IEEE
TRANSACTIONS ON
MEDICAL IMAGING, VOL.
11.
NO.
1,
DECEMBER
1992
457
Plana and Cylind ical Ac i e Mic owa e
Tempe a u e Imaging: Nume ical
Simula ions
J.
M.
Rius,
S uden
Membe , IEEE,
C.
Picho ,
L.
Jo e,
Membe , IEEE,
J.
C.
Bolomey,
N.
Joachimowicz,
A.
B oque as,
Membe , IEEE,
and
M.
Fe ando,
Membe , IEEE
Abs uc -
Ac i e mic owa e imaging sys ems o biomedical
appli ca ions ha e been gaining a en ion in ecen yea s. The need
o i de nal s uc u e in o ma ion imposes he use o di ac ion
omog aphy app oaches.
Two
di e en geome ies, plana and
cylinci ical, ha e been used o his pu pose. In his pape
we p esen a compa a i e s udy a
2.45
GHz
conce ning bo h
measw emen and econs uc ion pa ame e s o he wo con igu-
a ions. Fo he sake o compa ison, a nume ical model consis ing
o w 3 nonconcen ic cylinde s is conside ed and econs uc ed
using bo h geome ies om simula ed expe imen al da a. The
sca e ed ields and econs uc ed images allow o ex ac e y
use ul in o ma ion abou dynamic ange, sensi i i y, esolu ion
and quan i a i e image accu acy o he choice o he con igu a-
ion ,in a pa icula applica ion.
Th l geome ies chosen a e plana and cylind ical a ay con-
igu ai ions. Bo h geome ies can measu e o wa d and backwa d
sca e ed ields. The backsca e ing measu emen imp o es he
image esolu ion and econs uc ion in lossy mediums, bu , on he
o he hmd, has se e al dynamic ange di icul ies. This adeo
be ween only o wa d and o wa d-backwa d ield measu emen
is analyzed using a plana a ay o o wa d sca e ing and a
cylind icaE one o o wa d-backwa d.
As
di e en ial empe a u e imaging is a weakly sca e ing
p oblem, Bo n app oxima ion algo i hms can be used. The sim-
plici y o Ba n econs uc ion algo i hms and he use o FFT
make hem e y a ac i e o eal- ime biomedical imaging
sys ems
[19].
I.
INTRODUCTION
IFFERENT ypes o adia ion a e ac ually used o non
D
in asi e es ing o biological bodies. Each o hem has i s
own speci ic a ibu es, and he econs uc ed image depends
on he way in which he adia ion in e ac s wi h ma e
[l]
The mic owa e egion o he elec omagne ic spec um o -
e s he access o an in o ma ion which is ela ed
o
he elec ic
p ope ies o he ma e . These p ope ies, usually exp essed as
Manusc ip ecei ed Deembe
3,
1990;
e ised Ap il 27, 1992. This wo k
was suppo ed by CAICYT (Spanish Commi ee o Scien i ic and Technical
Resea ch unde G an
116S-84),
FISS (Spanish Na ional Ins i u e o Heal h
unde G an 84/21 12), Spanish-F ench Coope a ion P og am (G an 301135)
and Spanish-B i ish Coope a ion P og am (G an 171173). This wo k was
also
suppo ed by he Eu opean Commission
o
he
CEE
in he ame
o he Collabo a ion COMAC-BME P ojec “Op imiza ion
o
Hype he mia
Technology and Assessmen
o
Clinical E icacy in T ea men
o
Cance .”
J.M. Rius,
L.
Jo e, A. B oque as,
and
M. Fe ando a e wi h E.T.S.E.
Telecommunicaci6, Uni e si a Poli kcnica de Ca alunya, Apdo. co eos
30002,
08080
Ba celona, Spain.
C. Picho , J. C. Bolomey, and N. Joachimowicz, a e wi h Labo a oi e des
Signaux e Sys emes (CNRSIESE), Ecole SupC ieu e d’Elec ici e, Pla eau de
Moulon, F91192 Gi -su -Y e e Cedex, Pa is, F ance.
IEEE Log Numbe 9202263.
a complex pe mi i i y (pe mi i i y and conduc i i y), ansla e
in mac oscopic e ms he ine s uc u e o he issues a
he molecula le el, and he e o e depend on i s physical o
physiological ac o s as well as he in e oga ion equency
[2].
The mic owa e p opaga ion in biological issues shows
some speci ic p ope ies.
-The a enua ion is high, in pa icula o he issues wi h
a high wa e con en as muscles, li e , kidney. Howe e , so
issues such as a , lungs and bones p esen lowe mic owa e
abso p ion
[3].
I
is in hose so issues whe e ul asound
ene gy su e s he g ea es a enua ion.
-A he same equency, he wa eleng h in he issues a e
educed up
o
eigh imes compa ed
o
he ai
[4].
-Unlike X- ay imaging, whe e he wa eleng h is e y
small as compa ed
o
he magni ude o in e es ing objec
de ails, o mic owa es, e ec s o wa e p opaga ion can
no
be explained in e ms o ays, and di ac ion e ec s ha e
o
be conside ed
[5].
