A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

Abstract

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.

Fig. 1

(a) Vertical displacement applied to the indenter. (b) Reaction force measured on the indenter.

Fig. 2

(a) Approximation of the articular cartilage layer with one and two embedded families of fibers in the superficial and middle zones respectively. θ and κ characterize the mean orientations and the dispersion of the collagen fibers. (b) Example of undeformed finite element model of axisymmetric cartilage indentation. The indenter radius r’ = 0.25 mm, the cartilage radius R = 6 mm, and the thickness h varied in the range 0.41–0.6 mm. The heavy line indicates the boundary between the mesh for the superficial and middle zones.

Fig. 3

Measured and FE optimized reaction force using the BFPHVE model for one of the five samples considered in the study. Err1 = 1.2%, Err2 = 1.1%.

Fig. 4

Comparison of measured force with that predicted by the BFPHVE model: (a) only considering Err1 = 1.3% [Eq. (7)]; (b) only considering Err2 = 1.3% [Eq. (8)]

Fig. 5

Comparison of predicted cartilage response for nonlinear biphasic fiber-reinforced viscohyperelastic model (i.e., without using fiber reorientation and dispersion) to experiments

Fig. 6

Comparison of normalized (i.e., to the equilibrium value) indentation force measured by DiSilvestro and Suh [14] (r’ = 1.53 mm, h = 1.28 mm, r = 6.12 mm) with that predicted by the BFPHVE model using the optimized-FE material properties

Fig. 7

Comparison of (a) normalized (i.e., to the equilibrium value) reaction force and (b) lateral displacement (h = 1.28 mm, r = 1.5 mm) measured in the unconfined compression tests of DiSilvestro and Suh [14], with corresponding quantities predicted by the BFPHVE model using the same optimized-FE material properties as used in Fig. 5