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Mathematics of Photoacoustic and Thermoacoustic Tomography

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Handbook of Mathematical Methods in Imaging

Abstract

The chapter surveys the mathematical models, problems, and algorithms of the thermoacoustic tomography (TAT) and photoacoustic tomography (PAT). TAT and PAT represent probably the most developed of the several novel “hybrid” methods of medical imaging. These new modalities combine different physical types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a way that the resolution and contrast of the resulting method are much higher than those achievable using only acoustic or electromagnetic measurements.

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References and Further Reading

  1. Agranovsky M, Berenstein C, Kuchment P (1996) Approximation by spherical waves in L p-spaces. J Geom Anal 6(3):365–383

    Article  MathSciNet  MATH  Google Scholar 

  2. Agranovsky M, Finch D, Kuchment P (2009) Range conditions for a spherical mean transform. Inverse Probl Imaging 3(3):373–38

    Article  MathSciNet  MATH  Google Scholar 

  3. Agranovsky M, Kuchment P (2007) Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed. Inverse Probl 23:2089–2102

    Article  MathSciNet  MATH  Google Scholar 

  4. Agranovsky M, Kuchment P, Kunyansky L (2009) On reconstruction formulas and algorithms for the thermoacoustic and photoacoustic tomography, Chapter 8. In: Wang LH (ed) Photoacoustic imaging and spectroscopy. CRC Press, Boca Raton, pp 89–101

    Chapter  Google Scholar 

  5. Agranovsky M, Kuchment P, Quinto ET (2007) Range descriptions for the spherical mean Radon transform. J Funct Anal 248: 344–386

    Article  MathSciNet  MATH  Google Scholar 

  6. Agranovsky M, Nguyen L (2009) Range conditions for a spherical mean transform and global extension of solutions of Darboux equation. Preprint arXiv:0904.4225 To appear in J d’Analyse Mathematique

    Google Scholar 

  7. Agranovsky M, Quinto ET (1996) Injectivity sets for the Radon transform over circles and complete systems of radial functions. J Funct Anal 139:383–414

    Article  MathSciNet  MATH  Google Scholar 

  8. Ambartsoumian G, Kuchment P (2005) On the injectivity of the circular radon transform. Inverse Probl 21:473–485

    Article  MathSciNet  MATH  Google Scholar 

  9. Ambartsoumian G, Kuchment P (2006) A range description for the planar circular Radon transform. SIAM J Math Anal 38(2):681–692

    Article  MathSciNet  Google Scholar 

  10. Ammari H (2008) An Introduction to mathematics of emerging biomedical imaging. Springer, Berlin

    MATH  Google Scholar 

  11. Ammari H, Bonnetier E, Capdebosq Y, Tanter M, Fink M (2008) Electrical impedance tomography by elastic deformation. SIAM J Appl Math 68(6):1557–1573

    Article  MathSciNet  MATH  Google Scholar 

  12. Ammari H, Bossy E, Jugnon V, Kang H. Quantitative photo-acoustic imaging of small absorbers. SIAM Review, to appear 

    Google Scholar 

  13. Anastasio MA, Zhang J, Modgil D, Rivière PJ (2007) Application of inverse source concepts to photoacoustic tomography Inverse Probl 23:S21–S35

    MATH  Google Scholar 

  14. Anastasio MA, Zhang J, Sidky EY, Zou Z, Dan X, Pan X (2005) Feasibility of half-data image reconstruction in 3-D reflectivity tomography with a spherical aperture. IEEE Trans Med Imaging 24(9):1100–1112

    Article  Google Scholar 

  15. Anastasio M, Zhang J, Pan X, Zou Y, Ku G, Wang LV (2005) Half-time image reconstruction in thermoacoustic tomography. IEEE Trans Med Imaging 24:199–210

    Article  Google Scholar 

  16. Andersson L-E (1988) On the determination of a function from spherical averages. SIAM J Math Anal 19(1):214–232

    Article  MathSciNet  MATH  Google Scholar 

  17. Andreev V, Popov D et al (2002) Image reconstruction in 3D optoacoustic tomography system with hemispherical transducer array. Proc SPIE 4618:137–145

    Article  Google Scholar 

  18. Bal G, Jollivet A, Jugnon V (2010) Inverse transport theory of photoacoustics. Inverse Probl 26:025011, doi:10.1088/0266-5611/26/2/025011

