Abstract

The inverse problem in three-layered scattering media is investigated using simulations. Instead of using the common diffusion approximation, the light propagation is modeled using the spherical harmonics ($P_N$) approximation in the time domain. The optical properties are determined by fitting the $P_3$ approximation to solutions obtained by the $P_7$ approximation, representing an almost exact solution to the radiative transfer equation. Poisson noise is added to the data to simulate time-correlated single photon counting measurements. It is shown that, with simulated two-distance measurements, it is possible to derive the optical properties accurately, especially the absorption coefficient of the third layer and the reduced scattering coefficient of the upper layer. In the case of unknown layer thicknesses, solutions with different parameters but very similar reflectance curves are found, which can lead to larger errors in the identified optical properties. For known layer thicknesses, the retrieval of all properties improves. Altogether, the use of the spherical harmonics approximation enables identifying optical properties from time domain reflectance data much more accurately than using the diffusion equation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
    [Crossref]
  2. T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
    [Crossref]
  3. S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).
  4. W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
    [Crossref]
  5. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28(12), 2210–2215 (1989).
    [Crossref]
  6. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
    [Crossref]
  7. 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. A 11(10), 2727–2741 (1994).
    [Crossref]
  8. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
    [Crossref]
  9. I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
    [Crossref]
  10. A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37(4), 779–791 (1998).
    [Crossref]
  11. G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37(31), 7401–7409 (1998).
    [Crossref]
  12. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18(4), 821–830 (2001).
    [Crossref]
  13. Y.-K. Liao and S.-H. Tseng, “Reliable recovery of the optical properties of multi-layer turbid media by iteratively using a layered diffusion model at multiple source-detector separations,” Biomed. Opt. Express 5(3), 975–989 (2014).
    [Crossref]
  14. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37(28), 6852–6862 (1998).
    [Crossref]
  15. 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]
  16. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25(20), 1508–1510 (2000).
    [Crossref]
  17. F. Martelli, S. D. Bianco, and G. Zaccanti, “Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measure ments,” Opt. Lett. 28(14), 1236–1238 (2003).
    [Crossref]
  18. 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(3), 879–896 (2001).
    [Crossref]
  19. F. Martelli, S. D. Bianco, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29(17), 2037–2039 (2004).
    [Crossref]
  20. F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
    [Crossref]
  21. A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
    [Crossref]
  22. E. Alerstam, T. Svensson, S. Andersson-Engels, L. Spinelli, D. Contini, A. D. Mora, A. Tosi, F. Zappa, and A. Pifferi, “Single-fiber diffuse optical time-of-flight spectroscopy,” Opt. Lett. 37(14), 2877–2879 (2012).
    [Crossref]
  23. A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, and S. L. Jacques, “Time-resolved photon emission from layered turbid media,” Appl. Opt. 35(4), 719–728 (1996).
    [Crossref]
  24. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
    [Crossref]
  25. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
    [Crossref]
  26. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
    [Crossref]
  27. A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29(7), 1475–1481 (2012).
    [Crossref]
  28. A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3(1), 2018 (2013).
    [Crossref]
  29. A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
    [Crossref]
  30. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the ${\textrm {P}}_{{3}}$P3 approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).
    [Crossref]
  31. A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
    [Crossref]
  32. L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
    [Crossref]
  33. L. Liu, W. Wan, J. Li, H. Zhao, and F. Gao, “Simultaneous recovery of a full set of optical properties in turbid media using incomplete P5 approximation to CW radiance,” Opt. Lett. 43(17), 4188–4191 (2018).
    [Crossref]
  34. S. Agarwal, K. Mierle, and et al., “Ceres Solver,” http://ceres-solver.org .
  35. S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
    [Crossref]
  36. D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).
  37. A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15(2), 025002 (2010).
    [Crossref]
  38. A. Liemert and A. Kienle, “Explicit solutions of the radiative transport equation in the P3 approximation,” Med. Phys. 41(11), 111916 (2014).
    [Crossref]
  39. 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]
  40. H. García, G. Baez, and J. Pomarico, “Simultaneous retrieval of optical and geometrical parameters of multilayered turbid media via state-estimation algorithms,” Biomed. Opt. Express 9(8), 3953–3973 (2018).
    [Crossref]

