Abstract

Backscattered light was used to reconstruct cross-sectional images of absorption distributions in diffuse media. For efficient and accurate reconstruction, the inverse problem was solved for one dimension, thereby yielding the absorption distribution in a depth direction. A cross-sectional image or three- dimensional structure is reconstructed by shifting a source–detector pair along the object surface. The object is divided into imaginary layers to solve the inverse problem. This solution’s accuracy is further improved by solving the problem for two groups of layers successively instead of solving for all layers simultaneously. The technique’s effectiveness was verified using solid phantoms and biological tissues.

© 2009 Optical Society of America

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2008

Y. Minagawa-Kawai, K. Mori, J. C. Hebden, and E. Dupoux, “Optical imaging of infants' neurocognitive development: Recent advances and perspectives,” Dev. Neurobiol. 68, 712-728 (2008).
[PubMed]

T. Namita, Y. Kato, and K. Shimizu, “CT imaging of biological tissue using backscattered light,” in Topical Meeting Biomedical Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper BMD35.

2007

2006

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

2005

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

2004

2003

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties,” Appl. Opt. 42, 4612-4620 (2003).
[CrossRef] [PubMed]

J. Mobley and T. Vo-Vinh, “Optical properties of tissue,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC, 2003), pp. 2-1-2-75.

2002

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by using full time-resolved data,” Appl. Opt. 41, 778-791 (2002).
[CrossRef] [PubMed]

A. Awata, Y. Kato, and K. Shimizu, “Cross-sectional imaging of absorption distribution in biological tissue using backscattered light,” IEICE Trans. E85-D, 124-132 (2002).

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

F. Gao, Y. Tanikawa, H. Zhao, and Y. Yamada, “Semi-three-dimensional algorithm for time-resolved diffuse optical tomography by using the generalized pulse spectrum technique,” Appl. Opt. 41, 7346-7358 (2002).
[CrossRef] [PubMed]

2001

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Determining changes in NIR absorption using a layered model of the human head,” Phys. Med. Biol. 46, 879-896 (2001).
[CrossRef] [PubMed]

Y. Tsuchiya, “Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law,” Phys. Med. Biol. 46, 2067-2084 (2001).
[PubMed]

2000

V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Optical Engineering Press, 2000).

1999

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and first in vivo applications,” Appl. Opt. 38, 2927-2943 (1999).
[CrossRef]

S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

1998

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vision Comput. 16, 703-712 (1998).
[CrossRef]

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

1995

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

1993

S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” Proc. SPIE 2035, 218-229 (1993).
[CrossRef]

S. R. Arridge, “Inverse methods for optical tomography,” in Information Processing in Medical Imaging '93, H. H. Barrett and A. F. Gmitro, eds. (Springer, 1993), pp. 259-277.
[CrossRef]

1990

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317-1334 (1990).
[CrossRef] [PubMed]

1989

1987

P. van der Zee and D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo method,” Adv. Exp. Med. Biol. 215, 179-191 (1987).
[PubMed]

Andersson-Engels, S.

Arridge, S. R.

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628-3638 (2007).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699-1717 (1999).
[CrossRef] [PubMed]

S. R. Arridge, “Inverse methods for optical tomography,” in Information Processing in Medical Imaging '93, H. H. Barrett and A. F. Gmitro, eds. (Springer, 1993), pp. 259-277.
[CrossRef]

S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” Proc. SPIE 2035, 218-229 (1993).
[CrossRef]

Austin, T.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Awata, A.

A. Awata, Y. Kato, and K. Shimizu, “Cross-sectional imaging of absorption distribution in biological tissue using backscattered light,” IEICE Trans. E85-D, 124-132 (2002).

Bassi, A.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Benaron, D. A.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Berger, A. J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Bevilacqua, F.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Boas, D. A.

Butler, J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Cerussi, A. E.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Chance, B.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331-2336(1989).
[CrossRef] [PubMed]

Cheong, W.-F.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Choe, R.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

Colak, S. B.

S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

Cubeddu, R.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Culver, J. P.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

Dale, A. M.

Dam, J. S.

D'Andrea, C.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Delpy, D. T.

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628-3638 (2007).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

P. van der Zee and D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo method,” Adv. Exp. Med. Biol. 215, 179-191 (1987).
[PubMed]

Douek, M.

Duckworth, J. L.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Dupoux, E.

Y. Minagawa-Kawai, K. Mori, J. C. Hebden, and E. Dupoux, “Optical imaging of infants' neurocognitive development: Recent advances and perspectives,” Dev. Neurobiol. 68, 712-728 (2008).
[PubMed]

Durduran, T.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

Eda, H.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Endoh, R.

