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-

0018-9464/94$04.00 994 IEEE



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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.



5064

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



SEPULVEDA et

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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-



5066

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-



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.

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Magnetic Susceptibility Tomography