Abstract

A model of pulsed photothermal radiometry (PPTR) based on optical diffusion theory is presented for a turbid, two-layer, semi-infinite medium containing a surface layer whose optical absorption and scattering properties differ from that of the underlying layer. Assuming one-dimensional geometry, we develop expressions for the depth-dependent fluence distributions and radiant-energy-density profiles and for the time dependence of the PPTR signal. Experimental tests of the PPTR model in a series of layered phantoms of varying optical properties are described. The results of these tests are consistent with the model predictions.

© 1995 Optical Society of America

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References

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  1. J. A. Parrish, B. C. Wilson, “Current and future trends in laser medicine,” Photochem. Photobiol. 53, 731–738 (1991).
    [PubMed]
  2. L. Goldman, ed., Laser Non-Surgical Medicine (Technomic, Lancaster, Pa., 1991), Chap. 1, p. 1.
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.
  4. W. P. Leung, A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
    [CrossRef]
  5. R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
    [CrossRef]
  6. R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. Jacques, J. A. Parrish, “Pulsed photothermal radiometry in turbid media: Internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256–2262 (1989).
    [CrossRef] [PubMed]
  7. S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
    [CrossRef] [PubMed]
  8. S. L. Jacques, J. S. Nelson, W. H. Wright, T. E. Milner, “Pulsed photothermal radiometry of port-wine-stain lesions,” Appl. Opt. 32, 2439–2446 (1993).
    [CrossRef] [PubMed]
  9. D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
    [CrossRef]
  10. F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
    [CrossRef]
  11. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, p. 14.
  12. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4–5, p. 103.
  13. M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
    [CrossRef]
  14. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
    [CrossRef] [PubMed]
  15. W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979), Chap. 3, p. 49.
  16. M. Keijzer, W. M. Star, P. R. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  17. T. J. Farrell, M. S. Patterson, B. C. 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]
  18. M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 9, p. 178.
  20. S. Takatani, “On the theory and development of a noninvassive tissue reflectance oximeter,” Ph.D. dissertation (Case Western Reserve University, Cleveland, Ohio, 1978), p. 265.
  21. J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “Multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).
    [CrossRef] [PubMed]
  22. A. Cohen, R. L. Longini, “Theoretical determination of the blood's relative oxygen saturation in vivo,” Med. Biol. Eng. 9, 61–69 (1971).
    [CrossRef] [PubMed]
  23. M. Keijzer, “Light transport for medical laser treatments,” Ph.D. dissertation (Technische Universiteit Delft, The Netherlands, 1993), Chap. 4, p. 141.
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A, p. 477.
  25. W. Verkuysse, J. W. Pickering, J. F. Beek, M. Keijzer, M. J. C. van Gemert, “Modeling the effect of wavelength on the pulsed dye laser treatment of port wine stains,” Appl. Opt. 32, 393–398 (1993).
    [CrossRef]

1993 (2)

1992 (2)

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. 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]

1991 (2)

J. A. Parrish, B. C. Wilson, “Current and future trends in laser medicine,” Photochem. Photobiol. 53, 731–738 (1991).
[PubMed]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

1990 (1)

1989 (2)

1988 (1)

1987 (1)

F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
[CrossRef]

1986 (1)

D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
[CrossRef]

1984 (2)

W. P. Leung, A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[CrossRef]

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

1971 (1)

A. Cohen, R. L. Longini, “Theoretical determination of the blood's relative oxygen saturation in vivo,” Med. Biol. Eng. 9, 61–69 (1971).
[CrossRef] [PubMed]

Anderson, R. R.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. Jacques, J. A. Parrish, “Pulsed photothermal radiometry in turbid media: Internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256–2262 (1989).
[CrossRef] [PubMed]

F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
[CrossRef]

Balageas, D. L.

D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
[CrossRef]

Beck, H.

Beek, J. F.

Birch, D. S. J.

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A, p. 477.

Brewster, M. Q.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992).

Bruggemann, U.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. Jacques, J. A. Parrish, “Pulsed photothermal radiometry in turbid media: Internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256–2262 (1989).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, p. 14.

Cielo, P.

D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
[CrossRef]

Cohen, A.

A. Cohen, R. L. Longini, “Theoretical determination of the blood's relative oxygen saturation in vivo,” Med. Biol. Eng. 9, 61–69 (1971).
[CrossRef] [PubMed]

Deutch, T. F.

F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4–5, p. 103.

Egan, W. G.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979), Chap. 3, p. 49.

Farinelli, W.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, B. C. 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]

Gilchrist, J. R.

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4–5, p. 103.

Hilgeman, T. W.

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979), Chap. 3, p. 49.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A, p. 477.

