Abstract

After analytical expressions for the time-resolved reflectance are introduced from the diffusion approximation under the three most commonly used boundary conditions, a novel algorithm is demonstrated for determining the reduced scattering and the absorption coefficients from time-resolved reflectance (or backscatter) measurements at two positions on the surface of biotissue. The algorithm is straightforward and fast and involves only some simple mathematical operations, avoiding complicated iterative nonlinear fitting to the time-resolved curve. The derived reduced scattering coefficient is not affected by whatever boundary condition is applied. The algorithm was verified with time-resolved data from the Monte Carlo model. Both a semi-infinite medium and a turbid slab medium were tested. In contrast to the nonlinear fitting method, this algorithm allows both the scattering and the absorption coefficients to be determined to a high accuracy.

© 1998 Optical Society of America

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References

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  1. See, for example, the five special journal issues on biomedical optics: Appl. Opt. 28, 2207–2342 (1989); Appl. Opt. 32, 372–627 (1993); Opt. Eng. 32, 216–360 (1993); Appl. Opt.35, 9–231 (1997); J. Opt. Soc. Am. A 14, 136–324 (1997).
  2. B. Chance, J. Leigh, H. Miyake, D. Smith, S. Nioka, R. Greenfield, M. Finlander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yodshioka, R. Boretsky, “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. (USA) 85, 4971–4975 (1988).
    [CrossRef]
  3. B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
    [CrossRef]
  4. 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]
  5. A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
    [CrossRef]
  6. 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]
  7. M. Keijzer, W. M. Star, P. R. M. Stochi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  8. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  9. F. Liu, K. M. Yoo, R. R. Alfano, “Should the photon flux or the photon density be used to describe the temporal profiles of scattered ultrashort laser pulses in random media,” Opt. Lett. 18, 432–434 (1993).
    [CrossRef] [PubMed]
  10. A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
    [CrossRef]
  11. D. Contini, F. Martelli, G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory,” Appl. Opt. 36, 4587–4599 (1997).
    [CrossRef] [PubMed]
  12. L. H. Wang, S. L. Jacques, L. Zheng, “MCML-Monte-Carlo modelling of light transport in multilayered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]

1997 (2)

1995 (2)

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

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

1994 (1)

1993 (1)

1989 (3)

1988 (2)

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

M. Keijzer, W. M. Star, P. R. M. Stochi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

Alfano, R. R.

Boretsky, R.

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

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]

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

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Cohen, P.

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

Contini, D.

Feng, T.

Finlander, M.

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

Greenfield, R.

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

Haskell, R. C.

Hefetz, Y.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Hielscher, A.

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Jacques, S. L.

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

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

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Kaufmann, K.

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

Keijzer, M.

Kienle, A.

Leigh, J.

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

Levy, W.

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

Liu, F.

Liu, H.

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Madsen, S.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Martelli, F.

McAdams, M. S.

Miyake, H.

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

Nioka, S.

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

Park, Y.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Patterson, M.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Patterson, M. S.

Smith, D.

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

Star, W. M.

Stochi, P. R. M.

Svaasand, L. O.

Tittel, F. K.

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Tromberg, B. J.

Tsay, T.

Wang, L.

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

Wang, L. H.

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

Wang, L.-H.

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

Wilson, B.

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Wilson, B. C.

Yodshioka, H.

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

Yoo, K. M.

Yound, M.

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

Zaccanti, G.

Zheng, L.

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

Appl. Opt. (4)

Comput. Methods Programs Biomed. (1)

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

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phy. Med. Biol. (1)

A. 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,” Phy. Med. Biol. 40, 1957–1975 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. (USA) (1)

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

Proc. SPIE (1)

B. Wilson, Y. Park, Y. Hefetz, M. Patterson, S. Madsen, S. L. Jacques, “The potential of time-resolved reflectance measurements for non-invasive determination of tissue optical properties,” Proc. SPIE 1064, 97–107 (1989).
[CrossRef]

Other (1)

A. H. Hielscher, H. Liu, L.-H. Wang, F. K. Tittel, B. Chance, S. L. Jacques, “Determination of the blood oxygenation in the brain by time-resolved reflectance spectroscopy (I): influence of skin, skull, and meninges,” in Biochemical Diagnostic Instrumentation A: Optical Diagnosis of Blood and Blood Components, R. F. Bonner, G. E. Cohn, T. M. Lave, A. V. Priezzhev, eds., Proc. SPIE2136, 15–25 (1994).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Ratios of reflectance predicted from Eq. (8) to those that refer to different values of the numerical aperture of the fiber used: upper curve, NA = 0.5; lower curve, NA = 0.7; μ a = 0.001 mm-1 and μ s ′ = 0.5 mm-1 are assumed in the calculations.

Fig. 2
Fig. 2

Comparison between the MC simulation (noise curve) and the theoretical predictions from diffusion approximation for the three boundary conditions applied: (a) NA = 1.0, (b) NA = 0.5.

