Finite Element Reconstruction for Microwave Imaging of the Breast

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Finite Element Reconstruction for Microwave Imaging of the Breast

  1. 1. 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#
  2. 2. 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&
  3. 3. 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.&
  4. 4. 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 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&
  5. 5. 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&
  6. 6. 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&&
  7. 7. 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&&
  8. 8. 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&
  9. 9. 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+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& Inverse&Problem&Formula<on&
  10. 10. 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&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& Inverse&Problem&Formula<on&
  11. 11. 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&
  12. 12. 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&
  13. 13. 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& k2 f = Pf k2 C Jf = Pf JC 13& ! Laplacian#matrix#defined#on#coarse#mesh#as#well.# Reconstruc<on&Algorithm& DualQMesh&scheme& Random#color# used#to# dis=nguish# each##coarse# element#
  14. 14. 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 = 2.5λg φscatterer = 1λg number#of#itera=on#=#10# Ini=al#distribu=on## equal#to# the#background#dielectric## proper=es# kref 2 Results&
  15. 15. 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 = 2.5λg 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# kref 2 Results&
  16. 16. 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 = 2.5λg 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# kref 2 Results&
  17. 17. 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 = 27.98 σb = 1.01 f=950#MHz# εscat = 56.1 σscat = 0.8 Results&
  18. 18. 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 = 2.5λg φscatterer1 = 0.4λg Results& num_iter#=10&
  19. 19. 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.#

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