Abstract

The Green’s function of the steady-state radiative-transport equation is derived in the P3 approximation for isotropic sources. It is demonstrated that the P3 approximation models the radiance in highly absorbing media or close to sources more accurately than does diffusion theory. Boundary conditions consistent with the P3 approximation are also developed for semi-infinite media bounded by a nonscattering medium. Expressions for the reflectance remitted from media interrogated by a normally incident pencil beam are derived and fitted to simulated and experimental reflectance data from media having optical properties typical of biological tissue. The reconstructed optical properties are accurate to within ±10% in absorption and scattering for source–detector separations as small as 0.43 mm and albedos as low as 0.59. Methods for simplifying these expressions given some a priori knowledge of the scattering phase function are discussed.

© 2001 Optical Society of America

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  1. T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  2. M. G. Nichols, E. L. Hull, T. H. Foster, “Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems,” Appl. Opt. 36, 93–104 (1997).
    [CrossRef] [PubMed]
  3. 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]
  4. J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
    [CrossRef]
  5. V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
    [CrossRef]
  6. F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999).
    [CrossRef]
  7. A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
    [CrossRef] [PubMed]
  8. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorbance,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, Oxford, UK, 1997), Chap. 7.
  10. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 2.
  11. B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1958), Chaps. 10–12.
  12. D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).
  13. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  14. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, A. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
    [CrossRef]
  15. S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
    [CrossRef] [PubMed]
  16. R. Marchesini, A. Bertoni, S. Andreola, E. Melloni, A. E. Sichirollo, “Extinction and absorption coefficients and scattering phase functions of human tissues in vitro,” Appl. Opt. 28, 2318–2324 (1989).
    [CrossRef] [PubMed]
  17. R. C. Haskell, L. O. Svaasand, T. Tsay, T. 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]
  18. M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
    [CrossRef] [PubMed]
  19. 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]
  20. W. M. Star, “Comparing the P3 approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Inst. Ser. IS5, 146–154 (1989).
  21. D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).
  22. A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. Chicago Press, Chicago, Ill., 1958), pp. 272–278.
  23. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  24. G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).
  25. A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissue,” Phys. Med. Biol. 40, 1957–1975 (1995).
    [CrossRef] [PubMed]
  26. L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  27. E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
    [CrossRef] [PubMed]
  28. E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
    [CrossRef] [PubMed]

1999 (2)

F. Bevilacqua, C. Depeursinge, “Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path,” J. Opt. Soc. Am. A 16, 2935–2945 (1999).
[CrossRef]

E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
[CrossRef] [PubMed]

1998 (3)

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, A. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

1997 (3)

1996 (2)

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

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

1995 (3)

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

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

L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991 (1)

1989 (3)

1987 (1)

S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

1891 (1)

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

Andreola, S.

Aronson, R.

Berndt, K. W.

Bertoni, A.

Bevilacqua, F.

Boas, D. A.

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

Bryan, G. H.

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 2.

Chance, B.

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]

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

Conover, D. L.

E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
[CrossRef] [PubMed]

Davison, B.

B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1958), Chaps. 10–12.

Depeursinge, C.

Durian, D. J.

Eick, A. A.

Fantini, S.

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Feng, T.

Fishkin, J. B.

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Flock, S. T.

S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Foster, T. H.

E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
[CrossRef] [PubMed]

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

M. G. Nichols, E. L. Hull, T. H. Foster, “Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems,” Appl. Opt. 36, 93–104 (1997).
[CrossRef] [PubMed]

Freyer, J. P.

Gratton, E.

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haskell, R. C.

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hielscher, A. H.

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, A. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

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

Hull, E. L.

E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
[CrossRef] [PubMed]

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

M. G. Nichols, E. L. Hull, T. H. Foster, “Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems,” Appl. Opt. 36, 93–104 (1997).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, Oxford, UK, 1997), Chap. 7.

Jacques, S. L.

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

L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Johnson, T. M.

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. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

Lakowicz, J. R.

Liu, H.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

Marchesini, R.

McAdams, M. S.

Melloni, E.

Moulton, J. D.

Mourant, J. R.

Nichols, M. G.

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

M. G. Nichols, E. L. Hull, T. H. Foster, “Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems,” Appl. Opt. 36, 93–104 (1997).
[CrossRef] [PubMed]

O’Leary, M. A.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

Patterson, M. S.

