Viscoelastic Anisotropic Finite Element Mixture Model of Articular Cartilage using Viscoelastic Collagen Fibers and Validation with Stress Relaxation Data

Abstract

Viscoelastic Anisotropic Finite Element Mixture Model of Articular Cartilage using Viscoelastic Collagen fibers and Validation with Stress Relaxation Data Matthew Alexander Griebel Experimental results show that collagen fibers exhibit stress relaxation under tension and a highly anisotropic distribution. To further develop the earlier model of Stender [1], the collagen constituent was updated to reflect its intrinsic viscoelasticity and anisotropic distribution, and integrated with an existing mixture model with glycosaminoglycans and ground substance matrix. A two-term Prony series expansion of the quasi-linear viscoelastic model was chosen to model the viscoelastic properties of the collagen fibers. Material parameters were determined by using the simplex method to minimize the sum of squared errors between model results and experimental stress relaxation data of tissue in tension. Collagen elastic fiber modulus was calculated by fitting to the equilibrium data and viscoelastic parameters were determined by fitting to the relaxation curve. Results of newborn (∼1-3 week old) untreated bovine articular cartilage explants from the patellar femoral groove as well as explants cultured in transforming growth factor-β1 (TGF-β1), from both the superficial (∼0-0.5 mm from the articular surface) and middle (∼0.5-1.0 mm from the articular surface) layers were compared to examine the effects of TGFβ1. TGF-β1 has been shown to maintain or even enhance mechanical properties of articular cartilage in compression and tension [2, 3] and this study continues with the hope that it may be used to improve tissue engineering of mature cartilage to better survive implantation in vivo for the successful repair of articular cartilage defects. Results show that TGF-β1 has a maturational effect on collagen, causing the tissue to become stiffer through an increase in elastic collagen fiber modulus and less viscous through shorter relaxation time and less stress relaxation (tissue retained a higher percentage of residual stress). The results of this study further advance the understanding of the effects of location and treatment with TGF-β1.