Thermoacoustic Tomography –Inherently 3D Reconstruction … · 2005-11-18 · Thermoacoustic Tomography –Inherently 3D Reconstruction TCT Changes During Chemo (TCT V2.4) Longitudinal

Thermoacoustic Tomography – Inherently 3D Reconstruction

Thermoacoustic Tomography

An Inherently 3D Generalized Radon Inversion Problem

G Ambartsoumian Texas A&M D Finch Oregon State UniversitySK Patch* GE Healthcare Rakesh U. Delaware


Outline

NIR diffuses. . .

Xrays propagate straight through, image recovery stable.

Sound waves also propagate, permitting stable inversion.

supp f• WHY TCT – images, ... • Ties to wave equation• Physics - forward problem • Inversion formulae for

complete data• Inverting incomplete data

Backup• Wave Fronts • Recon Background

– Xray CT– Spherical Transforms


TransScan R&D

EIT

US

MRI

slice

UW-MadisonTransScan R&D

Xray

projec

tion

Images across modalities

TCT

slice

Kruger/OptoSonics, Inc.

ART optica

lPrototype TCT already competes

w/conventional scans!


Ductal Carcinoma in situ (DCIS)

Images courtesy R. Kruger, OptoSonics, Inc.


TCT Changes During Chemo (TCT V2.4)

Longitudinal changes during primary chemotherapy. Tumor mass (arrows) appears to have decreased markedly.

Baseline 7 weeks Pre-Surgery



Fibrocystic Breast(Extra Dense Breast)

cyst

cysts



Thermoacoustics (Kruger, Wang, . . . )

• RF/NIR heating →• thermal expansion →

• pressure waves →• US signal

C t

C t

???

breast

waveguides

Kruger, Stantz, Kiser. Proc. SPIE 2002.


Measured Data

• Integrate f over spheres

• Centers of spheres on sphere

• Partial data only for mammography

S+ upper hemisphere

S- lower hemisphere

inadmissabletransducer

( )( ) ( ) ?pp?

drfrrfRTCT ∫=

+=1

2, θ


imaging object

ρ−filtered inversion (complete data)

• Backproject data (thanks to V. Palamodov!)

• Switch order of integration

• Use δ−manifold identity (4x!) →

• f ∗Riesz potential →• f after high-pass filter

{ }( ) zdzzzhzdzh

zzMn

R

n

Mz n

))(()()(

0)(|1 φδφ

φ

∫∫ ∇=

==−

∈

Use co-area formula


ρ−filtered inversion (complete data)

( ) ( ) ( ) pypypxpyypxp y

ddfR

∫ ∫= ∈

−−−−

−=

1

22 21

3

δ

( )( ) ppxppxp

dfRTCT −−∫

=

,1

1

x

p

( ) p?pxpxppxp ?

ddf∫ ∫= =

−−+

−=

1

2

1

1θ

( ) ( ) yppxpyypy

ddfR

−−−= ∫∫

=∈ 1

2223

δ

( ) ( ) ( ) yppxpypypy

ddfRR

−−−−= ∫∫

∈∈ 33

222 12 δδ


ρ−filtered inversion (δ-identity)

∫−=+ −−

=22

221 1

21

11

hpp

dshxy

( )( ) ( )y

xyy

ppxppx

yp

df

dfRR

TCT ∫∫∈= −

=−− 3

2,1

1

π

x

y

h21 h−

( ) ( ) ( )∫∫−=−∈

−−

=−−−−pypxp

ppxy

ppxpyp 223222 11

123

ddR

δδδ

xy −=

π2


Inversion Formulae (complete data)

[ ] )(8

1)( 2,2 xRfxf sz∗∆

−= xπ

( )( ) ppxppxp

x dfRTCT −−

∆−

= ∫=

,1

81

12π

ρ-filtered

:( ) ( ) ppxp

pxp

dfRTCT −″−

−= ∫

=

,1

81

12π

:FBP


Numerical Results (G Ambartsoumian)256x256 images from (Nφ,Nθ,Nr) = (400,200,200)

