Abstract

Surface emission profiles and related functions are computed for particles (photons) migrating within a semi-infinite medium containing a surface layer whose absorbance differs from that of the underlying layer. Photons are assumed to be inserted at a single point on the surface. In certain cases distinct features appear in the emission profiles which enable determination of the thickness of the top layer and of the absorption coefficients of both layers. Computations are performed to provide estimates of parameter ranges for which the presence of one layer distorts photon emission profiles from the other. Several ancillary functions are calculated, including the absorbance profile as a function of depth, the expected path length of photons that are reemitted at a distance ρ from the point of insertion, and the average depth probed by those reemitted photons.

© 1988 Optical Society of America

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References

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  1. R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423 (1987).
    [CrossRef] [PubMed]
  2. R. Bonner, R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. 20, 2097 (1981).
    [CrossRef] [PubMed]
  3. R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
    [CrossRef]
  4. G. H. Weiss, R. J. Rubin, “Random Walks, Theory, and Selected Applications,” Adv. Chem. Phys. 52, 363 (1983).
    [CrossRef]
  5. W. E. Blumberg, “Light Propagation in Human Tissues,” Biophys. J. 51, No. 2, Part 2), 288a (1987).
  6. D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light Dosimetry in Tissue, Application to Photoradiation Therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, Eds. (Plenum, New York, 1983).
    [CrossRef]
  7. C. Kittrell, R. L. Willett, C. Santos-Pacheo, N. B. Ratliff, J. R. Kramer, E. G. Malk, M. S. Feld, “Diagnosis of Fibrous Arterial Atherosclerosis Using Fluorescence,” Appl. Opt. 24, 2280 (1985).
    [CrossRef] [PubMed]
  8. D. Ben-Avraham, S. Havlin, “Diffusion of Percolation Clusters at Criticality,” J. Phys. A 15, L691 (1982); S. Havlin, D. Ben-Avraham, “Diffusion in Disordered Media,” Adv. Phys. 36, 695 (1987).
    [CrossRef]
  9. S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
    [CrossRef]
  10. G. H. Weiss, R. Nossal, R. J. Bonner, “Statistics of Penetration Depth of Photons Re-Emitted from Irradiated Tissue,” Mod. Optics (in press).
  11. M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
    [CrossRef] [PubMed]
  12. P. J. Kolari, “Penetration of Unfocused Laser Light into the Skin,” Arch. Dermatol. Res. 277, 342 (1985).
    [CrossRef] [PubMed]
  13. B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327 (1986).
    [CrossRef] [PubMed]

1987 (2)

1986 (2)

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327 (1986).
[CrossRef] [PubMed]

1985 (2)

1984 (1)

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

1983 (1)

G. H. Weiss, R. J. Rubin, “Random Walks, Theory, and Selected Applications,” Adv. Chem. Phys. 52, 363 (1983).
[CrossRef]

1982 (1)

D. Ben-Avraham, S. Havlin, “Diffusion of Percolation Clusters at Criticality,” J. Phys. A 15, L691 (1982); S. Havlin, D. Ben-Avraham, “Diffusion in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

1981 (1)

Ben-Avraham, D.

D. Ben-Avraham, S. Havlin, “Diffusion of Percolation Clusters at Criticality,” J. Phys. A 15, L691 (1982); S. Havlin, D. Ben-Avraham, “Diffusion in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

Blumberg, W. E.

W. E. Blumberg, “Light Propagation in Human Tissues,” Biophys. J. 51, No. 2, Part 2), 288a (1987).

Bonner, R.

Bonner, R. F.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Bonner, R. J.

G. H. Weiss, R. Nossal, R. J. Bonner, “Statistics of Penetration Depth of Photons Re-Emitted from Irradiated Tissue,” Mod. Optics (in press).

Bowen, P. D.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Bowman, R. L.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Clem, T. R.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

Deutsch, T. F.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Dishon, M.

