thermoacoustic tomography – inherently 3d reconstruction transscan r&d eit us mri slice...

Thermoacoustic Tomography – Inherently 3D Reconstruction

TransScan R&D

EIT

US

MRI

slice

UW-MadisonTransScan R&D

Xray

proje

ction

Images across modalities

TCT

slice

Kruger/OptoSonics, Inc.

ART opticalPrototype TCT

already competes w/conventional

scans!

Thermoacoustic TomographySK Patch, UW-Milwaukee

4cm

qualitative potentially quantitative


Outline

NIR diffuses. . .

Xrays propagate straight through, image recovery stable.

Sound waves also propagate, permitting stable inversion.

supp f• Intro

– Physics – Measured data

• Ties to wave equation• Recon Background

– Xray CT– Spherical Transforms

• Inversion Formulae for complete data– -filtered– FBP

• Inverting incomplete data


Thermoacoustics (Kruger, Wang, . . . )

RF/NIR heating thermal expansion

pressure waves US signal

C t

C t

breast

waveguides

Kruger, Stantz, Kiser. Proc. SPIE 2002.

???contrast =

c(RF absorbtion,

contrast = c(RF absorbtion,

tissue elastic

ity,

more)

contrast = c(RF absorbtion,

tissue elastic

ity,


Measured Data (assumes constant soundspeed)

• Integrate f over spheres

• Centers of spheres on sphere

• Partial data only for mammography

S+ upper hemisphere

S- lower hemisphere

inadmissable transducer

θppθ

drfrrfR rTCTrr

1

1 ,


Ties to the Wave Equation

)()0,(

0)0,(

0),(2

2

xx

x

xx

fu

u

tut

t

in Rn

t

tfR

ttnTCT

n

n

),(

2

1

22

2

3

1

x

n-odd

t

r

TCTn

n

n

n

n

drrfRrrttn

tu0

22

322

2

2

1

),)(()!2(

1),( xx

t=0

xRn

t

supp f


Outline

Xray CT – measure line integrals

• 2D • 3D

– Grangeat• line plane • plane recon’d image

– Katsevich line recon’d image

supp f

supp f

• Intro– Physics – Measured data






Outline

Spherical Transforms• 2D

– Circles centered on lines– Circles through a point

• 3D – Spheres centered on plane– Spheres through a point


• Ties to wave equation • Recon Background





Outline






Spherical Transforms• 2D

Circles centered on circles (Norton)

• 3D Spheres centered on

sphere(Norton & Linzer, approximate inversion for complete data)


imaging object

filtered inversion (complete data)

• Backproject data (thanks to V. Palamodov!)

• Switch order of integration

• Use manifold identity (4x!)

• f Riesz potential • f after high-pass filter

zdzzzhzdzh

zzMn

R

n

Mz n

))(()()(

0)(|1

Use co-area formula


filtered inversion (complete data)

pypypxpyypxp y

ddfR

1

222

1

3

ppxppxp

dfRTCT

,1

1

x

p

pθpxpxppxp θ

ddf

1

2

1

1

yppxpyypy

ddfR

1

222

3

yppxpypypy

ddfRR

33

22212


filtered inversion (-identity)

22

221 1

21

11

hpp

dshxy

y

xy

yppxp

pxyp

df

dfRR

TCT

3

2,1

1

x

y

h21 h

pypxp

ppxy

ppxpyp 2232221

112

3

ddR

xy

2


Inversion Formulae (complete data)

)(8

1)( 2,2

xRfxf sz

x

ppxppxp

x dfRTCT

,1

8

1

12

-filtered

ppxp

pxp

dfRTCT

,1

8

1

12

FBP


Implementation

pppx

xpxp

drfRfr

TCT

,

1

8

1

12

• differentiate and interpolate wrt r → r = x/2

• integrate over p=p()S2

Gaussian wrt [0,)

Trapezoid wrt [0,2)

“r determines resolution; determine artifact”


Numerical Results 256x256 images from (N,N,Nr) = (800,400,512)

FBP with 1/weighting

greyscale =[0.9, 1.1]

greyscale =[0.9, 1.8]without 1/weighting

Gaik

Ambartsoumian


Partial Scan Reconstructions &

Low Contrast Detectability

FBP with ½ data ingrayscale = [0.4, 1.0]

FBP with full data ingrayscale = [0.9, 1.1]


