Abstract

Light-sheet microscopy has been widely used in high-speed fluorescence imaging with low phototoxicity, while the trade-off between the field-of-view and optical sectioning capability limits its application in large-scale imaging. Although Bessel beam light-sheet microscopy greatly enhances the light-sheet length with the self-healing ability, it suffers from the strong side-lobe effect. To solve these problems, we introduce the photobleaching imprinting technique in Bessel beam light-sheet microscopy. By extracting the non-linear photobleaching-induced fluorescence decay, we get rid of the large concentric side lobe structures of the Bessel beam to achieve uniform isotropic resolution across a large field-of-view for large-scale fluorescence imaging. Both numerical simulations and experimental results on various samples are demonstrated to show our enhanced resolution and contrast over traditional Bessel-beam light-sheet microscopy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

D. Wang, Y. Jin, R. Feng, Y. Chen, and L. Gao, “Tiling light sheet selective plane illumination microscopy using discontinuous light sheets,” Opt. Express 27(23), 34472–34483 (2019).
[Crossref]

2017 (1)

F. Strobl, A. Schmitz, and E. H. K. Stelzer, “Improving your four-dimensional image: traveling through a decade of light-sheet-based fluorescence microscopy research,” Nat. Protoc. 12(6), 1103–1109 (2017).
[Crossref]

2016 (1)

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

2015 (3)

P. J. Keller and M. B. Ahrens, “Visualizing whole-brain activity and development at the single-cell level using light-sheet microscopy,” Neuron 85(3), 462–483 (2015).
[Crossref]

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

2014 (3)

K. M. Dean and R. Fiolka, “Uniform and scalable light-sheets generated by extended focusing,” Opt. Express 22(21), 26141–26152 (2014).
[Crossref]

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

2013 (1)

2012 (4)

2011 (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

2010 (4)

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with hilo background rejection,” J. Biomed. Opt. 15(1), 016027 (2010).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

F. O. Fahrbach and A. Rohrbach, “A line scanned light-sheet microscope with phase shaped self-reconstructing beams,” Opt. Express 18(23), 24229–24244 (2010).
[Crossref]

2005 (1)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref]

1999 (1)

E. J. Peterman, S. Brasselet, and W. Moerner, “The fluorescence dynamics of single molecules of green fluorescent protein,” J. Phys. Chem. A 103(49), 10553–10560 (1999).
[Crossref]

1995 (1)

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

1993 (1)

A. H. Voie, D. Burns, and F. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993).
[Crossref]

1992 (1)

1987 (1)

Ahrens, M. B.

P. J. Keller and M. B. Ahrens, “Visualizing whole-brain activity and development at the single-cell level using light-sheet microscopy,” Neuron 85(3), 462–483 (2015).
[Crossref]

Alessandri, K.

Amat, F.

R. Tomer, K. Khairy, F. Amat, and P. J. Keller, “Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy,” Nat. Methods 9(7), 755–763 (2012).
[Crossref]

Aravind, A.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Artigas, D.

Bao, Z.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Bjorkman, P. J.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Brasselet, S.

E. J. Peterman, S. Brasselet, and W. Moerner, “The fluorescence dynamics of single molecules of green fluorescent protein,” J. Phys. Chem. A 103(49), 10553–10560 (1999).
[Crossref]

Bria, A.

Brown, D.

Burns, D.

A. H. Voie, D. Burns, and F. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993).
[Crossref]

Cai, L.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Cai, R.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Chan, K. Y.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Chen, X.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Chen, Y.

Cheng, K. W.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Cho, N. H.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Cižmár, T.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Coll-Lladó, C.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Dalgarno, H. I.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Dean, K. M.

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref]

Deverman, B. E.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Dholakia, K.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Dichgans, M.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Durnin, J.

Eberly, J.

Ertürk, A.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Fahrbach, F. O.

F. O. Fahrbach, V. Gurchenkov, K. Alessandri, P. Nassoy, and A. Rohrbach, “Light-sheet microscopy in thick media using scanned bessel beams and two-photon fluorescence excitation,” Opt. Express 21(11), 13824–13839 (2013).
[Crossref]

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3(1), 632 (2012).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

F. O. Fahrbach and A. Rohrbach, “A line scanned light-sheet microscope with phase shaped self-reconstructing beams,” Opt. Express 18(23), 24229–24244 (2010).
[Crossref]

Feng, R.

Feng, S.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Ferrier, D. E.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Fiolka, R.

Flytzanis, N. C.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Fowlkes, C. C.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Galland, R.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Gao, L.