    Article  MathSciNet  Google Scholar 

  19. Bell AG (1880) On the production and reproduction of sound by light. Am J Sci 20: 305–324

    Google Scholar 

  20. Beylkin G (1984) The inversion problem and applications of the generalized Radon transform. Commun Pur Appl Math 37:579–599

    Article  MathSciNet  MATH  Google Scholar 

  21. Bowen T (1981) Radiation-induced thermoacoustic soft tissue imaging. Proc IEEE Ultrason Symp 2:817–822

    Google Scholar 

  22. Burgholzer P, Grün H, Haltmeier M, Nuster R, Paltauf G (2007) Compensation of acoustic attenuation for high-resolution photoacoustic imaging with line detectors using time reversal. In: Proceedings of the SPIE number 6437–75 Photonics West, BIOS 2007, San Jose

    Google Scholar 

  23. Burgholzer P, Hofer C, Paltauf G, Haltmeier M, Scherzer O (2005) Thermoacoustic tomography with integrating area and line detectors. IEEE Trans Ultrason Ferroelectr Freq Control 52(9):1577–1583

    Article  Google Scholar 

  24. Burgholzer P, Hofer C, Matt GJ, Paltauf G, Haltmeier M, Scherzer O (2006) Thermoacoustic tomography using a fiber-based Fabry–Perot interferometer as an integrating line detector. Proc SPIE 6086:434–442

    Google Scholar 

  25. Clason C, Klibanov M (2007) The quasi-reversibility method in thermoacoustic tomography in a heterogeneous medium. SIAM J Sci Comput 30:1–23

    Article  MathSciNet  MATH  Google Scholar 

  26. Colton D, Paivarinta L, Sylvester J (2007) The interior transmission problem. Inverse Probl 1(1):13–28

    Article  MathSciNet  MATH  Google Scholar 

  27. Courant R, Hilbert D (1962) Methods of mathematical physics. Partial differential equations, vol II. Interscience, New York

    Google Scholar 

  28. Cox BT, Arridge SR, Beard PC (2007) Photoacoustic tomography with a limited aperture planar sensor and a reverberant cavity. Inverse Probl 23:S95–S112

    Article  MathSciNet  MATH  Google Scholar 

  29. Cox BT, Arridge SR, Beard PC (2009) Estimating chromophore distributions from multiwavelength photoacoustic images. J Opt Soc Am A 26:443–455

    Article  Google Scholar 

  30. Cox BT, Laufer JG, Beard PC (2009) The challenges for quantitative photoacoustic imaging. Proc SPIE 7177:717713

    Article  Google Scholar 

  31. Diebold GJ, Sun T, Khan MI (1991) Photoacoustic monopole radiation in one, two, and three dimensions. Phys Rev Lett 67(24):3384–3387

    Article  Google Scholar 

  32. Egorov Yu V, Shubin MA (1992) Partial differential equations I. Encyclopaedia of mathematical sciences, vol 30. Springer, Berlin, pp 1–259

    Google Scholar 

  33. Faridani A, Ritman EL, Smith KT (1992) Local tomography. SIAM J Appl Math 52(4):459–484

    Article  MathSciNet  MATH  Google Scholar 

  34. Fawcett JA (1985) Inversion of n-dimensional spherical averages. SIAM J Appl Math 45(2):336–341

    Article  MathSciNet  MATH  Google Scholar 

  35. Finch D, Haltmeier M, Rakesh (2007) Inversion of spherical means and the wave equation in even dimensions. SIAM J Appl Math 68(2):392–412

    Article  MathSciNet  MATH  Google Scholar 

  36. Finch D, Patch S, Rakesh (2004) Determining a function from its mean values over a family of spheres. SIAM J Math Anal 35(5):1213–1240

    Article  MathSciNet  MATH  Google Scholar 

  37. Finch D, Rakesh (2006) Range of the spherical mean value operator for functions supported in a ball. Inverse Probl 22:923–938