Imhof, R. E.

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 9, p. 178.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.

Jacques, S.

Jacques, S. L.

Keijzer, M.

Krapez, J. C.

D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
[CrossRef]

Leung, W. P.

W. P. Leung, A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[CrossRef]

Long, F. H.

F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
[CrossRef]

Longini, R. L.

A. Cohen, R. L. Longini, “Theoretical determination of the blood's relative oxygen saturation in vivo,” Med. Biol. Eng. 9, 61–69 (1971).
[CrossRef] [PubMed]

Milner, T. E.

Nelson, J. S.

Parrish, J. A.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, B. C. 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]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

Pickering, J. W.

Prahl, S. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Schmitt, J. M.

Star, W. M.

Storchi, P. R.

Strivens, T. A.

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

Takatani, S.

S. Takatani, “On the theory and development of a noninvassive tissue reflectance oximeter,” Ph.D. dissertation (Case Western Reserve University, Cleveland, Ohio, 1978), p. 265.

Tam, A. C.

W. P. Leung, A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[CrossRef]

Thornley, F. R.

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

van Gemert, M. J. C.

Verkuysse, W.

Vitkin, I. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Walker, E. C.

Wall, R. T.

Wilson, B. C.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, B. C. 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]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

J. A. Parrish, B. C. Wilson, “Current and future trends in laser medicine,” Photochem. Photobiol. 53, 731–738 (1991).
[PubMed]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

Wright, W. H.

Wyman, D. R.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

Zhou, G. X.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

F. H. Long, R. R. Anderson, T. F. Deutch, “Pulsed photothermal radiometry for depth profiling of layered media,” Appl. Phys. Lett. 51, 2076–2078 (1987).
[CrossRef]

J. Appl. Phys. (2)

W. P. Leung, A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[CrossRef]

D. L. Balageas, J. C. Krapez, P. Cielo, “Pulsed photothermal modelling of layered materials,” J. Appl. Phys. 59, 348–357 (1986).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. E (1)

R. E. Imhof, D. S. J. Birch, F. R. Thornley, J. R. Gilchrist, T. A. Strivens, “Optothermal transient emission radiometry,” J. Phys. E 17, 521–525 (1984).
[CrossRef]

Lasers Med. Sci. (1)

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue I. Models of radiation transport and their applications,” Lasers Med. Sci. 6, 155–168 (1991).
[CrossRef]

Med. Biol. Eng. (1)

A. Cohen, R. L. Longini, “Theoretical determination of the blood's relative oxygen saturation in vivo,” Med. Biol. Eng. 9, 61–69 (1971).
[CrossRef] [PubMed]

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, B. C. 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]

Photochem. Photobiol. (1)

J. A. Parrish, B. C. Wilson, “Current and future trends in laser medicine,” Photochem. Photobiol. 53, 731–738 (1991).
[PubMed]

Phys. Med. Biol. (1)

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Other (10)

W. G. Egan, T. W. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979), Chap. 3, p. 49.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, p. 14.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4–5, p. 103.

L. Goldman, ed., Laser Non-Surgical Medicine (Technomic, Lancaster, Pa., 1991), Chap. 1, p. 1.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.

M. Q. Brewster, Thermal Radiative Transfer and Properties (Wiley, New York, 1992).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, Chap. 9, p. 178.

S. Takatani, “On the theory and development of a noninvassive tissue reflectance oximeter,” Ph.D. dissertation (Case Western Reserve University, Cleveland, Ohio, 1978), p. 265.

M. Keijzer, “Light transport for medical laser treatments,” Ph.D. dissertation (Technische Universiteit Delft, The Netherlands, 1993), Chap. 4, p. 141.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A, p. 477.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the semi-infinite, two-layer model.

Fig. 2
Fig. 2

Predicted fluence distribution profiles (solid curves) showing (a)–(c) the effect of scattering on the top layer and (d)–(f) the effect of the top-layer thickness. The right ordinate of each graph shows the corresponding radiant-energy-density profiles (dotted curves). All curves were generated with a refractive-index mismatch at x = 0 (n0/n1 = 1/1.33) and an index match at x = d (n1/n2 = 1). The optical coefficients are in units of cm−1.

Fig. 3
Fig. 3

PPTR signal profiles obtained from Eq. (18): (a) PPTR signals for the fluence curves of Figs. 2(a)–2(c). (b) PPTR signals for the fluence curves of Figs. 2(d)–2(f). For these and subsequent model curves, the thermal diffusivity is equal to 0.0013 cm2 s−1.

Fig. 4
Fig. 4

Diagram of the experimental PPTR apparatus.