2018 (3)

2017 (1)

A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
[Crossref]

2014 (2)

2013 (3)

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3(1), 2018 (2013).
[Crossref]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref]

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
[Crossref]

2012 (2)

2010 (3)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

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

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref]

2009 (1)

2008 (1)

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

2007 (2)

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
[Crossref]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

2006 (1)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
[Crossref]

2004 (2)

2003 (1)

2001 (3)

2000 (2)

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]

1998 (3)

1996 (1)

1994 (1)

1992 (2)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
[Crossref]

1990 (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

1989 (1)

Agarwal, S.

S. Agarwal, K. Mierle, and et al., “Ceres Solver,” http://ceres-solver.org .

Alerstam, E.

Alexandrakis, G.

Andersson-Engels, S.

Arridge, S. R.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Baez, G.

Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref]

Bassi, A.

Bays, R.

Bianco, S. D.

Bykov, A.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Chin, L. C. L.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref]

Contini, D.

E. Alerstam, T. Svensson, S. Andersson-Engels, L. Spinelli, D. Contini, A. D. Mora, A. Tosi, F. Zappa, and A. Pifferi, “Single-fiber diffuse optical time-of-flight spectroscopy,” Opt. Lett. 37(14), 2877–2879 (2012).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Cova, S.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Cubeddu, R.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

F. Martelli, S. D. Bianco, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29(17), 2037–2039 (2004).
[Crossref]

Culver, J. P.

Dalla Mora, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Dayan, I.

I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
[Crossref]

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Dögnitz, N.

Durduran, T.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref]

Farrell, T. J.

Feng, T.-C.

Foster, T. H.

Gao, F.

García, H.

Gardner, A. R.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
[Crossref]

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]

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37(28), 6852–6862 (1998).
[Crossref]

Harju, M.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Haskell, R. C.

Havlin, S.

I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
[Crossref]

Hielscher, A. H.

Hull, E. L.

Ishimaru, A.

Jacques, S. L.

Kienle, A.

A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
[Crossref]

A. Liemert and A. Kienle, “Explicit solutions of the radiative transport equation in the P3 approximation,” Med. Phys. 41(11), 111916 (2014).
[Crossref]

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3(1), 2018 (2013).
[Crossref]

A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29(7), 1475–1481 (2012).
[Crossref]

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

A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37(28), 6852–6862 (1998).
[Crossref]

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

Kim, A. D.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
[Crossref]

Kiviniemi, V.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Korhonen, V.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Li, J.

Liao, Y.-K.

Liemert, A.

A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
[Crossref]

A. Liemert and A. Kienle, “Explicit solutions of the radiative transport equation in the P3 approximation,” Med. Phys. 41(11), 111916 (2014).
[Crossref]

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3(1), 2018 (2013).
[Crossref]

A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29(7), 1475–1481 (2012).
[Crossref]

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

Liu, H.

Liu, L.

Machida, M.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
[Crossref]

Markel, V. A.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
[Crossref]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
[Crossref]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
[Crossref]

Martelli, F.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
[Crossref]

F. Martelli, S. D. Bianco, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29(17), 2037–2039 (2004).
[Crossref]

F. Martelli, S. D. Bianco, and G. Zaccanti, “Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measure ments,” Opt. Lett. 28(14), 1236–1238 (2003).
[Crossref]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25(20), 1508–1510 (2000).
[Crossref]

McAdams, M. S.

Meglinski, I.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Mierle, K.

S. Agarwal, K. Mierle, and et al., “Ceres Solver,” http://ceres-solver.org .

Mora, A. D.

Myllylä, T.

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

Nieto-Vesperinas, M.

Ntziachristos, V.

O’Connor, D. V.

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

Obrig, H.

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(3), 879–896 (2001).
[Crossref]

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
[Crossref]

Panasyuk, G. Y.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
[Crossref]

Pattanayak, D. N.

Patterson, M. S.

Pham, T. H.

Phillips, D.

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

Pifferi, A.

Pomarico, J.

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Reitzle, D.

A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
[Crossref]

Rinneberg, H.

