Abstract

Using a Cramer-Rao analysis, we study the theoretical performances of a time and spatially resolved fDOT imaging system for jointly estimating the position and the concentration of a point-wide fluorescent volume in a diffusive sample. We show that the fluorescence lifetime is a critical parameter for the precision of the technique. A time resolved fDOT system that does not use spatial information is also considered. In certain cases, a simple steady-state configuration may be as efficient as this time resolved fDOT system.

© 2011 OSA

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    [CrossRef] [PubMed]
  6. G. Y. Panasyuk, Z.-M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33, 1744 (2008).
    [CrossRef] [PubMed]
  7. D. Hall, G. Ma, F. Lesage, and Y. Wang, “Simple time-domain optical method for estimating the depth and concentration of a fluorescent inclusion in a turbid medium,” Opt. Lett. 29, 2258 (2004).
    [CrossRef] [PubMed]
  8. P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. M. Boffety, M. Allain, A. Sentenac, M. Massonneau, and R. Carminati, “Analysis of the depth resolution limit of luminescence diffuse optical imaging,” Opt. Lett. 33, 2290 (2008).
    [CrossRef] [PubMed]
  14. F. Leblond, H. Dehghani, D. Kepshire, and B. W. Pogue, “Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations,” J. Opt. Soc. Am. A 26, 1444 (2009).
    [CrossRef]
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  16. A. B. Milstein, M. D. Kennedy, P. S. Low, C. A. Bouman, and K. J. Webb, “Statistical approach for detection and localization of a fluorescing mouse tumor in Intralipid,” Appl. Opt. 44, 2300 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. The temporal pulse of the excitation is assumed to be a Dirac function. However, any pulse shape s(t) for the excitation can be considered since its contribution can be incorporated into R(t) if the temporal response s(t) ★tR(t) is considered instead of R(t) in Eq. (1).
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    [CrossRef]
  26. A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science 330, 353 (2010).
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    [CrossRef] [PubMed]
  29. M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963 (1994).
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    [CrossRef]
  32. For the TD-fDOT configuration considered in Sec. 2, the total number of measurements is N = K × P × Ns with Ns the number of excitation sources. For the CW-fDOT and ITD-fDOT setups, the number of measurements reduces to N = K × Ns and N = P × Ns, respectively.
  33. For Poisson noise, both the mean and the variance are equal so that the signal to noise ratio (SNR) the j-th data bin is 〈mj〉1/2 —i.e. the SNR increases with the expected number of detection in the bin.
  34. P. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004), Sec. 7.8.
  35. Provided that we restrict our analysis to the class of unbiased estimators (i.e., to estimator that are free of systematic errors), the Cramer-Rao bounds does not depend on the method used to estimate the parameters; see [34] for details.
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  38. M. G. Kendall and A. Stuart, The Advanced Theory of Statistics , Vol. 1, (Griffin, 1963), p. 57.
  39. Since the average number of detected photons for the considered fDOT systems is proportional to η × σ [cf. Eq. (3), (4) and (5)], the precision limits are scaled by the product (η × σ)−1/2. As a result, changing the parameters η and/or σ would only scale the normalized CRB shown in Fig. 4 and 5.
  40. A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence-lifetime-based tomography for turbid media,” Opt. Lett. 30, 3347 (2005).
    [CrossRef]
  41. J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
    [CrossRef] [PubMed]
  42. A. T. N. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470 (2008).
    [CrossRef] [PubMed]
  43. A. T. N. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 8, 1152 (2008).
    [CrossRef]
  44. A. Laidevant, A. Da Silva, M. Berger, J. Boutet, J.-M. Dinten, and A. C. Boccara, “Analytical method for localizing a fluorescent inclusion in a turbid medium,” Appl. Opt. 46, 2131 (2007).
    [CrossRef] [PubMed]
  45. A. B. Milstein, J. J. Stott, S. Oh, D. A. Boas, and R. P. Millane , “Fluorescence optical diffusion tomography using multiple-frequency data,” J. Opt. Soc. Am. A 21, 1035 (2004).
    [CrossRef]
  46. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5537 (1998).
    [CrossRef]
  47. R. Roy and E. M. Sevick-Muraca, “Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration,” Appl. Opt. 40, 2206 (2001).
    [CrossRef]

2010 (4)

2009 (4)

V. Y. Soloviev, C. D’Andrea, G. Valentini, R. Cubeddu, and S. R. Arridge, “Combined reconstruction of fluorescent and optical parameters using time-resolved data,” Appl. Opt. 48, 28 (2009).
[CrossRef]

F. Leblond, H. Dehghani, D. Kepshire, and B. W. Pogue, “Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations,” J. Opt. Soc. Am. A 26, 1444 (2009).
[CrossRef]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. 25, 3010 (2009).
[CrossRef]

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

2008 (6)

