The viscoelastic standard nonlinear solid model: predicting the response of the lumbar intervertebral disk to low-frequency vibrations

Stress relaxation behavior of the transition zone in the intervertebral disc

The stress relaxation of the TZ region, located at the interface of the Annulus Fibrosus (AF) and Nucleus Pulposus (NP) of the disc, and how its stress is relaxed compared to the adjacent regions is unknown. The current study aimed to identify the TZ stress relaxation properties under different strain magnitudes (0.2, 0.4, and 0.6 mm/mm) and compared the TZ stress relaxation characteristics to the NP and inner AF (IAF) regions at a specific strain magnitude (0.6 mm/mm). The results of the current study revealed that the TZ region exhibited different stress relaxation properties under various strain magnitudes with significantly higher initial (p < 0.008) and reduced stresses (marginally; p = 0.06) at higher strains. Our experimental stress relaxation data revealed a significantly higher equilibrium stress for the IAF compared to the TZ and NP regions (p < 0.001) but not between the TZ and NP regions (p = 0.7). We found that NP radial stress relaxed significantly faster (p < 0.04) than the TZ and NP. Additionally, the current study proposed a simple mathematical model and identified that, consistent with experimental data, the overall effect of region on both the level of decayed stress and the rate at which stress is relaxed was significant (p < 0.006). The current study found a similar stress relaxation characteristic between the NP and TZ regions, while IAF exhibited different stress relaxation properties. It is possible that this mismatch in stress relaxation acts as a shape transformation mechanism triggered by viscoelastic behavior. STATEMENT OF SIGNIFICANCE: Our understanding of the biomechanical properties of the transition zone (TZ) in the IVD, a region at the interface of the Nucleus Pulposus (NP) and Annulus Fibrosus (AF), is sparse. Unfortunately, there are no current studies that investigate the TZ stress relaxation properties and how stress is relaxed in the TZ compared to the adjacent regions. For the first time, the current study characterized the stress relaxation properties of the TZ and described how the TZ stress is relaxed compared to its adjacent regions.

Strain rate-dependent failure mechanics of the intervertebral disc under tension/compression and constitutive analysis

BACKGROUND

The intervertebral disc exhibits not only strain rate dependence (viscoelasticity), but also significant asymmetry under tensile and compressive loads, which is of great significance for understanding the mechanism of lumbar disc injury under physiological loads.

OBJECTIVE

In this study, the strain rate sensitive and tension-compression asymmetry of the intervertebral disc were analyzed by experiments and constitutive equation.

METHOD

The Sheep intervertebral disc samples were divided into three groups, in order to test the strain rate sensitive mechanical behavior, and the internal displacement as well as pressure distribution.

RESULTS

The tensile stiffness is one order of magnitude smaller than the compression stiffness, and the logarithm of the elastic modulus is approximately linear with the logarithm of the strain rate, showing obvious tension-compression asymmetry and rate-related characteristics. In addition, the sensitivity to the strain rate is the same under these two loading conditions. The stress-strain curves of unloading and loading usually do not coincide, and form a Mullins effect hysteresis loop. The radial displacement distribution is opposite between the anterior and posterior region, which is consistent with the stress distribution. By introducing the damage factor into ZWT constitutive equation, the rate-dependent viscoelastic and weakening behavior of the intervertebral disc can be well described.

Constitutive Equations for Analyzing Stress Relaxation and Creep of Viscoelastic Materials Based on Standard Linear Solid Model Derived with Finite Loading Rate

The viscoelastic properties of materials such as polymers can be quantitatively evaluated by measuring and analyzing the viscoelastic behaviors such as stress relaxation and creep. The standard linear solid model is a classical and commonly used mathematical model for analyzing stress relaxation and creep behaviors. Traditionally, the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model are derived using the assumption that the loading is a step function, implying that the loading rate used in the loading process of stress relaxation and creep tests is infinite. Using such constitutive equations may cause significant errors in analyses since the loading rate must be finite (no matter how fast it is) in a real stress relaxation or creep experiment. The purpose of this paper is to introduce the constitutive equations for analyzing stress relaxation and creep behaviors based on the standard linear solid model derived with a finite loading rate. The finite element computational simulation results demonstrate that the constitutive equations derived with a finite loading rate can produce accurate results in the evaluation of all viscoelastic parameters regardless of the loading rate in most cases. It is recommended that the constitutive equations derived with a finite loading rate should replace the traditional ones derived with an infinite loading rate to analyze stress relaxation and creep behaviors for quantitatively evaluating the viscoelastic properties of materials.

