Unconfined Compression of Articular Cartilage: Nonlinear Behavior and Comparison With a Fibril-Reinforced Biphasic Model

[+] Author and Article Information
M. Fortin

Institute of Biomedical Engineering, Ecole Polytechnique, Montreal, Quebec, Canada

J. Soulhat, A. Shirazi-Adl

Department of Mechanical Engineering, Ecole Polytechnique, Montreal, Quebec, Canada

E. B. Hunziker

ME Müller Institute for Biomechanics, University of Bern, Bern, Switzerland

M. D. Buschmann

Department of Chemical Engineering, Institute of Biomedical Engineering, Ecole Polytechnique, Montreal, Quebec, Canadae-mail: mike@grbb.polymtl.ca

J Biomech Eng 122(2), 189-195 (Oct 18, 1999) (7 pages) doi:10.1115/1.429641 History: Received October 01, 1998; Revised October 18, 1999
Copyright © 2000 by ASME
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Load and position as a function of time for a series of ramp displacements for one of the samples
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Comparison of the fit of a composite biphasic model and the corresponding experimental data for two disks undergoing a 5 s ramp displacement of 5 μm amplitude at 4 and 9 percent compression offset. The matrix Poisson’s ratio νm=0 for all fits. With the 0.0001–0.01 Hz fitting interval, the model parameter values for the first sample were: Em(matrix modulus)=0.73 MPa,Ef(fibril network modulus)=5.7 MPa and k(matrix permeability)=3.4×10−15 m4/(N-s) at 4 percent compression offset (A) and Em=0.68 MPa,Ef=9.1 MPa and k=1.2×10−15 m4/(N-s) at 9 percent compression offset (C). For the second sample, the parameter values were: Em(matrix modulus)=0.65 MPa,Ef(fibril network modulus)=6.6 MPa , and k(matrix permeability)=3.2×10−15 m4/(N-s) at 4 percent compression offset (B) and Em=0.66 MPa,Ef=18.7 MPa, and k=0.63×10−15 m4/(N-s) at 9 percent compression offset (D).
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Ef, fibril network modulus (A), Em, the matrix modulus (A) and k, the permeability (B) versus axial compression offset strain at the beginning of each step assuming matrix Poisson’s ratio νm=0, as obtained by fitting the linear fibril-reinforced isotropic biphasic model to each of the stress relaxation profiles (examples in Fig. 2). The mean and standard error are shown.
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Dynamic stiffness (normalized to Em for each disk), phase and total harmonic distortion for five disks (mean and standard error) tested with sinusoids of 5, 10, and 15 μm amplitude at frequencies from 0.001 to 1 Hz. MANOVA performed with frequencies of 0.833 and 0.01 Hz showed that dynamic stiffness amplitude and phase and total harmonic distortion differed with amplitude (p=0.07, 0.06, and 0.03, respectively).
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Stress relaxation responses to ramp compression and ramp release displacements of 5, 10, and 15 μm amplitude and constant ramp time of 5 s imposed at a 10 percent strain offset. The stress relaxation profiles minus the equilibrium value at 10 percent offset were normalized to the equilibrium stress increment Emε0 where ε0 was the imposed surface-to-surface strain. The mean values and standard errors from five disks are shown. The response to release is smaller than the response to compression (p=0.01, 0.015, and 0.021 for amplitudes of 5, 10, and 15 μm).
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Data of Fig. 5 grouped into release and compression tests to investigate differences within each group. The response to compression was similar for the three amplitudes (p=0.29) but differed with amplitude for release (p=0.07).
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Dynamic stiffness (normalized to Em) and phase obtained directly from sinusoidal tests, from time-domain tests followed by numerical conversion of stress-relaxation 20 and from a fibril-reinforced biphasic model. The model parameters used were those obtained by fitting the data from stress-relaxation curves: νm(matrix Poisson’s ratio)=0,Em(matrix modulus)=0.54 MPa,Ef(fibers modulus)=10.4 MPa, and k(matrix permeability)=0.97×10−15 m4/(N-s). The data from sinusoidal tests is different from the one from time domain tests for the stiffness amplitude (p=0.0014) but it is equal for the phase (p=0.42).
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Normalized system function in the Laplace domain for five disks compressed by a 5 μm ramp achieved with 1 μm/s at a 10 percent strain offset. The same disks were punched to diameters of 3.6, 2.7, and 1.7 mm and tested at each radius. The data is shown both before and after rescaling the frequency according to frequency(rescaled)=frequency(original)×(diameter/3.6 mm)2. Biphasic and poroelastic models predict perfect superposition of rescaled curves in the lower panel. The curves of the upper panel are statistically different (p=0.0021) but they become statistically indifferent (p=0.27) after rescaling.
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Direct sinusoidal tests of the stiffness function before (A, C) and after (B, D) frequency rescaling described in the caption to Fig. 8. Current models predict superposition of curves in panels B and D. The curves of panels A and C are statistically different (p=0.0027 and 0.082) but they become statistically indifferent after rescaling of the frequency axis (p=0.53 and 0.40).




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