Research Papers

Finite Element Formulation of Multiphasic Shell Elements for Cell Mechanics Analyses in FEBio

[+] Author and Article Information
Jay C. Hou, Gerard A. Ateshian

Department of Mechanical Engineering,
Columbia University,
New York, NY 10027

Steve A. Maas, Jeffrey A. Weiss

Department of Bioengineering,
University of Utah,
Salt Lake City, UT 84112

Manuscript received April 9, 2018; final manuscript received July 17, 2018; published online September 25, 2018. Assoc. Editor: Raffaella De Vita.

J Biomech Eng 140(12), 121009 (Sep 25, 2018) (16 pages) Paper No: BIO-18-1171; doi: 10.1115/1.4041043 History: Received April 09, 2018; Revised July 17, 2018

With the recent implementation of multiphasic materials in the open-source finite element (FE) software FEBio, three-dimensional (3D) models of cells embedded within the tissue may now be analyzed, accounting for porous solid matrix deformation, transport of interstitial fluid and solutes, membrane potential, and reactions. The cell membrane is a critical component in cell models, which selectively regulates the transport of fluid and solutes in the presence of large concentration and electric potential gradients, while also facilitating the transport of various proteins. The cell membrane is much thinner than the cell; therefore, in an FE environment, shell elements formulated as two-dimensional (2D) surfaces in 3D space would be preferred for modeling the cell membrane, for the convenience of mesh generation from image-based data, especially for convoluted membranes. However, multiphasic shell elements are yet to be developed in the FE literature and commercial FE software. This study presents a novel formulation of multiphasic shell elements and its implementation in FEBio. The shell model includes front- and back-face nodal degrees-of-freedom for the solid displacement, effective fluid pressure and effective solute concentrations, and a linear interpolation of these variables across the shell thickness. This formulation was verified against classical models of cell physiology and validated against reported experimental measurements in chondrocytes. This implementation of passive transport of fluid and solutes across multiphasic membranes makes it possible to model the biomechanics of isolated cells or cells embedded in their extracellular matrix (ECM), accounting for solvent and solute transport.

