Abstract

Fluorescence resonance energy transfer (FRET) is a nonradiative energy transfer process based on dipole-dipole interaction between donor and acceptor fluorophores that are spatially separated by a distance of a few nanometers. FRET has proved to be of immense value in the study of cellular function and the underlying cause of disease due to, for example, protein misfolding (of consequence in Alzheimer’s disease). The standard parameterization in intramolecular FRET is the lifetime and yield, which can be related to the donor-acceptor (DA) distance. FRET imaging has thus far been limited to in vitro or near-surface microscopy because of the deleterious effects of substantial scatter. We show that it is possible to extract the microscopic FRET parameters in a highly scattering environment by incorporating the FRET kinetics of an ensemble of DA molecules connected by a flexible or rigid linker into an optical diffusion tomography (ODT) framework. We demonstrate the efficacy of our approach for extracting the microscopic DA distance through simulations and an experiment using a phantom with scattering properties similar to tissue. Our method will allow the in vivo imaging of FRET parameters in deep tissue, and hence provide a new vehicle for the fundamental study of disease.

© 2009 Optical Society of America

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2008

2006

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

2005

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858-870 (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2004

2003

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081-3094 (2003).
[CrossRef] [PubMed]

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

C. Dobson, “Protein folding and misfolding,” Nature 426, 884-890 (2003).
[CrossRef] [PubMed]

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

2002

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
[CrossRef] [PubMed]

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95-97 (2002).
[CrossRef]

S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983-1993 (2002).
[CrossRef]

2001

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909-922 (2001).
[CrossRef]

K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
[CrossRef]

C. Dobson, “The structural basis of protein folding and its links with human disease,” Phil. Trans. R. Soc. Lond. B 356, 133-145 (2001).
[CrossRef]

A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001).
[CrossRef]

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

2000

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

1999

1998

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

1997

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

1996

R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
[CrossRef] [PubMed]

1989

J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113-146 (1989).
[CrossRef]

1973

1948

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Phys. 2, 55 (1948).
[CrossRef]

Adams, J.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Adams, J. C.

J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113-146 (1989).
[CrossRef]

J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colorado, 1991).

Aelst, L.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Allen, R.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Asano, T.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Bacskai, B.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

Bartoli, M.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Baty, J.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Berg, L.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Bernacchi, S.

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

Berry, G.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Boas, D.

Boas, D. A.

Bouman, C.

Bouman, C. A.

Bourg, N.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Bremer, C.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

Bullock, A.

A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001).
[CrossRef]

Chen, H.

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

Cheng, J.

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

Coats, M.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Davoust, J.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Dobson, C.

C. Dobson, “Protein folding and misfolding,” Nature 426, 884-890 (2003).
[CrossRef] [PubMed]

C. Dobson, “The structural basis of protein folding and its links with human disease,” Phil. Trans. R. Soc. Lond. B 356, 133-145 (2001).
[CrossRef]

Fersht, A.

A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001).
[CrossRef]

Förster, T.

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Phys. 2, 55 (1948).
[CrossRef]

Foschum, F.

Gearing, M.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Goate, A. M.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Gordon, G.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Haas, E.

E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858-870 (2005).
[CrossRef]

Hale, G.

Harvey, C.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[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-R43 (2005).
[CrossRef] [PubMed]

Heikal, A.

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

Heim, R.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Helm, A. P. G.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Herman, B.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Hickey, G.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

Hyman, B.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

Ikura, M.

K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
[CrossRef]

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Inukai, K.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Kienle, A.

Kramer, F.

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
[CrossRef] [PubMed]

S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
[CrossRef] [PubMed]

Kummer, S.

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 2nd ed. (Kluwer Academic, 1999).

Linder, K.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Llopis, J.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Low, P.

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

Mahajan, N.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Marras, S.

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
[CrossRef] [PubMed]

McCaffery, J.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

McKeel, D. W.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Melon, D.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Mely, Y.

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

Michels, R.

Millane, R.

Millane, R. P.

Miller, J. P.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Mills, J.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Milstein, A.

Milstein, A. B.

Mirra, S. S.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Mitra, R. D.

R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

Miyawaki, A.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Moerner, W.

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

Morris, J. C.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Norton, J.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Ntziachristos, V.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

Oh, S.

Ohkura, R.

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

Okonkwo, D.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Ozawa, T.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Price, J. L.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Querry, M.

