Abstract

Quantitative photoacoustic tomography is an imaging modality in which distributions of optical parameters inside tissue are estimated from photoacoustic images. This optical parameter estimation is an ill-posed problem and it needs to be approached in the framework of inverse problems. In this work, utilising surface light measurements in quantitative photoacoustic tomography is studied. Estimation of absorption and scattering is formulated as a minimisation problem utilising both internal quantitative photoacoustic data and surface light data. The image reconstruction problem is studied with two-dimensional numerical simulations in various imaging situations using the diffusion approximation as the model for light propagation. The results show that quantitative photoacoustic tomography augmented with surface light data can improve both absorption and scattering estimates when compared to the conventional quantitative photoacoustic tomography.

© 2017 Optical Society of America

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2017 (1)

J. Brunker, J. Yao, J. Laufer, and S. E. Bohndiek, “Photoacoustic imaging using genetically encoded reporters: a review,” J. Biomed. Opt. 22, 070901 (2017).
[Crossref]

2016 (6)

L. V. Wang and J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” Nature Methods 13, 627–638 (2016).
[Crossref] [PubMed]

J. Weber, P. C. Beard, and S. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nature Methods 13, 639–650 (2016).
[Crossref] [PubMed]

A. Hannukainen, N. Hyvönen, H. Majander, and T. Tarvainen, “Efficient inclusion of total variation type priors in quantitative photoacoustic tomography,” SIAM J. Imag. Sci. 9, 1132–1153 (2016).
[Crossref]

M. Venugopal, P. van Es, S. Manohar, D. Roy, and R. M. Vasu, “Quantitative photoacoustic tomography by stochastic search: direct recovery of the optical absorption field,” Opt. Lett. 41, 4202–4205 (2016).
[Crossref] [PubMed]

R. Hochuli, S. Powell, S. Arridge, and B. Cox, “Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radiance,” J. Biomed. Opt. 21, 126004 (2016).
[Crossref] [PubMed]

A. Pulkkinen, B. T. Cox, S. R. Arridge, H. Goh, J. P. Kaipio, and T. Tarvainen, “Direct estimation of optical parameters from photoacoustic time series in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 35, 2497–2508 (2016).
[Crossref]

2015 (5)

M. Haltmeier, L. Neumann, and S. Rabanser, “Single-stage reconstruction algorithm for quantitative photoacoustic tomography,” Inv. Probl. 31, 065005 (2015).
[Crossref]

H. Gao, J. Feng, and L. Song, “Limited-view multi-source quantitative photoacoustic tomography,” Inv. Probl. 31, 065004 (2015).
[Crossref]

T. Ding, K. Ren, and S. Vallélian, “A one-step reconstruction algorithm for quantitative photoacoustic imaging,” Inv. Probl. 31, 095005 (2015).
[Crossref]

A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “Quantitative photoacoustic tomography using illuminations from a single direction,” J. Biomed. Opt. 20, 036015 (2015).
[Crossref] [PubMed]

E. Malone, S. Powell, B. T. Cox, and S. Arridge, “Reconstruction-classification method for quantitative photoacoustic tomography,” J. Biomed. Opt. 20, 126004 (2015).
[Crossref] [PubMed]

2014 (7)

N. Song, C. Deumié, and A. Da Silva, “Considering sources and detectors distributions for quantitative photoacoustic tomography,” Biomed. Opt. Express 5, 3960–3974 (2014).
[Crossref] [PubMed]

A. Pulkkinen, V. Kolehmainen, J. P. Kaipio, B. T. Cox, S. R. Arridge, and T. Tarvainen, “Approximate marginalization of unknown scattering in quantitative photoacoustic tomography,” Inv. Probl. Imag. 8, 811–829 (2014).
[Crossref]

W. Naetar and O. Scherzer, “Quantitative photoacoustic tomography with piecewise constant material parameters,” SIAM J. Imaging Sci. 7, 1755–1774 (2014).
[Crossref]

X. Zhang, W. Zhou, X. Zhang, and H. Gao, “Forward-backward splitting method for quantitative photoacoustic tomography,” Inv. Probl. 30, 125012 (2014).
[Crossref]

A. V. Mamonov and K. Ren, “Quantitative photoacoustic imaging in radiative transport regime,” Comm. Math. Sci. 12, 201–234 (2014).
[Crossref]

A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “A Bayesian approach to spectral quantitative photoacoustic tomography,” Inv. Probl. 30, 065012 (2014).
[Crossref]

J. Xia and L. V. Wang, “Small-animal whole-body photoacoustic tomography: a review,” Phys. Med. Biol. 61, 1380–1389 (2014).

