Abstract

Statistical properties of the expected amount of time spent by a photon at different depths of a semi-infinite turbid medium are derived with formalism based on the continuous-time random walk. The formalism is applied to the study of both cw and time-gated experiments. Earlier analytical results relating to cw experiments are reproduced with a single approximation, rather than the more complicated approach used in earlier research based on the discrete-time random walk. The distribution of the occupancy of different depths in a time-gated experiment is found to have a convenient scaling form.

© 1998 Optical Society of America

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  1. B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
    [CrossRef]
  2. J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
    [CrossRef] [PubMed]
  3. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  4. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  5. A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in OpticsE. Wolf, ed. (North-Holland, Amsterdam, 1995), Vol. 34, pp. 333–401.
    [CrossRef]
  6. L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
    [CrossRef] [PubMed]
  7. I. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef] [PubMed]
  8. D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [CrossRef]
  9. E. W. Montroll, G. H. Weiss, “Random walks on lattices. II,” J. Math. Phys. 6, 167–181 (1965).
    [CrossRef]
  10. G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).
  11. G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
    [CrossRef]
  12. G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
    [CrossRef]
  13. W. J. Cui, N. Wang, B. Chance, “Study of photon migration depths with time-resolved spectroscopy,” Opt. Lett. 16, 1632–1635 (1991).
    [CrossRef] [PubMed]
  14. T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
    [CrossRef] [PubMed]
  15. A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
    [CrossRef]
  16. M. J. Lighthill, Fourier Analysis and Generalised Functions (Cambridge U Press, Cambridge, UK, 1958).
    [CrossRef]
  17. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
    [CrossRef] [PubMed]
  18. V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
    [CrossRef] [PubMed]

1998

G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
[CrossRef]

1997

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

D. J. Durian, J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
[CrossRef]

1996

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
[CrossRef] [PubMed]

V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
[CrossRef] [PubMed]

1994

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

1992

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

1991

1989

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

I. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989).
[CrossRef] [PubMed]

1987

1965

E. W. Montroll, G. H. Weiss, “Random walks on lattices. II,” J. Math. Phys. 6, 167–181 (1965).
[CrossRef]

Alfano, R. R.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Arridge, S. R.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Bonner, R. F.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

Chance, B.

Chernomordik, V.

V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
[CrossRef] [PubMed]

Cui, W. J.

Das, B. B.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Durian, D. J.

Feld, M. S.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Gandjbakhche, A. H.

V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in OpticsE. Wolf, ed. (North-Holland, Amsterdam, 1995), Vol. 34, pp. 333–401.
[CrossRef]

Havlin, S.

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

Ishimaru, I.

Itzkan, I.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Lighthill, M. J.

M. J. Lighthill, Fourier Analysis and Generalised Functions (Cambridge U Press, Cambridge, UK, 1958).
[CrossRef]

Liu, F.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Masoliver, J.

G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
[CrossRef]

Montroll, E. W.

E. W. Montroll, G. H. Weiss, “Random walks on lattices. II,” J. Math. Phys. 6, 167–181 (1965).
[CrossRef]

Nossal, R.

V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

Page, D. L.

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
[CrossRef] [PubMed]

Perelman, L. T.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Porrà, J. M.

G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
[CrossRef]

Rudnick, J.

Sevick-Muraca, E. M.

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
[CrossRef] [PubMed]

Troy, T. L.

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
[CrossRef] [PubMed]

Wang, N.

Weiss, G. H.

G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
[CrossRef]

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

E. W. Montroll, G. H. Weiss, “Random walks on lattices. II,” J. Math. Phys. 6, 167–181 (1965).
[CrossRef]

G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in OpticsE. Wolf, ed. (North-Holland, Amsterdam, 1995), Vol. 34, pp. 333–401.
[CrossRef]

Wu, J.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Appl. Opt.

J. Biomed. Opt.

T. L. Troy, D. L. Page, E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–345 (1996).
[CrossRef] [PubMed]

J. Math. Phys.

E. W. Montroll, G. H. Weiss, “Random walks on lattices. II,” J. Math. Phys. 6, 167–181 (1965).
[CrossRef]

J. Mod. Opt.

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

J. Opt. Soc. Am. A

J. Stat. Phys.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Med. Phys.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply imbedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

V. Chernomordik, R. Nossal, A. H. Gandjbakhche, “Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments,” Med. Phys. 23, 1857–1861 (1996).
[CrossRef] [PubMed]

Opt. Commun.

G. H. Weiss, J. M. Porrà, J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 14, 268–276 (1998).
[CrossRef]

Opt. Lett.

Phys. Med. Biol.

J. C. Hebden, S. R. Arridge, D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825–840 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Phys. Rev. Lett.

L. T. Perelman, J. Wu, I. Itzkan, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
[CrossRef] [PubMed]

Rep. Prog. Phys.

B. B. Das, F. Liu, R. R. Alfano, “Time-resolved fluorescence and photon migration studies in biomedical and model random media,” Rep. Prog. Phys. 60, 227–292 (1997).
[CrossRef]

Other

A. H. Gandjbakhche, G. H. Weiss, “Random walk and diffusion-like models of photon migration in turbid media,” in Progress in OpticsE. Wolf, ed. (North-Holland, Amsterdam, 1995), Vol. 34, pp. 333–401.
[CrossRef]

G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland, Amsterdam, 1994).

