Problems that led me to Gunther Uhlmann David Isaacson RPI 1. - - PowerPoint PPT Presentation

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Problems that led me to Gunther Uhlmann David Isaacson RPI 1. - - PowerPoint PPT Presentation

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Problems that led me to Gunther Uhlmann David Isaacson RPI 1. Inverse problem in electrocardiography. 2. Inverse boundary value problem for conductivity. GU Can we improve the diagnosis and treatment of heart disease? How does the heart

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Problems that led me to Gunther Uhlmann

David Isaacson RPI

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  • 1. Inverse problem in

electrocardiography.

  • 2. Inverse boundary value problem for

conductivity. GU

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Can we improve the diagnosis and treatment of heart disease?

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How does the heart work?

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The heart is an electro-mechanical pump.

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How does the pump work?

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How does the electrical part work?

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How is the electrical function diagnosed? Electrocardiograms (ECG, or EKG) 1887 - Waller, 1892 - Einthoven

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.1mv

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

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

Model Standard The

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ctor. density ve current Ohmic t) (x, J heart. in sources

  • f

density Current t) (x, J ty Conductivi t) (x, Field Magnetic t) B(x, Field Electric t) E(x, density. charge t) (x, ctor. density ve current Total t) J(x,

O H

        

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0). ( t B/

  • E

law; s Farady' E J , J J J law; s Ohm' 0). ( / Charge;

  • f
  • n

Conservati 0. t / ; ion approximat Static

O H O

                    t J

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SLIDE 31

H O

J E J potential. electrical

  • r

Voltage U U

  • E

t B/

  • E

                       U  

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Standard Forward Model

S.

  • n

U, V find , J and Given S.

  • n

0, U/ B in , J

H H

              U

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Since 1887 we’ve measured V(x,t)=U(x,t)

  • n the chest S.
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Forward problem: given JH and σ, find V. Inverse (physician’s) problem: given V, find JH (and σ). Warning – not unique!

F    

H H

J J

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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.

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

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H.

  • n

U, v Find S.

  • n

0, U/ and , V U S. and H between 0, U given; i.e. V, map surface body from v potentials epicardial t Reconstruc              

Colli-Franzone

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  • R. Macleod
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Problems :

  • 1. How to find conductivity σ?

Goes back to Schlumberger – 1912

  • 2. How to get pumping information?
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Electrical Impedance Tomography and Spectroscopy

David Isaacson Jonathan Newell Gary Saulnier RPI

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With help from

D.G.Gisser, M.Cheney J.Mueller,S.Siltanen

and

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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.

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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?

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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
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Reasons

TISSUE Conductivity S/M Resistivity Ohm-Cm Blood .67 150 Cardiac Muscle .2 500 Lung .05 2000 Normal Breast .03 3000 Breast Carcinoma .2 500

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

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Apply I’s – Measure V’s Reconstruct iwe

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Apply current density j;

) , ( (x)e t) W(x, Hz. 100 /2 for , j(x)e t) (x, j Hz. 100 /2 for 0, ) , ( J S.

  • n

j, W/ B in , J

t i t i H H

t x U U x W

H

              

w w

 w  w w   

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Main Equation

    U 

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S

  • n

/ j U     

B in     U 

S B 

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Forward Problem: Given conductivity  and current density j find v = U on S.

(S) 1/2 H (S) 1/2

  • H

: ) R( Where . : map Dirichlet to Neuman the Find i.e.

v )j R(

 

σ

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Inverse Problem: Given R(σ) Find σ

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Does it have unique solution?

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Yes!

Langer – 1934 Calderon Kohn and Vogelius Sylvester and Uhlmann Nachman Astala and Paivarinta …

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  • 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!

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What can a linearization do?

Noser – a 2-D reconstruction Toddler – a 3-D reconstruction (both assume conductivity differs

  • nly a little from a constant.)

FNoser - Fast ,20 frames/sec Real time imaging of Cardiac and Lung function shown in the following examples.

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

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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

) (                         

  

       

 

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

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

                             

    

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 

        

   

k , ) , (

k k

) ( ) , ( solve; to need

  • nly

Thus ) ( ) ( )}, ( { , BASIS Choose ) , (

k C k n m M n m Data

dx u u x C n m Data x C x x dx u u m n Data

n m B k k k k k n m B

    

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Does it work?

Test by experiment

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ACT 3

  • 32 Current sources
  • 32 Voltmeters
  • 32 Electrodes
  • 30 KHZ
  • 20 Frames / Sec
  • Accuracy > .01%
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Real-time acquisition and display

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Can it image heart and lung function?

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Phantom

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Reconstructions

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2 – D

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ACT 3 imaging blood as it leaves the heart ( blue) and

fills the lungs (red) during systole.

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Show 2D Ventilation and Perfusion Movie

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3D Electrode Placement

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3-D Chest Images

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Heart Lung Static Image

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Show Heart Lung View from

  • ther source
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Ventilation in 3D

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3D Human Results

  • Images showing conductivity changes with

respiration

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Cardiac in 3D

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

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Nachman’s D-Bar method.

  • J. Mueller , S. Siltanen, D.I.

Special thanks to A. Nachman.

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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 0 of these solutions to

recover and display the conductivity in the region

  • f interest.
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Problem: Find the Conductivity σ from the measured Dirichlet to Neumann map

B.

  • f
  • d

neighborho a in 1 B.

  • n

u/ V B.

  • n

V u B. inside : Assume                 

u

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B.

