Abstract

The images method is widely used to solve the diffusion equation for the propagation of light within a semi-infinite medium (z>0) in which both scattering and absorption occur. The results are widely used both to analyze data and to illustrate the failings of the diffusion approximation. We emphasize that the images method does not properly obey the extrapolation boundary conditions of diffusion theory, namely, (1-ze d/dz)φ(z)|z=0=0; instead, it approximates these by φ(-ze)=0. The images method also does not describe the source of diffusing photons produced by the exponential attenuation of a collimated incident beam. By correcting these defects and comparing with a simulation, we show that most of the error in the images solution is due to the source and boundary treatment rather than to the diffusion approximation. Our prediction for backscattered intensity versus distance from the source point should be useful for improved analysis of experimental data, especially in cases of nonzero wall reflectivity.

© 1999 Optical Society of America

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1998

D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 33, 1502–504 (1998).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

1997

1996

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian, “Two-stream theory of diffusing-light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

1995

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

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

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy 22, 787–817 (1995).
[CrossRef]

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

1994

D. J. Durian, “The influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (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]

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

1993

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

J. Masoliver, J. M. Porra, G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

1992

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

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knuttel, R. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
[CrossRef] [PubMed]

1991

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1990

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).
[CrossRef] [PubMed]

1989

1988

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

1983

1978

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Alfano, R. R.

A. Y. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
[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).
[CrossRef] [PubMed]

Aronson, R.

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

R. Aronson, N. Corngold, “The photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A (to be published).

Bassani, M.

Boas, D. A.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Boretsky, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, UK, 1959).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967).

Chance, B.

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
[CrossRef]

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[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]

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Cohen, P.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Contini, D.

Corngold, N.

R. Aronson, N. Corngold, “The photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A (to be published).

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Durduran, T.

Durian, D. J.

D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 33, 1502–504 (1998).
[CrossRef]

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

D. J. Durian, J. Rudnick, “Photon migration at short times and distances, and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
[CrossRef]

M. U. Vera, P.-A. Lemieux, D. J. Durian, “The angular distribution of diffusely backscattered light,” J. Opt. Soc. Am. A 14, 2800–2808 (1997).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

D. J. Durian, “Two-stream theory of diffusing-light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

D. J. Durian, “The influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

Eason, G.

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Farrell, T. J.

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

Feng, T.-C.

Ferwerda, H. A.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Finander, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Fishkin, J. B.

Fulton, R. C.

M. Lax, V. Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, Leningrad, June 1987, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987).

Furutsu, K.

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

Ganapol, B. D.

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy 22, 787–817 (1995).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Gratton, E.

Greenfeld, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Groenhuis, R. A. J.

Gutman, S.

Haskell, R. C.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Ishimaru, I.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, UK, 1959).

Kaufmann, K.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Knutson, R. R.

Knuttel, A.

Lax, M.

A. Y. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
[CrossRef]

M. Lax, V. Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, Leningrad, June 1987, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987).

Leigh, J. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

M. U. Vera, P.-A. Lemieux, D. J. Durian, “The angular distribution of diffusely backscattered light,” J. Opt. Soc. Am. A 14, 2800–2808 (1997).
[CrossRef]

Levy, W.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Liu, H.

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

Martelli, F.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Masoliver, J.

J. M. Porrà, J. Masoliver, G. H. Weiss, “When the te-legrapher’s equation furnishes a better approximation to the transport equation than the diffusion approximation,” Phys. Rev. E 55, 7771–7774 (1997).
[CrossRef]

J. Masoliver, J. M. Porra, G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

McAdams, M. S.

Miyake, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nayaranamurti, V.

M. Lax, V. Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, Leningrad, June 1987, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987).

Nioka, S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Nisbet, R. M.

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Patterson, M. J.

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

Patterson, M. S.

Pine, D. J.

Polishchuk, A. Y.

Pomraning, G. C.

G. C. Pomraning, B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy 22, 787–817 (1995).
[CrossRef]

Porra, J. M.

J. Masoliver, J. M. Porra, G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

Porrà, J. M.

J. M. Porrà, J. Masoliver, G. H. Weiss, “When the te-legrapher’s equation furnishes a better approximation to the transport equation than the diffusion approximation,” Phys. Rev. E 55, 7771–7774 (1997).
[CrossRef]

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B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
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Svaasand, L. O.

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Tromberg, B. J.

Tsay, T.-T.

Turnbull, F. W.

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Veitch, A. R.

