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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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References

Fung, Y. C., 1967, “Elasticity of Soft Tissues in Simple Elongation,” Am. J. Physiol., 213(6), pp. 1532–1544. [PubMed]
Humphrey, J. D., 2003, “Review Paper: Continuum Biomechanics of Soft Biological Tissues,” Proc. R. Soc. Lond. A, 459(2029), pp. 3–46. [CrossRef]
Humphrey, J. D., and Yin, F. C., 1989, “Constitutive Relations and Finite Deformations of Passive Cardiac Tissue II: Stress Analysis in the Left Ventricle,” Circ. Res., 65(3), pp. 805–817. [CrossRef] [PubMed]
Gasser, T. C., Ogden, R. W., and Holzapfel, G. A., 2006, “Hyperelastic Modelling of Arterial Layers With Distributed Collagen Fibre Orientations,” J. R. Soc. Interface, 3(6), pp. 15–35. [CrossRef] [PubMed]
Hu, J.-J., Baek, S., and Humphrey, J. D., 2007, “Stress–Strain Behavior of the Passive Basilar Artery in Normotension and Hypertension,” J. Biomech., 40(11), pp. 2559–2563. [CrossRef] [PubMed]
Wagner, H. P., and Humphrey, J. D., 2011, “Differential Passive and Active Biaxial Mechanical Behaviors of Muscular and Elastic Arteries: Basilar Versus Common Carotid,” ASME, J. Biomech. Eng., 133(5), p. 051009. [CrossRef]
Zahalak, G. I., Wagenseil, J. E., Wakatsuki, T., and Elson, E. L., 2000, “A Cell-Based Constitutive Relation for Bio-Artificial Tissues,” Biophys. J., 79(5), pp. 2369–2381. [CrossRef] [PubMed]
Marquez, J. P., Genin, G. M., Zahalak, G. I., and Elson, E. L., 2005, “The Relationship Between Cell and Tissue Strain in Three-Dimensional Bio-Artificial Tissues,” Biophys. J., 88(2), pp. 778–789. [CrossRef] [PubMed]
Marquez, J. P., Genin, G. M., Zahalak, G. I., and Elson, E. L., 2005, “Thin Bio-Artificial Tissues in Plane Stress: The Relationship between Cell and Tissue Strain, and an Improved Constitutive Model,” Biophys. J., 88(2), pp. 765–777. [CrossRef] [PubMed]
Barocas, V. H., and Tranquillo, R. T., 1997, “An Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics: The Interplay Among Cell Traction, Fibrillar Network Deformation, Fibril Alignment, and Cell Contact Guidance,” ASME, J. Biomech. Eng., 119(2), pp. 137–145. [CrossRef]
Stevenson, M. D., Sieminski, A. L., McLeod, C. M., Byfield, F. J., Barocas, V. H., and Gooch, K. J., 2010, “Pericellular Conditions Regulate Extent of Cell-Mediated Compaction of Collagen Gels,” Biophys. J., 99(1), pp. 19–28. [CrossRef] [PubMed]
Guilak, F., and Mow, V. C., 2000, “The Mechanical Environment of the Chondrocyte: A Biphasic Finite Element Model of Cell–Matrix Interactions in Articular Cartilage,” J. Biomech., 33(12), pp. 1663–1673. [CrossRef] [PubMed]
Breuls, R. G. M., Sengers, B. G., Oomens, C. W. J., Bouten, C. V. C., and Baaijens, F. P. T., 2002, “Predicting Local Cell Deformations in Engineered Tissue Constructs: A Multilevel Finite Element Approach,” ASME, J. Biomech. Eng., 124(2), pp. 198–207. [CrossRef]
StamenovićD., Fredberg, J. J., Wang, N., Butler, J. P., and Ingber, D. E., 1996, “A Microstructural Approach to Cytoskeletal Mechanics Based on Tensegrity,” J. Theor. Biol., 181(2), pp. 125–136. [CrossRef] [PubMed]
Wang, N., Naruse, K., Stamenović, D., Fredberg, J. J., Mijailovich, S. M., Tolić-Nørrelykke, I. M., Polte, T., Mannix, R., and Ingber, D. E., 2001, “Mechanical Behavior in Living Cells Consistent With the Tensegrity Model,” Proc. Natl. Acad. Sci., 98(14), pp. 7765–7770. [CrossRef]
Coughlin, M. F., and Stamenović, D., 1998, “A Tensegrity Model of the Cytoskeleton in Spread and Round Cells,” ASME, J. Biomech. Eng., 120(6), pp. 770–777. [CrossRef]
Stamenović, D., and Coughlin, M. F., 2000, “A Quantitative Model of Cellular Elasticity Based on Tensegrity,” ASME, J. Biomech. Eng., 122(1), pp. 39–43. [CrossRef]
Gibson, L. J., and Ashby, M. F., 1999, Cellular Solids: Structure and Properties, Cambridge University Press, New York.
