Abstract

Coherent light scattered by tissues brings structural and dynamic information, at depth, that standard imaging techniques cannot reach. Dynamics of cells or sub-cellular elements can be measured thanks to dynamic light scattering in thin samples (single scattering regime) or thanks to diffusive wave spectroscopy in thick samples (diffusion regime). Here, we address the intermediate regime and provide an analytical relationship between scattered light fluctuations and the distribution of cell displacements as a function of time. We illustrate our method by characterizing cell motility inside half millimeter thick multicellular aggregates.

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

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References

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2018 (2)

D. L. Marks, R. L. Blackmon, and A. L. Oldenburg, “Diffusion tensor optical coherence tomography,” Phys. Med. Biol. 63(2), 025007 (2018).
[Crossref]

T. Colin, G. Dechriste, J. Fehrenbach, L. Guillaume, V. Lobjois, and C. Poignard, “Experimental estimation of stored stress within spherical microtissues,” J. Math. Biol. 77(4), 1073–1092 (2018).
[Crossref]

2017 (4)

M. E. Dolega, M. Delarue, F. Ingremeau, J. Prost, A. Delon, and G. Cappello, “Cell-like pressure sensors reveal increase of mechanical stress towards the core of multicellular spheroids under compression,” Nat. Commun. 8(1), 14056 (2017).
[Crossref]

H. T. Nia, H. Liu, G. Seano, M. Datta, D. Jones, N. Rahbari, J. Incio, V. P. Chauhan, K. Jung, J. D. Martin, V. Askoxylakis, T. P. Padera, D. Fukumura, Y. Boucher, F. J. Hornicek, A. J. Grodzinsky, J. W. Baish, L. L. Munn, and R. K. Jain, “Solid stress and elastic energy as measures of tumour mechanopathology,” Nat. Biomed. Eng. 1(1), 0004 (2017).
[Crossref]

B. Brunel, C. Blanch, A. Gourrier, V. Petrolli, A. Delon, J.-F. Joanny, R. Carminati, R. Pierrat, and G. Cappello, “Structure and dynamics of multicellular assemblies measured by coherent light scattering,” New J. Phys. 19(7), 073033 (2017).
[Crossref]

A. Badon, A. C. Boccara, G. Lerosey, M. Fink, and A. Aubry, “Multiple scattering limit in optical microscopy,” Opt. Express 25(23), 28914–28934 (2017).
[Crossref]

2016 (1)

S. Monnier, M. Delarue, B. Brunel, M. E. Dolega, A. Delon, and G. Cappello, “Effect of an osmotic stress on multicellular aggregates,” Methods 94, 114–119 (2016).
[Crossref]

2015 (2)

2014 (1)

2013 (3)

K. Alessandri, B. R. Sarangi, V. V. Gurchenkov, B. Sinha, T. R. Kießling, L. Fetler, F. Rico, S. Scheuring, C. Lamaze, A. Simon, S. Geraldo, D. Vignjević, H. Doméjean, L. Rolland, A. Funfak, J. Bibette, N. Bremond, and P. Nassoy, “Cellular capsules as a tool for multicellular spheroid production and for investigating the mechanics of tumor progression in vitro,” Proc. Natl. Acad. Sci. U. S. A. 110(37), 14843–14848 (2013).
[Crossref]

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(6), 819–825 (2013).
[Crossref]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref]

2012 (4)

A. K. Dunn, “Laser speckle contrast imaging of cerebral blood flow,” Ann. Biomed. Eng. 40(2), 367–377 (2012).
[Crossref]

M. T. Janet, G. Cheng, J. A. Tyrrell, S. A. Wilcox-Adelman, Y. Boucher, R. K. Jain, and L. L. Munn, “Mechanical compression drives cancer cells toward invasive phenotype,” Proc. Natl. Acad. Sci. U. S. A. 109(3), 911–916 (2012).
[Crossref]

G. Hall, S. L. Jacques, K. W. Eliceiri, and P. J. Campagnola, “Goniometric measurements of thick tissue using monte carlo simulations to obtain the single scattering anisotropy coefficient,” Biomed. Opt. Express 3(11), 2707–2719 (2012).
[Crossref]

