Abstract

The lifetime of a fluorophore generally varies in different environments, making the molecule a sensitive indicator of tissue oxygenation, pH, and glucose. However, lifetime measurements are complicated when the fluorophore is embedded in an optically thick, highly scattering medium such as human tissue. We formulate the inverse problem for fluorescence lifetime tomography using diffuse photon density waves, and we demonstrate the technique by deriving spatial images of heterogeneous fluorophore distribution and lifetime, using simulated measurements in heterogeneous turbid media.

© 1996 Optical Society of America

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  1. S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
    [CrossRef]
  2. D. W. Piston, M. S. Kirby, H. P. Cheng, W. J. Lederer, W. W. Webb, Appl. Opt. 33, 662 (1994).
    [CrossRef] [PubMed]
  3. E. M. Sevick-Muraca, C. L. Burch, Opt. Lett. 19, 1928 (1994).See also C. L. Hutchinson, T. L. Troy, E. M. Sevick-Muraca, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 274 (1995).
    [CrossRef] [PubMed]
  4. M. S. Patterson, B. W. Pogue, Appl. Opt. 33, 1963 (1994).
    [CrossRef] [PubMed]
  5. B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.
  6. A. J. Durkin, S. Jaikumar, N. Ramanujam, R. Richards-Kortum, Appl. Opt. 33, 414 (1994).
    [CrossRef] [PubMed]
  7. J. Wu, M. S. Feld, R. P. Rave, Appl. Opt. 32, 3585 (1993).
    [CrossRef] [PubMed]
  8. A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
    [CrossRef]
  9. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
    [CrossRef]
  10. J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Desari, M. S. Feld, Opt. Lett. 20, 489 (1995).
    [CrossRef] [PubMed]
  11. X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).
  12. M. S. Patterson, B. Chance, B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  13. J. B. Fishkin, E. Gratton, J. Opt. Soc. Am. A 10, 127 (1993).
    [CrossRef] [PubMed]
  14. A. G. Yodh, B. Chance, Phys. Today 48(3), 34 (1995).
    [CrossRef]
  15. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
    [CrossRef]
  16. Eq. (2) is the Fourier transform of the time-domain equation presented by Sevick-Muraca et al.3 and was presented in a similar form by Patterson and Pogue.4
  17. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
    [CrossRef]
  18. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.
  19. X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

1995 (4)

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

A. G. Yodh, B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Desari, M. S. Feld, Opt. Lett. 20, 489 (1995).
[CrossRef] [PubMed]

1994 (5)

1993 (4)

J. B. Fishkin, E. Gratton, J. Opt. Soc. Am. A 10, 127 (1993).
[CrossRef] [PubMed]

S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
[CrossRef]

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

J. Wu, M. S. Feld, R. P. Rave, Appl. Opt. 32, 3585 (1993).
[CrossRef] [PubMed]

1992 (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

1989 (1)

Bambot, S. B.

S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
[CrossRef]

Barnes, R.

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

Beauvoit, B.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

Burch, C. L.

Chance, B.

A. G. Yodh, B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, Appl. Opt. 28, 2331 (1989).
[CrossRef] [PubMed]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

Chapman, C.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Cheng, H. P.

Desari, R. R.

Durkin, A. J.

Feld, M. S.

Fishkin, J. B.

Gratton, E.

Haskell, R. C.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Itzkan, I.

Jaikumar, S.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

Kirby, M. S.

Knutson, J. R.

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

Knuttel, A.

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

Lakowitz, J. R.

S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
[CrossRef]

Lederer, W. J.

Li, X. D.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

Madsen, S.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Nioka, S.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

Patterson, M. S.

Perelman, L.

Piston, D. W.

Pogue, B. W.

Ramanujam, N.

Rao, G.

S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
[CrossRef]

Rave, R. P.

Richards-Kortum, R.

Schmitt, J. M.

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

Sevick-Muraca, E. M.

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

Svaasand, L. O.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Tromberg, B. J.

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Wang, Y.

Webb, W. W.

White, R.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

Wilson, B. C.

Wu, J.

Yodh, A. G.