2013 (4)

T. Tarvainen, A. Pulkkinen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Bayesian image reconstruction in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 32, 2287–2298 (2013).
[Crossref]

X. Li and H. Jiang, “Impact of inhomogeneous optical scattering coefficient distribution on recovery of optical absorption coefficient maps using tomographic photoacoustic data,” Phys. Med. Biol. 58, 999–1011 (2013).
[Crossref] [PubMed]

K. Ren, H. Gao, and H. Zhao, “A hybrid reconstruction method for quantitative PAT,” SIAM J. Imag. Sci. 6, 32–55 (2013).
[Crossref]

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Inv. Probl. 29, 075006 (2013).
[Crossref]

2012 (8)

C. Xu, P. D. Kumavor, A. Aguirre, and Q. Zhu, “Investigation of a diffuse optical measurements-assisted quantitative photoacoustic tomographic method in reflection geometry,” J. Biomed. Opt. 17, 061213 (2012).
[Crossref] [PubMed]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Inv. Probl. 28, 084009 (2012).
[Crossref]

S. Bu, Z. Liu, T. Shiina, K. Kondo, M. Yamakawa, K. Fukutani, Y. Someda, and Y. Asao, “Model-based reconstruction integrated with fluence compensation for photoacoustic tomography,” IEEE Trans. Biomed. Eng. 59, 1354–1363 (2012).
[Crossref] [PubMed]

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 31, 1922–1928 (2012).
[Crossref]

P. Shao, T. Harrison, and R. J. Zemp, “Iterative algorithm for multiple illumination photoacoustic tomography (MIPAT) using ultrasound channel data,” Biomed. Opt. Express 3, 3240–3248 (2012).
[Crossref] [PubMed]

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imgaging: a review,” J. Biomed. Opt. 17, 061202 (2012).
[Crossref]

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in a diffusive regime,” Inv. Probl. 28, 025010 (2012).
[Crossref]

D. Razansky, “Multispectral optoacoustic tomography - volumetric color hearing in real time,” IEEE Sel.Top. Quantum Electron. 18, 1234–1243 (2012).
[Crossref]

2011 (3)

P. Beard, “Biomedical photoacoustic imaging,” Interface Focus 1, 602–631 (2011).
[Crossref]

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Probl. 27, 075003 (2011).
[Crossref]

X. Li, L. Xi, R. Jiang, L. Yao, and H. Jiang, “Integrated diffuse optical tomography and photoacoustic tomography: phantom validations,” Biomed. Opt. Express 2, 2348–2353 (2011).
[Crossref] [PubMed]

2010 (2)

2009 (5)

C. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. 54, R59–R97 (2009).
[Crossref] [PubMed]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am A 26, 443–455 (2009).
[Crossref]

D. Razansky, J. Baeten, and V. Ntziachristos, “Sensitivity of molecular target detection by multispectral optoacoustic tomography (MSOT),” Med. Phys. 36, 939–945 (2009).
[Crossref] [PubMed]

T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, and K. H. Englmeier, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009).
[Crossref]

L. Yao, Y. Sun, and H. Jiang, “Quantitative photoacoustic tomography based on the radiative transfer equation,” Opt. Lett. 34, 1765–1767 (2009).
[PubMed]

2008 (1)

2006 (2)

2005 (2)

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
[Crossref]

B. T. Cox and P. C. Beard, “Fast calculation of pulsed photoacoustic fields in fluids using k-space methods,” J. Acoust. Soc. Am. 117, 3616–3627 (2005).
[Crossref] [PubMed]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
[Crossref]

1995 (1)

1993 (1)

S. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[Crossref] [PubMed]

Aguirre, A.

C. Xu, P. D. Kumavor, A. Aguirre, and Q. Zhu, “Investigation of a diffuse optical measurements-assisted quantitative photoacoustic tomographic method in reflection geometry,” J. Biomed. Opt. 17, 061213 (2012).
[Crossref] [PubMed]

Arridge, S.

