Abstract

Collimated light sources in turbid media are difficult to describe within the diffusion approximation, because they do not meet the requirement of near isotropy. For precise calculation of light intensities close to the source, alternative descriptions of the light source are necessary. In this paper the transition of collimated light into diffusivity is studied by Monte Carlo simulations. On the basis of these simulations and the diffusion approximation a hybrid approach is designed and used to analyze approaches based on analytic source terms. The influence of boundaries to air is studied. The benefits of increased approximation orders are investigated. It is shown that, even in the presence of strong absorption, the diffusion approach can give satisfactory results if only the source terms are suitably chosen.

© 2000 Optical Society of America

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    [CrossRef] [PubMed]
  3. D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.
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    [CrossRef] [PubMed]
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  32. 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]
  33. J. A. Parrish, “New concepts in therapeutic photomedicine: photochemistry, optical targeting and the therapeutic window,” J. Invest. Dermatol. 77, 45–50 (1981).
    [CrossRef] [PubMed]
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1999

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

1998

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

1997

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42, 763–787 (1997).
[CrossRef] [PubMed]

1995

M. J. C. van Gemert, A. J. Welch, J. W. Pickering, O. T. Tan, G. H. M. Gijsbers, “Wavelengths for laser treatment of port wine stains and telangiectasia,” Lasers Surg. Med. 16, 147–155 (1995).
[CrossRef] [PubMed]

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).

I. S. Saidi, S. L. Jacques, F. K. Tittel, “Mie and Rayleigh modeling of visible-light scattering in neonatal skin,” Appl. Opt. 34, 7410–7418 (1995).
[CrossRef] [PubMed]

1994

1993

1990

M. N. Berberan-Santos, “Beer’s law revisited,” J. Chem. Educ. 67, 757–759 (1990).
[CrossRef]

1989

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]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1153 (1989).
[CrossRef] [PubMed]

1988

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues. I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

1986

B. C. Wilson, M. S. Patterson, “The physics of photodynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

1981

J. A. Parrish, “New concepts in therapeutic photomedicine: photochemistry, optical targeting and the therapeutic window,” J. Invest. Dermatol. 77, 45–50 (1981).
[CrossRef] [PubMed]

1979

1976

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

1941

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables (Dover, New York, 1974).

Anderson, R. E.

T. Spott, L. O. Svaasand, R. E. Anderson, P. F. Schmedling, “Application of optical diffusion theory to transcutaneous bilirubinometry,” in Laser-Tissue Interaction, Tissue Optics, and Laser Welding III, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, O. Svaasand, eds., Proc. SPIE3195, 234–245 (1997).
[CrossRef]

Ay, H.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

Berberan-Santos, M. N.

M. N. Berberan-Santos, “Beer’s law revisited,” J. Chem. Educ. 67, 757–759 (1990).
[CrossRef]

Berns, M. W.

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992).

Boas, D. A.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1996), http://ftp.osa.org/BioOptic/PDFS/BOAS/Disserta.htm .

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, 1981).

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Cheong, W.-F.

W.-F. Cheong, “Summary of optical properties,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. C. J. van Gemert, eds. (Plenum, New York, 1995), pp. 275–302.

Duck, F. A.

F. A. Duck, Physical Properties of Tissue (Academic, London, 1990).

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Feng, T.-C.

Fiskerstrand, E. J.

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, W. M. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

Gijsbers, G. H. M.

M. J. C. van Gemert, A. J. Welch, J. W. Pickering, O. T. Tan, G. H. M. Gijsbers, “Wavelengths for laser treatment of port wine stains and telangiectasia,” Lasers Surg. Med. 16, 147–155 (1995).
[CrossRef] [PubMed]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Guadette, T. J.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

Haskell, R. C.

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hornung, R.

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

Jacques, S. L.

L. H. Wang, S. L. Jacques, L. Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).

I. S. Saidi, S. L. Jacques, F. K. Tittel, “Mie and Rayleigh modeling of visible-light scattering in neonatal skin,” Appl. Opt. 34, 7410–7418 (1995).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1153 (1989).
[CrossRef] [PubMed]

Joseph, J. H.

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

Keefe, K. A.

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

Koroshetz, W. J.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

Lamarsh, J. R.

J. R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, Mass., 1975).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
[CrossRef]

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues. I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

McAdams, M. S.

Meador, W. E.

Nelson, J. S.

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

Norvang, L. T.

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

Parrish, J. A.

J. A. Parrish, “New concepts in therapeutic photomedicine: photochemistry, optical targeting and the therapeutic window,” J. Invest. Dermatol. 77, 45–50 (1981).
[CrossRef] [PubMed]

Patterson, M. S.

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]

B. C. Wilson, M. S. Patterson, “The physics of photodynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo—diffusion theory modelling of light distributions in tissue,” in Laser Interaction with Tissue, W. M. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

Pham, T. H.

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

Pickering, J. W.

