Abstract

Motion analysis of optically trapped objects is demonstrated using a simple 2D Fourier transform technique. The displacements of trapped objects are determined directly from the phase shift between the Fourier transform of subsequent images. Using end- and side-view imaging, the stiffness of the trap is determined in three dimensions. The Fourier transform method is simple to implement and applicable in cases where the trapped object changes shape or where the lighting conditions change. This is illustrated by tracking a fluorescent particle and a myoblast cell, with subsequent determination of diffusion coefficients and the trapping forces.

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References

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  1. J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  14. P. J. Rodrigo, I. R. Perch-Nielsen, C. A. Alonzo, and J. Glückstad, “GPC-based optical micromanipulation in 3D real-time using a single spatial light modulator,” Opt. Express 14(26), 13107–13112 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  19. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
    [CrossRef]
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    [CrossRef]

2011 (3)

J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
[CrossRef]

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

J. Fung, K. E. Martin, R. W. Perry, D. M. Kaz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19(9), 8051–8065 (2011).
[CrossRef] [PubMed]

2010 (5)

2008 (1)

2006 (2)

2004 (2)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

2003 (2)

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

2001 (2)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

2000 (1)

T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000).
[CrossRef] [PubMed]

1999 (1)

1996 (1)

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

Alonzo, C. A.

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Balci, M.

M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the Fourier domain,” IEEE Trans. Image Process. 15(7), 1965–1972 (2006).
[CrossRef] [PubMed]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Bowman, R.

Brito, J. M.

Chang, E. C.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

Chatterji, B. N.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

Cheong, F. C.

Cortelazzo, G. M.

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Czerwinski, F.

De Paul, S. M.

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Fienup, J. R.

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

Foroosh, H.

M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the Fourier domain,” IEEE Trans. Image Process. 15(7), 1965–1972 (2006).
[CrossRef] [PubMed]

Fung, J.

Gibson, G.

Glückstad, J.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
[CrossRef]

D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. 51(3), 249–265 (2010).
[CrossRef]

P. J. Rodrigo, I. R. Perch-Nielsen, C. A. Alonzo, and J. Glückstad, “GPC-based optical micromanipulation in 3D real-time using a single spatial light modulator,” Opt. Express 14(26), 13107–13112 (2006).
[CrossRef] [PubMed]

Gornall, J. L.

Grassman, T. J.

T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000).
[CrossRef] [PubMed]

Grier, D. G.

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Guizar-Sicairos, M.

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Kaz, D. M.

Keiding, S. R.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Keyser, U. F.

Knowles, M. K.

T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000).
[CrossRef] [PubMed]

Krishnatreya, B. J.

Kristensen, M. V.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Kylling, A. P.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Lindballe, T. B.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Lucchese, L.

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Manoharan, V. N.

Marcus, A. H.

T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000).
[CrossRef] [PubMed]

Martin, K. E.

Martucci, S. A.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

McGorty, R.

Monti, C. M.

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Oddershede, L. B.

Orchard, M. T.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

Otto, O.

Padgett, M.

Palima, D.

D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. 51(3), 249–265 (2010).
[CrossRef]

Palima, D. Z.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Pasche, S.

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Perch-Nielsen, I. R.

Perry, R. W.

Reddy, B. S.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

Rodrigo, P. J.

Saavedra, C.

Seidel, R.

Solano, P.

Spencer, N. D.

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Staforelli, J. P.

Stapelfeldt, H.

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

Stober, G.

Stone, H. S.

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

Textor, M.

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Thurman, S. T.

Torres, S.

Vera, E.

Voros, J.

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Biophys. J. (1)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. Palima and J. Glückstad, “Generalised phase contrast: microscopy, manipulation and more,” Contemp. Phys. 51(3), 249–265 (2010).
[CrossRef]

IEEE T Geosci. Remote (1)

H. S. Stone, M. T. Orchard, E. C. Chang, and S. A. Martucci, ““A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE T Geosci. Remote 39(10), 2235–2243 (2001).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Balci and H. Foroosh, “Subpixel estimation of shifts directly in the Fourier domain,” IEEE Trans. Image Process. 15(7), 1965–1972 (2006).
[CrossRef] [PubMed]

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5(8), 1266–1271 (1996).
[CrossRef] [PubMed]

J. Eur. Opt. Soc-Rapid. (1)

T. B. Lindballe, M. V. Kristensen, A. P. Kylling, D. Z. Palima, J. Glückstad, S. R. Keiding, and H. Stapelfeldt, “Three-dimensional imaging and force characterization of multiple trapped particles in low NA counterpropagating optical traps,” J. Eur. Opt. Soc-Rapid. 6, 110576 (2011).

