Abstract

A general linear model for time domain (TD) fluorescence tomography is presented that allows a lifetime-based analysis of the entire temporal fluorescence response from a turbid medium. Simulations are used to show that TD fluorescence tomography is optimally performed using two complementary approaches: A direct TD analysis of a few time points near the peak of the temporal response, which provides superior resolution; and an asymptotic multi-exponential analysis based tomography of the decay portion of the temporal response, which provides accurate localization of yield distributions for various lifetime components present in the imaging medium. These results indicate the potential of TD technology for biomedical imaging with lifetime sensitive targeted probes, and provide useful guidelines for an optimal approach to fluorescence tomography with TD data.

© 2006 Optical Society of America

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2006

2005

G. M. Turner, G Zacharakis, A. Sourbet, J. Ripoll, V. Ntziachristos, "Complete-angle projection diffuse optical tomography by use of early photons," Opt. Lett. 30, 409 (2005).
[CrossRef] [PubMed]

X. Lam, F. Lesage, X. Intes, "Time domain fluorescent diffuse optical tomography:analytical expressions," Opt. Express 13, 2263-2275 (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, 3347-3349 (2005).
[CrossRef]

Hintersteiner et al., "In-vivo detection of amyloid-beta deposits by near-infrared imaging using an oxazinederivative probe," Nat. Biotechnol. 23, 577 (2005).
[CrossRef] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, R. Weissleder, "Looking and listening to light," Nat. Biotechnol. 23, 314 (2005).

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[CrossRef]

2004

2003

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

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

A. Godavarty, E. M. Sevick-Muraca and M. J. Eppstein, "Three-dimensional fluorescence lifetime tomography," Med. Phys. 48, 1701-1720 (2003).
[CrossRef]

2001

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, "On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales," J. Phys. Chem. B. 105, 7847-7856 (2001).
[CrossRef]

J. P. Culver, V. Ntziachristos, M. J. Holboke and A. G. Yodh, "Optimization of optode arrangements for diffuse optical tomography: A singular-value analysis," Opt. Lett. 26, 701-703 (2001).
[CrossRef]

2000

K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, "Optical computed tomography in a turbid medium using early arriving photons," J. Biomed. Opt. 5144 (2000).
[CrossRef] [PubMed]

P. R. Sevin, "The renaissance of fluorescence resonance energy transfer," Nat. Struct. Biol. 7, 730-734, (2000).
[CrossRef]

1999

P. I. H. Bastiaens, and A. Squire, "Fluorescence lifetime imaging microscopy: spatial resolution biochemical processes in the cell," Trends Cell. Biol. 9, 48-52 (1999).
[CrossRef] [PubMed]

1997

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

J. Wu, L. Perelman, R. R. Dasari, ans M. S. Feld, "Fluorescence tomographic imaging in turbid media using early-arriving photons and Laplace transforms," Proc. Natl. Acad. Sci. USA 94, 8783 (1997).
[CrossRef] [PubMed]

1996

1992

1989

Achilefu, S

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

Alfano, R. R.

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

Allen, R.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Apreleva, S. V.

Bacskai, B. J.

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[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, 3347-3349 (2005).
[CrossRef]

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Bastiaens, P. I. H.

P. I. H. Bastiaens, and A. Squire, "Fluorescence lifetime imaging microscopy: spatial resolution biochemical processes in the cell," Trends Cell. Biol. 9, 48-52 (1999).
[CrossRef] [PubMed]

Bloch, S.

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

Boas, D. A.

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, 3347-3349 (2005).
[CrossRef]

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[CrossRef]

A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
[CrossRef]

M. A. Oleary, D. A. Boas, X.D. Li, B. Chance, and A. G. Yodh, "Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158 (1996).
[CrossRef]

Bouman, C. A.

A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
[CrossRef]

Champion, P. M.

A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, "On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales," J. Phys. Chem. B. 105, 7847-7856 (2001).
[CrossRef]

Chance, B.

Chen, K.

K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, "Optical computed tomography in a turbid medium using early arriving photons," J. Biomed. Opt. 5144 (2000).
[CrossRef] [PubMed]

Chernomordik, V.

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

Christian, J. F.

A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, "On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales," J. Phys. Chem. B. 105, 7847-7856 (2001).
[CrossRef]

Culver, J. P.