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. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, J. R. Lakowicz, “Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue,” Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

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]

S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Rudnick, J.

Shen, A.

Sichirollo, A. E.

Star, W. M.

W. M. Star, “Comparing the P3 approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Inst. Ser. IS5, 146–154 (1989).

Svaasand, L. O.

Tittel, F. K.

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

Tromberg, B. J.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T. Tsay, T. 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]

Tsay, T.

vandeVen, M. J.

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

Venugopalan, V.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Wang, L.

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

Wang, L. H.

L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Weinberg, A. M.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. Chicago Press, Chicago, Ill., 1958), pp. 272–278.

Wigner, E. P.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. Chicago Press, Chicago, Ill., 1958), pp. 272–278.

Wilson, B.

S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

Yodh, A. G.

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

You, J. S.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Zheng, L.

L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 2.

Appl. Opt. (5)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Br. J. Cancer (1)

E. L. Hull, D. L. Conover, T. H. Foster, “Carbogen-induced changes in rat mammary tumour oxygenation reported by near infrared spectroscopy,” Br. J. Cancer 79, 1709–1716 (1999).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

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

Med. Phys. (2)

S. T. Flock, B. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

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

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-simulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

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

Phys. Rev. E (2)

J. B. Fishkin, S. Fantini, M. J. vandeVen, E. Gratton, “Gigahertz photon density waves in a turbid medium: theory and experiments,” Phys. Rev. E 53, 2307–2319 (1996).
[CrossRef]

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source–detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

Proc. London Math. Soc. (1)

G. H. Bryan, “An application of the method of images to the conduction of heat,” Proc. London Math. Soc. 22, 424–430 (1891).

SPIE Inst. Ser. (1)

W. M. Star, “Comparing the P3 approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” SPIE Inst. Ser. IS5, 146–154 (1989).

Other (6)

D. A. Boas, H. Liu, M. A. O’Leary, B. Chance, A. G. Yodh, “Photon migration within the P3 approximation,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 240–247 (1995).

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactors (U. Chicago Press, Chicago, Ill., 1958), pp. 272–278.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Oxford U. Press, Oxford, UK, 1997), Chap. 7.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 2.

B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1958), Chaps. 10–12.

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996).

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Figures (18)

Fig. 1
Fig. 1

Infinite slab geometry used in derivation of the P3 Green’s function.

Fig. 2
Fig. 2

P3 transient attenuation coefficient ν+, asymptotic attenuation coefficient ν-, and μeff (the diffusion-theory attenuation coefficient) as a function of the transport albedo assuming μs=10.0 mm-1, Henyey–Greenstein scattering, and a scattering anisotropy g of 0.9. The corresponding value of the absorption coefficient is indicated on the top axis.

Fig. 3
Fig. 3

Asymptotic attenuation coefficients of the diffusion approximation μeff, the P3 approximation νP3-, and the P5 approximation νP5- as a function of the transport albedo for the scattering conditions used to generate Fig. 2.

Fig. 4
Fig. 4

Results of Monte Carlo simulations of isotropic point sources in infinite media. The data points are the absorbed fluence as a function of r, the distance from the source. Also shown are the diffusion-theory solution for the fluence (dotted curves), the asymptotic P3 solution for the fluence (contribution from ν- only, dashed curves), and the total P3 solution for the fluence (contributions from ν+ and ν-, solid curves). In the upper panel the medium optical properties were μa=0.01 mm-1, μs=10.0 mm-1, g=0.9. For this simulation the diffusion-theory solution and the asymptotic P3 solutions are identical. In the lower panel, the optical properties were μa=1.00 mm-1, μs=10.0 mm-1, g=0.9.

Fig. 5
Fig. 5

Reflectance generated from a pencil beam incident on a semi-infinite medium for which μa=0.1 mm-1 and μs=1.00 mm-1. Data were generated by Monte Carlo simulation. Five different cases, all using Henyey–Greenstein scattering, are shown: g=0.99, 0.90, 0.80, 0.70, and 0.50. For the full range of source–detector separations, the results for g0.7 are nearly indistinguishable.

Fig. 6
Fig. 6

Parameters γ and δ [defined by γ(1-g2)/(1-g1) and δ(1-g3)/(1-g1)] as a function of the scattering anisotropy for the Henyey–Greenstein phase function. The magnitudes of γ and δ vary significantly over the full range of g1. However, in the range 0.7g10.99, γ=1.85±0.15 (±8% variation) and δ=2.6±0.4 (±15% variation).