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Vera, M. U.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

M. U. Vera, P.-A. Lemieux, D. J. Durian, “The angular distribution of diffusely backscattered light,” J. Opt. Soc. Am. A 14, 2800–2808 (1997).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

Weiss, G. H.

J. M. Porrà, J. Masoliver, G. H. Weiss, “When the te-legrapher’s equation furnishes a better approximation to the transport equation than the diffusion approximation,” Phys. Rev. E 55, 7771–7774 (1997).
[CrossRef]

J. Masoliver, J. M. Porra, G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

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J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
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T. J. Farrell, M. J. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance from the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

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

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Yodh, A. J.

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

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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).
[CrossRef] [PubMed]

Yoshioka, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Young, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

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H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

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J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
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[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

A. Y. Polishchuk, S. Gutman, M. Lax, R. R. Alfano, “Photon-density modes beyond the diffusion approximation: scalar wave-diffusion equation,” J. Opt. Soc. Am. A 14, 230–234 (1997).
[CrossRef]

D. J. Durian, J. Rudnick, “Photon migration at short times and distances, and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
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J. M. Schmitt, A. Knuttel, R. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
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J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

M. U. Vera, P.-A. Lemieux, D. J. Durian, “The angular distribution of diffusely backscattered light,” J. Opt. Soc. Am. A 14, 2800–2808 (1997).
[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).
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R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

A. G. Yodh, B. Tromberg, E. Sevick-Muraca, D. J. Pine, eds., feature issue on diffusing photons in turbid media, J. Opt. Soc. Am. A 14, 136–342 (1997);Appl. Opt. 36, 5–231 (1997).
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T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

J. Phys. D

G. Eason, A. R. Veitch, R. M. Nisbet, F. W. Turnbull, “The theory of the backscattering of light by blood,” J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Med. Phys.

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

Opt. Lett.

Phys. Med. Biol.

H. Liu, D. A. Boas, Y. Zhang, A. J. Yodh, B. Chance, “Determination of optical properties of blood oxygenation in tissues using continuous NIR light,” Phys. Med. Biol. 40, 1983–1993 (1995).
[CrossRef] [PubMed]

Phys. Rev. A

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

J. M. Porrà, J. Masoliver, G. H. Weiss, “When the te-legrapher’s equation furnishes a better approximation to the transport equation than the diffusion approximation,” Phys. Rev. E 55, 7771–7774 (1997).
[CrossRef]

D. J. Durian, “The influence of boundary reflection and refraction on diffusive photon transport,” Phys. Rev. E 50, 857–866 (1994).
[CrossRef]

M. U. Vera, D. J. Durian, “The angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215–3224 (1996).
[CrossRef]

J. Masoliver, J. M. Porra, G. H. Weiss, “Solution to the telegrapher’s equation in the presence of reflecting and partly reflecting boundaries,” Phys. Rev. E 48, 939–944 (1993).
[CrossRef]

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

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing-light spectroscopies outside the diffusive limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Phys. Rev. Lett.

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).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
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D. J. Durian, “Two-stream theory of diffusing-light spectroscopies,” Physica A 229, 218–235 (1996).
[CrossRef]

Proc. Natl. Acad. Sci. USA

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Other

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

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M. Lax, V. Nayaranamurti, R. C. Fulton, “Classical diffusive photon transport in a slab,” in Proceedings of the Symposium on Laser Optics of Condensed Matter, Leningrad, June 1987, J. L. Birman, H. Z. Cummins, A. A. Kaplyanskii, eds. (Plenum, New York, 1987).

R. Aronson, N. Corngold, “The photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A (to be published).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

Total diffuse backscattering probability versus absorption μa=l*/la for a unit source deposited into semi-infinite slabs with angle-independent boundary reflectivity. Circles, diamonds, squares, and triangles are simulation results for R=0, 0.5, 0.75, and 0.875, respectively; solid and dashed curves are for the corresponding images and telegrapher predictions, Eqs. (2.3) and (3.4), respectively. Note that the images method deviates more strongly and incorrectly converges to exp(-3)/2=0.08846 for strong absorption.

Fig. 2
Fig. 2

Predictions for the spatial distribution of backscattered photons as the slab reflectivity R is systematically increased from top to bottom; the corresponding extrapolation length ratios are ze=(2/3)(1+R)/(1-R) as labeled. The dotted curves are for the images method [Eq. (2.2)], the dashed curves are for a point source with correct boundary conditions [Eq. (3.15)], and the solid curves are for the full telegrapher theory with correct boundary conditions and an exponential distribution of sources [Eq. (3.16)].