Fig. 3
Fig. 3

Time-resolved results at r equal to 5, 10, and 20 mm from a homogeneous semi-infinite medium with μ a = 0.001 mm-1, μ s ′ = 0.5 mm-1, and g = 0.9. Noise curves (jagged) are from the MC model, and the smooth curves are predicted from diffusion equations based on the EPB condition in which the extracted values of μ a and μ s ′ from the proposed method are used.

Fig. 4
Fig. 4

Time-resolved results at r equal to 5, 10, and 20 mm from a 20-mm-thick, infinite turbid slab with μ a = 0.0045 mm-1, μ s ′ = 0.6 mm-1, and g = 0.9. Noise curves (jagged) are from the MC model, and the smooth curves are predicted from diffusion equations based on the EPB condition in which the extracted values of μ a and μ s ′ from the proposed method are used.

Tables (2)

Tables Icon

Table 1 Results of Optical Properties Obtained for a Semi-Infinite Medium

Tables Icon

Table 2 Results of Optical Properties Extracted for a Turbid Slab (d = 20 mm)

Equations (20)

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ϕ r ,   z ,   t = c 4 π Dct 3 / 2 exp - μ a ct exp - r 2 4 Dct × exp - z - z 0 2 4 Dct + exp - z + z 0 2 4 Dct - 2 z b 0 d l   exp - l z b exp - z + z 0 + l 2 4 Dct ,
z b = 2 D   1 + R eff 1 - R eff ,
L r ,   z ,   s ˆ ,   t = 1 4 π   ϕ r ,   z ,   t + 3 4 π   F r ,   z ,   t s ˆ ,
R r ,   t = Ω d Ω 1 - R Fres θ 1 4 π ϕ r ,   z = 0 ,   t + 3 D   ϕ r ,   z ,   t z z = 0   cos θ cos θ ,
= Ω d Ω 1 - R Fres θ 1 4 π ϕ r ,   z = 0 ,   t + 3 2 1 - R eff 1 + R eff   ϕ r ,   z = 0 ,   t cos θ cos θ ,
R PCB r ,   t = 0.1644 ϕ r ,   z = 0 ,   t = 0.3288 c 4 π Dct 3 / 2 exp - μ a ct × exp - r 2 4 Dct exp - z 0 2 4 Dct - 1 z b 0 d l × exp - 1 z b   exp - z 0 + l 2 4 Dct .
ϕ r ,   z = 0 ,   t = c 4 π Dct 3 / 2 exp - μ a ct × exp - r 2 4 Dct exp - z - z 0 2 4 Dct - exp - z + z 0 + 2 z b 2 4 Dct ,
F r ,   t = 0.5 4 π Dc 3 / 2 t 5 / 2 exp - μ a ct exp - r 2 4 Dct × z 0   exp - z 0 2 4 Dct + z 0 + 2 z b ×   exp - z 0 + 2 z b 2 4 Dct .
R EPB r ,   t = 0.1101 ϕ r ,   z = 0 ,   t + 0.3001 F r ,   t = 1 4 π Dct 3 / 2 exp - μ a ct exp - r 2 4 Dct × 0.15 z 0 t + 0.1101 c exp - z 0 2 4 Dct + 0.15 z 0 + 2 z b t - 0.1101 c × exp - z 0 + 2 z b 2 4 Dct ,
R ZB r ,   t = 0.3001 F r ,   t = 0.3001 z 0 4 π Dct 3 / 2 t exp - μ a ct × exp - r 2 4 Dct exp - z 0 2 4 Dct .
R PCB = 0.08676 ϕ r ,   z = 0 ,   t ,
R ZB = 0.16612 F r ,   t ,
R EPB r ,   t = 0.05666 ϕ r ,   z = 0 ,   t + 0.16612 F r ,   t .
R r ,   t = f r ,   t f t = exp - r 2 4 Dct × 1 4 π Dct 3 / 2 exp - μ a ct f D ,   t ,
f D ,   t = 0.3288 c exp - z 0 2 4 Dct - 1 z b 0 d l   exp - l z b exp - z 0 + l 2 4 Dct ,
f D ,   t = 0.15 z 0 t + 0.1101 c exp - z 0 2 4 Dct + 0.15 z 0 + 2 z b t - 0.1101 c × exp - z 0 + 2 z b 2 4 Dct ,
f D ,   t = 0.3001 z 0 t exp - z 0 2 4 Dct .
R r 1 ,   t R r 2 ,   t = exp - r 1 2 - r 2 2 4 Dct .
μ s = 4 ct 3 r 2 2 - r 1 2 ln R r 1 ,   t R r 2 ,   t .
- μ a ct + A = ln R r ,   t + r 2 4 Dct - ln 4 π Dct - 3 / 2 - ln f D ,   t ,

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