Abstract

Transient optical transport in highly scattering media such as tissues is usually modeled as a diffusion process in which the energy flux is assumed proportional to the fluence (intensity averaged over all solid angles) gradients. Such models exhibit an infinite speed of propagation of the optical signal, and finite transmission values are predicted even at times smaller than those associated with the propagation of light. If the hyperbolic, or wave, nature of the complete transient radiative transfer equation is retained, the resulting models do not exhibit such drawbacks. Additionally, the hyperbolic equations converge to the solution at a faster rate, which makes them very attractive for numerical applications in time-resolved optical tomography.

© 1996 Optical Society of America

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References

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  1. Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).
  2. J. C. Hebden, K. S. Wong, “Time-resolved optical tomography,” Appl. Opt. 32, 372–380 (1993).
    [CrossRef] [PubMed]
  3. R. L. Fork, J. Y. Brito, C. H. Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compression,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  4. A. Ishimaru, Y. Kuga, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [CrossRef]
  5. E. A. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216–2222 (1989).
    [CrossRef] [PubMed]
  6. A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef] [PubMed]
  7. 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]
  8. S. L. Jacques, “Time-resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
    [CrossRef] [PubMed]
  9. R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
    [CrossRef]
  10. 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]
  11. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 34–40 (1995).
    [CrossRef]
  12. C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).
  13. R. D. Skocypec, R. O. Buckius, “Photon path length distributions for an isotropically scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 28, 425–439 (1982).
    [CrossRef]
  14. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  15. S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
    [CrossRef]
  16. Y. Hasegawa, Y. Yamada, M. Tamura, Y. Nomura, “Monte Carlo simulations of light transmission through living tissues,” Appl. Opt. 31, 4515–4520 (1991).
    [CrossRef]
  17. A. H. Gandjbakche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
    [CrossRef]
  18. S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.
  19. S. Kumar, K. Mitra, “Transient radiative transfer,” in Proceedings of the First International Symposium on Radiative Heat Transfer, M. P. Mengüc, ed., (Begell House, New York, 1995), pp. 488–504.
  20. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Academic, New York, 1976).
  21. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.
  22. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  23. A. Vedavarz, S. Kumar, M. K. Moallemi, “Significance of non-Fourier heat waves,” J. Heat Transfer 116, 221–224 (1993).
    [CrossRef]
  24. S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
    [CrossRef]

1995 (2)

Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

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

1994 (1)

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

1993 (3)

1991 (1)

1990 (2)

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]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

1989 (5)

1987 (1)

1983 (2)

A. Ishimaru, Y. Kuga, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
[CrossRef]

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

1982 (1)

R. D. Skocypec, R. O. Buckius, “Photon path length distributions for an isotropically scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 28, 425–439 (1982).
[CrossRef]

1978 (1)

Alfano, R. R.

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]

Andersson-Engels, S.

R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
[CrossRef]

Becker, P. C.

Berg, R.

R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
[CrossRef]

Bonner, R. F.

Brito, J. Y.

Buckius, R. O.

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

R. D. Skocypec, R. O. Buckius, “Photon path length distributions for an isotropically scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 28, 425–439 (1982).
[CrossRef]

Chance, B.

Cheung, R. L. T.

Cruz, C. H.

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Academic, New York, 1976).

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
[CrossRef]

Fork, R. L.

Furutsu, K.

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

Gandjbakche, A. H.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Academic, New York, 1976).

Hasegawa, Y.

Hebden, J. C.

Ishimaru, A.

Jacques, S. L.

Jarlman, O.

R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
[CrossRef]

Kuga, Y.

Kumar, S.

A. Vedavarz, S. Kumar, M. K. Moallemi, “Significance of non-Fourier heat waves,” J. Heat Transfer 116, 221–224 (1993).
[CrossRef]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

S. Kumar, K. Mitra, “Transient radiative transfer,” in Proceedings of the First International Symposium on Radiative Heat Transfer, M. P. Mengüc, ed., (Begell House, New York, 1995), pp. 488–504.

S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.

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]

Majumdar, A.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

Mitra, K.

S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.

S. Kumar, K. Mitra, “Transient radiative transfer,” in Proceedings of the First International Symposium on Radiative Heat Transfer, M. P. Mengüc, ed., (Begell House, New York, 1995), pp. 488–504.

Moallemi, M. K.