T. Stylianopoulos, J. D. Martin, V. P. Chauhan, S. R. Jain, B. Diop-Frimpong, N. Bardeesy, B. L. Smith, C. R. Ferrone, F. J. Hornicek, Y. Boucher, L. L. Munn, and R. K. Jain, “Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors,” Proc. Natl. Acad. Sci. U. S. A. 109(38), 15101–15108 (2012).
[Crossref]

2011 (2)

F. Montel, M. Delarue, J. Elgeti, L. Malaquin, M. Basan, T. Risler, B. Cabane, D. Vignjevic, J. Prost, G. Cappello, and J.-F. Joanny, “Stress clamp experiments on multicellular tumor spheroids,” Phys. Rev. Lett. 107(18), 188102 (2011).
[Crossref]

D. D. Nolte, R. An, J. Turek, and K. Jeong, “Holographic tissue dynamics spectroscopy,” J. Biomed. Opt. 16(8), 087004 (2011).
[Crossref]

2010 (2)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7(8), 603–614 (2010).
[Crossref]

F. Hirschhaeuser, H. Menne, C. Dittfeld, J. West, W. Mueller-Klieser, and L. A. Kunz-Schughart, “Multicellular tumor spheroids: an underestimated tool is catching up again,” J. Biotechnol. 148(1), 3–15 (2010).
[Crossref]

2009 (3)

M. Basan, T. Risler, J.-F. Joanny, X. Sastre-Garau, and J. Prost, “Homeostatic competition drives tumor growth and metastasis nucleation,” HFSP J. 3(4), 265–272 (2009).
[Crossref]

J. Friedrich, C. Seidel, R. Ebner, and L. A. Kunz-Schughart, “Spheroid-based drug screen: considerations and practical approach,” Nat. Protoc. 4(3), 309–324 (2009).
[Crossref]

V. Rajan, B. Varghese, T. G. van Leeuwen, and W. Steenbergen, “Review of methodological developments in laser doppler flowmetry,” Lasers Med. Sci. 24(2), 269–283 (2009).
[Crossref]

2008 (6)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2(2), 110–115 (2008).
[Crossref]

M. Suissa, C. Place, E. Goillot, and E. Freyssingeas, “Internal dynamics of a living cell nucleus investigated by dynamic light scattering,” Eur. Phys. J. E: Soft Matter Biol. Phys. 26(4), 435–448 (2008).
[Crossref]

M. Xu, T. T. Wu, and J. Y. Qu, “Unified mie and fractal scattering by cells and experimental study on application in optical characterization of cellular and subcellular structures,” J. Biomed. Opt. 13(2), 024015 (2008).
[Crossref]

Z. J. Smith and A. J. Berger, “Integrated raman-and angular-scattering microscopy,” Opt. Lett. 33(7), 714–716 (2008).
[Crossref]

R. Pierrat, N. B. Braham, L. F. Rojas-Ochoa, R. Carminati, and F. Scheffold, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281(1), 18–22 (2008).
[Crossref]

R. Pierrat, “Transport equation for the time correlation function of scattered field in dynamic turbid media,” J. Opt. Soc. Am. A 25(11), 2840 (2008).
[Crossref]

2007 (2)

J. Huisken and D. Y. R. Stainier, “Even fluorescence excitation by multidirectional selective plane illumination microscopy (mspim),” Opt. Lett. 32(17), 2608–2610 (2007).
[Crossref]

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello, and E. H. K. Stelzer, “High-resolution three-dimensional imaging of large specimens with light sheet–based microscopy,” Nat. Methods 4(4), 311–313 (2007).
[Crossref]

2006 (2)

M. J. Rust, M. Bates, and X. Zhuang, “Stochastic optical reconstruction microscopy (storm) provides sub-diffraction-limit image resolution,” Nat. Methods 3(10), 793–796 (2006).
[Crossref]

F. Jaillon, S. E. Skipetrov, J. Li, G. Dietsche, G. Maret, and T. Gisler, “Diffusing-wave spectroscopy from head-like tissue phantoms: influence of a non-scattering layer,” Opt. Express 14(22), 10181 (2006).
[Crossref]

2005 (2)