Ac i e mic owa e imaging sys ems ha e no been de el-
oped o use in biosys ems due
o
a numbe o appa en
limi a ions ela ed wi h he esolu ion and pene a ion.
Al-
hough he e was a i s a emp a he beginning o he
70’s [6],
in ac hey we e he i s expe imen s o La sen and Jacobi
[7],
[8],
a he end o he decade, ha show he po en iali y o he
me hod. The basic p inciple employed consis s o imme sing
he a ge in wa e and wo king in he lowe egion o he
mic owa e spec um
(1-10
GHz.).
The mos signi ican poin s
o he p ocedu e a e he ollowing.
a) Wa e , being a medium wi h dielec ic cons an close
o
hose o human body, imp o es he coupling and pene a ion
o he elec omagne ic adia ion in o he body.
b) The esolu ion o di ac ion-limi ed imaging sys ems is
de e mined by he wa eleng h o he in e oga ion adia ion.
When ope a ing in wa e , wi h a ela i e pe mi i i y close
o
75,
he e is a con ac ion in wa eleng h and he e o e in
esolu ion close
o
9.
c) The high a enua ion o wa e a hese equencies a oids
he mul ipa h p opaga ion ex e nal
o
he in e oga ed body.
Wa e is no he only possible ma ching medium. Fo
ins ance, a issues could be be e ma ched using lowe
pe mi i i y liquids
[9].
The choice he ope a ing equency is a adeo be ween
esolu ion and dep h o pene a ion. The equi emen o pen-
e a ion wi h deep-lying o gans a gues o a lowe equency
0278-0062/92$(13.00
0
19’92
IEEE
458
IEEE
TRANSACTIONS
ON
MEDICAL IMAGING,
VOL.
1
I,
NO.
4,
DECEMBER
1992
o
ope a ion. In con as , di ac ion-limi ed imaging sys ems
equi e high ope a ing equencies o enhanced esolu ion.
The
1
o
6
GHz egion appea o be op imum o mic owa e
imaging o he human body [lo].
The i s s udies we e de o ed o p ojec ion imaging
[7],
[
111,
[
121. Bu , apidly, omog aphic econs uc ion p ocesses
we e conside ed. A i s app oach
(131,
close o X- ays
CAT, ailed o s ongly inhomogeneous media. Howe e , he
di ac ion e ec s ha e o be aken in o accoun [14],
[15]
in
o de
o
ob ain good quali y images.
The
use
o mic owa e adia ion in imaging biological
s uc u es has been gaining a en ion ecen ly. The in e es ely
on i s low cos and low complexi y echnology, as well as in i s
capaci y o disc imina e di e en issues (pe mi i i ies anges
om
5
o
50,
compa ed o he ew pe cen o densi ies in
so issues) and o de ec physiological changes,
as
solu e
concen a ion, pe usion, blood low a e and empe a u e
[
161-[19]. Mo eo e , he pho on ene gy o mic owa es is
su icien ly low o p e en any ioniza ion o occu in biological
issue a o dina y in ensi y le els.
The illumina ion and measu emen geome y depend on
he pa icula applica ion and on he echnology a ailable.
In his pape , by using a compu e model, we compa e he
mos signi ican pa ame e s o he wo basic con igu a ions
ha ha e mainly been used in ac i e mic owa e imaging: he
plana geome y
[18]
and he cylind ical geome y
[20].
The
model consis s o wo nonconcen ic cylinde s wi h di e en
pe mi i i y alues, simula ing a gene al biological body.
Sec ion
I1
ou lines he basics o he o mula ion o he wo
geome ies.
An
e o has been made ying o enhance he
common s eps o bo h solu ions.
The hi d sec ion is de o ed o compa e he di e en mea-
su emen and econs uc ion pa ame e s o he wo geome ies
on he basis o a common simula ed s uc u e.
11.
TOMOGRAPHIC FORMULATION
Imaging echniques consis o illumina ing he objec wi h
an inciden adia ion, measu ing he di ac ed ields, and
ob aining an image o he dis ibu ion o he p ope ies
o
he
objec in e ac ing wi h he illumina ing ene gy. Two kinds
o
mic owa e omog aphic sys ems will be o mula ed, di e ing
on he geome y o inciden wa e and he measu emen
an enna.
Tomog aphy wi h di ac ing ene gy, such as mic owa es,
mus
be modelled wi h a second-o de linea di e en ial
equa ion, he inhomogeneous wa e equa ion, which desc ibes
he elec omagne ic ield dis ibu ion in he objec o be
imaged. Since compu ing he objec image om he sca e ed
ield measu emen s is an ex emely di icul p oblem o sol e,
a i s -o de app oxima ion has o be used,
so
ha only weakly
sca e ing objec s can be econs uc ed wi h easonable e o .