M. J. C. van Gemert, A. J. Welch, J. W. Pickering, O. T. Tan, G. H. M. Gijsbers, “Wavelengths for laser treatment of port wine stains and telangiectasia,” Lasers Surg. Med. 16, 147–155 (1995).
[CrossRef] [PubMed]

Prahl, S. A.

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (The University of Texas at Austin, Austin, Texas, 1988), http://omlc.ogi.edu/pubs/prahl-pubs/index.html .

Saidi, I. S.

Schmedling, P. F.

T. Spott, L. O. Svaasand, R. E. Anderson, P. F. Schmedling, “Application of optical diffusion theory to transcutaneous bilirubinometry,” in Laser-Tissue Interaction, Tissue Optics, and Laser Welding III, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, O. Svaasand, eds., Proc. SPIE3195, 234–245 (1997).
[CrossRef]

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, 1981).

Siegel, A.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

Sorenson, A. G.

D. A. Boas, L. Wang, T. J. Guadette, A. Siegel, H. Ay, A. G. Sorenson, W. J. Koroshetz, “Optical monitoring of cerebral perfusion as perturbed by an ischemic stroke in an adult human,” in Digest of Topical Meeting on Biomedical Optics: New Concepts in Therapeutic Laser Applications; Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics; Advances in Optical Imaging, Photon Migration, and Tissue Optics (Optical Society of America, Washington, D.C., 1999), pp. 261–263.

Spott, T.

T. Spott, L. O. Svaasand, R. E. Anderson, P. F. Schmedling, “Application of optical diffusion theory to transcutaneous bilirubinometry,” in Laser-Tissue Interaction, Tissue Optics, and Laser Welding III, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, O. Svaasand, eds., Proc. SPIE3195, 234–245 (1997).
[CrossRef]

Star, W. M.

W. M. Star, “Light dosimetry in vivo,” Phys. Med. Biol. 42, 763–787 (1997).
[CrossRef] [PubMed]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1153 (1989).
[CrossRef] [PubMed]

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues. I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 6, pp. 131–206.
[CrossRef]

W. M. Star, “Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry,” in Dosimetry of Laser Radiation in Medicine and Biology, (Society for Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989), Vol. IS5, pp. 146–154.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables (Dover, New York, 1974).

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1153 (1989).
[CrossRef] [PubMed]

Stopps, E. K. S.

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

Svaasand, L. O.

L. O. Svaasand, L. T. Norvang, E. J. Fiskerstrand, E. K. S. Stopps, M. W. Berns, J. S. Nelson, “Tissue parameters determining the visual appearance of normal skin and port-wine stains,” Lasers Med. Sci. 10, 55–65 (1995).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. 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]

T. Spott, L. O. Svaasand, R. E. Anderson, P. F. Schmedling, “Application of optical diffusion theory to transcutaneous bilirubinometry,” in Laser-Tissue Interaction, Tissue Optics, and Laser Welding III, G. P. Delacretaz, G. Godlewski, R. Pini, R. W. Steiner, O. Svaasand, eds., Proc. SPIE3195, 234–245 (1997).
[CrossRef]

Tadir, Y.

R. Hornung, T. H. Pham, K. A. Keefe, M. W. Berns, Y. Tadir, B. J. Tromberg, “Quantitative near-infrared spectroscopy of cervical dysplasia in vivo,” Hum. Reprod. 14, 2908–2916 (1999).
[CrossRef] [PubMed]

Tan, O. T.

M. J. C. van Gemert, A. J. Welch, J. W. Pickering, O. T. Tan, G. H. M. Gijsbers, “Wavelengths for laser treatment of port wine stains and telangiectasia,” Lasers Surg. Med. 16, 147–155 (1995).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Semi-infinitely extended turbid medium, irradiated by a plane wave.

Fig. 2
Fig. 2

Zeroth- and first-order moments of the source photon distribution after 2, 5, 10, and 20 scattering interactions in a semi-infinite medium with μ a = 167 m-1, μs = 3920 m-1, and g = 0.77. Values are normalized with respect to Φ 0.

Fig. 3
Fig. 3

Approximation of the zeroth- and the first-order moment of the source photon distribution after 20 scattering interactions by Gaussian distributions (12) and (13). Values are normalized with respect to Φ 0.

Fig. 4
Fig. 4

Zeroth- and first-order moments of the source photon distribution of the hybrid method (ten scattering interactions in the first stage), δ-E(2), P1 approximation and isotropic source approach. Values are normalized with respect to Φ 0.

Fig. 5
Fig. 5

Fluence rate and diffuse reflectance γ in semi-infinite medium with optical properties corresponding to dermis of newborns, calculated with different source functions.

Fig. 6
Fig. 6

Reflectance versus reduced scattering albedo, calculated with different source terms.

Fig. 7
Fig. 7

Reflectance versus wavelength, calculated for semi-infinite medium with optical properties corresponding to dermis of newborns, with different source terms.

