The degeneration of lumbar intervertebral discs (IVDs) has been implicated as a possible cause of low back pain which affects more than 600 million people worldwide. Abnormal mechanical loading is thought to be a primary etiological factor leading to degenerative changes in the discs. The abnormal mechanical loading could initiate disc degeneration through two pathways:​ causing the failure of disc structures, and disturbing the balance between cellular anabolic and catabolic activities. The latter one is thought to be the main pathway. However, the mechanism of extracellular mechanotransduction is not fully understood, neither the abnormal mechanical loading is quantified. The objectives of this dissertation were to develop mathematical models to characterize the biomechanics and mechanobiology of discs in order to quantify the abnormal mechanical loading and delineate how the cells perceive and respond to mechanical stimuli.

A new constitutive model for hydration-dependent aggregate modulus of soft and hydrated materials was developed based on the biphasic theory and transport models, and it was validated with experimental results on hydrogel and cartilage tissues. An anisotropic multiphysics model was developed based on the continuum mixture theory and employed to characterize the nonlinear couplings among anisotropic and large solid deformation, anisotropic transport of interstitial water and solutes, and the electro-osmotic effect in the disc. Numerical simulations demonstrated that this model is capable of systematically predicting the mechanical and electrochemical signals within the disc under various loading conditions. This anisotropic multiphysics model was then further developed by incorporating a continuum damage model to investigate the initiation and propagation of damage in the annulus fibrosus (AF). The simulated results showed that damages initiate in different regions under compression, flexion, and compression-flexion. The posterior AF was shown to be more susceptible to structure failures as it may be damaged under all three loading conditions with sufficient high magnitudes.

A cell volume dependent glycosaminoglycan (GAG) synthesis mathematical model was developed to quantitatively describe the response of cell biosynthetic behaviors to extracellular mechanical stimuli. It was found that our proposed mathematical model is able to describe the change of GAG synthesis rate in isolated cells and in cartilage with variations of the osmotic loading or mechanical loading. Then, a multiscale mathematical model was developed based on the cell volume dependent GAG synthesis model and biphasic theory to quantify the effect of mechanical loading on GAG synthesis. This multiscale mathematical model was shown to be capable of predicting the effect of static load (creep load) on GAG synthesis in bovine tail discs. This model was also used to investigate the effect of static and diurnal loads on GAG synthesis.

These models developed in this dissertation provide numerical approaches to quantify the abnormal mechanical loading which may cause disc degeneration. These mathematical models are important in understanding the etiology of disc degeneration, as well as in preventing and treating disc degeneration. These models are also important in designing scaffolds and optimizing loading conditions to grow engineered disc tissues.