Abstract

Multi-photon scanning microscopy provides a robust tool for optical sectioning, which can be used to capture fast biological events such as blood flow, mitochondrial activity, and neuronal action potentials. For many studies, it is important to visualize several different focal planes at a rate akin to the biological event frequency. Typically, a microscope is equipped with mechanical elements to move either the sample or the objective lens to capture volumetric information, but these strategies are limited due to their slow speeds or inertial artifacts. To overcome this problem, remote focusing methods have been developed to shift the focal plane axially without physical movement of the sample or the microscope. Among these methods is liquid lens technology, which adjusts the focus of the lens by changing the wettability of the liquid and hence its curvature. Liquid lenses are inexpensive active optical elements that have the potential for fast multi-photon volumetric imaging, hence a promising and accessible approach for the study of biological systems with complex dynamics. Although remote focusing using liquid lens technology can be used for volumetric point scanning multi-photon microscopy, optical aberrations and the effects of high energy laser pulses have been concerns in its implementation. In this paper, we characterize a liquid lens and validate its use in relevant biological applications. We measured optical aberrations that are caused by the liquid lens, and calculated its response time, defocus hysteresis, and thermal response to a pulsed laser. We applied this method of remote focusing for imaging and measurement of multiple in-vivo specimens, including mesenchymal stem cell dynamics, mouse tibialis anterior muscle mitochondrial electrical potential fluctuations, and mouse brain neural activity. Our system produces 5 dimensional (x,y,z,λ,t) data sets at the speed of 4.2 volumes per second over volumes as large as 160 x 160 x 35 µm3.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

2018 (6)

M.-C. Kim, Y. R. Silberberg, R. Abeyaratne, R. D. Kamm, and H. H. Asada, “Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration,” Proc. Natl. Acad. Sci. U.S.A. 115(3), E390–E399 (2018).
[Crossref] [PubMed]

B. N. Ozbay, G. L. Futia, M. Ma, V. M. Bright, J. T. Gopinath, E. G. Hughes, D. Restrepo, and E. A. Gibson, “Three dimensional two-photon brain imaging in freely moving mice using a miniature fiber coupled microscope with active axial-scanning,” Sci. Rep. 8(1), 8108 (2018).
[Crossref] [PubMed]

K. F. Tehrani, E. G. Pendleton, W. M. Southern, J. A. Call, and L. J. Mortensen, “Two-photon deep-tissue spatially resolved mitochondrial imaging using membrane potential fluorescence fluctuations,” Biomed. Opt. Express 9(1), 254–259 (2018).
[Crossref] [PubMed]

M. A. Taylor, T. Nöbauer, A. Pernia-Andrade, F. Schlumm, and A. Vaziri, “Brain-wide 3D light-field imaging of neuronal activity with speckle-enhanced resolution,” Optica 5(4), 345–353 (2018).
[Crossref]

K. Dobek, “Motionless microscopy with tunable thermal lens,” Opt. Express 26(4), 3892–3902 (2018).
[Crossref] [PubMed]

M. Bawart, A. Jesacher, and M. Ritsh-Marte, “Remote focusing in confocal microscopy by means of a modified Alvarez lens,” J. Microsc. 271(3), 337–344 (2018).

2017 (6)

2016 (3)

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

P. Rupprecht, A. Prendergast, C. Wyart, and R. W. Friedrich, “Remote z-scanning with a macroscopic voice coil motor for fast 3D multiphoton laser scanning microscopy,” Biomed. Opt. Express 7(5), 1656–1671 (2016).
[Crossref] [PubMed]

W. Yang, J. E. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref] [PubMed]

2015 (3)

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Y. Zou, W. Zhang, F. S. Chau, and G. Zhou, “Miniature adjustable-focus endoscope with a solid electrically tunable lens,” Opt. Express 23(16), 20582–20592 (2015).
[Crossref] [PubMed]

M.-C. Kim, J. Whisler, Y. R. Silberberg, R. D. Kamm, and H. H. Asada, “Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network,” PLOS Comput. Biol. 11(10), e1004535 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (2)

2012 (2)

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

A. H. Pham, J. M. McCaffery, and D. C. Chan, “Mouse lines with photo-activatable mitochondria to study mitochondrial dynamics,” Genesis 50(11), 833–843 (2012).
[Crossref] [PubMed]

2011 (4)

