Abstract

Almost all reconstruction methods in photoacoustic tomography (PAT) have been developed by assuming that sound propagation is linear, which is valid for ordinary PAT applications but would become inappropriate when the sound amplitude is higher than a certain threshold level. In the current study, we investigate the effect of nonlinear sound propagation on PAT by using a numerical method which utilizes the time-reversal (TR) technique. In the forward stage, the Euler equations are solved to simulate nonlinear sound propagation, and the flow variables (pressure, velocity and density) are recorded by an array of virtual sensors. The recorded data are used to reconstruct the initial fields within the computational domain by using both linear and nonlinear TR techniques. Furthermore, TR results constructed with and without the recorded flow velocity field, which is difficult to measure for practical applications, have also been compared. The current results show that nonlinear reconstructions produce images with superior clarity, resolution and contrast compared to those reconstructed by the linear method, particularly when the recorded velocity field is used in the reconstruction.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (1)

L. V. Wang and J. J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” Nature Methods 13, 627 (2016).
[Crossref] [PubMed]

2014 (1)

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

2012 (3)

K. Wilson, K. Homan, and S. Emelianov, “Biomedical photoacoustics beyond thermal expansion using triggered nanodroplet vaporization for contrast-enhanced imaging,” Nature Communications 3, 618 (2012).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs,” Science 335, 1458–1462 (2012).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335, 1458–1462 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

C. Tao and X. J. Liu, “Reconstruction of high quality photoacoustic tomography with a limited-view scanning,” Opt. Express 18, 2760–2766 (2010).
[Crossref] [PubMed]

B. E. Treeby, E. Z. Zhang, and B. T. Cox, “Photoacoustic tomography in absorbing acoustic media using time reversal,” Inverse Problems 26, 115003 (2010).
[Crossref]

2009 (3)

C. Hedberg, “Basics of nonlinear time reversal acoustics,” AIP Conference Proceedings 1106, 164–172 (2009).
[Crossref]

C. H. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. 54, R59 (2009).
[Crossref] [PubMed]

A. Elia, P. M. Lugara, C. Di Franco, and V. Spagnolo, “Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,” Sensors 9, 9616–9628, (2009).
[Crossref] [PubMed]

2008 (1)

L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 171–179 (2008).
[Crossref]

2007 (2)

P. Burgholzer and G. J. Matt, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E 75, 046706 (2007).
[Crossref]

N. N. Cao and A. Nehorai, “Tumor localization using diffuse optical tomography and linearly constrained minimum variance beamforming,” Opt. Express 15, 897–909 (2007).
[Crossref]

2006 (3)

M. H. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101 (2006).
[Crossref]

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology 24, 848–851 (2006).
[Crossref] [PubMed]

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

2005 (1)

M. H. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
[Crossref]

2004 (1)

Y. Xu and L. V. Wang, “Time reversal and its application to tomography with diffracting sources,” Phys. Rev. Lett. 92, 033902 (2004).
[Crossref] [PubMed]

2003 (1)

G. Ashcroft and X. Zhang, “Optimized prefactored compact schemes,” J. Comput. Phys. 190, 459–477 (2003).
[Crossref]

2001 (1)

K. B. Cunningham, M. F. Hamilton, A. P. Brysev, and L. M. Krutyansky, “Time-reversed sound beams of finite amplitude,” J. Acoustical Soc. Am. 109, 2668–2674 (2001).
[Crossref]

2000 (2)

M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
[Crossref]

J. M. Sun, B. S. Gerstman, and B. Li, “Bubble dynamics and shock waves generated by laser absorption of a photoacoustic sphere,” J. Appl. Phys. 88, 2352–2362 (2000).
[Crossref]

1996 (2)

F. Q. Hu, M. Y. Hussaini, and J. Manthey, “Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics,” J. Comput. Phys. 124, 177–191 (1996).
[Crossref]

G. S. Jiang and C. W. Shu, “Efficient implementation of weighted ENO schemes,” J. Comput.l Phys. 126, 202–228 (1996).
[Crossref]

1991 (1)

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67, 3384–3387 (1991).
[Crossref] [PubMed]

1987 (1)

R. G. Stearns and G. S. Kino, “High-frequency photoacoustics in air,” IEEE Transactions on Ultrasonics, Ferro-electrics, and Frequency Control 2, 179–190 (1987).
[Crossref]

1974 (1)

L. A. Shepp and B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nuclear Science 3, 21–43 (1974).
[Crossref]

1953 (1)

L. S. G. Kovasznay, “Turbulence in supersonic flow,” J. Aeronautical Sci. 20, 657–682 (1953).
[Crossref]

Ashcroft, G.