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(3), 879–896 (2001).
[Crossref]

Ripoll, J.

Sassaroli, A.

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
[Crossref]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25(20), 1508–1510 (2000).
[Crossref]

Schotland, J. C.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26(5), 1291–1300 (2009).
[Crossref]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
[Crossref]

Spinelli, L.

E. Alerstam, T. Svensson, S. Andersson-Engels, L. Spinelli, D. Contini, A. D. Mora, A. Tosi, F. Zappa, and A. Pifferi, “Single-fiber diffuse optical time-of-flight spectroscopy,” Opt. Lett. 37(14), 2877–2879 (2012).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Spott, T.

Steinbrink, J.

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(3), 879–896 (2001).
[Crossref]

Svaasand, L. O.

Svensson, T.

Taroni, P.

Tittel, F. K.

Torricelli, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

F. Martelli, S. D. Bianco, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29(17), 2037–2039 (2004).
[Crossref]

Tosi, A.

E. Alerstam, T. Svensson, S. Andersson-Engels, L. Spinelli, D. Contini, A. D. Mora, A. Tosi, F. Zappa, and A. Pifferi, “Single-fiber diffuse optical time-of-flight spectroscopy,” Opt. Lett. 37(14), 2877–2879 (2012).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Tromberg, B. J.

Tsay, T.-T.

Tseng, S.-H.

van den Bergh, H.

Venugopalan, V.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
[Crossref]

Villringer, A.

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(3), 879–896 (2001).
[Crossref]

Vitkin, I. A.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

Wabnitz, H.

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(3), 879–896 (2001).
[Crossref]

Wagnières, G.

Wan, W.

Weiss, G. H.

I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
[Crossref]

Welch, A. J.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

Whelan, W. M.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

Worthington, A. E.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

Yamada, Y.

Yodh, A. G.

Zaccanti, G.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
[Crossref]

F. Martelli, S. D. Bianco, G. Zaccanti, A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, and R. Cubeddu, “Phantom validation and in vivo application of an inversion procedure for retrieving the optical properties of diffusive layered media from time-resolved reflectance measurements,” Opt. Lett. 29(17), 2037–2039 (2004).
[Crossref]

F. Martelli, S. D. Bianco, and G. Zaccanti, “Procedure for retrieving the optical properties of a two-layered medium from time-resolved reflectance measure ments,” Opt. Lett. 28(14), 1236–1238 (2003).
[Crossref]

F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Method for measuring the diffusion coefficient of homogeneous and layered media,” Opt. Lett. 25(20), 1508–1510 (2000).
[Crossref]

Zappa, F.

E. Alerstam, T. Svensson, S. Andersson-Engels, L. Spinelli, D. Contini, A. D. Mora, A. Tosi, F. Zappa, and A. Pifferi, “Single-fiber diffuse optical time-of-flight spectroscopy,” Opt. Lett. 37(14), 2877–2879 (2012).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Zhao, H.

Appl. Opt. (6)

Biomed. Opt. Express (2)

IEEE J. Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26(12), 2166–2185 (1990).
[Crossref]

J. Biomed. Opt. (2)

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis,” J. Biomed. Opt. 12(6), 064027 (2007).
[Crossref]

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

J. Biophotonics (1)

T. Myllylä, M. Harju, V. Korhonen, A. Bykov, V. Kiviniemi, and I. Meglinski, “Assessment of the dynamics of human glymphatic system by near-infrared spectroscopy,” J. Biophotonics 11(8), e201700123 (2018).
[Crossref]

J. Mod. Opt. (1)

I. Dayan, S. Havlin, and G. H. Weiss, “Photon Migration in a Two-layer Turbid Medium a Diffusion Analysis,” J. Mod. Opt. 39(7), 1567–1582 (1992).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Phys. A: Math. Gen. (1)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A: Math. Gen. 39(1), 115–137 (2006).
[Crossref]

J. Phys. A: Math. Theor. (1)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A: Math. Theor. 43(6), 065402 (2010).
[Crossref]

Med. Phys. (1)

A. Liemert and A. Kienle, “Explicit solutions of the radiative transport equation in the P3 approximation,” Med. Phys. 41(11), 111916 (2014).
[Crossref]

