the applied field, spatial information is encoded
so
that,
from the measured spectrum, the density of hydrogen at-
oms (and also th e nature of their chem ical environment)
can be determined as a function of position. Recent ad-
vances in NMR microscopy
[7]
have improved the spatial
resolution of this technique to about 100 pm. Microwave
imaging
[8],
electrical impedance tomography (EIT)
[9]
and ultrasound tomography [101 are still considered to be
under development, although good qualitative images a re
obtained with echo ultrasoun d. Acoustic tomography has
also found applications outside medical science in studies
of the Earth’s mantle
[ l l ]
and oceans
1121.
Although magnetic susceptibility tomography has not
previously been attempted, there are several reports of the
measurement and imaging of magnetized susceptible ma-
terials, mainly for biomedical applications. Bauman and
Harris
[
131 used an iron-core transformer with an air gap
to attempt to quantify the iron content of the liver in rab-
bits in vitro and rats in vivo, but this method lacked suf-
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MAGNETIC
SUSCEPTIBILITY
TOMOGRAPHY 5063
ficient sensitivity fo r reliable measurements, even o n small
animals. Th e earliest susceptibility m easurement tech -
nique that utilized a superconducting quantum interfer-
ence device (SQUID) was termed magnetic susceptibility
plethysmography [14 [15 which imaged the cardiac
cycle by distinguishing noninvasively between th e sus-
ceptibilities of blood and tissue in humans. To ad dress the
problem of diagnosing iron overload in the human liver,
Farrell
et al. [16]
and Bastuscheck and Williamson
[17]
constructed SQUID susceptometers which measured dc
and ac susceptibility, respectively. Instrumental devel-
opment culminated in the Ferritometer
[18],
a custom
SQUID biosusceptometer system, installed in a hospital
for clinical use [191.
The principal ad vance offered by the present technique
over these and other recent susceptibility mappings of
phantoms [20], post mortem specimens [21], and entire
human bodies [22] is in the analysis of the data. Whereas
previous experiments have produced susceptibility-per-
turbed field maps or model-dependent solutio ns, magnetic
susceptibility imaging
[23]
involves an actual solution of
the inverse problem. This is relatively straightforward for
planar samples [24], but problems with nonuniqueness are
encountered with three-dimensional susceptibility distri-
butions. This paper is concerned with magnetic suscepti-
bility tomography, which addresses these problems.
In magnetic susceptibility tom ograp hy, the internal dis-
tribution of magnetic susceptibility of an object is deter-
mined by applying various configurations of magnetic
fields and measuring how the object perturbs them. This
may be compared with CAT scans where the distribution
of density is determined by irradiating the object with
X-rays and analyzing the absorption.
In Section 11, the theory of magnetic susceptibility is
described and equations for the magnetic forward and in-
verse problems, as they apply in this case, are derived.
The algorithm is explained in Section I11 and its perform-
ance under three different experimental strategies is dis-
cussed in Section IV. The potential applications of this
technique and further work for its development are sum-
marized in Section V.
11.
MAGNETIC
H E O R Y
A . Magnetic Susceptibility
The magnetic susceptibility
x
(i.e. , volume suscepti-
bility, a dimensionless quantity in S.I. units) of a material
defines the magnetization M that it develops when it is
exposed to a magnetic field of strength
H.
In general, the
applied field is not uniform and susceptibility in a com-
posite sample is a function of position
r‘
so that
M(r’ )
= x(r’)H(r’) . (1)
The magnetic flux density
B
(which will, hereafter, be
referred to a s the ‘magnetic field’) is given by
B(r ’ ) =
o
( H ( r ’ ) W r ’ } ,
(2)
where po = 4 x T - m -A-’ is the magnetic perme-
ability of free space. By substituting (1) into
(2)
and using
the definitions of relative permeability
p
and absolute
permeability
p
p r ( r ’ ) =
1
~ r ’ ) ,
d r ’ )
= P o p r ( r ‘ ) ,
B ( r ’ )
= p ( r ’ ) H ( r ’ ) . (3)
the magnetic field may be expressed as
Diamagnetic materials generate a weak magnetization
in opposition to the applied field x - except for
superconductors where
x
=
-1).
