, f. dughiero , a. savini , p. di barba m. e. mognaschi v ... thermal field.pdf · case study Γ 2...
TRANSCRIPT
Non
-inva
sive
The
rmom
etry
for
the
The
rmal
A
blat
ion
of L
iver
Tum
or:
a C
ompu
tatio
nal M
etho
dolo
gy
* D
epar
tmen
tof E
lect
rical
Engi
neer
ing
Uni
vers
ity o
f Pad
ova,
Ital
y∆
Dep
artm
ento
f Ele
ctric
alEn
gine
erin
gU
nive
rsity
of P
avia
, Ita
ly
V. D
’Am
bros
io* ,
P. D
i Bar
ba∆ ,
F.D
ughi
ero* ,
M. E
. Mog
nasc
hi∆ ,
A. S
avin
i∆
INT
RO
DU
CT
ION
Dur
ing
the
ther
mal
trea
tmen
t, it
is n
ot p
ossi
ble
to m
easu
re th
e te
mpe
ratu
re re
ache
d in
the
tissu
es b
y m
eans
of a
non
-inva
sive
tech
niqu
e.
From
the
clin
ical
vie
wpo
int,
know
ing
the
inte
rnal
ther
mal
fiel
d is
of u
tmos
t int
eres
t in
orde
r to
cont
rol p
ower
dep
ositi
on fo
r hav
ing
an
effe
ctiv
e tre
atm
ent.
Rad
iofr
eque
ncy
ther
mal
abl
atio
n ha
s bec
ame
an im
porta
nt tr
eatm
ent o
f pr
imar
y an
d m
etas
tatic
liver
tum
or.
Rad
iofr
eque
ncy
gene
rato
r45
0 kH
zN
eedl
edel
ectro
deEn
ergy
deliv
ered
toth
e tu
mor
Ener
gyde
liver
edto
the
tum
or
Ther
apeu
ticte
mpe
ratu
re
(50-
100
°C)
PRO
BL
EM
ST
AT
EM
EN
T (1
/2)
Ω0
Ω
Γ 1 (c
ontin
uous
, tem
pera
ture
kno
wn
)
Γ 2 (
disc
rete
, tem
pera
ture
kno
wn)
Γ 3 (c
ontin
uous
, con
vect
ion
exch
ange
)
Γ 0 (c
ontin
uous
)
ΩΩ
⊂0
Γ 1∩
Γ 2=0
Γ 1∩
Γ 3=0
Γ 2∩
Γ 3=Γ
2
Γ 1: e
lect
rical
ly g
roun
ded
elec
trode
, Ω0 : t
arge
t vol
ume
(i.e.
the
tum
oral
mas
s)
Γ 2: p
ositi
on o
f the
tem
pera
ture
pro
bes
f)u
k(=
∇⋅
∇−
Dir
ect t
herm
alpr
oble
m:
u: te
mpe
ratu
re (°
C),
k: th
erm
al c
ondu
ctiv
ity (W
m-1
°C-1
), f:
sour
ce te
rm (W
m-3
),σ:
elec
tric
cond
uctiv
ity (S
m-1
), E:
impr
esse
d el
ectri
c fie
ld (V
m-1
), c b
: spe
cific
hea
t of t
he b
lood
(J k
g-1°C
-1),
wb:
mas
s flo
w ra
te (k
g s-1
), u b
: tem
pera
ture
of t
he b
lood
(°C
), h:
con
vect
ion
coef
ficie
nt n
: nor
mal
uni
t vec
tor.
)u
u(w
cE
fb
bb
2−
−σ
= u =
Ual
ong
Γ 1
alon
g Γ 3
Bou
ndar
y co
nditi
ons: )
uu(h
nuk
0−
=∂∂
−
in Ω
,with
Line
ar, w
ell-p
osed
pro
blem
.So
lutio
n: e
stim
ated
tem
pera
ture
fie
ld a
ll ov
er th
e do
mai
n Ω
.
PRO
BL
EM
ST
AT
EM
EN
T (2
/2)
Inve
rse
prob
lem
:
Kno
win
g th
e ge
omet
ry o
f the
dom
ain
Ω, t
he ti
ssue
pro
pert
ies
(k, σ
, cb,
wb,
h), t
he so
urce
term
f, t
he b
ound
ary
cond
ition
s alo
ng Γ
1
and
Γ 3 a
nd th
e su
pple
men
tary
con
ditio
n al
ong
Γ 2, f
ind
the
tem
pera
ture
fiel
d u
in th
e do
mai
n Ω
\Ω0
and
alon
g th
e bo
unda
ry Γ
0 .
Solu
tion:
reco
nstru
cted
tem
pera
ture
fiel
d ov
er th
e do
mai
n Ω
\Ω0
The
unce
rtain
ty o
f tis
sue
data
and
the
stea
dy-s
tate
typo
logy
of t
he d
irect
pr
oble
m m
ake
the
cont
inua
tion
prob
lem
ill-p
osed
, be
caus
e it
is im
poss
ible
to g
uara
ntee
the
uniq
uene
ss o
f a so
lutio
n a
prio
ri.
