Abstract

Photon density and photon flux are widely used to model the measurable quantity in diffuse optical tomography problems. However, it is not these two quantities that are actually measured, but rather the radiance accepted by the detection system. We provide a theoretical analysis of the model deviations related to the choice of the measurable quantity—either photon density or flux. By using the diffusion approximation to the radiative transfer equation and its solution with extrapolated boundary conditions, an exact analytical expression of the measurable quantity has been obtained. This expression has been employed as a reference to assess model deviation when considering the photon density or the photon flux as the measurable quantity. For the case of semi-infinite geometry and for both continuous wave and time domains, we show that the photon density approximates the measurable quantity better than the photon flux. We also demonstrate that the validity of this approximation strongly depends on the optical parameters.

© 2008 Optical Society of America

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References

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  1. D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
    [CrossRef] [PubMed]
  2. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  3. X. D. Li, M. A. O'Leary, D. A. Boas, and B. Chance, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746-3758 (1996).
    [CrossRef] [PubMed]
  4. E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
    [CrossRef]
  5. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical-properties,” Appl. Opt. 28, 2331-2336 (1989).
    [CrossRef] [PubMed]
  6. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
    [CrossRef] [PubMed]
  7. F. Liu, K. M. Yoo, and 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]
  8. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
    [CrossRef] [PubMed]
  9. A. Kienle and 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]
  10. D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51, 3733-3746 (2006).
    [CrossRef] [PubMed]
  11. T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).
  12. A. Laidevant, A. da Silva, M. Berger, and J. M. Dinten, “Effects of the surface boundary on the determination of the optical properties of a turbid medium with time-resolved reflectance,” Appl. Opt. 45, 4756-4764 (2006).
    [CrossRef] [PubMed]
  13. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727-2741 (1994).
    [CrossRef]
  14. A. Kienle, “Light diffusion through a turbid parallelepiped,” J. Opt. Soc. Am. A 22, 1883-1888 (2005).
    [CrossRef]
  15. R. Pierrat, L. J. Greffet, and R. Carminati, “Photon diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 23, 1106-1110 (2006).
    [CrossRef]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  17. J. D. Moulton, “Diffusion modelling of picosecond pulse propagation in turbid media,” Master's thesis (McMaster University, Hamilton, Ontario, 1990).
  18. R. Aronson, “Boundary conditions for diffusion of light.” J. Opt. Soc. Am. A 12, 2532-2539 (1995).
    [CrossRef]
  19. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671-1681 (2000).
    [CrossRef]
  20. T. Farrell, M. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879-888 (1992).
    [CrossRef] [PubMed]

2006 (3)

2005 (2)

A. Kienle, “Light diffusion through a turbid parallelepiped,” J. Opt. Soc. Am. A 22, 1883-1888 (2005).
[CrossRef]

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

2000 (1)

1999 (2)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

1995 (1)

1994 (2)

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727-2741 (1994).
[CrossRef]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (2)

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

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

1989 (1)

Alfano, R. R.

Aronson, R.

Arridge, S. R.

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671-1681 (2000).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Berger, M.

Boas, D. A.

X. D. Li, M. A. O'Leary, D. A. Boas, and B. Chance, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746-3758 (1996).
[CrossRef] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Carminati, R.

Chance, B.

Chen, G. Y.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Comsa, D. C.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51, 3733-3746 (2006).
[CrossRef] [PubMed]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

da Silva, A.

Dehghani, H.

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

Dinten, J. M.

Farrell, T.

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

Farrell, T. J.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51, 3733-3746 (2006).
[CrossRef] [PubMed]

Feng, T.-C.

Godavarty, A.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Greffet, L. J.

Haskell, R. C.

Houston, J. P.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Ishimaru, A.

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

Kienle, A.

Kuwana, E.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Laidevant, A.

Li, X. D.

Liu, F.

Martelli, F.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

McAdams, M. S.

Moulton, J. D.

J. D. Moulton, “Diffusion modelling of picosecond pulse propagation in turbid media,” Master's thesis (McMaster University, Hamilton, Ontario, 1990).

Nieto-Vesperinas, M.

O'Leary, M. A.