Although this effect is
also present in predominantely paramagnetic materials, it
is not seen because unpaired electrons become aligned,
producing a much stronger magnetization that enhances
the applied field x
-
Both of these effects are very small co mpa red with fer-
romagnetism, whereby cooperative forces maintain the
parallel alignment of atomic magnetic mom ents over mac-
roscopic regions. Ferromagnets are, in general, magne-
tized eve n in the absence of an applied field and, wh en a
field is applied, its effect on the magnetization of a fer-
romagnetic obje ct is very stron g, nonlinear and dependent
on the previous magnetic history of the sample. Thus, the
susceptibility is variable and large, typically in the range
+ l o 3 +lo5,so that
p
= 1 + x
=
x. 4)
Because the susceptibility of ferromagnetic materials is so
high, the magnetization in
(2)
makes a much greater
contribution to the magnetic field B than does the applied
field strength H. Consequently, the magnetization at one
location is determined both by the applied field and by the
additional field strength at that location resulting from th e
magnetization at all other locations in the sample. A self-
consistent solution to
(2)
can only be obtained by simul-
taneously computing
H
and M everywhere and, fo r a fer-
romagnetic object
of
complex shape, this presents severe
difficulties.
However, for diamagnetic and paramagnetic materials,
the magnitude of the induced magnetization M is less than
about of the original applie d field
Ha ,
so that, in
contrast with (4),
(5)
Furthermore, the susceptibility is independent of the ap-
plied field at relatively low field strengths, so the mag-
netization is linear and nonhysteretic. Th e remainder of
this paper will be concerned only with this weak-field
limit, known as the Born approximation: at each location
in the sample, the contribution to the applied field from
the magnetization elsewhere may be ignored.
In general, the total magnetic field at a location r’ in
the sample is the sum
of
the applied field
Bo,
the local
field
BI
due to the magnetization at that location , and the
distant field Bd due to the magnetization of the rest of the
p r =
1
x
= 1.
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I EEE TRANSACTIONS
ON MAGNETICS, VOL. 30, NO.
6,
NOVE M BE R
1994
sample. In the Bo m approximation, Bd is the negligible,
yielding the truly local equation,
Z
B(r’) = pOH(r’)
poM(r ’ )
B, (r’ )
+
Bi(r ’ ) .
( 6 )
Furthermore, since B, = poH to one part in IO5 for a
diamagnetic material with
x
=
o-’,
and to one part in
lo3 for a paramagnetic material with
x
=
+
O p 3
the lo-
cal field is given by
Bi r’) P O W ~ ’ )pOx(r ’ )H , (r ‘ ) ,
X(r’) = ___
However, in a real experiment to study diamagnetic or
paramagnetic materials,
BI
r‘) cannot be measured di-
rectly. Instead, the magnetometer is placed at a distant
so that
(7)
i
( r ’ )
C loK(r ‘ )
field point r and a forward problem must be developed
and then inverted to determine
M ( r ’ )
and, hence, ~ r ’ ) ,
For isotropic materials,
x
is given by a single value and,
depending o n which type of magnetism is dom inant,
M
is
either parallel or antiparallel with H a . A susceptibility
tensor is required for anisotropic materia ls, and that topic
will not be considered here.
B. The Forward
Problem
SQUID
magnetometers have sufficient sensitivity and
resolution to measure the very smal l perturbations that are
superimposed upon th e stronger applied field, so that weak
diamagnetism or paramagnetism can be detected. In their
normal mode of ope ration, they a re sensitive to local
magnetic field variations, whereas a uniform background
field is not measured. For this reason , the applied field
B ,
may be ignored and the perturbation field
BI
alone will be
considered.
