Abstract

To calculate the diffuse reflectance from a semi-infinite slab of tissue, we introduce a probability distribution function, fn(g), that a photon will escape from the tissue after n scattering events. This approach permits the separation of the phase dependence of scattering, described by the anisotropy coefficient, g, from the absorption, μa, and scattering, μs coefficients in the calculation of diffuse reflectance. We demonstrate that fn(g) and g are related to each other through a universal probability function. The analytical form of this probability function is explored and used to obtain the diffuse reflectance from tissue. The diffuse reflectance calculated with this method is in excellent agreement with Monte Carlo simulations over the parameter range typically found in human tissue, even for the values in which diffusion theory is a poor approximation.

© 1993 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1.
  2. B. C. Wilson, S. L. Jacques, “Optical reflectance and transmittance of tissue: principles and applications,” IEEE J. Quantum Electron. 26, 2186–2199 (1990).
    [CrossRef]
  3. W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  4. S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
    [CrossRef] [PubMed]
  5. R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tisue,” Appl. Opt. 28, 2238–2244 (1989).
    [CrossRef] [PubMed]
  6. See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
    [PubMed]
  7. R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).
  8. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).
  9. M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
    [CrossRef] [PubMed]
  10. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).
  11. A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
    [CrossRef] [PubMed]
  12. L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  13. G. Yoon, S. A. Prahl, A. J. Welch “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
    [CrossRef] [PubMed]
  14. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
    [CrossRef]
  15. S. Jacques, M. D. Anderson Cancer Center, Houston, Tex. 77030. (personal communication).
  16. J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
    [CrossRef]

1991

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).

1990

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

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

1989

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tisue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

G. Yoon, S. A. Prahl, A. J. Welch “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
[CrossRef] [PubMed]

1987

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1976

J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Baraga, J. J.

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).

Bonner, R. F.

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tisue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

Cheong, W.

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

Feld, M. S.

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Fitzmaurice, M.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Havlin, S.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

Ishimaru, A.

Jacques, S.

S. Jacques, M. D. Anderson Cancer Center, Houston, Tex. 77030. (personal communication).

Jacques, S. L.

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

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Johnson, C.

Joseph, J. H.

J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

Nossal, R.

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tisue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Petras, R. E.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Prahl, S. A.

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

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

G. Yoon, S. A. Prahl, A. J. Welch “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

Rava, R. P.

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Reynolds, L.

Richards-Kortum, R.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Sivak, M.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Tong, L. L.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

van Gemert, M. J. C.

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Weinman, J. A.

J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Weiss, G. H.

R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tisue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

Welch, A. J.

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

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

G. Yoon, S. A. Prahl, A. J. Welch “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
[CrossRef] [PubMed]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

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

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Wiscomb, W. J.

J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

Yoon, G.

G. Yoon, S. A. Prahl, A. J. Welch “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2255 (1989).
[CrossRef] [PubMed]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Appl. Opt.

IEEE J. Quantum Electron.

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

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

IEEE Trans. Biomed. Eng.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef] [PubMed]

J. Atmos. Sci.

J. H. Joseph, W. J. Wiscomb, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

J. Opt. Soc. Am.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. 4, 423–432 (1987).
[CrossRef]

Lasers Life Sci.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Lasers Surg. Med.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical models for light distribution in laser-irradiated tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Photochem. Photobiol.

See, for example, R. Richards-Kortum, R. P. Rava, R. E. Petras, M. Fitzmaurice, L. L. Tong, M. Sivak, M. S. Feld, “Spectroscopic diagnosis of colonic dysplasia,” Photochem. Photobiol. 53, 777–786 (1991).
[PubMed]

Spectrochim. Acta

R. P. Rava, J. J. Baraga, M. S. Feld, “Near infrared Fourier transform Raman spectroscopy of human artery,” Spectrochim. Acta 47A, 509–512 (1991).

Other

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

S. Jacques, M. D. Anderson Cancer Center, Houston, Tex. 77030. (personal communication).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Vol. 1.

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Figures (6)

Fig. 1
Fig. 1

Plot of k(g) determined by fitting Eq. (11b) versus 1 − g.

Fig. 2
Fig. 2

Correlation of reflectance determined from Eq. (12) (solid curve) to Monte Carlo simulations (solid circles) for g = 0.95.

Fig. 3
Fig. 3

Monte Carlo simulation for fn(g), g = 0.75, and g = 0.95.

Fig. 4
Fig. 4

Results of Monte Carlo simulations of fn(g) for several values of g: (a) reciprocal of the peak position of fn(g) versus (1 − g), (b) peak value of fn(g) versus (1 − g).