B. F. Grewe, F. F. Voigt, M. van ’t Hoff, and F. Helmchen, “Fast two-layer two-photon imaging of neuronal cell populations using an electrically tunable lens,” Biomed. Opt. Express 2(7), 2035–2046 (2011).
[Crossref] [PubMed]

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

D. Trung-Dung, P. Cheol-Woo, and K. Gyu-Man, “Fabrication of Focus-Variable Fluidic Microlens Using Single Casting,” Jpn. J. Appl. Phys. 50(6S), 06GM15 (2011).
[Crossref]

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref] [PubMed]

2010 (2)

P. A. Kirkby, K. M. N. Srinivas Nadella, and R. A. Silver, “A compact acousto-optic lens for 2D and 3D femtosecond based 2-photon microscopy,” Opt. Express 18(13), 13720–13745 (2010).
[Crossref] [PubMed]

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

2009 (1)

2007 (1)

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[Crossref] [PubMed]

2005 (1)

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

2003 (2)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online 2(1), 13 (2003).
[Crossref] [PubMed]

2002 (1)

2000 (1)

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

1976 (1)

Abdelfattah, A. S.

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

Abeyaratne, R.

M.-C. Kim, Y. R. Silberberg, R. Abeyaratne, R. D. Kamm, and H. H. Asada, “Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration,” Proc. Natl. Acad. Sci. U.S.A. 115(3), E390–E399 (2018).
[Crossref] [PubMed]

Abrahamsson, S.

Agard, D.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[Crossref] [PubMed]

Agostinho, A.

Ahrens, M. B.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Ankrum, J. A.

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

Araki, S.

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

Arisaka, K.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref] [PubMed]

Asada, H. H.

M.-C. Kim, Y. R. Silberberg, R. Abeyaratne, R. D. Kamm, and H. H. Asada, “Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration,” Proc. Natl. Acad. Sci. U.S.A. 115(3), E390–E399 (2018).
[Crossref] [PubMed]

M.-C. Kim, J. Whisler, Y. R. Silberberg, R. D. Kamm, and H. H. Asada, “Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network,” PLOS Comput. Biol. 11(10), e1004535 (2015).
[Crossref] [PubMed]

Bargmann, C. I.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Barnstedt, O.

Bawart, M.

M. Bawart, A. Jesacher, and M. Ritsh-Marte, “Remote focusing in confocal microscopy by means of a modified Alvarez lens,” J. Microsc. 271(3), 337–344 (2018).

Berge, B.

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

Bernhem, K.

Bifano, T.

Blom, H.

Boetto, S. E.

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

Booth, M. J.

Brase, J. M.

Bright, V. M.

B. N. Ozbay, G. L. Futia, M. Ma, V. M. Bright, J. T. Gopinath, E. G. Hughes, D. Restrepo, and E. A. Gibson, “Three dimensional two-photon brain imaging in freely moving mice using a miniature fiber coupled microscope with active axial-scanning,” Sci. Rep. 8(1), 8108 (2018).
[Crossref] [PubMed]

O. D. Supekar, B. N. Ozbay, M. Zohrabi, P. D. Nystrom, G. L. Futia, D. Restrepo, E. A. Gibson, J. T. Gopinath, and V. M. Bright, “Two-photon laser scanning microscopy with electrowetting-based prism scanning,” Biomed. Opt. Express 8(12), 5412–5426 (2017).
[Crossref] [PubMed]

Brismar, H.

Call, J. A.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

K. F. Tehrani, E. G. Pendleton, W. M. Southern, J. A. Call, and L. J. Mortensen, “Two-photon deep-tissue spatially resolved mitochondrial imaging using membrane potential fluorescence fluctuations,” Biomed. Opt. Express 9(1), 254–259 (2018).
[Crossref] [PubMed]

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H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
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B. N. Ozbay, G. L. Futia, M. Ma, V. M. Bright, J. T. Gopinath, E. G. Hughes, D. Restrepo, and E. A. Gibson, “Three dimensional two-photon brain imaging in freely moving mice using a miniature fiber coupled microscope with active axial-scanning,” Sci. Rep. 8(1), 8108 (2018).
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W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
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H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
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W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
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Müller, M.

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Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
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Nakano, M.