J. Li, G. Dietsche, D. Iftime, S. E. Skipetrov, G. Maret, T. Elbert, B. Rockstroh, and T. Gisler, “Noninvasive detection of functional brain activity with near-infrared diffusing-wave spectroscopy,” J. Biomed. Opt. 10(4), 044002 (2005).
[Crossref]

K. A. Lidke, B. Rieger, T. M. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express 13(18), 7052–7062 (2005).
[Crossref]

2004 (2)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref]

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92(21), 213903 (2004).
[Crossref]

2003 (1)

P. Yu, M. Mustata, J. J. Turek, P. M. W. French, M. R. Melloch, and D. D. Nolte, “Holographic optical coherence imaging of tumor spheroids,” Appl. Phys. Lett. 83(3), 575–577 (2003).
[Crossref]

2001 (4)

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21(3), 195–201 (2001).
[Crossref]

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2000 (1)

D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20(3), 469–477 (2000).
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1999 (1)

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1998 (2)

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1997 (2)

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1995 (1)

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1994 (1)

1992 (1)

1991 (2)

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1990 (3)

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1988 (1)

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1981 (3)

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M. T. Janet, G. Cheng, J. A. Tyrrell, S. A. Wilcox-Adelman, Y. Boucher, R. K. Jain, and L. L. Munn, “Mechanical compression drives cancer cells toward invasive phenotype,” Proc. Natl. Acad. Sci. U. S. A. 109(3), 911–916 (2012).
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T. Stylianopoulos, J. D. Martin, V. P. Chauhan, S. R. Jain, B. Diop-Frimpong, N. Bardeesy, B. L. Smith, C. R. Ferrone, F. J. Hornicek, Y. Boucher, L. L. Munn, and R. K. Jain, “Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors,” Proc. Natl. Acad. Sci. U. S. A. 109(38), 15101–15108 (2012).
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Braham, N. B.

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K. Alessandri, B. R. Sarangi, V. V. Gurchenkov, B. Sinha, T. R. Kießling, L. Fetler, F. Rico, S. Scheuring, C. Lamaze, A. Simon, S. Geraldo, D. Vignjević, H. Doméjean, L. Rolland, A. Funfak, J. Bibette, N. Bremond, and P. Nassoy, “Cellular capsules as a tool for multicellular spheroid production and for investigating the mechanics of tumor progression in vitro,” Proc. Natl. Acad. Sci. U. S. A. 110(37), 14843–14848 (2013).
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A. Fercher and J. D. Briers, “Flow visualization by means of single-exposure speckle photography,” Opt. Commun. 37(5), 326–330 (1981).
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B. Brunel, C. Blanch, A. Gourrier, V. Petrolli, A. Delon, J.-F. Joanny, R. Carminati, R. Pierrat, and G. Cappello, “Structure and dynamics of multicellular assemblies measured by coherent light scattering,” New J. Phys. 19(7), 073033 (2017).
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F. Montel, M. Delarue, J. Elgeti, L. Malaquin, M. Basan, T. Risler, B. Cabane, D. Vignjevic, J. Prost, G. Cappello, and J.-F. Joanny, “Stress clamp experiments on multicellular tumor spheroids,” Phys. Rev. Lett. 107(18), 188102 (2011).
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Cappello, G.

M. E. Dolega, M. Delarue, F. Ingremeau, J. Prost, A. Delon, and G. Cappello, “Cell-like pressure sensors reveal increase of mechanical stress towards the core of multicellular spheroids under compression,” Nat. Commun. 8(1), 14056 (2017).
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B. Brunel, C. Blanch, A. Gourrier, V. Petrolli, A. Delon, J.-F. Joanny, R. Carminati, R. Pierrat, and G. Cappello, “Structure and dynamics of multicellular assemblies measured by coherent light scattering,” New J. Phys. 19(7), 073033 (2017).
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S. Monnier, M. Delarue, B. Brunel, M. E. Dolega, A. Delon, and G. Cappello, “Effect of an osmotic stress on multicellular aggregates,” Methods 94, 114–119 (2016).
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F. Montel, M. Delarue, J. Elgeti, L. Malaquin, M. Basan, T. Risler, B. Cabane, D. Vignjevic, J. Prost, G. Cappello, and J.-F. Joanny, “Stress clamp experiments on multicellular tumor spheroids,” Phys. Rev. Lett. 107(18), 188102 (2011).
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Carminati, R.