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Blinov, M. L. , Schaff, J. C. , Vasilescu, D. , Moraru, I. I. , Bloom, J. E. , and Loew, L. M. , 2017, “ Compartmental and Spatial Rule-Based Modeling With Virtual Cell,” Biophys. J., 113(7), pp. 1365–1372. [CrossRef] [PubMed]
Moraru, I. I. , Schaff, J. C. , Slepchenko, B. M. , and Loew, L. M. , 2002, “ The Virtual Cell: An Integrated Modeling Environment for Experimental and Computational Cell Biology,” Ann. N. Y. Acad. Sci., 971, pp. 595–596. [CrossRef] [PubMed]
Schaff, J. , Fink, C. C. , Slepchenko, B. , Carson, J. H. , and Loew, L. M. , 1997, “ A General Computational Framework for Modeling Cellular Structure and Function,” Biophys. J., 73(3), pp. 1135–1146. [CrossRef] [PubMed]
Keener, J. P. , and Sneyd, J. , 2009, Mathematical Physiology (Interdisciplinary Applied Mathematics, Vol. 8), 2nd ed., Springer, New York.
Kedem, O. , and Katchalsky, A. , 1958, “ Thermodynamic Analysis of the Permeability of Biological Membranes to Non-Electrolytes,” Biochim. Et Biophys. Acta, 27, pp. 229–246. [CrossRef]
Katzir-Katchalsky, A. , and Curran, P. F. , 1965, Nonequilibrium Thermodynamics in Biophysics (Harvard Books in Biophysics, Vol. 1), Harvard University Press, Cambridge, UK.
Truesdell, C. , and Toupin, R. , 1960, The Classical Field Theories (Handbuch der Physik, Vol. III/1), Springer, Berlin.
Bowen, R. , 1976, Theory of Mixtures (Continuum Physics, Vol. 3), Academic Press, New York.
Ateshian, G. A. , Likhitpanichkul, M. , and Hung, C. T. , 2006, “ A Mixture Theory Analysis for Passive Transport in Osmotic Loading of Cells,” J. Biomech., 39(3), pp. 464–475. [CrossRef] [PubMed]
Lai, W. M. , Hou, J. S. , and Mow, V. C. , 1991, “ A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage,” ASME J. Biomech. Eng., 113(3), pp. 245–258. [CrossRef]
Gu, W. Y. , Lai, W. M. , and Mow, V. C. , 1993, “ Transport of Fluid and Ions Through a Porous-Permeable Charged-Hydrated Tissue, and Streaming Potential Data on Normal Bovine Articular Cartilage,” J. Biomech., 26(6), pp. 709–23. [CrossRef] [PubMed]
Huyghe, J. M. , and Janssen, J. D. , 1997, “ Quadriphasic Mechanics of Swelling Incompressible Porous Media,” Int. J. Eng. Sci., 35(8), pp. 793–802. [CrossRef]
Gu, W. Y. , Lai, W. M. , and Mow, V. C. , 1998, “ A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors,” ASME J. Biomech. Eng., 120(2), pp. 169–180. [CrossRef]
Maas, S. A. , Ellis, B. J. , Ateshian, G. A. , and Weiss, J. A. , 2012, “ Febio: Finite Elements for Biomechanics,” ASME J. Biomech. Eng., 134(1), p. 011005. [CrossRef]
Ateshian, G. A. , Albro, M. B. , Maas, S. , and Weiss, J. A. , 2011, “ Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation,” ASME J. Biomech. Eng., 133(8), p. 081005. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2013, “ Multiphasic Finite Element Framework for Modeling Hydrated Mixtures With Multiple Neutral and Charged Solutes,” ASME J. Biomech. Eng., 135(11), p. 111001. [CrossRef]
Ateshian, G. A. , Nims, R. J. , Maas, S. , and Weiss, J. A. , 2014, “ Computational Modeling of Chemical Reactions and Interstitial Growth and Remodeling Involving Charged Solutes and Solid-Bound Molecules,” Biomech. Model. Mechanobiol., 13(5), pp. 1105–1120. [CrossRef] [PubMed]
Halloran, J. , Sibole, S. , V., Donkelaar , C., Van Turnhout , M., Oomens , C. W., Weiss , J., Guilak , F. , and Erdemir, A. , 2012, “ Multiscale Mechanics of Articular Cartilage: Potentials and Challenges of Coupling Musculoskeletal, Joint, and Microscale Computational Models,” Ann. Biomed. Eng., 40(11), pp. 2456–2474.
Wilkins, R. , Browning, J. , and Ellory, J. , 2000, “ Surviving in a Matrix: Membrane Transport in Articular Chondrocytes,” J. Membr. Biol., 177(2), pp. 95–108. [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]