When inciden adia ion is concen a ed
on
a slice o he
objec , and wi h objec slowly a ying in he hi d dimension,
a bidimensional model can be used: dielec ic objec s and
elec omagne ic ields will be assumed o ha e p ope ies
a ying only o e he
2-y
plane c oss sec ion. Using e ically
pola ized an ennas in he z-axis di ec ion, all elec ic ield and
cu en ec o s will be z-di ec ed, and acco dingly we can use
scala ield equa ions. The wo dimensional model is alid
only i he objec is ac ually uni o m in he z-di ec ion inside
he slice in which illumina ing ene gy is concen a ed. The
econs uc ed objec will be he
(T.
y)
image o his slice.
A.
Spec al
Fo niula ion
o
Di ac ion P oblems
The physical p ope ies o he objec which in e ac wi h
mic owa e adia ion a e he elec ical pe mi i i y and con-
-
duc i i y. Since di ac ion heo y is based on cohe en iel.ds,
phaso o mula ion o ields and cu en s will be used wi h he
ime dependence
pJw .
Complex pe mi i i y and wa enumbe a e hen de in ed as
ollows:
-complex dielec ic pe mi i i y o he objec unde . es :
-complex dielec ic pe mi i i y o ex e nal medium:
-complex wa enumbe in he ex e nal medium:
=
wm,
whe e
po
is he acuum magne ic pe meabili y.
Le
E(?')
ep esen he o al elec ic ield and
E'(
F)
he
inciden ield, ha is
o
say, he ield in he absence
o
he
objec . The sca e ed ield is de ined as
4.');
0(F);
ES(F)
=
E(?')
-
I?(?')
1)
and can be conside ed as he ield gene a ed by he eqni alen
elec ic cu en adia ing in he ex e nal medium:
whe e
G(F-
7)
is he G een's unc im, and
(3)
he equi alen induced elec ic cu en .
en is de ined h ough he equa ion
The bidimensional Fou ie ans o m o he :qui alen cu -
(4)
I
he objec complex pe mi i i y
is
dose
o
ha o he
su ounding medium, he sca e ed ield is negligible in on o
he inciden one (weak sca e ing) and he Bo n app oxima ion
can be applied:
(F)
N
€0
--
E(?')
N
I?(?).
(5)
When he objec is illumina ed by a plane wa e inciden a
an angle
00,
using a uni ec o
00
in his di ec ion, we ha e
(6)
E";)
=
e-3Ko&
7.
Equa ion (4), e alua ed a
I
I?
I
=
KO,
can be exp essed as:
ql')
=
-jw oC
-(
I?
-
K0ijO
11-1
K
=
K"
(7)
RlUS
e
U/.:
MICROWAVE TEMPERATURE IMAGING: NUMERICAL SIMULATIONS
459
Fig.
1.
Fou ie domain o an inciden plane wa e.
Unde
Bo n app oxima-
ion, he bidimensional Fou ie ans o m o he con as p o ile is ob ained
in he Fou ie domain o e a ci cum e ence shi cd
on
he di ec ion opposi e
o he inciden
plane
wa e.
whe e
C(g
-
Kndo)
FT[C(F)]
C(~) 'I(I;.---oH ,)
i(jF
K
=
KO
(8)
=U
1-1
is he bidimensional Fou ie ans o m o he con as p o ile
de ined
as
(F)
C(F)
=
1
-
~
0
(9)
e alua ed o e ci cum e ences o adius
Kc,
and cen e
-KoHo
in
he Fou ie space (Fig.
1).
When he su ounding medium is lossy, he wa enumbe
KO
is complex, and he Fou ie ans o m becomes a Laplace
one. Howe e , in o de o use he
FFT
algo i hm he complex
wa enumbe mus be app oxima ed by i s eal pa . This
in oduces an a enua ion o he highe spa ial equencies,
which p oduces "high- equency" noise in he econs uc ed
image.
Fo
ha eason, some low-pass il e ing o he econ-
s uc ed image is necessa y when backsca e ing in o ma ion
is p ocessed, because
i
is associa ed o he highe spa ial
equencies in he objec spec um.
As
he bidimensiona: Fou ie ans o m
o
he equi alen
cu en e alua ed a
I
K
I=
KO
can be easily ob ained om
sca e ed ield measu emen s, he con as p o ile
o
a
weakly
sca e ing objec will be econs uc ed h ough
(7).
The econs uc ion
(7)
gi es in o ma ion o he
2-D
Fou ie
ans o m o he con as p o ile only o e
a
ci cum e ence
shi ed in di ec ion opposi e
o
he p opaga ion o he inciden
ield. Thus, di e en di ec ions o incidence mus be consid-
e ed in o de o ob ain a bidimensional sampling inside a ci cle
in he spec al domain (Fig.
2).
Acco dingly, he econs uc ed
Fig.
2.
Mul i iew Fou ie domain.
I
mul iple iews a e aken o di e en
di ec ions
o
illumina ion, he bidimensional Fou ie ans o m o he con as
p o ile can be ob ained inside a ci cle o adius
21io
cen e ed a he o igin
o he Fou ie domain
inciden
plane
wa e
Fig.
3.
Plana sys em measu emen geome y ( ansmisison con igu a ion).
