Abstract

Accurately determining the optical properties of multi-layer turbid media using a layered diffusion model is often a difficult task and could be an ill-posed problem. In this study, an iterative algorithm was proposed for solving such problems. This algorithm employed a layered diffusion model to calculate the optical properties of a layered sample at several source-detector separations (SDSs). The optical properties determined at various SDSs were mutually referenced to complete one round of iteration and the optical properties were gradually revised in further iterations until a set of stable optical properties was obtained. We evaluated the performance of the proposed method using frequency domain Monte Carlo simulations and found that the method could robustly recover the layered sample properties with various layer thickness and optical property settings. It is expected that this algorithm can work with photon transport models in frequency and time domain for various applications, such as determination of subcutaneous fat or muscle optical properties and monitoring the hemodynamics of muscle.

© 2014 Optical Society of America

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  1. A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
    [CrossRef] [PubMed]
  2. S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express17(17), 14599–14617 (2009).
    [CrossRef] [PubMed]
  3. A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol.44(11), 2689–2702 (1999).
    [CrossRef] [PubMed]
  4. V. Toronov, A. Webb, J. H. Choi, M. Wolf, L. Safonova, U. Wolf, and E. Gratton, “Study of local cerebral hemodynamics by frequency-domain near-infrared spectroscopy and correlation with simultaneously acquired functional magnetic resonance imaging,” Opt. Express9(8), 417–427 (2001).
    [CrossRef] [PubMed]
  5. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A11(10), 2727–2741 (1994).
    [CrossRef] [PubMed]
  6. S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
    [CrossRef] [PubMed]
  7. S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett.30(23), 3165–3167 (2005).
    [CrossRef] [PubMed]
  8. B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
    [CrossRef] [PubMed]
  9. T. J. Farrell, M. S. Patterson, and M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt.37(10), 1958–1972 (1998).
    [CrossRef] [PubMed]
  10. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
    [CrossRef] [PubMed]
  11. A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt.15(2), 025002 (2010).
    [CrossRef] [PubMed]
  12. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt.37(4), 779–791 (1998).
    [CrossRef] [PubMed]
  13. T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt.39(25), 4733–4745 (2000).
    [CrossRef] [PubMed]
  14. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
    [CrossRef]
  15. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt.40(22), 3810–3821 (2001).
    [CrossRef] [PubMed]
  16. A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl.25(3), 161–165 (2010).
    [CrossRef]
  17. S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
    [CrossRef] [PubMed]

2012 (1)

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

2010 (2)

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt.15(2), 025002 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl.25(3), 161–165 (2010).
[CrossRef]

2009 (2)

S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express17(17), 14599–14617 (2009).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
[CrossRef] [PubMed]

2007 (1)

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

2005 (1)

2003 (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

2001 (2)

2000 (2)

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt.39(25), 4733–4745 (2000).
[CrossRef] [PubMed]

1999 (1)

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol.44(11), 2689–2702 (1999).
[CrossRef] [PubMed]

1998 (2)

1995 (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
[CrossRef]

1994 (1)

Alexandrakis, G.

Bargo, P.

Bays, R.

Busch, D. R.

Butler, J.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Cerussi, A.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Choi, J. H.

Del Bianco, S.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

Dognitz, N.

Durkin, A.

S. H. Tseng, P. Bargo, A. Durkin, and N. Kollias, “Chromophore concentrations, absorption and scattering properties of human skin in-vivo,” Opt. Express17(17), 14599–14617 (2009).
[CrossRef] [PubMed]

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

Durkin, A. J.

S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett.30(23), 3165–3167 (2005).
[CrossRef] [PubMed]

Espinoza, J.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Essenpreis, M.

Faris, G. W.

Farrell, T. J.

Feng, T. C.

Glanzmann, T.

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol.44(11), 2689–2702 (1999).
[CrossRef] [PubMed]

Gratton, E.

Haskell, R. C.

Hayakawa, C.

S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett.30(23), 3165–3167 (2005).
[CrossRef] [PubMed]

Hsiang, D.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
[CrossRef]

Jakeman, P.

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

Kienle, A.

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt.15(2), 025002 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl.25(3), 161–165 (2010).
[CrossRef]

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol.44(11), 2689–2702 (1999).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt.37(4), 779–791 (1998).
[CrossRef] [PubMed]

Kollias, N.