B. Brunel, C. Blanch, A. Gourrier, V. Petrolli, A. Delon, J.-F. Joanny, R. Carminati, R. Pierrat, and G. Cappello, “Structure and dynamics of multicellular assemblies measured by coherent light scattering,” New J. Phys. 19(7), 073033 (2017).
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R. Pierrat, N. B. Braham, L. F. Rojas-Ochoa, R. Carminati, and F. Scheffold, “The influence of the scattering anisotropy parameter on diffuse reflection of light,” Opt. Commun. 281(1), 18–22 (2008).
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H. T. Nia, H. Liu, G. Seano, M. Datta, D. Jones, N. Rahbari, J. Incio, V. P. Chauhan, K. Jung, J. D. Martin, V. Askoxylakis, T. P. Padera, D. Fukumura, Y. Boucher, F. J. Hornicek, A. J. Grodzinsky, J. W. Baish, L. L. Munn, and R. K. Jain, “Solid stress and elastic energy as measures of tumour mechanopathology,” Nat. Biomed. Eng. 1(1), 0004 (2017).
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T. Stylianopoulos, J. D. Martin, V. P. Chauhan, S. R. Jain, B. Diop-Frimpong, N. Bardeesy, B. L. Smith, C. R. Ferrone, F. J. Hornicek, Y. Boucher, L. L. Munn, and R. K. Jain, “Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors,” Proc. Natl. Acad. Sci. U. S. A. 109(38), 15101–15108 (2012).
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Cheng, G.

M. T. Janet, G. Cheng, J. A. Tyrrell, S. A. Wilcox-Adelman, Y. Boucher, R. K. Jain, and L. L. Munn, “Mechanical compression drives cancer cells toward invasive phenotype,” Proc. Natl. Acad. Sci. U. S. A. 109(3), 911–916 (2012).
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D. A. Benaron, S. R. Hintz, A. Villringer, D. Boas, A. Kleinschmidt, J. Frahm, C. Hirth, H. Obrig, J. C. van Houten, E. L. Kermit, W.-F. Cheong, and D. K. Stevenson, “Noninvasive functional imaging of human brain using light,” J. Cereb. Blood Flow Metab. 20(3), 469–477 (2000).
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Climov, M.

J. Lee, H. Radhakrishnan, W. Wu, A. Daneshmand, M. Climov, C. Ayata, and D. A. Boas, “Quantitative imaging of cerebral blood flow velocity and intracellular motility using dynamic light scattering–optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(6), 819–825 (2013).
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Datta, M.

H. T. Nia, H. Liu, G. Seano, M. Datta, D. Jones, N. Rahbari, J. Incio, V. P. Chauhan, K. Jung, J. D. Martin, V. Askoxylakis, T. P. Padera, D. Fukumura, Y. Boucher, F. J. Hornicek, A. J. Grodzinsky, J. W. Baish, L. L. Munn, and R. K. Jain, “Solid stress and elastic energy as measures of tumour mechanopathology,” Nat. Biomed. Eng. 1(1), 0004 (2017).
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B. Brunel, C. Blanch, A. Gourrier, V. Petrolli, A. Delon, J.-F. Joanny, R. Carminati, R. Pierrat, and G. Cappello, “Structure and dynamics of multicellular assemblies measured by coherent light scattering,” New J. Phys. 19(7), 073033 (2017).
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M. E. Dolega, M. Delarue, F. Ingremeau, J. Prost, A. Delon, and G. Cappello, “Cell-like pressure sensors reveal increase of mechanical stress towards the core of multicellular spheroids under compression,” Nat. Commun. 8(1), 14056 (2017).
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W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
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J. A. Izatt, M. A. Choma, and A.-H. Dhalla, “Theory of optical coherence tomography,” Optical Coherence Tomography: Technology and Applications pp. 65–94 (2015).

