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# A Nonlinear Constituent Based Viscoelastic Model for Articular Cartilage and Analysis of Tissue Remodeling Due to Altered Glycosaminoglycan-Collagen Interactions

[+] Author Affiliations
Gregory C. Thomas

Department of Mechanical Engineering, California Polytechnic State University, San Luis Obispo, CA 93407

Anna Asanbaeva, Robert L. Sah

Department of Bioengineering, University of California-San Diego, La Jolla, CA 92093

Pasquale Vena

Department of Structural Engineering, Laboratory of Biological Structure Mechanics, Politecnico di Milano, 20133, Milan, Italy

Stephen M. Klisch1

Department of Mechanical Engineering, California Polytechnic State University, San Luis Obispo, CA 93407, sklisch@calpoly.edu

These differences are discussed in more detail in Secs. 2,4.

These assumptions are discussed further in Sec. 4.

In Eq. 4, stresses are VE stresses; a superscript $e$ will be used to designate elastic stresses.

Equation numbers preceded by “A” and “B” refer to those in Appendices .

VE models usually include a term for strain rate (i.e., $dE/dt$) inside the convolution integral to account for the dependence on strain rate. As stress is proportional to strain, the dependence on strain rate can also be achieved with a stress rate term $dS/dt$(23).

Note that as $t→∞$, $S(t)$ is the equilibrium (elastic) stress $Se(0)+ΔSe$.

Note that relative to a zero stress-zero strain configuration, $R$ defined by Eq. 19 is strain independent, which is also a characteristic of quasilinear viscoelasticity.

The derivation is nearly identical to that in Ref. 23 and the reader is referred to that paper for full details.

This assumption is discussed further in Sec. 4.

Equation 25 defines a viscous material constant that is independent of strain, decoupling elastic and viscous properties and is preferable to Eq. 24, which couples elastic and viscous properties, for statistical analysis of COL viscous properties.

For GD-85 specimens, the COL VE parameters ($τ1COL$, $g1COL$) were independent of initial guesses.

Also, our own pilot simulations using a poroviscoelastic model (results of which are not presented) justified the assumption that fluid flow-dependent viscoelasticity can be neglected for out UT protocols.

It can be seen from Eq. 8 that when a COL fiber direction is not stretched in tension, it is “turned off” and does not contribute to $SeBIM$.

1

Corresponding author.

J Biomech Eng 131(10), 101002 (Sep 01, 2009) (11 pages) doi:10.1115/1.3192139 History: Received November 15, 2008; Revised June 03, 2009; Published September 01, 2009

## abstract

A constituent based nonlinear viscoelastic (VE) model was modified from a previous study (Vena, , 2006, “A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments  ,” J. Biomech. Eng., 128, pp. 449–457) to incorporate a glycosaminoglycan (GAG)-collagen (COL) stress balance using compressible elastic stress constitutive equations specific to articular cartilage (AC). For uniaxial loading of a mixture of quasilinear VE constituents, time constant and relaxation ratio equations are derived to highlight how a mixture of constituents with distinct quasilinear VE properties is one mechanism that produces a nonlinear VE tissue. Uniaxial tension experiments were performed with newborn bovine AC specimens before and after ∼55% and ∼85% GAG depletion treatment with guanidine. Experimental tissue VE parameters were calculated directly from stress relaxation data, while intrinsic COL VE parameters were calculated by curve fitting the data with the nonlinear VE model with intrinsic GAG viscoelasticity neglected. Select tissue and intrinsic COL VE parameters were significantly different from control and experimental groups and correlated with GAG content, suggesting that GAG-COL interactions exist to modulate tissue and COL mechanical properties. Comparison of the results from this and other studies that subjected more mature AC tissue to GAG depletion treatment suggests that the GAGs interact with the COL network in a manner that may be beneficial for rapid volumetric expansion during developmental growth while protecting cells from excessive matrix strains. Furthermore, the underlying GAG-COL interactions appear to diminish as the tissue matures, indicating a distinctive remodeling response during developmental growth.

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## Figures

Figure 1

The stress balance hypothesis. The GAG and COL constituents have their own reference configurations, κ0GAG and κ0COL. The GAG constituent has the same configuration in κ0P and κ0, which is equivalent to F0GAG=I and SeGAG(0)=−α1I. To balance this swelling stress, the COL network supports a tensile prestress produced by an initial COL deformation gradient tensor F0COL, which yields the initial collagen stress tensor SeCOL(0)=α1I. After a deformation F is applied, the constituent stresses are calculated relative to their respective reference configurations and using FGAG=FF0GAG and FCOL=FF0COL.

Figure 2

Stress response to a step increase in strain and, consequently, a step increase in elastic stress ΔSe at time tstep>0 with initial equilibrium elastic stress Se(0)

Figure 3

The effects of GAG depletion treatment on experimental tissue VE parameters (Eexp, τexp, and Rexp): mean+1 standard deviation values shown; GD-0 is the control group with no GAG depletion; GD-55 and GD-85 are experimental groups with ∼55% and 85% GAG depletion, respectively;  ∗ is the significant difference between experimental and control group values and  ∗∗ is the significant difference between experimental group values (ANOVA with posthoc Tukey testing p<0.05)

Figure 4

Experimental tissue VE parameters (Eexp, τexp, and Rexp) versus GAG and COL contents: GD-0 is the control group with no GAG depletion; GD-55 and GD-85 are experimental groups with ∼55% and 85% GAG depletion, respectively. The linear regression results were only shown for significant correlations (t-test analysis of regression slope, p<0.05).

Figure 5

Specimen-specific curve-fits of the constituent based VE model: GD-0 is the group with no GAG depletion; GD-55 and GD-85 are experimental groups with ∼55% and 85% GAG depletion, respectively.

Figure 6

The effects of GAG depletion treatment on COL VE parameters (γ1, τ1COL, g1COL, τ2COL, g2COL, τACOL, and RACOL); mean+1 standard deviation values were shown; GD-0 is the control group with no GAG depletion; GD-55 and GD-85 are experimental groups with ∼55% and 85% GAG depletion, respectively; τ2COL and g2COL values were not reported for GD-85 specimens because only one Prony series term was used in the COL relaxation function for that group. For γ1, τACOL, and RACOL:  ∗ is the significant difference between experimental and control group values and  ∗∗ is the significant difference between experimental group values (ANOVA with posthoc Tukey testing p<0.05). For τ1COL, g1COL, τ2COL, and g2COL:  ∗ is the significant difference between GD-0 and GD-55 group values (paired t-test p<0.05).

Figure 7

COL VE parameters (γ1, τ1COL, g1COL, τ2COL, g2COL, τACOL, and RACOL) versus GAG and COL contents: GD-0 is the control group with no GAG depletion; GD-55 and GD-85 are experimental groups with ∼55% and 85% GAG depletion, respectively. The linear regression results were only shown for significant correlations (t-test analysis of regression slope p<0.05).

Figure 8

The parameter study results were shown to differentiate between the different mechanisms arising from GAG-COL interactions on the experimental tissue VE response: absolute (top) and normalized (bottom) stress results for UT simulations with progressive changes in untreated (GD-0) group parameters due to treatment (GD-55, GD-85); parameters used are listed in Table 1. The stress-time curves from the first three simulations for each treatment were indistinguishable in the normalized stress plots. GAG and COL elastic parameters modulate the peak and equilibrium stresses (top) but not the relaxation behavior (bottom). COL viscous parameters modulate peak and equilibrium stresses (top) and relaxation behavior (bottom).

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