R. Endoh, A. Suzuki, M. Fujii, and K. Nakayama, “Fundamental study on diffuse reflective optical tomography,” Phys. Med. Biol. 49, 1881-1889 (2004).
[CrossRef] [PubMed]

Enfield, L. C.

Everdell, N.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Everdell, N. L.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317-1334 (1990).
[CrossRef] [PubMed]

Fujii, M.

R. Endoh, A. Suzuki, M. Fujii, and K. Nakayama, “Fundamental study on diffuse reflective optical tomography,” Phys. Med. Biol. 49, 1881-1889 (2004).
[CrossRef] [PubMed]

Gao, F.

Gibson, A.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Gibson, A. P.

Giusto, A.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Grosenick, D.

Hebden, J. C.

Y. Minagawa-Kawai, K. Mori, J. C. Hebden, and E. Dupoux, “Optical imaging of infants' neurocognitive development: Recent advances and perspectives,” Dev. Neurobiol. 68, 712-728 (2008).
[PubMed]

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628-3638 (2007).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
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A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Hillman, E. M. C.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Hintz, S. R.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

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J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
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Holcombe, R. F.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
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S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
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S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

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H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
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A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957-1975 (1995).
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Jakubowski, D.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
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M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation,” J. Biomed. Opt. 12, 034019 (2007).
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T. Namita, Y. Kato, and K. Shimizu, “CT imaging of biological tissue using backscattered light,” in Topical Meeting Biomedical Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper BMD35.

A. Awata, Y. Kato, and K. Shimizu, “Cross-sectional imaging of absorption distribution in biological tissue using backscattered light,” IEICE Trans. E85-D, 124-132 (2002).

Keshtgar, M.

Kuijpers, F. A.

S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

Liebert, A.

Liu, F. W. H.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Macdonald, R.

Maniewski, R.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation,” J. Biomed. Opt. 12, 034019 (2007).
[CrossRef] [PubMed]

Meek, J. H.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
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Y. Minagawa-Kawai, K. Mori, J. C. Hebden, and E. Dupoux, “Optical imaging of infants' neurocognitive development: Recent advances and perspectives,” Dev. Neurobiol. 68, 712-728 (2008).
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J. Mobley and T. Vo-Vinh, “Optical properties of tissue,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC, 2003), pp. 2-1-2-75.

Moesta, K. T.

Möller, M.

Mori, K.

Y. Minagawa-Kawai, K. Mori, J. C. Hebden, and E. Dupoux, “Optical imaging of infants' neurocognitive development: Recent advances and perspectives,” Dev. Neurobiol. 68, 712-728 (2008).
[PubMed]

Nakayama, K.

R. Endoh, A. Suzuki, M. Fujii, and K. Nakayama, “Fundamental study on diffuse reflective optical tomography,” Phys. Med. Biol. 49, 1881-1889 (2004).
[CrossRef] [PubMed]

Namita, T.

T. Namita, Y. Kato, and K. Shimizu, “CT imaging of biological tissue using backscattered light,” in Topical Meeting Biomedical Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper BMD35.

Niu, H.

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vision Comput. 16, 703-712 (1998).
[CrossRef]

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J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

Obrig, H.

Oda, I.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Oda, M.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Ohmae, E.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

Oikawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

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V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317-1334 (1990).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331-2336(1989).
[CrossRef] [PubMed]

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317-1334 (1990).
[CrossRef] [PubMed]

Pifferi, A.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Richardson, C.

Rinneberg, H.

Rinneberg, H. H.

Sassaroli, A.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Sawosz, P.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation,” J. Biomed. Opt. 12, 034019 (2007).
[CrossRef] [PubMed]

Schlag, P. M.

Schweiger, M.

L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628-3638 (2007).
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S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” Proc. SPIE 2035, 218-229 (1993).
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Shah, N.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Shimizu, K.

T. Namita, Y. Kato, and K. Shimizu, “CT imaging of biological tissue using backscattered light,” in Topical Meeting Biomedical Optics, OSA Technical Digest Series (Optical Society of America, 2008), paper BMD35.

A. Awata, Y. Kato, and K. Shimizu, “Cross-sectional imaging of absorption distribution in biological tissue using backscattered light,” IEICE Trans. E85-D, 124-132 (2002).

Slemp, A.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
[CrossRef] [PubMed]

Spilman, S. D.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

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A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Steinbrink, J.

Stevenson, D. K.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Suzuki, A.

R. Endoh, A. Suzuki, M. Fujii, and K. Nakayama, “Fundamental study on diffuse reflective optical tomography,” Phys. Med. Biol. 49, 1881-1889 (2004).
[CrossRef] [PubMed]

Suzuki, T.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

Swartling, J.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of spatially and temporally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties,” Appl. Opt. 42, 4612-4620 (2003).
[CrossRef] [PubMed]

Takeda, M.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

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H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Tanikawa, Y.