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Figures (6)

Fig. 1.
Fig. 1. Schematic view of the experimental setup. Light scattering dectection was added to a commercial microscope. The illumination part was composed of a laser (${850}\,{\rm{nm}}$) attenuated (neutral density) and narrowed (L1 and L2). The detection part was composed of two lenses L3 and L4 positioned so that the front focal plane of L3 is imaged on the CCD2 camera.
Fig. 2.
Fig. 2. (b) q-resolved diffraction pattern as acquired on CCD2. $\varphi$ is the azimuthal angle. (a) Time evolution of the scattered intensity for a given scattering vector $\boldsymbol {q}$, and (c) its autocorrelation function. (d) The autocorrelation function is averaged over $\varphi$, i.e. over pixels inbetween the two white circles in (b).
Fig. 3.
Fig. 3. (a) Diagram of the light propagation model inside a spheroid (large circle). Photons arrive from the laser beam on the left with a direction vector $\boldsymbol{{u_i}}$. Then, they follow a random walk trajectory (red lines) with changes in direction vectors ($\boldsymbol{u'}$, $\boldsymbol{u}$) following an angular probability (red shapes). The last scattering event emits a wave, observed in the direction $\boldsymbol{{u_s}}$. (b,c) Proportion in the scattered light of photons scattered between $1$ and $8$ times, for a Gaussian beam with a waist of $w={192 \pm 12}\,{{\mathrm{\mu}} {\rm m} }$ and a spheroid of diameter $L={250}\,{{\mathrm{\mu}} {\rm m} }$ (b) and $L={470}\,{{\mathrm{\mu}} {\rm m} }$ (c). (d,e) Correlation function $C$ as a function of the scattering vector norm $q$, calculated using Eq. (3) (solid orange line) or a Monte Carlo simulation (dash black line), for virtual homogeneous spheres of $L={250}\,{{\mathrm{\mu}} {\rm m} }$ (d) or $L={470}\,{{\mathrm{\mu}} {\rm m} }$ (e), containing scatterers whose displacement distribution follows Eq. (11)
Fig. 4.
Fig. 4. (a) Autocorrelation functions $C(q,\tau )$ measured for a $L={413}\,{{\mathrm{\mu}} {\rm m} }$ spheroid, with a delay going from $6$ to ${24}\,{\textrm{min}}$ (markers). Data was fitted (lines) using Eqs. (4), 3 and 11. Fitting parameters are $\sigma ={0.79}\,{\mathrm{\mu} {\rm m}}$, ${1.40}\mathrm{\mu}\rm{m}$, ${2.16}\mathrm{\mu}\rm{m}$ and ${2.87}\,{{\mathrm{\mu}} {\rm m} }$. (b) Cell displacements magnitude distribution ($D(\Delta r,\tau$) multiplied by $4 \pi |\Delta r|^{2}$) deduced from the fit of (a). (c) Phase function $p$ measured for a spheroid of 146 µm in diameter, averaged azimuthally. Data was fitted using Eqs. (4) and 8. (d) Mean cells speed inside spheroids as a function of the pressure applied, for spheroids of respectively ${250}\,{{\mathrm{\mu}} {\rm m} }$ and ${470}\,{{\mathrm{\mu}} {\rm m} }$ in diameter (markers). Data was fitted by an exponential decay (lines).
Fig. 5.
Fig. 5. (a) Diagram representing a spheroid and the area illuminated by the laser positioned on the side of the spheroid. Arrows indicate radial (white) and tangential (black) directions. (b) Example of a diffraction pattern obtained for an off-center illuminated spheroid. Radial and tangential directions are indicated similarly to (a). For a given direction and a given $q$ norm, pixels included inside white sections are considered. (c) Cell velocity $\boldsymbol{v}/v_0$ as a function of the laser position $r/R$, normalized by the speed at the center $v_0$.
Fig. 6.
Fig. 6. Diagram showing the trajectory of a photon scattered twice (bottom) inside a spheroid, compared with the phase reference trajectory (up). $\boldsymbol{{u_{i}}}$, $\boldsymbol{{u_{1}}}$ and $\boldsymbol{{u_{s}}}$ are the directions of the photon incoming, scattered once and outcoming respectively. Scaterring events are located at position $\boldsymbol{r_1}$ and $\boldsymbol{r_2}$ with ${\mathbf{\delta }}=\boldsymbol{r_2}-\boldsymbol{r_1}$.

Tables (1)

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Table 1. Pressure exerted by Dextran molecules in solution depending on their mass concentration.