On
plana geome y sys ems, he objec is illumina ed by a plane wa e and
he sca e ed ield
is
measu ed along a p obing line pa allel o he inciden
wa e on . I ano he p obing line is loca ed a
-,I,",
bo h o wa d and
backsca e ed ields can he measu ed.
con as p o ile ob ained h ough in e sion o
C(k)
will be
a
low-pass
il e ed e sion o he o iginal one.
R.
Plana
Geome y
A omog aphic sys em wi h plana geome y
uses
a plane
wa e inciden ield
Ei
(80;
F)
p opaga ing on di ec ion
00,
o
simplici y
Ho
=
Oo,
and he o al ield
E(z,.y)
is measu ed
along he line
z
=
icg
pa allel
o
he inciden wa e on
(Fig.
3).
Mechanical o a ion o he objec a ound z-axis
p o ides di e en di ec ions o incidence ela i e
o
he objec .
The Fou ie ans o m o he equi alen cu en can be
ob ained om he measu ed o al ield h ough he Fou ie
ans o m o
(2)
along y-di ec ion. A e some manipula ion,
see
[
181,
he ollowing exp ession can be ob ained:
E"(: ;o.
K,y)
=
-jw,aoG(
J-)
,'(
JW,
K,)
(10)
460
IEEE
TRANSACTIONS
ON
MEDICAL
IMAGING,
VOL.
1
I,
NO.
4,
DECEMBER
1992
whe e
i”(Ky)
and a e he angula spec um
o he sca e ed ield and he G een’s unc ion, espec i ely,
Ula)
(1lb)
being
K,
and
Ky
he Ca esian componen s o he eal
app oxima ion o he p opaga ion ec o ela ed o each o
he plane wa e componen s o he angula spec um
I?
=
(K,.K,)
K,
=
KocosQ
K,
KO
sin
0.
Unde Bo n app oxima ion, he 2-D Fou ie ans o m o
he con as p o ile is ob ained h ough
(7),
(lo), and (11):
When he plana omog aphic sys em measu es only he
o wa d sca e ed ield h ough he objec , only he componen s
o he angula spec um o he sca e ed ield p opaga ing
owa ds posi i e
z(Kx
>
0)
a i e a he measu e line. These
spec al componen s a e associa ed
o
he igh hal o he
ci cle
(K,
>
0)
in he spec al domain (Fig.
4).
Thus, o a ion
o he objec p o ides he 2-D Fou ie ans o m o he con as
p o ile inside o ci cle o adius
iK0
(Fig.
5)
[18],
[19].
I
ano he p obing line is loca ed a
-20
(Fig.
3),
bo h o wa d
and backsca e ed ields can be measu ed. In his case, which
is no analyzed in his pape , o a ion
o
he objec p o ides
2-D Fou ie ans o m o he con as p o ile inside a ci cle
o adius
2K0
as indica ed in Fig.
2.
The simplici y o he
econs uc ion algo i hm and he use o
FFT
make
i
e y
a ac i e o eal- ime imaging sys ems
[
191.
I
4-
-KoOo
Fig.
4.
Fou ie domain o
1
iew plana sys em. I only o wa d sca e ed
ield is measu ed on plana geome y sys ems, he bidimensional Fou ie
ans o m o he con as p o ile is ob ained only o e hal o he shi ed ci -
cum e ences. I backsca e ed ield is also measu ed, he whole ci cum e ence
is ob ained.
-JZK
C.
Cylind ical
Geome y
In cylind ical omog aphic sys ems he measu emen is pe -
o med no by a s aigh line, bu by a ci cula a ay o iso opic
an ennas (Fig.
6).
Di e en di ec ions o illumina ion can be
achie ed by ansmi ing wi h each one o he omnidi ec ional
elemen s,
so
ha mechanical o a ion o he objec is a oided.
Al hough he inciden ield is no a plane wa e, bu a
cylind ical one, and he measu e an enna is a ci cula a ay, he
spec al o mula ion de eloped o plana geome y imaging
sys ems can be applied i a syn he ic ape u e app oach is
made.
As
he ci cula a ay can measu e, in p inciple, bo h he
o wa d and backwa d sca e ed ield, he Fou ie ans o m
o
he con as p o ile is ob ained o e he en i e shi ed ci cle
in he spec al domain, and be e esolu ion is achie ed wi h
o wa d-backsca e ing algo i hms due
o
a less igo ous low-
pass il e ing han wi h o wa d sca e ing sys ems. Howe e ,
in p ac ice, backsca e ed ields a e di icul
o
measu e nea
Fig.
5.
Fou ie domain o
8
iew ansmission plana sys em. When mul i-
ple iews a e aken on a ansmission plana geome y sys em, he bidimen-
sional Fou ie ans o m
o
he con as p o ile
is
ob ained inside a ci cle o
adius
&lio
ins ead o
21;”.
he ansmi ing elemen , and esolu ion is no as good as
heo e ically expec ed.
