Abstract

The traditional diffusion theory, often used for isotropic sources, becomes inaccurate at short source–detector spacings and cannot be applied to media with large absorption or with small scattering strengths. We show that for any type of source anisotropy, a Green’s-function-based procedure can remove these limitations. The accuracy of the new approach is examined through a comparison with the numerical solution to the radiative transfer equation.

© 2005 Optical Society of America

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References

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  1. A. Yodh and B. Chance, Phys. Today 48(3), 34 (1995).
    [CrossRef]
  2. See articles in Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE, Bellingham, Wash., 2002).
  3. B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
    [CrossRef]
  4. D. J. Durian and J. Rudnick, J. Opt. Soc. Am. A 14, 235 (1997).
    [CrossRef]
  5. W. M. Star, in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J.C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.
    [CrossRef]
  6. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation. (University of Texas at Austin, Austin, Tex., 1998).
  7. V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
    [CrossRef]
  8. C. K. Hayakawa, B. Y. Hill, J. S. You, F. Bevilacqua, J. Spanier, and V. Venugopalan, Appl. Opt. 43, 4677 (2004).
    [CrossRef] [PubMed]
  9. S. Menon, Q. Su, and R. Grobe, “Determination of gand? using multiply scattered light in turbid media,” Phys. Rev. Lett. (to be published).
  10. For a modified spherical harmonics method, see V. A. Markel, Waves Random Media 14, L13 (2004).
    [CrossRef]
  11. E. L. Hull and T. H. Foster, J. Opt. Soc. Am. A 18, 584 (2001).
    [CrossRef]
  12. A. C. Selden, Phys. Med. Biol. 49, 3017 (2004).
    [CrossRef] [PubMed]
  13. J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
    [CrossRef]
  14. L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
    [CrossRef] [PubMed]
  15. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, Opt. Express 10, 159 (2002).
    [CrossRef] [PubMed]

2004

C. K. Hayakawa, B. Y. Hill, J. S. You, F. Bevilacqua, J. Spanier, and V. Venugopalan, Appl. Opt. 43, 4677 (2004).
[CrossRef] [PubMed]

For a modified spherical harmonics method, see V. A. Markel, Waves Random Media 14, L13 (2004).
[CrossRef]

A. C. Selden, Phys. Med. Biol. 49, 3017 (2004).
[CrossRef] [PubMed]

2002

2001

1998

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

1997

B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
[CrossRef]

D. J. Durian and J. Rudnick, J. Opt. Soc. Am. A 14, 235 (1997).
[CrossRef]

1996

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

1995

L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
[CrossRef] [PubMed]

A. Yodh and B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

Alfano, R. R.

B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
[CrossRef]

Bevilacqua, F.

Boas, D. A.

Chance, B.

A. Yodh and B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

Culver, J. P.

Das, B. B.

B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
[CrossRef]

Dunn, A. K.

Durian, D. J.

Fantini, S.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

Fishkin, J. B.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

Foster, T. H.

Gratton, E.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

Grobe, R.

S. Menon, Q. Su, and R. Grobe, “Determination of gand? using multiply scattered light in turbid media,” Phys. Rev. Lett. (to be published).

Hayakawa, C. K.

Hill, B. Y.

Hull, E. L.

Jacques, S. L.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
[CrossRef] [PubMed]

Liu, F.

B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
[CrossRef]

Markel, V. A.

For a modified spherical harmonics method, see V. A. Markel, Waves Random Media 14, L13 (2004).
[CrossRef]

Menon, S.

S. Menon, Q. Su, and R. Grobe, “Determination of gand? using multiply scattered light in turbid media,” Phys. Rev. Lett. (to be published).

Prahl, S. A.

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation. (University of Texas at Austin, Austin, Tex., 1998).

Rudnick, J.

Selden, A. C.

A. C. Selden, Phys. Med. Biol. 49, 3017 (2004).
[CrossRef] [PubMed]

Spanier, J.

Star, W. M.

W. M. Star, in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J.C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.
[CrossRef]

Stott, J. J.

Su, Q.

S. Menon, Q. Su, and R. Grobe, “Determination of gand? using multiply scattered light in turbid media,” Phys. Rev. Lett. (to be published).

Tromberg, B. J.

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

van de Ven, M. J.

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

Venugopalan, V.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
[CrossRef] [PubMed]

Yodh, A.

A. Yodh and B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

You, J. S.

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
[CrossRef] [PubMed]

Appl. Opt.

Comput. Methods Programs Biomed.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, Comput. Methods Programs Biomed. 47, 131 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Express

Phys. Med. Biol.

A. C. Selden, Phys. Med. Biol. 49, 3017 (2004).
[CrossRef] [PubMed]

Phys. Rev. E

J. B. Fishkin, S. Fantini, M. J. van de Ven, and E. Gratton, Phys. Rev. E 53, 2307 (1996).
[CrossRef]

V. Venugopalan, J. S. You, and B. J. Tromberg, Phys. Rev. E 58, 2395 (1998).
[CrossRef]

Phys. Today

A. Yodh and B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

Rep. Prog. Phys.