The dipole moment of a magnetized volume element
dv‘
at a location r’ (where dv‘ is small enough that its
magnetization M ( r ‘ ) can be assumed to be uniform) is
dm(r ’)= M(r’ ) dv‘
and its contribution to th e magnetic field at the field point
r is given by the dip ole field equation [ 2 5 ]
po [ 3 d m ( r ’ )
( r
r’)
r r’)
47r
Ir
rrI5 ( r r’I3
dm (r ’ )
dB(r) =
(9)
The solution to the forward problem is then obtained by
integrating the field from the magnetization associated
with all the elemental d ipoles constituting the magnetized
object,
3M(r ‘ )
r r’)
r
r‘)
47r
Fig.
1.
Simulated experimen ts to image three-dimensional susceptibility
distributions. (a) The vertical B component
of
the magnetic field
1 mm
above a homogeneous cube
x
=
1.2
X
IO-’,
S.I . )
exposed
to
an applied
magnetic field B,
=
B o k . (b) The Bz field component 1 mm above an in-
homogeneous cube (in S.I. units, xwhlle
=
+
1.0
x
IO-’,
xahaded + 1.O
x
IO-’) exposed
to
an applied magnetic field Bo
=
B,, i
+ BA.
and, from (8) ,
3 x ( r ’ ) H u ( r ’ )
r r ’ )
r
r’)
for the entire distribution.
Fig.
1
illustrates the complexity of imaging three-di-
mensional magnetic field sources. It shows the simulated
magnetic field patterns (calculated using
(10))
that would
be measured in a square plane of side 12mm placed cen-
trally,
1
mm above the upper surface of two different
cubes, both of side 3 mm, under two different conditions
of a uniform applied field B, = p o H , . In Fig. l(a) , the
cube is homogeneous with a susceptibility
x
= +
1.2 x
IOp5 (in
S.I.
units) and the applied field is in the positive
z-direction with a magnitude B, = 100 pT. The general
shape of the resulting field map can be predicted easily.
The solution in Fig. l(b ) is not so intuitively obvious. In
this case, the cube consists of sixty-four equal cubic ele-
ments of which the eight com er ones are assigned a
sus-
ceptibility
x
=
+ 1
.O x ( S . I . ) , while the remainder
of the sample has
x
= 1.0 x lo-’ (S.I .) . The applied
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MAGNETIC
SUSCEPTIBILITY
TOMOGRAPHY
field is directed diagonally, parallel to the xy-plane and,
hence, parallel to the upper surface of the cube, again
with a magnitude
B,
=
100
p T .
C . The Inverse Problem
This section describes a general solution to the inverse
problem for situations in which the Born approximation
is valid. Consider an object that consists of a number
of
magnetic dipoles, eac h of which generates a magnetic field
given by (9),
dB(r) =
r r’)
___
47r
r r’I5
If each dipole moment dm(r’) arises from the magneti-
zation of an elemental volume
d v ’
n an applied field
H a ( r ’ )
which, in general, may be nonun iform), it follows
that
dm(r’)
= x ( r f ) H a ( r ’ ) v’,
so that, for a single source and a single measurement,
Equation
(1 1)
can be written a s
dB(r) =
G r,r’ , H a)x(r ’ )
dv’,
(12)
where the vector Green’s function is defined as
If the location
r ’
of a source, which is a single mag-
netized volume element, and the strength and direction of
the applied field at that point
Ha(r’)
are known, a single
measurement of the magnetic field
B
at the field point
r
is sufficient to determine
~ ( r ’ ) .
nly one component of
B ( r )
is required as long as that component is non-zero.
The problem becomes m ore complex when the location of
the volume element is unknown, or when there are either
multiple sources o r a continuous distribution of suscepti-
ble matter (a magnetization distribution). In these cases,
(12)
must be integrated over the entire sample.