A. Vedavarz, S. Kumar, M. K. Moallemi, “Significance of non-Fourier heat waves,” J. Heat Transfer 116, 221–224 (1993).
[CrossRef]

Nomura, Y.

Nossal, R.

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
[CrossRef]

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]

Profio, E. A.

Rackmil, C. I.

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

Shank, C. V.

S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
[CrossRef]

R. L. Fork, J. Y. Brito, C. H. Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compression,” Opt. Lett. 12, 483–485 (1987).
[CrossRef] [PubMed]

Shimizu, K.

Skocypec, R. D.

R. D. Skocypec, R. O. Buckius, “Photon path length distributions for an isotropically scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 28, 425–439 (1982).
[CrossRef]

Svanberg, O.

R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
[CrossRef]

Tamura, M.

Tien, C. L.

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

Vedavarz, A.

A. Vedavarz, S. Kumar, M. K. Moallemi, “Significance of non-Fourier heat waves,” J. Heat Transfer 116, 221–224 (1993).
[CrossRef]

S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
[CrossRef]

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]

Wong, K. S.

Yamada, Y.

Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

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

Y. Hasegawa, Y. Yamada, M. Tamura, Y. Nomura, “Monte Carlo simulations of light transmission through living tissues,” Appl. Opt. 31, 4515–4520 (1991).
[CrossRef]

S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.

Yodh, A.

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

Yoo, K. M.

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]

Ann. Rev. Fluid Mech. Heat Transfer (1)

Y. Yamada, “Light–tissue interaction and optical imaging in biomedicine,” Ann. Rev. Fluid Mech. Heat Transfer 6, 1–59 (1995).

Appl. Opt. (7)

IEEE Trans. Bio-Med. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” IEEE Trans. Bio-Med. Eng. 36, 1162–1168 (1989).
[CrossRef]

J. Heat Transfer (2)

A. Vedavarz, S. Kumar, M. K. Moallemi, “Significance of non-Fourier heat waves,” J. Heat Transfer 116, 221–224 (1993).
[CrossRef]

S. Kumar, A. Majumdar, C. L. Tien, “The differential-discrete-ordinate method for solutions of the equation of radiative transfer,” J. Heat Transfer 112, 424–429 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

R. D. Skocypec, R. O. Buckius, “Photon path length distributions for an isotropically scattering planar medium,” J. Quant. Spectrosc. Radiat. Transfer 28, 425–439 (1982).
[CrossRef]

Numer. Heat Transfer (1)

C. I. Rackmil, R. O. Buckius, “Numerical solution technique for the transient equation of transfer,” Numer. Heat Transfer 6, 135–153 (1983).

Opt. Lett. (1)

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

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]

Phys. Today (1)

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

Other (5)

R. Berg, S. Andersson-Engels, O. Jarlman, O. Svanberg, “Time-resolved transillumination for medical diagnostics,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 110–119 (1991).
[CrossRef]

S. Kumar, K. Mitra, A. Vedavarz, Y. Yamada, “Short-pulse radiation transport in participating media,” in Proceedings of the ASME National Heat Transfer Conference, Y. Bayazitoglu, D. Kaminski, P. D. Jones, eds., Heat Transfer Division (American Society of Mechanical Engineers, New York, 1995), Vol. 315, pp. 45–51.

S. Kumar, K. Mitra, “Transient radiative transfer,” in Proceedings of the First International Symposium on Radiative Heat Transfer, M. P. Mengüc, ed., (Begell House, New York, 1995), pp. 488–504.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Academic, New York, 1976).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the model of radiation transport through scattering–absorbing media.

Fig. 2
Fig. 2

Nondimensional average intensity inside the medium at the nondimensional time of 0.5 plotted as a function of distance for both the hyperbolic and parabolic models.

Fig. 3
Fig. 3

Plot of the nondimensional transmittance through a 5-mm sample as a function of time for both the hyperbolic and parabolic models.

Fig. 4
Fig. 4

Plot of the normalized transmittance through a 30-mm sample as a function of time for both the hyperbolic and parabolic models.

Fig. 5
Fig. 5

Plot of the computational time versus the spatial nodes for a particular value of time.

Fig. 6
Fig. 6

Normalized transmittance through a 5-mm sample plotted as a function of time for the hyperbolic P 1, two-flux, and 12-quadrature discrete-ordinate methods.

