Biphasic Finite Element Modeling Reconciles Mechanical Properties of Tissue-Engineered Cartilage Constructs Across Testing Platforms

Abstract

Cartilage tissue engineering is emerging as a promising treatment for osteoarthritis, and the field has progressed toward utilizing large animal models for proof of concept and preclinical studies. Mechanical testing of the regenerative tissue is an essential outcome for functional evaluation. However, testing modalities and constitutive frameworks used to evaluate in vitro grown samples differ substantially from those used to evaluate in vivo derived samples. To address this, we developed finite element (FE) models (using FEBio) of unconfined compression and indentation testing, modalities commonly used for such samples. We determined the model sensitivity to tissue radius and subchondral bone modulus, as well as its ability to estimate material parameters using the built-in parameter optimization tool in FEBio. We then sequentially tested agarose gels of 4%, 6%, 8%, and 10% weight/weight using a custom indentation platform, followed by unconfined compression. Similarly, we evaluated the ability of the model to generate material parameters for living constructs by evaluating engineered cartilage. Juvenile bovine mesenchymal stem cells were seeded (2 × 10⁷ cells/mL) in 1% weight/volume hyaluronic acid hydrogels and cultured in a chondrogenic medium for 3, 6, and 9 weeks. Samples were planed and tested sequentially in indentation and unconfined compression. The model successfully completed parameter optimization routines for each testing modality for both acellular and cell-based constructs. Traditional outcome measures and the FE-derived outcomes showed significant changes in material properties during the maturation of engineered cartilage tissue, capturing dynamic changes in functional tissue mechanics. These outcomes were significantly correlated with one another, establishing this FE modeling approach as a singular method for the evaluation of functional engineered and native tissue regeneration, both in vitro and in vivo.

Keywords: cartilage tissue engineering; mechanical evaluation; mesenchymal stem cells.

FIG. 1.

(A) Schematic of indentation testing setup. (B) Indentation test of a 4 mm diameter artificial chondral defect repaired with a hydrogel. (C) Front view of the 5° quasi-axisymmetric FEBio model of an indentation test. Color images available online at www.liebertpub.com/tea

FIG. 2.

(Left) FEBio model of (A) indentation and (B) unconfined compression tests showing peak fluid pressure distribution, (center) experimental testing protocols, and (right) representative curve fit of experimental data (second step shown for indentation test). Color images available online at www.liebertpub.com/tea

FIG. 3.

Sensitivity analysis for an FE model of indentation of a cartilage disk with height of 1 mm and radii of 2, 3, 4, or 5 mm using a spherical indenter with diameter of 2 mm at a rate of 0.1%/s. (A) Fluid pressure as a function of time and distance from the center as the radius of tissue increases from 2 to 5 mm. (B) Reaction force (R_Z) for 2 through 5 mm radius cartilage cylinders over three stress-relaxation steps. Note the near-identical force response at 2, 3, 4, and 5 mm radii. (C) Peak reaction force of the model as a function of sample radius. FE, finite element. Color images available online at www.liebertpub.com/tea

FIG. 4.

Peak fluid pressure distribution for (A) an osteochondral sample without bone and (B) an osteochondral sample with subchondral bone modeled. (C) Reaction force (R_Z) during two steps of stress relaxation indentation as a function subchondral bone moduli. (D) Change in peak reaction force for a range of subchondral bone moduli. Color images available online at www.liebertpub.com/tea

FIG. 5.

Experimental reaction force (R_Z) from indentation testing of a 4 mm diameter chondral plug from the trochlear groove of a juvenile bovine knee. Indentation tests were performed at rates of (A) 0.05%/s, (B) 0.25%/s, (C) 0.50%/s, and (D) 1.00%/s on the same sample. Results from the FEBio parameter optimization at each loading rate showed that (E) the estimated Young's modulus was lower for fits to experimental data acquired at 1.00%/s, while (F) the value of ksi increased for fits to data acquired at 0.05%/s. The permeability (G) was the most variable parameter. (H) Calculation of % error showed that error was reduced when optimizations were performed on the second or third step of the indentation test.

FIG. 6.

(A) Example curve fit from native porcine samples and the (B) Young's modulus, (C) ksi, and (D) permeability of those samples (n = 6) outputted from the model. Color images available online at www.liebertpub.com/tea

FIG. 7.

Maturation of engineered cartilage grown in vitro for up to 9 weeks. (A) Histological analysis (Alcian blue [left] and Picrosirius red [right]) as a function of time in culture (scale bar = 1 mm). (B) GAG and (C) collagen content as well as (D) equilibrium and dynamic modulus (E) derived from unconfined compression testing increased with duration of culture (*p < 0.05; **p < 0.01; ***p < 0.001; n = 10). Scatter plots showing the two individual studies are provided in Supplementary Figure S1. GAG, glycosaminoglycan. Color images available online at www.liebertpub.com/tea

FIG. 8.

Mechanical properties as determined from FE parameter optimization of indentation testing (white bars) and unconfined compression (black bars). Young's modulus, ksi, and permeability were computed for paired testing of (A) acellular agarose hydrogels of 4%, 6%, 8%, and 10% weight/weight (n = 6) and (B) engineered cartilage constructs grown in vitro for 3, 6, and 9 weeks (n = 10). *p < 0.05; **p < 0.01; ***p < 0.001. Scatter plots for agarose hydrogels are shown in Supplementary Figure S2 and scatter plots showing the two individual engineered cartilage studies are shown in Supplementary Figure S3.

FIG. 9.

FEBio optimization results from unconfined compression (FEU) versus results from indentation testing (FEI) on paired (A) acellular agarose hydrogels and (B) engineered cartilage constructs after varying periods of in vitro culture. Constitutive model parameters for Young's modulus (E_Y), ksi, and permeability significantly correlated between the two testing modalities, although greater variation was observed in the living engineered constructs. The regression lines between sample types were not significantly different. Color images available online at www.liebertpub.com/tea