Das, B. B.

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

Dasari, R. R.

K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, "Optical computed tomography in a turbid medium using early arriving photons," J. Biomed. Opt. 5144 (2000).
[CrossRef] [PubMed]

J. Wu, L. Perelman, R. R. Dasari, ans M. S. Feld, "Fluorescence tomographic imaging in turbid media using early-arriving photons and Laplace transforms," Proc. Natl. Acad. Sci. USA 94, 8783 (1997).
[CrossRef] [PubMed]

Demidov, A. A.

A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, "On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales," J. Phys. Chem. B. 105, 7847-7856 (2001).
[CrossRef]

Dunn, A. K.

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, 3347-3349 (2005).
[CrossRef]

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[CrossRef]

Eppstein, M. J.

A. Godavarty, E. M. Sevick-Muraca and M. J. Eppstein, "Three-dimensional fluorescence lifetime tomography," Med. Phys. 48, 1701-1720 (2003).
[CrossRef]

Feld, M. S.

K. Chen, L. T. Perelman, Q. G. Zhang, R. R. Dasari, and M. S. Feld, "Optical computed tomography in a turbid medium using early arriving photons," J. Biomed. Opt. 5144 (2000).
[CrossRef] [PubMed]

Gallant, P.

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

Gandjbakche, A.

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

Gandjbakhche, A.

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

Gannot, I.

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

Gao, F.

Godavarty, A.

A. Godavarty, E. M. Sevick-Muraca and M. J. Eppstein, "Three-dimensional fluorescence lifetime tomography," Med. Phys. 48, 1701-1720 (2003).
[CrossRef]

Graves, E. E.

Hall, D.

Hammond, F. L.

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[CrossRef]

Haselgrove, J. C.

Hattery, D.

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

Hickey, G. A.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Hintersteiner,

Hintersteiner et al., "In-vivo detection of amyloid-beta deposits by near-infrared imaging using an oxazinederivative probe," Nat. Biotechnol. 23, 577 (2005).
[CrossRef] [PubMed]

Holboke, M. J.

Hyman, B. T.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Intes, X.

Kumar, A. T. N.

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, 3347-3349 (2005).
[CrossRef]

A. T. N. Kumar, J. Skoch, F. L. Hammond, A. K. Dunn, D. A. Boas and B. J. Bacskai, "Time resolved fluorescence imaging in diffuse media," Proc. of SPIE 600960090Y, (2005).
[CrossRef]

A. T. N. Kumar, L. Zhu, J. F. Christian, A. A. Demidov, and P. M. Champion, "On the Rate Distribution Analysis of Kinetic Data Using theMaximum EntropyMethod: Applications toMyoglobin Relaxation on the Nanosecond and Femtosecond Timescales," J. Phys. Chem. B. 105, 7847-7856 (2001).
[CrossRef]

Lam, X.

Leigh, J. S.

Lesage, F.

X. Lam, F. Lesage, X. Intes, "Time domain fluorescent diffuse optical tomography:analytical expressions," Opt. Express 13, 2263-2275 (2005).
[CrossRef] [PubMed]

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

D. Hall, G. Ma, F. Lesage, 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]

Li, X.D.

Liang, K.

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

Liu, F.

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

Loew, M.

D. Hattery, V. Chernomordik, M. Loew, I. Gannot, A. Gandjbakhche, J. Opt. Soc. Am. A, "Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue," J. Opt. Soc. Am. 18, 1523- 1530 (2001).
[CrossRef]

Ma, G.

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

D. Hall, G. Ma, F. Lesage, 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]

Mackintosh, L.

S. Bloch, F. Lesage, L. Mackintosh, A. Gandjbakche, K. Liang, S Achilefu, "Whole-body fluorescence lifetime imaging of a tumor-targeted near-infrared molecular probe in mice," J. Biomed. Opt. 10, 054003-1 (2005).
[CrossRef]

McIntosh, L.

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

Millane, R. P.

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[CrossRef]

Milstein, A.

A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
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Mincu, N.

G. Ma, N. Mincu, F. Lesage, P. Gallant, and L. McIntosh, "System irf impact on fluorescence lifetime fitting in turbid medium," Proc. SPIE 5699, 263 (2005).
[CrossRef]

Mycek, M.

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Oh, S.