Tables (1)

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Table 1 Nomenclature

Equations (25)

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1 c I ( x , μ , t ) t + μ I ( x , μ , t ) x = σ e I ( x , μ , t ) + σ s 2 × 1 1 I ( x , μ , t ) p ( μ μ ) d μ + S ( x , μ , t ) ,
I c ( x , μ , t ) = I 0 exp ( σ e x ) × { H [ t ( x / c ) ] H [ ( t t p ) ( x / c ) ] } δ ( μ 1 ) .
S ( x , μ , t ) = σ s 2 1 1 I c ( x , μ , t ) p ( μ μ ) d μ ,
= σ s 2 I 0 exp ( σ e x ) { H [ t ( x / c ) ] } H { t [ t p ( x / c ) ] } p ( 1 μ ) .
4 π c ϕ ( x , t ) t + J ( x , t ) x = 4 π σ a ϕ ( x , t ) + 2 π 1 1 S ( x , μ , t ) d μ ,
1 c J ( x , t ) t + 2 π 1 1 μ 2 I ( x , μ , t ) x d μ = σ e J ( x , t ) + σ s 2 2 π 1 1 1 1 I ( x , μ , t ) p ( μ μ ) μ d μ d μ + 2 π 1 1 S ( x , μ , t ) μ d μ ,
ϕ ( x , t ) = 1 4 π 4 π I ( x , μ , t ) d ω = 1 2 1 1 I ( x , μ , t ) d μ ,
J ( x , t ) = 4 π I ( x , μ , t ) μ d ω = 2 π 1 1 I ( x , μ , t ) μ d μ .
I ( x , μ , t ) = ϕ ( x , t ) + 3 4 π J ( x , t ) μ .
4 π c ϕ ( x , t ) t + J ( x , t ) x = 4 π σ a ϕ ( x , t ) + 2 π 1 1 S ( x , μ , t ) d μ ,
1 c J ( x , t ) t + 4 π 3 ϕ ( x , t ) x = σ e J ( x , t ) + σ s g J ( x , t ) + 2 π 1 1 S ( x , μ , t ) μ d μ ,
g = 3 4 1 1 1 1 p ( μ μ ) μ μ d μ d μ ,
2 π 1 1 S ( x , μ , t ) d μ = 2 π σ s I 0 exp ( σ e x ) × { H [ t ( x / c ) ] H [ ( t t p ) ( x / c ) ] } ,
2 π 1 1 S ( x , μ , t ) μ d μ = 2 π σ s I 0 exp ( σ e x ) × { H [ t ( x / c ) ] H [ ( t t p ) ( x / c ) ] } p * , p * = 1 2 1 1 p ( 1 μ ) μ d μ .
J + ( x = 0 , t ) = 2 π 0 1 I ( 0 , μ , t ) μ d μ = 0 ,
ϕ ( 0 , t ) = 1 2 π J ( 0 , t ) ,
J ( x = L , t ) = 2 π 1 0 I ( L , μ , t ) μ d μ = 0 ,
ϕ ( L , t ) = 1 2 π J ( L , t ) .
3 c 2 2 ϕ t 2 2 ϕ x 2 + 3 c [ σ a + σ e σ s g ] ϕ t + 3 [ σ e σ s g ] σ a ϕ = [ σ e σ s g ] 3 2 1 1 S d μ 3 2 1 1 S x μ d μ + 3 2 1 1 1 c S t d μ .
1 c ϕ t D 2 ϕ x 2 + σ a ϕ = 1 4 π 4 π S d ω ,
D = 1 3 [ ( 1 g ) ] σ s + σ a , J = 4 π D ϕ x .
G 1 t 1 / 2 exp [ 3 4 ( σ a + σ e σ s g ) x 2 c t ( σ e σ s g ) ( σ a + σ e σ s g ) σ a c t ] .
G 1 t 1 / 2 exp [ x 2 4 Dct σ a c t ] .
τ ( L , t ) 1 t 3 / 2 exp [ 3 ( σ e σ s g ) 4 E σ a t ] × { 4 ( L x 0 ) E exp [ E t ( L x 0 ) 2 ] 4 ( L + x 0 ) E exp [ E t ( L + x 0 ) 2 ] + 4 ( 3 L x 0 ) E exp [ E t ( 3 L x 0 ) 2 ] 4 ( 3 L + x 0 ) E exp [ E t ( 3 L + x 0 ) 2 ] } ,
E = [ 3 4 c ( σ a + σ e σ s g ) ] ,

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