Fig. 5
Fig. 5

Comparison of experimental and predicted PPTR signals for a two-layered turbid medium; d = 90 μm.

Fig. 6
Fig. 6

Experimental and theoretical PPTR signals as a function of the thickness of the top layer. Theoretical PPTR profiles are obtained from Eq. (18) with (μa, μs)1 = 30, 65 cm−1 and (μa, μs)2 = 50, 90 cm−1

Fig. 7
Fig. 7

Comparison of a theoretical model with in vivo PPTR data, from Ref. 8. The data (points) are from a subsurface skin lesion known as a port wine stain. We generated the fit (curve) assuming optical and thermal properties from Refs. 8 and 25 (see text). The data and the fit were independently normalized to their respective delayed peaks for ease of comparison. (Normalization to the t = 0 peak was not used because of uncertainty in the data at early times.)

Equations (18)

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D d 2 ϕ ( x ) d x 2 + μ a ϕ ( x ) = s ( x ) ,
s 1 ( x ) = μ s 1 ϕ 0 exp ( μ t 1 x ) ,
ϕ 1 2 k D 1 d ϕ 1 d x = 0 , x = 0 ,
s 2 ( x ) = ϕ 0 exp ( μ t 1 d ) T μ s 2 exp [ μ t 2 ( x d ) ] ,
D 1 d ϕ 1 d x = D 2 d ϕ 2 d x , x = d ,
ϕ 1 ϕ 2 = F , x = d .
ϕ i ( x ) = A i exp ( μ eff i x ) + B i exp ( + μ eff i x ) + C i exp ( μ t i x ) ,
ϕ coll 1 ( x ) = ϕ 0 exp ( μ t 1 x ) .
ϕ coll 2 ( x ) = ϕ 0 exp ( μ t 1 d ) T exp [ μ t 2 ( x d ) ] .
C 1 = ( 3 μ s 1 3 μ a 1 μ t 1 + 1 ) ϕ 0 = 2 ϕ 0 9 μ a 1 D 1 1 ,
C 2 = ( 3 μ s 2 3 μ a 2 μ t 2 + 1 ) ϕ 0 T exp ( Δ μ t d ) = 2 ϕ 0 T exp ( Δ μ t d ) 9 μ a 2 D 2 1 ,
a 11 = 1 + 2 k D 1 μ eff 1 , a 21 = D 1 μ eff 1 exp ( μ eff 1 d ) , a 31 = exp ( μ eff 1 d ) , a 12 = 1 2 k D 1 μ eff 1 , a 22 = D 1 μ eff 1 exp ( μ eff 1 d ) , a 32 = exp ( μ eff 1 d ) , a 13 = 0 , a 23 = D 2 μ eff 2 exp ( μ eff 2 d ) , a 33 = F exp ( μ eff 2 d ) .
b 1 = C 1 ( 1 + 2 k D 1 μ t 1 ) , b 2 = C 1 D 1 μ t 1 exp ( μ t 1 d ) C 2 D 2 μ t 2 exp ( μ t 2 d ) , b 3 = C 1 exp ( μ t 1 d ) + F C 2 exp ( μ t 2 d ) ,
R d = D d ϕ d x , x = 0 .
R d = D 1 ( μ eff 1 A 1 μ eff 1 B 1 μ t 1 C 1 ) ,
R d = ϕ ( x = 0 ) ϕ 0 2 k .
S ( t ) = K μ a ϕ 0 [ erfc ( μ a 2 α t ) 1 / 2 exp ( μ a 2 α t ) ] ,
S ( t ) = K ( μ a 1 A 1 exp ( μ eff 1 2 α t ) { erf [ d 2 ( α t ) 1 / 2 + μ eff 1 ( α t ) 1 / 2 ] erf [ μ eff 1 ( α t ) 1 / 2 ] } + μ a 1 B 1 exp ( μ eff 1 2 α t ) × { erf [ d 2 ( α t ) 1 / 2 μ eff 1 ( α t ) 1 / 2 ] + erf [ μ eff 1 ( α t ) 1 / 2 ] } + μ a 1 C 1 exp ( μ t 1 2 α t ) { erf [ d 2 ( α t ) 1 / 2 + μ t 1 ( α t ) 1 / 2 ] erf [ μ t 1 ( α t ) 1 / 2 ] } + μ a 2 A 2 exp ( μ eff 2 2 α t ) × { erfc [ d 2 ( α t ) 1 / 2 ] + μ eff 2 ( α t ) 1 / 2 } + μ a 2 C 2 exp ( μ t 2 2 α t ) × { erfc [ d 2 ( α t ) 1 / 2 + μ t 2 ( α t ) 1 / 2 ] } ) ,

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