Quantification of long-term nonlinear stress relaxation of bovine trabecular bone

Abstract

The reliability of computational models in orthopedic biomechanics depends often on the accuracy of the bone material properties. It is widely recognized that the mechanical response of trabecular bone is time-dependent, yet it is often ignored for the sake of simplicity. Previous investigations into the viscoelastic properties of trabecular bone have not explored the relationship between nonlinear stress relaxation and bone mineral density. The inclusion of this behavior could enhance the accuracy of simulations of orthopedic interventions, such as of primary fixation of implants. Although methods to quantify the viscoelastic behavior are known, the time period during which the viscoelastic properties should be investigated to obtain reliable predictions is currently unclear. Therefore, this study aimed to: 1) Investigate the duration of stress relaxation in bovine trabecular bone; 2) construct a material model that describes the nonlinear viscoelastic behavior of uniaxial stress relaxation experiments on trabecular bone; and 3) implement bone density into this model. Uniaxial compressive stress relaxation experiments were performed with cylindrical bovine femoral trabecular bone samples (n = 16) with constant strain held for 24 h. Additionally, multiple stress relaxation experiments with four ascending strain levels with a holding time of 30 min, based on the results of the 24-h experiment, were executed on 18 bovine bone cores. The bone specimens used in this study had a mean diameter of 12.80 mm and a mean height of 28.70 mm. A Schapery and a Superposition model were used to capture the nonlinear stress relaxation behavior in terms of applied strain level and bone mineral density. While most stress relaxation happened in the first 10 min (up to 53 %) after initial compression, the stress relaxation continued even after 24 h. Up to 69 % of stress relaxation was observed at 24 h. Extrapolating the results of 30 min of experimental data to 24 h provided a good fit for accuracy with much improved experimental efficiency. The Schapery and Superposition model were both capable of fitting the repeated stress relaxation in a sample-by-sample approach. However, since bone mineral density did not influence the time-dependent behavior, only the Superposition model could be used for a group-based model fit. Although the sample-by-sample approach was more accurate for an individual specimen, the group based approach is considered a useful model for general application.

Keywords: Bone mineral density; Nonlinear viscoelasticity; Schapery model; Stress relaxation; Superposition model; Time-dependent; Trabecular bone.