G. Ashcroft and X. Zhang, “Optimized prefactored compact schemes,” J. Comput. Phys. 190, 459–477 (2003).
[Crossref]

Aubry, J.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

Boccara, A.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

Bossy, E.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

Brysev, A. P.

K. B. Cunningham, M. F. Hamilton, A. P. Brysev, and L. M. Krutyansky, “Time-reversed sound beams of finite amplitude,” J. Acoustical Soc. Am. 109, 2668–2674 (2001).
[Crossref]

Burgholzer, P.

P. Burgholzer and G. J. Matt, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E 75, 046706 (2007).
[Crossref]

Cao, N. N.

N. N. Cao and A. Nehorai, “Tumor localization using diffuse optical tomography and linearly constrained minimum variance beamforming,” Opt. Express 15, 897–909 (2007).
[Crossref]

Coulouvrat, F.

M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
[Crossref]

Cox, B. T.

B. E. Treeby, E. Z. Zhang, and B. T. Cox, “Photoacoustic tomography in absorbing acoustic media using time reversal,” Inverse Problems 26, 115003 (2010).
[Crossref]

Cunningham, K. B.

K. B. Cunningham, M. F. Hamilton, A. P. Brysev, and L. M. Krutyansky, “Time-reversed sound beams of finite amplitude,” J. Acoustical Soc. Am. 109, 2668–2674 (2001).
[Crossref]

Daoudi, K.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

Di Franco, C.

A. Elia, P. M. Lugara, C. Di Franco, and V. Spagnolo, “Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,” Sensors 9, 9616–9628, (2009).
[Crossref] [PubMed]

Diebold, G. J.

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67, 3384–3387 (1991).
[Crossref] [PubMed]

Elia, A.

A. Elia, P. M. Lugara, C. Di Franco, and V. Spagnolo, “Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,” Sensors 9, 9616–9628, (2009).
[Crossref] [PubMed]

Emelianov, S.

K. Wilson, K. Homan, and S. Emelianov, “Biomedical photoacoustics beyond thermal expansion using triggered nanodroplet vaporization for contrast-enhanced imaging,” Nature Communications 3, 618 (2012).
[Crossref] [PubMed]

Fink, M.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
[Crossref]

Gerstman, B. S.

J. M. Sun, B. S. Gerstman, and B. Li, “Bubble dynamics and shock waves generated by laser absorption of a photoacoustic sphere,” J. Appl. Phys. 88, 2352–2362 (2000).
[Crossref]

Hamilton, M. F.

K. B. Cunningham, M. F. Hamilton, A. P. Brysev, and L. M. Krutyansky, “Time-reversed sound beams of finite amplitude,” J. Acoustical Soc. Am. 109, 2668–2674 (2001).
[Crossref]

Hedberg, C.

C. Hedberg, “Basics of nonlinear time reversal acoustics,” AIP Conference Proceedings 1106, 164–172 (2009).
[Crossref]

Hirschberg, A.

S. W. Rienstra and A. Hirschberg, “An Introduction to Acoustics” (Eindhoven University of Technology, 2017).

Homan, K.

K. Wilson, K. Homan, and S. Emelianov, “Biomedical photoacoustics beyond thermal expansion using triggered nanodroplet vaporization for contrast-enhanced imaging,” Nature Communications 3, 618 (2012).
[Crossref] [PubMed]

Hu, F. Q.

F. Q. Hu, M. Y. Hussaini, and J. Manthey, “Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics,” J. Comput. Phys. 124, 177–191 (1996).
[Crossref]

Hu, S.

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335, 1458–1462 (2012).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs,” Science 335, 1458–1462 (2012).
[Crossref] [PubMed]

Hussaini, M. Y.

F. Q. Hu, M. Y. Hussaini, and J. Manthey, “Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics,” J. Comput. Phys. 124, 177–191 (1996).
[Crossref]

Jiang, G. S.