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W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
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O. D. Supekar, B. N. Ozbay, M. Zohrabi, P. D. Nystrom, G. L. Futia, D. Restrepo, E. A. Gibson, J. T. Gopinath, and V. M. Bright, “Two-photon laser scanning microscopy with electrowetting-based prism scanning,” Biomed. Opt. Express 8(12), 5412–5426 (2017).
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H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
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[Crossref] [PubMed]

Pnevmatikakis, E.

W. Yang, J. E. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref] [PubMed]

Pologruto, T. A.

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online 2(1), 13 (2003).
[Crossref] [PubMed]

Portera-Cailliau, C.

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref] [PubMed]

Poyneer, L. A.

Preibisch, S.

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

Prendergast, A.

Qiao, W.

Qualls, A. E.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

Renders, C. A.

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Restrepo, D.

B. N. Ozbay, G. L. Futia, M. Ma, V. M. Bright, J. T. Gopinath, E. G. Hughes, D. Restrepo, and E. A. Gibson, “Three dimensional two-photon brain imaging in freely moving mice using a miniature fiber coupled microscope with active axial-scanning,” Sci. Rep. 8(1), 8108 (2018).
[Crossref] [PubMed]

O. D. Supekar, B. N. Ozbay, M. Zohrabi, P. D. Nystrom, G. L. Futia, D. Restrepo, E. A. Gibson, J. T. Gopinath, and V. M. Bright, “Two-photon laser scanning microscopy with electrowetting-based prism scanning,” Biomed. Opt. Express 8(12), 5412–5426 (2017).
[Crossref] [PubMed]

Ritsh-Marte, M.

M. Bawart, A. Jesacher, and M. Ritsh-Marte, “Remote focusing in confocal microscopy by means of a modified Alvarez lens,” J. Microsc. 271(3), 337–344 (2018).

Rupprecht, P.

Saalfeld, S.

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

Sabatini, B. L.

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online 2(1), 13 (2003).
[Crossref] [PubMed]

Sato, M.

Schindelin, J.

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

Schlumm, F.

Schmid, B.

Schreiter, E. R.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Schwiegerling, J.

Sciuto, T. E.

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

Sedat, J. W.

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[Crossref] [PubMed]

Shain, W. J.

Silberberg, Y. R.

M.-C. Kim, Y. R. Silberberg, R. Abeyaratne, R. D. Kamm, and H. H. Asada, “Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration,” Proc. Natl. Acad. Sci. U.S.A. 115(3), E390–E399 (2018).
[Crossref] [PubMed]

M.-C. Kim, J. Whisler, Y. R. Silberberg, R. D. Kamm, and H. H. Asada, “Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network,” PLOS Comput. Biol. 11(10), e1004535 (2015).
[Crossref] [PubMed]

Silver, R. A.

Simms, K.

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

Southern, W. M.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

K. F. Tehrani, E. G. Pendleton, W. M. Southern, J. A. Call, and L. J. Mortensen, “Two-photon deep-tissue spatially resolved mitochondrial imaging using membrane potential fluorescence fluctuations,” Biomed. Opt. Express 9(1), 254–259 (2018).
[Crossref] [PubMed]

Srinivas Nadella, K. M. N.

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Sun, Y.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Supekar, O. D.

Svoboda, K.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online 2(1), 13 (2003).
[Crossref] [PubMed]

Tang, J.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Taylor, M. A.

Tehrani, K. F.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

K. F. Tehrani, E. G. Pendleton, W. M. Southern, J. A. Call, and L. J. Mortensen, “Two-photon deep-tissue spatially resolved mitochondrial imaging using membrane potential fluorescence fluctuations,” Biomed. Opt. Express 9(1), 254–259 (2018).
[Crossref] [PubMed]

Teo, G. S. L.

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
[Crossref] [PubMed]

Teramoto, T.

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

Tomancak, P.

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

Trung-Dung, D.

D. Trung-Dung, P. Cheol-Woo, and K. Gyu-Man, “Fabrication of Focus-Variable Fluidic Microlens Using Single Casting,” Jpn. J. Appl. Phys. 50(6S), 06GM15 (2011).
[Crossref]

Tsegaye, G.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Tukker, T. W.

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

van ’t Hoff, M.

Van As, M. A. J.

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Vaziri, A.

Vickers, N. A.

Voigt, F. F.

Waddell, S.

Walpita, D.

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Wang, C.

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Whisler, J.

M.-C. Kim, J. Whisler, Y. R. Silberberg, R. D. Kamm, and H. H. Asada, “Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network,” PLOS Comput. Biol. 11(10), e1004535 (2015).
[Crossref] [PubMed]

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Wu, J.