Fig. 5
Fig. 5

Fit of Eq. (20) for fn(g) to Monte Carlo simulations for g = 0.75 and g = 0.95.

Fig. 6
Fig. 6

Plot of Ra, μs, g) versus μsa for g = 0.7 and g = 0.95; solid circles, Monte Carlo simulations; solid curve, photon-migration theory of Eq. (21b); dotted curve, diffusion theory of Eq. (6).

Equations (29)

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

p ( L ) = μ t exp ( - μ t L ) .
p ( A ) = μ a μ a + μ s
p ( θ , ϕ ) = 1 4 π ( 1 - g 2 ) ( 1 + g 2 - 2 g cos θ ) 3 / 2 .
U d ( z ) = C exp ( - μ t z ) + C 2 exp ( - κ d z ) ,
C = - Q 0 μ t 2 - κ d 2 , C 2 = - C ( 1 + μ t h ) ( 1 + κ d h ) - Q 1 2 π ( 1 + κ d h ) , Q 0 = ( 3 μ s μ t r + 3 μ s μ t g ) 4 π F 0 , Q 1 = μ s g μ t r F 0 , μ t = μ s + μ a , μ t r = ( 1 - g ) μ s + μ a , κ d = { 3 μ a [ μ a + ( 1 - g ) μ s ] } 1 / 2 , h = 2 3 μ t r .
F d ( z ) = μ s g μ t r F 0 exp ( - μ t z ) - 4 π 3 μ t r d d z U d ( z ) .
R = - μ s g μ t r - 4 π 3 μ t r ( C μ t + C 2 κ d ) .
R ( μ a , μ s , g ) = n = 1 a n f n ( g ) .
n = 1 f n ( g ) = 1
f 1 = 1 6 1 ,             f 2 = 4 6 2 ,             f 3 = 17 6 3 ,             f 4 = 76 6 4 , f 5 = 354 6 5 ,             f 6 = 1988 6 6 ,             f 7 = 10821 6 7 , .
f n ( g ) = k ( g ) exp [ - k ( g ) n ] ,
R ( μ a , μ s , g ) = 0 a n k ( g ) exp [ - k ( g ) n ] d n ,
R ( μ a , μ s , g ) = k ( g ) k ( g ) - ln ( a ) .
R ( μ a , μ s , g ) = k ( g ) μ s k ( g ) μ s + μ a .
k ( g ) 1 - g = k ( g ) 1 - g = S ,
R ( μ a , μ s , g ) = 1 1 - ( 1 / S ) [ ln ( a ) / 1 - g ]
f n ( g ) ( 1 - g ) = W [ ( 1 - g ) n ] .
f n ( 0 ) = ( 3 2 π ) 1 / 2 n - 3 / 2 .
f n ( 0.95 ) = ( 0.05 ) - 1 / 2 ( 3 2 π ) 1 / 2 n - 3 / 2
f n ( 0 ) = W ( n ) = F ( n ) ( 3 2 π ) 1 / 2 n - 3 / 2 .
W ( n ) = [ 1 - exp ( - A n ) ] 2 ( 3 2 π ) 1 / 2 n - 3 / 2 .
f n ( g ) = { 1 - exp [ - 0.45 ( 1 - g ) n ] } 2 [ 3 2 π ( 1 - g ) ] 1 / 2 n - 3 / 2 .
R ( μ a , μ s , g ) = 0 a n { 1 - exp [ - 0.45 ( 1 - g ) n ] } 2 × [ 3 2 π ( 1 - g ) ] 1 / 2 n - 3 / 2 d n ,
R ( μ a , μ s , g ) = 6 { 2 [ 0.45 - ln ( a ) ( 1 - g ) ] 1 / 2 - [ - ln ( a ) ( 1 - g ) ] 1 / 2 - [ 0.9 - ln ( a ) ( 1 - g ) ] 1 / 2 } .
J 3 + α J 2 + β J + γ = 0 ,
α = 3 A - 9 16 R 2 , β = 2 A 2 + 3 32 ( 4 A 2 - 12 R 2 A + R 4 ) , γ = - ( 4 A 2 - 12 R 2 + R 4 ) 2 256 R 2 ,
J 3 + 3 α J + 2 β = 0 ,
α = 1 3 ( β - α 2 3 ) ,             β = 1 2 ( γ - α β 3 + 2 α 3 27 ) .
J = - [ β + ( β 2 + α 3 ) 1 / 2 ] 1 / 3 - [ β - ( β 2 + α 3 ) 1 / 2 ] 1 / 3 .

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