Analysis of Nonlinear Responses of Adherent Epithelial Cells Probed by Magnetic Bead Twisting: A Finite Element Model Based on a Homogenization Approach

[+] Author and Article Information
Jacques Ohayon, Philippe Tracqui

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut de l’Ingénierie et de l’Information de Santé, Faculté de Médecine, 38706 La Tronche Cedex, France

Redouane Fodil, Sophie Féréol, Valérie M. Laurent

INSERM, UMR 492, Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

Emmanuelle Planus

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut del’ Ingénierie et de l’Information de Santé Faculté de Médecine, 38706 La Tronche, Cedex France INSERM, UMR 492, Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

Daniel Isabey

INSERM, UMR 492 Physiopathologie et Thérapeutique Respiratoires, Faculté de Médecine, 8, rue du Général Sarrail, 94010 Créteil cedex, France

J Biomech Eng 126(6), 685-698 (Feb 04, 2005) (14 pages) doi:10.1115/1.1824136 History: Received October 09, 2003; Revised July 26, 2004; Online February 04, 2005
Copyright © 2004 by ASME
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Grahic Jump Location
Representative cross section of cell–bead geometry (bead #25, Table 1) in a plane running through the center of the bead and perpendicular to the substrate. This cross section was obtained from a 3D reconstruction of confocal images performed after staining F-actin with fluorescent phalloidin. The bead-embedding half-angle α is estimated as the mean of the two bead-embedding half-angles α1 and α2, and hu is the measured cell thickness under the bead. Scaling is given by the bead diameter, i.e., 3.2 μm.
Grahic Jump Location
Finite element mesh of the half-representative cell volume element (RCVE) measuring a/2×a×h. The bead surface and the portion of the cell surface (area bounded by white lines) that may come into contact with the bead during bead rotation have been meshed with contact elements. The arrow indicates the orientation of the torque applied at the center of the bead. Free-boundary conditions are applied to the RCVE surfaces, except for the displacement conditions indicated in the figure.
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Values of cell material parameters obtained from the MTC data. (A) The RCVE homogeneous model response (solid line) gives a very good fit to the experimental data (open circles ±SD) for the two optimal material parameters a1=2.25 Pa and a2=50 Pa. The mean response in the 25 individual simulations performed with the geometrical features of each bead listed in Table 1 is superimposed (full triangles ±SD). (B) Representative curve of the Young modulus Ecell derived from the two-parameter Yeoh strain-energy function and computed from Eq. (B1) in Appendix B with a1=2.25 Pa and a2=50 Pa.
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3D color maps showing the deformed cell shapes and the spatial distributions of the Lagrangian strains near the bead. These numerical results were obtained when a torque T equal to 1010.5 pN⋅μm was applied to the bead, inducing a bead rotation θ of 29.6° and a translation Δy along the Oy axis of 0.77 μm. The bead was removed to visualize more clearly the spatial strain distributions on the cell–bead contact area. The Lagrangian strain fields shown in the figure are (A) the normal strain Eyy; (B) the normal strain Ezz; (C) the shear strain Eyz, and (D) the effective strain eeff. The Std set of parameter values was used for these simulations.
Grahic Jump Location
2D color maps showing the spatial distribution of the secant modulus Es (in Pa) in the plane of symmetry (see Fig. 1) and in the vicinity of the bead. Increasing values of the applied torque T are considered: (A) T=31.7 pN⋅μm, bead rotation θ=5.2°, and translation along the Oy axis Δy=−0.11 μm; (B) T=87.7 pN⋅μm, θ=9.6°, and Δy=−0.22 μm; (C) T=385 pN⋅μm, θ=19.6°, and Δy=−0.49 μm; and (D) T=1010.5 pN⋅μm, θ=29.6°, and Δy=−0.77 μm. The Std set of parameter values was used for these simulations. Propagation of the strain from the bead contact area to the substrate induced a dramatic increase in the secant modulus from (A) to (D). The inserts correspond to the same figure but with the color code rescaled between 0 and 100 Pa.
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Influence of the Yeoh strain-energy function parameters a1 and a2 (in Pa) on the bead rotation amplitude θ (A), (C) and on the associated normalized translation Δy*y/R (B), (D). The Std curve is the control simulation obtained with the Std set of parameter values. In each simulation, except for the parameter under study, the parameter values were those of the Std set.
Grahic Jump Location
Influence of the geometrical parameters R (A), (B), α (C), (D), and h (E), (F) on the bead rotation amplitude θ and on the associated normalized translation Δy*y/R. The response curves simulated with the Std set of parameter values is given in each case as the reference (Std curves). In each simulation, except for the parameter under study, the parameter values were those of the Std set.
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Influence on the cell mechanical response of two schematic cell adhesion patterns, PF1 and PF2. Top left: Top view of the two models PF1 and PF2. The white areas correspond to absence of cell adhesion. The arrows indicate the orientation of the torque applied at the center of the bead. Bottom left: Cross-section view of the two models PF1 and PF2. The 2D color maps show the heterogeneous spatial distribution of the effective strain fields eeff in the plane of symmetry when the bead rotation is equal to 30°. Right: Computed cell response obtained for each cell adhesion pattern, compared to the control simulation (full cell adhesion). All computations were done using the Std set of parameter values.
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Variation in normalized apparent cell stiffness [Gθ/Ecell, with Gθ=T/(6Vbθ), where Vb is the bead volume] with increasing values of the bead-immersion half-angle α and in the special case of a semi-infinite medium. The numerical results given by the linear approximation of the hyperelastic model are compared to the analytical expression proposed by Laurent et al. [Gθ/Ecell=sin3(α)/6; see Ref. 17] and to the numerical results reported by Mijailovitch et al. 42. The geometrical parameter values were chosen in agreement with those of Mijailovitch et al., i.e., R=2.25 μm and h=20 μm.
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(A) Mechanical response when considering the cell as a linear elastic material with increasing Young modulus (E=6a1) values from 13.5 to 135 Pa. In each simulation, except for the parameter under study, the parameter values were those of the Std set. (B) Nonlinear effect induced by the arc cosine function on the apparent mechanical response when considering the cell as a linear elastic material with increasing Young modulus (E=6a1) values from 30 to 240 Pa. For each value of E, we calculated the torque-apparent bead rotation response (i.e., T–θ* curves) according to the procedure described in the text. Note that open circles (±SD) are the experimental MTC data and that all simulations were performed using a neo-Hookean strain-energy function.




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