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

Wyart, C.

Yagi, S.

Yang, W.

W. Yang, J. E. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref] [PubMed]

Yin, A.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

Yin, H.

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

Yu, Y.

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

Yuste, R.

W. Yang, J. E. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref] [PubMed]

Zhang, W.

Zhao, Y.

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

Zhou, G.

Zhou, Z.

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Zohrabi, M.

Zou, Y.

Žurauskas, M.

Appl. Opt. (1)

Biomed. Eng. Online (1)

T. A. Pologruto, B. L. Sabatini, and K. Svoboda, “ScanImage: flexible software for operating laser scanning microscopes,” Biomed. Eng. Online 2(1), 13 (2003).
[Crossref] [PubMed]

Biomed. Opt. Express (7)

S. Abrahamsson, H. Blom, A. Agostinho, D. C. Jans, A. Jost, M. Müller, L. Nilsson, K. Bernhem, T. J. Lambert, R. Heintzmann, and H. Brismar, “Multifocus structured illumination microscopy for fast volumetric super-resolution imaging,” Biomed. Opt. Express 8(9), 4135–4140 (2017).
[Crossref] [PubMed]

P. Rupprecht, A. Prendergast, C. Wyart, and R. W. Friedrich, “Remote z-scanning with a macroscopic voice coil motor for fast 3D multiphoton laser scanning microscopy,” Biomed. Opt. Express 7(5), 1656–1671 (2016).
[Crossref] [PubMed]

M. Žurauskas, O. Barnstedt, M. Frade-Rodriguez, S. Waddell, and M. J. Booth, “Rapid adaptive remote focusing microscope for sensing of volumetric neural activity,” Biomed. Opt. Express 8(10), 4369–4379 (2017).
[Crossref] [PubMed]

M. Sato, Y. Motegi, S. Yagi, K. Gengyo-Ando, M. Ohkura, and J. Nakai, “Fast varifocal two-photon microendoscope for imaging neuronal activity in the deep brain,” Biomed. Opt. Express 8(9), 4049–4060 (2017).
[Crossref] [PubMed]

B. F. Grewe, F. F. Voigt, M. van ’t Hoff, and F. Helmchen, “Fast two-layer two-photon imaging of neuronal cell populations using an electrically tunable lens,” Biomed. Opt. Express 2(7), 2035–2046 (2011).
[Crossref] [PubMed]

K. F. Tehrani, E. G. Pendleton, W. M. Southern, J. A. Call, and L. J. Mortensen, “Two-photon deep-tissue spatially resolved mitochondrial imaging using membrane potential fluorescence fluctuations,” Biomed. Opt. Express 9(1), 254–259 (2018).
[Crossref] [PubMed]

O. D. Supekar, B. N. Ozbay, M. Zohrabi, P. D. Nystrom, G. L. Futia, D. Restrepo, E. A. Gibson, J. T. Gopinath, and V. M. Bright, “Two-photon laser scanning microscopy with electrowetting-based prism scanning,” Biomed. Opt. Express 8(12), 5412–5426 (2017).
[Crossref] [PubMed]

bioRxiv (1)

W. M. Southern, A. S. Nichenko, K. F. Tehrani, M. J. McGranahan, L. Krishnan, A. E. Qualls, N. J. Jenkins, L. J. Mortensen, H. Yin, A. Yin, R. E. Guldberg, S. M. Greising, and J. A. Call, “PGC-1α overexpression partially rescues impaired oxidative and contractile pathophysiology following volumetric muscle loss injury,” bioRxiv 9(1), 535328 (2019).
[Crossref]

eLife (1)

H. Dana, B. Mohar, Y. Sun, S. Narayan, A. Gordus, J. P. Hasseman, G. Tsegaye, G. T. Holt, A. Hu, D. Walpita, R. Patel, J. J. Macklin, C. I. Bargmann, M. B. Ahrens, E. R. Schreiter, V. Jayaraman, L. L. Looger, K. Svoboda, and D. S. Kim, “Sensitive red protein calcium indicators for imaging neural activity,” eLife 5, e12727 (2016).
[Crossref] [PubMed]

Eur. Phys. J. E (1)

B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000).
[Crossref]

Genesis (1)

A. H. Pham, J. M. McCaffery, and D. C. Chan, “Mouse lines with photo-activatable mitochondria to study mitochondrial dynamics,” Genesis 50(11), 833–843 (2012).
[Crossref] [PubMed]

J. Microsc. (2)

M. Bawart, A. Jesacher, and M. Ritsh-Marte, “Remote focusing in confocal microscopy by means of a modified Alvarez lens,” J. Microsc. 271(3), 337–344 (2018).

