Nonlinear Lumped-Parameter Model of the Lumbar Intervertebral Disc: A Study on Viscoelastic Deformation and Three-Dimensional Modeling of the Spine

Abstract

Due to the mathematical complexity of current musculoskeletal spine models, there is a need for computationally-efficient models of the intervertebral disc (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 and three-dimensional quasi-static loading as well as maintain computational efficiency for use in future musculoskeletal spine models.

A viscoelastic standard nonlinear solid (SNS) model is introduced within this study. It was developed to predict the axial response of the human lumbar IVD subjected to low-frequency vibration. Nonlinear axial behavior of the SNS model was simulated by a strain-dependent elastic modulus on the standard linear solid (SLS) model.

The SNS model was able to predict the dynamic modulus of the IVD for frequencies of 0.01, 0.1, and 1 Hz. Furthermore, the model was able to quantitatively predict the load relaxation at a frequency of 0.01 Hz. However, model performance was unsatisfactory when predicting load relaxation and hysteresis at higher frequencies (0.1 Hz and 1 Hz). Results suggest that the standard solid model may require strain-dependent elastic and viscous behavior to represent the dynamic response to compressive strain.

The SNS model was expanded to a three-dimensional elastic model by adding a matrix of spring elements in parallel with the SNS model. The geometry and orientation of the added elements represent the regional variations in stiffness and physiologic fiber angle. Results suggest that lordotic posture may be advantageous when modeling the intervertebral joint (IVJ) behavior.