    Article  MathSciNet  MATH  Google Scholar 

  38. Finch D, Rakesh. Recovering a function from its spherical mean values in two and three dimensions. In [94], pp 77–88

    Google Scholar 

  39. Finch D, Rakesh (2007) The spherical mean value operator with centers on a sphere. Inverse Probl 23(6):S37–S50

    Article  MathSciNet  MATH  Google Scholar 

  40. Gebauer B, Scherzer O (2009) Impedance-acoustic tomography. SIAM J Appl Math 69(2):565–576

    Article  MathSciNet  Google Scholar 

  41. Gelfand I, Gindikin S, Graev M (2003) Selected topics in integral geometry. Transl Math Monogr vol 220, American Mathematical Society, Providence

    Google Scholar 

  42. Grün H, Haltmeier M, Paltauf G, Burgholzer P (2007) Photoacoustic tomography using a fiber based Fabry-Perot interferometer as an integrating line detector and image reconstruction by model-based time reversal method. Proc SPIE 6631:663107

    Article  Google Scholar 

  43. Haltmeier M, Burgholzer P, Paltauf G, Scherzer O (2004) Thermoacoustic computed tomography with large planar receivers. Inverse Probl 20:1663–1673

    Article  MathSciNet  MATH  Google Scholar 

  44. Haltmeier M, Scherzer O, Burgholzer P, Nuster R, Paltauf G (2007) Thermoacoustic tomography and the circular radon transform: exact inversion formula. Math Mod Methods Appl Sci 17(4):635–655

    Article  MathSciNet  MATH  Google Scholar 

  45. Helgason S (1980) The Radon transform. Birkh äuser, Basel

    MATH  Google Scholar 

  46. Hörmander L (1983) The analysis of linear partial differential operators, vols 1 and 2. Springer, New York

    Book  Google Scholar 

  47. Hristova Y (2009) Time reversal in thermoacoustic tomography: error estimate. Inverse Probl 25:1–14

    Article  MathSciNet  Google Scholar 

  48. Hristova Y, Kuchment P, Nguyen L (2008) On reconstruction and time reversal in thermoacoustic tomography in homogeneous and non-homogeneous acoustic media. Inverse Probl 24:055006

    Article  MathSciNet  Google Scholar 

  49. Isakov V (2005) Inverse problems for partial differential equations, 2nd edn. Springer, Berlin

    Google Scholar 

  50. Jin X, Wang LV (2006) Thermoacoustic tomography with correction for acoustic speed variations. Phys Med Biol 51:6437–6448

    Article  Google Scholar 

  51. John F (1971) Plane waves and spherical means applied to partial differential equations. Dover, New York

    Google Scholar 

  52. Kowar R, Scherzer O, Bonnefond X. Causality analysis of frequency dependent wave attenuation, preprint arXiv:0906.4678

    Google Scholar 

  53. Kruger RA, Liu P, Fang YR, Appledorn CR (1995) Photoacoustic ultrasound (PAUS)reconstruction tomography. Med Phys 22:1605–1609

    Article  Google Scholar 

  54. Kuchment P, Lancaster K, Mogilevskaya L (1995) On local tomography. Inverse Probl 11:571–589

    Article  MathSciNet  MATH  Google Scholar 

  55. Kuchment P, Kunyansky L (2008) Mathematics of thermoacoustic tomography. Eur J Appl Math 19(02):191–224

    Article  MathSciNet  MATH  Google Scholar 

  56. Kuchment P, Kunyansky L, Synthetic focusing in ultrasound modulated tomography. Inverse Probl Imaging, to appear

    Google Scholar 

  57. Kunyansky L (2007) Explicit inversion formulae for the spherical mean Radon transform. Inverse probl 23:737–783

    Article  Google Scholar 

  58. Kunyansky L (2007) A series solution and a fast algorithm for the inversion of the spherical mean Radon transform. Inverse Probl 23:S11–S20

    Article  MathSciNet  MATH  Google Scholar 

  59. Kunyansky L (2008) Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm. Inverse Probl 24(5):055021

    Article  MathSciNet  Google Scholar 

  60. Lin V, Pinkus A (1994) Approximation of multivariate functions. In: Dikshit HP, Micchelli CA (eds) Advances in computational mathematics. World Scientific, Singapore, pp 1–9

    Google Scholar 

  61. Louis AK, Quinto ET (2000) Local tomographic methods in Sonar. In: Surveys on solution methods for inverse problems. Springer, Vienna, pp 147–154

    Chapter  Google Scholar 

  62. Maslov K, Zhang HF, Wang LV (2007) Effects of wavelength-dependent fluence attenuation on the noninvasive photoacoustic imaging of hemoglobin oxygen saturation in subcutaneous vasculature in vivo. Inverse Probl 23:S113–S122