X. D. Li, M. A. O'Leary, D. A. Boas, and B. Chance, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746-3758 (1996).
[CrossRef] [PubMed]

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Patterson, M.

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

Patterson, M. S.

Pierrat, R.

Ripoll, J.

Roy, R.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Sassaroli, A.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Svaasand, L. O.

Thompson, A. B.

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

Tian, J. G.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Tromberg, B. J.

Tsay, T.-T.

Wilson, B.

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

Wilson, B. C.

Xu, T.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Yamada, Y.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

Yodh, A. G.

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Yoo, K. M.

Zaccanti, G.

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

Zhang, C. P.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Zhang, G. Y.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Zhao, C. M.

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Appl. Opt. (3)

Chin. Phys. (1)

T. Xu, C. P. Zhang, G. Y. Chen, J. G. Tian, G. Y. Zhang, and C. M. Zhao, “Theoretical and experimental study of the intensity distribution in biological tissues,” Chin. Phys. 14, 1813-1820 (2005).

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Opt. Soc. Am. A (6)

Med. Phys. (1)

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

Opt. Lett. (1)

Phys. Med. Biol. (3)

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue—temporal and frequency-analysis,” Phys. Med. Biol. 37, 1531-1560 (1992).
[CrossRef] [PubMed]

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257-1275 (1999).
[CrossRef] [PubMed]

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. 51, 3733-3746 (2006).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

D. A. Boas, M. A. O'Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887-4891 (1994).
[CrossRef] [PubMed]

Other (3)

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

J. D. Moulton, “Diffusion modelling of picosecond pulse propagation in turbid media,” Master's thesis (McMaster University, Hamilton, Ontario, 1990).

E. M. Sevick-Muraca, E. Kuwana, A. Godavarty, J. P. Houston, A. B. Thompson, and R. Roy, “Near infrared fluorescence imaging and spectroscopy in random media and tissues,” in Biomedical Photonics Handbook, J.Vo-Dinh, ed. (CRC Press, 2003), Chap. 33.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Position of the problem and notations. The lower semi-infinite medium is absorbing and scattering. The position of the source S expressed in cylindrical coordinates is ( ρ s , z s ) , the position of the detector D is ( ρ , z ) . The relative position between S and D is represented by Δ z and Δ ρ .

Fig. 2
Fig. 2

Determination of the MQ. s ̂ n ̂ = cos θ . θ and θ are related through the Snell–Descartes law. The radiance is integrated over all refracted angles within the acceptance of the detection system. All θ smaller than θ m must be considered.

Fig. 3
Fig. 3

CW deviations in tomographic conditions. MQ is approximated by the photon density (left) and by the photon flux (right). z s varies from 1 μ s to 5 cm and Δ ρ from 0 to 4 cm . Here μ a = 0.05 cm 1 , μ s = 2 cm 1 , n = 1.54 , and NA = 0.7 . The closer to the surface the source is, the larger the deviations are.

Fig. 4
Fig. 4

TR deviations in tomographic conditions. MQ is approximated by the photon density (left) and by the photon flux (right). z s varies from 1 μ s , to 5 cm . TR deviations do not depend on Δ ρ . Here μ a = 0.05 cm 1 , μ s = 5 cm 1 , n = 1.54 , and NA = 0.7 . The earlier the time, the larger the deviations.

Tables (3)

Tables Icon

Table 1 Indicating Values of Coupling Coefficients for Different NA and Refractive Indices n a

Tables Icon

Table 2 CW Deviations at z s = 1 / μ s cm for μ a = 0.05 cm 1 and NA = 0.7

Tables Icon

Table 3 TR Deviations at μ s = 5 cm 1 , μ a = 0.05 cm 1 , and NA = 0.7

Equations (29)