In the numerical app roach, the source object is discre-
tized into m elements of volume
v f ,
where
1
m ,
such that the field at point r is given by
m
B(r)
=
x
( r ,
r: ,
H,)x(r , )vf .
(14)
r = l
A
single measurement of
B(r)
is now inadequate to deter-
mine the susceptibility values for the m elements. In fact,
it is necessary to make measurements at
n
locations
r j ,
where
1
n and n m .
5065
The analysis can be simplified by converting to matrix
notation. The vector Green’s function G will be repre-
sented by the
( n
X
m )
matrix
6
that contains, as each of
its rows, the Green’s functions that relate a single mea-
surement to every so urce element. If the n field measure-
ments are written a s the n elements of the (n
X
1) column
matrix 63, then the magnetic susceptibility of each of the
m source elements will be given by the ( m X
1)
column
matrix
3c
in
63 = s3c,
(15)
where the volume of each source element
U,
has been in-
corporated into the
6
matrix.
The ability to solve this set
of
simultaneous equations
is determined by the complexity of the source, by mea-
surement noise, and by how well the source space
is
spanned by both the applied field and the measurements.
If
n = m ,
the system of equations will be exactly deter-
mined, but a solution will not be possible unless the ma-
trix s is nonsingular. Even then, digitization errors and
uncertainties in 63 (measurement noise) are likely to ren-
der the matrix singular. The approach adopted in this pa-
per (because it is suitable for practical implementation) is
to take extra measurements
so
that
n >> m
and the system
is overdetermined, and then use singular value decom-
position (S.V.D.)
[26]
to find the most probable solution.
However, th e Green’s functions must still be m ade suffi-
ciently different from each other that the matrix remains
well-conditioned and this is achieved by varying the rel-
ative geometries of the applied field, the source, and the
measurement locations.
111. DESCRIPTION
F
THE ALGORITHM
In summary, the gen eral approach for determining the
susceptibility distribution
~ r ’ )
f a three-dimensional
sample from a measured component of the magnetic field
follows:
Divide the sample into
m
small elements, each of
volume U : .
Determine the centroid
r[
of each element.
Given the known applied m agnetic fieId distribution
B , ( r i ) =
p o H a ( r : ) ,
calculate the field strength
H , ( r ’ )
experienced by each element (this step is
trivial if the applied field is uniform).
For a given volum e element r: and measurement lo-
cation
rj,
calculate the corresponding Green’s func-
tion using (13),
Calculate the Green’s functions
other volume elements to rj, so
that relate all the
that the measure-
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IEEE TRANSACTIONS ON MAGNETICS, V O L.
30,
NO. 6 , NOVEMBER
1994
ment B ( r j ) s given by (14)
i
m
B(rj) = x 1 G y v l x 2 G 2 j v 2
+ x m G m j v , , , iGi iv i .
i =
6) Calculate the summed Green’s functions for the re-
mainder of the
n
locations at which measurements
have been made (where n >> m , to generate a
complete set of simultaneous equations which may
be written in matrix notation as
(15)
63 =
SX,
where the matrix 9 has dimensions n
x
m ,
n
being
the number of data in the column matrix 63 and
m
being the number of effective dipole sources in the
7) Use singular value decomposition [26 ] to Solve the
system of equations given by (15), by finding the sus-
ceptibility distribution X hat minimizes the function,
column matrix
X.
Fig.
2.
Perspective view
of
the @-element cube implemented
for
testing
the algorithm.
S . I .
susceptibility values are x = f 2 . 5 x IO-’
for
the ten
shaded elements and
x
= 2.0
X
for the remainder
of
the cube.
7
F
=
(gX I 2 .
(16)
To guide future instrument and experiment design , the
algorithm was used to investigate three practical strate-
gies by simulation. Any one, or a combination,
of
these
strategies could be used in practice.