    Article  MathSciNet  MATH  Google Scholar 

  63. Natterer F (1986) The mathematics of computerized tomography. Wiley, New York

    MATH  Google Scholar 

  64. Nguyen L (2009) A family of inversion formulas in thermoacoustic tomography. Inverse Probl Imaging 3(4):649–675

    Article  MathSciNet  MATH  Google Scholar 

  65. Nguyen LV. On singularities and instability of reconstruction in thermoacoustic tomography, preprint arXiv:0911.5521v1

    Google Scholar 

  66. Norton SJ (1980) Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution. J Acoust Soc Am 67:1266–1273

    Article  MathSciNet  MATH  Google Scholar 

  67. Norton SJ, Linzer M (1981) Ultrasonic reflectivity imaging in three dimensions: exact inverse scattering solutions for plane, cylindrical, and spherical apertures. IEEE Trans Biomed Eng 28:200–202

    Article  Google Scholar 

  68. Olafsson G, Quinto ET (eds) The radon transform, inverse problems, and tomography. American Mathematical Society Short Course January 3–4, 2005, Atlanta, Georgia, Proc Symp Appl Math, vol 63, AMS, RI, 2006

    Google Scholar 

  69. Oraevsky AA, Jacques SL, Esenaliev RO, Tittel FK (1994) Laser-based ptoacoustic imaging in biological tissues. Proc SPIE 2134A:122–128

    Google Scholar 

  70. Palamodov VP (2004) Reconstructive integral geometry. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  71. Palamodov V (2007) Remarks on the general Funk–Radon transform and thermoacoustic tomography. Preprint arxiv: math.AP/0701204

    Google Scholar 

  72. Paltauf G, Nuster R, Burgholzer P (2009) Weight factors for limited angle photoacoustic tomography. Phys Med Biol 54:3303–3314

    Article  Google Scholar 

  73. Paltauf G, Nuster R, Haltmeier M, Burgholzer P (2007) Thermoacoustic computed tomography using a Mach–Zehnder interferometer as acoustic line detector. Appl Opt 46(16):3352–3358

    Article  Google Scholar 

  74. Paltauf G, Nuster R, Haltmeier M, Burgholzer P (2007) Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors. Inverse Probl 23:S81–S94

    Article  MathSciNet  MATH  Google Scholar 

  75. Paltauf G, Viator JA, Prahl SA, Jacques SL (2002) Iterative reconstruction algorithm for optoacoustic imaging J. Acoust Soc Am 112(4):1536–1544

    Article  Google Scholar 

  76. Paltauf G, Nuster R, Burgholzer P (2009) Characterization of integrating ultrasound detectors for photoacoustic tomography. J Appl Phys 105:102026

    Article  Google Scholar 

  77. Passechnik VI, Anosov AA, Bograchev KM (2000) Fundamentals and prospects of passive thermoacoustic tomography. Crit Rev Biomed Eng 28(3–4):603–640

    Google Scholar 

  78. Patch SK (2004) Thermoacoustic tomography – consistency conditions and the partial scan problem. Phys Med Biol 49:1–11

    Article  Google Scholar 

  79. Patch S (2009) Photoacoustic or thermoacoustic tomography: consistency conditions and the partial scan problem, in [94], 103–116

    Google Scholar 

  80. Patch SK, Haltmeier M (2006) Thermoacoustic tomography – ultrasound attenuation artifacts. IEEE Nucl Sci Sym Conf 4:2604–2606

    Google Scholar 

  81. Popov DA, Sushko DV (2002) A parametrix for the problem of optical-acoustic tomography. Dokl Math 65(1):19–21

    MATH  Google Scholar 

  82. Popov DA, Sushko DV (2004) Image restoration in optical-acoustic tomography. Probl Inform Transm 40(3):254–278

    Article  MathSciNet  MATH  Google Scholar 

  83. La Rivière PJ, Zhang J, Anastasio MA (2006) Image reconstruction in optoacoustic tomography for dispersive acoustic media. Opt Lett 31(6):781–783

    Article  Google Scholar 

  84. Shubin MA (2001) Pseudodifferential operators and spectral theory. Springer, Berlin

    Book  MATH  Google Scholar 

  85. Stefanov P, Uhlmann G (2008) Integral geometry of tensor fields on a class of non-simple Riemannian manifolds. Am J Math 130(1):239–268