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ϕ ( r , t ) 4 π L ( s ̂ , r , t ) d s ̂ .
ϕ ( r , r s , t ) = ν ( 4 π ν D t ) 3 2 exp ( μ a ν t Δ ρ 2 4 ν D t ) [ exp ( Δ z 2 4 ν D t ) exp ( ( z + 2 d e + z s ) 2 4 ν D t ) ] ,
F ( r , t ) ϕ ( r , t ) n ̂ .
F ( r , r s , t ) = 1 2 D t ( 4 π ν D t ) 3 2 exp ( μ a ν t Δ ρ 2 4 ν D t ) { ( z + 2 d e + z s ) exp [ ( z + 2 d e + z s ) 2 4 ν D t ] Δ z exp ( Δ z 2 4 ν D t ) } .
M Q ( r , r s , t ) Ω d e t T F ( s ̂ ) L ( r , r s , s ̂ , t ) s ̂ n ̂ ( r ) d s ̂ ,
T F ( θ ) = 1 1 2 ( n cos θ n 0 cos θ n cos θ + n 0 cos θ ) 2 1 2 ( n cos θ n 0 cos θ n cos θ + n 0 cos θ ) 2 ,
L ( r , r s , s ̂ , t ) = 1 4 π ϕ ( r , r s , t ) 3 D 4 π ϕ ( r , r s , t ) s ̂ .
M Q ( r , r s , t ) ϕ ( r , r s , t ) + 3 D C F ( r , r s , t ) .
C = 0 θ m T F ( θ ) sin θ ( 1 n 0 2 n 2 sin 2 θ ) d θ 0 θ m T F ( θ ) sin θ ( 1 n 0 2 n 2 sin 2 θ ) 1 2 d θ ,
ϵ d , C W ( r , r s ) = α ϕ ( r , r s , t ) d t M Q ( r , r s , t ) d t M Q ( r , r s , t ) d t ,
ϵ f , C W ( r , r s ) = β F ( r , r s , t ) d t M Q ( r , r s , t ) d t M Q ( r , r s , t ) d t .
ϵ d , T R ( r , r s , t ) = α ϕ ( r , r s , t ) M Q ( r , r s , t ) M Q ( r , r s , t ) ,
ϵ f , T R ( r , r s , t ) = β F ( r , r s , t ) M Q ( r , r s , t ) M Q ( r , r s , t ) .
L lim z s + ϕ ( r , r s , t ) d t F ( r , r s , t ) d t ,
ϕ ( r , r s , t ) d t = 1 4 π D [ exp ( k r 1 ) r 1 exp ( k 2 ) r 2 ] ,
ϕ ( r , r s , t ) d t = 1 4 π D [ ( z s + 2 d e ) ( k r 2 + 1 ) r 2 3 exp ( k r 2 ) + z s ( k r 1 + 1 ) r 1 3 exp ( k r 1 ) ] ,
ϕ ( r , r s , t ) d t = 1 4 π D exp ( k z s ) z s [ 1 exp ( 2 d e k ) ] A ( r , r s ) ,
F ( r , r s , t ) d t = k 4 π D exp ( k z s ) z s [ 1 + exp ( 2 d e k ) ] B ( r , r s ) ,
L = 1 exp ( 2 d e k ) 1 + exp ( 2 d e k ) 1 k .
lim t 0 + ϵ d , T R ( r , r s , t ) = lim t 0 + α ϕ ( r , r s , t ) M Q ( r , r s , t ) 1 ,
lim t 0 + ϵ f , T R ( r , r s , t ) = lim t 0 + β F ( r , r s , t ) M Q ( r , r s , t ) 1 .
ϕ ( r , r s , t ) = ν ( 4 π ν D t ) 3 2 exp ( Δ ρ + z s 2 4 ν D t ) I ( r , r s , t ) ,
F ( r , r s , t ) = z s 2 D ( 4 π ν D t ) 3 2 1 t exp ( Δ ρ + z s 2 4 ν D t ) J ( r , r s , t ) ,
M Q ( r , r s , t ) = 3 C z s 2 ( 4 π ν D t ) 3 2 1 t exp ( Δ ρ + z s 2 4 ν D t ) K ( r , r s , t ) ,
lim t 0 + ϵ d , T R ( r , r s , t ) = 1 ,
lim t 0 + ϵ f , T R ( r , r s , t ) = β 3 C D 1 .
lim z s β F ( r , r s , t ) d t M Q ( r , r s , t ) d t 1 = 0 .
β = L + 3 C D .
lim t 0 + ϵ f , T R ( r , r s , t ) = L 3 C D > 0 .

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