Strategy
1)
Vary the direction of the applied field while
maintaining the sample and the magn etometer array fixed.
Strategy
2)
Vary the orientation of the sample while
maintaining the magnetometer array and the applied field
fixed.
Strategy 3)
Vary the location of the magnetometer ar-
ray while maintaining the applied field and the sample
fixed.
A cube of side
3
mm was divided into sixty-four equal
volume elements. Ten randomly selected elements (as il-
lustrated in Fig.
2)
were assigned a magnetic susceptibil-
ity of x = +2.5 X (S. I . ) , while the remainder of
the cube was assigned a v alue of x =
+2.0 X lop5 S. I . ) .
In each of three separate experiments, the forward
problem was solved several times while varying the ge-
ometry of either the applied field, the sample or the mea-
surement plane (representing the magnetometer array).
The square measurement plane, extending from - 6 mm
to +6
mm in two orthogonal directions, was placed par-
allel to and 1 mm from one of the cube’s faces, and the
field component normal to that face was calculated. The
separation between adjacent data points was 0.6 mm
so
that each calculation generated 441 data values. These
simulated magnetic field values were then substituted for
the measured data in order to test the algorithm’s ability
to determine the susceptibility distribution.
A . Varying the Direction
o
the Applied Field
With the m easurement plane fixed 1 mm from the pos-
itive z-face of the cube (see Fig. 2) the
B ,
component of
X A
b) B, = BQ
a) B. = B.i
[e)
B,=
Ha(- i
k)
(I)
B.
= H.(-i t t )
(h)
B.
= B,(i t )
g)
B.
= B.(i
t
t
)
Fig.
3 .
Simulated magnetic fields B, corresponding to eight directions of
the applied field.
the field was calculated for each of eight applied field di-
rections (as illustrated in Fig.
3).
Although the results of
the forward problem are consistent with the experiment (a
paramagnetic object exp osed to the various applied fields),
it is not possible to infer any information about the fine
structure of the susceptibility distribution within the cube
from a purely visual inspection. The simulated data and
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MAGNETIC SUSCEPTIBILITY TOMOGRAPHY
5067
details of each applied field were passed to the algorithm
and S.V.D. was used to estimate the susceptibility of each
of the sixty-four volume elements. Th e computation was
carried out with high-precision data (about on e part in lo9,
limited by the single-precision arithmetic of the SUN
4/260 computer) and with data truncated to five signifi-
cant figures to simulate the effect
of
measurement noise.
For each level of precision, the algorithm was tested,
firstly, with just six directions of the applied field (Fig.
3(a)-(f)) an d, second ly, with all eight directions.
B .
Varying the Orientation ofthe Sample
The uniform applied field Ba =
kB,
(where
B, =
100
pT) and measurement plane (1 mm from the positive
z-face) were fixed while the cube was placed in each of
seven orientations, which were obtained with 90
or
180
rotations about a single axis. The normal field component
(B,) was calculated and, as in the previous section, both
high-precision and truncated data were then used to esti-
mate the susceptibility distribution. The algorithm at-
tempted a reconstruction using data simulated with just
five orientations of the cube and with all seven.
C. Varying the L ocation of the Magnetometer Array
Finally, the applied field (100 pT in the positive z-di-
rection) and the cube were fixed, while the measurement
plane (representing the magnetometer array) was moved
successively to locations 1 mm from each of the cube's
faces. T he normal f ield component in each case was c al-
culated and the algorithm was then tested with high-pre-
cision and truncated data.
IV. RESULTS N D
DISCUSSION
The performance o f the algorithm is summarized in Ta-
ble I . In each experiment, the deviation between the ac-
tual and predicted susceptibilities was calculated fo r each
volume element. The root mean sq uare (r .m.s.) error for
the sixty-four elements is taken as a figure of merit for a
particular ex periment, and the percentage error is calcu-
lated by dividing the r.m.s. error by the mean S.I. sus-
ceptibility of the sample. The percentage error
for
the
worst-case volume element is also quoted.