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J. Wu, L. Perelman, R. R. Dasari, ans M. S. Feld, "Fluorescence tomographic imaging in turbid media using early-arriving photons and Laplace transforms," Proc. Natl. Acad. Sci. USA 94, 8783 (1997).
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A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
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A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
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V. Ntziachristos, J. Ripoll, L. V. Wang, R. Weissleder, "Looking and listening to light," Nat. Biotechnol. 23, 314 (2005).

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A. Milstein, J. J. Stott, S. Oh, D. A. Boas, R. P. Millane, C. A. Bouman and K. J. Webb, "Fluorescence optical diffusion tomography using multiple-frequency data," J. Opt. Soc. Am A 21, 1035-1049 (2004).
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[CrossRef] [PubMed]

Proc. of SPIE

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[CrossRef]

Proc. SPIE

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Other

Equation (4) is a generalized form of a semi-analytical expression for an infinite medium given in Reference 9, as can be checked by setting t′ →t−t′ and using the analytical expression for the Greens functions with μx a,s =μma,s.

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[CrossRef]

22. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

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Figures (7)

Fig. 1.
Fig. 1.

Simulations to elucidate the diffuse and pure fluorescent decay components in the diffuse fluorescence temporal response, and to demonstrate the accuracy the time domain fluorescence model presented in Eq. (11) in the text. The medium was an infinite slab of thickness 2cm (left panel) and 10cm (right panel), with optical properties µsx =µsm =10/cm, µax =µam =0.1/cm. The fluorescence signal was calculated for a single source detector pair, with a small fluorescent inclusion at the center. The signal calculated using the conventional approach in Eqs.(1–2), (+ symbol) is compared with that calculated using an effective-absorption based model, viz., Eqs. (11–12) (solid black line). The convolved medium diffusion, A(t) (dotted blue line) and asymptotic fluorescence decay (dashed red line) are also delineated for both cases.

Fig. 2.
Fig. 2.

Measurement geometry and arrangement of sources (*) and detectors (o) used for the simulations. The medium was assumed to be a diffusive slab of thickness 2cm. The targets used in the tomographic reconstructions are shown as gray shaded areas. (a) shows the topdown view and laterally separated (perpendicular to the source-detector axis) targets and (b) shows the side view, and targets located axially, i.e., along the source-detector axis.

Fig. 3.
Fig. 3.

(a) Singular value spectra of the TD weight matrix Wn [Eq. (10)] evaluated for a single temporal measurement. The time point for which the individual spectra are plotted are indicated as vertical lines in the inset, along with a representative DFTR. (b) Number of useful singular values, Nσ , as a function of the number of time points used in the weight matrix Wn . The weight matrix was optimized separately for each combination of time points. The noise threshold for evaluating Nσ was 10-14 (horizontal dotted line in panel (a)). The inset shows the optimal location of the five most significant time points on the DFTR (filled circles) along with Nσ for a single-time weight matrix as a function of the chosen time point along the DFTR (dashed line, right Y-axis).

Fig. 4.
Fig. 4.

Plot of the contrast-to-noise ratio (CNR) vs full-width-half-maximum (FWHM) for reconstructions using CW (solid black), optimal direct TD using 4 most significant time points near the rise and peak of the DFTR (red), and the asymptotic approach (blue circles). The cross talk for the 1ns inclusion, viz., the false yield amplitude at the location of the 1ns lifetime component due the 1.5ns component is also shown for the TD (red dash-dot line) and asymptotic (blue dashed line) cases. The simulations used the measurement setup shown in Fig 2, with two 2mm3 fluorescent targets 7mm apart, having distinct lifetimes of 1ns and 1.5ns.

Fig. 5.
Fig. 5.

X-Z slices of the 3-D Reconstructions of two targets with separations of 7mm, 5mm and 3mm, and with lifetimes of 1ns and 1.5ns, located transverse to the S-D axis, using CW (a-c), direct TD (d-f) and asymptotic (g-i) data sets. The measurement geometry used is shown in Fig. 2. The true location of the inclusions (2mm3) in each case is indicated by the gray shaded area. The reconstructions were regularized such that log 10(CNR) is near unity for all the cases. The images were generated by setting the red and green components of the RGB colormap to be the scaled yield reconstructions for the 1ns and 1.5ns lifetime components, respectively. This way, the cross-talk between the two components is easily visualized as mixture of the two colors (thus, yellow indicates 100% crosstalk). Quantitative plots of the yield along the X axis, at the fixed depth of the inclusions are shown for separations of 7mm (j), 5mm (k) and 3mm (l) (CW- black line; direct TD - 1ns, dashed-dot red and 1.5ns, dashed-dot green; asymptotic - 1ns, solid red and 1.5ns, solid green.)