Reynolds, J. S.

Richard, I.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Rubin, D.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Rubin, E. H.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Sato, M.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Saunders, A. M.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Schwille, P.

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

Sillon, G.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Silva, C. M.

R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

Skoch, J.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

Sobczyk, A.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Stockholm, D.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Stone, J.

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

Storandt, M.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Stott, J.

Sutoh, K.

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

Suzuki, Y.

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

Svoboda, K.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Truong, K.

K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
[CrossRef]

Tsien, R.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Tung, C.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

Tyagi, S.

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
[CrossRef] [PubMed]

S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
[CrossRef] [PubMed]

Umezawa, Y.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Vlahov, I.

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

Wakabayashi, T.

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

Webb, K.

Webb, K. J.

Webb, W.

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

Weissleder, R.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

Yang, J.

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

Yasuda, R.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Yasunaga, T.

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

Ye, J. C.

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909-922 (2001).
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J. C. Ye, K. J. Webb, C. A. Bouman, and R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400-2412 (1999).
[CrossRef]

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R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

Zhang, Q.

Zhong, H.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
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L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Chem. Phys.

E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858-870 (2005).
[CrossRef]

Curr. Op. Struct. Biol.

K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
[CrossRef]

Gene

R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

IEEE Trans. Image Process.

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909-922 (2001).
[CrossRef]

J. Biomed. Opt.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. 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]

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

J. Mol. Biol.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Nat. Biotechnol.

S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
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[CrossRef] [PubMed]

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Nat. Med.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
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Nature

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Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
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R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Nucleic Acids Res.

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
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J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
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Figures (8)

Fig. 1
Fig. 1

(a) The r F = 4.2 nm isosurface plot of the true image for a fixed linker. Also shown are the locations of the sources (top circles) and detectors (bottom circles) used to produce the simulated data. (b) Discretized spherical geometry with radius 2.5 mm (with 33 × 33 × 33 image resolution). For a rigid linker with r F = 4.2 nm and η = 0.025 : (c) reconstructed r F , (d) reconstructed η. For a flexible linker with a = 5.42 × 10 17 m 2 and b = 5.45 nm , with the geometry in (b): (e) reconstructed a, (f) reconstructed b. The average detector SNR was 30 dB .

Fig. 2
Fig. 2

Piecewise polynomial and exponential fitting of the real and imaginary parts of ζ ( r F ) when R 0 = 4.7 nm and τ D = 3.8 ns : (a) real part, and (b) imaginary part, for a modulation frequency of 80 MHz .

Fig. 3
Fig. 3

A phantom having two spheres, each of radius 2 mm , containing DA molecules connected by a flexible linker. (a) True image of a. For the top right sphere, a = 3.5 × 10 17 m 2 , and for bottom left sphere, a = 7.45 × 10 17 m 2 . (b) Reconstructed a. (c) True image of b. For top right sphere, b = 5.95 nm , and for bottom left sphere, b = 3.65 nm . (d) Reconstructed b. (e) True image of η. For top right and bottom left spheres, η = 0.025 . (f) Reconstructed η. The average detector SNR = 30 dB .

Fig. 4
Fig. 4

(a) and (b) Experimental setup: a 3 mW 488 nm argon-ion laser (Uniphase), an x y scanning mirror system, a Plexiglas box of size 8.8 cm ( L ) × 8.8 cm ( H ) × 3.4 cm (W), a cylindrical plastic vial (length 3 cm , inner-diameter 0.65 cm , outer-diameter 0.75 cm ), to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod 1 mm in diameter and 1.5 cm in length, a 520 nm narrow bandpass filter (Edmund Optics) with FWHM 10 nm [31], and a 105 mm , f/2.8 lens (AF micro Nikkor, Nikon) to focus the 4.8 cm × 4.8 cm image of the scattering medium on a 512 × 512   pixel Peltier cooled CCD Camera (PI-MAX, Roper Scientific).

Fig. 5
Fig. 5

Slices along the x y plane of the reconstructed image of η for the plastic vial containing the donor mixed with Intralipid. (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . Reconstructed η 0.0013 cm 1 .

Fig. 6
Fig. 6

Slices along the x y plane of the true r F , true location and shape of the plastic vial suspended inside the Intralipid scattering medium (see Fig. 3 for xyz axes orientation). (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . (d) Isosurface plot of the DA distance at r F = 3.6 nm , which is the distance estimated without the scattering medium (see Section 5E).