Fig. 7
Fig. 7

Extrapolation lengths as a function of μa for the asymptotic (left axis) and transient (right axis) P3 solutions. The diffusion-theory extrapolation length is also plotted on the left axis. A relative refractive-index mismatch of 1.4 and Henyey–Greenstein scattering with μs=10.0 and g=0.9 were assumed.

Fig. 8
Fig. 8

Coordinates and unit vectors used in calculation of the reflectance in the P3 approximation.

Fig. 9
Fig. 9

Reflectance data generated by Monte Carlo simulation (points) for an isotropic point source at z0=0.99 mm in a semi-infinite medium having optical properties of μa=0.01 mm-1, μs=10.0 mm-1, and g=0.9 (a=0.99) with Henyey–Greenstein scattering: solid curve, P3 prediction for the reflectance computed by using Eq. (44); dashed curve, complete diffusion-theory solution [Eq. (41)].

Fig. 10
Fig. 10

Reflectance data generated by Monte Carlo simulation (points) for an isotropic point source at z0=0.5 mm in a semi-infinite medium having optical properties of μa=1.1 mm-1, μs=10.0 mm-1, and g=0.9 (a=0.5) with Henyey–Greenstein scattering: solid curve, P3 prediction for the reflectance computed by using Eq. (44); dashed curve, complete diffusion-theory solution [Eq. (41)].

Fig. 11
Fig. 11

Monte Carlo reflectance generated by a pencil beam normally incident on a semi-infinite medium having optical properties of μa=1.00 mm-1, μs=10.0 mm-1, and g=0.9 mm-1: solid curve, two-source beam model, calculated by using the P3 Green’s function and PCBC’s; dash–dotted curve, two-source beam model calculated with the P3 Green’s function and extrapolated boundary conditions; long-dashed curve, a hybrid Green’s function obtained by using the the diffusion-theory Green’s function, but with μeff replaced by ν- and D replaced by μa/(ν-)2; short-dashed curve, standard single-source diffusion-theory solution.

Fig. 12
Fig. 12

Values of μa (upper figure) and μs (lower figure) resulting from fits of the DTfull (solid circles and solid curves), DTflux (open circles and dashed curves), and P3 (solid squares and dotted curves) solutions to simulated reflectance data for a medium with optical properties of μa=0.01 mm-1, μs=1.00 mm-1, and nrel=1.4.

Fig. 13
Fig. 13

Values of μa (upper panel) and μs (lower panel) resulting from fits of the DTfull (solid circles and solid curves), DTflux (open circles and dashed curves), and P3 (solid squares and dotted–dashed curves) reflectance expressions to simulated reflectance data for a medium with optical properties of μa=0.50 mm-1, μs=1.00 mm-1, and nrel=1.4.

Fig. 14
Fig. 14

Fitted values of μa returned from fits to a series of index-mismatched Monte Carlo data sets. In the simulations, μs was equal to 1.00 mm-1 and μa ranged from 0.0001 to 5.0 mm-1: diagonal line, line of exact agreement between fitted and actual absorption coefficients; open circles, DTflux solution with ρmin=1.65 mm; open triangles, DTfull solution with ρmin=1.65 mm; solid squares, P3 solution with ρmin=0.45 mm.

Fig. 15
Fig. 15

Fitted values of μs returned from fits to a series of index-mismatched Monte Carlo data sets. In the simulations μs was equal to 1.00 mm-1 and μa ranged from 0.0001 to 5.0 mm-1: horizontal line, value of μs used in the simulations; open triangles, DTfull solution with ρmin=1.65 mm; open circles, DTflux solution with ρmin=1.65 mm; solid squares, P3 solution with ρmin=0.45 mm.

Fig. 16
Fig. 16

Absorption spectra reconstructed from reflectance data collected from a phantom of MnTPPS in an aqueous suspension of polystyrene spheres 0.511 μm in diameter: solid curve, actual phantom absorption spectrum computed from the cuvette spectrum of MnTPPS and the known MnTPPS concentration; open circles, results from fitting the DTfull solution with ρmin=0.432 mm and ρmax=3.75 mm; solid squares, results from fitting the P3 solution with identical values of ρmin and ρmax.

