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TECHNICAL PAPERS

The Influence of the Fixed Negative Charges on Mechanical and Electrical Behaviors of Articular Cartilage Under Unconfined Compression

[+] Author and Article Information
D. D. Sun, X. E. Guo, M. Likhitpanichkul, W. M. Lai, V. C. Mow

Orthopaedic Research Laboratory + Bone Bioengineering Laboratory, Department of Biomedical Engineering, Orthopaedic Surgery and Mechanical Engineering, Columbia University

J Biomech Eng 126(1), 6-16 (Mar 09, 2004) (11 pages) doi:10.1115/1.1644562 History: Received June 20, 2002; Revised August 28, 2003; Online March 09, 2004
Copyright © 2004 by ASME
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Figures

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The schematic representation of the unconfined compression test configuration for stress relaxation; a Heaviside step-function displacement is imposed.
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Comparison of histories of the normalized radial displacement of the lateral edge of the cylindrical explant after an imposed step-vertical displacement. The solid line is the case with the initial FCD of 0.2 mEq/ml (i.e., the triphasic case); the dash line is the case with FCD=0 (i.e., the biphasic case); the magnitude of the suddenly applied 10% axial strain is ε0=ε(0+)=−0.1. The following numerical parameters were used in the calculations: the tissue radius a=1.5 mm and the height h=1.0 mm, the cation diffusivity D+=0.5×10−9 m2/s, the anion diffusivity D=0.8×10−9 m2/s, the external concentration c* =0.15 M, the shear modulus of the solid matrix μs=0.15 MPa, the porosity=0.75 and the drag coefficient α=0.7×1015 Ns/m4. Due to the osmotic (swelling) pressure, the lateral strain for a charged tissue recoils at a slower rate than that for an uncharged tissue for all intrinsic Poisson’s ratio νs of the solid matrix.
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Comparison of histories of the total compressive load on the loading platens of charged and uncharged tissues of the same intrinsic Young’s modulus but with three Poisson’s ratios (0.0, 0.2 and 0.5) following a sudden step application of compression. The solid line is the case with initial FCD of 0.2 mEq/ml (i.e., the triphasic case); the dash line is the case with FCD=0 (i.e., the biphasic case). Other parameter values are the same as in Fig. 2.
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(a) The fluid pressure history at the center of the tissue. The solid line is for the triphasic case with initial FCD of 0.2 mEq/ml; the dash line is for the biphasic case (FCD=0); the Poisson’s ratio νs=0.2. The intrinsic Young’s modulus=0.36 MPa. Other parameter values are the same as in Fig. 2 Here p0 is the pre-stress at the free swollen state due to the FCD (p0=0 in the uncharged case). With current parameter values, the pre-stress p0=0.148 MPa in the charged case. (b) The radial distribution of the fluid pressure inside the explant after the sudden application of loading. No distinction can be made on the distribution p−p0 inside the tissue between the charged and uncharged cases at t=0 sec(p−p0=0 in both charged and uncharged cases) and at t=0+ sec(p−p0=same constant in both charged and uncharged cases).
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The radial distribution of the FCD inside the tissue after application of a suddenly applied vertical load. The initial FCD is 0.2 mEq/ml, and the Poisson’s ratio νs=0.2. Other parameters are the same as in Fig. 2. The FCD is related to the dilatation of the tissue, e, as cF=c0F/(1+e/ϕ0w). Vertical compression would eventually increase the FCD inside the tissue explant in a uniform manner (as predicted in this homogeneous model).
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(a) The electrical potential time-history at the lateral edge inside the tissue. The external bathing solution is taken as the zero (reference) electrical potential. At any given instant, the electrical potential is not continuous across the lateral interface boundary between the external bathing solution and the tissue. (b) The radial distribution of the electrical potential inside the explant after the application of a suddenly applied step displacement. The initial FCD is 0.2 mEq/ml and the Poisson’s ratio νs=0.2. Other parameters are the same as in Fig. 2. A time step of 0.1 s is used to calculate the response below 1 s.
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(a) The electrical potential at the lateral edge and at the center of the tissue calculated at 10 s vs. the intrinsic Young’s modulus of the charged solid matrix. (b) The electrical potential difference between the center and the lateral edge inside the tissue and its streaming potential and diffusion potential components calculated at 10 s vs. the Young’s modulus of the solid matrix. The initial FCD of the solid matrix is 0.2 mEq/ml and the Poisson’s ratio of the solid matrix is 0.2. Other parameter values are the same as in Fig. 2. The electrical potential difference is the electrical potential at the center of the tissue minus the one at the lateral edge inside the tissue. This electrical potential difference is zero when Es=0.42 MPa.
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The electrochemical potential response on the Ag/AgCl electrode pairs located at the center and the lateral edge of the tissue sample. It is actually the difference of electrochemical potential of Cl at these two positions. The parameter values are the same as in Fig. 2.
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The apparent Poisson’s ratio of the tissue at equilibrium vs. the intrinsic Poisson’s ratio of the solid matrix with the FCD as a parameter. The intrinsic shear modulus is fixed as 0.15 MPa, other parameter values are the same as in Fig. 2. Due to the increased osmotic pressure, the charged tissue has larger lateral expansion than the non-charged tissue whose equilibrium lateral deformation is determined only by the intrinsic Poisson’s ratio of the solid matrix.
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The apparent Young’s modulus of the charged and uncharged tissues at equilibrium vs. the intrinsic Young’s modulus of the solid matrix with the same intrinsic Poisson’s ratio. For the uncharged case, the apparent Young’s modulus is the same as the intrinsic Young’s modulus of the solid matrix and is independent of the intrinsic Poisson’s ratio of the solid matrix. Due to the increased osmotic pressure, the charged tissue has larger apparent Young’s modulus than the uncharged tissue, and the enhancement of the apparent Young’s modulus is more remarkable at the low value of the intrinsic Poisson’s ratio of the solid matrix, i.e., greater apparent compressibility of the charged matrix.
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The potential response (calculated at 1.0 s) and the maximum potential value during the time course vs. the initial FCD of the tissue with the Young’s modulus of the solid matrix as a parameter. The potential difference is from the Ag/AgCl electrode pairs located at the center and the lateral edge of the tissue sample. The Poisson’s ratio of the solid matrix is set as 0.2. Other parameter values are the same as in Fig. 2.

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