In all experimen ts, the predictions ba sed o n truncated
data were worse than those based on the high-precision
data, emphasizing the importance of minimizing mea-
surement noise. When the direction of the applied field
was varied (strategy
l ) ,
reasonable predictions were ob-
tained from the precise data but the truncated data caused
the algorithm to fail, yielding meaningless results. How-
ever, with the other experimental strategies the algorithm
proved to be robust to noise, producing errors
of
a few
percent while predictions based on precise data were in
error by only a few hundredths of a percent. Even the
worst-case elem ents were predicted with an error less than
35 when the sample was moved (strategy
2)
and less
than 20% when the measurement plane was moved (strat-
egy 3).
TABLE I
SUMMARYF T HE ALGORITHM'SREDICTIONSOR THREE XPERIMENTAL
STRATEGIES. TH ER .M .S . ER R O ROR ALL SIXTY-FOURLEMENTSND THE
ERROR
OR THE
WORST-CASE ELEMENTRE GIVEN
S
PERCENTAGES
OF
T HE
MEANSUSCEPTIBILITY. * SIGNIFIES THAT THE ERROR AS SO LARGEHAT A
PERCENTAGER R O R O U LD N O TBEEANINGFUL
High Precisio n Data Truncated Data
Root Worst-
Root Worst-
Mean Case Mean Case
Variable Square Element
Square Element
1)
Applied field:
1.6 7 . 8 -* -
- * -
6
Directions
2 . 1 % 1 4 . 1 % -*-
- * -
8 Directions
5 Orientations 0.016%
0.064
5 . 5 3 4 . 4
7
Orientations 0.009% 0.046 5 . 3
3 2 . 0 %
Plane:
6 Locations
0 . 0 1 3
0 .0 5 9 3 . 3 % 1 8 . 7 %
2) Sample:
3) Measurement
Variation of the ap plied field alone left the sam ple fixed
with respect to the measurement space and the results
demonstrate that relative motion between these two sys-
tems is essential. Without it, one subset
of
elements
is
always closest to the m agnetome ter and another subset is
always furthest away. In all four reconstructions based on
applied field variation, the fourteen most-accurately pre-
dicted elements resided in the layer closest to the mag-
netometer, and all sixteen closest elements were always
ranked in the top twenty.
When relative motion b etween the sample and m agne-
tometer was introduced either by rotating the sample
or
by moving the magnetometer, the algorithm performed
very well. When seven sample orientations were used in-
stead of five, there were significant improvements in the
predictions with both high-precision and truncated data,
as expected. In principle, any number of different orien-
tations could be used to improve performance, but the
coding would be more complicated if , for instance, the
sample were placed at an obliqu e angle to the measure-
ment plane. In both these experiments, it was the central
block of eight elements that was most difficult to predict.
In all six reconstructions, the fou r largest errors, and at
least six of the top eight, were associated with elements
from this central subset. This phenomenon is clearly il-
lustrated in Fig. 4, in which a gray-scale
(xwhite
+ 2 . 0
represent the susceptibilities in the actual and predicted
(truncated data with strategy
3)
distributions. Th e four
central elements in each of the two central layers suffer
the largest errors.
x s.1. with
Xblack
=
+2.5 x
s.1.) iS used to
V .
CONCLUDINGEMARKS
Magnetic susceptibility tomography is an imaging tech-
nique involving the recording of multiple scan sets while
varying the relative geometries of the three independent
systems: th e applied field, the sam ple and the measure-
ment plane. In the experiments described in this paper,
the matrix was overdetermined by taking 441 measure-
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5068
1 E E E T R A N S A C T I O N S O N M A G N E T I C S , V O L .
30, N O . 6,
NOVEMBER
1994
Fig. 4. Gray-scale representation of the actual susceptibility distribution
(uppe r sequence) and the susceptibility distribution predicted by the algo-
rithm based on truncated data obtained with strategy 3 (lower sequence).
