Abstract

To model the effect of a tissue–air boundary on time-resolved optical measurements, a convolution picture is presented based on the photon migration picture. It is demonstrated that the boundary conditions (either index matched or index mismatched) can be formulated as a spatiotemporal convolution of two terms, with the first being identical to the solution in the infinite medium and the second being independent of the original light source. The conditions under which the spatial convolution part in the second term becomes negligible are also determined, thus permitting the complete separation of the two terms by use of a temporal Laplace or Fourier transformation. This result is promising, since it suggests a method of removing the effect of the boundary conditions in the applications of time-resolved optical imaging (in both the time domain and the frequency domain) in biological tissues.

© 1997 Optical Society of America

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References

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  1. For a recent overview, see Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389, B. Chance, R. Alfano, eds. (1995), and references therein.
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Chap. 9, p. 175.
  3. S. Glasstone, Principles of Nuclear Reactor Engineering (Van Nostrand, New York, 1955), Chap. 3, p. 137.
  4. M. Keijzer, M. W. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  5. M. S. Patterson, B. Chance, 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. 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]
  7. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  8. J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
    [CrossRef] [PubMed]
  9. J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
    [CrossRef] [PubMed]
  10. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Photon migration in turbid media using path integrals,” Phys. Rev. Lett. 72, 1341–1344 (1994).
    [CrossRef] [PubMed]
  11. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
    [CrossRef]
  12. J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescence objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
    [CrossRef] [PubMed]
  13. J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
    [CrossRef] [PubMed]
  14. J. Wu, “Photon migration in turbid media: time-resolved optical imaging in tissuelike phantom,” Ph.D. dissertation (Division of Health Sciences and Technology, MIT, Cambridge, Mass., 1996), Chap. 4, p. 63.
  15. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, London, 1959), Chap. 14, p. 353.
  16. R. A. Groenhuis, A. A. Ferwerda, J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  17. G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994).
    [CrossRef] [PubMed]
  18. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
    [CrossRef]
  19. R. Mattuck, A Guide to Feynman Diagram in the Many-Body Problems (McGraw-Hill, New York, 1976), Chap. 1, p. 1.

1995 (4)

1994 (4)

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]

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

G. Mitic, J. Kolzer, J. Otto, E. Plies, G. Solkner, W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

1993 (2)

1989 (1)

1988 (1)

1983 (1)

Aronson, R.

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

Bosch, J. J. T.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, London, 1959), Chap. 14, p. 353.

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Dasari, R. R.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescence objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
[CrossRef] [PubMed]

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

Feld, M. S.

Feng, T. C.

Ferwerda, A. A.

Glasstone, S.

S. Glasstone, Principles of Nuclear Reactor Engineering (Van Nostrand, New York, 1955), Chap. 3, p. 137.

Groenhuis, R. A.

Haskell, R. C.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Chap. 9, p. 175.

Itzkan, I.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
[CrossRef] [PubMed]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescence objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

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

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, London, 1959), Chap. 14, p. 353.

Keijzer, M.

Kolzer, J.

Mattuck, R.

R. Mattuck, A Guide to Feynman Diagram in the Many-Body Problems (McGraw-Hill, New York, 1976), Chap. 1, p. 1.

McAdams, M. S.

Mitic, G.

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

Otto, J.

Partovi, F.

Patterson, M. S.

Perelman, L.

Perelman, L. T.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

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

Plies, E.

Rava, R. P.

Solkner, G.

Star, M. W.

Storchi, P. R. M.

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Wang, Y.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescence objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
[CrossRef] [PubMed]

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

Wilson, B. C.

Wu, J.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescence objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Three-dimensional imaging of objects embedded in turbid media with fluorescence and Raman spectroscopy,” Appl. Opt. 34, 3425–3430 (1995).
[CrossRef] [PubMed]

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

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

J. Wu, F. Partovi, M. S. Feld, R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[CrossRef] [PubMed]

J. Wu, “Photon migration in turbid media: time-resolved optical imaging in tissuelike phantom,” Ph.D. dissertation (Division of Health Sciences and Technology, MIT, Cambridge, Mass., 1996), Chap. 4, p. 63.

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

Zinth, W.

Appl. Opt. (7)

J. Lumin. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60–61, 281–286 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phys. Rev. E (1)

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (6)

For a recent overview, see Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389, B. Chance, R. Alfano, eds. (1995), and references therein.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978), Chap. 9, p. 175.

S. Glasstone, Principles of Nuclear Reactor Engineering (Van Nostrand, New York, 1955), Chap. 3, p. 137.

J. Wu, “Photon migration in turbid media: time-resolved optical imaging in tissuelike phantom,” Ph.D. dissertation (Division of Health Sciences and Technology, MIT, Cambridge, Mass., 1996), Chap. 4, p. 63.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, London, 1959), Chap. 14, p. 353.

R. Mattuck, A Guide to Feynman Diagram in the Many-Body Problems (McGraw-Hill, New York, 1976), Chap. 1, p. 1.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagrams of the boundary conditions in photon migration. (a) Infinite medium, (b) index-matched, semi-infinite medium, and (c) index-mismatched, semi-infinite medium.

Fig. 2
Fig. 2

[defined in Eq. (9)] calculated as a function of s with the diffusion equation solutions for various values of z0 (=10, 25, 40 mm) and ρ (=0 mm).