Fig. 7
Fig. 7

Slices along the x y plane of the reconstructed image of the DA distance. Expected r F = 3.6 nm . (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . (d) Isosurface plot of the reconstructed DA distance at r F = 3.4 nm .

Fig. 8
Fig. 8

Acquired image of plastic vials without the scattering medium. (a) Image of vial containing donor Bodipy-FL mixed with Intralipid at concentration C D = 250 nM . The total detected power is assumed to be the integral of the intensity in the dotted rectangle, which gives P D = γ 7.3 × 10 8 W . (b) Image of the vial containing the FRET chemical mixed with Intralipid at concentration C DA = 3 μ M . The detected power, using the intensity in the dotted rectangle, is P DA = γ 4.2 × 10 8 W . Using (20) and R 0 = 5.8 nm gives r F 3.6 nm

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

1 τ DA = 1 τ D [ 1 + ( R 0 r F ) 6 ] = k r + k n r + k F
η DA = k r k r + k n r + k F .
[ D x ( r ) φ x ( r , ω ) ] [ μ a x ( r ) i ω c ] φ x ( r , ω ) = S x ( r ; ω )
[ D m ( r ) φ m ( r , ω ) ] [ μ a m ( r ) i ω c ] φ m ( r , ω ) = φ x ( r , ω ) S f ( r ; ω ) ,
S f ( r ; ω ) = r min r max η DA μ a D ( r ) 1 i ω τ DA ( r ) p ( r F ( r ) ) d r F ( r ) = r min r max η ( r ) ζ ( r F ( r ) ) p ( r F ( r ) ) d r F ( r ) ,
φ ( r s k , r d m ; ω ) = g m ( r , r d m ; ω ) S ( r ; ω ) g x ( r s k , r ; ω ) d 3 r ,
x F = [ η ( r 1 ) η ( r N ) , a ( r 1 ) a ( r N ) , b ( r 1 ) b ( r N ) ] T
x F = [ η ( r 1 ) η ( r N ) , r F ( r 1 ) r F ( r N ) ] T
{ x ̂ F } MAP = arg max x F 0 , s , d { q = 1 Q p ( y F q | x F , s q , d q ) p ( x F ) } ,
p ( y F q | x F , s q , d q ) = 1 ( π α F ) P | Λ F q | 1 exp [ y F q f F q ( x F , s q , d q ) Λ F q 2 α F ] ,
f F q ( x F , s q , d q ) = [ s 1 q d 1 q φ ( r s 1 , r d 1 ; ω , x F ) s 1 q d M q φ ( r s 1 , r d M ; ω , x F ) , s 2 q d 1 q φ ( r s 2 , r d 1 ; ω , x F ) s K q d M q φ ( r s K , r d M ; ω , x F ) ] T ,
p ( x k ) = 1 σ k N z ( ρ k ) exp ( 1 ρ k σ ρ k { i , j } N k b i j | x k i x k j | ρ k ) ,
φ ( r s k , r d m ; ω , x F ) = j = 1 N V [ g m ( r j , r d m ; ω ) S ( r j ; ω , x F ) g x ( r s k , r j ; ω ) ] ,
f F q ( x F , s q , d q ) = G ω q S ω q ( x F ) ,
( { x ̂ F } MAP , α ̂ F , s ̂ , d ̂ ) = arg min x F 0 , s , d min α F { 1 α F q = 1 Q y F q G ω q S ω q ( x F ) Λ F q 2 + P log α F log [ p ( x F ) ] } .
x ̂ k i arg min x k i 0 { 1 α ̂ F q = 1 Q y F q [ G ω q ] * ( i ) S ( r i ; ω q , x F ) Λ F q 2 + 1 ρ k σ k ρ k j N k i b i j | x k i x ̂ k j | ρ k } ,
R 0 6 = 9000 ln ( 10 ) κ 2 η D 128 π 5 N n 4 λ min λ max f D ( λ ) ε A ( λ ) λ 4 d λ ,
y km cal = y km uncal . y km syn y km base ,
P I . d S ,
P D P DA = η D C D ( η D C DA ) [ 1 + ( R 0 r F ) 6 ] ,
r F = R 0 ( C DA P D P DA C D 1 ) 1 6 ,

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