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

Langmuir (1)

S. Pasche, S. M. De Paul, J. Voros, N. D. Spencer, and M. Textor, “Poly(L-lysine)-graft-poly(ethylene glycol) assembled monolayers on niobium oxide surfaces: A quantitative study of the influence of polymer interfacial architecture on resistance to protein adsorption by ToF-SIMS and in situ OWLS,” Langmuir 19(22), 9216–9225 (2003).
[CrossRef]

Nat. Photonics (1)

J. Glückstad, “Optical manipulation sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

T. J. Grassman, M. K. Knowles, and A. H. Marcus, “Structure and dynamics of fluorescently labeled complex fluids by fourier imaging correlation spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(66 Pt B), 8245–8257 (2000).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (2)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75(3), 594–612 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).
[CrossRef] [PubMed]

Other (1)

C. D. Kuglin and D. C. Hines, “The phase correlation image alignment method,” Proc. Int. Conf. Cybernetics Society 163–5 (1975).

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Figures (5)

Fig. 1
Fig. 1

Comparison of traces of particle displacements along the x-axis obtained through the FTMA method (red) and a reference tracking algorithm (blue). The error bars indicate the uncertainty of the tracking algorithms. The inset shows an example of the phase plane corresponding to the displacement at 0.12 seconds, indicated by the green arrow. The displacement is shown in both µm and pixels. A threshold filter was applied to remove phase data corresponding to low spatial frequencies (DC-components) and spatial frequencies with low spectral amplitude. The phase plane is down-sloped in the x-axis direction indicating a displacement along the negative x-axis

Fig. 2
Fig. 2

A reference tracking algorithm (a) and the FTMA method (b) and (c) are used on the same 400 fps movie of a trapped 10 μm polystyrene bead for comparison. (a) Power spectrum of the particle movement (black line). The red line is the fitted power spectrum |Sx|2 (b) Power spectrum for the particle displacement along the x-axis (black line). The red line is the fitted power spectrum of the displacement |SΔx|2 as given by Eq. (5). The low frequency noise floor in (b) is 7 × 10−8 µm2/Hz. (c) Histogram showing distribution of displacements along the x-axis (black dots) and the fitted Gaussian distribution. In (a) and (b) sharp noise peaks at 56 and 76 Hz caused by mechanical noise in the laboratory was removed from the data before fitting the power spectra. The experimental data points were averaged (10 point adjacent averaging) before plotting, but after fitting.

Fig. 3
Fig. 3

(a) Normalized fluorescence intensity for an optically trapped 10 μm diameter dyed polystyrene bead as a function of time, t. At t = 0.6 seconds an excitation beam is turned on. (b) The corresponding particle displacement along the x-axis (red) and the y-axis (black). The curves were shift 100 nm in each direction for clarity.

Fig. 4
Fig. 4

(a) Image of the trapped myoblast cell indicating asymmetric illumination and background. (b) 2D Fourier transform of the image in (a). The DC background in the image results in the two lines at kx = kz = 0 μm−1. (c) A 2D filter removes the DC-values around kx = kz = 0 μm−1 and applies a threshold in Fourier amplitude. The mask has a value of 1 in the black area and a value of 0 in the white area. (d) After application of the filtered image in (c) to the 2D Fourier transform in (b) a much clearer image of the relevant frequencies are obtained. This justifies the following use of the filter on the phase space image in order to obtain the displacements. The images of the Fourier transform amplitudes in (b) and (d) are both shown on a logarithmic scale. Red corresponds to the highest amplitude and blue to the lowest.

Fig. 5
Fig. 5

(a) Frequency analysis of the x-displacement of the myoblast cell shown in Fig. 4. The fitted curve corresponds to a corner frequency of 0.62 Hz for the x-direction. Data points were averaged after fitting as in Fig. 2. (b) Distribution of the x-displacements obtained by tracking the motion of the myoblast cells using the FTMA method. The diffusion coefficient is Dx = 0.015(1) µm2s−1. Combined with the corner frequency obtained in (a) the trap stiffness can be determined to be κx = 1.07 pN/µm.

Equations (6)

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f 2 ( x+Δx,y+Δy )= f 1 ( x,y ).
F 2 ( k x , k y ) F 1 ( k x , k y ) = A( k x , k y ) e i( k x ( x+Δx )+ k y ( y+Δy ) ) A( k x , k y ) e i( k x x+ k y y ) = e i( k x Δx+ k y Δy ) = e iΔθ( k x , k y ) .
Δθ( k x , k y )=Δx k x +Δy k y .
Δ 2 =2Dτ( 1+πτ f c )
| S Δx ( f ) | 2 = τ 2 f 2 | S x ( f ) | 2 =2 τ 2 f 2 D f 2 + f c 2 .
g( x,y )= 1 [ F 2 ( k x , k y ) F 1 ( k x , k y ) ]= 1 [ e i( k x Δx+ k y Δy ) ]=δ( xΔx,yΔy ).

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