Z. Kam, P. Kner, D. Agard, and J. W. Sedat, “Modelling the application of adaptive optics to wide-field microscope live imaging,” J. Microsc. 226(1), 33–42 (2007).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

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

Jpn. J. Appl. Phys. (1)

D. Trung-Dung, P. Cheol-Woo, and K. Gyu-Man, “Fabrication of Focus-Variable Fluidic Microlens Using Single Casting,” Jpn. J. Appl. Phys. 50(6S), 06GM15 (2011).
[Crossref]

Nat. Biotechnol. (1)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003).
[Crossref] [PubMed]

Nat. Methods (3)

A. Cheng, J. T. Gonçalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different depths with spatiotemporal multiplexing,” Nat. Methods 8(2), 139–142 (2011).
[Crossref] [PubMed]

L. Kong, J. Tang, J. P. Little, Y. Yu, T. Lämmermann, C. P. Lin, R. N. Germain, and M. Cui, “Continuous volumetric imaging via an optical phase-locked ultrasound lens,” Nat. Methods 12(8), 759–762 (2015).
[Crossref] [PubMed]

S. Preibisch, S. Saalfeld, J. Schindelin, and P. Tomancak, “Software for bead-based registration of selective plane illumination microscopy data,” Nat. Methods 7(6), 418–419 (2010).
[Crossref] [PubMed]

Neuron (1)

W. Yang, J. E. Miller, L. Carrillo-Reid, E. Pnevmatikakis, L. Paninski, R. Yuste, and D. S. Peterka, “Simultaneous Multi-plane Imaging of Neural Circuits,” Neuron 89(2), 269–284 (2016).
[Crossref] [PubMed]

Opt. Express (7)

Opt. Lett. (1)

Opt. Rev. (1)

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3), 255–259 (2005).
[Crossref]

Optica (1)

PLOS Comput. Biol. (1)

M.-C. Kim, J. Whisler, Y. R. Silberberg, R. D. Kamm, and H. H. Asada, “Cell Invasion Dynamics into a Three Dimensional Extracellular Matrix Fibre Network,” PLOS Comput. Biol. 11(10), e1004535 (2015).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

M.-C. Kim, Y. R. Silberberg, R. Abeyaratne, R. D. Kamm, and H. H. Asada, “Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration,” Proc. Natl. Acad. Sci. U.S.A. 115(3), E390–E399 (2018).
[Crossref] [PubMed]

Sci. Rep. (1)

B. N. Ozbay, G. L. Futia, M. Ma, V. M. Bright, J. T. Gopinath, E. G. Hughes, D. Restrepo, and E. A. Gibson, “Three dimensional two-photon brain imaging in freely moving mice using a miniature fiber coupled microscope with active axial-scanning,” Sci. Rep. 8(1), 8108 (2018).
[Crossref] [PubMed]

Science (2)

Y. Zhao, S. Araki, J. Wu, T. Teramoto, Y.-F. Chang, M. Nakano, A. S. Abdelfattah, M. Fujiwara, T. Ishihara, T. Nagai, and R. E. Campbell, “An expanded palette of genetically encoded Ca2+ indicators,” Science 333(6051), 1888–1891 (2011).
[Crossref] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Stem Cells (1)

G. S. L. Teo, J. A. Ankrum, R. Martinelli, S. E. Boetto, K. Simms, T. E. Sciuto, A. M. Dvorak, J. M. Karp, and C. V. Carman, “Mesenchymal stem cells transmigrate between and directly through tumor necrosis factor-α-activated endothelial cells via both leukocyte-like and novel mechanisms,” Stem Cells 30(11), 2472–2486 (2012).
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K. Ogata, System Dynamics, Fourth ed. (Pearson/Prentice Hall, 2004).

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Supplementary Material (4)

NameDescription
» Visualization 1       Volumetric imaging of a single mesenchymal stem cell in suspension. The cell is isolated from a mouse and stained with DiD membrane dye.
» Visualization 2       Multi-color volumetric imaging of a mouse tibialis anterior (TA) muscle.
» Visualization 3       3D visualization of blood flow in the skull of a mouse ubiquitously expressing Dendra2 in mitochondria. Hollow particles seen in the blood flow are red blood cells.
» Visualization 4       Volumetric imaging of calcium activity in the barrel cortex of a mouse transfected with an R-GECO calcium indicator.