Lanning, R.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Leahy, S.

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

Liemert, A.

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt.15(2), 025002 (2010).
[CrossRef] [PubMed]

A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl.25(3), 161–165 (2010).
[CrossRef]

Martelli, F.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

McAdams, M. S.

McCreesh, K.

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

Mehta, R.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

O’Neill, C.

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

Patterson, M. S.

Pham, T.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Pham, T. H.

Safonova, L.

Sassaroli, A.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

Shah, N.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Spanier, J.

S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett.30(23), 3165–3167 (2005).
[CrossRef] [PubMed]

Spott, T.

Svaasand, L.

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Svaasand, L. O.

Toomey, C.

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

Toronov, V.

Tromberg, B. J.

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, B. J. Tromberg, J. Spanier, and A. J. Durkin, “Quantitative spectroscopy of superficial turbid media,” Opt. Lett.30(23), 3165–3167 (2005).
[CrossRef] [PubMed]

T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt.39(25), 4733–4745 (2000).
[CrossRef] [PubMed]

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary Conditions for the Diffusion Equation in Radiative Transfer,” J. Opt. Soc. Am. A11(10), 2727–2741 (1994).
[CrossRef] [PubMed]

Tsay, T. T.

Tseng, S. H.

van den Bergh, H.

Wagnieres, G.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
[CrossRef]

Webb, A.

Wolf, M.

Wolf, U.

Yamada, Y.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

Zaccanti, G.

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
[CrossRef]

Appl. Opt. (4)

Comput. Meth. Prog. Bio. (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml - Monte-Carlo Modeling of Light Transport in Multilayered Tissues,” Comput. Meth. Prog. Bio.47(2), 131–146 (1995).
[CrossRef]

J. Biomed. Opt. (2)

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt.15(2), 025002 (2010).
[CrossRef] [PubMed]

S. H. Tseng, C. Hayakawa, J. Spanier, and A. J. Durkin, “Investigation of a probe design for facilitating the uses of the standard photon diffusion equation at short source-detector separations: Monte Carlo simulations,” J. Biomed. Opt.14(5), 054043 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Med. Laser Appl. (1)

A. Liemert and A. Kienle, “Bioluminescent light diffusion in a four-layered turbid medium,” Med. Laser Appl.25(3), 161–165 (2010).
[CrossRef]

Neoplasia (1)

B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, and J. Butler, “Non-invasive in vivo characterization of breast tumors using photon migration spectroscopy,” Neoplasia2(1/2), 26–40 (2000).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol.44(11), 2689–2702 (1999).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solution of the time-dependent diffusion equation for layered diffusive media by the eigenfunction method,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(5), 056623 (2003).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg, “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Natl. Acad. Sci. U.S.A.104(10), 4014–4019 (2007).
[CrossRef] [PubMed]

Ultrasound Med. Biol. (1)

S. Leahy, C. Toomey, K. McCreesh, C. O’Neill, and P. Jakeman, “Ultrasound measurement of subcutaneous adipose tissue thickness accurately predicts total and segmental body fat of young adults,” Ultrasound Med. Biol.38(1), 28–34 (2012).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Recovered (a) absorption coefficients and (b) reduced scattering coefficients of the top layer (squares) and bottom layer (triangles) at various source-detector separations. (c) Recovered top layer thickness L1 (asterisks) at various source-detector separations. Solid lines represent benchmark values.

Fig. 2
Fig. 2

χ2 computed at various top layer μa1 and μs1' combinations for (a) top layer thickness = 1 mm, and (b) top layer thickness = 1.5 mm, at source detector separation of 6 mm.

Fig. 3
Fig. 3

χ2 computed at various top layer μa1 and μs1' combinations for (a) top layer thickness = 1 mm, and (b) top layer thickness = 1.5 mm, at source detector separation of 13 mm.

Fig. 4
Fig. 4

Flowchart of the iterative algorithm for solving the two-layered sample problem.

Fig. 5
Fig. 5

Recovered (a) absorption coefficients and (b) reduced scattering coefficients of the top layer (squares) and bottom layer (triangles), as well as (c) recovered top layer thickness L1 (asterisks) versus iteration number for a dermis-fat structure. Solid lines represent benchmark values.

