Problems that led me to Gunther Uhlmann. David Isaacson RPI

Transcription

1 Problems that led me to Gunther Uhlmann David Isaacson RPI

2 1. Inverse problem in electrocardiography. 2. Inverse boundary value problem for conductivity. GU

3 Can we improve the diagnosis and treatment of heart disease?

4 How does the heart work?

5 The heart is an electro-mechanical pump.

6 How does the pump work?

7

8

9

10

11

12 How does the electrical part work?

13

14

15

16

17 How is the electrical function diagnosed? Electrocardiograms (ECG, or EKG) Waller, Einthoven

18

19

20 .1mv

21

22

23

24

25

26

27 How does the heart produce the voltages on the bodies surface? Forward problem.

28 TheStandardModel B=body ν=unit normal H=heart S=surface

29 J(x,t) (x,t) E(x,t) B(x, (x,t) J J H O t) (x,t) (x,t) Totalcurrent density vector. chargedensity. Electric Field Magnetic Field Conductivity Current density of sourcesin heart. Ohmic current density vector.

30 Static approximation ; / t. Conservation of Charge; J / t ( ). Ohm's law; J J O J H, J O E Farady' s law; E - B/ t ( ).

31 E -B/ t E -U U Voltage or electricalpotential. J O E U J H

32 Standard Forward Model U J H, in B U/, on S. Given and J H, find V U, on S.

33 Since 1887 we ve measured V(x,t)=U(x,t) on the chest S.

34 Forward problem: given J H and σ, find V. Inverse (physician s) problem: given V, find J H (and σ). Warning not unique! J H J H F

35 How to find clinically useful solutions to the inverse problem? Sylvester s solution; Simulate ECGs by solving many forward problems with special J s and σ s.

36

37

38 Can we do better? Body Surface mapping 1963 Taccardi 1978 Colli-Franzone

39

40

41 Colli-Franzone Reconstruct epicardial potentials v frombody surfacemap V, i.e. given; U, between H and S. U V, and U/, on S. Find v U, on H.

42

43

44 R. Macleod

45

46

47 Problems : 1. How to find conductivity σ? Goes back to Schlumberger How to get pumping information?

48 Electrical Impedance Tomography and Spectroscopy David Isaacson Jonathan Newell Gary Saulnier RPI

49 With help from D.G.Gisser, M.Cheney J.Mueller,S.Siltanen and

50 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.

51 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?

52 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

53 TISSUE Reasons Conductivity S/M Blood Cardiac Muscle.2 5 Lung.5 2 Normal Breast.3 3 Resistivity Ohm-Cm Breast Carcinoma.2 5

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

55 Apply I s Measure V s Reconstruct iwe

56 Apply current density j; W J H, in B W/ j, on S. J H ( x, w ), for w /2 1 Hz. j(x,t) j(x)e iwt, for w /2 1 Hz. W(x,t) U (x)e iwt U H ( x, t)

57 Main Equation U

58 U in B B U / j on S S

59 Forward Problem: Given conductivity and current density j find v = U on S. i.e. Find the Neuman to Dirichlet R( )jv Where R( σ):h-1/2(s) H1/2(S). map:

60 Inverse Problem: Given R(σ) Find σ

61 Does it have unique solution?

62 Yes! Langer 1934 Calderon Kohn and Vogelius Sylvester and Uhlmann Nachman Astala and Paivarinta

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

64 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,2 frames/sec Real time imaging of Cardiac and Lung function shown in the following examples.

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

66 n n m m n m j u j u u u / / 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 ) (

67 ) ( ) ( ), ( ) ( then If ) ( Data(n,m) )) ( ) (,( 2 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

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

69 Does it work? Test by experiment

70 ACT 3 32 Current sources 32 Voltmeters 32 Electrodes 3 KHZ 2 Frames / Sec Accuracy >.1%

71 Real-time acquisition and display

72 Can it image heart and lung function?

73 Phantom

74 Reconstructions

75 2 D

76

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

78 Show 2D Ventilation and Perfusion Movie

79

80 3D Electrode Placement

81

82 3-D Chest Images

83

84 Heart Lung Static Image

85

86 Show Heart Lung View from other source

87 Ventilation in 3D

88 3D Human Results Images showing conductivity changes with respiration

89

90 Cardiac in 3D

91

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

93 Nachman s D-Bar method. J. Mueller, S. Siltanen, D.I. Special thanks to A. Nachman.

94 Nachman s D-Bar 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 of these solutions to recover and display the conductivity in the region of interest.

