Electrical Impedance Tomography

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1 Electrical Impedance Tomography David Isaacson Jonathan Newell Gary Saulnier RPI

2 Part 1. Adaptive Current Tomography and Eigenvalues

3 With help from D.G.Gisser, M.Cheney, E. Somersalo, J.Mueller, S.Siltanen and

4 Denise Angwin, B.S. Greg Metzger, B.S. Hiro Sekiya, B.S. Steve Simske, M.S. Kuo-Sheng Cheng, M.S. Luiz Felipe Fuks, Ph.D. Adam Stewart Andrew Ng, B.S. Frederick Wicklin, M.S. Scott Beaupre, B.S. Andrew Kalukin, B.S. Tony Chan, B.S. Matt Uyttendaele, M.S. Steve Renner, M.S. Laurie Christian, B.S. Van Frangopoulos, M.S. Tim Gallagher, B.S. Lewis Leung, B.S. Jeff Amundson, B.S. Kathleen Daube, B.S. Candace Meindl Matt Fisher Audrey Dima, M.Eng. Skip Lentz Nelson Sanchez, M.S. Clark Hochgraf John Manchester Erkki Somersalo, Ph.D. Hung Chung Molly Hislop Steve Vaughan Joyce Aycock Laurie Carlyle, M.S. Paul Anderson, M.S. John Goble, Ph.D. Dan Kacher Chris Newton, M.S. Brian Gery Qi Li Ray Cook, Ph.D. Paul Casalmir Dan Zeitz, B.S. Kris Kusche, M.S. Carlos Soledade, B.S. Daneen Frazier Leah Platenik Xiaodan Ren, M.S. David Ng, Ph. D. Brendan Doerstling, Ph.D. Mike Danyleiko, B.S. Cathy Caldwell, Ph.D. Nasriah Zakaria Peter M. Edic, Ph. D. Bhuvanesh Abrol Julie Andreasson, B.S. Jim Kennedy, B.S. Trisha Hayes, B.S. Seema Katakkar Yi Peng Elias Jonsson, Ph. D. Pat Tirino M.S. Hemant Jain, Ph. D. Rusty Blue, Ph. D. Julie Larson-Wiseman, Ph. D.

5 Main Problem! Can we improve the Diagnosis and Treatment of Disease, especially Heart Disease and Breast Cancer with Electromagnetic Fields?

6 Example of E&M for Diagnosis EKG _ V +

9 Example of E&M for Treatment- Defibrillation! 3000 V DC, 20A for 5ms V

10 Impedance Imaging Problem; How can one make clinically useful images of the electrical conductivity and permittivity inside a body from measurements on a body s surface?

11 Potential Applications I. Continuous Real Time Monitoring of Function of: 1. Heart 2. Lung 3. Brain 4. Stomach 5. Temperature II. Screening: 1. Breast Cancer 2. Prostate Cancer III. Electrophysiological Data for Inverse problems in: 1. EKG 2. EEG 3. EMG

12 TISSUE Reasons Conductivity S/M Blood Cardiac Muscle Lung Normal Breast Resistivity Ohm-Cm Breast Carcinoma.2 500

13 Procedure For Imaging Heart and Lung Function in 3D Electrical Impedance Tomography

14 Apply I s Measure V s Reconstruct s+iwe

15 Procedure used by T-Scan for Electrical Impedance Mammography

16 T-Scan = Only Commercial System 0 V Apply V Meas. I s Tumor 2 V Display I s V=0 volts Current I s in ma. V=2 volts

17 Is it useful? Test by blinded Trials!

18 Sensitivity = # predicted to have cancer Total # that have cancer Specificity = # predicted NOT to have cancer Total # that do NOT have cancer X 100

20 How can one increase the sensitivity and specificity?

21 Increase Resolution?

22 What determines resolution? FIRST LIMIT IS THE NUMBER OF ELECTRODES because # Degrees of freedom (voxels) reconstructed < L(L-1)/2 Where L=# of Electrodes

23 How to get Higher Resolution? Let L

24 Problem Theorem The ability to distinguish different conductivity distributions 0 as L for fixed # of I gen s.

25 Solve the Problems: 1. Should we apply currents or voltages? 2. Can we distinguish σ from? 3. Which patterns of currents or voltages should we apply? 4. How many electrodes? 5. What size electrodes? 6. How can we reconstruct useful images?

