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Research Papers

Simulating the Growth of Articular Cartilage Explants in a Permeation Bioreactor to Aid in Experimental Protocol Design

[+] Author and Article Information
Timothy P. Ficklin, Andrew Davol

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

Stephen M. Klisch1

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

1

Corresponding author.

J Biomech Eng 131(4), 041008 (Feb 03, 2009) (11 pages) doi:10.1115/1.3049856 History: Received February 22, 2008; Revised October 01, 2008; Published February 03, 2009

Recently a cartilage growth finite element model (CGFEM) was developed to solve nonhomogeneous and time-dependent growth boundary-value problems (Davol, 2008, “A Nonlinear Finite Element Model of Cartilage Growth,” Biomech. Model. Mechanobiol., 7, pp. 295–307). The CGFEM allows distinct stress constitutive equations and growth laws for the major components of the solid matrix, collagens and proteoglycans. The objective of the current work was to simulate in vitro growth of articular cartilage explants in a steady-state permeation bioreactor in order to obtain results that aid experimental design. The steady-state permeation protocol induces different types of mechanical stimuli. When the specimen is initially homogeneous, it directly induces homogeneous permeation velocities and indirectly induces nonhomogeneous solid matrix shear stresses; consequently, the steady-state permeation protocol is a good candidate for exploring two competing hypotheses for the growth laws. The analysis protocols were implemented through the alternating interaction of the two CGFEM components: poroelastic finite element analysis (FEA) using ABAQUS and a finite element growth routine using MATLAB . The CGFEM simulated 12 days of growth for immature bovine articular cartilage explants subjected to two competing hypotheses for the growth laws: one that is triggered by permeation velocity and the other by maximum shear stress. The results provide predictions for geometric, biomechanical, and biochemical parameters of grown tissue specimens that may be experimentally measured and, consequently, suggest key biomechanical measures to analyze as pilot experiments are performed. The combined approach of CGFEM analysis and pilot experiments may lead to the refinement of actual experimental protocols and a better understanding of in vitro growth of articular cartilage.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 12

Contours of the final COL volume fraction for the shear growth trigger normalized to the initial volume fraction and plotted on the released specimen. The average dimension of the elements are 25.33×63.85 μm2. The curvature deformation is scaled by a factor of 5.

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Figure 11

Contours of the final PG volume fraction for the shear growth trigger normalized to the initial volume fraction and plotted on the released specimen. The average dimension of the elements are 25.33×63.85 μm2. The curvature deformation is scaled by a factor of 5.

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Figure 10

GAG COL and masses normalized to WWI to represent content and WWF to represent concentration for permeation and shear growth triggers. Shear values are averaged throughout the entire depth while shear-S, shear-M, and shear-D values are averaged throughout the supply side, middle, and drain side slices, respectively.

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Figure 9

Geometric changes for permeation and shear growth triggers on day 12 relative to day 0 values: height, diameter, and edge lift

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Figure 8

Confined compression properties on day 0 and day 12 for permeation and shear growth triggers. (a) Aggregate modulus (HA) at 15%, 30%, and 45% strains. (b) Initial permeability coefficients at 0% strain (k0). Shear values are averaged throughout the entire depth while shear-S, shear-M, and shear-D values are averaged throughout the supply side, middle, and drain side slices, respectively.

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Figure 7

Evolution of pore pressure and Tresca stress profiles for steady-state permeation resulting from growth with the shear stress trigger. (a) Total pore pressure drop and gradient increase during growth. (b) Tresca shear stress profiles do not change substantially during growth and the trigger value is exceeded only in the three rows of elements on the drain side.

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Figure 6

After growth and CC testing, the specimen is (a) released from the CC chamber for geometric measurements and (b) cut into three slices (S, M, and D) of equal thickness for biochemical testing. The FEA analysis models these steps by changing the boundary conditions on the specimen and allowing the model to reach equilibrium in step (a) and the results are postprocessed in MATLAB to obtain GAG and COL masses and tissue WW for each of the three slices in step (b).

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Figure 5

After growth in the permeation bioreactor, the grown cartilage disk is (a) pushed from the fluid permeation (FP) chamber to the CC chamber, (b) measured for a new equilibrium height, and (c) tested in CC. The FEA analysis models these steps by changing the boundary conditions on the specimen and allowing the model to reach equilibrium following axial expansion in step (b) and application of confined compression strains in step (c).

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Figure 4

The meshed quarter-disk model and the displacement boundary conditions. The dimensions of each element are 24.17×61.54 μm2. Symmetry boundary conditions were applied on the planes of symmetry, the outer edge of the disk was fixed in the radial direction, and the fluid drain side surface was fixed in the axial (i.e., z or three) direction. These displacement boundary conditions were applied using cylindrical coordinates (with the z and three axes coincident) in the initial compression, permeation, chamber transfer, and confined compression analyses and then removed in the release analysis.

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Figure 3

(a) The cartilage disk is precompressed between porous platens as it is loaded into the permeation bioreactor and (b) grown under steady-state permeation loading. The permeation velocity across the specimen is prescribed, the supply side pressure is measured, and the drain side is maintained at atmospheric pressure. If the specimen is initially homogeneous, this protocol directly induces homogeneous permeation velocities and indirectly induces nonhomogenous SM shear stresses.

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Figure 2

The total specimen growth BVP for one increment (n) of growth in the CGFEM. Each SM finite element from the total specimen FEM is (i) unloaded from κnR to κnr by relieving residual stress, (ii) grown from κnr to κng, (iii) deformed back into the pregrowth compatible configuration κnR, and (iv) loaded via residual stress to the equilibrium configuration κnG of the total specimen FEM after growth.

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Figure 1

The kinematics of the incremental growth BVP for a solid matrix (SM) element. The total deformation gradient tensor F maps a SM element from a SM stress-free reference configuration κr to a SM stress-free grown configuration κg. The growth tensors (Fgp,Fgc) describe mass addition (or removal) at constant constituent stress and density. The elastic growth tensors (Fegp,Fegc) describe the stress- and density-changing deformation needed to maintain the SM immobility constraint F=FegpFgp=FegcFgc.

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