Opt. Lett. (5)

Phys. Med. Biol. (5)

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(3), 879–896 (2001).
[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]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

F. Martelli, A. Sassaroli, S. D. Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to study photon migration through a simplified adult head model,” Phys. Med. Biol. 52(10), 2827–2843 (2007).
[Crossref]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref]

Phys. Rev. E (1)

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87(6), 063308 (2013).
[Crossref]

Phys. Rev. Lett. (1)

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[Crossref]

Rep. Prog. Phys. (1)

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73(7), 076701 (2010).
[Crossref]

Sci. Rep. (2)

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3(1), 2018 (2013).
[Crossref]

A. Liemert, D. Reitzle, and A. Kienle, “Analytical solutions of the radiative transport equation for turbid and fluorescent layered media,” Sci. Rep. 7(1), 3819 (2017).
[Crossref]

Waves Random Media (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
[Crossref]

Other (3)

S. Agarwal, K. Mierle, and et al., “Ceres Solver,” http://ceres-solver.org .

D. V. O’Connor and D. Phillips, Time-correlated Single Photon Counting (Academic Press, 1984).

S. Chandrasekhar, Radiative Transfer (Dover Publications, 1960).

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Figures (3)

Fig. 1.
Fig. 1. Schematic view of the three-layered medium used in the $P_N$ approximation. The first and second layers have fixed thicknesses $l_1$ and $l_2$, respectively. The third layer extends infinitely. All three layers have the same refractive index $n$ but differ in the absorption $\mu _{\mathrm {a}i}$ and scattering $\mu ^\prime _{\mathrm {s}i}$ coefficients ($i \in \{1,2,3\}$). The signal is detected at the surface at a distance $\rho$ from the incident beam.
Fig. 2.
Fig. 2. The distribution of the absolute values $e_{\mathrm r}$ of the relative errors in $\mu _{\mathrm {a} {3}}$ for “good fits”. Relative fits to reflectance data were performed.
Fig. 3.
Fig. 3. Reflectance curves for different sets of parameters (see Table 4) at distances $\rho _1= {8}\,{\textrm{mm}}$ and $\rho _2={16}\,{\textrm{mm}}$. For the noisy data, only every fifth data point is plotted.

Tables (7)

Tables Icon

Table 1. The optical parameters and layer thicknesses are randomly chosen, between min and max. The start value for the fit-algorithm is also presented.

Tables Icon

Table 2. Comparison of the mean, the standard deviation $\sigma$ and the median of the absolute values of the relative errors in the fit parameters for “good fits”. The layer thicknesses are fitted.

Tables Icon

Table 3. Comparison of the mean, the standard deviation $\sigma$ and the median of the absolute values of the relative errors in the fit parameters for “good fits”. The layer thicknesses are held constant.

Tables Icon

Table 4. Differences between the found values and the values used for the forward calculation for an example with higher errors. The reduced scattering and the absorption coefficients are in units mm−1, the layer thicknesses are in units mm. The scaling factors $k_1$ and $k_2$ correspond to distances of 8 mm and 16 mm, respectively. The values represent one example for the problems, where ambiguity in the reflectance curves appears.

Tables Icon

Table 5. Comparison of the mean, the standard deviation $\sigma$ and the median of the absolute values of the relative errors in the fit parameters for “good fits”. The layer thicknesses are fitted ($R_{\mathrm {max}}= 1\times 10^{6}$).

Tables Icon

Table 6. Comparison of the mean, the standard deviation $\sigma$ and the median of the absolute values of the relative errors in the fit parameters for “good fits”. The layer thicknesses are held constant ($R_{\mathrm {max}}= 1\times 10^{6}$).

Tables Icon

Table 7. Comparison of the mean, the standard deviation $\sigma$ and the median of the absolute values of the relative errors in the fit parameters for “good fits”, assuming relative measurements. Diffusion theory is used ($R_{\mathrm {max}}= 1\times 10^{5}$).

Equations (3)

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

w(ρ)=2πρw2exp(2ρ2ρw2),
χ2=1MPj=1M(Rs,jRt,jwj)2,
er=|pfprpr|,