Abstract

An efficient method is proposed for the evaluation of the absorption and the transport scattering coefficients from a time-resolved reflectance or transmittance distribution. The procedure is based on a library of Monte Carlo simulations and is fast enough to be used in a nonlinear fitting algorithm. Tests performed against both Monte Carlo simulations and experimental measurements on tissue phantoms show that the results are significantly better than those obtained by fitting the data with the diffusion approximation, especially for low values of the scattering coefficient. The method requires an a priori assumption on the value of the anisotropy factor g. Nonetheless, the transport scattering coefficient is rather independent of the exact knowledge of the g value within the range 0.7 < g < 0.9.

© 1998 Optical Society of America

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References

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  1. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  2. S. L. Jacques, “Time resolved reflectance spectroscopy in turbid tissues,” IEEE Trans. Biomed. Eng. 36, 1155–1161 (1989).
    [CrossRef] [PubMed]
  3. B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
    [CrossRef]
  4. S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
    [CrossRef] [PubMed]
  5. R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
    [CrossRef]
  6. K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).
  7. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  8. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  9. A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
    [CrossRef] [PubMed]
  10. M. H. Eddowes, T. N. Mills, D. T. Delpy, “Monte Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissues in vivo,” Appl. Opt. 34, 2261–2267 (1995).
    [CrossRef] [PubMed]
  11. A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. 41, 2221–2227 (1996).
  12. A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.
  13. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  14. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  15. L. Wang, S. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” Rep. (Laser Biology Research Laboratory, M.D. Anderson Cancer Center, University of Texas, 1515 Holcombe Boulevard, Houston, Tex., 1992).
  16. R. Berg, “Laser-based cancer diagnosis and therapy—tissue optics considerations,” Ph.D. dissertation (Division of Atomic Physics, Lund Institute of Technology, Lund, Sweden, 1995).
  17. L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
    [CrossRef]

1997 (1)

1996 (1)

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. 41, 2221–2227 (1996).

1995 (2)

1994 (3)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

1992 (1)

1990 (2)

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).

1989 (2)

1987 (1)

1978 (1)

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).

Andersson-Engels, S.

A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
[CrossRef] [PubMed]

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Berg, R.

A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
[CrossRef] [PubMed]

R. Berg, “Laser-based cancer diagnosis and therapy—tissue optics considerations,” Ph.D. dissertation (Division of Atomic Physics, Lund Institute of Technology, Lund, Sweden, 1995).

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Bonner, R. F.

Chance, B.

Cubeddu, R.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

Delpy, D. T.

Eddowes, M. H.

Feng, T.-C.

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Haskell, R. C.

Havlin, S.

Hefetz, Y.

Jacques, S.

L. Wang, S. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” Rep. (Laser Biology Research Laboratory, M.D. Anderson Cancer Center, University of Texas, 1515 Holcombe Boulevard, Houston, Tex., 1992).

Jacques, S. L.

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

S. L. Jacques, “Time resolved reflectance spectroscopy in turbid tissues,” IEEE Trans. Biomed. Eng. 36, 1155–1161 (1989).
[CrossRef] [PubMed]

Jain, A.

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Kienle, A.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. 41, 2221–2227 (1996).

Lasdon, L. S.

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).

Madsen, S. J.

McAdams, M. S.

Mills, T. N.

Musolino, M.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

Nilsson, A. M. K.

Nossal, R.

Park, Y. D.

Patterson, M. S.

Pifferi, A.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Ratner, M.

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Svaasand, L. O.

Taroni, P.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Tromberg, B. J.

Tsay, T.-T.

Valentini, G.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

Wang, L.

L. Wang, S. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” Rep. (Laser Biology Research Laboratory, M.D. Anderson Cancer Center, University of Texas, 1515 Holcombe Boulevard, Houston, Tex., 1992).

Waren, A. D.

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Weiss, G. H.

Wilson, B. C.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).

ACM Trans. Math. Software (1)

L. S. Lasdon, A. D. Waren, A. Jain, M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Software 4, 34–50 (1978).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (2)

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, G. Valentini, “Time resolved reflectance: a systematic study for the application to the optical characterization of tissue,” IEEE J. Quantum Electron. 30, 2421–2430 (1994).
[CrossRef]

B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissues: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. L. Jacques, “Time resolved reflectance spectroscopy in turbid tissues,” IEEE Trans. Biomed. Eng. 36, 1155–1161 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Phys. Med. (1)

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. 41, 2221–2227 (1996).

Phys. Rev. E (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?”Phys. Rev. Lett.64, 2647–2650 (1990); 65, 2210–2211 (1990).

Other (3)

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, Vol. 2 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

L. Wang, S. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” Rep. (Laser Biology Research Laboratory, M.D. Anderson Cancer Center, University of Texas, 1515 Holcombe Boulevard, Houston, Tex., 1992).

R. Berg, “Laser-based cancer diagnosis and therapy—tissue optics considerations,” Ph.D. dissertation (Division of Atomic Physics, Lund Institute of Technology, Lund, Sweden, 1995).

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Figures (7)

Fig. 1
Fig. 1

Scattering coefficient determines the distance between two scattering sites, whereas the shape of the photon path is fixed solely by the g value. The TPSF of a medium with a certain μ s can be derived when the TPSF of a medium with a different μ s is scaled and if both are characterized by the same g value.

Fig. 2
Fig. 2

Absorption coefficient reduces the survival probability of photons for elapsing time. The TPSF of photons reemitted from a turbid medium can be derived when the TPSF of a medium with the same scattering and no absorption is weighed.

Fig. 3
Fig. 3

Experimental setup for the time-resolved transmittance measurements on tissue phantoms.

Fig. 4
Fig. 4

Relative error on the fitted μ a for (a)–(c) simulated reflectance or (d)–(f) transmittance curves. The method used for the fitting is (a) and (d) the Monte Carlo, (b) and (e) the diffusion with the full fitting range, or (c) and (f) the diffusion with the reduced fitting range.

Fig. 5
Fig. 5

Relative error on the fitted μ s ′ for (a)–(c) simulated reflectance or (d)–(f) transmittance curves. The method used for the fitting is (a) and (d) the Monte Carlo, (b) and (e) the diffusion with the full fitting range, or (c) and (f) the diffusion with the reduced fitting range.

Fig. 6
Fig. 6

Relative error on the fitted μ a for experimental transmittance curves for which (a) the Monte Carlo, (b) the diffusion with the full fitting range, or (c) the diffusion with the reduced fitting range were used.

Fig. 7
Fig. 7

Relative error on the fitted μ s ′ for experimental transmittance curves for which (a) the Monte Carlo, (b) the diffusion with the full fitting range, or (c) the diffusion with the reduced fitting range were used.

Equations (5)

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

R μ s ,   μ a = 0 ,   ρ ,   t = k 3 R μ s 0 ,   μ a = 0 ,   k ρ ,   kt ,
R μ s ,   μ a ,   ρ ,   t = k 3 R μ s 0 ,   μ a = 0 ,   k ρ ,   kt exp - μ a vt ,
R ρ ,   t = 4 π Dc / n - 3 / 2 z o t - 5 / 2 × exp - μ a ct / n exp - ρ 2 + z o 2 4 Dct / n ,
T ρ ,   d ,   t = 4 π Dct / n - 3 / 2 t - 5 / 2 × exp - μ a vt exp - ρ 2 4 Dct / n × n = 1 , k = 2 n - 1 n = kd - z 0 exp - kd - z 0 2 4 Dct / n - kd + z 0 exp - kd + z 0 2 4 Dct / n ,
ε = | μ f - μ e | μ e ,

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