Research Papers

Simulation of the Mechanical Response of Cells on Micropost Substrates

[+] Author and Article Information
William Ronan

Department of Mechanical and
Biomedical Engineering,
National University of Ireland Galway,
University Road,
Galway 78746, Ireland

Amit Pathak

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106-5070

Vikram S. Deshpande

Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK

Robert M. McMeeking

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106-5070
School of Engineering,
University of Aberdeen King's College,
Aberdeen AB24 3UE, Scotland

J. Patrick McGarry

Department of Mechanical and
Biomedical Engineering,
National University of Ireland Galway,
University Road,
Galway 78746, Ireland
e-mail:  patrick.mcgarry@nuigalway.ie

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received April 16, 2013; final manuscript received July 4, 2013; accepted manuscript posted July 29, 2013; published online September 20, 2013. Assoc. Editor: Guy M. Genin.

J Biomech Eng 135(10), 101012 (Sep 20, 2013) (10 pages) Paper No: BIO-13-1192; doi: 10.1115/1.4025114 History: Received April 16, 2013; Revised July 04, 2013; Accepted July 29, 2013

Experimental studies where cells are seeded on micropost arrays in order to quantify their contractile behavior are becoming increasingly common. Interpretation of the data generated by this experimental technique is difficult, due to the complexity of the processes underlying cellular contractility and mechanotransduction. In the current study, a coupled framework that considers strain rate dependent contractility and remodeling of the cytoskeleton is used in tandem with a thermodynamic model of tension dependent focal adhesion formation to investigate the biomechanical response of cells adhered to micropost arrays. Computational investigations of the following experimental studies are presented: cell behavior on different sized arrays with a range of post stiffness; stress fiber and focal adhesion formation in irregularly shaped cells; the response of cells to deformations applied locally to individual posts; and the response of cells to equibiaxial stretching of micropost arrays. The predicted stress fiber and focal adhesion distributions; in addition to the predicted post tractions are quantitatively and qualitatively supported by previously published experimental data. The computational models presented in this study thus provide a framework for the design and interpretation of experimental micropost studies.

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Grahic Jump Location
Fig. 1

Schematic diagram showing simulation approach. The cell is simulated using the active SF material formulation that predicts SF formation in every direction at each point in the cell. Microposts are modeled as linear springs attached to rigid circular surfaces. Contact between the cell and posts is simulated using the FA interface model.

Grahic Jump Location
Fig. 2

(a) and (c) Predicted stress fiber (SF) formation and (b) and (d) focal adhesion (FA) formation in cells adhered to 5 × 5 and 13 × 13 arrays of microposts. Microposts have a 1 μm radius and 4 μm center-to-center spacing and a stiffness of 18.16 nN/μm. SFs and FAs are shown have reached a steady state following 600 s of signal driven formation.

Grahic Jump Location
Fig. 3

(a) and (c) Predicted stress fiber (SF) formation and (b) and (d) focal adhesion (FA) formation in cells adhered to 13 × 13 arrays of microposts. Microposts have a 1 μm radius and 4 μm center to center spacing and a stiffness of (a) and (b) 1556 nN/μm and (c) and (d) 1.9 nN/μm. SFs and FAs are shown having reached a steady state following 600 s of signal driven formation. (e) Individual post forces are shown for different levels of post stiffness as a function of distance from cell center.

Grahic Jump Location
Fig. 4

(a) Average force per post for and (b) total force per cell for individual cells adhered to micropost arrays of different stiffness as a function of both cell area (top axis) and number of posts (bottom axis). (c) and (d) Experimental observations by [19] of average and total force for endothelial cells, reproduced with permission.

Grahic Jump Location
Fig. 5

(a) Computed focal adhesion (FA) area for each cell for different post stiffness as a function of cell size (area—top axis, number of posts—bottom axis). (c) The experimental observations of FA area as a function of cell size by Fu et al. [10] are reproduced for comparison. (b) The total force (i.e., the sum of the magnitudes of the individual post forces) is shown as a function of the total adhesion area for a range of cells adhered to micropost arrays with different post stiffness. A linear trend line is fitted to each group of cells on a given micropost stiffness (solid lines). (d) The slope of these trend lines, which is also known as the “FA stress,” is presented as a function of post stiffness. (e) Finally, the experimental results of Fu et al. [10] (reproduced with permission) are presented for comparison; note that the experimentally observed FA stress is computed in the same way.

Grahic Jump Location
Fig. 6

(a) Predicted stress fiber formation in an irregularly shaped cell geometry. The orientation of the dominant, or most activated, fiber at each point is shown as a vector plot. The length of each vector corresponds to the SF activation level in the dominant direction.

Grahic Jump Location
Fig. 7

Comparison of (a) computed FA formation and (b) experimental observations of focal adhesion formation. Predictions show the dimensionless concentration of high affinity integrins (ξh). Experimental images are reproduced from Tan et al. [8]. In both the computed and experimental FA images, the reaction force exerted by the cell on each post is shown as a vector superimposed over the image. In order to highlight the distinctive horseshoe shaped adhesions, magnified views (150% of original size) of two regions identified in purple in (a) are shown in (c) for both the experimental and predicted results. For each individual post adhered to the cell, the computed force and FA area are shown in (d) together with the corresponding experimental forces and FA areas observed by Tan et al. [8] (reproduced with permission).

Grahic Jump Location
Fig. 8

(a) Focal adhesion areas for a cell on an array of microposts where a single post is subjected to an applied displacement. FA areas are normalized by the steady state area immediately prior to stretch are presented for the moved post and the remaining unmoved posts. In four separate simulations, a post at either the corner of the cell or midway along a cell edge is moved either inwards or outwards, as shown in (b). (c) For comparison, the experimental results of Sniadecki et al. [11] are reproduced with permission.

Grahic Jump Location
Fig. 9

Cells are simulated adhered to a micropost array which is subjected to biaxial stretch, as shown in (a). (b) Post forces are present for three separate simulations with different stretch activated signals: a constant signal triggered by the applied stretch at 600 s; an exponentially decaying signal, also initiated at 600 s; and with no stretch activated signal. Post forces are normalized by the equilibrium post force immediately prior to the application of the 7% strain at 600 s. (c) Simulations are also performed for a 7% strain using different material formulations to represent the passive cytoplasm. Forces are presented for the equilibrium force prior to stretch, the peak force following the stretch, and the force 3600 s after the stretch is applied. (d) For the case of the Maxwell model viscoelastic cytoplasm, simulations are performed for three different strain magnitudes. The experimentally observed post forces following biaxial strain are also adapted and reproduced for comparison [12].



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