B. B. Das, F. Liu, and R. R. Alfano, Rep. Prog. Phys. 60, 227 (1997).
[CrossRef]

Waves Random Media

For a modified spherical harmonics method, see V. A. Markel, Waves Random Media 14, L13 (2004).
[CrossRef]

Other

S. Menon, Q. Su, and R. Grobe, “Determination of gand? using multiply scattered light in turbid media,” Phys. Rev. Lett. (to be published).

See articles in Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE, Bellingham, Wash., 2002).

W. M. Star, in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch and M. J.C. van Gemert, eds. (Plenum, New York, 1995), pp. 131–206.
[CrossRef]

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation. (University of Texas at Austin, Austin, Tex., 1998).

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Figures (1)

Fig. 1
Fig. 1

Angle-integrated fluence Φ ( r ) as a function of distance from the source r for the highly absorbing ( μ a = 0.1 μ s ) and the weakly absorbing ( μ a = 0.001 μ s ) medium for an anisotropic source ( γ = 0.7 ) . The scattering anisotropy parameter is g = 0.9 ( f = 0.81 and g ̃ = 0.47 for the analytical results). The circles denote the Monte Carlo results, the solid curves are the generalized diffusion solution, and the dotted curves are the traditional diffusion solution. Inset, emission profile of the source for γ = 0.7 . For comparison the traditional diffusion solution for an isotropic source ( γ = 0 ) is shown by the dotted–dashed curves.

Equations (13)

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( Ω + μ ω ) I ( r , Ω , ω ) = μ s d Ω p ( Ω , Ω ) I ( r , Ω , ω ) + S ( r , Ω , ω ) .
( Ω + μ ω ) G ( r , Ω , Ω 0 ) = μ s d Ω p ( Ω , Ω ) G ( r , Ω , Ω 0 ) + δ ( r ) δ ( Ω - Ω 0 ) ,
G c ( r , Ω , Ω 0 ) = δ ( Ω 1 r ) δ ( Ω 2 r ) exp ( μ ω Ω r ) Θ ( Ω r ) δ ( Ω Ω 0 ) ,
( Ω + μ ω ) G ¯ ( r , Ω , Ω 0 ) = μ s d Ω p ( Ω , Ω ) G ¯ ( r , Ω , Ω 0 ) + S ¯ ( r , Ω , Ω 0 ) ,
G ¯ 1 ( r , Ω 0 ) = ( μ ω μ s ) G ¯ 0 ( r , Ω 0 ) + q 0 ( r , Ω 0 ) ,
G ¯ 0 ( r , Ω 0 ) = ( 3 i ω c + 1 D ) G ¯ 1 ( r , Ω 0 ) + 3 g q 0 ( r , Ω 0 ) Ω 0 ,
( 2 α 2 ) G ¯ 0 ( r , Ω 0 ) = ( 3 i ω c + 1 D 3 g Ω 0 ) q 0 ( r , Ω 0 ) ,
G ¯ 0 ( r , Ω 0 ) = μ s [ 1 D + 3 g μ T + ( 1 + g ) 3 i ω c ] d ζ exp ( μ ω ζ ) exp ( α r ) 1 4 π r 3 μ s g exp ( α r ) 1 4 π r ,
Φ ( r ) d Ω I ( r , Ω ) = 1 γ 2 ( 1 + γ 2 2 γ cos θ ) 3 2 exp ( μ ω r ) 1 4 π r 2 3 μ s g exp ( α r ) 1 4 π r + [ 1 D + 3 g μ T + 3 i ( 1 + g ) ω c ] μ s 4 π n ( 2 n + 1 ) γ n P n ( cos θ ) B n ( α r ) .
B n ( α r ) α k n ( α r ) 0 r d ζ exp ( μ ω ζ ) i n ( α ζ ) + i n ( α r ) r d ζ exp ( μ ω ζ ) k n ( α ζ ) ,
Φ iso ( r ) = exp ( μ ω r ) 1 4 π r 2 3 g μ s 4 π r exp ( α r ) + 3 μ s 8 π α r [ μ T + g μ a + i ( 1 + g ) ω c ] { exp ( α r ) E 1 [ ( μ ω + α ) r ] exp ( α r ) E 1 [ ( μ ω α ) r ] + exp ( α r ) ln ( μ ω + α μ ω α ) } ,
Φ Diff ( r ) = 1 4 π D r exp ( α r ) + 3 γ cos θ ( α + 1 r ) 1 4 π r exp ( α r ) .
Φ ( r ) = ( μ s ( 1 D + 3 g μ T ) { 1 2 α ln ( μ T + α μ T α ) + 3 γ cos θ α [ sinh ( α μ T ) α μ T 1 ] } 3 μ s g ) 1 4 π r exp ( α r ) .

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