Kim, E. , Guilak, F. , and Haider, M. A. , 2008, “ The Dynamic Mechanical Environment of the Chondrocyte: A Biphasic Finite Element Model of Cell-Matrix Interactions Under Cyclic Compressive Loading,” ASME J. Biomech. Eng., 130(6), p. 061009. [CrossRef]
Cao, L. , Guilak, F. , and Setton, L. A. , 2009, “ Pericellular Matrix Mechanics in the Anulus Fibrosus Predicted by a Three-Dimensional Finite Element Model and in Situ Morphology,” Cell Mol. Bioeng., 2(3), pp. 306–319. [CrossRef] [PubMed]
Cao, L. , Guilak, F. , and Setton, L. A. , 2011, “ Three-Dimensional Finite Element Modeling of Pericellular Matrix and Cell Mechanics in the Nucleus Pulposus of the Intervertebral Disk Based on in Situ Morphology,” Biomech. Model. Mechanobiol., 10(1), pp. 1–10. [CrossRef] [PubMed]
Tanska, P. , Mononen, M. E. , and Korhonen, R. K. , 2015, “ A Multi-Scale Finite Element Model for Investigation of Chondrocyte Mechanics in Normal and Medial Meniscectomy Human Knee Joint During Walking,” J. Biomech., 48(8), pp. 1397–1406. [CrossRef] [PubMed]
Ateshian, G. A. , Costa, K. D. , and Hung, C. T. , 2007, “ A Theoretical Analysis of Water Transport Through Chondrocytes,” Biomech. Model. Mechanobiol., 6(1–2), pp. 91–101. [CrossRef] [PubMed]
Moo, E. K. , Herzog, W. , Han, S. K. , Abu Osman, N. A. , Pingguan-Murphy, B. , and Federico, S. , 2012, “ Mechanical Behaviour of In-Situ Chondrocytes Subjected to Different Loading Rates: A Finite Element Study,” Biomech. Model. Mechanobiol., 11(7), pp. 983–993. [CrossRef] [PubMed]
Alexopoulos, L. G. , Haider, M. A. , Vail, T. P. , and Guilak, F. , 2003, “ Alterations in the Mechanical Properties of the Human Chondrocyte Pericellular Matrix With Osteoarthritis,” ASME J. Biomech. Eng., 125(3), pp. 323–333. [CrossRef]
Alexopoulos, L. G. , Williams, G. M. , Upton, M. L. , Setton, L. A. , and Guilak, F. , 2005, “ Osteoarthritic Changes in the Biphasic Mechanical Properties of the Chondrocyte Pericellular Matrix in Articular Cartilage,” J. Biomech., 38(3), pp. 509–517. [CrossRef] [PubMed]
Trickey, W. R. , Baaijens, F. P. , Laursen, T. A. , Alexopoulos, L. G. , and Guilak, F. , 2006, “ Determination of the Poisson's Ratio of the Cell: Recovery Properties of Chondrocytes After Release From Complete Micropipette Aspiration,” J. Biomech., 39(1), pp. 78–87. [CrossRef] [PubMed]
Cao, L. , Youn, I. , Guilak, F. , and Setton, L. A. , 2006, “ Compressive Properties of Mouse Articular Cartilage Determined in a Novel Micro-Indentation Test Method and Biphasic Finite Element Model,” ASME J. Biomech. Eng., 128(5), pp. 766–771. [CrossRef]
Nguyen, B. V. , Wang, Q. G. , Kuiper, N. J. , El Haj, A. J. , Thomas, C. R. , and Zhang, Z. , 2010, “ Biomechanical Properties of Single Chondrocytes and Chondrons Determined by Micromanipulation and Finite-Element Modelling,” J. R. Soc. Interface, 7(53), pp. 1723–33. [CrossRef] [PubMed]
Darling, E. , Zauscher, S. , and Guilak, F. , 2006, “ Viscoelastic Properties of Zonal Articular Chondrocytes Measured by Atomic Force Microscopy,” Osteoarthr. Cartilage, 14(6), pp. 571–579. [CrossRef]
Darling, E. M. , Wilusz, R. E. , Bolognesi, M. P. , Zauscher, S. , and Guilak, F. , 2010, “ Spatial Mapping of the Biomechanical Properties of the Pericellular Matrix of Articular Cartilage Measured In Situ Via Atomic Force Microscopy,” Biophys. J., 98(12), pp. 2848–2856. [CrossRef] [PubMed]
Nguyen, T. D. , and Gu, Y. , 2014, “ Determination of Strain-Rate-Dependent Mechanical Behavior of Living and Fixed Osteocytes and Chondrocytes Using Atomic Force Microscopy and Inverse Finite Element Analysis,” ASME J. Biomech. Eng., 136(10), p. 101004. [CrossRef]
Haider, M. A. , and Guilak, F. , 2000, “ An Axisymmetric Boundary Integral Model for Incompressible Linear Viscoelasticity: Application to the Micropipette Aspiration Contact Problem,” ASME J. Biomech. Eng., 122(3), pp. 236–244. [CrossRef]
