Finite Element Reconstruction for Microwave Imaging of the Breast

FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
FEM&Based&Image&Reconstruc<on&for&&
Microwave&Imaging&of&the&Breast&
E.#A.#A%ardo##(1),#A.#Borsic#(1),#P.M.#Meaney#(1),#and#G.#Vecchi#(2)#
(2)#Dipar=mento#di#Ele%ronica,#Politecnico#di#Torino,#Turin,#Italy#
(1)#Thayer#School#of#Engineering,#Dartmouth#College,#Hanover,#NH,#US#

Outline&
1.  Background:&
!  Microwave#Imaging#Tomography#for#breast#cancer#detec=on#
#
2.  Formula<on:&
!  Inverse#Problem#Formula=on#
!  FEM#Forward#Solu=on#
!  Reconstruc=on#algorithm#
#
3.&Results:&
!  Simula=on#experiments#
!  Reconstruc=on#on#test#phantom#
&
4.&Conclusions/future&works&
&
&
&
2&

Microwave&Imaging&Tomography&for&breast&cancer&detec<on&
3&
!  Microwave# Imaging# (MI)# has# developed# into# a#
promising# technique# in# breast# cancer# detec=on#
based# on# the# diﬀerent# response# of# normal# and#
malignant# breast# =ssue# to# electromagne=c#
waves.#
!  Reconstruc=on# of# =ssue# proper=es# through# an#
inverse&problem&formula<on.&
Microwave#Func=onal#Spectrum##
Tomography#Prototype#at#
Dartmouth#College:#
P.M.#Meaney,#and#P.#Robbie#
!  MI# is# based# on# using# a# set# of# antennas# to#
propagate# the# electromagne=c# ﬁelds# in# the#
breast#and#sense#sca%ered#responses.&

Inverse&Problem&Formula<on&
! # # Star=ng# from# the# knowledge# of# the# fields# outside& the# breast# and# try# to# minimize# a#
func=onal# such# that# the# difference# between# the# measured# and# computed# fields# is# lesser#
than#a#selected#tolerance.#
Parameter#
Es=ma=on#
Emeas
k0
2
krec
2
Ecalc
(kn
2
)
‖Emeas
− Ecalc
(krec
2
)‖2
< 
FEM&forward&solu<on&
with#
! #The#parameter#to#es=mate#is#represented#by#the#squared#wavenumber#k:#
k2
(r) = ω2
µ0
(r)− jωµ0
σ(r)
(r)
σ(r)
permiWvity#and#conduc=vity#to#be#es=mated#
4&

FEM&forward&solu<on&&
!  A# longXstanding# research# program# has# been#
developed#at#Dartmouth#College,#including#a#system##
for#clinical#use.#
!  Trials#have#been#conducted#at#Dartmouth#Hitchcock#
Medical#Center.#
5&

6&
Empty#Tank#with#moving#antennas#
#(monopole#antennas#have#been#used)## 80:20#glycerinXwater#as#ﬂuid#bath#
Coupling#medium#necessary#to:#
!  reduce#the#unwanted#reﬂec=ons#from#the#walls;#
!  #promote#the#signalXcoupling;#
!  ensure#a#good#=ssue#contact.#

Tank&
PML&
Reconstruc<on&&
domain&
7&
CrossXcut#view#
#Mesh#generated#
#by#using##
NETGEN®#
!  Modeling#the#MI#system#

Empty#Tank#with#(only#coupling#liquid)###
!  f1=#900#MHz;#
!  All#antennas#in#one#selected#plane;#
!  Num.#of#ant.#=#16#
"  for#each#TX#are#available#15#RX#
!  Total#num.#of#meas.#=#240#
Simulated:#magnitude#of#received#voltage# Measured:#magnitude#of#received#voltage#
rbk
= 28.9
σrbk
= 0.96
@f1#
15.24#cm#
8&

k2
= arg min ‖Emeas
− Ecalc
(k2
)‖22
! #The#func=onal#to#minimize#is#expressed#as:#
! #The#calcula=on#of#the#forward#solu=on#################is#based#on#3D#formula=on#of#Maxwell’s#
equa=on#yielding#a#nonlinear&op=miza=on#problem#for#which#NewtonQRaphson&method#is#
applied.#
Ecalc
(k2
)
! #The#nonlinear#expression#for#the#field#can#be#approximated#(locally)#by#firstXorder#Taylor#
expansion#as:#
E(kn+1
2
) = Ecalc
(kn
) + J(kn
2
)·Δkn
2 with# Δkn
2
= k2
n+1
− k2
n
kn+1
2
= arg min ‖Emeas
−(Ecalc
(kn
2
) + J(kn
2
)·Δkn
2
)‖2
{ }
!  The#minimiza=on#problem##is#now:#
#
Emeas Samples&of&&
Measured&fields&
J Jacobian&matrix(1)&& J((s,r),τ)
=
∂E(r)
∂kτ
2
= Ψτ
(r)·Es
(r)Er
(r)dΩ
Ω
∫∫∫
(1)#K.D.#Paulsen,#P.M.#Meaney#“#Alterna4ve$Breast$Imaging”,#Springer,#2005#
9&