    Article  MathSciNet  MATH  Google Scholar 

  86. Stefanov P, Uhlmann G (2009) Thermoacoustic tomography with variable sound speed. Inverse Probl 25:075011

    Article  MathSciNet  Google Scholar 

  87. Steinhauer D. A uniqueness theorem for thermoacoustic tomography in the case of limited boundary data, preprint arXiv:0902.2838

    Google Scholar 

  88. Tam AC (1986) Applications of photoacoustic sensing techniques. Rev Mod Phys 58(2):381–431

    Article  Google Scholar 

  89. Tuchin VV (ed) (2002) Handbook of optical biomedical diagnostics. SPIE, Bellingham

    Google Scholar 

  90. Vainberg B (1975) The short-wave asymptotic behavior of the solutions of stationary problems, and the asymptotic behavior as t → of the solutions of nonstationary problems. Russ Math Surv 30(2):1–58

    Article  MathSciNet  MATH  Google Scholar 

  91. Vainberg B (1982) Asymptotics methods in the equations of mathematical physics. Gordon & Breach, New York

    Google Scholar 

  92. Vo-Dinh T (ed) (2003) Biomedical photonics handbook. CRC Press, Boca Raton

    Google Scholar 

  93. Wang K, Anastasio MA. Photoacoustic and thermoacoustic tomography: image formation principles, Chapter 28 in this volume

    Google Scholar 

  94. Wang L (ed) (2009) Photoacoustic imaging and spectroscopy. CRC Press, Boca Raton

    Google Scholar 

  95. Wang LV, Wu H (2007) Biomedical optics. Principles and imaging. Wiley, New York

    Google Scholar 

  96. Xu M, Wang L-HV (2002) Time-domain reconstruction for thermoacoustic tomography in a spherical geometry. IEEE Trans Med Imaging 21:814–822

    Article  Google Scholar 

  97. Xu M, Wang L-HV (2005) Universal back-projection algorithm for photoacoustic computed tomography. Phys Rev E71:016706

    Article  Google Scholar 

  98. Xu Y, Feng D, Wang L-HV (2002) Exact frequency-domain reconstruction for thermoacoustic tomography: I Planar geometry. IEEE Trans Med Imag 21:823–828

    Article  Google Scholar 

  99. Xu Y, Xu M, Wang L-HV (2002) Exact frequency-domain reconstruction for thermoacoustic tomography: II Cylindrical geometry. IEEE Trans Med Imaging 21:829–833

    Article  Google Scholar 

  100. Xu Y, Wang L, Ambartsoumian G, Kuchment P (2004) Reconstructions in limited view thermoacoustic tomography. Med Phys 31(4):724–733

    Article  Google Scholar 

  101. Xu Y, Wang L, Ambartsoumian G, Kuchment P (2009) Limited view thermoacoustic tomography, Ch. 6. In: Wang LH (ed) Photoacoustic imaging and spectroscopy. CRC Press, Boca Raton, pp 61–73

    Google Scholar 

  102. Zangerl G, Scherzer O, Haltmeier M (2009) Circular integrating detectors in photo and thermoacoustic tomography. Inverse Probl Sci Eng 17(1):133–142

    Article  MathSciNet  MATH  Google Scholar 

  103. Yuan Z, Zhang Q, Jiang H (2006) Simultaneous reconstruction of acoustic and optical properties of heterogeneous media by quantitative photoacoustic tomography. Opt Express 14(15):6749

    Article  Google Scholar 

  104. Zhang J, Anastasio MA (2006) Reconstruction of speed-of-sound and electromagnetic absorption distributions in photoacoustic tomography. Proc SPIE 6086:608619

    Article  Google Scholar 

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Acknowledgments

The work of both authors was partially supported by the NSF DMS grant 0908208. The first author was also supported by the NSF DMS grant 0604778 and by the KAUST grant KUS-CI-016-04 through the IAMCS. The work of the second author was partially supported by the DOE grant DE-FG02-03ER25577. The authors express their gratitude to NSF, DOE, KAUST, and IAMCS for the support.

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Kuchment, P., Kunyansky, L. (2011). Mathematics of Photoacoustic and Thermoacoustic Tomography. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_19

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