G. S. Jiang and C. W. Shu, “Efficient implementation of weighted ENO schemes,” J. Comput.l Phys. 126, 202–228 (1996).
[Crossref]

Jose, J.

Khan, M. I.

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67, 3384–3387 (1991).
[Crossref] [PubMed]

Kino, G. S.

R. G. Stearns and G. S. Kino, “High-frequency photoacoustics in air,” IEEE Transactions on Ultrasonics, Ferro-electrics, and Frequency Control 2, 179–190 (1987).
[Crossref]

Kovasznay, L. S. G.

L. S. G. Kovasznay, “Turbulence in supersonic flow,” J. Aeronautical Sci. 20, 657–682 (1953).
[Crossref]

Krutyansky, L. M.

K. B. Cunningham, M. F. Hamilton, A. P. Brysev, and L. M. Krutyansky, “Time-reversed sound beams of finite amplitude,” J. Acoustical Soc. Am. 109, 2668–2674 (2001).
[Crossref]

Larson-Smith, K.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Li, B.

J. M. Sun, B. S. Gerstman, and B. Li, “Bubble dynamics and shock waves generated by laser absorption of a photoacoustic sphere,” J. Appl. Phys. 88, 2352–2362 (2000).
[Crossref]

Li, C. H.

C. H. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. 54, R59 (2009).
[Crossref] [PubMed]

Liu, X. J.

Logan, B. F.

L. A. Shepp and B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nuclear Science 3, 21–43 (1974).
[Crossref]

Lombardo, M.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Lugara, P. M.

A. Elia, P. M. Lugara, C. Di Franco, and V. Spagnolo, “Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,” Sensors 9, 9616–9628, (2009).
[Crossref] [PubMed]

Manohar, S.

Manthey, J.

F. Q. Hu, M. Y. Hussaini, and J. Manthey, “Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics,” J. Comput. Phys. 124, 177–191 (1996).
[Crossref]

Maslov, K.

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology 24, 848–851 (2006).
[Crossref] [PubMed]

Matt, G. J.

P. Burgholzer and G. J. Matt, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E 75, 046706 (2007).
[Crossref]

Matula, T.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Montaldo, G.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

Nehorai, A.

N. N. Cao and A. Nehorai, “Tumor localization using diffuse optical tomography and linearly constrained minimum variance beamforming,” Opt. Express 15, 897–909 (2007).
[Crossref]

O’Donnel, M.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Pelivanov, I.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Perez, C.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Piras, D.

Pozzo, D.

C. W. Wei, M. Lombardo, K. Larson-Smith, I. Pelivanov, C. Perez, J. Xia, T. Matula, D. Pozzo, and M. O’Donnel, “Nonlinear contrast enhancement in photoacoustic molecular imaging with gold nanosphere encapsulated nanoemulsions,” J. Appl. Phys. 104, 033701 (2014).

Resink, S.

Rienstra, S. W.

S. W. Rienstra and A. Hirschberg, “An Introduction to Acoustics” (Eindhoven University of Technology, 2017).

Shepp, L. A.

L. A. Shepp and B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nuclear Science 3, 21–43 (1974).
[Crossref]

Shu, C. W.

G. S. Jiang and C. W. Shu, “Efficient implementation of weighted ENO schemes,” J. Comput.l Phys. 126, 202–228 (1996).
[Crossref]

Slump, C. H.

Spagnolo, V.

A. Elia, P. M. Lugara, C. Di Franco, and V. Spagnolo, “Photoacoustic techniques for trace gas sensing based on semiconductor laser sources,” Sensors 9, 9616–9628, (2009).
[Crossref] [PubMed]

Stearns, R. G.

R. G. Stearns and G. S. Kino, “High-frequency photoacoustics in air,” IEEE Transactions on Ultrasonics, Ferro-electrics, and Frequency Control 2, 179–190 (1987).
[Crossref]

Steenbergen, W.

Stoica, G.

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology 24, 848–851 (2006).
[Crossref] [PubMed]

Sun, J. M.

J. M. Sun, B. S. Gerstman, and B. Li, “Bubble dynamics and shock waves generated by laser absorption of a photoacoustic sphere,” J. Appl. Phys. 88, 2352–2362 (2000).
[Crossref]

Sun, T.

