Abstract

We describe a new light transport model, which was applied to three-dimensional lifetime imaging of Förster resonance energy transfer in mice in vivo. The model is an approximation to the radiative transfer equation and combines light diffusion and ray optics. This approximation is well adopted to wide-field time-gated intensity-based data acquisition. Reconstructed image data are presented and compared with results obtained by using the telegraph equation approximation. The new approach provides improved recovery of absorption and scattering parameters while returning similar values for the fluorescence parameters.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  21. V. Y. Soloviev and S. R. Arridge, “Fluorescence lifetime optical tomography in weakly scattering media in the presence of highly scattering inclusions,” J. Opt. Soc. Am. A 28, 1513–1523 (2011).
    [CrossRef]
  22. H. C. van de Hulst, in Light Scattering by Small Particles(Dover, 1981).
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    [CrossRef]

2011

2010

2009

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

J. McGinty, V. Y. Soloviev, K. B. Tahir, R. Laine, D. W. Stuckey, J. V. Hajnal, A. Sardini, P. M. W. French, and S. R. Arridge, “Three-dimensional imaging of Förster resonance energy transfer in heterogeneous turbid media by tomographic fluorescent lifetime imaging,” Opt. Lett. 34, 2772–2774(2009).
[CrossRef] [PubMed]

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

V. Gaind, K. J. Webb, S. Kularatne, and C. A. Bouman, “Towards in vivo imaging of intramolecular fluorescence resonance energy transfer parameters,” J. Opt. Soc. Am. A 26, 1805–1813 (2009).
[CrossRef]

2007

2006

2005

2004

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[CrossRef]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

1999

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. R. Lakowicz, in Principles of Fluorescence Spectroscopy(Plenum, 1999).

J. Nocedal and S. J. Wright, Numerical Optimization(Springer-Verlag, 1999).
[CrossRef]

1996

1990

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

1985

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255(1985).
[CrossRef] [PubMed]

1981

H. C. van de Hulst, in Light Scattering by Small Particles(Dover, 1981).

1968

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

1963

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

Achilefu, S.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

Akers, W.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

Andrews, D. L.

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[CrossRef]

Arridge, S. R.

J. McGinty, D. W. Stuckey, V. Y. Soloviev, R. Laine, M. Wylezinska-Arridge, D. J. Wells, S. R. Arridge, P. M. W. French, J. V. Hajnal, and A. Sardini, “In vivo fluorescence lifetime tomography of a FRET probe expressed in mouse,” Biomed. Opt. Express 2, 1907–1917 (2011).
[CrossRef] [PubMed]

V. Y. Soloviev and S. R. Arridge, “Optical tomography in weakly scattering media in the presence of highly scattering inclusions,” Biomed. Opt. Express 2, 440–451 (2011).
[CrossRef] [PubMed]

V. Y. Soloviev and S. R. Arridge, “Fluorescence lifetime optical tomography in weakly scattering media in the presence of highly scattering inclusions,” J. Opt. Soc. Am. A 28, 1513–1523 (2011).
[CrossRef]

V. Y. Soloviev, C. D’Andrea, P. S. Mohan, G. Valentini, R. Cubeddu, and S. R. Arridge, “Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme,” Biomed. Opt. Express 1, 998–1013 (2010).
[CrossRef]

J. McGinty, V. Y. Soloviev, K. B. Tahir, R. Laine, D. W. Stuckey, J. V. Hajnal, A. Sardini, P. M. W. French, and S. R. Arridge, “Three-dimensional imaging of Förster resonance energy transfer in heterogeneous turbid media by tomographic fluorescent lifetime imaging,” Opt. Lett. 34, 2772–2774(2009).
[CrossRef] [PubMed]

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, “Fluorescence lifetime imaging by using time gated data acquisition,” Appl. Opt. 46, 7384–7391 (2007).
[CrossRef] [PubMed]

V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. 32, 2034–2036(2007).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

Bacskai, B. J.

Boas, D. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Bouman, C. A.

Boverman, G.

Bradshaw, D. S.

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[CrossRef]

Chance, B.

Cubeddu, R.

Culver, J. P.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

D’Andrea, C.

Davis, D. M.

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Dunn, A. K.

Dunsby, C.

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Elson, D. S.

V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, “Fluorescence lifetime imaging by using time gated data acquisition,” Appl. Opt. 46, 7384–7391 (2007).
[CrossRef] [PubMed]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

French, P. M. W.

