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Research Papers

# A Cartilage Growth Mixture Model With Collagen Remodeling: Validation Protocols

[+] Author and Article Information
Stephen M. Klisch1

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

Anna Asanbaeva, Robert L. Sah

Department of Bioengineering, and Whitaker Institute of Biomedical Engineering, University of California-San Diego, La Jolla, CA 92093

Sevan R. Oungoulian

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

Koichi Masuda

Departments of Orthopedic Surgery and Biochemistry, Rush University Medical Center, Chicago, IL 60612

Eugene J.-MA. Thonar

Departments of Orthopedic Surgery and Biochemistry, and Department of Internal Medicine, Rush University Medical Center, Chicago, IL 60612

Andrew Davol

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

In Ref. 16, an additional constituent representing the other noncollagenous proteins, which is assumed to not directly affect the mechanical properties of the SM, is included.

The fluid stress at equilibrium is assumed to be zero everywhere due to homogeneity assumptions.

For exact and computational solutions of nonhomogeneous problems, the SM element would correspond to a virtual configuration and a finite element, respectively, and additional elastic compatibility deformations would be required. See Ref. 18 for details.

The introduction of $Fr$ is similar to previous tensorial descriptions of tissue remodeling (38,51). In nonhomogeneous problems, $Fr$ may differ between adjacent elements and a compatibility deformation may be introduced to ensure continuity of the SM.

The effect of anisotropic growth is presented in the discussion and has been studied in Refs. 14,16.

In Ref. 14, it is shown how these growth laws can be used to determine the “incremental growth tensor” using a first order Taylor series approximation.

Calculations show that tissue density changes by less than 2% during these growth protocols.

The reader is referred to Ref. 54 for full details.

1

Corresponding author.

J Biomech Eng 130(3), 031006 (Apr 25, 2008) (11 pages) doi:10.1115/1.2907754 History: Received January 22, 2007; Revised December 12, 2007; Published April 25, 2008

## Abstract

A cartilage growth mixture (CGM) model is proposed to address limitations of a model used in a previous study. New stress constitutive equations for the solid matrix are derived and collagen (COL) remodeling is incorporated into the CGM model by allowing the intrinsic COL material constants to evolve during growth. An analytical validation protocol based on experimental data from a recent in vitro growth study is developed. Available data included measurements of tissue volume, biochemical composition, and tensile modulus for bovine calf articular cartilage (AC) explants harvested at three depths and incubated for $13days$ in 20% fetal borine serum (FBS) and 20% $FBS+β$-aminopropionitrile. The proposed CGM model can match tissue biochemical content and volume exactly while predicting theoretical values of tensile moduli that do not significantly differ from experimental values. Also, theoretical values of a scalar COL remodeling factor are positively correlated with COL cross-link content, and mass growth functions are positively correlated with cell density. The results suggest that the CGM model may help us to guide in vitro growth protocols for AC tissue via the a priori prediction of geometric and biomechanical properties.

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## Figures

Figure 3

The coordinate system and experimental specimen orientations in relation to anatomical directions. The unit vector E1 is parallel to the local split-line direction, the unit vector E3 is perpendicular to the articular surface, and the unit vector E2 is perpendicular to the split-line direction and parallel to the surface. The rectangular slices labeled S, M1, and M2 represent ∼0.4mm, 0.25mm, and 0.25mm thick specimens used in the control and experimental groups.

Figure 4

Steps in the analysis procedure to model the in vitro growth protocols. Step 1 defines the reference configuration by determining the SM material constants and the COL swelling strain F0c. Step 2 determines the SM deformation gradient tensor due to mass deposition Fm. Steps 3 and 4 determine the SM deformation gradient tensor due to COL remodeling Fr needed to match final tissue volume. Step 5 determines the mechanical properties relative to the grown configuration by solving boundary-value problems defined by the applied SM deformation gradient tensor Fl.

Figure 8

Linear regression analysis of the theoretical remodeling factor (χ) versus cross-link content (PYR/WWF). Left: trend is not significant when all groups are considered (p=0.20, n=6). Right: trend is significant when only S and M1 layers are considered (p<0.0001, n=4).

Figure 9

Linear regression analysis of the calculated mass growth functions cp and cc (% mass increase per day) versus cell density (number of cells normalized by initial tissue WW). Left: trend for cp is significant (p<0.001, n=6). Right: trend for cc is not significant (p=0.22, n=6).

Figure 10

Description of the mechanical response of the M2 control group predicted by the PG-COL stress equations. Left: Cauchy stress versus Biot strain for uniaxial tension (UT) and unconfined compression (UCC). Right: Poisson’s ratios (νij) at 15% strain for UT and UCC; i=loading direction, j=direction of transverse strain component.

Figure 11

Description of stress softening behavior in UCC for the M2 control group predicted by the PG-COL stress equations. Left: Cauchy stress versus Biot strain for UT and UCC. Right: secant UCC modulus versus UCC Biot strain.

Figure 7

Experimental and theoretical values of secant tensile modulus at 20% strain in the 2-direction. Experimental values measured before (D0-exp.) and after (D13-exp.) growth for three layers (S, M1, and M2) and two growth medium types (FBS and BAPN); error bars represent ±1 standard deviations. Theoretical values represent predictions of D13 values using the CGM model. No significant difference is detected between the theoretical and experimental D13 values using a paired t-test (p=0.40, n=6)

Figure 6

Determinants of the overall SM deformation gradient tensor (J) calculated from experimental measurements of tissue WW, and determinants of SM deformation gradient tensors due to mass deposition (Jm) and COL remodeling (Jr) predicted from the CGM model. Jr>Jm (p>0.05, n=6)

Figure 5

Determinants of the constituent growth tensors Jgp and Jgc calculated from experimental measurements of PG (p) and COL (c) masses: Jgp>Jgc (p<0.001, n=6)

Figure 2

Kinematics of growth and remodeling for a homogeneous stress-free solid matrix (SM) element. Fm is the SM deformation gradient tensor due to mass deposition. The growth tensors Fgp and Fgc describe differential mass deposition of the PG (p) and COL (c) constituents. The elastic growth tensors Fegp and Fegc ensure continuity of the SM during mass deposition. Fr is the SM deformation gradient tensor due to COL remodeling.

Figure 1

The major components of the articular cartilage solid matrix: proteoglycans (PG), collagen (COL), and pyridinoline (PYR) cross-links

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