Satcher, R. L., Jr., and Dewey, C. F., Jr., 1996, “Theoretical Estimates of Mechanical Properties of the Endothelial Cell Cytoskeleton,” Biophys. J., 71(1), pp. 109–118. [CrossRef] [PubMed]
Susilo, M. E., Roeder, B. A., Voytik-Harbin, S. L., Kokini, K., and Nauman, E. A., 2010, “Development of a Three-Dimensional Unit Cell to Model the Micromechanical Response of a Collagen-Based Extracellular Matrix,” Acta Biomater., 6(4), pp. 1471–1486. [CrossRef] [PubMed]
Isambert, H., and Maggs, A. C., 1996, “Dynamics and Rheology of Actin Solutions,” Macromolecules, 29(3), pp. 1036–1040. [CrossRef]
Storm, C., Pastore, J. J., MacKintosh, F. C., Lubensky, T. C., and Janmey, P. A., 2005, “Nonlinear Elasticity in Biological Gels,” Nature, 435(7039), pp. 191–194. [CrossRef] [PubMed]
Palmer, J. S., and Boyce, M. C., 2008, “Constitutive Modeling of the Stress–Strain Behavior of F-Actin Filament Networks,” Acta Biomater., 4(3), pp. 597–612. [CrossRef] [PubMed]
Stylianopoulos, T., and Barocas, V. H., 2007, “Multiscale, Structure-Based Modeling for the Elastic Mechanical Behavior of Arterial Walls,” ASME, J. Biomech. Eng., 129(4), pp. 611–618. [CrossRef]
Lake, S. P., Hadi, M. F., Lai, V. K., and Barocas, V. H., 2012, “Mechanics of a Fiber Network Within a Non-Fibrillar Matrix: Model and Comparison With Collagen-Agarose Co-Gels,” Ann. Biomed. Eng., 40(10), pp. 2111–2121. [CrossRef] [PubMed]
Zhang, L., Lake, S. P., Lai, V. K., Picu, C. R., Barocas, V. H., and Shephard, M. S., 2013, “A Coupled Fiber-Matrix Model Demonstrates Highly Inhomogeneous Microstructural Interactions in Soft Tissues Under Tensile Load,” ASME, J. Biomech. Eng., 135(1), p. 011008. [CrossRef]
Hadi, M. F., Sander, E. A., and BarocasV. H., 2012, “Multiscale Model Predicts Tissue-Level Failure From Collagen Fiber-Level Damage,” ASME, J. Biomech. Eng., 134(9), p. 091005. [CrossRef]
Hashin, Z., 1962, “The Elastic Moduli of Heterogeneous Material,” J. Appl. Mech., 29(1), pp. 143–150. [CrossRef]
Chandran, P. L., and Barocas, V. H., 2007, “Deterministic Material-Based Averaging Theory Model of Collagen Gel Micromechanics,” ASME, J. Biomech. Eng., 129(2), pp. 137–147. [CrossRef]
Stylianopoulos, T., and Barocas, V. H., 2007, “Volume-Averaging Theory for the Study of the Mechanics of Collagen Networks,” Comput. Meth. Appl. Mech. Eng., 196(31–32), pp. 2981–2990. [CrossRef]
Fung, Y., 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York.