Using a ecip oci y heo em o mula ion, he bidimensional
Fou ie ans o m o he equi alen cu en can be easily
ob ained om he sca e ed ield measu ed along an a bi a ily
shaped cylind ical an enna, p o iding ha plane wa es in any
di ec ion can be syn hesized by cu en s on he ecei ing
an enna, ac ing as ansmi e . The ecip oci y heo em s a es
ha
RIUS
e
al.:
MICROWAVE
TEMPERATURE
IMAGING: NUMERICAL
SIMULATIONS
~
46
1
Y
ecei e
ansmi e '
e&
Fig.
6.
Cylind ical geome y sys em.
Fo
cylind ical geome y sys ems, he
measu emen
is
achie ed wi h a ci cula a ay
o
omnidi ec ional an ennas.
Di e en di ec ions
o
illumina ion can
be
achie ed
by
ansmi ing wi h each
one
o
he omnidi ec ional elemen s.
whe e and
51
a e he elec ic cu en s exis ing in olumes
71,
and
'ub,
which adia 2elec ic ields and
,!?h,
espec i ely.
I we assume ha
Jh
is he elec ic equi alen cu en; in
he objec , induced by he inciden ield
E'
(e
Ho),
and
J,,
is
he elec ic cu en on he an enna ac ing as ansmi e ha
adia es a plane wa e izld p opaga ing along he uni Xec o
8,
hen we ha e ha
-7,
adia es he sca e ed ield
E"(?),
which is measu ed along he an enna, and ha
.xL
adia es he
ield
e-3~oo.~.
F om (3) and (9) we ge
subs i u ing
unde Bo n app oxima ion
in (13), and conside ing
(7)
and (8), we ob ain
C(Ko(8
-
Fo a ci cula an enna o adius
R,
cen e ed in he
:E-y
plane,
(15) can be exp essed as
C(K"(Q
-
whe e
1(0
-
a)
is he ampli ude o he cu en dis ibu ion
a angula posi ion
0
along he an enna, ha , when ac ing
as ansmi e , adia es
inside
he
sou ce
ing
a plane wa e
p opaga ing a an angle
0
in he di ec ion
8
When he inciden ield is a cylind ical wa e, we can expand
he plane wa e
E'(<
0")
as a supe posi ion o cylind ical
wa es
E'(<
a")
gene a ed by line sou ces
I(&
-
00)
loca ed
as posi ions
00
along he ci cula an enna. Then i ollows ha
Ez
(<
8,)
=
1
I(&
-
a")E'(( ';
0o)R
dao
(18)
and he e o e he ield sca e ed om an inciden plane wa e,
measu ed along he ci cula an enna can be exp essed as
E"(c ;&)
=
i
I(&
-
ao)E"(c~:o~)Rdc~
2.
2T
(19)
whe e
E"(a;
00)
is he sca e ed ield measu ed a posi ion
c ,
gene a ed by an inciden cylind ical wa e p oduced by a uni
sou ce loca ed a posi ion
00
(Fig.
5).
Finally, om
(16)
and
(19)
we ob ain
.
1(0
-
a)I(0o
-
00)
d0dao.
(20)
Equa ion (20) has he ollowing physical in e p e a ion:
E"(0:
00)
is con ol ed wi h he cu en ha , exis ing along
a ci cula a ay, gene a es a plane wa e. This con olu ion
ans o ms he ci cula geome y in o a Ca esian one, o
bo h ansmi e and ecei e a iables
o
and
00.
As
wi h
plana con igu a ion algo i hm, his equa ion can be easily
implemen ed by means o Fas Fou ie T ans o ms (FFT),
which allows eal- ime econs uc ions
[
191, [20]. Fu he
in o ma ion abou he algo i hm heo y and implemen a ion
can be ound in [21].
111.
NUMERICAL
RESULTS
This sec ion compa es he ela i e pe o mance o he
o wa d and o wa d-backsca e ing measu emen using, e-
spec i ely, plana and cylind ical geome y econs uc ion
algo i hms. The sca e ed ields ha e been compu ed in bo h
cases by momen me hod, sol ing he linea sys em o equa-
ions by conjuga e-g adien i e a i e algo i hm.
The so wa e used
o
compu e sca e ed ields, o bo h
geome ies, and o econs uc he objec image o plana
geome y ha e been de eloped a Labo a o ie des Signaux
e Sys kmes, Ecole SupC ieu e d'Elec ici C, Gi -su -Y e e,
while he cylind ical geome y econs uc ion p og am has
been de eloped a
AMR
g oup, Depa men o Signal Theo y
and Communica ions, Poly echnic Uni e si y o Ca alonia,
Ba celona.
Pos p ocessing algo i hms ha e been applied o imp o e
image quali y when necessa y.
-Ge chbe g-Papoulis algo i hm has been applied o plana
geome y econs uc ions when he numbe o iews is small.
-Cylind ical geome y econs uc ions mus be low-pass
il e ed in o de o emo e high- equency noise due o
backsca e ed ield measu emen s.
A.
Nume ical Model
One o he possible applica ions o mic owa e omog aphy
conce ns medical imaging.