FBP with 1/ρ weighting FBP w/experimental weighting


FBP ρ−filtered

Simulated data sans noise (Nφ,Nθ,Nr) = (800,400,512)

machine precision,

as it should be


full scan data w/o noise

(Nφ,Nθ,Nr) = (800,400,512)

Low Contrast Detectability

σabs = 0.002σabs = 0.004σabs = 0.006

Thermoacoustic Tomography – Inherently 3D Reconstruction Low Contrast Detectability


Partial Scan Reconstructions (Nφ,Nθ,Nr) = (400,200,200)

FBP with ½ data in θ ρ−filtered with ½ data in φ


Consistency Conditions –Necessary, but maybe not sufficient

( ) ( )rfRrfR TCTTCT −= ,, pp even wrt r

( ) ( ) ( ) xxpxpp dfdrrfRrMomnR

k

R

TCTk

k ∫∫ −==+

,

( ) ( ) ( )ppp kSk QMom n =−∈ 12

( ) ( ) ( ) xxppxxp dfMomnR

k

k ∫ +•−= 222 2


Consistency Conditions –Implications

( ) ( ) ( )∫−=

≡1

12 ,~ˆ

rTCTkk drrfRrPc pp

( ) ( ) ( )∑∞

=

=0

2~

,n

kkTCT 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 ExpansionAccuracy

High-order expansions required!!!

( ) ( )

( ) ( )∑∞

=

=

=

02

3321

~

,,,,

nkk

TCTTCT

rPc

rpfRrpppfR

p

deg 8

deg 16deg 24

R TCTf

(3/8

,r)

r

f=1

f=0


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


additive white noiseσ = 0.01

½ scan only ½ scan+deg-10 extrap


full scan data w/o noise

(Nφ,Nθ,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 = 0.3window width = 0.2

Thermoacoustic Tomography – Inherently 3D Reconstruction 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


Attenuation Blurs

xb

e ∆− τα

DISCLAIMER - WIP

•

•

•

• soundspeed c=1500m/s

tce τ1.0− PNT Wells, Biomedical Ultrasonics

where

• τ ∼ t are dual Fourier variables

• b ~ 1

• α ~ 0.1 MHz-1 cm-1

∆x= 1

∆x= 2

∆x= 4 ∆x= 6


Heuristic Image Quality Impact

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

use 2D xray transform & exploit projection-sliceDISCLAIMER - WIP


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”– Palamodov– Davison & Grunbaum

• Attenuation – expect blurring

GOAL #1

Incorporate

fundamental

physics

GOAL #2

incorporate

hardware

constraints

– Anastasio et al , Xu et al – OK inside

• cos θ transducer response – Finch


Back-Up slides


Simulated data sans noise (Nφ,Nθ,Nr) = (800,400,512)

FBP ρ−filtered

Thermoacoustic Tomography – Inherently 3D Reconstruction ½ scan reconstruction –

0.2% “absolute” noise

zero-filling vs.data extension

window width = 0.8window width = 0.8


Wave Fronts in 2-D Standard Radon

so

∫=•

=sx

xdxfsRfo

o 1)(),(

measure

w/resolution comparable to that of surface parameter s

Recover image edges tangent to measurement surface edges

( ) ( )oo ,)( σσ ∧∧ = RfRfprojection-slice theorem


Wave Fronts in TCT

f=1

“indirect” information about vertical edges.

“direct” information about horizontal edges;


Recon Background

Xray CT – line integrals

• 2D • 3D

– Grangeat• line ∫ → plane ∫• plane ∫ → recon’d image

– Katsevichline ∫ → recon’d image

supp f

supp f

Spherical Transforms• 2D

– Circles centered on lines– Circles through a point

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


Recon Background

• 2D– Circles centered on lines– Circles through a point

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

• 2DCircles centered on circles

(Norton)

• 3D Spheres centered on

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

Spherical Transforms

Thermoacoustic Tomography –Inherently 3D Reconstruction … · 2005-11-18 · Thermoacoustic Tomography –Inherently 3D Reconstruction TCT Changes During Chemo (TCT V2.4) Longitudinal

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