Haider, M. A. , and Guilak, F. , 2002, “ An Axisymmetric Boundary Integral Model for Assessing Elastic Cell Properties in the Micropipette Aspiration Contact Problem,” ASME J. Biomech. Eng., 124(5), pp. 586–595. [CrossRef]
Baaijens, F. P. , Trickey, W. R. , Laursen, T. A. , and Guilak, F. , 2005, “ Large Deformation Finite Element Analysis of Micropipette Aspiration to Determine the Mechanical Properties of the Chondrocyte,” Ann. Biomed. Eng., 33(4), pp. 494–501. [CrossRef] [PubMed]
Haider, M. A. , and Guilak, F. , 2007, “ Application of a Three-Dimensional Poroelastic Bem to Modeling the Biphasic Mechanics of Cell–Matrix Interactions in Articular Cartilage,” Comput. Method Appl. Mech. Eng., 196(31–32), pp. 2999–3010. [CrossRef]
Kim, E. , Guilak, F. , and Haider, M. A. , 2010, “ An Axisymmetric Boundary Element Model for Determination of Articular Cartilage Pericellular Matrix Properties in Situ Via Inverse Analysis of Chondron Deformation,” ASME J. Biomech. Eng., 132(3), p. 031011. [CrossRef]
Guilak, F. , Alexopoulos, L. G. , Upton, M. L. , Youn, I. , Choi, J. B. , Cao, L. , Setton, L. A. , and Haider, M. A. , 2006, “ The Pericellular Matrix as a Transducer of Biomechanical and Biochemical Signals in Articular Cartilage,” Ann. N.Y. Acad. Sci., 1068(1), pp. 498–512. [CrossRef]
Guilak, F. , Haider, M. A. , Setton, L. A. , Laursen, T. A. , and Baaijens, F. P. T. , 2006, Multiphasic Models of Cell Mechanics. Cambridge Texts in Biomedical Engineering, Cambridge University Press, Cambridge, UK, pp. 84–102.
Haider, M. A. , Schugart, R. C. , Setton, L. A. , and Guilak, F. , 2006, “ A Mechano-Chemical Model for the Passive Swelling Response of an Isolated Chondron Under Osmotic Loading,” Biomech. Model. Mechanobiol., 5(2–3), pp. 160–171. [CrossRef] [PubMed]
Bischoff, M. , Bletzinger, K.-U. , Wall, W. , and Ramm, E. , 2004, “ Models and Finite Elements for Thin-Walled Structures,” Encyclopedia of Computational Mechanics, Vol. 2, E. Stein, R. Borst, de, and T. Hughes, eds., Wiley, Chichester, UK.
Simo, J. , Rifai, M. , and Fox, D. , 1990, “ On a Stress Resultant Geometrically Exact Shell Model—Part IV: Variable Thickness Shells With Through-the-Thickness Stretching,” Comput Method Appl. Mech. Eng., 81(1), pp. 91–126. [CrossRef]
Ahmad, S. , Irons, B. M. , and Zienkiewicz, O. , 1970, “ Analysis of Thick and Thin Shell Structures by Curved Finite Elements,” Int. J. Numer. Methods Eng., 2(3), pp. 419–451. [CrossRef]
Parisch, H. , 1995, “ A Continuum-Based Shell Theory for Non-Linear Applications,” Int. J. Numer. Methods Eng., 38(11), pp. 1855–1883. [CrossRef]
Miehe, C. , 1998, “ A Theoretical and Computational Model for Isotropic Elastoplastic Stress Analysis in Shells at Large Strains,” Comput. Method. Appl. Mech. Eng., 155(3–4), pp. 193–233. [CrossRef]
Klinkel, S. , Gruttmann, F. , and Wagner, W. , 1999, “ A Continuum Based Three-Dimensional Shell Element for Laminated Structures,” Comput. Struct., 71(1), pp. 43–62. [CrossRef]
Vu-Quoc, L. , and Tan, X. , 2003, “ Optimal Solid Shells for Non-Linear Analyses of Multilayer Composites—I: Statics,” Comput. Method Appl. Mech. Eng., 192(9–10), pp. 975–1016. [CrossRef]
MacNeal, R. H. , 1978, “ A Simple Quadrilateral Shell Element,” Comput. Struct., 8(2), pp. 175–183. [CrossRef]
Bathe, K.-J. , and Dvorkin, E. N. , 1986, “ A Formulation of General Shell Elements—The Use of Mixed Interpolation of Tensorial Components,” Int. J. Numer. Methods Eng., 22(3), pp. 697–722. [CrossRef]
Betsch, P. , and Stein, E. , 1995, “ An Assumed Strain Approach Avoiding Artificial Thickness Straining for a Non-Linear 4-Node Shell Element,” Int. J. Numer. Methods Biomed. Eng., 11(11), pp. 899–909.
Bischoff, M. , and Ramm, E. , 1997, “ Shear Deformable Shell Elements for Large Strains and Rotations,” Int. J. Numer. Methods Eng., 40(23), pp. 4427–4449. [CrossRef]