As
human body is composed
mainly o wa e , and he complex dielec ic pe mi i i y o

462
IEEE TRANSACTIONS ON MEDICAL IMAGING,
VOL.
1
I,
NO.
4,
DECEMBER
1992
5
cm
-
SCATTERED
FIELD
>
X
Fig.
7.
Simula ed model o biological objec . Fo compu e simula ion
pu pose, a e y ough biological objec
is
modeled
by
a ci cula egion
o
wa e
o
adius
4
cm a
37OC
imme sed in wa e a
25OC.
A
local empe a u e
change inside he body
is
simula ed by a sma!le . nonconcen ic cylinde
o
wa e
o
adius
1
cm a
38OC
o
41OC.
TABLE
I
Complex pe mi i i y and con as p o ile
o
wa e
Complex pe mi i i y Con as p o ilc
(2.jo)
=
77.27
-
,j9.29
~(37")
=
73.49
-
jG.35
~(38")
=
73.13
-
,jG.35
~(41")
=
72.34
-
j.j.88
IC(2Y)J
=
0
(C(37")I
=
0.0591
lC'(3s")I
=
0.OG4G
IC(4l")i
=
0.07G5
wa e a ies linea ly wi h empe a u e, empe a u e a ia ions
inside human body can be de ec ed and localized. This sug-
ges s an applica ion
o
hype he mia moni o ing and umo
de ec ion.
A
comple e se o compu e simula ions has been pe o med
o
es he capabili y
o
bo h algo i hms
o
de ec localized
empe a u e a ia ions inside wa e .
A
e y ough biological
objec is modelled by a ci cula egion
o
wa e a 37°C
imme sed in wa e a 25°C.
A
local empe a u e change inside
he body is simula ed by a smalle , nonconcen ic, cylinde o
wa e a 38°C o 41°C (Fig. 7). We ha e used an ope a ing
equency o 2.45 GHz
[19],
[20],
wi h a wa eleng h o
1.39
cm in wa e a 25"C, which is a good adeo be ween
a enua ion and esolu ion.
1)
Model Pa ame e s:
Ou model is a weakly sca e ing
objec , because
i s
con as p o ile is low and i s dimension
no
oo
la ge compa ed o he wa eleng h. Thus, o al ield
is simila
o
inciden ield, and Bo n app oxima ion can be
applied. The complex pe mi i i y and con as p o ile o wa e
a 2.45 GHz o di e en empe a u es a e lis ed in Table
I.
In his ange, he pe mi i i y sensi i i y wi h empe a u e is
almos linea , equal
o
a€'/€;
-
=
-3.9876
10-"c-'
AT
2)
Plana Geome y Simula ion Pa ame e s:
Leng h o measu emen line:
Numbe o ecei ing an ennas: 67
Numbe o iews:
64
Spacing be ween ecei ing an ennas:
31.6 cm
4.7
mm
=
X/3
-
-Inc.
y
axis
deg
CYLINDRICAL
Fig.
8.
Sca e ed ield. Magni ude
o
he p obing sca e ed ield in dB
o
bo h plana and Cylind ical geome y
o
local empe a u e change inside he
cylinde
o
4'.
Two di e en di ec ions o incidence ha e
been
conside ed,
along . -axis
(0')
and !/-axis
(OOO).
A
minimum dynamic ange o
30
dB
o he cylind ical case and
45
dB o he plana
one
is needed o de ec he
di e ence be ween he wo iews.
Dis ance
o
he cen e o he objec : 5 cm
Dimension o econs uc ed image ma ix:
67
X
67
3)
Cylind ical Geome y Simula ion Pa ame e s:
Numbe o ansmi ing an ennas:
64
Numbe o ecei ing an ennas: 64
Radius
o
he ci cula a ay: 12.5 cm
5.625"
=
1.23x
64
x
64
Spacing be ween ecei ing an ennas:
Dimension
o
econs uc ed image ma ix:
B.
Sca e ed
Field
Fo compa ison pu pose he inciden ield a he cen e
o
he
objec is ixed: 0.5 V/m in ampli ude, ze o deg ee in phase.
Fig.
8
shows he magni ude
o
he p obing sca e ed ields o
bo h geome ies. Two di e en iews
o
he objec ha e been
plo ed o di e en di ec ions o incidence along he x-axis
(0")
and y-axis
(90").
Fo he plana case he wo cu es o e lap, because he
p obing line is always in he same ela i e posi ion
o
he
inciden wa e, while o he cylind ical case he wo plo ed
cu es a e simila in shape, al hough shi ed
90".
Fig.
9
shows he magni ude o he di e en ial ields, i.e.
he di e ence be ween he sca e ed ields o a empe a u e
change o 4°C and he sca e ed ields o no empe a u e
change.
463
RIUS
e
al.:
MICROWAVE TEMPERATURE IMAGING. NUMERICAL SIMULATIONS
DIFFERENTIAL
FIELD
(41
C
-
37
C)
bo
-10
-20
53
4
I/,',
,,
,
,
,
,
,
,,
, ,,
,
,
,
,
,,
-18
-12
a
4
o
4
8
12
is
-SI
PLANAR
cm
50
o
60
120
en
240
m
360
a
'
,,,.,'
..I.
m...