!  The#applica=on#of#NewtonXRaphson#on#the#func=onal#produces:#
Newton&Q&direc<on&Jn
T
Jn
Δkn
2
= Jn
T
·(Emeas
− Ecalc
(kn
2
))
!  Since#the#problem#is#illXcondionated#Tikhonov#(with#regulariza=on#)#is#needed:#
(Jn
T
Jn
+ αLT
L)·Δkn
2
= [Jn
T
·(Emeas
− Ecalc
(kn
2
))− αLT
L(kn
2
− kref
2
)]
kn+1
2
= kn
2
+ βΔkn
2
Reference#wavenumber#
Samples#of#simulated#electric#ﬁeld#
resul=ng#from#forward#solver#(FEM)#
#
Tikhonov#factor#
scale#factor#(0,1]##(line#search)#
Regulariza=on#matrix#(Laplacian#ﬁlter)#
where:#
β
kref
2
E(kn
2
)
α
L
10&

DualQMesh&scheme&
!  Typical#forward#mesh##
"  #304,818#Nodes#
"  #1,623,725#Tetrahedra#
"  #3.5#Million#unknowns#
!  A# subXvolume# of# the# mesh,# of# 9,700#
elements# is# used# for# reconstruc=on#
and#fi%ed#to#the#data.#
!  The# number# of# material# parameters#
to# be# fi%ed# is# further# reduced#
adop=ng# a# coarse/fine# interpola<on&
scheme&
Reconstruc<on&Algorithm&
11&

!  The#midpoint#of#every#tetrahedra#is#
used# to# compute# the# distance# to#
the#closest#seed#point#(indicated#in#
red)#
!  By# cycling# on# all# tetrahedra# it# is#
possible#to#associate#each#of#them#
to#a#seed#point,#and#to#form#groups#
of# tetrahedra# that# form# a# coarse#
pixel#in#the#reconstruc=on#
Seed&points&
1#seed&point&=#Nf#ﬁne#FEM#elements#
!  Interpola=on# scheme# to# link# ﬁne#
with#coarse#mesh#by#using#certain#
points# on# coarse# mesh# (seed&
points)(1)#
(1)#A.#Borsic,#R.#Halter,#Y.#Wan,#A.#Hartov,#K.#Paulsen#X##“Electrical$impedance$tomography$reconstruc4on$for$three>dimensional$imaging$$
$$$$$of$the$prostate,$Phys.Measurement,#2010$
12&

Pf
Pc
Fine&
Mesh&
Coarse&
Mesh&
! # Jacobian,# # wavenumber# are# represented# in# the# coarse# mesh,# naturally,# we# can# change# this#
representa=on#by#means#of#interpola=on#matrix###
#
P
Rendering&of&unknowns&&
on&coarse&mesh& Laplacian&ﬁlter&
k2
f
= Pf
k2
C
Jf
= Pf
JC
13&
! Laplacian#matrix#deﬁned#on#coarse#mesh#as#well.#
DualQMesh&scheme&
Random#color#
used#to#
dis=nguish#
each##coarse#
element#

Preliminary&results&on&simulated&data&(1)&
14&
f=950#MHz#
φdom
= 2.5λg
φscatterer
= 1λg
number#of#itera=on#=#10#
Ini=al#distribu=on##
equal#to#
the#background#dielectric##
proper=es#
kref
2
Results&

15&
f=950#MHz#
φdom
= 2.5λg
φscatterer1
= 0.5λg
φscatterer2
= 0.27λg
equal#to#
proper=es#
kref
2
Results&

16&
f=950#MHz#
φdom
= 2.5λg
φscatterer1
= 1.7λg
φscatterer2
= 0.27λg
equal#to#
proper=es#
kref
2
Results&

Preliminary&results&on&REAL&data&&
17&
Data&were&acquired&by&using&
&MIS&at&Dartmouth&College&
εb
= 27.98
σb
= 1.01
f=950#MHz#
εscat
= 56.1
σscat
= 0.8
Results&

Preliminary&results&on&REAL&data&&
18&
f=950#MHz#
φdom
= 2.5λg
φscatterer1
= 0.4λg
Results&
num_iter#=10&

"  include#a#mul=frequency#approach#
"  #include#a#mul=Xitera=ve#procedure#
"  #increase#the#number#of#views#
19&
! Promising#technique#to#be#used#in#parallel#with#mammography#
#
#
Conclusions&
! To#improve#the#reconstruc=on#quality#(especially#on#real#data)#we#need#to:#
! By#using#FEM#technique#the#discre=za=on#density#can#be#adjusted#in#the#domain#
! High#accuracy#can#be#achieved#in#predic=ng#the#true#measurements#
Future&works&
! Using# GPU# to# accelerate# the# reconstruc=on# algorithm# (already# done# in# Electric#
Impedance#Tomography#for#the#Jacobian#with#a#speed#up#equal#to#35x#with#respect#to##
CPU#computa=on(1))#
E.#A.A%ardo.,#A.#Borsic.,#R.#Halter#(2011)#–#Jacobian#Op=miza=on#for#3D#Electric#Impedance#Tomography#via#GPU#accelera=on,#In:#12th#
Interna=onal#Conference#in#Electrical#Impedance#Tomography,University#of#Bath,#Bath,#UK,##May##4X7#2011.#

Finite Element Reconstruction for Microwave Imaging of the Breast

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Finite Element Reconstruction for Microwave Imaging of the Breast