D. Wang, Y. Jin, R. Feng, Y. Chen, and L. Gao, “Tiling light sheet selective plane illumination microscopy using discontinuous light sheets,” Opt. Express 27(23), 34472–34483 (2019).
[Crossref]

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Ghasemigharagoz, A.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Gradinaru, V.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Greenbaum, A.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Grenci, G.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Gualda, E. J.

Gunn-Moore, F. J.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Gurchenkov, V.

Hellal, F.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Helmchen, F.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref]

Hennink, E.

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

Huang, B.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Huang, H.

Iannello, G.

Jin, Y.

Keller, P. J.

P. J. Keller and M. B. Ahrens, “Visualizing whole-brain activity and development at the single-cell level using light-sheet microscopy,” Neuron 85(3), 462–483 (2015).
[Crossref]

R. Tomer, K. Khairy, F. Amat, and P. J. Keller, “Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy,” Nat. Methods 9(7), 755–763 (2012).
[Crossref]

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Khairy, K.

R. Tomer, K. Khairy, F. Amat, and P. J. Keller, “Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy,” Nat. Methods 9(7), 755–763 (2012).
[Crossref]

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Kim, J.

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with hilo background rejection,” J. Biomed. Opt. 15(1), 016027 (2010).
[Crossref]

Ladinsky, M. S.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Leonetti, M. D.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Li, C.

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

Licea-Rodriguez, J.

Lignell, A.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Lin, Y.

Lord, S. J.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Lourbopoulos, A.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Loza-Alvarez, P.

Matryba, P.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Mayer, J.

Mertz, J.

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with hilo background rejection,” J. Biomed. Opt. 15(1), 016027 (2010).
[Crossref]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Moerner, W.

E. J. Peterman, S. Brasselet, and W. Moerner, “The fluorescence dynamics of single molecules of green fluorescent protein,” J. Phys. Chem. A 103(49), 10553–10560 (1999).
[Crossref]

Mullins, R. D.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Nassoy, P.

Nylk, J.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Olarte, O. E.

Palero, J. A.

Pan, C.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Pavone, F.

Pessino, V.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Peterman, E. J.

E. J. Peterman, S. Brasselet, and W. Moerner, “The fluorescence dynamics of single molecules of green fluorescent protein,” J. Phys. Chem. A 103(49), 10553–10560 (1999).
[Crossref]

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Plesnila, N.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Quacquarelli, F. P.

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Rangel-Rojo, R.

Rocha-Mendoza, I.

Rohrbach, A.

F. O. Fahrbach, V. Gurchenkov, K. Alessandri, P. Nassoy, and A. Rohrbach, “Light-sheet microscopy in thick media using scanned bessel beams and two-photon fluorescence excitation,” Opt. Express 21(11), 13824–13839 (2013).
[Crossref]

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3(1), 632 (2012).
[Crossref]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

F. O. Fahrbach and A. Rohrbach, “A line scanned light-sheet microscope with phase shaped self-reconstructing beams,” Opt. Express 18(23), 24229–24244 (2010).
[Crossref]

Sacconi, L.

Santella, A.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Schmidt, A. D.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Schmitz, A.

F. Strobl, A. Schmitz, and E. H. K. Stelzer, “Improving your four-dimensional image: traveling through a decade of light-sheet-based fluorescence microscopy research,” Nat. Protoc. 12(6), 1103–1109 (2017).
[Crossref]

Seka, W.

Sharpe, J.

Sibarita, J.-B.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Silvestri, L.

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Song, L.

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

Spelman, F.

A. H. Voie, D. Burns, and F. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993).
[Crossref]

Stelzer, E. H.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Stelzer, E. H. K.

F. Strobl, A. Schmitz, and E. H. K. Stelzer, “Improving your four-dimensional image: traveling through a decade of light-sheet-based fluorescence microscopy research,” Nat. Protoc. 12(6), 1103–1109 (2017).
[Crossref]

Strobl, F.

F. Strobl, A. Schmitz, and E. H. K. Stelzer, “Improving your four-dimensional image: traveling through a decade of light-sheet-based fluorescence microscopy research,” Nat. Protoc. 12(6), 1103–1109 (2017).
[Crossref]

Studer, V.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Stuurman, N.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Swoger, J.

Tanke, H. J.

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

Tomer, R.

R. Tomer, K. Khairy, F. Amat, and P. J. Keller, “Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy,” Nat. Methods 9(7), 755–763 (2012).
[Crossref]

Treweek, J. B.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Vettenburg, T.

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Viasnoff, V.

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

Voie, A. H.

A. H. Voie, D. Burns, and F. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993).
[Crossref]

Wang, D.

Wang, L. V.

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

Wang, Y.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Wittbrodt, J.

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

Xiao, C.

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Xie, D.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Xu, L.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

Yang, B.

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Young, I. T.

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

Zhu, L.