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67, 3384–3387 (1991).
[Crossref] [PubMed]

Tanter, M.

E. Bossy, K. Daoudi, A. Boccara, M. Tanter, J. Aubry, G. Montaldo, and M. Fink, “Time reversal of photoacoustic waves,” Appl. Phys. Lett. 89, 184108 (2006).
[Crossref]

M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
[Crossref]

Tao, C.

Thomas, J. L.

M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
[Crossref]

Treeby, B. E.

B. E. Treeby, E. Z. Zhang, and B. T. Cox, “Photoacoustic tomography in absorbing acoustic media using time reversal,” Inverse Problems 26, 115003 (2010).
[Crossref]

van Hespen, J. C. G.

van Leeuwen, T. G.

Wang, L. V.

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Y. Xu and L. V. Wang, “Time reversal and its application to tomography with diffracting sources,” Phys. Rev. Lett. 92, 033902 (2004).
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Xu, M. H.

M. H. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101 (2006).
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M. H. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
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Xu, Y.

Y. Xu and L. V. Wang, “Time reversal and its application to tomography with diffracting sources,” Phys. Rev. Lett. 92, 033902 (2004).
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Yao, J. J.

L. V. Wang and J. J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” Nature Methods 13, 627 (2016).
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J. J. Yao and L. V. Wang, “Photoacoustic tomography: fundamentals, advances and prospects,” Contrast Media Molecular Imaging 6332–345 (2011).
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J. J. Yao and L. V. Wang, “Photoacoustic tomography: fundamentals, advances and prospects,” Contrast Media Molecular Imaging 6332–345 (2011).
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B. E. Treeby, E. Z. Zhang, and B. T. Cox, “Photoacoustic tomography in absorbing acoustic media using time reversal,” Inverse Problems 26, 115003 (2010).
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H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nature Biotechnology 24, 848–851 (2006).
[Crossref] [PubMed]

Nature Communications (1)

K. Wilson, K. Homan, and S. Emelianov, “Biomedical photoacoustics beyond thermal expansion using triggered nanodroplet vaporization for contrast-enhanced imaging,” Nature Communications 3, 618 (2012).
[Crossref] [PubMed]

Nature Methods (1)

L. V. Wang and J. J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” Nature Methods 13, 627 (2016).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Med. Biol. (1)

C. H. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. 54, R59 (2009).
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M. Tanter, J. L. Thomas, F. Coulouvrat, and M. Fink, “Breaking of time reversal invariance in nonlinear acoustics,” Phys. Rev. E 64, 016602 (2000).
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M. H. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
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Y. Xu and L. V. Wang, “Time reversal and its application to tomography with diffracting sources,” Phys. Rev. Lett. 92, 033902 (2004).
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M. H. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77, 041101 (2006).
[Crossref]

Science (2)

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335, 1458–1462 (2012).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs,” Science 335, 1458–1462 (2012).
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Figures (4)

Fig. 1
Fig. 1

The reconstructed pressure fluctuation fields of the Shepp-Logan phantom with the amplitude Ap = 0.1p0. (a) nonlinear result with p and u; (b) nonlinear result with p; (c) linear result with p and u; (d) linear result only p.

Fig. 2
Fig. 2

Instantaneous distributions of the pressure (a) and density (b) fields at t = 1.5 for the Ap = 0.1p0 case. The level of the pressure is p ∈ (0.71, 0.75); and the level of the density field is ρ ∈ (0.85, 1.05).

Fig. 3
Fig. 3

The reconstructed pressure fields of the Shepp-Logan phantom with the amplitude Ap = 0.5p0. The description for the figures is the same as Fig. 1.

Fig. 4
Fig. 4

The tendency of the reconstruction error with the initial pressure amplitude for the 4 reconstruction methods.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

ρ t + u ρ + ρ u = 0 , u t + u u + p ρ = 0 , p t + u p + γ p u = 0 ,
ρ = ρ 0 + ρ , p = p 0 + p , u = u 0 + u = u .
ρ t + ρ 0 u = 0 , u t + p ρ 0 = 0 , p t + γ p 0 u = 0 .
= 1 A p Ω | p c ( x ) p i ( x ) | d S ,

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