26 How do we solve these problems?

27 Use Mathematics and Electromagnetic theory to design system to reconstruct and display conductivity inside the body in 3D

28 What are the Equations?

29 Maxwell s H J + D / t E B / t D B 0

30 Assume t i t i t i t i t i e x J t x J e x B t x B e x D t x D e x H t x H e x E t x E w w w w w ) ( ), ( ) ( ), (, ) ( ), ( ) ( ), (, ) ( ), (

31 Constitutive Relations D B e E H J s E

32 Thus H ( s + iw e) E E i w H

33 H ( s + iw e) E 0 E i w H

34 Assume w 0 s E 0 E 0

35 E 0 E U s Thus U 0

36 Main Equation s U 0

37 s U 0 in B B s U / j on S S

38 Forward Problem: Given conductivity s and current density j find v = U on S. i.e. Find the Neuman to Dirichlet R( s )jv Where R( σ):h-1/2(s) H+ 1/2(S). map:

39 Inverse Problem: Given R(σ) Find σ

40 Apply Currents or Voltages?

41 1. Apply currents measure voltages since R(s) is smoothing but L(s) R(s) 1 Is desmoothing! { L( s ) : H + 1/2 ( S) H 1/ 2 ( S)}

42 Properties of R R has a complete orthonormal set of eigenfunctions in L 2 ( S) and its eigenvalues are decreasing to 0. See Board.

43 Can we distinguish different conductivities?

44 2.We can distinguish different conductivities when using infinite precision! Calderon Kohn and Vogelius Sylvester and Uhlmann Nachman

45 How to distinguish different conductivities when precision is finite?

46 3. Apply the best current patterns to distinguish conductivities; These are eigenfunctions of N-D map.

47 Let f, g f ( x) g ( x) ds x S f 2 f, f

48 The distinguishability of s from by current density j is denoted by d(s,;j) Given by d ( s, ; j) ( R( s) R( )) j / j

49 We find the best Current density by finding the max over j of <(R(s)R())j, (R(s)R())j> <j,j>

50 From Rayleigh the max distinguishability is d(s,;j) where (R(s)R()) j=j is the largest eigenvalue of the absolute value of R(s)R()

51 Thus s is distinguishable from by measurements of precision e iff max d(s,;j)=d(s,;j)= > e

52 What is the size of the smallest inhomogeniety?

53 r1 r0 s 0 s 0 s 1 s ) cos( Best j ). (. ) ;, ( Distinguishable iff ). ) /( ( 0; 1/ ). /(1 )2 0 / ( ) ;, ( e e s d s s s s s s d O j r r r j + Example

55 How do we find best current patterns when we don t know what is inside? Secret Adaptive Process!

56 Adaptive method to find best current; Guess Measure(or Let If j 1 j 1 - ( - R( j 0 j 0 s )j compute)- 0 Let j R( 0 j )j 1 0 ) / R( and s R( )j s 0, )j 0 R( try again! R( )j 0 )j 0

58 How do we find ALL the Current patterns that are useful?

60 Does it Work?

68 Can optimal currents improve sensitivity and specificity in screening for breast cancer?

69 5x5 Electrode Array Tank

70 Absolute Value of Power Change Distinguishability Study Results Power Distinguishability for Big Conductive Target Using 5x5 Array 8% 7% Iopt_d 6% 5% 4% V_con I_apt 3% 2% Iopt_s 1% 0% Distance from Main Electrode Array Face to Target ( cm ) Figure 3. Optimal Current Pattern Iopt_d has better power distinguishability at all distances. For this experiment, the experimental noise level of the magnitude of the power change is 0.02%, above which targets may be distinguishable.