Fig. 17
Fig. 17

Theoretical transport-scattering spectrum calculated from Mie theory (thin solid line) for the phantom described in Fig. 16: open circles, transport-scattering coefficients reconstructed by using the DTfull solution; solid squares, transport-scattering coefficients reconstructed by using the P3 solution.

Fig. 18
Fig. 18

Concentrations resulting from fits of the MnTPPS cuvette spectrum to absorption spectra reconstructed from reflectance measurements of a phantom comprised of an aqueous suspension of polystyrene spheres and various concentrations of MnTPPS: solid line, actual MnTPPS concentration; solid squares, concentrations resulting from fits to absorption spectra reconstructed from P3 theory; open circles, concentrations resulting from fits to absorption spectra reconstructed with diffusion theory. The maximum absorption coefficient of the phantoms ranged from 0.01 mm-1 ([MnTPPS]=4.3 mM) to 0.6 mm-1 ([MnTPPS]=256.8 mM). There are no diffusion-theory data points for the two largest concentrations because the diffusion-theory spectra failed in the manner illustrated in Figs. 16 and 17.

Tables (1)

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Table 1 Coefficients of the Terms in Eq. (45 ) for Various Detector Types and Refractive-Index Mismatches

Equations (77)

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1cL(r, sˆ, t)t=-L(r, sˆ, t)sˆ-μtL(r, sˆ, t)+μs4πL(r, sˆ, t)f (sˆ, s^)dΩ+S(r, sˆ, t),
L(r, sˆ, t)=l=0Nm=-ll2l+14π1/2ϕlm(r, t)Ylm(sˆ),
S(r, sˆ, t)=l=0Nm=-ll2l+14π1/2σlm(r, t)Ylm(sˆ),
ϕ00(r, t)=4πL(r, t)dsˆ.
f (sˆs^)=l=0N2l+14π glPl(sˆs^),
f (sˆs^)=f (cos θ)=14π1-g2(1+g2-2g cos θ).
ΦG(r)=14πDexp(-μeffr)r,
L(z, η )=m=032m+14π φm(z)Pm(η ),
S(z, η )=m=032m+14π qm(z)Pm(η),
f (sˆs^)=m=032m+14π gmPm(sˆs^),
12l+1l ddz φl-1+(l+1) ddz φl+1+(μt-μsgl)φl=ql.
μaφ0 +φ1=q0,
13φ0+μt(1)φ1+ 23φ2=q1,
15φ1 +μt(2)φ2+35φ3=q2,
17φ2+μt(3)φ3=q3,
φl(z)=j=03Aljexp(νjz)
ν[(l+1)Al+1+lAl-1]+(2l+1)μt(l)Al=0
(l=0 ,, 3).
ν=±ν+=±β+β2-γ181/2,
ν=±ν-=±β-β2-γ181/2,
β27μaμt(1)+28μaμt(3)+35μt(2)μt(3),
γ3780μaμt(1)μt(2)μt(3).
 Alj=A0jh1(νj);h1(νj)-μaνj,
A2j=A0jh2(νj);h2(νj)-12+3μaμt(1)2νj2,
A3j=A0jh3(νj);h3(νj)-9μaμt(1)14μt(3)νj+3νj14μt(3).
L(z, η)=l=032l+14π φl(z)Pl(η),