Billiar, K. L., and Sacks, M. S., 2000, “Biaxial Mechanical Properties of the Native and Glutaraldehyde-Treated Aortic Valve Cusp: Part II—A Structural Constitutive Model,” ASME, J. Biomech. Eng., 122(4), pp. 327–335. [CrossRef]
Evans, M. C., and Barocas, V. H., 2009, “The Modulus of Fibroblast-Populated Collagen Gels is not Determined by Final Collagen and Cell Concentration: Experiments and an Inclusion-Based Model,” ASME, J. Biomech. Eng., 131(10), p. 101014. [CrossRef]
Bonet, J., and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, New York.
Roy, S., Silacci, P., and Stergiopulos, N., 2005, “Biomechanical Properties of Decellularized Porcine Common Carotid Arteries,” Am. J. Physiol. Heart Circ. Physiol., 289(4), pp. H1567–H1576. [CrossRef] [PubMed]
Lynch, H. A., Johannessen, W., Wu, J. P., Jawa, A., and Elliott, D. M., 2003, “Effect of Fiber Orientation and Strain Rate on the Nonlinear Uniaxial Tensile Material Properties of Tendon,” ASME, J. Biomech. Eng., 125(5), pp. 726–731. [CrossRef]
Lake, S. P., Miller, K. S., Elliott, D. M., and Soslowsky, L. J., 2010, “Tensile Properties and Fiber Alignment of Human Supraspinatus Tendon in the Transverse Direction Demonstrate Inhomogeneity, Nonlinearity, and Regional Isotropy,” J. Biomech., 43(4), pp. 727–732. [CrossRef] [PubMed]
Cheng, V. W. T., and Screen, H. R. C., 2007, “The Micro-Structural Strain Response of Tendon,” J. Mater. Sci., 42(21), pp. 8957–8965. [CrossRef]
Lai, V. K., Frey, C. R., Kerandi, A. M., Lake, S. P., Tranquillo, R. T., and Barocas, V. H., 2012, “Microstructural and Mechanical Differences Between Digested Collagen–Fibrin Co-Gels and Pure Collagen and Fibrin Gels,” Acta Biomater., 8(11), pp. 4031–4042. [CrossRef] [PubMed]
Nachtrab, S., Kapfer, S. C., Arns, C. H., Madadi, M., Mecke, K., and Schröder-Turk, G. E., 2011, “Morphology and Linear-Elastic Moduli of Random Network Solids,” Adv. Mater., 23(22–23), pp. 2633–2637. [CrossRef] [PubMed]
Lai, V. K., Lake, S. P., Frey, C. R., Tranquillo, R. T., and Barocas, V. H., 2012, “Mechanical Behavior of Collagen-Fibrin Co-Gels Reflects Transition From Series to Parallel Interactions With Increasing Collagen Content,” ASME, J. Biomech. Eng., 134(1), p. 011004. [CrossRef]
Burridge, K., Fath, K., Kelly, T., Nuckolls, G., and Turner, C., 1988, “Focal Adhesions: Transmembrane Junctions Between the Extracellular Matrix and the Cytoskeleton,” Ann. Rev. Cell Biol., 4(1), pp. 487–525. [CrossRef]
Goffin, J. M., Pittet, P., Csucs, G., Lussi, J. W., Meister, J.-J., and Hinz, B., 2006, “Focal Adhesion Size Controls Tension-Dependent Recruitment of α-Smooth Muscle Actin to Stress Fibers,” J. Cell. Biol., 172(2), pp. 259–268. [CrossRef] [PubMed]
Durrant, L. A., Archer, C. W., Benjamin, M., and Ralphs, J. R., 1999, “Organisation of the Chondrocyte Cytoskeleton and its Response to Changing Mechanical Conditions in Organ Culture,” J. Anat., 194(3), pp. 343–353. [CrossRef] [PubMed]