R. Hochuli, S. Powell, S. Arridge, and B. Cox, “Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radiance,” J. Biomed. Opt. 21, 126004 (2016).
[Crossref] [PubMed]

E. Malone, S. Powell, B. T. Cox, and S. Arridge, “Reconstruction-classification method for quantitative photoacoustic tomography,” J. Biomed. Opt. 20, 126004 (2015).
[Crossref] [PubMed]

S. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[Crossref] [PubMed]

B. Cox, T. Tarvainen, and S. Arridge, “Multiple illumination quantitative photoacoustic tomography using transport and diffusion models,” in “Tomography and Inverse Transport Theory (Contemporary Mathematics),”, G. Bal, D. Finch, P. Kuchment, J. Schotland, P. Stefanov, and G. Uhlmann, eds., vol. 559, pp. 1–12, (American Mathematical Society, Providence, 2011).
[Crossref]

Arridge, S. R.

A. Pulkkinen, B. T. Cox, S. R. Arridge, H. Goh, J. P. Kaipio, and T. Tarvainen, “Direct estimation of optical parameters from photoacoustic time series in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 35, 2497–2508 (2016).
[Crossref]

A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “Quantitative photoacoustic tomography using illuminations from a single direction,” J. Biomed. Opt. 20, 036015 (2015).
[Crossref] [PubMed]

A. Pulkkinen, V. Kolehmainen, J. P. Kaipio, B. T. Cox, S. R. Arridge, and T. Tarvainen, “Approximate marginalization of unknown scattering in quantitative photoacoustic tomography,” Inv. Probl. Imag. 8, 811–829 (2014).
[Crossref]

A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “A Bayesian approach to spectral quantitative photoacoustic tomography,” Inv. Probl. 30, 065012 (2014).
[Crossref]

T. Tarvainen, A. Pulkkinen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Bayesian image reconstruction in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 32, 2287–2298 (2013).
[Crossref]

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Inv. Probl. 29, 075006 (2013).
[Crossref]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Inv. Probl. 28, 084009 (2012).
[Crossref]

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imgaging: a review,” J. Biomed. Opt. 17, 061202 (2012).
[Crossref]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am A 26, 443–455 (2009).
[Crossref]

B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866–1875 (2006).
[Crossref] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl. 15, R41–R93 (1999).
[Crossref]

S. R. Arridge and M. Schweiger, “Direct calculation of the moments of the distribution of photon time of flight in tissue with a finite-element method,” Appl. Opt. 34, 2683–2687 (1995).
[Crossref] [PubMed]

Asao, Y.

S. Bu, Z. Liu, T. Shiina, K. Kondo, M. Yamakawa, K. Fukutani, Y. Someda, and Y. Asao, “Model-based reconstruction integrated with fluence compensation for photoacoustic tomography,” IEEE Trans. Biomed. Eng. 59, 1354–1363 (2012).
[Crossref] [PubMed]

Baeten, J.

D. Razansky, J. Baeten, and V. Ntziachristos, “Sensitivity of molecular target detection by multispectral optoacoustic tomography (MSOT),” Med. Phys. 36, 939–945 (2009).
[Crossref] [PubMed]

Bagchi, S.

Bal, G.

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in a diffusive regime,” Inv. Probl. 28, 025010 (2012).
[Crossref]

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Probl. 27, 075003 (2011).
[Crossref]

Banerjee, B.

Beard, P.

Beard, P. C.

J. Weber, P. C. Beard, and S. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nature Methods 13, 639–650 (2016).
[Crossref] [PubMed]

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X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 31, 1922–1928 (2012).
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R. Hochuli, S. Powell, S. Arridge, and B. Cox, “Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radiance,” J. Biomed. Opt. 21, 126004 (2016).
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A. Pulkkinen, B. T. Cox, S. R. Arridge, H. Goh, J. P. Kaipio, and T. Tarvainen, “Direct estimation of optical parameters from photoacoustic time series in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 35, 2497–2508 (2016).
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E. Malone, S. Powell, B. T. Cox, and S. Arridge, “Reconstruction-classification method for quantitative photoacoustic tomography,” J. Biomed. Opt. 20, 126004 (2015).
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X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 31, 1922–1928 (2012).
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M. Haltmeier, L. Neumann, and S. Rabanser, “Single-stage reconstruction algorithm for quantitative photoacoustic tomography,” Inv. Probl. 31, 065005 (2015).
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A. Pulkkinen, B. T. Cox, S. R. Arridge, H. Goh, J. P. Kaipio, and T. Tarvainen, “Direct estimation of optical parameters from photoacoustic time series in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 35, 2497–2508 (2016).
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A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “Quantitative photoacoustic tomography using illuminations from a single direction,” J. Biomed. Opt. 20, 036015 (2015).
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T. Tarvainen, A. Pulkkinen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Bayesian image reconstruction in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 32, 2287–2298 (2013).
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A. Pulkkinen, V. Kolehmainen, J. P. Kaipio, B. T. Cox, S. R. Arridge, and T. Tarvainen, “Approximate marginalization of unknown scattering in quantitative photoacoustic tomography,” Inv. Probl. Imag. 8, 811–829 (2014).
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B. A. Kaplan, J. Buchmann, S. Prohaska, and J. Laufer, “Monte-Carlo-based inversion scheme for 3D quantitative photoacoustic tomography,” in “Photons Plus Ultrasound: Imaging and Sensing 2017, Proc of SPIE,”, A. Oraevsky and L. Wang, eds., vol. 10064, pp. 100645J, (2017).

