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Research Papers

A Multiscale Approach to Modeling the Passive Mechanical Contribution of Cells in Tissues

[+] Author and Article Information
Victor K. Lai

Department of Chemical Engineering and Materials Science,
University of Minnesota–Twin Cities,
421 Washington Avenue Southeast,
Minneapolis, MN 55455

Mohammad F. Hadi

Department of Biomedical Engineering,
University of Minnesota–Twin Cities,
7-105 Nils Hasselmo Hall,
312 Church Street Southeast,
Minneapolis, MN 55455

Robert T. Tranquillo

Department of Chemical Engineering and Materials Science,
University of Minnesota–Twin Cities,
421 Washington Avenue Southeast,
Minneapolis, MN 55455;
Department of Biomedical Engineering,
University of Minnesota–Twin Cities,
7-105 Nils Hasselmo Hall,
312 Church Street Southeast,
Minneapolis, MN 55455

Victor H. Barocas

Department of Biomedical Engineering,
University of Minnesota–Twin Cities,
7-105 Nils Hasselmo Hall,
312 Church Street Southeast,
Minneapolis, MN 55455
e-mail: baroc001@umn.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received January 17, 2013; final manuscript received April 16, 2013; accepted manuscript posted April 30, 2013; published online June 11, 2013. Assoc. Editor: Keith Gooch.

J Biomech Eng 135(7), 071007 (Jun 11, 2013) (9 pages) Paper No: BIO-13-1025; doi: 10.1115/1.4024350 History: Received January 17, 2013; Revised April 16, 2013; Accepted April 30, 2013

In addition to their obvious biological roles in tissue function, cells often play a significant mechanical role through a combination of passive and active behaviors. This study focused on the passive mechanical contribution of cells in tissues by improving our multiscale model via the addition of cells, which were treated as dilute spherical inclusions. The first set of simulations considered a rigid cell, with the surrounding ECM modeled as (1) linear elastic, (2) Neo-Hookean, and (3) a fiber network. Comparison with the classical composite theory for rigid inclusions showed close agreement at low cell volume fraction. The fiber network case exhibited nonlinear stress–strain behavior and Poisson's ratios larger than the elastic limit of 0.5, characteristics similar to those of biological tissues. The second set of simulations used a fiber network for both the cell (simulating cytoskeletal filaments) and matrix, and investigated the effect of varying relative stiffness between the cell and matrix, as well as the effect of a cytoplasmic pressure to enforce incompressibility of the cell. Results showed that the ECM network exerted negligible compression on the cell, even when the stiffness of fibers in the network was increased relative to the cell. Introduction of a cytoplasmic pressure significantly increased the stresses in the cell filament network, and altered how the cell changed its shape under tension. Findings from this study have implications on understanding how cells interact with their surrounding ECM, as well as in the context of mechanosensation.

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Figures

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Fig. 1

(a) Schematic representation of cells within a tissue organized in a periodic lattice, based on the assumption that cells are dilute, noninteracting, and spherical in shape. (b) Finite-element mesh showing boundary conditions and three symmetry planes. In the multiscale formulation, each Gauss point in every element is associated with a unique representative volume element (RVE) comprised of a random, interconnected network.

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Fig. 2

Undeformed meshes, as well as Cauchy stress (σ11) distributions after 10% strain (averaged over three runs for each), for the cases of a linear elastic matrix, Neo-Hookean matrix, and a fiber network at rigid cell volume fractions of 5%, 10%, and 15%. The rigid cells were removed for clarity. In general, larger stresses were observed with increasing cell volume fraction. Variations in cell stresses in the fiber network cases were due to the uniqueness of Voronoi networks used for each element.

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Fig. 3

Cauchy stress σ11 versus engineering strain for the linear elastic matrix, Neo-Hookean matrix, and fiber network cases at rigid cell volume fractions of 0%, 5%, 10%, and 15%. Stress–strain curves for the linear elastic and Neo-Hookean matrix cases appeared linear and largely coincided with each other. The fiber network case exhibited nonlinear stress–strain behavior, similar to that of soft tissues. Error bands in the fiber network cases are 95% confidence intervals, n = 3 for each case.

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Fig. 4

Plots at 10% stretch of (a) composite elastic modulus E*, (b) composite Poisson's ratio ν*, and (c) composite Poisson's ratio normalized with Poisson's ratio of the matrix, ν*/νm, compared with the Hashin model. The * represents statistical significance at the 95% level. Model results showed close agreement with the Hashin solution at lower volume fractions. Unlike the linear elastic and Neo-Hookean matrix cases, the Poisson's ratio for the fiber network case did not decrease with increasing cell volume fraction. Error bars for the fiber network cases are 95% confidence intervals, n = 3 for each case.

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Fig. 5

Cauchy stress (σ11) distributions at 10% stretch of the 10× stiffer cell (first row), equal cell and matrix stiffness (second row), and the 10× stiffer matrix (third row) cases, for both the compressible and incompressible cells (averaged over three runs for each case). Stress distributions around the cell differed depending on the relative stiffness of cell and matrix. In all cases, introduction of a cytoplasmic pressure to enforce incompressibility increased the stresses borne by the cell filament networks.

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Fig. 6

(a) Representative cell filament network at 1%, 5%, and 10% strain for the compressible and incompressible cell cases, showing the distribution of filament stretches in the networks. Larger filament stretches were observed for the incompressible cell case. (b) Average filament orientation in the 1- (Ω11) and 3- (Ω33) directions versus strain for the compressible and incompressible cell cases. Introduction of a cytoplasmic pressure inhibited filament rotation into the direction of stretch, such that the filaments were less oriented in the incompressible cell. Error bars are 95% confidence intervals, n = 112 (total number of cell elements).

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Fig. 7

Composite Poisson's ratio, ν*, at 10% stretch for the compressible and incompressible cell cases, with different relative stiffness of cell and matrix. The * and # represent statistical significance at the 95% level. No significant differences were observed between the different relative stiffness cases for the compressible cell. The 10× stiffer cell case had significantly higher ν* than the 10× stiffer matrix (p = 0.0036) and equal stiffness (p = 0.0285) cases for the incompressible cell. Error bars represent 95% confidence intervals, with n = 3 for each case.

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Fig. 8

Cell surface traction at 10% stretch of the 10× stiffer cell (first row), equal cell and matrix stiffness (second row), and the 10× stiffer matrix (third row) cases, for both the compressible and incompressible cells (averaged over three runs for each case). In the compressible cell cases, large tensile stresses were observed in the direction of tension (region A). The cell surface region under the Poisson effect (regions B and C) did not exhibit large compressive stresses. Addition of pressure increased the surface traction from the cell filament network, and slightly increased the overall compressive stress in regions B and C.

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Fig. 9

(a)–(b): Normalized ratio of cell dimension to composite dimension in the 3-direction versus the 1-direction. Addition of pressure increased the cell proportion in the 3-direction for all cases. (c)–(h): Schematic drawings showing differences in cell shape, and the stresses exerted on the cell surface at equilibrium; dotted lines represent the equal cell and matrix stiffness case without pressure. In the compressible cell cases (c), (e), (g), matrix tension in the 1-direction was always balanced by cell tension. In the incompressible cell cases, (d), (f), (h), the outward-exerting pressure pushed out against the cell to increase cell proportion in the 3-direction.

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