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53, 5797 (2008).
[CrossRef] [PubMed]

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

A. T. N. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 8, 1152 (2008).
[CrossRef]

A. T. N. Kumar, S. B. Raymond, B. J. Bacskai, and D. A. Boas, “Comparison of frequency-domain and time-domain fluorescence lifetime tomography,” Opt. Lett. 33, 470 (2008).
[CrossRef] [PubMed]

G. Y. Panasyuk, Z.-M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33, 1744 (2008).
[CrossRef] [PubMed]

M. Boffety, M. Allain, A. Sentenac, M. Massonneau, and R. Carminati, “Analysis of the depth resolution limit of luminescence diffuse optical imaging,” Opt. Lett. 33, 2290 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (5)

A. B. Milstein, M. D. Kennedy, P. S. Low, C. A. Bouman, and K. J. Webb, “Statistical approach for detection and localization of a fluorescing mouse tumor in Intralipid,” Appl. Opt. 44, 2300 (2005).
[CrossRef] [PubMed]

A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence-lifetime-based tomography for turbid media,” Opt. Lett. 30, 3347 (2005).
[CrossRef]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313 (2005).
[CrossRef] [PubMed]

A. H. Hielscher, “Optical Tomographic Imaging of small animals,” Curr. Opin. Biotechnol. 16, 79 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901 (2003).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91, 10 (2003).
[CrossRef]

2001 (2)

2000 (1)

L. V. Wang and S. L. Jacques, “Source of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61, 163 (2000).
[CrossRef]

1998 (1)

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5537 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
[CrossRef] [PubMed]

1989 (1)

1963 (1)

M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963 (1994).

1928 (1)

E. Sevick-Muraca and C. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928 (1994).

Allain, M.

Arridge, S. R.

V. Y. Soloviev, C. D’Andrea, G. Valentini, R. Cubeddu, and S. R. Arridge, “Combined reconstruction of fluorescent and optical parameters using time-resolved data,” Appl. Opt. 48, 28 (2009).
[CrossRef]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. 25, 3010 (2009).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[CrossRef] [PubMed]

Bacskai, B. J.

Belenkov, A.

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Berger, M.

Birk, U.

Boas, D.

Boas, D. A.

Boccara, A. C.

Boffety, M.

Bouman, C. A.

Boutet, J.

Burch, C.

E. Sevick-Muraca and C. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928 (1994).

Carminati, R.

Chance, B.

Chen, J.

Comsa, D. C.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53, 5797 (2008).
[CrossRef] [PubMed]

Contini, D.

Cubeddu, R.

D’Andrea, C.

Da Silva, A.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

A. Laidevant, A. Da Silva, M. Berger, J. Boutet, J.-M. Dinten, and A. C. Boccara, “Analytical method for localizing a fluorescent inclusion in a turbid medium,” Appl. Opt. 46, 2131 (2007).
[CrossRef] [PubMed]

Dehghani, H.

Deliolanis, N.

Delpy, D. T.

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
[CrossRef] [PubMed]

Dinten, J.-M.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

A. Laidevant, A. Da Silva, M. Berger, J. Boutet, J.-M. Dinten, and A. C. Boccara, “Analytical method for localizing a fluorescent inclusion in a turbid medium,” Appl. Opt. 46, 2131 (2007).
[CrossRef] [PubMed]

Ducros, N.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

Dunn, A. K.

A. T. N. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 8, 1152 (2008).
[CrossRef]

A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence-lifetime-based tomography for turbid media,” Opt. Lett. 30, 3347 (2005).
[CrossRef]

Farrell, T. J.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53, 5797 (2008).
[CrossRef] [PubMed]

Favicchio, R.

Firbank, M.

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
[CrossRef] [PubMed]

Fisher, R. A.

R. A. Fisher, Statistical Methods and Scientific Inference (Oliver & Boyd, 1956).

Gaiduk, A.

A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science 330, 353 (2010).
[CrossRef] [PubMed]

Gallant, P.

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[CrossRef] [PubMed]

Graves, E. E.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901 (2003).
[CrossRef] [PubMed]

Hall, D.

D. Hall, G. Ma, F. Lesage, and Y. Wang, “Simple time-domain optical method for estimating the depth and concentration of a fluorescent inclusion in a turbid medium,” Opt. Lett. 29, 2258 (2004).
[CrossRef] [PubMed]

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Hall, D. J.

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
[CrossRef] [PubMed]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[CrossRef] [PubMed]

J. C. Hebden, D. J. Hall, M. Firbank, and D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl Opt. 34, 8038 (1995).
[CrossRef] [PubMed]

Hervé, L.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

Hielscher, A. H.

A. H. Hielscher, “Optical Tomographic Imaging of small animals,” Curr. Opin. Biotechnol. 16, 79 (2005).
[CrossRef] [PubMed]

Hu, H.

