Summary
The aim of this chapter is to review attenuation models in photoacoustic imaging and discuss their causality properties. We also derive integro-differential equations which the attenuated waves are satisfying and highlight the ill–conditionness of the inverse problem for calculating the unattenuated wave from the attenuated one, which has been discussed in Chap. 3.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Agranovsky, P. Kuchment, Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed, Inv. Probl. 23(5), 2089–2102 (2007)
E.J. Beltrami, M.R. Wohlers, Distributions and the Boundary Values of Analytic Functions. Academic Press, New York and London (1966)
P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors. Inverse Probl. 23(6), 65–80 (2007)
P. Burgholzer, H. Grün, M. Haltmeier, R. Nuster, G. Paltauf, in Compensation of acoustic attenuation for high-resolution photoacoustic imaging with line detectors. ed. by A.A. Oraevsky, L.V. Wang, Photons Plus Ultrasound: Imaging and Sensing 2007: The Eighth Conference on Biomedical Thermoacoustics, Optoacoustics, and Acousto-optics. vol. 6437 Proceedings of SPIE, p. 643724. SPIE (2007)
P. Burgholzer, H. Roitner, J. Bauer-Marschallinger, G. Paltauf, Image Reconstruction in Photoacoustic Tomography Using Integrating Detectors Accounting for Frequency-Dependent Attenuation. vol. 7564 Proc. SPIE p. 75640O. (2010)
W. Chen, S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115(4), 1424–1430 (2004)
W.F. Cheong, S.A. Prahl, A.J. Welch, A review of the optical properties of biological tissues. IEEE J. Quantum Electron 26(12), 2166–2185 (1990)
R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1, Springer, New York (2000)
R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 2, Springer, New York (2000)
R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5, Springer, New York (2000)
D. Finch, S. Patch, Rakesh, Determining a function from its mean values over a family of spheres. Siam J. Math. Anal. 35(5), 1213-1240 (2004)
C. Gasquet, P. Witomski, Fourier Analysis and Applications. Springer, New York (1999)
V.E. Gusev, A.A. Karabutov, Laser Optoacoustics. American Institute of Physics, New York (1993)
M. Haltmeier, O. Scherzer, P. Burgholzer, P. Paltauf, Thermoacoustic imaging with large planar receivers. Inverse Probl. 20(5), 1663-1673 (2004)
A. Hanyga, M. Seredynska, Power-law attenuation in acoustic and isotropic anelastic media. Geophys. J. Int. 155, 830-838 (2003)
H. Heuser, Gewöhnliche Differentialgleichungen. 2nd edn. Teubner, Stuttgart (1991)
L. Hörmander, The Analysis of Linear Partial Differential Operators I. 2nd edn. Springer, New York (2003)
Y. Hristova, P. Kuchment, L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media. Inverse Probl. 24(5), 055006 (25pp) (2008)
F. John, Partial Differential Equations. Springer, New York (1982)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, in Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. vol. 204 (Elsevier Science B.V., Amsterdam 2006)
L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics. Wiley, New York (2000)
R. Kowar, O. Scherzer, X. Bonnefond, Causality analysis of frequency-dependent wave attenuation. Math. Meth. Appl. Sci., 22, 108–124 (2011). DOI: 10.1002/mma.1344
R.A. and Kiser, W. L. and Miller, K. D. and Reynolds, H. E.: Thermoacoustic CT: imaging principles. Proc. SPIE, vol. 3916, pp. 150–159 (2000)
G. Ku, X. Wang, G. Stoica, L.V. Wang, Multiple-bandwidth photoacoustic tomography. Phys. Med. Biol. 49, 1329–1338 (2004)
L.A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform. Inverse Probl. 23, 373–383 (2007)
P. Kuchment, L.A. Kunyansky, Mathematics of thermoacoustic and photoacoustic tomography. Eur. J. Appl. Math. 19, 191–224 (2008)
L.D. Landau, E.M. Lifschitz, Lehrbuch der theoretischen Physik, Band VII: Elastizitätstheorie. Akademie, Berlin (1991)
M.J. Lighthill, Introduction to Fourier Analysis and Generalized Functions. Student’s Edition. Cambridge University Press, London (1964)
A.I. Nachman, J.F. Smith III, R.C. Waag, An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88(3) 1584–1595 (1990)
Oraevsky, A. and Wang, L.V., editors: Photons Plus Ultrasound: Imaging and Sensing 2007: The Eighth Conference on Biomedical Thermoacoustics, Optoacoustics, and Acousto-optics, (SPIE Publishing, Bellingham, WA, 2007), Vol. 6437, p. 82