Eggli, P. S., Hunzinker, E. B., and Schenk, R. K., 1988, “Quantitation of Structural Features Characterizing Weight- and Less-Weight-Bearing Regions in Articular Cartilage: A Stereological Analysis of Medical Femoral Condyles in Young Adult Rabbits,” Anat. Rec., 222(3), pp. 217–227. [CrossRef] [PubMed]
Janmey, P. A., and McCulloch, C. A., 2007, “Cell Mechanics: Integrating Cell Responses to Mechanical Stimuli,” Ann. Rev. Biomed. Eng., 9(1), pp. 1–34. [CrossRef]
Ofek, G., Wiltz, D. C., and Athanasiou, K. A., 2009, “Contribution of the Cytoskeleton to the Compressive Properties and Recovery Behavior of Single Cells,” Biophys. J., 97(7), pp. 1873–1882. [CrossRef] [PubMed]
Eastwood, M., MuderaV. C., McgroutherD. A., and Brown, R. A., 1998, “Effect of Precise Mechanical Loading on Fibroblast Populated Collagen Lattices: Morphological Changes,” Cell Motil. Cytoskel., 40(1), pp. 13–21. [CrossRef]
Huang, D., Chang, T. R., Aggarwal, A., Lee, R. C., and Ehrlich, H. P., 1993, “Mechanisms and Dynamics of Mechanical Strengthening in Ligament-Equivalent Fibroblast-Populated Collagen Matrices,” Ann. Biomed. Eng., 21(3), pp. 289–305. [CrossRef] [PubMed]
Chan, C. E., and Odde, D. J., 2008, “Traction Dynamics of Filopodia on Compliant Substrates,” Science, 322(5908), pp. 1687–1691. [CrossRef] [PubMed]
Yeung, T., Georges, P. C., Flanagan, L. A., Marg, B., Ortiz, M., Funaki, M., Zahir, N., Ming, W., Weaver, V., and Janmey, P. A., 2005, “Effects of Substrate Stiffness on Cell Morphology, Cytoskeletal Structure, and Adhesion,” Cell Motil. Cytoskel., 60(1), pp. 24–34. [CrossRef]
Subramanian, A., and Lin, H.-Y., 2005, “Crosslinked Chitosan: Its Physical Properties and the Effects of Matrix Stiffness on Chondrocyte Cell Morphology and Proliferation,” J. Biomed. Mater. Res. A, 75A(3), pp. 742–753. [CrossRef]
Engler, A. J., Sen, S., Sweeney, H. L., and Discher, D. E., 2006, “Matrix Elasticity Directs Stem Cell Lineage Specification,” Cell, 126(4), pp. 677–689. [CrossRef] [PubMed]
Hadjipanayi, E., Mudera, V., and Brown, R. A., 2009, “Guiding Cell Migration in 3D: A Collagen Matrix With Graded Directional Stiffness,” Cell Motil. Cytoskel., 66(3), pp. 121–128. [CrossRef]
Legant, W. R., Miller, J. S., Blakely, B. L., Cohen, D. M., Genin, G. M., and Chen, C. S., 2010, “Measurement of Mechanical Tractions Exerted by Cells in Three-Dimensional Matrices,” Nature Methods, 7(12), pp. 969–971. [CrossRef] [PubMed]
Pizzo, A. M., Kokini, K., Vaughn, L. C., Waisner, B. Z., and Voytik-Harbin, S. L., 2005, “Extracellular Matrix (ECM) Microstructural Composition Regulates Local Cell-ECM Biomechanics and Fundamental Fibroblast Behavior: A Multidimensional Perspective,” J. Appl. Physiol., 98(5), pp. 1909–1921. [CrossRef] [PubMed]

Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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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