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

A. G. Yodh, B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Opt. Lett. 20, 426 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

Appl. Opt. (5)

J. Lumin. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994);D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. Lett. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

Phys. Today (1)

A. G. Yodh, B. Chance, Phys. Today 48(3), 34 (1995).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

X. D. Li, B. Beauvoit, R. White, S. Nioka, B. Chance, A. G. Yodh, Proc. Soc. Photo-Opt. Instrum. Eng. 2389, 789 (1995).

Rev. Sci. Instrum. (1)

A. Knuttel, J. M. Schmitt, R. Barnes, J. R. Knutson, Rev. Sci. Instrum. 64, 638 (1993).
[CrossRef]

Trends Biotechnol. (1)

S. B. Bambot, J. R. Lakowitz, G. Rao, Trends Biotechnol. 13, 106 (1993).
[CrossRef]

Other (4)

B. J. Tromberg, S. Madsen, C. Chapman, L. O. Svaasand, R. C. Haskell, in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Eq. (2) is the Fourier transform of the time-domain equation presented by Sevick-Muraca et al.3 and was presented in a similar form by Patterson and Pogue.4

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. (to be published).

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Figures (2)

Fig. 1
Fig. 1

(a) Scanning geometry consists of a source and a detector scanning every 0.2 cm about the four sides of a 7.0 cm × 7.0 cm square in an infinite medium. The reconstruction area is a slab of area 5.0 cm × 5.0 cm and height 1.0 cm. The optical properties of the media are μ s = 10 cm 1 everywhere, μa (chromophore) = 0.03 cm−1 everywhere for both the excitation and the emission wavelength, and μa (fluorophore, inside the sphere) = 0.02 cm−1 at the excitation wavelength and 0.01 cm−1 at the emission wavelength. The source modulation frequency is 50 MHz. (b) Reconstruction of the fluorophore concentration for the system described in (a). The lifetime inside the 1.0-cm-diameter sphere is 0.6 ns. The gray scale ranges from 0 cm−1 (white) to 0.007 cm−1 (black). 2500 SIRT iterations were performed with a constraint on both concentration (0.0 ≤ η ≤ 0.1 cm−1) and lifetime (0 ≤ τ ≤ 10 ns). (c) Using this setup, we varied the actual lifetime of the fluorophore from 0.5 to 1.5 ns, and the average reconstructed lifetime was calculated. The open squares are derived from a reconstruction of both fluorophore concentration and lifetime, and the filled circles are derived from the lifetime reconstruction only. (d), (e) Same as (b), (c), except that the source modulation frequency has been increased to f = 150 MHz. The gray scale in (d) ranges from 0 cm−1 (white) to 0.009 cm−1 (black).

Fig. 2
Fig. 2

Scanning geometry for a system with a background fluorophore. μa of the background fluorophore is 0.001 cm−1 at the excitation wavelength and 0.005 cm−1 at the emission wavelength, and the lifetime is 1.0 ns. The setup is the same as in Fig. 1(a) with the addition of the background fluorophore and a second source 0.6 cm from the first. (b) Reconstruction of the heterogeneous fluorophore distribution for the system described in (a). The lifetime inside the 1.5-cm-diameter sphere is 0.6 ns. The gray scale ranges from 0 cm−1 (white) to 0.017 cm−1 (black). 2500 SIRT iterations were performed with a constraint on both concentration (0.0 ≤ η ≤ 0.1 cm−1) and lifetime (0 ≤ τ ≤10 ns). (c) Average reconstructed lifetime for a series of reconstructions, with the setup in (a). The open squares are derived from a reconstruction of both fluorophore concentration and lifetime. The filled circles are derived from the lifetime reconstruction only. (d), (e) Same as (b), (c), except that the source modulation frequency has been increased to f = 150 MHz. The gray scale in (d) ranges from 0 cm−1 (white) to 0.020 cm−1 (black).

Equations (5)

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δ u fl ( r , r s , r d ) = U 0 ( r s r , k λ 1 ) η ( r ) 1 i ω τ ( r ) υ D λ 2 × G ( r d r , k λ 2 ) d 3 r .
U fl ( r s , r d ) =         d 3 r δ u fl ( r , r s , r d ) .
U fl ( r s i , r d i ) i = j = 1 N voxels δ u fl ( r j , r s i , r d i ) d 3 r j .
U fl ( r s , r d ) = all space d 3 r U 0 ( r r s , k λ 1 ) × η 0 1 i ω τ 0 υ D λ 2 G ( r d r , k λ 2 ) + heterogeneity d 3 r × U 0 ( r r s , k λ 1 ) [ η 1 i ω τ ( r ) η 0 1 i ω τ 0 ] × υ D λ 2 G ( r d r , k λ 2 ) .
Δ U fl ( r s , r d ) = d 3 r U 0 ( r r s , k λ 1 ) × υ η ( r ) 1 i ω τ ( r ) υ D λ 2 G ( r d r , k λ 2 ) .

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