I
,,I
CYLINDRICAL
d=g
Fig.
9.
Di e en ial sca e ed ield
(4loC-37OC).
Magni ude
o
he di e -
ence be ween he sca e ed ield measu ed o a
4OC
empe a u e change
and
he sca e ed ields o no empe a u e change. Two iews along
.I'
and axis
a e
shown.
A
minimum dynamic ange
o
35
dB is needed
o
he cylind ical
case, and
60
dB
o
he plana one. The la e can be educed down o
40
dB
wi h
a
sho e p obing line.
Fo p ac ical pu poses, only he p obing ield co esponding
o
a ce ain dynamic ange, i.e.,
30
dB, has a clea in luence
on
he econs uc ed image. Because o he di e en beha io
o he wo measu ed pa e ns, he use ul angula ex end
o
he
measu ed ield o he cylind ical geome y will be la ge han
he equi alen one o he plana case.
This can be explained as ollows.
The di e en ial ield can be conside ed as he ield sca e ed
by he empe a u e change. Fo he plana case, he empe a u e
change loca ion is close
o
he cen e o he measu e line and
a away om he ends. The maximum di e ence o dis ances
is abou
13
cm, leading
o
an a enua ion di e ence o abou
30
dB [Fig. lO(a)].
In
he cylind ical case, he empe a u e change loca ion
is nea he cen e
o
he ci cula a ay, and he e o e he
dis ances
o
each one o he ecei ing elemen s a e simila . The
g ea es dis ance di e ence is abou
4
cm, and he a enua ion
di e ence is abou
9
dB [Fig. 10(b)].
On he o he hand, he la ge numbe o elemen s in he
cylind ical a ay which measu e ields wi hin he speci ied
dynamic ange, p o ide a la ge amoun o in o ma ion
o
be
p ocessed, which esul s in a sligh ly be e spec al padding
and image quali y, a he p ice o a longe p obing line.
C. Recons uc ed Images
I)
Absolu e Recons uc ions:
Fig.
11
shows he econ-
I
(
h)
Fig.
10.
Sca e ed ield a enua ion. The dis ance om he empe a u e
change loca ion o each one
o
he ecei ing elemen s
o
he measu e a ay
is app oxima ely cons an in he cylind ical geome y sys em [Fig. 10(b), bu
no in he plana one [Fig.
lO(a)].
Thus, he sca e ed ield a i ing a he
elemen s close o he ends o he measu e
line
is s ongly a enua ed compa ed
o he ield a he cen e o he a ay. Fo ha eason, he plana geome y
sys em equi es
a
dynamic ange
o
abou
20
dB g ea e han he cylind ical
geome y one.
s uc ed con as p o iles o empe a u e changes o
0,
1,
and
4°C.
Plana geome y econs uc ion and cylind ical one a e
e y simila .
I
mus be no iced ha cylind ical con igu a ion
has a la ge spec um co e age due
o
he back-sca e ed ield
measu emen : ci cle o adius
2Ko
while only
iKo
o he
plana one. Acco dingly, in heo y he spa ial esolu ion o an
ideal backsca e ing sys em is be e by a
i
ac o .
Howe e , images ob ained by he cylind ical con igu a ion
om o wa d and backwa d sca e ing ield measu emen s a e
no
so
be e [see Fig.
111
o he ollowing easons.
-The backsca e ed ield a elemen s e y close
o
he
ansmi ing one canno be measu ed, and is app oxima ed by
ze o.
As
backsca e ed ields co espond
o
high equencies
in he Fou ie domain o he econs uc ed objec (Figs.
1
and
2),
some spa ial esolu ion is los .
-When he FFT algo i hm is used
o
compu e he
2-D
in e se Fou ie ans o m o he objec spec um
C
(K)
,
he
wa enumbe
I?
is assumed o be eal alued. I he medium
is lossy, some e o is in oduced in he highe equencies
o he objec spec um. As he backwa d ield measu emen s
a e associa ed wi h his highe spa ial equencies, a low-pass
image il e mus be applied and some esolu ion is los again.
This il e ing is no necessa y when only o wa d sca e ing
measu emen is pe o med.
464
IEEE TRANSACTIONS ON MEDICAL IMAGING,
VOL.
11,
NO.
4,
DECEMBER
1992
m
75
74
73
CONTRAST
PROFILE
~
23
0.018
L
0.018
1
;',
72
71
0.06
0.05
0.M
0.m
0.m
0.01
0
$1
$1
'" ''-
PLANAR
an
-10
a
B
4
I
o
z
4
e
e
10
CYLINDRICAL
an
CONTRAST
PROFILE
Fig.
11.
Recons uc ion o con as p o ile o empe s u e changes o
OOC,
l0C,
and
4OC.
Plana geome y and cylind ical geome y econs uc ions a e
e y simila -al hough he cylind ical geome y should be in p inciple be e
by a ac o
&
due o he backsca e ed ield measu emen .