Simo, J. C. , and Rifai, M. , 1990, “ A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes,” Int. J. Numer. Methods Eng., 29(8), pp. 1595–1638. [CrossRef]
Simo, J. , Armero, F. , and Taylor, R. , 1993, “ Improved Versions of Assumed Enhanced Strain Tri-Linear Elements for 3D Finite Deformation Problems,” Comput. Method Appl. Mech. Eng., 110(3–4), pp. 359–386. [CrossRef]
Sun, D. N. , Gu, W. Y. , Guo, X. E. , Lai, W. M. , and Mow, V. C. , 1999, “ A Mixed Finite Element Formulation of Triphasic Mechano-Electrochemical Theory for Charged, Hydrated Biological Soft Tissues,” Int. J. Numer. Methods Eng., 45(10), pp. 1375–1402. [CrossRef]
Bonet, J. , and Wood, R. D. , 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge, New York.
Maas, S. A. , Rawlins, D. , Weiss, J. A. , and Ateshian, G. A. , 2018, “ Febio 2.7 Theory Manual,” accessed Aug. 16, 2018, https://help.febio.org/FEBio/FEBio_tm_2_7/index.html
Mow, V. C. , Kuei, S. C. , Lai, W. M. , and Armstrong, C. G. , 1980, “ Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments,” ASME J. Biomech. Eng., 102(1), pp. 73–84. [CrossRef]
Guilak, F. , Erickson, G. R. , and Ting-Beall, H. P. , 2002, “ The Effects of Osmotic Stress on the Viscoelastic and Physical Properties of Articular Chondrocytes,” Biophys. J., 82(2), pp. 720–727. [CrossRef] [PubMed]
Chao, P. G. , Tang, Z. , Angelini, E. , West, A. C. , Costa, K. D. , and Hung, C. T. , 2005, “ Dynamic Osmotic Loading of Chondrocytes Using a Novel Microfluidic Device,” J. Biomech., 38(6), pp. 1273–1281. [CrossRef] [PubMed]
Albro, M. B. , Petersen, L. E. , Li, R. , Hung, C. T. , and Ateshian, G. A. , 2009, “ Influence of the Partitioning of Osmolytes by the Cytoplasm on the Passive Response of Cells to Osmotic Loading,” Biophys. J., 97(11), pp. 2886–2893. [CrossRef] [PubMed]
Alberts, B. , 2002, Molecular Biology of the Cell, 4th ed., Garland Science, New York.
Albro, M. B. , Chahine, N. O. , Caligaris, M. , Wei, V. I. , Likhitpanichkul, M. , Ng, K. W. , Hung, C. T. , and Ateshian, G. A. , 2007, “ Osmotic Loading of Spherical Gels: A Biomimetic Study of Hindered Transport in the Cell Protoplasm,” ASME J. Biomech. Eng., 129(4), pp. 503–510. [CrossRef]
Mauck, R. L. , Soltz, M. A. , Wang, C. C. , Wong, D. D. , Chao, P. H. , Valhmu, W. B. , Hung, C. T. , and Ateshian, G. A. , 2000, “ Functional Tissue Engineering of Articular Cartilage Through Dynamic Loading of Chondrocyte-Seeded Agarose Gels,” ASME J. Biomech. Eng., 122(3), pp. 252–260. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2010, “ Finite Element Algorithm for Frictionless Contact of Porous Permeable Media Under Finite Deformation and Sliding,” ASME J. Biomech. Eng., 132(6), p. 061006. [CrossRef]
Maas, S. A. , Rawlins, D. , Weiss, J. A. , and Ateshian, G. A. , 2018, "Febio 2.8 User Manual," accessed Aug. 16, 2018, https://help.febio.org/FEBio/FEBio_um_2_8/index.html
Hoffmann, E. K. , Lambert, I. H. , and Pedersen, S. F. , 2009, “ Physiology of Cell Volume Regulation in Vertebrates,” Physiol. Rev., 89(1), pp. 193–277. [CrossRef] [PubMed]
Tsuga, K. , Tohse, N. , Yoshino, M. , Sugimoto, T. , Yamashita, T. , Ishii, S. , and Yabu, H. , 2002, “ Chloride Conductance Determining Membrane Potential of Rabbit Articular Chondrocytes,” J. Membr. Biol., 185(1), pp. 75–81. [CrossRef] [PubMed]
Wilson, J. R. , Duncan, N. A. , Giles, W. R. , and Clark, R. B. , 2004, “ A Voltage-Dependent k+ Current Contributes to Membrane Potential of Acutely Isolated Canine Articular Chondrocytes,” J. Physiol., 557(1), pp. 93–104. [CrossRef] [PubMed]
Lewis, R. , Asplin, K. E. , Bruce, G. , Dart, C. , Mobasheri, A. , and Barrett-Jolley, R. , 2011, “ The Role of the Membrane Potential in Chondrocyte Volume Regulation,” J. Cell Physiol., 226(11), pp. 2979–2986. [CrossRef] [PubMed]
