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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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Figures

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. 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. 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. 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. 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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