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B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imgaging: a review,” J. Biomed. Opt. 17, 061202 (2012).
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A. Hannukainen, N. Hyvönen, H. Majander, and T. Tarvainen, “Efficient inclusion of total variation type priors in quantitative photoacoustic tomography,” SIAM J. Imag. Sci. 9, 1132–1153 (2016).
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M. Haltmeier, L. Neumann, and S. Rabanser, “Single-stage reconstruction algorithm for quantitative photoacoustic tomography,” Inv. Probl. 31, 065005 (2015).
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X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 31, 1922–1928 (2012).
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H. Gao, H. Zhao, and S. Osher, “Bregman methods in quantitative photoacoustic tomography,” UCLA CAM Report10–24 (2010).

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R. Hochuli, S. Powell, S. Arridge, and B. Cox, “Quantitative photoacoustic tomography using forward and adjoint Monte Carlo models of radiance,” J. Biomed. Opt. 21, 126004 (2016).
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E. Malone, S. Powell, B. T. Cox, and S. Arridge, “Reconstruction-classification method for quantitative photoacoustic tomography,” J. Biomed. Opt. 20, 126004 (2015).
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B. A. Kaplan, J. Buchmann, S. Prohaska, and J. Laufer, “Monte-Carlo-based inversion scheme for 3D quantitative photoacoustic tomography,” in “Photons Plus Ultrasound: Imaging and Sensing 2017, Proc of SPIE,”, A. Oraevsky and L. Wang, eds., vol. 10064, pp. 100645J, (2017).

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A. Pulkkinen, B. T. Cox, S. R. Arridge, H. Goh, J. P. Kaipio, and T. Tarvainen, “Direct estimation of optical parameters from photoacoustic time series in quantitative photoacoustic tomography,” IEEE Trans. Med. Imag. 35, 2497–2508 (2016).
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A. Pulkkinen, B. T. Cox, S. R. Arridge, J. P. Kaipio, and T. Tarvainen, “Quantitative photoacoustic tomography using illuminations from a single direction,” J. Biomed. Opt. 20, 036015 (2015).
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T. Ding, K. Ren, and S. Vallélian, “A one-step reconstruction algorithm for quantitative photoacoustic imaging,” Inv. Probl. 31, 095005 (2015).
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A. V. Mamonov and K. Ren, “Quantitative photoacoustic imaging in radiative transport regime,” Comm. Math. Sci. 12, 201–234 (2014).
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J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005).
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T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, and K. H. Englmeier, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009).
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T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Inv. Probl. 29, 075006 (2013).
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W. Naetar and O. Scherzer, “Quantitative photoacoustic tomography with piecewise constant material parameters,” SIAM J. Imaging Sci. 7, 1755–1774 (2014).
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T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, and K. H. Englmeier, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009).
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S. Bu, Z. Liu, T. Shiina, K. Kondo, M. Yamakawa, K. Fukutani, Y. Someda, and Y. Asao, “Model-based reconstruction integrated with fluence compensation for photoacoustic tomography,” IEEE Trans. Biomed. Eng. 59, 1354–1363 (2012).
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Figures (4)

Fig. 1
Fig. 1

Absorption (columns 1–3) and scattering (columns 4–6) distributions in different size domains: 20 mm ×20 mm (first row), 40 mm×40 mm (second row) and 60 mm×60 mm (third row). Columns from left to right: simulated absorption (first column), reconstructed absorption obtained using the augmented (second column) and conventional (third column) QPAT, simulated scattering (fourth column), reconstructed scattering obtained using augmented (fifth column) and conventional (sixth column) QPAT.

Fig. 2
Fig. 2

Mean relative errors for absorption Eμa (left image) and scattering E μ s (right image) against iteration obtained with augmented QPAT and conventional QPAT in 20 mm ×20 mm domain.