Fig. 8
Fig. 8

Reflectance versus scattering anisotropy g, for a reduced scattering albedo of a′ = 0.9, calculated with different source functions.

Equations (51)

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·sLr, s=-μtLr, s+μs4π Lr, sps, sdΩ+Qr, s,
Lz, s=14π φd+34πjdz·s.
pHGs, s=14π1-g21+g2-2gs·s3/2=k2k+14π gkPks·s,
d2φdzdz2-φdzδ2=-q0zζ+3 dq1zdz,
φdz=C1 exp-z/δ+C2 expz/δ+δ×z0z-q0τζ+3 dq1τdτsinhz-τδdτ.
jd|z0+=Aφd|z0+.
φtot=φd+φq.
γ=-jd|z0+Φ0.
Qz=0, θ=Φ0δ1-cos θδz=Φ0k2k+12 Pkcos θδz,
q˜0zk=Φ0NkNd,
q˜1zk=Φ0Ndl=1Nksk,l·n,
q0z=α02πσ0 exp-z-ξ022σ02,
q1z=α12πσ1 exp-z-ξ122σ12+α22πσ2 exp-z-ξ222σ22,
φdz=C1 exp-z/δ+C2 expz/δ-δ4ζt0-z+t0+z+34-t1-z+t1+z-t2-z+t2+z,
tν-z=αν exp2ξνδ+σν22δ2erfξν-zδ+σν22 δσνexp-zδ,  tν+z=αν×exp-2ξνδ+σν22δ2erf-ξν+zδ+σν22 δσνexpzδ.
jdz=ζδC1 exp-zδ-C2 expzδ+u1z+u2z+-t0-z+t0+z4+34ζδ-t1-z+t1+z-t2-z+t2+z,
uνz=-3ανζ2πσν exp-ξν-z22σν2.
C2=α0δ4ζ exp-2ξ0δ+σ022δ2-34α1 exp-2ξ1δ+σ122δ2+α2 exp-2ξ2δ+σ222δ2.
φqzk=Φ0AkNd,
γtot=γd+γq,
Φz=Φ0 exp-μtz.
-ddzΦz=μtΦ0 exp-μtz=μa+μsΦ0 exp-μtz,
Qz=pHGn·sμsΦ0 exp-μtz=μsk=0N2k+14π gkPkn·sΦ0 exp-μtz,
qkz=μsgkΦ0 exp-μtz.
φdz=C1 exp-z/δ+C2 expz/δ+Φ0δ2μs1-μt2δ21ζ+3gμtexp-μtz.
φqz=Φ0 exp-μtz.
pδ-EN+1n·s=12π fδ1-n·s+1-fk=0N2k+14π g˜kPkn·s.
k=0N+12k+14π gkPkn·s=f k=0N+12k+14π Pkn·s+1-fk=0N2k+14π g˜kPkn·s
g˜k=gk-f1-f,
f=gN+1.
-ddzΦ˜z=μa+μ˜sΦ˜z.
μ˜sΦ˜z=1-fμsΦ˜z,
Qz=μ˜spEn·sΦ0 exp-μ˜tz,
qkz=μ˜sg˜kΦ0 exp-μ˜tz.
φdz=C1 exp-z/δ+C2 expz/δ+Φ0δ2μ˜s1-μ˜t2δ2×1ζ+3 gμ˜t1+gexp-μ˜tz.
φqz=Φ0 exp-μ˜tz.
q0z=μsΦ0 exp-μtrz
φdz=C1 exp-z/δ+C2 expz/δ+Φ0δ2μsζ1-μtr2δ2 exp-μtrz.
φqz=Φ0 exp-μtrz.
Lz, s=k2k+14π ϕkzPks·n,
Qz, s=k2k+14π qkzPks·n,
μtr0ϕ0z+ddz ϕ1z=q0z,  13ddz ϕ0z+μtr1ϕ1z+23ddz ϕ2z=q1z,  25ddz ϕ1z+μtr2ϕ2z+35ddz ϕ3z=q2z,  37ddz ϕ2z+μtr3ϕ3z=q3z,
μaφdz+ddz jdz=q0z,
13ddz φdz+μtrjdz=q1z,
s·n<0 Lz0+, sPns·-ndΩ=s·n>0 RFresns·nLz0+, sPns·ndΩ,  n odd,
4-2R1+1ϕ0+83R2+1ϕ1+5-12R3+4R1+1ϕ2+285R4-3R2ϕ3=0,
-20R3+12R1-1ϕ0+125R4-3R2ϕ1+5-30R5+28R3-6R1+1ϕ2+2175R6-210R4+63R2+4ϕ3=0,
Rk=0π/2 RFresnθcosk θ sin θdθ.
-2R1+1ϕ0+23R2+1ϕ1=0.
jd=-Aφd,
ϕkz=l=03 Ckl expνlz+Dk exp-μtz,  k=0, 1, 2, 3.

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