J. McGinty, D. W. Stuckey, V. Y. Soloviev, R. Laine, M. Wylezinska-Arridge, D. J. Wells, S. R. Arridge, P. M. W. French, J. V. Hajnal, and A. Sardini, “In vivo fluorescence lifetime tomography of a FRET probe expressed in mouse,” Biomed. Opt. Express 2, 1907–1917 (2011).
[CrossRef] [PubMed]

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

J. McGinty, V. Y. Soloviev, K. B. Tahir, R. Laine, D. W. Stuckey, J. V. Hajnal, A. Sardini, P. M. W. French, and S. R. Arridge, “Three-dimensional imaging of Förster resonance energy transfer in heterogeneous turbid media by tomographic fluorescent lifetime imaging,” Opt. Lett. 34, 2772–2774(2009).
[CrossRef] [PubMed]

V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. 32, 2034–2036(2007).
[CrossRef] [PubMed]

V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, “Fluorescence lifetime imaging by using time gated data acquisition,” Appl. Opt. 46, 7384–7391 (2007).
[CrossRef] [PubMed]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Gaind, V.

Galletly, N.

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Hajnal, J. V.

Hares, J. D.

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

Kellett, P. A.

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

Krasnosselskaia, L. V.

Kularatne, S.

Kumar, A. T.

Kumar, A. T. N.

Laine, R.

Lakowicz, J. R.

J. R. Lakowicz, in Principles of Fluorescence Spectroscopy(Plenum, 1999).

Lanigan, P. M. P.

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Li, X. D.

Low, P. S.

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

McCann, F.

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

McGinty, J.

J. McGinty, D. W. Stuckey, V. Y. Soloviev, R. Laine, M. Wylezinska-Arridge, D. J. Wells, S. R. Arridge, P. M. W. French, J. V. Hajnal, and A. Sardini, “In vivo fluorescence lifetime tomography of a FRET probe expressed in mouse,” Biomed. Opt. Express 2, 1907–1917 (2011).
[CrossRef] [PubMed]

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

J. McGinty, V. Y. Soloviev, K. B. Tahir, R. Laine, D. W. Stuckey, J. V. Hajnal, A. Sardini, P. M. W. French, and S. R. Arridge, “Three-dimensional imaging of Förster resonance energy transfer in heterogeneous turbid media by tomographic fluorescent lifetime imaging,” Opt. Lett. 34, 2772–2774(2009).
[CrossRef] [PubMed]

V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, “Fluorescence lifetime imaging by using time gated data acquisition,” Appl. Opt. 46, 7384–7391 (2007).
[CrossRef] [PubMed]

V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. 32, 2034–2036(2007).
[CrossRef] [PubMed]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Mohan, P. S.

Munro, I.

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Neil, M. A. A.

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

V. Y. Soloviev, K. B. Tahir, J. McGinty, D. S. Elson, M. A. A. Neil, P. M. W. French, and S. R. Arridge, “Fluorescence lifetime imaging by using time gated data acquisition,” Appl. Opt. 46, 7384–7391 (2007).
[CrossRef] [PubMed]

V. Y. Soloviev, J. McGinty, K. B. Tahir, M. A. A. Neil, A. Sardini, J. V. Hajnal, S. R. Arridge, and P. M. W. French, “Fluorescence lifetime tomography of live cells expressing enhanced green fluorescent protein embedded in a scattering medium exhibiting background autofluorescence,” Opt. Lett. 32, 2034–2036(2007).
[CrossRef] [PubMed]

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization(Springer-Verlag, 1999).
[CrossRef]

Nothdurft, R. E.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

Ntziachristos, V.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

O’Leary, M. A.

Onfelt, B.

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

Patwardhan, S. V.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

Perona, P.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell. 12, 629–639 (1990).
[CrossRef]

Raymond, S. B.

Requejo-Isidro, J.

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J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
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C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
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S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
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[CrossRef]

J. Biomed. Opt.

R. E. Nothdurft, S. V. Patwardhan, W. Akers, Y. Ye, S. Achilefu, and J. P. Culver, “In vivo fluorescence lifetime tomography,” J. Biomed. Opt. 14, 024004 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Phys. D

C. Dunsby, P. M. P. Lanigan, J. McGinty, D. S. Elson, J. Requejo-Isidro, I. Munro, N. Galletly, F. McCann, B. Treanor, B. Onfelt, D. M. Davis, M. A. A. Neil, and P. M. W. French, “An electronically tunable ultrafast laser source applied to fluorescence imaging and fluorescence lifetime imaging microscopy,” J. Phys. D 37, 3296–3303 (2004).
[CrossRef]

J. McGinty, J. Requejo-Isidro, I. Munro, C. B. Talbot, P. A. Kellett, J. D. Hares, C. Dunsby, M. A. A. Neil, and P. M. W. French, “Signal-to-noise characterization of time-gated intensifiers used for wide-field time-domain FLIM,” J. Phys. D 42, 135103 (2009).
[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Meshes of four mice legs used in experiments. Legs (a) and (c) were transfected with unlinked eGFP and mCherry fluorophores (control mice). Legs (b) and (d) were transfected with FRET construct, in which eGFP and mCherry are coupled by a flexible linker.

