GE Medical Systems Template - Confidential

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Thermoacoustic Tomography SK Patch, UW-Milwaukee
Images across modalities
Prototype TCT already competes w/conventional
scans!
EIT
TransScan RD
qualitative potentially quantitative
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Outline

• Intro
• Physics
• Measured data
• Ties to wave equation
• Recon Background
• Xray CT
• Spherical Transforms
• Inversion Formulae for complete data
• r-filtered
• FBP
• Inverting incomplete data

NIR diffuses. . .
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Thermoacoustics (Kruger, Wang, . . . )
breast
waveguides
contrast c(RF absorbtion, tissue elasticity,
more)
contrast c(RF absorbtion, tissue elasticity,

Kruger, Stantz, Kiser. Proc. SPIE 2002.
? RF/NIR heating ? ? thermal expansion ? ?
pressure waves ? ? US signal
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Measured Data (assumes constant soundspeed)
S upper hemisphere
inadmissable transducer

• Integrate f over spheres
• Centers of spheres on sphere
• Partial data only for mammography

S- lower hemisphere
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Ties to the Wave Equation
t
t0 x?Rn
supp f
in Rn
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Outline
supp f

• Intro
• Physics
• Measured data
• Ties to wave equation
• Recon Background
• Xray CT
• Spherical Transforms
• Inversion Formulae for complete data
• r-filtered
• FBP
• Inverting incomplete data

• Xray CT measure line integrals
• 2D
• 3D
• Grangeat
• line ? ? plane ?
• plane ? ? recond image
• Katsevich
• line ? ? recond image

supp f
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Outline

• Spherical Transforms
• 2D
• Circles centered on lines
• Circles through a point

• Intro
• Physics
• Measured data
• Ties to wave equation
• Recon Background
• Xray CT
• Spherical Transforms
• Inversion Formulae for complete data
• r-filtered
• FBP
• Inverting incomplete data

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

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Outline

• Spherical Transforms
• 2D
• Circles centered on circles (Norton)

• Intro
• Physics
• Measured data
• Ties to wave equation
• Recon Background
• Xray CT
• Spherical Transforms
• Inversion Formulae for complete data
• r-filtered
• FBP
• Inverting incomplete data

• 3D
• Spheres centered on sphere(Norton Linzer,
  approximate inversion for complete data)

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r-filtered inversion (complete data)

• Backproject data (thanks to V. Palamodov!)
• Switch order of integration
• Use d-manifold identity (4x!) ?
• f ?Riesz potential ?
• f after high-pass filter

imaging object
Use co-area formula
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r-filtered inversion (complete data)
x
p
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r-filtered inversion (d-identity)
y
x
h
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Inversion Formulae (complete data)
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Implementation

• differentiate and interpolate wrt r ? Dr Dx/2
• integrate over pp(q,f)?S2
• Gaussian wrt q?0,p)
• Trapezoid wrt f?0,2p)

Dr determines resolution Dq, Df determine
artifact
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Numerical Results 256x256 images from
(Nf,Nq,Nr) (800,400,512)
FBP with 1/r weighting
without 1/r weighting
Gaik Ambartsoumian
greyscale 0.9, 1.1
greyscale 0.9, 1.8
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Partial Scan Reconstructions Low Contrast
Detectability
FBP with ½ data in q grayscale 0.4, 1.0
FBP with full data in q grayscale 0.9, 1.1
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Consistency Conditions Necessary, but not
sufficient
trivial derivation, nontrivial implication
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Consistency Conditions Implications
measure ck on S- evaluate ck on S
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Polynomial Extrapolation Stability
f1
f0
Measure over p3?-1,0)

• rescale so q3?-1,1)
• fit measurements to another set of Leg. polys

deg 24
P2
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½ scan reconstruction zero-filling vs. data
extension with respect to z-only
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½ scan FBP reconstruction 0.2 absolute
additive white noise zero-filling vs.data
extension
window width 1.2
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Heuristic Image Quality Impact
DISCLAIMER WIP sans dispersion
use 2D xray transform exploit projection-slice
Ideal Object/Full Scan
Attenuation-Full Scan
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XR beam hardening vs US attenuation
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Image Quality Impact
Recon of Markus attenuated data using
son-of-Gaiks code
Xray CT beam hardening vs TCT pulse softening
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Attenuation Dispersion
b1.10
b1.05
Waters, et al, JASA 2000
b1
Cfat 1460 Cblood 1560 Cskull 4080
a0 10-7 MHz -1 cm-1 w0 107 MHz
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GOAL biannual screening TCT for small
low-contrast masses xrays miss. Xrays for
precursors (microcalcs)

• 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

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GOAL 1 Incorporate fundamental physics GOAL 2
incorporate hardware constraints

• Math Conclusions
• FBP type inversion formulae
• Partial scan - unstable outside of audible
  zone, OK inside
• Palamodov
• Davison Grunbaum

• Anastasio et al , Xu et al OK inside

• SVD perhaps better way Alexsei???

• Attenuation expect blurring, dispersion may be
  a factor

• cos q transducer response Finch

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References Incomplete List

• 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.

• 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).

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measure
Wave Fronts in Standard CT
so
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Wave Fronts in TCT