  • f
  • d

neighborho a in q and B

  • n

/ B in q

  • Then

q(p) q u, ) (p, ; Let

1/2 1/2

  • 1/2

                         

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. | p | as 1 where ) (p, p) exp(i ) (p, Let ik k k where i) k(1, take R In 0. where , | p | as p) exp(i satisfy that B

  • utside

q with ) 2 (n R

  • f

all

  • n

Solutions for Look

2 1 2 n

                              

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identical. are they at 1 and both Since q

  • (p,0)

q (p,0)

  • :

Reason ). , ( lim ) , ( ) ( 1/2 hat; property t by the from recover may We . | p | as 1 and ) 2 (- that Observe

1/2 1/2 1/2

                                     p p p q i

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 

. | p | as p) exp(i G , G

  • function

Greens Faddeev the is G(p) where t t)w(t)ds

  • G(p

(Sw)(p)

  • perator

the denotes S Here B.

  • n

) exp( )]

  • S(

[I solving by B

  • n

hence and find First 1. ? find Given : Problem Main

B 1

                

    

 

p i

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. Display . 5 ) ( ) (p, lim Take . 4 ) , ( ) ) ( exp( ) ( 4 1 / ); (p, for equation the Solve . 3 ) ( ) ( ) ( p) exp(i k) t( transform scattering " unphysical " the Compute 2.

1/2 k 1

           

p k p p i k t k k p ds p

B

            

 

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Does it Work?

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Numerical Simulation

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Phantom tank with saline and agar

First D-Bar Reconstruction Results from Experimental Data

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First D-Bar Cardiac Images

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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.

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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.

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Regions of interest: lungs and heart

Anterior Posterior Left Right

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Regions of interest: lungs and heart

Anterior Posterior Left Right

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Admittivity of the heart region.

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Admittivity of the lung region.

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

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Tracheal Divider

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Regions of interest over the right and left lungs.

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Left Right Ventilation of: Both Left Right

Admittivity of the left and right lungs during ventilation of both lungs, then left lung only, then right lung only.

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Regions of interest over the lung.

Balloon No Balloon

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

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

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

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Thank you! Especially

  • S. McD., P.S., A.V., L.W., M.Z,

and

G.U.

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Lunch time!

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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?

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

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Breast Cancer Problem

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HS14R HS10L HS21R HS25L

Which ones have cancer ?

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

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How to measure Impedance Spectra.

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) L A Place sample in this box.

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How we plot electrical impedance spectra in each voxel.

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σ real part of admittivity Siemens/Meter ωε imaginary part

  • f admittivity

Siemens/Meter σ(ω)+iωε(ω) Electrical Impedance Spectra, EIS Plot, of admittivity, σ(ω)+iωε(ω) , for 5kHz <ω<1MHz.

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SLIDE 129

Rensselaer Polytechnic Institute April 2007

Admittance Loci: format for summaries of EIS data

Results of in-vitro studies of excised breast tissue. Jossinet & Schmitt 1999

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Rensselaer Polytechnic Institute April 2007

13

Electrical Impedance Tomography with Tomosynthesis for Breast Cancer Detection

Jonathan Newell Rensselaer Polytechnic Institute

With: David Isaacson Gary J. Saulnier Tzu-Jen Kao Greg Boverman Richard Moore* Daniel Kopans* And: Rujuta Kulkarni Chandana Tamma Dave Ardrey Neha Pol

*Massachusetts General Hospital

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SLIDE 131

Rensselaer Polytechnic Institute April 2007

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.

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SLIDE 132

Rensselaer Polytechnic Institute April 2007

EIT Instrumentation

ACT 4 with Tomosynthesis unit Radiolucent electrode array

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SLIDE 133

Rensselaer Polytechnic Institute April 2007

Co-registration of EIT and Tomo Images

To find the he elect ctrode posit ition, display the he slic ice co contain inin ing the he electrodes. Sup uperim impose the he mesh grid id with co corre rect sca cale. . Slic ice 15 of 91 The hen select the he desire ired tomosynthesis layer. r. Slic ice 50 of 91

HS_14R Norma mal

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SLIDE 134

Rensselaer Polytechnic Institute April 2007

120 EIS plots for a normal breast (HS14_Right)

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SLIDE 135

Rensselaer Polytechnic Institute April 2007

HS25_L: Invasive Ductal Carcinoma

ROI 1 ROI 2

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SLIDE 136

Rensselaer Polytechnic Institute April 2007

Linear Correlation Measure –LCM

m m

Y Y Y Y LCM , 1 1  

1 2 3 4 5 1 2 3 4 5 6

Y Ym compute for each voxel LCM Image 700

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SLIDE 137

Rensselaer Polytechnic Institute April 2007

LCM Image of invasive ductal CA (HS25_L)

Gray scale image of LCM 700

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SLIDE 138

Rensselaer Polytechnic Institute April 2007

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.

300

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SLIDE 139

Rensselaer Polytechnic Institute April 2007

LCM for the regions of interest in 4 patients

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.

Hyalinized Fibroadenoma Invasive ductal carcinoma Normal Normal

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SLIDE 140

Rensselaer Polytechnic Institute April 2007

LCM on the same scale Normal Breast Fibroadenoma Invasive Ductal Carcinoma Invasive Ductal Carcinoma

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SLIDE 141

HS14R HS10L HS21R HS25L

Which ones have cancer ?

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SLIDE 142

Which ones have cancer ?

HS14L LCM=137 HS21L LCM=328 HS25R LCM=709 HS10R LCM=1230

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SLIDE 143

HS14R HS10L HS21R HS25L

Which ones have cancer?

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SLIDE 144

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

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Questions and Suggestions Happily Received by

isaacd@rpi.edu

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SLIDE 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  