Fig. 2
Fig. 2

Reconstruction results using the approximated radiative transfer model. The first column displays meshes of four legs with cuts at heights where slices are taken. The first and third rows show reconstruction of parameters of legs transfected with unlinked eGFP and mCherry. The second and fourth rows display parameters of legs transfected with FRET construct. The second and third columns show the absorption and scattering coefficients in mm 1 . The fourth and fifth columns display the quantum yield and lifetime reconstructions. The lifetime is given in nanoseconds.

Fig. 3
Fig. 3

Reconstructed absorption and scattering coefficients overlayed with MRI slices. The first row [(a)–(d)] shows absorption coefficients for four mouse legs. The second row [(e)–(h)] shows scattering coefficients. Red indicates peak values.

Fig. 4
Fig. 4

Reconstruction results obtained with the TE approximation. The layout of the figure is the same as in Fig. 2.

Equations (33)

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s · I + μ ˜ I = μ s u .
u = 1 4 π ( 4 π ) I ( s ) d 2 s .
Λ u = 1 4 π μ s u 0 ,
Λ = · κ + μ a + i ω / c ,
κ = 1 / 3 μ ˜ .
s · I 0 + μ ˜ I 0 = q 0 δ ( r r 0 ) δ ( s s 0 ) ,
I 0 ( r , s 0 ) = q 0 exp ( 0 l μ ˜ ( r 0 + s 0 l ) d l ) ,
I ( r , s ) = 0 l max μ s ( r s l ) u ( r s l ) × exp ( 0 l μ ˜ ( r s l ) d l ) d l ,
s · I 0 + μ ˜ I 0 = 1 4 π B 0 ,
B 0 ( r ) = μ a η 1 + i ω τ ( u + I 0 ) .
I 0 ( r , s ) = 1 4 π 0 l max B 0 ( r s l ) × exp ( 0 l μ ˜ ( r s l ) d l ) d l .
s · I + μ ˜ I = μ s u ,
Λ u = B 0 .
I ( r , s ) = I 0 ( r , s ) + 0 l max μ s ( r s l ) × u ( r s l ) exp ( 0 l μ ˜ ( r s l ) d l ) d l .
F = ς ( ω ) ( E + L + D ) d ω + ϒ .
E = | s | = 1 ξ ( s ) d 2 s V χ ( r ) ( | I E I | 2 + | I F I | 2 ) d 3 r ,
ξ ( s ) = 0 n < N δ ( s s n ) ,
χ ( r ) = a p 0 m < M δ ( r r m ) , ς ( ω ) = 0 l < L δ ( ω ω l ) ,
L = Re | s | = 1 ξ ( s ) J , s · I + μ ˜ I μ s B d 2 s + Re | s | = 1 ξ ( s ) J , s · I + μ ˜ I μ s B B 0 / 4 π d 2 s ,
D = Re | s | = 1 ξ ( s ) ψ , Λ u μ s u 0 d 2 s + Re | s | = 1 ξ ( s ) ψ , Λ u B 0 d 2 s .
x = ( μ s , μ a , η , τ ) T ,
ϒ = j = 1 4 α j Δ x j 2 .
δ F ( I , I , J , J , u , u , ψ , ψ , x ) = 0.
s n · J + μ ˜ J = 2 χ ( r ) ( I E I ) ,
s n · J * + μ ˜ J * = 2 χ ( r ) ( I F I ) .
Λ ψ * = μ s J * ,
Λ ψ * = μ s J * + μ a η 1 + i ω τ ( ψ * + J * ) ,
x k + 1 j = x k j + ( 1 / α j ) f j ,
f 1 Re | s | = 1 ξ ( s ) [ ( u I ) J * + ( u I ) J * ] d 2 s + Re | s | = 1 ξ ( s ) [ 3 κ 2 ( ψ * · u + ψ * · u ) ] d 2 s ,
f 2 Re | s | = 1 ξ ( s ) [ J * I J * I ψ * u ψ * u + η u 1 + i ω τ ( J * + ψ * ) ] d 2 s ,
f 3 = Re | s | = 1 ξ ( s ) μ a u 1 + i ω τ ( J * + ψ * ) d 2 s ,
f 4 = ω Im | s | = 1 ξ ( s ) η μ a u ( 1 + i ω τ ) 2 ( J * + ψ * ) d 2 s .
E TE = | s | = 1 ξ ( s ) d 2 s V χ ( r ) ( | u E u | 2 + | u F u | 2 ) d 3 r ,

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