..-.
E
0
>
DIELECTRIC
PERMllTlVlTY
- --?
P
::
,'.,,,
I/
-
'
x
(an)
Fig.
12.
Recons uc ion o dielec ic pe mi i i y on cylind ical sys em.
As
a conclusion, he image esolu ion ob ained wi h he
o wa d sca e ing plana con igu a ion and he cylind ical one
a e e y simila , al hough he la ge spec um co e age due o
he backsca e ing measu emen in he cylind ical case.
Cylind ical geome y econs uc ion is pe o med aking
in o accoun he lossy medium wi h complex
KO
in (ZO),
so
ha he co ec alue o he objec spec um
6'
is ob ained
and quan i a i e alues o he con as p o ile a e ob ained
wi h e y good app oxima ion. The complex pe mi i i y o
he image can be easily de i ed om
(9)
(see Fig. 12),
and a empe a u e map image can be ob ained om he
app oxima ely linea ela ionship be ween con as p o ile and
empe a u e (Fig.
13).
The esul s o Fig.
11
ha e been ob ained by he plana
algo i hm using
K,
eal alued in (12). Thus, he highly
a enua ed ampli ude o he sca e ed ield due o he lossy
medium is supposed o be caused by a less di ac ing objec
and he econs uc ed alue o he con as p o ile is lowe
han he o iginal. The e o e, he plana geome y algo i hm
wi h
K,
eal in
(12)
p o ides only a quali a i e image o he
con as p o ile (Fig.
11).
2)
Loss
Co ec ion in Plana Geome y Algo i hm:
When a
quan i a i e econs uc ion in a lossy medium is desi ed, a loss
co ec ion mus be pe o med in plana geome y algo i hm.
This
is
done by aking
K,
complex alued in (12). Wi h his
co ec ion, each componen o he angula spec um o he
measu ed sca e ed ield is ampli ied by a exponen ial ac o
equal
o
he a enua ion in he lossy medium. This may be a
majo p oblem in an expe imen al sys em, because he noise
ela ed o signals a i ing a bo h ends o he a ay is ampli ied
RlUS
e
al.:
MICROWAVE TEMPERATURE IMAGING: NUMERICAL SIMULATIONS
465
78-
n-
78-
75
74
73-
72,o
TEMPERATURE MAP
COMPLEX
PERMITTIVITY
model
-30
c
-
~
-'I
"
.a
d
4
-2
0
2
4
6
8
10
-104
6
4
-2
0
2
4
8
e
10
CYUNDRICAL
cm
Fig.
13.
Recons uc ion
o
empe a u e map
on
cylind ical sys em.
As
he
dielec ic pe mi i i y
o
wa e a ies linea y wi h empe a u e, empe a u e
maps ob ained wi h he cylind ical sys em.
by he
loss
co ec ion exponen ial ac o . In gene al, he
loss
co ec ion is e y sensi i e
o
he ampli ica ion o noise o
he highe spa ial equencies.
Figs.
14
and
15
show he econs uc ed images wi h bo h he
plana and cylind ical geome y algo i hms, o a empe a u e
change o
1"C,
loca ed a
z
=
0
cm.
A
loss
co ec ion has
been pe o med o he plana case, while o he cylind ical
one he loss co ec ion is implici aking complex
KO
in
(20).
Dielec ic pe mi i i y is econs uc ed p ope ly wi h bo h
algo i hms, bu he plana geome y image has been dis o ed
by losses co ec ion, as shown in Fig.
14.
A empe a u e map
o bo h geome ies is also shown in Fig.
15.
The main eason o he be e esul s achie ed wi h he
cylind ical algo i hm compa ed
o
he plana one wi h
loss
co ec ion is ha in lossless media he o wa d sca e ed ield
is much la ge han he backsca e ed,
bu
in lossy media
he ela i e impo ance o he backsca e ed ield inc eases
wi h he
loss
angen ,
so
ha he cylind ical algo i hm akes
ad an age o e he plana one. This poin has been alida ed
by nume ical simula ion: i in he cylind ical algo i hm only
he o wa d sca e ing in o ma ion is used, he esul s a e e y
simila
o
hose o he plana wi h losses co ec ion.
As
a conclusion, i can be no iced in Figs.
14
and
15
ha he
backsca e ed ield measu emen in lossy media imp o es he
image quali y when a quan i a i e econs uc ion is ob ained.
3)
Di e en ial Recons uc ions:
An image o a localized
empe a u e change in he objec can be ob ained sub ac ing
he image o he objec om he image o he objec wi h he
72
-loa
8
4
-2
o
2
4
e
8
10
PIANAR
w'kh
LOSSES CORRECTION
cm
(a)
CONTRAST PROFILE
plana
wi h
losses co ec ion
c lind ical
-.z.
P
(
b)
Fig.
14.
Recons uc ion
o
dielec ic pe mi i i y wi h losses co ec ion.
Dielec ic pe mi i i y is econs uc ed p ope ly wi h bo h algo i hms bu
plana geome y image has been dis o ed by
loss
co ec ion.