SOL
UT
ION
ST
RA
TE
GY
Min
imiz
atio
n st
rate
gy
Def
inin
g th
e er
ror f
unct
iona
l(
)(
)2
*
22
uu
FΓ
Γ−
λ=
λ, s
tarti
ng fr
om a
gue
ss so
lutio
n,
find
λ∗su
ch th
at(
)(
) λ=
λλ
Fin
fF
*(
) 2u
Γλ
whe
re
is t
he re
cons
truct
ed fi
eld
alon
g Γ 2
2
* uΓ
is th
e su
pple
men
tary
con
ditio
n.
1ℜ
∈λ
Γ 0su
ppos
ed to
be
isot
herm
al
Giv
en a
val
ue o
f λ, t
he d
irect
pro
blem
is w
ell-p
osed
: a fi
nite
-ele
men
t ana
lysi
s of
the
follo
win
g pr
oble
m is
per
form
ed a
nd th
e er
ror f
unct
iona
l F(λ
) is u
pdat
ed.
f)u
k(=
∇⋅
∇−
in Ω
\Ω0
u =
U
alon
g Γ 1
)u
u(hnu
k0
−=
∂∂−
alon
g Γ 3
u =
λal
ong
Γ 0
CA
SE S
TU
DY
Γ 2
Γ 1Ω0 U
= 3
7°C
Mes
h: 2
,990
nod
es a
nd 5
,848
firs
t ord
er-tr
iang
ular
ele
men
ts
The
elec
tric
pote
ntia
l of t
he n
eedl
ed
elec
trode
is se
t to
50 V
with
resp
ect t
o th
e gr
ound
ed e
lect
rode
Γ1
Γ 3: c
ondi
tion
of ta
ngen
tial e
lect
ric fi
eld
Bas
ic p
robl
em:
conv
ectio
n co
effic
ient
h
= 12
W m
-2°C
-1
Fini
te-e
lem
ent m
odel
Elec
tric
pote
ntia
l lin
es [V
]
Live
rFa
tM
uscl
eTu
mor
0.41
60.
360.
4516
.67
Perf
usio
nco
effic
ient
w(k
g s-1
m-3
)
Elec
trom
agne
ticpr
oble
mSo
urce
term
of th
e th
erm
alpr
oble
m
NU
ME
RIC
AL
RE
SUL
TS
ℓ2er
ror o
ver a
dom
ain
D:
()
21
er
Te
r)]
uu(
)u
u[(
D−
−=
ε
u r(i) i
= 1
,..,n
p: re
cons
truct
ed te
mpe
ratu
re fi
eld,
ue(i
):es
timat
ed te
mpe
ratu
re fi
eld
n p: nu
mbe
r of n
odes
of t
he g
rid d
iscr
etiz
ing
D.
ε 1 ≡
ε(Ω
) ε
2≡
ε(Ω
\Ω0)
Var
iatio
nε 1
(°C
)ε 2
(°C
)R
econ
stru
cted
te
mpe
ratu
re o
f Γ0
(°C
)
Bas
ic p
robl
em
6.12
101.
1637
40.8
951
h =
20 (W
m-2
°C-1
)6.
1324
1.16
5940
.889
8
h=
30 (W
m-2
°C-1
)6.
1415
1.16
7740
.885
3
w’=
0.5
w (k
g s-1
m-3
)9.
1634
2.50
3143
.864
5
w’’
= 2
w (
kg s-1
m-3
)3.
6620
0.44
5839
.048
3
Star
ting
tem
pera
ture
of
Γ0
(°C
)ε 1
(°C
)ε 2
(°C
)R
econ
stru
cted
te
mpe
ratu
re
of Γ
0(°
C)
Fina
l res
idua
l of t
he
erro
r fun
ctio
nal (
°C)
106.
1210
1.16
3740
.895
11.
5827
10-4
306.
1064
1.16
1140
.895
21.
7150
10-4
706.
1066
1.16
1140
.895
21.
7118
10-4
956.
1197
1.16
3540
.895
11.
5831
10-4
Dep
eden
ceof
num
eric
al re
sults
on
the
star
ting
poin
t of t
he m
inim
izat
ion
Num
eric
al re
sults
var
ying
som
e th
erm
ophy
sica
lpro
perti
es
Estim
ated
te
mpe
ratu
res (
°C)
Rec
onst
ruct
ed
tem
pera
ture
(°C
)
40.1
91
42.5
68
44.8
82
42.2
07
42.9
96
40.1
26
41.0
69
42.9
15
42.6
21
42.6
55
40.8
95
Bas
ic p
robl
em
CO
NC
LU
SIO
N
•The
pro
blem
of d
eter
min
ing
the
tem
pera
ture
in a
sub
dom
ain
and
alon
g its
bo
unda
ry
has
been
fo
rmul
ated
as
an
op
timis
atio
n pr
oble
m, m
inim
izin
g a
suita
ble
func
tiona
l.
•Con
verg
ence
of
th
e nu
mer
ical
so
lutio
n ha
s be
en
obta
ined
, st
artin
g fr
om v
ario
us g
uess
solu
tions
for t
he te
mpe
ratu
re a
long
the
boun
dary
of t
he su
bdom
ain.
•The
met
hodo
logy
can
be
usef
ully
app
lied
for t
he p
redi
ctio
n of
the
tem
pera
ture
of t
umor
altis
sues
dur
ing
ther
mal
abl
atio
n tre
atm
ent.