finite element reconstruction for microwave imaging of the breast
DESCRIPTION
TRANSCRIPT
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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#
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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.&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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
k02
krec2
Ecalc(kn2)
‖Emeas −Ecalc(krec2 )‖2<
FEM&forward&solu<on&
with#
! #The#parameter#to#es=mate#is#represented#by#the#squared#wavenumber#k:#
k2(r) = ω2µ0(r)− jωµ
0σ(r)
(r)σ(r)
permiWvity#and#conduc=vity#to#be#es=mated#
4&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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.#
FEM&forward&solu<on&&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Tank&
PML&
Reconstruc<on&&domain&
7&
CrossXcut#view##Mesh#generated#
#by#using##NETGEN®#
! Modeling#the#MI#system#FEM&forward&solu<on&&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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#
FEM&forward&solu<on&&
8&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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+12 ) = Ecalc(k
n)+ J(k
n2)·Δk
n2 with# Δk
n2 = k2
n+1− k2
n
kn+12 = arg min ‖Emeas −(Ecalc(kn2)+ J(kn2)·Δkn2)‖2{ }
! The#minimiza=on#problem##is#now:##
Emeas Samples&of&&Measured&fields&
J Jacobian&matrix(1)&& J
((s,r),τ)=∂E(r)∂k
τ2
= Ψτ(r)·E
s(r)E
r(r)dΩ
Ω
∫∫∫
(1)#K.D.#Paulsen,#P.M.#Meaney#“#Alterna4ve$Breast$Imaging”,#Springer,#2005#
9&
Inverse&Problem&Formula<on&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
! The#applica=on#of#NewtonXRaphson#on#the#func=onal#produces:#
Newton&Q&direc<on&JnT JnΔk
n2 = J
nT·(Emeas −Ecalc(k
n2))
! Since#the#problem#is#illXcondionated#Tikhonov#(with#regulariza=on#)#is#needed:#
(JnT Jn+ αLTL)·Δk
n2 = [J
nT·(Emeas −Ecalc(k
n2))−αLTL(k
n2 − k
ref2 )]
kn+12 = k
n2 + βΔk
n2
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
kref2
E(kn2)
α
L
10&
Inverse&Problem&Formula<on&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
! 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&Reconstruc<on&Algorithm&
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&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
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&
k2f = P
fk
2C
Jf= P
fJ
C
13&
! Laplacian#matrix#defined#on#coarse#mesh#as#well.#
Reconstruc<on&Algorithm&DualQMesh&scheme&
Random#color#used#to#dis=nguish#each##coarse#element#
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Preliminary&results&on&simulated&data&(1)&
14&
f=950#MHz#
φdom= 2.5λ
g
φscatterer
=1λg
number#of#itera=on#=#10#
Ini=al#distribu=on##equal#to#the#background#dielectric##proper=es#
kref2
Results&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
15&
f=950#MHz#
φdom= 2.5λ
g
φscatterer1
= 0.5λg
number#of#itera=on#=#10#
φscatterer2
= 0.27λg
Preliminary&results&on&simulated&data&(2)&
Ini=al#distribu=on##equal#to#the#background#dielectric##proper=es#
kref2
Results&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
16&
f=950#MHz#
φdom= 2.5λ
g
φscatterer1
=1.7λg
number#of#itera=on#=#10#
φscatterer2
= 0.27λg
Preliminary&results&on&simulated&data&(3)&
Ini=al#distribu=on##equal#to#the#background#dielectric##proper=es#
kref2
Results&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Preliminary&results&on&REAL&data&&
17&
Data&were&acquired&by&using&&MIS&at&Dartmouth&College&
εb = 27.98σb =1.01
f=950#MHz#
εscat = 56.1σscat = 0.8
Results&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Preliminary&results&on&REAL&data&&
18&
f=950#MHz#φdom= 2.5λ
g
φscatterer1
= 0.4λg
Results&
num_iter#=10&
FEM$Based$Image$Reconstruc4on$for$Microwave$Imaging$of$the$Breast$
Antenna&and&EMC&Lab&Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
Politecnico&di&Torino&&&&&&&&&&&&&&&&&&&&&&
" 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.#