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

Dorion, D. R.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light Dosimetry in Tissue, Application to Photoradiation Therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, Eds. (Plenum, New York, 1983).
[CrossRef]

Feld, M. S.

Havlin, S.

R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423 (1987).
[CrossRef] [PubMed]

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

D. Ben-Avraham, S. Havlin, “Diffusion of Percolation Clusters at Criticality,” J. Phys. A 15, L691 (1982); S. Havlin, D. Ben-Avraham, “Diffusion in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

Kiefer, J. E.

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

Kittrell, C.

Kolari, P. J.

P. J. Kolari, “Penetration of Unfocused Laser Light into the Skin,” Arch. Dermatol. Res. 277, 342 (1985).
[CrossRef] [PubMed]

Kramer, J. R.

Malk, E. G.

Margolis, R.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Mathews-Roth, M. M.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Nossal, R.

Oseroff, A. R.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Parrish, J. A.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Patterson, M. S.

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327 (1986).
[CrossRef] [PubMed]

Prince, M. R.

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

Profio, A. E.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light Dosimetry in Tissue, Application to Photoradiation Therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, Eds. (Plenum, New York, 1983).
[CrossRef]

Ratliff, N. B.

Rubin, R. J.

G. H. Weiss, R. J. Rubin, “Random Walks, Theory, and Selected Applications,” Adv. Chem. Phys. 52, 363 (1983).
[CrossRef]

Santos-Pacheo, C.

Svaasand, L. O.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light Dosimetry in Tissue, Application to Photoradiation Therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, Eds. (Plenum, New York, 1983).
[CrossRef]

Weiss, G. H.

R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423 (1987).
[CrossRef] [PubMed]

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

G. H. Weiss, R. J. Rubin, “Random Walks, Theory, and Selected Applications,” Adv. Chem. Phys. 52, 363 (1983).
[CrossRef]

G. H. Weiss, R. Nossal, R. J. Bonner, “Statistics of Penetration Depth of Photons Re-Emitted from Irradiated Tissue,” Mod. Optics (in press).

Willett, R. L.

Wilson, B. C.

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327 (1986).
[CrossRef] [PubMed]

Adv. Chem. Phys. (1)

G. H. Weiss, R. J. Rubin, “Random Walks, Theory, and Selected Applications,” Adv. Chem. Phys. 52, 363 (1983).
[CrossRef]

Appl. Opt. (2)

Arch. Dermatol. Res. (1)

P. J. Kolari, “Penetration of Unfocused Laser Light into the Skin,” Arch. Dermatol. Res. 277, 342 (1985).
[CrossRef] [PubMed]

Biophys. J. (1)

W. E. Blumberg, “Light Propagation in Human Tissues,” Biophys. J. 51, No. 2, Part 2), 288a (1987).

J. Clin Invest. (1)

M. R. Prince, T. F. Deutsch, M. M. Mathews-Roth, R. Margolis, J. A. Parrish, A. R. Oseroff, “Preferential Light Absorption in Atheromas in vitro,” J. Clin Invest. 78, 295 (1986).
[CrossRef] [PubMed]

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

J. Phys. A (2)

D. Ben-Avraham, S. Havlin, “Diffusion of Percolation Clusters at Criticality,” J. Phys. A 15, L691 (1982); S. Havlin, D. Ben-Avraham, “Diffusion in Disordered Media,” Adv. Phys. 36, 695 (1987).
[CrossRef]

S. Havlin, G. H. Weiss, J. E. Kiefer, M. Dishon, “Exact Enumeration of Random Walks with Traps,” J. Phys. A 17, L347 (1984).
[CrossRef]

Phys. Med. Biol. (1)

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327 (1986).
[CrossRef] [PubMed]

Other (3)

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser Doppler Real-Time Monitor of Pulsatile and Mean Blood Flow in Tissue Microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, 1981).
[CrossRef]

G. H. Weiss, R. Nossal, R. J. Bonner, “Statistics of Penetration Depth of Photons Re-Emitted from Irradiated Tissue,” Mod. Optics (in press).