L. V. Wang and H. Hu, Biomedical Optics (Wiley, 2007), Sec. 7.7.

Hyde, D.

Intes, X.

Ishimaru, A.

A. Ishimaru, “Wave propagation and scattering in random media,” (IEEE Press, 1997), p. 179.

Jacques, S. L.

S. L. Jacques and B. W. Pogue, “Tutorial on diffuse light transport,” J. Biomed. Opt. 13, 041302 (2008).
[CrossRef] [PubMed]

L. V. Wang and S. L. Jacques, “Source of error in calculation of optical diffuse reflectance from turbid media using diffusion theory,” Comput. Meth. Prog. Bio. 61, 163 (2000).
[CrossRef]

Jiang, H.

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. 37, 5537 (1998).
[CrossRef]

Kendall, M. G.

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics , Vol. 1, (Griffin, 1963), p. 57.

Kennedy, M. D.

Kepshire, D.

Kumar, A. T. N.

Laidevant, A.

Lasser, T.

Leblond, F.

Lesage, F.

V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189 (2010).
[CrossRef] [PubMed]

D. Hall, G. Ma, F. Lesage, and Y. Wang, “Simple time-domain optical method for estimating the depth and concentration of a fluorescent inclusion in a turbid medium,” Opt. Lett. 29, 2258 (2004).
[CrossRef] [PubMed]

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Li, X.

Low, P. S.

Ma, G.

D. Hall, G. Ma, F. Lesage, and Y. Wang, “Simple time-domain optical method for estimating the depth and concentration of a fluorescent inclusion in a turbid medium,” Opt. Lett. 29, 2258 (2004).
[CrossRef] [PubMed]

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Mamalaki, C.

Markel, V. A.

Martelli, F.

Massonneau, M.

McIntosh, L.

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

Millane, R. P.

Milstein, A. B.

Niedre, M.

Ntziachristos, V.

M. Niedre and V. Ntziachristos, “Comparison of fluorescence tomographic imaging in mice with early-arriving and quasi-continuous-wave photons,” Opt. Lett. 35, 369 (2010).
[CrossRef] [PubMed]

N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. 32, 382 (2007).
[CrossRef] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313 (2005).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91, 10 (2003).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901 (2003).
[CrossRef] [PubMed]

V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation,” Opt. Lett. 26, 893 (2001).
[CrossRef]

O’Leary, M.

Oh, S.

Orrit, M.

A. Gaiduk, M. Yorulmaz, P. V. Ruijgrok, and M. Orrit, “Room-temperature detection of a single molecule’s absorption by photothermal contrast,” Science 330, 353 (2010).
[CrossRef] [PubMed]

Panasyuk, G. Y.

Patterson, M.

Patterson, M. S.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53, 5797 (2008).
[CrossRef] [PubMed]

M. S. Patterson and B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963 (1994).

Peyrin, F.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. Three dimensional reconstructions,” Phys. Med. Biol. 54, 7107 (2009).
[CrossRef] [PubMed]

Pogue, B. W.

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V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23, 313 (2005).
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S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. 25, 3010 (2009).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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A. T. N. Kumar, J. Skoch, B. J. Bacskai, D. A. Boas, and A. K. Dunn, “Fluorescence-lifetime-based tomography for turbid media,” Opt. Lett. 30, 3347 (2005).
[CrossRef]

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Science (1)

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[CrossRef] [PubMed]

Other (12)

A. Ishimaru, “Wave propagation and scattering in random media,” (IEEE Press, 1997), p. 179.

The temporal pulse of the excitation is assumed to be a Dirac function. However, any pulse shape s(t) for the excitation can be considered since its contribution can be incorporated into R(t) if the temporal response s(t) ★tR(t) is considered instead of R(t) in Eq. (1).

L. V. Wang and H. Hu, Biomedical Optics (Wiley, 2007), Sec. 7.7.

P. Gallant, A. Belenkov, G. Ma, F. Lesage, Y. Wang, D. Hall, and L. McIntosh, “A quantitative time-domain optical imager for small animals in vivo fluorescence studies,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper WD2.

For the TD-fDOT configuration considered in Sec. 2, the total number of measurements is N = K × P × Ns with Ns the number of excitation sources. For the CW-fDOT and ITD-fDOT setups, the number of measurements reduces to N = K × Ns and N = P × Ns, respectively.

For Poisson noise, both the mean and the variance are equal so that the signal to noise ratio (SNR) the j-th data bin is 〈mj〉1/2 —i.e. the SNR increases with the expected number of detection in the bin.

P. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004), Sec. 7.8.

Provided that we restrict our analysis to the class of unbiased estimators (i.e., to estimator that are free of systematic errors), the Cramer-Rao bounds does not depend on the method used to estimate the parameters; see [34] for details.