Equations (45)

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C ( q , τ ) = I ( q , t ) I ( q , t + τ ) I ( q , t ) φ I ( q , t + τ ) φ t 1.
p s ( k 0 ( u u ) ) = I s ( k 0 ( u u ) ) I s ( k 0 ( u u ) ) k 0 2 d u .
C ( q , τ ) = | k α k [ p s ( q ) D ~ ( q , τ ) ] k 1 [ p s ( q ) D ~ ( q , τ ) ] p ( q ) | 2 ,
p ( q ) = k α k [ p s k 1 p s ] ( q ) ,
P b ( k , s ) = s k s k k ! exp [ s s ] .
α k = 0 2 π 0 P b ( k , s ( r ) ) I l ( r , θ ) r d r d θ 0 2 π 0 I l ( r , θ ) r d r d θ .
α k = α k 1 α 0 .
p s [ k 0 ( u u ) ] = a 2 π q 0 2 exp [ [ k 0 ( u u ) ] 2 2 q 0 2 ] + 1 a 4 π k 0 2 1 b 2 [ 1 + b 2 2 b u u ] 3 / 2 ,
g = k 0 p s ( k 0 ( u u ) ) u u d u = 0.92.
= s 1 g = 950 μ m .
D ( Δ r , τ ) = 1 ( 2 π ) 3 / 2 σ τ 3 exp ( Δ r 2 2 σ τ 2 ) ,
[ u r + 1 s ] g ( 1 ) ( r , u , τ ) = 1 s k 0 2 p s ( k 0 ( u u ) ) h ~ ~ ( k 0 | u u | τ ) g ( 1 ) ( r , u , τ ) d u
h ~ ( w ) = δ [ w w v w ] h ( v ) d v .
h ( v ) = D ( τ v , τ ) τ .
C = | g ( 1 ) | 2 .
D ( Δ r , τ ) = 8 ( π τ ) 3 v ¯ r v ¯ t 2 exp [ 4 π { ( Δ r r τ v ¯ r ) 2 + ( Δ r t τ v ¯ t ) 2 + ( Δ r z τ v ¯ t ) 2 } ]
ϕ = k 0 ( u i . r 1 + u 1 . δ u s . r 2 ) = k 0 ( u i . r 1 + u 1 . ( r 2 r 1 ) u s . r 2 ) = q 1 . r 1 + k 0 ( u 1 u s ) . r 2 = q 1 . r 1 + ( q 1 q ) . r 2
ϕ ( q , t ) = m = 1 k ( q m q m 1 ) . r m ( t )
E ( q , t ) = j = 1 N E j ( q ) e i ϕ j ( t )
I ( q , t ) t = | E ( q , t ) | 2 t = E ( q , t ) E ( q ¯ , t ) t = ( j N E j ( q ) e i ϕ j ( q , t ) ) ( m N E m ( q ) e i ϕ m ( q , t ) ) t = j , m N E j ( q ) e i ϕ j ( q , t ) E m ( q ) e i ϕ m ( q , t ) t = j , m N E j ( q ) E m ( q ) e i ϕ j ( q , t ) e i ϕ m ( q , t ) t
e i ϕ j ( q , t ) e i ϕ m ( q , t ) t = { e i ϕ j ( q , t ) t e i ϕ m ( q , t ) t = 0 if j m 1 if j = m }
I ( q , t ) t = j N E j 2 ( q )
I ( q , t ) t = k N α k E j 2 ( q ) j ( k )
E j 2 ( q ) = I 0 p s ( q q k 1 ( j ) ) .
E j 2 ( q ) j ( 1 ) = I 0 p s ( q )
E j 2 ( q ) j ( 2 ) = I 0 p s ( q 1 ) p s ( q q 1 ) d q 1 = I 0 [ p s p s ] ( q )
E j 2 ( q ) j ( k ) = I 0 [ p s k p s ] ( q )
I ( q , t ) t = N I 0 k α k [ p s k p s ] ( q )
I ( q , t ) t d q = N I 0 k α k [ p s k p s ] ( q ) d q = N I 0 k α k = N I 0
p ( q ) = I ( q , t ) t N I 0 = k α k [ p s k p s ] ( q )
C ( q , τ ) = I ( q , t ) I ( q , t + τ ) t I ( q ) 2 1
Δ r ( t , τ ) = r ( t + τ ) r ( t )
Δ ϕ ( q , t , τ ) = ϕ ( q , t + τ ) ϕ ( q , t ) = m = 1 k ( q m q m 1 ) . ( r m ( t + τ ) r m ( t ) ) = j = m k ( q m q m 1 ) . Δ r m ( t , τ )