M. J. Lighthill, Fourier Analysis and Generalised Functions (Cambridge U Press, Cambridge, UK, 1958).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Plot of the relative average depth and associated standard deviation probed by a photon detected at time τ at a distance ρ from the source that introduces the photon into the medium. The abscissa is the parameter a = ρ[3/(2τ)]3/2.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

ψ t d t = Pr t T i < t + d t = k   exp - kt d t .
U ξ , τ Z ;   ρ = exp - ν τ 6 ρ ξ τ exp - τ - τ × g ρ ,   Z ,   ξ | 0 ,   1 ,   0 g ρ ,   1 ,   τ | ρ ,   Z ,   ξ d τ ,
E Z ρ ,   τ = 0 τ   U ξ , τ Z ;   ρ d ξ .
g ρ ,   1 ,   τ | ρ ,   Z ,   ξ = g ρ - ρ ,   Z ,   τ - ξ | 0 ,   1 ,   0 , τ ξ ,
U ξ , τ Z ;   ρ = exp - ν τ 6 ρ ξ τ exp - τ - τ × g ρ ,   Z ,   ξ | 0 ,   1 ,   0 × g ρ - ρ ,   Z ,   τ - ξ | 0 ,   1 ,   0 d τ .
g ρ ,   Z ,   ξ | 0 ,   1 ,   0 = exp - ξ I x ξ 3 I y ξ 3 × I Z - 1 ξ 3 - I Z + 1 ξ 3 ,
Γ x τ ,   ξ = x = -   I x ξ 3 I x - x τ - ξ 3 .
Γ x τ ,   ξ = I x τ 3 .
U ξ , τ Z ;   ρ = exp - 1 + ν τ 6 ξ τ   I x τ 3 I y τ 3 M Z ξ 3 × M Z τ - ξ 3 d τ ,
E Z ρ = 0 d τ   0 τ   U ξ , τ Z ;   ρ d ξ = 1 6 0 exp - 1 + ν τ d τ 0 τ   M Z ξ 3 d ξ × ξ τ   I x τ 3 I y τ 3 M Z τ - ξ 3 d τ
E Z ρ = 1 6 1 + ν 0 exp - 1 + ν τ I x τ 3 I y τ 3 d τ × 0 τ   M Z ξ 3 M Z τ - ξ 3 d ξ .
E ˆ Z ω = ρ   E Z ρ exp i ω · ρ ,
E ˆ Z ω = 1 6 1 + ν 0 exp - 1 + ν - c / 3 τ d τ × 0 τ   M Z ξ 3 M Z τ - ξ 3 d ξ ,
E ˆ Z ω = 1 6 1 + ν   M ˆ Z 2 1 + ν - c 3 ,
M ˆ s = 6 1 3 s + 9 s 2 - 1 1 / 2 Z .
E ˆ Z ω = 6   exp - 2 Z θ 1 + ν .
1 + ν - 1 3 cos   ω 1 + cos   ω 2 1 3 + ν + ω 2 6 .
cosh   θ 1 + 3 ν + ω 2 2 .
E ˆ Z ω 6 1 + ν exp - 2 Z ω 2 + 6 ν 1 / 2 .
E Z ρ 1 4 π 2   -   E ˆ Z ω exp - i ω · ρ d 2 ω = 1 2 π 0   ω J 0 ω ρ E Z ω d ω 3 π 1 + ν 0   ω J 0 ω ρ exp - 2 Z ω 2 + 6 ν 1 / 2 d ω = 6 Z π 1 + ν 1 ρ 2 + 4 Z 2 3 / 2 + 6 ν ρ 2 + 4 Z 2 × exp - 6 ν ρ 2 + 4 Z 2 1 / 2 ,
E Z ρ ,   τ = 0 τ   U ξ , τ Z ;   ρ d ξ .
E ˆ Z ω ,   s = ρ exp i ω · ρ 0 exp - s τ E Z ρ ,   τ d τ = 1 6 1 + ν + s 0 exp - 1 + ν + s - c / 3 τ d τ × 0 τ   M Z ξ 3 M Z τ - ξ 3 d ξ = 1 6 1 + ν + s   M ˆ Z 2 1 + ν + s - c 3 ,
E Z ρ ,   τ 6 Z π exp - ν τ - 1 1 R 3 + 6 s R 2 exp - R 6 s = 6 Z π 27 2 π τ 5 1 / 2 exp - 3 2 τ ρ 2 + 4 Z 2 - ν τ ρ 2 + 4 Z 2 1 / 2 ,
p Z | ρ ,   τ = E Z ρ ,   τ 0   E Z ρ ,   τ d Z = 8 Z   3 2 π τ exp - 3 2 τ ρ 2 + 4 Z 2 erfc ρ 3 2 τ ρ 2 + 4 Z 2 1 / 2 .
h Γ = 2 a π Γ   exp - a 2 1 + Γ 2 erfc a 1 + Γ 2 1 / 2 ,
Γ max = 2 Z max ρ = 1 2 1 + 2 a 2 1 / 2 - 1 1 / 2 .
Z max 2 ρ τ / 48 ,
Z 0.321 ρ 0.1107 τ 0.5 , σ Z 0.170 τ 0.5 ,
G β ,   ω = x = -   I x β exp i ω x .
I x β = 1 2 π - π π exp β   cos   ν + i ν x d ν ,
l = - exp 2 il η = l = -   δ η π - l .
G β ,   ω = exp β   cos   ω .
x = -   Γ x τ ,   ξ exp i ω x = G ξ 3 ,   ω G τ - ξ 3 ,   ω = exp τ / 3 cos   ω .

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