Fig. 3
Fig. 3

Absorption (columns 1–3) and scattering (columns 4–6) parameters with different distributions of the optical parameters (rows 1–4). Columns from left to right: simulated absorption (first column), reconstructed absorption obtained using the augmented (second column) and conventional (third column) QPAT, simulated scattering (fourth column), reconstructed scattering obtained using the augmented (fifth column) and conventional (sixth column) QPAT.

Fig. 4
Fig. 4

Absorption (first row), scattering (second row) and Grüneisen (third row) parameters. Columns from left to right: simulated distributions (first column), reconstructions obtained using the augmented QPAT (second column), and reconstructions obtained using the conventional QPAT with a constant (third column) and correct (fourth column) values of the Grüneisen parameter.

Tables (5)

Tables Icon

Table 1 The number of nodes and elements in the FE-discretisations used in simulating the data.

Tables Icon

Table 2 The number of nodes and elements in the FE-discretisations used in the reconstructions.

Tables Icon

Table 3 The mean and standard deviation of the prior distributions for the absorption ημa (mm−1), σμa (mm−1), scattering η μ s (mm−1), σ μ s (mm−1) and Grüneisen parameter ηG (unitless), σG (unitless), and the characteristic length scale ξ(mm) used in the simulations.

Tables Icon

Table 4 Mean relative errors for absorption Eμa (%) and scattering E μ s (%) obtained with augmented QPAT and conventional QPAT.

Tables Icon

Table 5 Mean relative errors for absorption Eμa (%), scattering E μ s (%) and Grüneisen parameter EG (%) obtained with augmented QPAT and conventional QPAT.

Equations (44)