Hall, A. , Starks, I. , Shoults, C. , and Rashidbigi, S. , 1996, “ Pathways for k+ Transport Across the Bovine Articular Chondrocyte Membrane and Their Sensitivity to Cell Volume,” Am. J. Physiol.—Cell Ph., 270(5), pp. C1300–C1310. [CrossRef]
Freeman, M. , 1979, Adult Articular Cartilage, 2nd ed., Pitman Medical, Kent, UK.
Guilak, F. , 2000, “ The Deformation Behavior and Viscoelastic Properties of Chondrocytes in Articular Cartilage,” Biorheology, 37(1), pp. 27–44. [PubMed]
Oswald, E. S. , Chao, P.-H. G. , Bulinski, J. C. , Ateshian, G. A. , and Hung, C. T. , 2008, “ Dependence of Zonal Chondrocyte Water Transport Properties on Osmotic Environment,” Cell. Mol. Bioeng., 1(4), pp. 339–348. [CrossRef] [PubMed]
Maidhof, R. , Jacobsen, T. , Papatheodorou, A. , and Chahine, N. O. , 2014, “ Inflammation Induces Irreversible Biophysical Changes in Isolated Nucleus Pulposus Cells,” PLoS One, 9(6), p. e99621. [CrossRef] [PubMed]
Sánchez, J. C. , and Wilkins, R. J. , 2004, “ Changes in Intracellular Calcium Concentration in Response to Hypertonicity in Bovine Articular Chondrocytes,” Comp. Biochem. Phys. A, 137(1), pp. 173–182. [CrossRef]
Soltz, M. A. , and Ateshian, G. A. , 2000, “ A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage,” ASME J. Biomech. Eng., 122(6), pp. 576–586. [CrossRef]
Ateshian, G. A. , Rajan, V. , Chahine, N. O. , Canal, C. E. , and Hung, C. T. , 2009, “ Modeling the Matrix of Articular Cartilage Using a Continuous Fiber Angular Distribution Predicts Many Observed Phenomena,” ASME J. Biomech. Eng., 131(6), p. 061003. [CrossRef]
Hou, C. , and Ateshian, G. A. , 2016, “ A Gauss-Kronrod-Trapezoidal Integration Scheme for Modeling Biological Tissues With Continuous Fiber Distributions,” Comput. Method Biomech., 19(8), pp. 883–893. [CrossRef]
Wilusz, R. E. , Sanchez-Adams, J. , and Guilak, F. , 2014, “ The Structure and Function of the Pericellular Matrix of Articular Cartilage,” Matrix Biol., 39, pp. 25–32. [CrossRef] [PubMed]
Hille, B. , 2001, Ion Channels of Excitable Membranes, 3rd ed., Sinauer, Sunderland, MA.
Weiss, T. F. , 1996, Cellular Biophysics, MIT Press, Cambridge, MA.
Ateshian, G. A. , 2007, “ On the Theory of Reactive Mixtures for Modeling Biological Growth,” Biomech. Model. Mechanobiol., 6(6), pp. 423–445. [CrossRef] [PubMed]
Tieleman, D. P. , Marrink, S.-J. , and Berendsen, H. J. , 1997, “ A Computer Perspective of Membranes: Molecular Dynamics Studies of Lipid Bilayer Systems,” Biochim. Biophys. Acta (BBA)-Rev. Biomembr., 1331(3), pp. 235–270. [CrossRef]
Feller, S. E. , 2000, “ Molecular Dynamics Simulations of Lipid Bilayers,” Curr. Opin. Colloid Interface Sci., 5(3–4), pp. 217–223. [CrossRef]
Phillips, J. C. , Braun, R. , Wang, W. , Gumbart, J. , Tajkhorshid, E. , Villa, E. , Chipot, C. , Skeel, R. D. , Kale, L. , and Schulten, K. , 2005, “ Scalable Molecular Dynamics With NAMD,” J. Comput. Chem., 26(16), pp. 1781–1802. [CrossRef] [PubMed]
Lindahl, E. , and Sansom, M. S. , 2008, “ Membrane Proteins: Molecular Dynamics Simulations,” Curr. Opin. Struct. Biol., 18(4), pp. 425–431. [CrossRef] [PubMed]
Ateshian, G. A. , and Weiss, J. A. , 2010, “ Anisotropic Hydraulic Permeability Under Finite Deformation,” ASME J. Biomech. Eng., 132(11), p. 111004. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2012, “ Solute Transport Across a Contact Interface in Deformable Porous Media,” J. Biomech., 45(6), pp. 1023–1027. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Contact force exerted by the rigid plane compressing the elastic sphere for (a) neo-Hookean (NH) and (b) Mooney–Rivlin (MR) materials, using linear or quadratic hexahedral (H8, H20) or tetrahedral (T4, T10) solid elements, optionally combined with linear (T3, Q4) or quadratic (T6, Q8) shell elements