71 Distance from Electrodes to Target (cm) The depth below mammography electrodes at which an inhomogeneity can be detected by.4% accuracy Constant Voltage Israel Group (T-SCAN) Maximum Detected Distance Optimal Current Patterns Single Current Source Russian Group (BCDD) R.P.I. Group (ACT 3) Electrode Array 5x5, Array Size: 8cm Square, Electrode Size: 14.8mm Square Target Size: 24 mm Cube From Poster by Tzu-Jen Kao

72 How many electrodes to use?

73 L Number of electrodes e Measurement precision Then L O ( e -1/2 ) in 2D, O( e -1/3 ) in 3D.

74 How many current sources to use? As many as we have electrodes. See Proof that as L distinguishability 0 for a fixed number of current sources.

75 What size should we make the electrodes? Space filling!

76 4. How can we reconstruct useful images? 1. Linearization (Noser 2-D,Toddler 3-D); Fast, useful, not accurate for large contrast conductivities. 2. Optimization (Regularized Gauss-Newton); Slow, more accurate, iterative methods 3. Direct methods (Layer stripping, D-Bar); Solve full non-linear problem, no iteration!

77 End Part II How do we make images?

78 Electrical Impedance Tomography David Isaacson Jonathan Newell Gary Saulnier RPI

79 Part 2. Electrical Impedance Imaging Algorithms.

80 Solve the Problems: 1. Should we apply currents or voltages? 2. Can we distinguish σ from? 3. Which patterns of currents or voltages should we apply? 4. How many electrodes? 5. What size electrodes? 6. How can we reconstruct useful images?

81 4. How can we reconstruct useful images? 1. Linearization (Noser 2-D,Toddler 3-D); Fast, useful, not accurate for large contrast conductivities. 2. Optimization (Gauss Newton, Adaptive Kacmarcz); Slow, more accurate, iterative methods. 3. Direct methods (Layer stripping, CGO, dbar); Solve full non-linear problem, no iteration!

82 What can a linearization do? Noser a 2-D reconstruction Toddler a 3-D reconstruction (both assume conductivity differs only a little from a constant.) FNoser - Fast,20 frames/sec Real time imaging of Cardiac and Lung function shown in the following examples.

83 Linearizations NOSER (S.Simske, ) FNOSER(P.Edic, ) TODDLER(R.Blue, )

84 Main Equation s U 0

85 s U 0 in B B s U / j on S S

86 Forward Problem: Given conductivity s and current density j find v = U on S. i.e. Find the Neuman to Dirichlet R( s )jv Where R( σ):h-1/2(s) H+ 1/2(S). map:

87 Inverse Problem: Given R(σ) Find σ

88 n n m m n m j u j u u u / / 0 0 s s s s dx u u ds u u u u dx u u u u u u u u n m B S n m m n n m m n n m m n ) ( s s s s s s s s

89 ) ( ) ( ), ( ) ( then If ) ( Data(n,m) )) ( ) (,( ds ds s s d s s s s ds s s s s s s O dx u u dx u u m n Data O u u dx u u j R R j ds j u j u ds u u u u n m B n m B m m n m B n m S n m m n S n m m n + + >

90 Data( n, m) Data( m, n) ds( x) k ds u k m 0 u C ( x) Thus only need to solve; ( x) u m 0 n 0 Choose BASIS,{ ( x)}, C B k B k k k k dx u n 0 dx Data( m, n) M k m, n C k k

