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&&&&&&&&&&&&&&&&&&&&&&

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

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&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#FEM&forward&solu<on&&




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+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&





!  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&





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&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&




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#




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&




15&

f=950#MHz#

φdom= 2.5λ

g

φscatterer1

= 0.5λg


φscatterer2

= 0.27λg



kref2

Results&




16&

f=950#MHz#

φdom= 2.5λ

g

φscatterer1

=1.7λg


φscatterer2

= 0.27λg



kref2

Results&




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&




Preliminary&results&on&REAL&data&&

18&

f=950#MHz#φdom= 2.5λ

g

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

Health & Medicine

breas jacobian

breas background

prendering laplacianlterk2f