Grahic Jump Location
Fig. 2

(a) Axisymmetric finite element model for indentation of spherical cell with rigid spherical AFM probe. (b) Interstitial fluid flux w in the cytoplasm at t = 0.48 s. Arrows display the direction and color contours display the magnitude of w.

Grahic Jump Location
Fig. 3

AFM indentation of chondrocyte: (a) The relative volume J, shown here at t = 60 s, is not uniform throughout the cytoplasm. (b) The AFM contact force predicted from the FEBio model (solid curve) can be successfully fitted to the experimental data of Darling et al. [32], with suitably chosen values for cytoplasmic Young's modulus and hydraulic permeability.

Grahic Jump Location
Fig. 4

(a) Spherically symmetric cell model consisting of two finite elements: a hex8 element to represent the cytoplasm and a quad4 element to represent the membrane. (b) The FEBio response for the membrane potential Δψ versus [Cl−]e agrees with the Nernst potential and experimental data from Tsuga et al. [69].

Grahic Jump Location
Fig. 5

(a) FEBio steady-state results for the cell volume ratio, Jcell = V/V r, plotted versus cer/ce, produce an identical straight line for osmotic loading by either NaCl or glucose, consistent with the Boyle–van't Hoff (Boyle) relation for an ideal osmometer. Model results agree with experimental data for osmotic loading of a chondrocyte with NaCl by Albro et al. [62] (Albro09). (b) Cell membrane potential Δψ versus time, for hyper-osmotic loading starting from homeostatic conditions (cer/ce = 1/2, 1≤t≤500  s) followed by unloading (cer/ce = 1, t > 500 s), using either NaCl or sucrose. Both osmolytes produce a Δψ response that achieves a steady-state value; steady-state Δψ remains unchanged for NaCl, but increases with hyper-osmotic loading with sucrose. (c) Equilibrium membrane potential Δψ over a broad range of external NaCl and sucrose osmolarities cer/ce; results for sucrose demonstrate increasing depolarization with increasing osmolarity ce.

Grahic Jump Location
Fig. 6

(a) Finite element mesh for idealized in situ model of chondrocyte surrounded by its PCM and ECM. (b)–(d) Variation in average electrical potential ψ and membrane potential Δψ, and intracellular and extracellular osmolarities and relative volume J, in response to substitution of [Cl]i with [InA−]i (0≤t<1), followed with ramping of PCM and ECM fixed-charge density from zero to final desired value (1≤t<11) and additional time to achieve steady-state (11<t≤41). Transport properties are artificially increased over this initial phase to accelerate the steady-state response.

Grahic Jump Location
Fig. 7

Transient response of idealized in situ model of chondrocyte to compressive loading by 30% (41≤t<241) followed by stress-relaxation (241≤t≤6000). (a)–(c) Relative volume J, net osmolarity, electrical potential, and membrane potential, averaged over the cell, PCM, and ECM. All compartments reduce in volume and increase in osmolarity due to fluid exudation; the cell is further depolarized with sustained compression.

Grahic Jump Location
Fig. 8

(a) Finite element mesh for image-based in situ model of chondrocytes surrounded by their PCM and ECM. (b) Transient response of the cell relative volumes J and membrane potential Δψ to compressive loading of the tissue by 30% (41≤t<241) followed by stress-relaxation (241≤t≤6000). The responses of the three cells in this image-based model (Jexp and Δψexp) are substantially similar to those of the idealized model (Jideal, Δψideal) in Fig. 7.



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