Fig. 6.
Fig. 6.

Reconstructions for targets located axially, i.e., along the S-D axis, as shown in Fig. 2(b). The X-Z slice of the 3-D reconstructions are shown for (a) CW (b) lifetime-based direct TD and (c) lifetime-based asymptotic reconstructions. The colormap scheme used is the same as in Fig. 5. (d) shows quantitative plots of the yield along the depth Z, for the fixed X location of the inclusions. (CW - black line; direct TD - 1ns, dashed-dot red and 1.5ns, dashed-dot green; asymptotic - 1ns, solid red and 1.5ns, solid green.)

Fig. 7.
Fig. 7.

Illustration of the enhancement of direct TD reconstructions in the presence of lifetime contrast. The colormap scheme used is direct and reflects the actual reconstructed yield, in contrast to that used in Figs (5) and (6). (a) Yield reconstructions with two inclusions separated by 7mm, with both having the same lifetime of 1ns and (b-c) yield reconstructions for inclusions with distinct lifetimes of 1ns and 1.5ns. (d) 1-D plot of the reconstructed yield along the X axis at the actual depth of the inclusions, for no lifetime contrast (black) and with the inclusions having 1ns (red) and 1.5ns (green) lifetimes. (e) Dependence of cross talk for the direct TD reconstructions on the mean lifetime. The two inclusions had a fixed lifetime separation of 0.5ns, while the mean lifetime was varied between 0.75ns and 3.25ns.

Equations (17)

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U F ( r s , r d , t ) = Ω d 3 r W ( r s , r d , r , t ) η ( r ) ,
W ( r s , r d , r , t ) = 0 t d t 0 t d t " G m ( r d r , t t ) e Γ ( r ) ( t t " ) G x ( r r s , t " ) ,
W B ( r s , r d , r , t ) = 0 t d t " G x ( r s , r , t t " ) G m ( r , r d , t " ) ,
W ( r s , r d , r , t ) = 0 t d t W B ( r s , r d , r , t ) e Γ ( r ) ( t t ) .
W n ( r s , r d , r , t ) = 0 t d t W B ( r s , r d , r , t ) e Γ n ( t t ) ,
U F ( r s , r d , t ) = n Ω d 3 r W n ( r s , r d , r , t ) η n ( r ) .
( s · + 1 v t + μ a ( x , m ) ( r ) + μ s ( x , m ) ( r ) ) G ( x , m ) ( r , s , t )
= μ s ( x , m ) ( r ) Ω Θ ( s , s ) G ( x , m ) ( r , s , t ) d s ,
G ( x , m ) ( r , t ) = G 0 ( x , m ) ( r , t ) e v μ a ( x , m ) ( r ) t ,
G n ( x , m ) ( r , t ) = G ( x , m ) ( r , t ) μ a Γ n v = G ( x , m ) ( r , t ) μ a e Γ n t .
W n ( r s , r d , r , t ) = e Γ n t 0 t d t W n B ( r s , r d , r , t ) ,
U F ( r s , r d , t ) = n A n ( r s , r d , t ) e Γ n t ,
A n ( r s , r d , t ) = d 3 r [ 0 t d t W n B ( r s , r d , r , t ) ] η n ( r ) .
lim t > τ D W n ( r s , r d , r , t ) e Γ n t W ¯ n ( r s , r d , r ) .
( · U F j ( t k ) U F j ( t k + 1 ) · ) = ( · · · · · W n j ( t k ) W n + 1 j ( t k ) · · W n j ( t k + 1 ) W n + 1 j ( t k + 1 ) · · · · · ) ( · η n ( r ) η n + 1 ( r ) · ) ,
( · A n j A n + 1 j · ) = ( · · · · · W ¯ n j 0 · · 0 W ¯ n + 1 j · · · · · ) ( · η n ( r ) η n + 1 ( r ) · ) .
X = V S ( S 2 + α λ I ) 1 U T y

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