Fig. 6
Fig. 6

Recovered (a) absorption coefficients, (b) reduced scattering coefficients, and (c) layer thickness of the top layer versus iteration number for a dermis-muscle structure. Solid lines represent benchmark values.

Fig. 7
Fig. 7

Recovered (a) absorption coefficients, (b) reduced scattering coefficients, and (c) layer thickness of the top layer at various top layer thickness for a dermis-fat structure. Solid lines represent benchmark values.

Fig. 8
Fig. 8

Recovered (a) absorption coefficients, (b) reduced scattering coefficients, and (c) layer thickness of the top layer versus top layer thickness for a dermis-muscle structure. Solid lines represent benchmark values.

Fig. 9
Fig. 9

Recovered (a) absorption coefficients and (b) reduced scattering coefficients of the top layer (squares) and bottom layer (triangles), as well as (c) recovered top layer thickness L1 (asterisks) versus top layer thickness for a dermis-muscle structure. Sample optical properties recovered using a semi-infinite diffusion model are depicted as empty circles. Solid lines represent benchmark values.

Fig. 10
Fig. 10

Recovered (a) absorption coefficients and (b) reduced scattering coefficients of the muscle layer (triangles) at various muscle layer absorption for a dermis-fat-muscle structure. Crosses are the results obtained with fixed top and middle layer thicknesses. Sample optical properties recovered using a semi-infinite diffusion model are depicted as empty circles. Solid lines represent benchmark values.

Tables (3)

Tables Icon

Table 1 Optical properties of top (μa1, μs1') and bottom (μa2, μs2') layers as well as top layer thickness (L1) recovered at various source-detector separation combinations for a dermis-fat structured sample. Recovery errors are listed below the values.

Tables Icon

Table 2 Optical properties of top (μa1, μs1') and bottom (μa2, μs2') layers as well as top layer thickness (L1) recovered at various source-detector separation combinations for a dermis-fat structured sample. Recovery errors are listed below the values.

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Table 3 Recovered properties of top (μa1, μs1', L1), middle (μa2, μs2', L2), and bottom (μa3, μs3') layers for a dermis-fat-muscle structured sample using the iterative algorithm. Recovery errors are listed below the values.

Equations (8)

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[ 1 c i t + μ ai [ D i (r)] ] Φ i (r,t)= S i (r,t),
Φ i (z, s 1 , s 2 ,t)= Φ i (x,y,z,t)exp[j( s 1 x+ s 2 y)] dxdy .
Φ 1 (z,s)= sinh[ α 1 ( z b1 + z 0 )] D 1 α 1 Z(z,s) N(z,s) sinh[ α 1 ( z 0 z)] D 1 α 1 ,
Z(z,s)= D 1 α 1 cosh[ α 1 ( l 1 z)]+( n 2 2 / n 1 2 ) D 2 α 2 sinh[ α 1 ( l 1 z)],
N(z,s)= D 1 α 1 cosh[ α 1 ( l 1 + z b1 )]+( n 2 2 / n 1 2 ) D 2 α 2 sinh[ α 1 ( l 1 + z b1 )].
Z(z,s)= D 1 α 1 [ D 2 α 2 cosh( α 2 l 2 )+( n 3 2 / n 2 2 ) D 3 α 3 sinh( α 2 l 2 ) ]cosh[ α 1 ( l 1 z)] +( n 2 2 / n 1 2 ) D 2 α 2 [ D 2 α 2 sinh( α 2 l 2 )+( n 3 2 / n 2 2 ) D 3 α 3 cosh( α 2 l 2 ) ] ×sinh[ α 1 ( l 1 z)],
N(z,s)= D 1 α 1 [ D 2 α 2 cosh( α 2 l 2 )+( n 3 2 / n 2 2 ) D 3 α 3 sinh( α 2 l 2 ) ]cosh[ α 1 ( l 1 + z b1 )] +( n 2 2 / n 1 2 ) D 2 α 2 [ D 2 α 2 sinh( α 2 l 2 )+( n 3 2 / n 2 2 ) D 3 α 3 cosh( α 2 l 2 ) ] ×sinh[ α 1 ( l 1 + z b1 )].
R(ρ)= 2π [1 R fres (θ)]cosθ( L 1 /4π) dΩ.

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