91 What is the Basis? SEE TRANSPARENCIES

92 Does it work? Test by experiment

93 ACT 3 32 Current sources 32 Voltmeters 32 Electrodes 30 KHZ 20 Frames / Sec Accuracy >.01%

94 Phantom

95 Reconstructions

96 Real-time acquisition and display

97 Can it image heart and lung function?

98 2 D

100 ACT 3 imaging blood as it leaves the heart ( blue) and fills the lungs (red) during systole.

101 Show 2D Ventilation and Perfusion Movie

102

103 3D Electrode Placement

104

105 3-D Chest Images

106

107 Heart Lung Static Image

108

109 Show Heart Lung View from other source

110 Ventilation in 3D

111 3D Human Results Images showing conductivity changes with respiration

112

113 Cardiac in 3D

114

115 How can one get more accurate values of the conductivity, less artifact,and still be fast?

116 D-Bar method for EIT J. Mueller, S.Siltanen, D.I. Special thanks to A.Nachman

117 D-Bar Reconstruction method Convert inverse conductivity problem to an Unphysical Inverse Scattering Problem for the Schrodinger Equation. Use the measured D-N map to solve a boundary integral equation for the boundary values of the exponentially growing Faddeev solutions. Compute the unphysical Scattering transform in the complex k-plane from these boundary values. Solve the D-Bar integral equation in the whole complex k-plane for the Faddeev solutions in the region of interest. Take the limit as k goes to 0 of these solutions to recover and display the conductivity in the region of interest.

118 Problem: Find the Conductivity σ from the measured Dirichlet to Assume: Neumann map Ls s u 0 inside B. u V on B. L s V s u/ on B. s 1 in a neighborhood of B.

119 Let ; (p, ) s 1/2 u, q q(p) s -1/2 s 1/2 Then - + q 0 in B L s / on B and q 0 in a neighborhood of B.

120 Look for Solutions on all of R q 0 outside B that satisfy exp(i In R 2 take k(1,i) wherek Let (p, ) exp(i where 1as p. p) as p, where 0. n k 1 + ik p) (p, ) (n 2 ) with 2

121 Observethat (- and 1as p. We may recover s from by theproperty that; Reason: - (p,0) + Since both s 2i ) + s 1/2 ( p) - s 1/2 0 q (p,0) 1/2 q ( p,0) + qs 1/2 0 0 and 1at lim ( p, ). 0 they are identical.

122 Main Problem:Given 1. First find [I+ S( L Here S denotes theoperator (Sw)(p) - G s -, L and hence 1 G )] B exp(i exp( i find ( ) G(p- t)w(t)dst on B by solving p) on B. whereg(p)is the FaddeevGreensfunction d L s? p) as p.

123 . Display 5. ) ( ) (p, 4.Takelim ), ( ) ) ( )exp( ( 4 1 / ); (p, equation for 3.Solve the ) ( ) ( ) p) ( exp(i k) t( " scattering transform 2.Compute the"unphysical 1/2 0 k 1 s s s p k p p i k t k k p ds p B + L L

124 Does it Work?

125 Numerical Simulation

126 First D-Bar Reconstruction Results from Experimental Data Phantom tank with saline and agar

127 First D-Bar Cardiac Images

First blood fills enlarging heart (red) while leaving lungs (blue).

128 Changes in conductivity as heart expands (diastole) and contracts (systole) from one fixed moment in cardiac cycle. First blood fills enlarging heart (red) while leaving lungs (blue). Then blood leaves contracting heart (blue) to fill lungs (red). Reconstruction by D-bar. Data by ACT3.

129 Click on the image at right to see a movie of changes in the conductivity inside a chest during the cardiac cycle. Difference s shown in the movie are all from one moment in the cycle. The movie starts with the heart filling and the lungs emptying. Reconstruction by D-Bar. Data from ACT3.

130 Problems: How to make D-bar method work better with experimental data? How to make it work in 3-D? How to make D-bar work with Optical,Acoustic, and Microwave Data?