P. Réfrégier, Noise Theory and Application to Physics: From Fluctuation to Information (Springer, 2004), p. 181.

R. A. Fisher, Statistical Methods and Scientific Inference (Oliver & Boyd, 1956).

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics , Vol. 1, (Griffin, 1963), p. 57.

Since the average number of detected photons for the considered fDOT systems is proportional to η × σ [cf. Eq. (3), (4) and (5)], the precision limits are scaled by the product (η × σ)−1/2. As a result, changing the parameters η and/or σ would only scale the normalized CRB shown in Fig. 4 and 5.

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Figures (5)

Fig. 1
Fig. 1

Schematic description of the reflection fDOT setup. L is the depth of the point-like fluorescent volume. The dots below the upper surface represent the 7×7 excitation laser sources. The detection is performed on a 32×32 pixel detector (CW-fDOT) or on a single detector with same area (TD-fDOT). The field of view is 25mm×25mm.

Fig. 2
Fig. 2

Expected counting signal in CW-fDOT (a) or in ITD-fDOT (c) obtained for a single excitation source (i.e. Ns = 1) located in s n = (0, 0, ) and a fluorescent volume located L = 3 mm in the slab —these signals are normalized by the incident excitation energy 0. The shaded area corresponds to the standard deviation due to photo noise. (b,d): Same as (a) and (c) for L = 4.5 mm (blue solid curve). The red dashed line is the expected signal for L = 3 mm, but with a concentration adjusted so that the maxima of the signals coincide for both depths. For the ITD configuration (c,d), the considered fluorophore lifetime is τ = 1 ns.

Fig. 3
Fig. 3

(Blue solid curve) Averaged temporal signal in the ITD configuration for a single excitation source (i.e. Ns = 1) located in s n = (0, 0, ) and a fluorescent volume located L = 5 mm with τ = 0 ns (a) or τ = 2 ns (b) —these signals are normalized by the incident excitation energy 0. The shaded area corresponds to the standard deviation due to photo noise. (Red dashed line) Averaged signal for a source in L = 2.5 mm with τ = 0 ns (a) or τ = 2 ns (b), but with a concentration adjusted so that the maxima of the averaged signals coincide for both depths.

Fig. 4
Fig. 4

(Black solid lines) Normalized precision limit for the estimation of L with different values of the fluorescence lifetime τ in the TD-fDOT situation, see Sec. 2 for details. (Red dashed line with circles) Normalized precision limit for τ = 0 ns in the ITD-fDOT setup. (Blue dashed line with triangles) Normalized precision limit for the CW-fDOT setup.

Fig. 5
Fig. 5

(Black solid lines) Normalized precision limit for different values of the fluorescence lifetime τ in the ITD-fDOT situation, see Sec. 2 for details. (Blue dashed line with triangles) Normalized precision limit for the CW-fDOT setup.

Equations (12)

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U ( r k , t ; s n ) = η σ 0 Ω slab G ( r k r , t ) t [ n ( r ) R ( t ; τ ) ] t G ( r s n , t ) d 3 r
m r k , t p , s n = 1 h ν Π p ( t ) Π k ( r ) D z b U ( r , t ; s n ) d 2 r d t
m r k , t p , s n TD ρ 0 × 𝒜 Δ T Δ Σ [ G ( r k r f , t p ) t R ( t p ; τ ) t G ( r f s n , t p ) ] = ρ 0 × g r k , t p , s n TD ( r f ; τ )
G CW ( r ) def . = 0 G ( r , t ) d t and G ITD ( t ; r ) def . = G ( r r , t ) d 2 r
m t p , s n ITD ρ 0 × 𝒜 Δ T [ G ITD ( t p ; r f ) t R ( t p ; τ ) t G ( r f s n , t p ) ] = ρ 0 × g t p , s n ITD ( r f ; τ )
m r k , s n CW ρ 𝒫 0 T × 𝒜 Δ Σ [ G CW ( r k r f ) G CW ( r f s n ) ] = ρ 𝒫 0 T × g r k , s n CW ( r f )
s n , k K p P m r k , t p , s n TD = p P m t p , s n ITD = k K m r k , s n CW
P ( m ; ρ , L ) = j = 1 N [ m j m j m j ! ] exp [ m j ] .
F = [ C L C L ρ C L ρ C ρ ] with { C L = ρ 0 Σ j [ L g j ( L ) ] 2 g j ( L ) C ρ = 0 ρ Σ j g j ( L ) C L ρ = 0 Σ j L g j ( L )
𝒫 L def. = CRB L δ L and 𝒫 ρ def. = CRB ρ δ ρ
CRB L def. = ( C L C L ρ 2 / C ρ ) 1 CRB ρ def. = ( C ρ C L ρ 2 / C L ) 1 .
𝒫 ˜ L def. = ( ρ 0 ) 1 / 2 × 𝒫 L

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