Fig. 3
Fig. 3

[defined in Eq. (9)] calculated as a function of z0 with the diffusion equation solutions for various values of ρ (=0, 5, 10 mm) and s (=6 ns-1).

Fig. 4
Fig. 4

[defined in Eq. (9)] calculated as a function of s with the Monte Carlo simulations for various values of z0 (=10, 25, 40 mm) and ρ (=0 mm).

Equations (36)

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Uinfinite(t, ρ, z0)=Usemi-infiniteindexmatch(t, ρ, z0)Ginfinite(t, ρ),
Usemi-infiniteindexmismatch(t, ρ, z0)=Usemi-infiniteindexmatch(t, ρ, z0)Gsemi-infiniteindexmismatch(t, ρ),
U˜˜infinite(st, ωρ, z0)=U˜˜semi-infiniteindexmatch(st, ωρ, z0)·G˜˜infinite(st, ωρ),
U˜˜semi-infiniteindexmismatch(st, ωρ, z0)=U˜˜semi-infiniteindexmatch(st, ωρ, z0)·G˜˜semi-infiniteindexmismatch(st, ωρ),
U˜˜infinite(st, ωρ, z0)=U˜˜semi-infinite(st, ωρ, z0)·G˜˜(st, ωρ)
U˜˜semi-infinite(st, ωρ, z0)=U˜˜infinite(st, ωρ, z0)·G˜˜(st, ωρ),
-Uinfinite(r, t)ct-D2Uinfinite(r, t)+μaUinfinite(r, t)=Sδ(z=z0)δ(t=0),
-Usemi-infinite(r, t)ct-D2Usemi-infinite(r, t)+μaUsemi-infinite(r, t)=Sδ(z=z0)δ(t=0),
hUsemi-infinite(r, t)=Usemi-infinite(r, t)zatz=0,
U˜infinite(r, s)=c2[Dc(s+μac)]1/2×exp{-[(s+μac)/Dc]1/2|z-z0|}=c2kqexp(-q|z-z0|),
U˜infinite(z=0, s)=c2kqexp(-qz0).
U˜semi-infinite(r, s)=c2kq{exp(-q|z-z0|)+exp[-q(z+z0)]}-hckq(q+h)×exp[-q(z+z0)],
U˜semi-infinite(z=0, s)=c2kqexp(-qz0)2-2hq+h
Uinfinite(t, z=0)=c(4πDct)-3/2×exp-ρ2+z024Dct-μact,
Usemi-infinite(t, z=0)=c(4πDct)-3/2 exp-ρ24Dct-μact×2 exp-z024Dct-2h0+ exp-hl-(z0+l)24Dctdl.
U˜˜infinite(s, ωx, ωy, z=0)=c4Dc[μac+s+16Dc(ωx2+ωy2)]1/2×exp{-[μa/D+s/DC+16(ωx2+ωy2)]1/2z0},
U˜˜semi-infinite(s, ωx, ωy, z=0)=c4Dc[μac+s+16Dc(ωx2+ωy2)]1/2×exp{-[μa/D+s/DC+16(ωx2+ωy2)]1/2z0}×2-2hh+[μa/D+s/DC+16(ωx2+ωy2)]1/2.
U˜˜infinite(z0)=U˜˜semi-infinite(z0)·G˜˜,
G˜˜=2-2hh+[μa/D+s/DC+16(ωx2+ωy2)]1/2-1.
Uinfinite(t, z=0)=c(4πDct)-3/2 exp-ρ2+z024Dct-μact,
Usemi-infinite(t, z=0)=z0t(4πDct)-3/2 exp-ρ2+z024Dct-μact,
U˜˜infinite(s, z=0)=c2{Dc[s+μac+16Dc(ωx2+ωy2)]}1/2×exp{-[s/Dc+μaD+16(ωx2+ωy2)]1/2z0},
U˜˜semi-infinite(s, z=0)=exp{-[s/Dc+μaD+16(ωx2+ωy2)]1/2z0}.
G˜˜=c2{Dc[s+μac+16Dc(ωx2+ωy2)]}1/2,
U˜infinite(z0)=U˜semi-infinite(z0)·G˜(z0),
G˜=2-2h0+ρ2+z02ρ2+(l+z0)21/2×exp{-hl-q[ρ2+(l+z0)2-ρ2+z02]}dl-1
[2+i(/t)-V]GV=δ(z-z0)δ(ρ)δ(t).
G˜V=G˜0+G˜0VG˜V=G˜0+G˜0VG˜0+G˜0VG˜0VG˜0+ ,
G˜V=G˜0(1+VG˜V).
G˜0=G˜V-G˜VVG˜0=G˜V-G˜VVG˜V+G˜VVG˜VVG˜V+ ,
G˜0=G˜V(1-VG˜0).
[-D2+(/ct)+V]Usemi-infinite=δ(z-z0)δ(ρ)δ(t)+δ(z+z0)δ(ρ)δ(t),
[-D2+(/ct)+V]Usemi-infinite±=δ(zz0)δ(ρ)δ(t).
U˜infinite=U˜semi-infininte+(1+VU˜infinite).
U˜infinite(z00)=U˜semi-infinite+(z00)×[1+2hDU˜infinite(00)],
U˜infinite=U˜semi-infinite++U˜semi-infinite+VU˜semi-infinite++U˜semi-infinite+VU˜semi-infinite+VU˜semi-infinite++ ,

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