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.