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1 c Φ ( r , t ) t 1 d ( μ a ( r ) + μ s ( r ) ) + Φ ( r , t ) + μ a ( r ) Φ ( r , t ) = 0 , r Ω
Φ ( r , t ) + 1 2 γ d 1 d ( μ a ( r ) + μ s ( r ) ) Φ ( r , t ) n ^ = { I s ( r , t ) γ d r j 0 , r Ω \ j
1 d ( μ a ( r ) + μ s ( r ) ) Φ ( r ) + μ a ( r ) Φ ( r ) = 0 , r Ω
Φ ( r ) + 1 2 γ d 1 d ( μ a ( r ) + μ s ( r ) ) Φ ( r ) n ^ = { I s ( r ) γ d , r j 0 , r Ω \ j
H ( r ) = μ a ( r ) Φ ( μ a ( r ) , μ s ( r ) )
p 0 ( r ) = p ( r , t = 0 ) = G ( r ) H ( r )
Γ + ( r , ) = 1 d ( μ a ( r ) + μ s ( r ) ) Φ ( μ a ( r ) , μ s ( r ) , ) n ^ = 2 γ n Φ ( μ a ( r ) , μ s ( r ) , ) .
i ω c Φ ( r , ω ) 1 d ( μ a ( r ) + μ s ( r ) ) Φ ( r , ω ) + μ a ( r ) Φ ( r , ω ) = 0 , r Ω
Φ ( r , ω ) + 1 2 γ d 1 d ( μ a ( r ) + μ s ( r ) ) Φ ( r , ω ) n ^ = { I s ( r , ω ) γ d , r j 0 , r Ω \ j
H meas = H ( μ a , μ s ) + e qpat
π ( μ a , μ s | H meas ) π ( H meas | μ a , μ s ) π ( μ a , μ s )
π ( μ a , μ s | H meas ) exp { 1 2 ( H meas H ( μ a , μ s ) η e , qpat ) T Γ e , qpat 1 ( H meas H ( μ a , μ s ) η e , qpat ) 1 2 ( μ a , η μ a ) T Γ μ a 1 ( μ a , η μ a ) 1 2 ( μ s η μ s ) T Γ μ s 1 ( μ s η μ s ) } .
( μ ^ a , μ ^ s ) = arg max ( μ a , μ s ) π ( μ a , μ s | H meas ) = arg min ( μ a , μ s ) { 1 2 L e , qpat ( H meas H ( μ a , μ s ) ) η e , qpat 2 2 + 1 2 L μ a ( μ a η μ a ) 2 2 + 1 2 L μ s ( μ s η μ s ) 2 2 }
Γ meas + = Γ + ( μ a , μ s ) + e dot
( μ ^ a , μ ^ s ) = arg min ( μ a , μ s ) { 1 2 L e , qpat ( H meas H ( μ a , μ s ) η e , qpat ) 2 2 + 1 2 L e , dot ( Γ meas + Γ + ( μ a , μ s ) η e , dot ) 2 2 + 1 2 L μ a ( μ a , η μ a ) 2 2 + L μ s ( μ s , η μ s ) 2 2 }
( μ ^ a , μ ^ s , G ^ ) = arg min ( μ a , μ s , G ) { 1 2 L e , p ( p 0 , meas p 0 ( μ a , μ s , G ) ) η e , p 2 2 + 1 2 L e , dot ( Γ meas + Γ + ( μ a , μ s ) η e , dot ) 2 2 + 1 2 L μ a ( μ a η μ a ) 2 2 + 1 2 L μ s ( μ s , η μ s ) 2 2 + 1 2 L G ( G η G ) 2 2 }
A qpat Φ qpat = b qpat
A dot Φ dot = b dot
K ( p , q ) = Ω 1 d ( μ a , μ s ) φ q ( r ) φ p ( r ) d r
C ( p , q ) = Ω μ a φ q ( r ) φ p ( r ) d r
R ( p , q ) = Ω 2 γ d φ q ( r ) φ p ( r ) d S
Z ( p , q ) = i ω c Ω φ q ( r ) φ p ( r ) d r
b qpat ( p ) = ε j 2 I s φ p ( r ) d S
b dot ( p ) = ε j 2 I s ( ω ) φ p ( r ) d S
x ( i + 1 ) = x ( i ) + s ( i ) ( J ( i ) T L e T L e J ( i ) + L x T L x ) 1 ( J ( i ) T L e T L e ( F meas F ( i ) η e ) L x T L x ( x ( i ) η x ) )
L e = ( L e , qpat 0 0 L e , dot ) , η e = ( η e , qpat η e , dot ) , L x = ( L μ a 0 0 L μ s ) , η x = ( η μ a η μ s ) .
J = ( J μ a qpat J μ s qpat J μ a dot J μ s dot )
j μ a qpat ( k ) = μ a ( k ) qpat A qpat 1 A qpat μ a k A qpat 1 b qpat + χ k qpat A qpat 1 b qpat
j μ a dot ( k ) = dot A dot 1 A dot μ a k A dot 1 b dot
j μ s qpat ( k ) = μ a ( k ) qpat A qpat 1 A qpat μ s k A qpat 1 b qpat
j μ s dot ( k ) = dot A dot 1 A dot μ s k A dot 1 b dot
A qpat μ a k ( p , q ) = A dot μ a k ( p , q ) = 1 d ( μ a k + μ s k ) 2 Ω k φ q ( r ) φ p ( r ) d r + Ω k φ q ( r ) φ p ( r ) d r
A qpat μ s k ( p , q ) = A dot μ s k ( p , q ) = 1 d ( μ a k + μ s k ) 2 Ω k φ q ( r ) φ p ( r ) d r
x ˜ ( i + 1 ) = x ˜ ( i ) + s ( i ) ( J ˜ ( i ) T L ˜ e T L ˜ T J ˜ ( i ) + L ˜ x T L ˜ x ) 1 ( J ˜ ( i ) T L ˜ e T L ˜ e ( F ˜ meas F ˜ ( i ) η ˜ e ) L ˜ x T L ˜ x ( x ˜ ( i ) η x ˜ ) )
L ˜ e = ( L e , p 0 0 L e , dot ) , η ˜ e = ( η e , p η e , dot ) , L ˜ x = ( L μ a 0 0 0 L μ s 0 0 0 L G ) , η x ˜ = ( η μ a η μ s η G ) .
J ˜ = ( J ˜ μ a qpat J ˜ μ s qpat J ˜ G qpat J μ a dot J μ s dot 0 )
j ˜ μ a qpat ( k ) = G ( k ) μ a ( k ) qpat A qpat 1 A qpat μ a k A qpat 1 b qpat + G ( k ) qpat A qpat 1 b qpat
j ˜ μ s qpat ( k ) = G ( k ) μ a ( k ) qpat A qpat 1 A qpat μ s k A qpat 1 b qpat
j ˜ G qpat ( k ) = μ a ( k ) qpat A qpat 1 b qpat
Γ = σ 2 Ξ
Ξ i j = exp ( r i r j / ξ ) ,
E μ a = 100 % μ ^ a μ a TRUE 2 μ a TRUE 2
E μ s = 100 % μ s μ s TRUE 2 μ s TRUE 2
E G = 100 % G ^ G TRUE 2 G TRUE 2