The Viscoelastic Standard Nonlinear Solid Model: Predicting the Response of the Lumbar Intervertebral Disk to Low-Frequency Vibrations

Due to the mathematical complexity of current musculoskeletal spine models, there is a need for computationally efficient models of the intervertebral disk (IVD). The aim of this study is to develop a mathematical model that will adequately describe the motion of the IVD under axial cyclic loading as well as maintain computational efficiency for use in future musculoskeletal spine models. Several studies have successfully modeled the creep characteristics of the IVD using the three-parameter viscoelastic standard linear solid (SLS) model. However, when the SLS model is subjected to cyclic loading, it underestimates the load relaxation, the cyclic modulus, and the hysteresis of the human lumbar IVD. A viscoelastic standard nonlinear solid (SNS) model was used to predict the response of the human lumbar IVD subjected to low-frequency vibration. Nonlinear behavior of the SNS model was simulated by a strain-dependent elastic modulus on the SLS model. Parameters of the SNS model were estimated from experimental load deformation and stress-relaxation curves obtained from the literature. The SNS model was able to predict the cyclic modulus of the IVD at frequencies of $0.01Hz$⁠, $0.1Hz$⁠, and $1Hz$⁠. Furthermore, the SNS model was able to quantitatively predict the load relaxation at a frequency of $0.01Hz$⁠. However, model performance was unsatisfactory when predicting load relaxation and hysteresis at higher frequencies (⁠$0.1Hz$ and $1Hz$⁠). The SLS model of the lumbar IVD may require strain-dependent elastic and viscous behavior to represent the dynamic response to compressive strain.