φ0in(z)=E cosh(zν-)+F cosh(zν+),
φ1in(z)=Eh1(ν-)sinh(|zν-|)+Fh1(ν+)sinh(|zν+|),
φ2in(z)=Eh2(ν-)cosh(zν-)+Fh2(ν+)cosh(zν+),
φ3in(z)=Eh3(ν-)sinh(|zν-|)+Fh3(ν+)sinh(|zν+|).
φ0out(z)=C exp[-ν-(|z|-b)]+D exp[-ν+(|z|-b)],
φ1out(z)=Ch1(-ν-)exp[-ν-(|z|-b)]+Dh1(-ν+)exp[-ν+(|z|-b)],
φ2out(z)=Ch2(-ν-)exp[-ν-(|z|-b)]+Dh2(-ν+)exp[-ν+(|z|-b)],
φ3out(z)=Ch3(-ν-)exp[-ν-(|z|-b)]+Dh3(-ν+)exp[-ν+(|z|-b)].
φ0(z, η)=q0/μa,φl(z, η)=0(l>0).
φ0in(±b)+q0μa=φ0out(±b),
φ1in(±b)=φ1out(±b),
φ2in(±b)=φ2out(±b),
φ3in(±b)=φ3out(±b),
C=bν-3[3μaμt(1)-ν+2]q03μa2μt(1)(1+bν-)(ν-2-ν+2),
D=bν+3[3μaμt(1)-ν-2]q03μa2μt(1)(1+bν+)(ν+2-ν-2).
C=ν-3[3μaμt(1)-ν+2]6μa2μt(1)(ν-2-ν+2),
D=ν+3[3μaμt(1)-ν-2]6μa2μt(1)(ν+2-ν-2).
φ0(z)=C exp(-ν-z)+D exp(-ν+z),
Ψ0sp (r)=-12πrddr [φ0pl(r)],
L(r, sˆ)=l=0N2l+14π Ψl(r)Pl(rˆsˆ).
Ψl(r)=j=0NBjhl(νj)Ql(νjr),
Ql(x)=Ql-2(x)-2l-1x Ql-1(x).
Q0(x)=exp(x)x,
Q1(x)=1-1xexp(x)x.
Ψ0(r)=-C(ν-)22πexp(-ν-r)(-ν-r)+-D(ν+)22πexp(-ν+r)(-ν+r)=CQ0(-ν-r)+DQ0(-ν+r).
L(r, sˆ)=l=032l+14π [Chl(-ν-)Ql(-ν-r)+Dhl(-ν+)Ql(-ν+r)]Pl(sˆrˆ).
Ψ1(r)=-D|Ψ0(r)|.
Ψ1(r)=Ch1(-ν-)Q1(-ν-r)+Dh1(-ν+)Q1(-ν+r)=-μa(ν-)2Q0(-ν-r)-μa(ν+)2Q0(-ν+r)-k|Ψ0(r)|,
γ(1-g2)(1-g1),δ(1-g3)(1-g1).
ψ0z=0=12AD ψ0z=0,
ΦGPCBC(ρ, z)
=14πDexp{-μeff [ρ2+(z-z0)2]1/2}[ρ2+(z-z0)2]1/2+exp{-μeff [ρ2+(z+z0)2]1/2}[ρ2+(z+z0)2]1/2-2zbl=0exp(-l/zb)×exp{-μeff [ρ2+(z+z0+l)2]1/2}[ρ2+(z+z0+l)2]1/2dl.
ΦGEBC(ρ, z)=ΦG[r1(ρ, z)]-ΦG[r2(ρ, z)]=14πDexp{-μeff [ρ2+(z-z0)2]1/2}[ρ2+(z-z0)2]1/2-exp{-μeff [ρ2+(z+z0+2zb)2]1/2}[ρ2+(z+z0+2zb)2]1/2,
Rdetected(ρ; z0)
=ΩdetectorTFresnel[cos-1(sˆnˆ)]L(r, sˆ; z0)nˆsˆdΩ,
R(ρ; z0)=-DΦGEBC(ρ, z; z0)(-zˆ)|z=0=14πz0μeff+1r1exp(-μeffr1)r12+(z0+2zb)μeff+1r2exp(-μeffr2)r22,
Rdetected(ρ; z0)=CΦΦGEBC(ρ; z0)+CjjzEBC(ρ; z0),
R(ρ; z0)=ΩdetectorTFresnel(θ)cos(θ)×l=032l+14πΨl(ρ; z0)Pl(μ)×sin (θ)dθdϕ,
μ=sˆrˆ=-z0cos θρ2+z02+ρ sin θ sin ϕρ2+z02
R(ρ; z0)=l=032l+14π Ψl(ρ; z0)Sl(ρ; z0),
S0(ρ; z0)=k1,
S1(ρ; z0)=k2z0(ρ2+z02)1/2,
S2(ρ; z0)=k33z022(ρ2+z02)1/2+k43ρ22(ρ2+z02)1/2-k112,
S3(ρ; z0)=k55z032(ρ2+z02)3/2+k615z0ρ22(ρ2+z02)3/2-k23z02(ρ2+z02)1/2.
S(z)=aμt4πexp(-μtz).
0zaμtexp(-μtz)dz=0zaδ (z-z0)dz
0zaμtexp(-μtz)dz
=0za12 [δ (z-z01)+δ (z-z02)]dz,
0z2aμtexp(-μtz)dz
=0z2a12 [δ (z-z01)+δ(z-z02)]dz
Rnorm(ρ)=R(ρ)R(ρnorm)

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