Tittel, F. K.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Torricelli, A.

A. Bassi, L. Spinelli, C. D'Andrea, A. Giusto, J. Swartling, A. Pifferi, A. Torricelli, and R. Cubeddu, “Feasibility of white-light time-resolved optical mammography,” J. Biomed. Opt. 11, 054035 (2006).
[CrossRef] [PubMed]

Tromberg, B. J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
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Y. Tsuchiya, “Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law,” Phys. Med. Biol. 46, 2067-2084 (2001).
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H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Tsunazawa, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
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V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Optical Engineering Press, 2000).

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Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

van der Linden, E. S.

S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

van der Mark, M. B.

S. B. Colak, M. B. van der Mark, G. W.'t Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Quantum. Electron. 5, 1143-1158 (1999).
[CrossRef]

van der Zee, P.

P. van der Zee and D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo method,” Adv. Exp. Med. Biol. 215, 179-191 (1987).
[PubMed]

van Houten, J. P.

S. R. Hintz, D. A. Benaron, J. P. van Houten, J. L. Duckworth, F. W. H. Liu, S. D. Spilman, D. K. Stevenson, and W.-F. Cheong, “Stationary headband for clinical time-of-flight optical imaging at the bedside,” Photochem. Photobiol. 68, 361-369 (1998).
[CrossRef] [PubMed]

Villringer, A.

Vo-Vinh, T.

J. Mobley and T. Vo-Vinh, “Optical properties of tissue,” in Biomedical Photonics Handbook, T. Vo-Dinh, ed. (CRC, 2003), pp. 2-1-2-75.

Wabnitz, H.

Wada, Y.

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Wilson, B. C.

Wyatt, J. S.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, and G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317-1334 (1990).
[CrossRef] [PubMed]

Yamada, Y.

Yamanaka, T.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

Yamashita, D.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
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Yamashita, Y.

Y. Ueda, T. Yamanaka, D. Yamashita, T. Suzuki, E. Ohmae, M. Oda, and Y. Yamashita, “Reflectance diffuse optical tomography: Its application to human brain mapping,” Jpn. J. Appl. Phys. 44, L1203-L1206 (2005).
[CrossRef]

H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takeda, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595-3602 (1999).
[CrossRef]

Yodh, A. G.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
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Yusof, R. M.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47, 4155-4166(2002).
[CrossRef] [PubMed]

Zhang, H.

F. Gao, H. Niu, H. Zhao, and H. Zhang, “The forward and inverse models in time-resolved optical tomography imaging and their finite-element method solutions,” Image Vision Comput. 16, 703-712 (1998).
[CrossRef]

Zhao, H.

Zolek, N.

M. Kacprzak, A. Liebert, P. Sawosz, N. Zolek, and R. Maniewski, “Time-resolved optical imager for assessment of cerebral oxygenation,” J. Biomed. Opt. 12, 034019 (2007).
[CrossRef] [PubMed]

Zubkov, L.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235-247 (2003).
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Acad. Radiol.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butler, R. F. Holcombe, and B. J. Tromberg, “Sources of absorption and scattering contrast for near-infrared optical mammography,” Acad. Radiol. 8, 211-218 (2001).
[CrossRef] [PubMed]

Adv. Exp. Med. Biol.

P. van der Zee and D. T. Delpy, “Simulation of the point spread function for light in tissue by a Monte Carlo method,” Adv. Exp. Med. Biol. 215, 179-191 (1987).
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Appl. Opt.

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331-2336(1989).
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F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by using full time-resolved data,” Appl. Opt. 41, 778-791 (2002).
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D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, and P. M. Schlag, “Development of a time-domain optical mammography and first in vivo applications,” Appl. Opt. 38, 2927-2943 (1999).
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L. C. Enfield, A. P. Gibson, N. L. Everdell, D. T. Delpy, M. Schweiger, S. R. Arridge, C. Richardson, M. Keshtgar, M. Douek, and J. C. Hebden, “Three-dimensional time-resolved optical mammography of the uncompressed breast,” Appl. Opt. 46, 3628-3638 (2007).
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F. Gao, Y. Tanikawa, H. Zhao, and Y. Yamada, “Semi-three-dimensional algorithm for time-resolved diffuse optical tomography by using the generalized pulse spectrum technique,” Appl. Opt. 41, 7346-7358 (2002).
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Figures (18)

Fig. 1
Fig. 1

Model of photon propagation in a layered scattering medium.

Fig. 2
Fig. 2

Principle of repetitive solution for two layer groups.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Example of measured N ( t ) .