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (1)

Biophys. J. (1)

L. Song, E. Hennink, I. T. Young, and H. J. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophys. J. 68(6), 2588–2600 (1995).
[Crossref]

J. Biomed. Opt. (1)

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with hilo background rejection,” J. Biomed. Opt. 15(1), 016027 (2010).
[Crossref]

J. Microsc. (1)

A. H. Voie, D. Burns, and F. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170(3), 229–236 (1993).
[Crossref]

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

J. Phys. Chem. A (1)

E. J. Peterman, S. Brasselet, and W. Moerner, “The fluorescence dynamics of single molecules of green fluorescent protein,” J. Phys. Chem. A 103(49), 10553–10560 (1999).
[Crossref]

J. R. Soc., Interface (1)

L. Gao, L. Zhu, C. Li, and L. V. Wang, “Nonlinear light-sheet fluorescence microscopy by photobleaching imprinting,” J. R. Soc., Interface 11(93), 20130851 (2014).
[Crossref]

Nat. Commun. (1)

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3(1), 632 (2012).
[Crossref]

Nat. Methods (8)

R. Galland, G. Grenci, A. Aravind, V. Viasnoff, V. Studer, and J.-B. Sibarita, “3d high-and super-resolution imaging using single-objective spim,” Nat. Methods 12(7), 641–644 (2015).
[Crossref]

B. Yang, X. Chen, Y. Wang, S. Feng, V. Pessino, N. Stuurman, N. H. Cho, K. W. Cheng, S. J. Lord, L. Xu, D. Xie, R. D. Mullins, M. D. Leonetti, and B. Huang, “Epi-illumination spim for volumetric imaging with high spatial-temporal resolution,” Nat. Methods 16(6), 501–504 (2019).
[Crossref]

R. Tomer, K. Khairy, F. Amat, and P. J. Keller, “Quantitative high-speed imaging of entire developing embryos with simultaneous multiview light-sheet microscopy,” Nat. Methods 9(7), 755–763 (2012).
[Crossref]

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

P. J. Keller, A. D. Schmidt, A. Santella, K. Khairy, Z. Bao, J. Wittbrodt, and E. H. Stelzer, “Fast, high-contrast imaging of animal development with scanned light sheet–based structured-illumination microscopy,” Nat. Methods 7(8), 637–642 (2010).
[Crossref]

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005).
[Crossref]

C. Pan, R. Cai, F. P. Quacquarelli, A. Ghasemigharagoz, A. Lourbopoulos, P. Matryba, N. Plesnila, M. Dichgans, F. Hellal, and A. Ertürk, “Shrinkage-mediated imaging of entire organs and organisms using udisco,” Nat. Methods 13(10), 859–867 (2016).
[Crossref]

Nat. Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Nat. Protoc. (2)

F. Strobl, A. Schmitz, and E. H. K. Stelzer, “Improving your four-dimensional image: traveling through a decade of light-sheet-based fluorescence microscopy research,” Nat. Protoc. 12(6), 1103–1109 (2017).
[Crossref]

J. B. Treweek, K. Y. Chan, N. C. Flytzanis, B. Yang, B. E. Deverman, A. Greenbaum, A. Lignell, C. Xiao, L. Cai, M. S. Ladinsky, P. J. Bjorkman, C. C. Fowlkes, and V. Gradinaru, “Whole-body tissue stabilization and selective extractions via tissue-hydrogel hybrids for high-resolution intact circuit mapping and phenotyping,” Nat. Protoc. 10(11), 1860–1896 (2015).
[Crossref]

Neuron (1)

P. J. Keller and M. B. Ahrens, “Visualizing whole-brain activity and development at the single-cell level using light-sheet microscopy,” Neuron 85(3), 462–483 (2015).
[Crossref]

Opt. Express (5)