I ( q , t ) I ( q , t + τ ) t = E ( q , t ) E ¯ ( q , t ) E ( q , t + τ ) E ¯ ( q , t + τ ) t = ( j N E j ( q ) e i ϕ j ( q , t ) ) ( l N E l ( q ) e i ϕ l ( q , t ) ) ( m N E m ( q ) e i ϕ m ( q , t + τ ) ) ( p N E p ( q ) e i ϕ p ( q , t + τ ) ) t = j , l , m , p N E j ( q ) E l ( q ) E m ( q ) E p ( q ) e i ( ϕ j ( q , t ) ϕ l ( q , t ) ) e i ( ϕ m ( q , t + τ ) ϕ p ( q , t + τ ) ) t = j , l , m , p N E j ( q ) E l ( q ) E m ( q ) E p ( q ) e i ( ϕ j ( q , t ) e i ϕ l ( q , t ) e i ϕ m ( q , t + τ ) e i ϕ p ( q , t + τ ) t
A = e i ( ϕ j ( q , t ) ϕ j ( q , t + τ ) ) t e i ( ϕ m ( q , t + τ ) ϕ m ( q , t ) ) t = e i Δ ϕ j ( q , t , τ ) t e i Δ ϕ m ( q , t , τ ) t
A = e i ( ϕ j ( q , t ) + ϕ j ( q , t + τ ) ) t e i ( ϕ p ( q , t ) + ϕ p ( q , t + τ ) ) t = 0
I ( q , t ) I ( q , t + τ ) t = j , m N E j 2 ( q ) E m 2 ( q ) + j , m N E j 2 ( q ) E m 2 ( q ) e i Δ ϕ j ( q , t , τ ) t e i Δ ϕ m ( q , t , τ ) t = I ( q ) 2 + ( j N E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t ) ( m N E m 2 ( q ) e i Δ ϕ m ( q , t , τ ) t ) = I ( q ) 2 + | j N E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t | 2
C ( q , τ ) = | j N E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t | 2 I ( q ) 2
j N E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t = k N α k E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t , j ( k )
E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t , j ( k ) = I 0 p s ( q q k 1 ) e i m = 1 k ( q m q m 1 ) . Δ r m ( t , τ ) t , j ( k ) = I 0 p s ( q q k 1 ) m = 1 k e i ( q m q m 1 ) . Δ r m ( t , τ ) t , j ( k )
E j 2 ( q ) e i Δ ϕ j ( q , t , τ ) t , j ( k ) = I 0 p s ( q q k 1 ) m = 1 k e i ( q m q m 1 ) . Δ r m ( t , τ ) t j ( k )
e i ( q j q j 1 ) . Δ r j ( t , τ ) t = D τ ( Δ r ) e i ( q j q j 1 ) . Δ r d Δ r = D τ ~ ( q j q j 1 )
E m 2 ( q ) e i Δ ϕ j ( q , t , τ ) t , j ( k ) = I 0 p s ( q q k 1 ) j = 1 k D ~ ( q j q j 1 ) j ( k ) = I 0 p s ( q 1 ) D τ ~ ( q 1 ) p s ( q 2 q 1 ) D τ ~ ( q 2 q 1 ) p s ( q q k 1 ) D τ ~ ( q q k 1 ) d q 1 d q k 1 = I 0 [ p s D τ ~ p s D τ ~ ] ( q 2 ) [ p s D τ ~ ] ( q 3 q 2 ) [ p s D τ ~ ] ( q q k 1 ) d q 2 d q k 1 = I 0 [ p s D τ ~ k 1 p s D τ ~ ] ( q )
C ( q , τ ) = | N I 0 k α k [ p s D τ ~ k 1 p s D τ ~ ] ( q ) | 2 I ( q ) 2
C ( q , τ ) = | 1 p ( q ) k α k [ p s D τ ~ k 1 p s D τ ~ ] ( q ) | 2

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