Fig. 5
Fig. 5

Example of a TPD of each layer obtained in Monte Carlo simulation.

Fig. 6
Fig. 6

Examples of estimated results. Broken and solid lines, respectively, show given and estimated distributions of absorption coefficient in the depth direction.

Fig. 7
Fig. 7

Extent of estimation error. Points and error bars show means and standard deviations of estimation error for 30 different configurations of absorption distributions.

Fig. 8
Fig. 8

Comparison of reconstruction with different layer division: (a) 6-layer division, (b) 12-layer division.

Fig. 9
Fig. 9

Arrangement of a source–detector pair and structure of solid phantom. S1–S5 indicate sampling lines for quantitative analysis.

Fig. 10
Fig. 10

Results of cross-sectional imaging: (a) structure of model phantom, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 11
Fig. 11

Depth profiles of absorption coefficient of Fig. 10 along lines S2 and S4 presented in Fig. 9: (a) profile of model phantom, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 12
Fig. 12

Comparison of estimation error: S1–S5 indicate sampling lines in Fig. 9. Bar graphs and error bars, respectively, show means and standard deviations of estimation errors for six layers.

Fig. 13
Fig. 13

Structure of tissue sample that consists of two gel absorbers in pork or chicken breast meat: (a) plane view, (b) cross- sectional view. S1–S5 indicate sampling lines for quantitative analysis.

Fig. 14
Fig. 14

CT images obtained with pork: (a) structure of specimen, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 15
Fig. 15

CT images obtained with chicken breast: (a) structure of specimen, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 16
Fig. 16

Depth profiles of absorption coefficient of Fig. 14 along line S4 presented in Fig. 13: (a) profile of specimen, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 17
Fig. 17

Depth profile of absorption coefficient of Fig. 15 along line S4 presented in Fig. 13: (a) profile of specimen, (b) simultaneous inversion of multilayers, (c) proposed technique.

Fig. 18
Fig. 18

Comparison of estimation error: (a) specimens with pork, (b) specimens with chicken breast. S1–S5 represent sampling lines in Fig. 13. Bar graphs and error bars, respectively, show means and standard deviations of estimation errors for six layers.

Equations (12)

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L k ( t ) = n = 1 N ( t ) l n k ,
I ( t ) = N ( t ) exp { k = 1 M μ k [ L k ( t ) N ( t ) ] } ,
N ( t ) ln [ N ( t ) I ( t ) ] = k = 1 M μ k L k ( t ) .
N ( t ) ln [ N ( t ) I ( t ) ] = μ 1 L 1 ( t ) + μ 2 L 2 ( t ) + μ 3 L 3 ( t ) .
{ 0 τ ln [ N ( t ) I ( t ) ] L 1 ( t ) d t = μ 1 κ 11 + μ 2 κ 12 + μ 3 κ 13 0 τ ln [ N ( t ) I ( t ) ] L 2 ( t ) d t = μ 1 κ 21 + μ 2 κ 22 + μ 3 κ 23 0 τ ln [ N ( t ) I ( t ) ] L 3 ( t ) d t = μ 1 κ 31 + μ 2 κ 32 + μ 3 κ 33 ,
L i ( t ) = L i ( t ) N ( t ) , κ i j = 0 τ L i ( t ) L j ( t ) d t , i , j = 1 , 2 , 3 .
N ( t ) ln [ N ( t ) I ( t ) ] = μ α L α ( t ) + μ β L β ( t ) ,
μ α = i = 1 k L i ( t ) μ i i = 1 k L i ( t ) , μ β = i = k + 1 M L i ( t ) μ i i = k + 1 M L i ( t ) L α ( t ) = i = 1 k L i ( t ) , L β ( t ) = i = k + 1 M L i ( t ) .
{ 0 τ ln N ( t ) I ( t ) L α ( t ) d t = μ α 0 τ L α 2 ( t ) d t + μ β 0 τ L α ( t ) L β ( t ) d t 0 τ ln N ( t ) I ( t ) L β ( t ) d t = μ α 0 τ L α ( t ) L β ( t ) d t + μ β 0 τ L β 2 ( t ) d t ,
L i = L i ( t ) N ( t ) .
μ α = i = 1 k μ i 0 τ L i ( t ) d t i = 1 k 0 τ L i ( t ) d t , μ β = i = k + 1 M μ i 0 τ L i ( t ) d t i = k + 1 M 0 τ L i ( t ) d t .
μ α = ( μ 1 0 τ L 1 ( t ) d t + μ 2 0 τ L 2 ( t ) d t ) / ( 0 τ L 1 ( t ) d t + 0 τ L 2 ( t ) d t )

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