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light Dosimetry in Tissue, Application to Photoradiation Therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, Eds. (Plenum, New York, 1983).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Two-dimensional projection of semi-infinite layered medium, approximated by a lattice of discrete scattering loci surrounded by continuously absorbing material. The scattering length here is understood to be identical in the two regions, and the absorption coefficients in the top and the bottom layers are μ1 and μ2.

Fig. 2
Fig. 2

Surface emission profiles as a function of distance from point of photon insertion for various values of top layer thickness . The absorption coefficient of the top layer is much greater than that of the bottom, namely, μ1 = 0.2, μ2 = 0.01: (a) the intensity in the rth ring Gr vs r; (b) logrGr vs r; (c) [logrGr + r(6μ2)1/2] vs r (skewed values, see text). Note that distinguishing features appear only in the tails of the emission profiles, where the intensity has fallen off by several decades.

Fig. 3
Fig. 3

Absorption as a function of depth for the same conditions as in Fig. 2: (a) logA(z) vs z; (b) [logA(z) + z(6μ2)1/2] (skewed values).

Fig. 4
Fig. 4

Expected path length 〈n|r〉, given that a photon emerges at a point separated by r scattering lengths from the point of incidence: (a) 〈n|r vs r; (b) normalized values, 〈n|r〉/〈n|r vs r, where 〈n|r are values expected when → ∞ (a homogeneous semi-infinite medium with the absorptive properties of the top layer). Conditions are identical to those stated in the caption of Fig. 2.

Fig. 5
Fig. 5

Expected value of the depth 〈z|r〉 probed by photons emerging at r. Conditions are as described in Fig. 2 caption.

Fig. 6
Fig. 6

LogGr vs r. The top layer absorption coefficient here is much smaller than that of the lower region (μ1 = 0.01, μ2 = 0.2).

Fig. 7
Fig. 7

Absorption profiles, logA(z), vs z for the same conditions as shown in Fig. 6.

Fig. 8
Fig. 8

(a) Expected path length, 〈n|r〉 vs r. Conditions identical to those of Fig. 6. (b) Same as (a), except the abscissa is expanded.

Fig. 9
Fig. 9

Expected depth, 〈z|r〉 vs r. Conditions identical to those of Fig. 6.

Fig. 10
Fig. 10

Surface emission profiles, logGr vs r, for media with layers having similar absorbance: (a) μ1 = 0.1, μ2 = 0.05; (b) μ1 = 0.05, μ2 = 0.1.

Fig. 11
Fig. 11

Average depth 〈z|r〉 probed by emergent photons. Conditions are the same as in Figs. 10(a) and (b). The thick solid lines correspond to the situation where the entire medium has absorption μ = 0.1; dotted lines correspond to the situation where the entire medium has absorption μ = 0.05.

Fig. 12
Fig. 12

Maximum distance along the surface for which emergent photons primarily probe the upper layer rmax as a function of . Numbers in parentheses are the values (μ1/μ2) corresponding to each curve. Data are determined by extrapolations of curves of 〈z|r〉, such as given in Fig. 11, to the point where 〈z|r〉 = (1 ± 0.05) 〈z|r.

Fig. 13
Fig. 13

rmax vs . Data obtained from extrapolations of surface emission curves, logGr, ●, and curves of [〈n|r〉/〈n|r − 1], ■, compared with data for 〈z|r〉 shown in Fig. 12, ▲.

Fig. 14
Fig. 14

Comparison between theoretical value of decay rate of A(z) in homogeneous media (6μm)1/2, as given by the theory developed in Ref. 1, and the rate obtained from direct enumeration of probability densities, as described in Sec. II. Solid line, theoretical value; dotted line, numerical computation.

Equations (1)

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γ ( ρ ) ~ ρ - 1 exp [ - ρ ( 6 μ ) 1 / 2 ] ,

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