Fig. 3
Fig. 3

Comparison of simulation (solid curves) with images method (dotted and middle-plot curves) and telegrapher theory (dashed and bottom-plot curves) predictions for the spatial distribution of backscattered photons as the slab thickness is systematically varied (as labeled from left to right).

Fig. 4
Fig. 4

Comparison of simulation (solid curves), with images method (dotted and middle-plot curves) and telegrapher theory (dashed and bottom-plot curves) predictions for the spatial distribution of backscattered photons as the scattering anisotropy is systematically varied. Results for l*/ls=1, 3, and 10 are indistinguishable in the upper plot, except for very small l*/ls, where B(ρ) increases monotonically with l*/ls. The corresponding average cosines are g=1-ls/l*=0, 2/3, and 9/10.

Fig. 5
Fig. 5

Comparison of simulation (solid curves) with images method (middle-plot curves) and telegrapher theory (dashed and bottom-plot curves) predictions for the spatial distribution of backscattered photons as the wall reflectivity is systematically varied (as labeled). The corresponding extrapolation length ratios are ze=2/3, 2, 14/3, and 10.

Fig. 6
Fig. 6

Comparison of simulation (solid curves) with images method (dotted and middle-plot curves) and telegrapher theory (dashed and bottom-plot curves) predictions for the spatial distribution of backscattered photons as the wall reflectivity is systematically varied (as labeled).

Equations (26)

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D02φ=(1+μa)φt+μa(1+μa)φ,
0=(1+zenˆ·)φ(r, t)|boundary,
ze=231+R21-R1withRn=01(n+1)μnRw(μ)dμ.
φ(r)=φ0exp(-μer)4πD0r,
B(ρ)=zp4πr13(1+μer1)exp(-μer1)+zp+2ze4πr23(1+μer2)exp(-μer2)zp+ze2πρ3(1+ρ3μa)exp(-ρ3μa)
forρzp+2ze,μa1,
B=0B(ρ)2πρ dρ=12{exp(-zpμe)+exp[-(zp+2ze)μe]}.
φ(0)dφ/dz|z=0=exp(-μer1)/r1-exp(-μer2)/r2(1+μer1)exp(-μer1)zp/r13-(1+μer2)exp(-μer2)(zp+2ze)/r23=ze[1+3(ze/ρ)2+O(1/ρ4)]forμa=0.
D02φ=D02φt2+(1+2D0μa)φt+μa(1+D0μa)φ.
0=(1+D0μa)+zenˆ·+D0tφ(r, t)boundary
J(r, t)=(D0/ze)φ(r, t)|boundary.
B=0Bzp exp[-zp(1+μa)]dzp(1+μa)=1+D0μa+ze[μa(1/D0+μa)]1/21+(D0+ze2/D0)μa+2ze[μa(1/D0+μa)]1/2×1+μa1+μa+[μa(1/D0+μa)]1/2.
u(x, y, z)=exp(iK·R)sin[kz+δ (k)],
λ=-D0(K2+k2).
uzz=0=αu(x, y, z=0),
δ (k)=arctan(k/α).
sin 2δ (k)=2αkα2+k2,
cos 2δ (k)=α2-k2α2+k2.
φ(x, y, z)=1(2π)3d2K dkexp(iK·R)×sin[kz+δ (k)]sin[kzp+δ (k)]D0(K2+k2+κ2).
sin[kz+δ (k)]sin[kzp+δ (k)]
=cos[k(z-zp)]-α+/zpα-/zpcos[k(z+zp)].
α-ddx-1f (x)=xf (w)exp[-α(w-x)]dw.
φ(x, y, z)=n(r-rp)+n(r-rpi)-2αzp exp[-α(w-zp)]n(r-rw)dw,
Bzp(ρ)=zpdss exp[-(s-zp)(1+D0μa)/ze]2πze/(1+D0μa)×exp{-[(s2+ρ2)μa(1/D0+μa)]1/2}×1+[(s2+ρ2)μa(1/D0+μa)]1/2(s2+ρ2)3/2.
B(ρ)=0dss{exp[-s(1+D0μa)/ze]-exp[-s(1+μa)]}2π[ze/(1+D0μa)-1/(1+μa)]×exp{-[(s2+ρ2)μa(1/D0+μa)]1/2}×1+[(s2+ρ2)μa(1/D0+μa)]1/2(s2+ρ2)3/2.
B(ρ)=S0,0(ρ)-S0,0(ρ/ze)/ze2π(ze-1).

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