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

96 Let ; (p, ) 1/2 u, q q(p) -1/2 1/2 Then - q in B / on B and q in a neighborhood of B.

97 Look for Solutions on all of R q 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. n k 1 ik p) (p, ) (n 2 ) with 2

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

99 Main Problem:Given 1. First find [I S( Here S denotes theoperator (Sw)(p) - G -, 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? p) as p.

100 . 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 k 1 p k p p i k t k k p ds p B

101 Does it Work?

102 Numerical Simulation

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

104 First D-Bar Cardiac Images

105 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.

106 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.

107 Regions of interest: lungs and heart Posterior Right Left Anterior

108 Regions of interest: lungs and heart Posterior Right Left Anterior

109 Admittivity of the heart region.

110 Admittivity of the lung region.

111 Admittivity of the lung region (blue) and heart region (red, inverted scale).

112 Tracheal Divider

113 Regions of interest over the right and left lungs.

114 Admittivity of the left and right lungs during ventilation of both lungs, then left lung only, then right lung only. Left Right Ventilation of: Both Left Right

115 Regions of interest over the lung. Balloon No Balloon

116 Changes in admittivity with deflation of a balloon in a branch of the pulmonary artery.

117 How to image σ better? The Holy Grail: How to image J H in real time at a microscopic scale?

118 Hybrid methods? CDI, MREIT,PAT,TAT, AMEIT New ideas are needed!

119 Thank you! Especially S. McD., P.S., A.V., L.W., M.Z, and G.U.

120 Lunch time!

121 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?

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

123 Breast Cancer Problem

124 Which ones have cancer? HS14R HS21R HS25L HS1L

125 Observation of Jossinet; Electrical Impedance Spectra can distinguish different tissues.

126 How to measure Impedance Spectra. L A Place sample in this box. V(t) I(t) Apply voltage, V(t) = V cos(ωt) = Re [ V exp(iωt) ]. Measure current, I(t) = V ( a cos(ωt) b sin(ωt) ) = Re [V(a+ib) exp(iωt) ]. σ+iωε (a+ib)(l/a)

127 How we plot electrical impedance spectra in each voxel.

128 Electrical Impedance Spectra, EIS Plot, of admittivity, σ(ω)+iωε(ω), for 5kHz <ω<1mhz. ωε imaginary part of admittivity σ(ω)+iωε(ω) Siemens/Meter σ real part of admittivity Siemens/Meter

129 Admittance Loci: format for summaries of EIS data Results of in-vitro studies of excised breast tissue. Jossinet & Schmitt 1999 Rensselaer Polytechnic Institute April 27

130 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 27 13

131 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 27

132 EIT Instrumentation ACT 4 with Tomosynthesis unit Radiolucent electrode array Rensselaer Polytechnic Institute April 27

133 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 5 of 91 Rensselaer Polytechnic Institute April 27

134 12 EIS plots for a normal breast (HS14_Right) Rensselaer Polytechnic Institute April 27

135 HS25_L: Invasive Ductal Carcinoma ROI 1 ROI 2 Rensselaer Polytechnic Institute April 27

136 Linear Correlation Measure LCM 5 LCM Image Y 1 Y m LCM 1 Y, Y Y 1 Y m m compute for each voxel Rensselaer Polytechnic Institute April 27

137 LCM Image of invasive ductal CA (HS25_L) 7 Gray scale image of LCM Rensselaer Polytechnic Institute April 27

138 LCM for 11 normal breasts 3 There are 12 EIS plots for layer 3 in each patient. The distribution of the LCM parameter in these plots is shown. Rensselaer Polytechnic Institute April 27

139 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 27

140 LCM on the same scale Normal Breast Fibroadenoma Invasive Ductal Carcinoma Invasive Ductal Carcinoma Rensselaer Polytechnic Institute April 27

141 Which ones have cancer? HS14R HS21R HS25L HS1L

142 Which ones have cancer? HS14L LCM=137 HS21L LCM=328 HS25R LCM=79 HS1R LCM=123

143 Which ones have cancer? HS14R HS21R HS25L HS1L

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

145 Questions and Suggestions Happily Received by

146 dx H H E E ds E H E H H H E E E H E H H E H E E H E H i i i i B S ) ( ) ( ] [ ) ( ) ( ] [,,,,, w w we we