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Research Papers

Multiphasic Finite Element Framework for Modeling Hydrated Mixtures With Multiple Neutral and Charged Solutes

[+] Author and Article Information
Gerard A. Ateshian

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

Jeffrey A. Weiss

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

A steady-state analysis is performed by setting time derivatives to zero in Eq. (2.15). This option is available in febio by setting a switch in the input file.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received October 17, 2012; final manuscript received June 2, 2013; accepted manuscript posted June 17, 2013; published online September 23, 2013. Assoc. Editor: Mohammad Mofrad.

J Biomech Eng 135(11), 111001 (Sep 23, 2013) (11 pages) Paper No: BIO-12-1493; doi: 10.1115/1.4024823 History: Received October 17, 2012; Revised June 02, 2013; Accepted June 17, 2013

Computational tools are often needed to model the complex behavior of biological tissues and cells when they are represented as mixtures of multiple neutral or charged constituents. This study presents the formulation of a finite element modeling framework for describing multiphasic materials in the open-source finite element software febio.1 Multiphasic materials may consist of a charged porous solid matrix, a solvent, and any number of neutral or charged solutes. This formulation proposes novel approaches for addressing several challenges posed by the finite element analysis of such complex materials: The exclusion of solutes from a fraction of the pore space due to steric volume and short-range electrostatic effects is modeled by a solubility factor, whose dependence on solid matrix deformation and solute concentrations may be described by user-defined constitutive relations. These solute exclusion mechanisms combine with long-range electrostatic interactions into a partition coefficient for each solute whose value is dependent upon the evaluation of the electric potential from the electroneutrality condition. It is shown that this electroneutrality condition reduces to a polynomial equation with only one valid root for the electric potential, regardless of the number and valence of charged solutes in the mixture. The equation of charge conservation is enforced as a constraint within the equation of mass balance for each solute, producing a natural boundary condition for solute fluxes that facilitates the prescription of electric current density on a boundary. It is also shown that electrical grounding is necessary to produce numerical stability in analyses where all the boundaries of a multiphasic material are impermeable to ions. Several verification problems are presented that demonstrate the ability of the code to reproduce known or newly derived solutions: (1) the Kedem–Katchalsky model for osmotic loading of a cell; (2) Donnan osmotic swelling of a charged hydrated tissue; and (3) current flow in an electrolyte. Furthermore, the code is used to generate novel theoretical predictions of known experimental findings in biological tissues: (1) current-generated stress in articular cartilage and (2) the influence of salt cation charge number on the cartilage creep response. This generalized finite element framework for multiphasic materials makes it possible to model the mechanoelectrochemical behavior of biological tissues and cells and sets the stage for the future analysis of reactive mixtures to account for growth and remodeling.

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Figures

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Fig. 1

Basic algorithmic scheme for finite element implementation of multiphasic materials

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Fig. 2

Cell volume response to osmotic loading: V is the current cell volume and Vr is its volume in the reference configuration. Black curves are the responses from the K–K model and gray curves are the finite element solutions. For the finite element model, the membrane thickness is 10 nm and its properties are ϕrs=0.3, μ=2.5kPa, κ=2.5MPa, k=5×10-7μm4/nN·s, d0p=1.43×10-3μm2/s, and κ∧p=1; the protoplasm has a radius of 10μm, and its properties are ϕrs=0.3, EY=2.5kPa, ν=0, k=5×10-3μm4/nN·s, d0p=d0n=14.3μm2/s, and κ∧p values are indicated on the graph. For the K–K model, the equivalent membrane properties are reported in Ref. [22]. For these analyses, c0=0.3M and c1=1M (see text).

Grahic Jump Location
Fig. 3

Donnan osmotic swelling of a triphasic material (ϕrs=0.2) with a neo-Hookean solid (λs=0, μs=0.25 MPa). The relative volume J=λ3 is plotted for four different values of crF (ranging from =100 to =400 mEq/L by increments of =100 mEq/L) over a range of values for the bath concentration c* (θ = 293 K). The numerical solution to Eq. (3.1) is given by the thin black curves, and the finite element solution is represented by the thicker gray curves.

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Fig. 4

Schematic for current flow in an electrolyte

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Fig. 5

One-dimensional current flow in NaCl. The concentration of NaCl along the entire length of the domain is provided at selected time points. Black curves represent the analytical solution and gray curves represent the finite element solution.

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Fig. 6

Schematic for configuration of current-generated stress analysis in cartilage

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Fig. 7

Current-generated stress. (a) Stress response σzz as a function of time, inclusive of the tare stress resulting from the initial Donnan osmotic pressure. (b) Concentrations c+ of Na+ and c of Cl along the axis z of the tissue at selected time points t (t = 0 s corresponds to the state immediately prior to current application; remaining time points increase from 100 s to 104 s by decade).

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Fig. 8

Effect of cation charge on the creep response of cartilage: The compressive engineering strain is reported for three solutions corresponding to a monovalent (NaCl), divalent (CaCl2), and trivalent (AlCl3) cation

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