S.K. Patch, O. Scherzer, Special section on photo- and thermoacoustic imaging. Inverse Probl. 23, S1–S122 (2007)
S.K. Patch, A. Greenleaf, Equations governing waves with attenuation according to power law. Technical report, Department of Physics, University of Wisconsin-Milwaukee (2006)
A. Papoulis, The Fourier Integral and its Applications. McGraw-Hill, New York (1962)
I. Podlubny, in Fractional Differential Equations, Mathematics in Science and Engineering. vol. 198 (Academic Press Inc., San Diego, CA 1999)
D. Razansky, M. Distel, C. Vinegoni, R. Ma, N. Perrimon, R.W. Köster, V. Ntziachristos, Multispectral opto-acoustic tomography of deep-seated fluorescent proteins in vivo. Nat. Photonics 3, 412-417 (2009)
P.J. La Riviére, J. Zhang, M.A. Anastasio, Image reconstruction in optoacoustic tomography for dispersive acoustic media. Opt. Lett. 31(6), 781–783 (2006)
O. Scherzer, H. Grossauer, F. Lenzen, M. Grasmair, M. Haltmeier, Variational Methods in Inmaging. Springer, New York (2009)
N.V. Sushilov, R.S.C. Cobbold, Frequency-domain wave equation and its time-domain solution in attenuating media. J. Acoust. Soc. Am. 115, 1431–1436 (2005)
T.L. Szabo, Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Amer. 96, 491–500 (1994)
T.L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power law. J. Acoust. Soc. Amer. 97, 14–24 (1995)
A.C. Tam, Applications of photoacoustic sensing techniques. Rev. Mod. Phys. 58(2), 381–431 (1986)
E.C. Titchmarch, Theory of Fourier Integrals. Clarendon Press, Oxford (1948)
K.R. Waters, M.S. Hughes, G.H. Brandenburger, J.G. Miller, On a time-domain representation of the Kramers-Krönig dispersion relation. J. Acoust. Soc. Amer. 108(5), 2114–2119 (2000)
K.R. Waters, J. Mobely, J.G. Miller, Causality-Imposed (Kramers-Krönig) Relationships Between Attenuation and Dispersion. IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 52(5) 822–833 (2005)
Webb, S., ed. The Physics of Medical Imaging. Institute of Physics Publishing, Bristol, Philadelphia (2000); reprint of the 1988 edition.
L.V. Wang, Prospects of photoacoustic tomography. Med. Phys. 35(12), 5758–5767 (2008)
X.D. Wang, Y.J. Pang, G. Ku, X.Y. Xie, G. Stoica, L.V. Wang, Noninvasive Laser-Induced Photoacoustic Tomography for Structural and Functional in Vivo Imaging of the Brain Nat. Biotechnol. 21(7), 803–806 (2003)
Y. Xu, D. Feng, L.V. Wang, Exact Frequency-Domain Reconstruction for Thermoacoustic Tomography – I: Planar Geometry. IEEE Trans. Med. Imag. 21(7) 823–828 (2002)
Y. Xu, M. Xu, L.V. Wang, Exact Frequency-Domain Reconstruction for Thermoacoustic Tomography – II: Cylindrical Geometry. IEEE Trans. Med. Imag. 21(7), 823–828 (2002)
M. Xu, Y. Xu, L.V. Wang, Time-Domain Reconstruction Algorithms and Numerical Simulation for Thermoacoustic Tomography in Various Geometries. IEEE Trans. Biomed. Eng. 50(9), 1086–1099 (2003)
Y. Xu, L.V. Wang, G. Ambartsoumian, P. Kuchment, Reconstructions in limited-view thermoacoustic tomography. Med. Phys. 31(4), 724–733 (2004)
Xu, M. and Wang, L. V.: Universal back-projection algorithm for photoacoustic computed tomography. Phys. Rev. E 71 (2005) [7 pages] Article ID 016706.
M. Xu, L.V. Wang, Photoacoustic imaging in biomedicine. Rev. Sci. Instrum. 77(4), 1–22 (2006) Article ID 041101.
Yosida, K.: Functional analysis. 5th edn., Springer, New York (1995)
Zhang, E.Z. and Laufer, J. and Beard, P.: Three-dimensional photoacoustic imaging of vascular anatomy in small animals using an optical detection system. (SPIE Publishing, Bellingham, WA, 2007), Vol. 6437, p. 82
H. Zhang, K. Maslov, G. Stoika, V.L. Wang, Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging. Nat. Biotechnol. 24, 848–851 (2006)
Acknowledgements
This work has been supported by the Austrian Science Fund (FWF) within the national research network Photoacoustic Imaging in Biology and Medicine, project S10505-N20.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kowar, R., Scherzer, O. (2012). Attenuation Models in Photoacoustics. In: Ammari, H. (eds) Mathematical Modeling in Biomedical Imaging II. Lecture Notes in Mathematics(), vol 2035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22990-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-22990-9_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22989-3
Online ISBN: 978-3-642-22990-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)