131 Can EIT Improve Sensitivity and Specificity in screening for Breast Cancer?

132 Breast Cancer Problem

133 Which ones have cancer? HS14R HS21R HS25L HS10L

134 Electrical Impedance Tomography with Tomosynthesis for Breast Cancer Detection Jonathan Newell With: David Isaacson Tzu-Jen Kao Richard Moore* Gary J. Saulnier Greg Boverman Daniel Kopans* And: Rujuta Kulkarni Chandana Tamma Dave Ardrey Neha Pol Rensselaer Polytechnic Institute *Massachusetts General Hospital Rensselaer Polytechnic Institute April

135 EIT electrodes added to mammography machine. 1 : 2 : 4 : 2 : 1 is the ratio of the mesh thicknesses. Only the center layer, III, is displayed in the results. Rensselaer Polytechnic Institute April 2007

136 EIT Instrumentation ACT 4 with Tomosynthesis unit Radiolucent electrode array Rensselaer Polytechnic Institute April 2007

Superimpose the mesh grid with correct scale.

137 Co-registration of EIT and Tomo Images To find the electrode position, display the slice containing the electrodes. Superimpose the mesh grid with correct scale. Slice 15 of 91 HS_14R Normal Then select the desired tomosynthesis layer. Slice 50 of 91 Rensselaer Polytechnic Institute April 2007

138 Admittance Loci: format for summaries of EIS data Results of in-vitro studies of excised breast tissue. Jossinet & Schmitt 1999 Rensselaer Polytechnic Institute April 2007

139 120 EIS plots for a normal breast (HS14_Right) Rensselaer Polytechnic Institute April 2007

140 HS25_L: Invasive Ductal Carcinoma ROI 1 ROI 2 Rensselaer Polytechnic Institute April 2007

141 Linear Correlation Measure LCM 5 LCM Image Y 1 0 Y m LCM 1 Y, Y Y 1 Y m m compute for each voxel Rensselaer Polytechnic Institute April 2007

142 LCM Image of invasive ductal CA (HS25_L) 700 Gray scale image of LCM Rensselaer Polytechnic Institute April

143 HS21_R: Fibroadenoma in the upper box ROI 1 ROI 2 Rensselaer Polytechnic Institute April 2007

LCM Image of fibroadenoma (HS21_R) 350 0 Gray scale

144 LCM Image of fibroadenoma (HS21_R) Gray scale image of LCM Rensselaer Polytechnic Institute April 2007

145 HS10_L: Invasive Ductal CA Proliferation is worrisome ROI 1 ROI 2 Rensselaer Polytechnic Institute April 2007

146 LCM Image of invasive ductal CA (HS10_L) Gray scale image of LCM Rensselaer Polytechnic Institute April 2007

147 LCM for 11 normal breasts There are 120 EIS plots for layer 3 in each patient. The distribution of the LCM parameter in these plots is shown. Rensselaer Polytechnic Institute April 2007

148 LCM for the regions of interest in 4 patients Hyalinized Fibroadenoma Invasive ductal carcinoma Normal Normal The distributions of the LCM for the regions of interest identified. Note the LCM values are much larger for voxels associated with the malignant lesions. Rensselaer Polytechnic Institute April 2007

149 LCM on the same scale Normal Breast Fibroadenoma Invasive Ductal Carcinoma Invasive Ductal Carcinoma Rensselaer Polytechnic Institute April 2007

150 Which ones have cancer? HS14R HS21R HS25L HS10L

151 Which ones have cancer? HS14L LCM=137 HS21L LCM=328 HS25R LCM=709 HS10R LCM=1230

152

153 Which ones have cancer? HS14R HS21R HS25L HS10L

154 Can EIT Improve Sensitivity and Specificity in screening for Breast Cancer?

155 Questions and Suggestions Happily Received by

156 Thank you, especially J.M., M.C., P.M.

Electromagnetic Inverse Problems in Biomedical Imaging CSU Workshop. David Isaacson RPI

Electromagnetic Inverse Problems in Biomedical Imaging CSU Workshop David Isaacson RPI 1. Inverse problem in electrocardiography. 2. Inverse boundary value problem for electrical conductivity. Can we improve