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the system setup. The galvanometer (GM), annular mask (AM) and the back pupil plane of the illumination objective (IO) are conjugated. The thin annular illumination on the back pupil plane of IO forms a Bessel beam within the sample in the $x$ direction and the scanning of GM introduces laterally scanning along the FOV in the $y$ direction to generate a light sheet. The emitted fluorescence is collected by the detection objective (DO) and a tube lens (TL) onto the sCMOS camera.
Fig. 2.
Fig. 2. Simulation results of effective excitation volumes. (a, b) Simulated excitation volumes with scanned Gaussian beams and Bessel beams (NA = 0.1 and $e$ = 0.9). (c, d) The corresponding first-order PIM component. (e, f) The corresponding fourth-order PIM component. Scale bar: 20$\mu$m. (g) Comparison of normalized equivalent illumination intensity along $z$-axis (dashed and solid lines in (a-f)). (h) Comparison of normalized equivalent illumination intensity along $x$-axis(dashed and solid lines in (a-f)).
Fig. 3.
Fig. 3. Measured excitation of Gaussian beam and Bessel beam. (a, b) Fluorescence line in the FITC gel excited by Gaussian beam and Bessel beam. (c, d) Fourth-order PIM component of the Gaussian beam and Bessel beam. Scale bar: 100$\mu$m. (e, f) Normalized intensity along $z$ direction at different $x=300\mu$m, $600\mu$m and $900\mu$m (dashed and solid lines in (a-d)).
Fig. 4.
Fig. 4. PIM component of Bessel beam with different number of time-lapsed images. 4th order PIM component using (a)10, (b)50 and (c)100 time-lapse images of Bessel beam in fluorescence dye. (d) Normalized intensity along $z$ direction at different $x=200\mu$m, $600\mu$m and $1000\mu$m. Scale bar: 100$\mu$m.
Fig. 5.
Fig. 5. Different order PIM terms with same time-lapsed images. (a)2nd, (b)4th and (c)8th PIM term extracted from 100 time-lapse images of Bessel beam in the fluorescence dye. (d) Normalized intensity along $z$ direction at different $x=200\mu$m, $600\mu$m and $1000\mu$m. Scale bar: 100$\mu$m.
Fig. 6.
Fig. 6. 3D image stack of fluorescent beads ($\bf {d=1\mu }$m) in solid agar with original Bessel beam and LSF-PIM-BB. Maximum intensity projection of the original image stack shown above and the maximum intensity projection of the fourth-order PIM component is shown below. (a, c) $x-y$ projection. Scale bar: 25$\mu$m. (b, d) $x-z$ projection. (e) Magnified images of the area marked by a dashed white rectangle in (b, d). Scale bar: 5$\mu$m. (f) Magnified images of the area marked by a solid white rectangle in (b, d). Scale bar: 5$\mu$m. (g) Statistics of the axial FWHM of 10 beads in the original image stack (cyan) and the fourth-order PIM component (red).
Fig. 7.
Fig. 7. Image of fluorescent beads ($\bf {d=1\mu }$m) in solid agar with Gaussian and Bessel beams. Image with (a) conventional scanned Gaussian beams and (b) conventional scanned Bessel beams are shown. Scale bar: 100$\mu$m. Fourth-order PIM component of (c) Gaussian beams and (d) Bessel beams are extracted. The max projection along $y$ direction of (c) and (d) is plotted in (e) and (f).
Fig. 8.
Fig. 8. Images of cleared mouse brain. (a) Light-sheet fluorescence image taken with scanned Bessel beams. (b) Fourth-order PIM component extracted from time-lapse images. Scale bar: 100$\mu$m. (c,d) Magnification of the part marked by a white box in the original image and the PIM image. Scale bar: 5$\mu$m. (e) Normalized intensity along dashed and solid lines in (c, d). (f,g) Magnification of the part marked by a dashed white box in the original image and the PIM image. Scale bar: 5$\mu$m. (h)Original time-lapse data of the point marked by an arrow and its fitting curve.

Equations (7)

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I ( x , y ) = C { μ a ( x , y , z ) ( F b ( z ) + F s ( x , y , z ) ) } P S F z ( x , y ) d z
μ a ( t ) = μ a 0 exp ( k t ) = μ a 0 exp ( B ( F b + F s ) t )
I ( x , y , t ) = C n = 0 t n ( B ) n n ! μ a 0 ( x , y , z ) ( F b ( z ) + F s ( x , y , z ) ) n + 1 P S F z ( x , y ) d z = n = 0 I n ( x , y ) t n
I n ( x , y ) = D n μ a 0 ( x , y , z ) ( F b ( z ) + F s ( x , y , z ) ) n + 1 P S F z ( x , y ) d z
I n ( x , y ) = D n μ a 0 ( x , y , z ) ( n k ) F b n + 1 k ( z ) F s k ( x , y , z ) P S F z ( x , y ) d z
I n ( x , y ) = D n F b n + 1 ( 0 ) { z 0 μ a 0 ( x , y , z ) [ F s ( x , y , z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z 0 z 0 μ a 0 ( x , y , z ) [ F b ( z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z 0 μ a 0 ( x , y , z ) [ F s ( x , y , z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z }
I n ( x , y ) = D n F b n + 1 ( 0 ) { z 1 μ a 0 ( x , y , z ) [ F s ( x , y , z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z 1 z b μ a 0 ( x , y , z ) [ F b ( z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z b z b μ a 0 ( x , y , z ) [ F b ( z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z b z 1 μ a 0 ( x , y , z ) [ F b ( z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z + z 1 μ a 0 ( x , y , z ) [ F s ( x , y , z ) F b ( 0 ) ] n + 1 P S F z ( x , y ) d z }

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