Thermoacoustic Tomography

1. Thermoacoustic Tomography G Ambartsoumian Texas A&M D Finch Oregon State University SK Patch* GE Healthcare Rakesh U. Delaware An Inherently 3D Generalized Radon Inversion Problem

2. Xrays propagate straight through, image recovery stable. Sound waves also propagate, permitting stable inversion. supp f Outline • 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 NIR diffuses. . .

3. MRI slice US TCT slice Xray projection optical ART Kruger/OptoSonics, Inc. UW-Madison TransScan R&D Images across modalities Prototype TCT already competes w/conventional scans! EIT TransScan R&D

4. Ductal Carcinoma in situ (DCIS) Images courtesy R. Kruger, OptoSonics, Inc.

5. TCT Changes During Chemo (TCT V2.4) Baseline 7 weeks Pre-Surgery Longitudinal changes during primary chemotherapy. Tumor mass (arrows) appears to have decreased markedly. Images courtesy R. Kruger, OptoSonics, Inc.

6. cyst cysts Fibrocystic Breast(Extra Dense Breast) Images courtesy R. Kruger, OptoSonics, Inc.

7. C t C t Thermoacoustics (Kruger, Wang, . . . ) breast waveguides Kruger, Stantz, Kiser. Proc. SPIE 2002.  RF/NIR heating   thermal expansion   pressure waves   US signal ???

8. Measured Data S+ upper hemisphere inadmissable transducer • Integrate fover spheres • Centers of spheres on sphere • Partial data only for mammography S- lower hemisphere

9. 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

10. r-filtered inversion (complete data) x p

11. r-filtered inversion (d-identity) y h x

12. r-filtered   FBP Inversion Formulae (complete data)

13. FBP w/experimentalweighting Numerical Results (G Ambartsoumian)256x256 images from (Nf,Nq,Nr) = (400,200,200) FBP with 1/r weighting

14. machine precision, as it should be FBP r-filtered Simulated data sans noise (Nf,Nq,Nr) = (800,400,512)

15. full scan data w/o noise (Nf,Nq,Nr) = (800,400,512) sabs = 0.002 sabs = 0.004 sabs = 0.006 Low Contrast Detectability

16. Low Contrast Detectability

17. r-filtered with ½ data in f Partial Scan Reconstructions (Nf,Nq,Nr) = (400,200,200) FBP with ½ data inq

18. even wrt r Consistency Conditions –Necessary, but maybe not sufficient

19. poly of degree k in p!! Consistency Conditions –Implications measure ck on S-; evaluate ck on S+

20. deg 24 deg 16 RTCTf(3/8,r) deg 8 r Polynomial Expansion Accuracy f=1 f=0 High-order expansions required!!!

21. P2 deg 8 Evaluate for p3 [0, 1)i.e.,q3[1,3) deg 16 Polynomial Extrapolation – Stability f=1 f=0 Measure over p3[-1,0) • rescale so q3[-1,1) • fit measurements to another set of Leg. polys deg 24 P2

22. extrapolated data measured data

23. ½ scan only ½ scan+deg-10 extrap additive white noise s = 0.01

24. ½ scan reconstruction – zero-filling vs. data extension with respect to z-only window width = 0.6 full scan data w/o noise (Nf,Nq,Nr) = (800,400,512) window width = 0.3 window width = 0.2 window width = 0.3

25. window width = 1.2 ½ scan FBP reconstruction – 0.2% “absolute” additive white noise zero-filling vs.data extension deg 12 deg 16 deg 4 window width = 1.3 window width = 0.6 window width = 0.6 window width = 0.6 deg 8

26. Dx= 1 Dx= 2 Dx= 4 Dx= 6 Attenuation Blurs DISCLAIMER - WIP PNT Wells, Biomedical Ultrasonics • soundspeed c=1500m/s • where • t ~ t are dual Fourier variables • b ~ 1 • a ~ 0.1 MHz-1 cm-1

27. Attenuation-Partial Scan Heuristic Image Quality Impact DISCLAIMER - WIP use 2D xray transform & exploit projection-slice Ideal Object/Full Scan Attenuation-Full Scan

28. 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

29. GOAL #1 Incorporate fundamental physics GOAL #2 incorporate hardware constraints Math Conclusions • FBP type inversion formulae • Partial scan - unstable outside of “audible zone” • Palamodov • Davison & Grunbaum • Anastasio et al , Xu et al – OK inside • Attenuation – expect blurring • cos qtransducer response – Finch

30. Back-Up slides

31. Simulated data sans noise (Nf,Nq,Nr) = (800,400,512) r-filtered FBP

32. ½ scan reconstruction – 0.2% “absolute” noise zero-filling vs.data extension window width = 0.8 window width = 0.8

33. projection-slice theorem Recover image edges tangent to measurement surface edges w/resolution comparable to that of surface parameter s measure Wave Fronts in 2-D Standard Radon so

34. f=1 “direct” information about horizontal edges; “indirect” information about vertical edges. Wave Fronts in TCT

35. supp f Spherical Transforms • 2D • Circles centered on lines • Circles through a point Xray CT – line integrals • 2D • 3D • Grangeat • line   plane  • plane   recon’d image • Katsevich line   recon’d image • 3D • Spheres centered on plane • Spheres through a point supp f Recon Background

36. Recon Background Spherical Transforms • 2D Circles centered on circles (Norton) • 2D • Circles centered on lines • Circles through a point • 3D • Spheres centered on sphere(Norton & Linzer, approximate inversion for complete data) • 3D • Spheres centered on plane • Spheres through a point