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Figures (8)

Fig. 1
Fig. 1 - Diagram of the microscope. BS and PBS are regular and polarizing beam splitters. BPP is the back pupil plane and IP is the image plane. L is lens, DP is achromatic doublet pair, TL is tube lens, MLA is microlens array, F is filter, DiM is dichroic mirror, FM is flip mirror, ND is neutral density filter, PD is photodiode, CCD is charge coupled device camera, and PMT is photon multiplier tube.
Fig. 2
Fig. 2 - Illustration of the derivation of the defocus term.
Fig. 3
Fig. 3 Characterization of the liquid lens. A-C show wavefront measurements with triangle input waveform produced with 1, 2, and 5 Hz frequencies, respectively. D shows the hysteresis curve of the defocus terms (Asc and Des show ascending and descending hysteresis behaviors respectively). E is the step response of the defocus produced by the liquid lens, and a fit applied to it to calculate the settling time. F-H show the thermal response of the lens in response to different 54 kW, 27 kW, and 9 kW peak pulse power, respectively. I shows long term measurement of the wavefront when the liquid lens was exposed to a 100mW 775nm beam, measured at 5 minute intervals. Average of every 15 minutes (T1-T4) and the corresponding errors are shown in J.
Fig. 4
Fig. 4 - Calibration of axial scanning using 200 nm fluorescent microspheres in a polyacrylamide gel. Left panels: mechanical stage scanning. Right panels: Remote focusing using the liquid lens. (A) panels show axial sections (x-z), and (B)-(D) panels show lateral sections (x-y). The scalebar is 5 μm.
Fig. 5
Fig. 5 - Volumetric imaging of a single cell in suspension. The MSC is isolated from a mouse and stained with DiD membrane dye. The field of view is 42 µm x 42 µm x 28 µm (x,y,z). 775 nm excitation light was used. Visualization 1 is available as supplementary.
Fig. 6
Fig. 6 - Multi-color volumetric imaging of a mouse TA muscle. In A green shows the ubiquitous expression of Dendra2 mitochondria, red is TMRE staining of mitochondria. The fluorescent fluctuations of mitochondria in A (i, ii, iii) are shown in B. The volume dimensions are 27 µm x 27 µm x 35 µm (x,y,z). 850 nm excitation light was used for imaging. Visualization 2 is available in the supplemental information.
Fig. 7
Fig. 7 - 3D visualization of blood flow in the skull of a mouse ubiquitously expressing Dendra2 (green) in mitochondria. Blue shows second harmonic generation of bone collagen fibers, and red shows Rhodamine B dextran conjugate present in the blood. Panel A shows an overview of a single WBC travelling in the vessel. B-D show different time points. The volume dimensions are 58 µm x 58 µm x 25 µm (x,y,z). Visualization 3 is available in the supplemental information.
Fig. 8
Fig. 8 - Volumetric imaging of calcium activity in the mouse barrel cortex. (A) shows a pre-processed volumetric frame from the timelapse. The volume dimensions are 80 µm x 80 µm x 25 µm (x,y,z). After 3D registration several ROIs were located in the data set using mean volumetric fluorescence intensity, shown in (B). From each ROI, the intensity fluctuation is derived and background normalized (F0) to obtain the DF/F0 calcium activity fluctuations, shown in (C). The red ticks (STIM) indicate the timing of an air blow stimulation on the contralateral whiskers to the recording site. 1050 nm excitation light was used for imaging. Visualization 4 is available in the supplemental information.

Equations (8)

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Δz= 1 M 2 f L3 2 f LL n,
z( r,θ )= c 3 3 ( 2 r 2 1 )
f=z+ r( 1 tan 2 α ) 2tanα
tanα= dz dr =4c 3 r
f=c 3 ( 2 r 2 1 )+ r( 1 ( 4c 3 r ) 2 ) 2( 4c 3 r ) .
f= 1 8c 3 ,
D= 1 f =8c 3 .
y( t )=A( 1 1 1 ζ 2 e ζ ω 0 t sin( ω 0 1 ζ 2 t+ cos 1 ζ ) ),

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