Consistency Conditions – Necessary, but not sufficient

rfRrfR TCTTCT ,, pp even wrt r

xxpxpp dfdrrfRrMomnR

k

R

TCTk

k

,

ppp kSk QMom n 12

xxppxxp dfMomnR

k

k 22

2 2

trivial

derivati

on,

nontrivial

implic

ation…


Consistency Conditions – Implications

1

1

2 ,~

ˆr

TCTkk drrfRrPc pp

0

2

~,

nkkTCT rPcrfR pp

1

1

2

0

,r

TCTl

k

l

kl drrfRrc p

pp l

k

l

klk Qcc

0

poly of degree k in p!!

measure ck on S- ;

evaluate ck on S+


Polynomial Extrapolation – Stability

deg 24

Measure over p3[-1,0)

• rescale so q3[-1,1)

• fit measurements to another set of Leg. polys

P2

f=1

f=0

deg 8

deg 16

Evaluate for p3 [0, 1) i.e., q3[1,3)

P2


measured data

extrapolated data


full scan data w/o noise

(N,N,Nr) = (800,400,512)

½ scan reconstruction –

zero-filling vs.

data extension

with respect to z-only

window width = 0.6

window width = 0.3window width = 0.2


½ scan FBP reconstruction –

0.2% “absolute” additive white noise

zero-filling vs.data extension

window width = 1.3deg 4

window width = 1.2

window width = 0.6deg 8 window width = 0.6deg 12 window width = 0.6deg 16


Heuristic Image Quality Impact

Ideal Object/Full Scan Attenuation-Full ScanAttenuation-Partial Scan

use 2D xray transform & exploit projection-sliceDISCLAIMER –

WIP sans

dispersion


XR beam hardening vs US attenuation


Image Quality Impact

Xray CT beam hardening vs TCT pulse softening

Recon of Markus’ attenuated data using son-of-Gaik’s code


Attenuation & Dispersion

1,302

tan11 11

bbfor

b

ccbo

bo

o

o

o

occb

ln

211

1

lim b=1.10

b=1.05

b=1

0 = 10-7 MHz -1 cm-1

0 = 107 MHz

Waters, et al, JASA 2000

Cfat = 1460 Cblood = 1560 Cskull = 4080


non-Math Conclusions

• Positives– cheap ??– non-ionizing – high-res (exploits hyperbolic physics)– 2x depth penetration of ultrasound, sans

speckle– detect masses

• Issues– will not detect microcalcifications– contrast mechanism not understood– fundamental physics (attenuation, etc)

and HW constraints will impact IQ

GOAL : biannual

screening

TCT for small

low-contrast

masses xrays

miss.

Xrays for

precursors

(microcalcs)


Math Conclusions

• FBP type inversion formulae• Partial scan - unstable outside of

“audible zone”, OK inside– Palamodov– Davison & Grunbaum

• Attenuation – expect blurring, dispersion may be a factor

GOAL #1

Incorporate

fundamental

physics

GOAL #2

incorporate

hardware

constraints

– Anastasio et al , Xu et al – OK inside

• cos transducer response – Finch

– SVD perhaps better way – Alexsei???


• Kruger, et. al.– Radiology 2000; 216.– Radiology 1999; 211.– Med. Phys. 1999; 26(9).

• Math/physics – Joines, Zhang, Li, Jirtle,

Med. Phys. 1994; 21(4).– Norton, Linzer, IEEE

Trans. BME 1981; BME-28.

– Norton, J. Acoust. Soc. Am. 1980; 67(4).

– Xu, Wang, IEEE TMI, 2002; 21(7).

– Louis, Quinto, Surveys on solution methods for inverse problems, 2000.

– Agranovsky, Quinto, J. Functional Analysis, 1996; 139.

– Agranovsky, Quinto, Duke Math J., 2001; 107(1).

– Uhlmann, Duke Math J., 1989; 58(1).

– Finch, Patch, Rakesh, SIAM J. Math Anal. 2004; 35(5).

– Patch, Phys in Med. & Bio, submitted.

References Incomplete List


so

sx

xdxfsRfo

o 1)(),(

measure

w/resolution comparable to that of surface parameter s

Recover image edges tangent to measurement surface edges

oo ,)( RfRfprojection-slice theorem

Wave Fronts in Standard CT


f=1

“indirect” information about vertical edges.

“direct” information about horizontal edges;

Wave Fronts in TCT

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