In each sequence, the uppermost layer of the cube (see Fig. 2 ) is depicted
on the left with consecutive layers to the right. x = + 2 . 0
x
(S.1.) is
represented by a white square and x = f 2 . 5 X IO- ’
(S .1 . )
by a black one.
When the predicted susceptibility value fell outside this range, it was coded
in gray-scale according to the magnitude of the error so that, for example ,
a prediction of 1.8 X
would be represented by the same shade of gray
a s 2 .2
X
if the correct value was 2.0
X
ments in each of
5-8
scan sets (a total of about
3,000
data
points) in order to find the susceptibilities of 64 volume
elements. Singular value decomposition was then used to
solve the system of equations.
Three strategies were investigated, each corresponding
to fixing two
of
the systems while varying the other one.
The algorithm performed best when the measurement
plane was moved and also gave accurate predictions when
the sample was moved. When only the direction of the
applied field was varied, the algorithm failed to find the
correct solution. However, this might be rectified with a
spatially nonuniform field because that would introduce
greater differences between the Green’s functions, mak-
ing
a
solution possible. A nonuniform field would prob-
ably improve the algorithm’s performance under sample
rotation, also.
In summary, with two of the three strategies investi-
gated, the algorithm succeeded in accurately distinguish-
ing between two materials, whose susceptibilities differ
by only
5
X
l op6 S . I . ) ,
even in the presence of noise. It
might be argued that a
25
contrast is high in comparison
with X-ray tomographic techniques. However, the full
range of susceptibility values of diamagnetic and para-
magnetic materials is from
op4
o -
so that
the contrast used here corresponds to a much smaller frac-
tional difference. In addition, this preliminary analysis has
used only a very small number of different arrangements
of the field, sample and measurement plane.
Magnetic susceptibility tomography may have appli-
cations in medical science, geo physics and nondestructive
evaluation of materials. Knowledge of the three-dimen-
sional susceptibility distribution of an object may provide
information about iron stores in the liver [
181,
different
rock types in sedimentary or volcanic samples
[24],
and
voids or unwanted inclusions in critical engineering com-
ponents [23]. Further simulations, utilizing nonuniform
applied fields and a larger number of movements, are re-
quired to investigate spatial resolution and ability to dis-
criminate between materials with marginally different
susceptibilities, and to provide a mo re stringent test of the
algorithm’s robustness to measurement noise.
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Nestor G. Sepulveda was born in Barbosa, Colomb ia in 1946. He received
the B.E.E.E. degree from the Universidad Distrital, Bogota in 1970, and
the M.S. and Ph.D. degrees in biomedical engineering from Tulane Uni-
versity in 1981 and 1984, respectively.
In 19 84, he was appointed as a Research Assistant Professor in the De-
partment
of
Physics and Astronomy at Vanderbilt University.
Ian
M.
Thomas
was born in London, England in 1961. He received the
B.Sc. degree in physics from Imperial College (University of Lo ndon) in
1985, the M.Sc. in bioengineering from the University of S trathclyde in
1986 and the Ph.D. in physics from the Open University in 1991.
His present appointment is that
of
Research Associate in the Department
of Physics and Astronomy at Vanderbilt University.
John P. Wikswo, Jr. was bom in Lynchburg, Virginia in 1949. He re-
ceived the B.A. degree in physics from the University of Virginia, Char-
lottesville in 19 70 and the M.S. and Ph.D . degrees in physics from Stan-
ford University in 197 3 and 19 75, respectively.
He was a Research Fellow in cardiology at the Stanford University School
of Medicine from 1975 to 19 77, at which time he joined the faculty of the
Department of Physics and Astronomy at Vanderbilt University. In 1992,
he was appointed the A. B. Learned Professor of Living State Physics.