Effects of cyclic compression on intervertebral disc metabolism using a whole-organ rat tail model

Many factors contribute to the development and progression of intervertebral disc (IVD) degeneration. This study was designed to assess the effects of compressive load magnitude on IVD metabolism. It was hypothesized that as load magnitude increased, there would be a significant increase in release of proinflammatory and degradative biomarkers, and a significant decrease in tissue proteoglycan (GAG) and collagen contents compared with unloaded controls. IVD whole organ functional spinal units (FSU) consisting of cranial and caudal body halves, cartilage endplates, and IVD (n = 36) were harvested from the tails of six Sprague Dawley rats, and FSUs were cultured at 0.0 MPa, 0.5 MPa, or 1.0 MPa at 0.5 Hz for 3 days. After culture, media were collected for biomarker analysis and FSUs were analyzed for extracellular matrix composition. Significant differences were determined using a one-way analysis of variance or Kruskal-Wallis test and post hoc analyses. Media concentrations of IFN-γ, IL-6, IL-1β, and MMP-8 were significantly higher in the 0.5 MPa compared with the 0.0 MPa group. Media concentrations of PGE2 and TIMP-1 were significantly higher in the 1.0 MPa group compared with the 0.0 MPa group, and media PGE2 was significantly higher in the 1.0 MPa group compared with the 0.5 MPa group. Media GAG content was significantly higher in the 1.0 MPa group compared with the 0.0 MPa group, and percent GAG in the tissue was significantly lower in 0.5 MPa and 1.0 MPa groups compared with the 0.0 MPa group. Clinical Significance: These data suggest that there are magnitude-dependent inflammatory and degradative IVD responses to cyclic loading, which may contribute to IVD degeneration.

Creep experimental study on the lumbar intervertebral disk under vibration compression load

The intervertebral disk cushions the load generated by human activity and absorbs energy to keep the spine moving steadily. Vibration condition is one of the important causes of disk degeneration. Creep experiments using the sheep lumbar intervertebral disk were carried out under vibration compression. Regularities of the strain of the disk with time were obtained and compared with those of static load. The influence of vibration frequency and time on the creep properties of the intervertebral disk was analyzed. An intervertebral disk three-parameter solid creep constitutive model considering vibration factors was established and the parameters in the model were identified. The results show that the strain of the lumbar intervertebral disk exhibits an exponential relationship with time and is unrelated to static compression or vibration load. Under the same vibration amplitude, the creep increases with vibration frequency and the relationship between them is nonlinear. The vibration frequency has a significant effect on the strain. The creep rate decreases gradually with time and is obviously influenced by vibration frequency at low vibration amplitudes. The creep prediction results obtained using the constitutive model with the time-varying material parameters are in good agreement with the experimental results. The two elastic moduli in the model decrease with time and the viscosity coefficient increases with time.

Hybrid Rigid-Deformable Model for Prediction of Neighboring Intervertebral Disk Loads During Flexion Movement After Lumbar Interbody Fusion at L3-4 Level

Knowledge of spinal loads in neighboring disks after interbody fusion plays an important role in the clinical decision of this treatment as well as in the elucidation of its effect. However, controversial findings are still noted in the literature. Moreover, there are no existing models for efficient prediction of intervertebral disk stresses within annulus fibrosus (AF) and nucleus pulposus (NP) regions. In this present study, a new hybrid rigid-deformable modeling workflow was established to quantify the mechanical stress behaviors within AF and NP regions of the L1-2, L2-3, and L4-5 disks after interbody fusion at L3-4 level. The changes in spinal loads were compared with results of the intact model without interbody fusion. The fusion outcomes revealed maximal stress changes (10%) in AF region of L1-2 disk and in NP region of L2-3 disk. The minimal stress change (1%) is noted at the NP region of the L1-2 disk. The validation of simulation outcomes of fused and intact lumbar spine models against those of other computational models and in vivo measurements showed good agreements. Thus, this present study may be used as a novel design guideline for a specific implant and surgical scenario of the lumbar spine disorders.

Load-relaxation properties of the human trunk in response to prolonged flexion: measuring and modeling the effect of flexion angle

Experimental studies suggest that prolonged trunk flexion reduces passive support of the spine. To understand alterations of the synergy between active and passive tissues following such loadings, several studies have assessed the time-dependent behavior of passive tissues including those within spinal motion segments and muscles. Yet, there remain limitations regarding load-relaxation of the lumbar spine in response to flexion exposures and the influence of different flexion angles. Ten healthy participants were exposed for 16 min to each of five magnitudes of lumbar flexion specified relative to individual flexion-relaxation angles (i.e., 30, 40, 60, 80, and 100%), during which lumbar flexion angle and trunk moment were recorded. Outcome measures were initial trunk moment, moment drop, parameters of four viscoelastic models (i.e., Standard Linear Solid model, the Prony Series, Schapery's Theory, and the Modified Superposition Method), and changes in neutral zone and viscoelastic state following exposure. There were significant effects of flexion angle on initial moment, moment drop, changes in normalized neutral zone, and some parameters of the Standard Linear Solid model. Initial moment, moment drop, and changes in normalized neutral zone increased exponentially with flexion angle. Kelvin-solid models produced better predictions of temporal behaviors. Observed responses to trunk flexion suggest nonlinearity in viscoelastic properties, and which likely reflected viscoelastic behaviors of spinal (lumbar) motion segments. Flexion-induced changes in viscous properties and neutral zone imply an increase in internal loads and perhaps increased risk of low back disorders. Kelvin-solid models, especially the Prony Series model appeared to be more effective at modeling load-relaxation of the trunk.