Constitutive Material Modeling of Cell: A Micromechanics Approach

[+] Author and Article Information
G. U. Unnikrishnan, V. U. Unnikrishnan

Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123

J. N. Reddy

Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123 and Engineering Science Programme, National University of Singapore, Block E3A, #04-17,9 Engineering Drive 1, Singapore 117576jnreddy@tamu.edu

J Biomech Eng 129(3), 315-323 (Nov 19, 2006) (9 pages) doi:10.1115/1.2720908 History: Received May 02, 2006; Revised November 19, 2006

The variations in mechanical properties of cells obtained from experimental and theoretical studies can be overcome only through the development of a sound mathematical framework correlating the derived mechanical property with the cellular structure. Such a formulation accounting for the inhomogeneity of the cytoplasm due to stress fibers and actin cortex is developed in this work. The proposed model is developed using the Mori-Tanaka method of homogenization by treating the cell as a fiber-reinforced composite medium satisfying the continuum hypothesis. The validation of the constitutive model using finite element analysis on atomic force microscopy (AFM) and magnetic twisting cytometry (MTC) has been carried out and is found to yield good correlation with reported experimental results. It is observed from the study that as the volume fraction of the stress fiber increases, the stiffness of the cell increases and it alters the force displacement behavior for the AFM and MTC experiments. Through this model, we have also been able to find the stress fiber as a likely cause of the differences in the derived mechanical property from the AFM and MTC experiments. The correlation of the mechanical behavior of the cell with the cell composition, as obtained through this study, is an important observation in cell mechanics.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

The cross section of a typical adherent cell showing the random distribution of actin stress fibers. The presence of stress fibers introduces considerable inhomogeneity in the cytoplasm.

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Figure 2

The stress-strain curve for the material after homogenization for different volume fractions of the fiber. With an increase in the volume fraction of stress fiber, the effect of fiber on the material property of the composite increases, thus making the material much stiffer than the matrix.

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Figure 3

The effect of stress fiber volume fraction on Poisson’s ratio of the composite. With an increase in the fiber volume fraction, Poisson’s ratio decreases from the nearly incompressible value of the matrix to reach that of the fiber.

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Figure 4

Half cell axisymmetric finite element model of the cell having a graded finer mesh towards the region of indentation. The cell geometry considered is 3.5μm in half width, 3.0μm in height with an actin cortical depth of 0.2μm with a nucleus of 0.9μm diameter at a height of 0.75μm from the base. The model is meshed using an axisymmetric linear element having a total of 2637 nodes with 2746 elements. The top layer forms the actin cortex, the inner layer of cytoplasm, having the nucleus as an inclusion in the inner cytoplasm.

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Figure 5

Strain distribution obtained from the finite element analysis of axisymmetric cell model due to an indentation of 0.5μm on the cell with a stress fiber volume fraction of 0.1%. The layer directly underneath the indenter suffers the maximum deformation and gradually fades away from the cell.

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Figure 6

Force deflection curve for the cytoplasm having stress fiber volume fraction of 0.1% and 1% and its comparison with experimentally obtained values for healthy and diseased cells. The experimental results are obtained from the following reference L929 (8), HCV29, and Hu 609 (9).

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Figure 7

Finite element mesh of the cell block selected for MTC simulation. The top layer is the actin cortical region having a thickness of 0.2μm below which forms the inner cytoplasm without the nucleus. The cell block geometry is 20×10×5μm3, with a bead of 4μm diameter making an indentation angle of 90deg.

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Figure 8

Strain distribution induced by bead displacement along 1-2 (a) and 2-2 (b) directions due to a lateral load of 500pN. In both the figures, the shear strain distribution is concentrated near the surface and decreases away from bead surface.

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Figure 9

The vertical displacement distribution due the action of the load at the center of the bead. Maximum displacements occur on the two ends of the bead contact region.

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Figure 10

The variation of the displacement of the bead center with the change in stress fiber volume fraction. Similar to the AFM study, the simulation of MTC is carried out with the material property at different volume fractions of stress fibers and the displacement of the bead center is noted after each analysis. As the volume fraction of the stress fiber increases the displacement of the center of the bead decreases considerably.

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Figure 11

The comparison of bead rotation obtained from simulation with the results published in (28) (indicated by *) for a torque applied at the center of the bead. The boundary condition in the finite element analysis was changed to model the torque instead of the lateral load at the bead center. The result shows that the computational model is able to predict the difference in the bead rotation for changes in the volume fraction of fiber, which affects the stiffness.



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