Mechanical response of a simple finite element model of the intervertebral disc under complex loading

A simple axisymmetric finite element model of a human spine segment containing two adjacent vertebrae and the intervening intervertebral disc was constructed. The bodies and disc were modeled by three substructures; one to represent each of the vertebral bodies, the annulus fibrosus, and the nucleus pulposus. A semi-analytic technique was used to maintain the computational economies of a two-dimensional analysis when non- axisymmetric loads were imposed on the model. The response of the model to compression, shear, torsion and bending loads applied to the superior vertebral body was examined to determine the effects of disc geometry and material properties on response. Comparisons of model responses with experimentally measured responses were made to estimate material property values for which model behaviors are in agreement with measured behaviors.     

The internal mechanics of the intervertebral disc under cyclic loading

The mechanics of the intervertebral disc (IVD) under cyclic loading are investigated via a one-dimensional poroelastic model and experiment. The poroelastic model, based on that of Biot (J. Appl. Phys. 12 (1941) 155; J. Appl. Mech. 23 (1956) 91), includes a power-law relation between porosity and permeability, and a linear relation between the osmotic potential and solidity. The model was fitted to experimental data of the unconfined IVD undergoing 5 cyclic loads of 20 min compression by an applied stress of 1MPa, followed by 40 min expansion. To obtain a good agreement between experiment and theory, the initial elastic deformation of the IVD, possibly associated with the bulging of the IVD into the vertebral bodies or laterally, was removed from the experimental data. Many combinations of the permeability-porosity relationship with the initial osmotic potential (pi(i)) were investigated, and the best-fit parameters for the aggregate modulus (H(A)) and initial permeability (k(i)) were determined. The values of H(A) and k(i) were compared to literature values, and agreed well especially in the context of the adopted high-stress testing regime, and the strain related permeability in the model.     

Application of the polynomial chaos expansion to approximate the homogenised response of the intervertebral disc

A possibility to simulate the mechanical behaviour of the human spine is given by modelling the stiffer structures, i.e. the vertebrae, as a discrete multi-body system (MBS), whereas the softer connecting tissue, i.e. the softer intervertebral discs (IVD), is represented in a continuum-mechanical sense using the finite-element method (FEM). From a modelling point of view, the mechanical behaviour of the IVD can be included into the MBS in two different ways. They can either be computed online in a so-called co-simulation of a MBS and a FEM or offline in a pre-computation step, where a representation of the discrete mechanical response of the IVD needs to be defined in terms of the applied degrees of freedom (DOF) of the MBS. For both methods, an appropriate homogenisation step needs to be applied to obtain the discrete mechanical response of the IVD, i.e. the resulting forces and moments. The goal of this paper was to present an efficient method to approximate the mechanical response of an IVD in an offline computation. In a previous paper (Karajan et al. in Biomech Model Mechanobiol 12(3):453–466, 2012), it was proven that a cubic polynomial for the homogenised forces and moments of the FE model is a suitable choice to approximate the purely elastic response as a coupled function of the DOF of the MBS. In this contribution, the polynomial chaos expansion (PCE) is applied to generate these high-dimensional polynomials. Following this, the main challenge is to determine suitable deformation states of the IVD for pre-computation, such that the polynomials can be constructed with high accuracy and low numerical cost. For the sake of a simple verification, the coupling method and the PCE are applied to the same simplified motion segment of the spine as was used in the previous paper, i.e. two cylindrical vertebrae and a cylindrical IVD in between. In a next step, the loading rates are included as variables in the polynomial response functions to account for a more realistic response of the overall viscoelastic intervertebral disc. Herein, an additive split into elastic and inelastic contributions to the homogenised forces and moments is applied.     

Degeneration of the intervertebral disc

The intervertebral disc is a cartilaginous structure that resembles articular cartilage in its biochemistry, but morphologically it is clearly different. It shows degenerative and ageing changes earlier than does any other connective tissue in the body. It is believed to be important clinically because there is an association of disc degeneration with back pain. Current treatments are predominantly conservative or, less commonly, surgical; in many cases there is no clear diagnosis and therapy is considered inadequate. New developments, such as genetic and biological approaches, may allow better diagnosis and treatments in the future.     

Modeling of time-dependent force response of fingertip to dynamic loading

An extended exposure to repeated loading on fingertip has been associated to many vascular, sensorineural, and musculoskeletal disorders in the fingers, such as carpal tunnel syndrome, hand-arm vibration syndrome, and flexor tenosynovitis. A better understanding of the pathomechanics of these sensorineural and vascular diseases in fingers requires a formulation of a biomechanical model of the fingertips and analyses to predict the mechanical responses of the soft tissues to dynamic loading. In the present study, a model based on finite element techniques has been developed to simulate the mechanical responses of the fingertips to dynamic loading. The proposed model is two-dimensional and incorporates the essential anatomical structures of a finger: skin, subcutaneous tissue, bone, and nail. The skin tissue is assumed to be hyperelastic and viscoelastic. The subcutaneous tissue was considered to be a nonlinear, biphasic material composed of a hyperelastic solid and an invicid fluid, while its hydraulic permeability was considered to be deformation dependent. Two series of numerical tests were performed using the proposed finger tip model to: (a) simulate the responses of the fingertip to repeated loading, where the contact plate was assumed to be fixed, and the bone within the fingertip was subjected to a prescribed sinusoidal displacement in vertical direction; (b) simulate the force response of the fingertip in a single keystroke, where the keyboard was composed of a hard plastic keycap, a rigid support block, and a nonlinear spring. The time-dependent behavior of the fingertip under dynamic loading was derived. The model predictions of the time-histories of force response of the fingertip and the phenomenon of fingertip separation from the contacting plate during cyclic loading agree well with the reported experimental observations.     

Reduced nucleus pulposus glycosaminoglycan content alters intervertebral disc dynamic viscoelastic mechanics

The intervertebral disc functions over a range of dynamic loading regimes including axial loads applied across a spectrum of frequencies at varying compressive loads. Biochemical changes occurring in early degeneration, including reduced nucleus pulposus glycosaminoglycan content, may alter disc mechanical behavior and thus may contribute to the progression of degeneration. The objective of this study was to determine disc dynamic viscoelastic properties under several equilibrium loads and loading frequencies, and further, to determine how reduced nucleus glycosaminoglycan content alters dynamic mechanics. We hypothesized that (1) dynamic stiffness would be elevated with increasing equilibrium load and increasing frequency, (2) the disc would behave more elastically at higher frequencies, and finally, (3) dynamic stiffness would be reduced at low equilibrium loads under all frequencies due to nucleus glycosaminoglycan loss. We mechanically tested control and chondroitinase ABC injected rat lumbar motion segments at several equilibrium loads using oscillatory loading at frequencies ranging from 0.05 to 5 Hz. The rat lumbar disc behaved non-linearly with higher dynamic stiffness at elevated compressive loads irrespective of frequency. Phase angle was not affected by equilibrium load, although it decreased as frequency was increased. Reduced glycosaminoglycan decreased dynamic stiffness at low loads but not at high equilibrium loads and led to increased phase angle at all loads and frequencies. The findings of this study demonstrate the effect of equilibrium load and loading frequencies on dynamic disc mechanics and indicate possible mechanical mechanisms through which disc degeneration can progress.     

Pathophysiology of the human intervertebral disc

Intervertebral disc degeneration is a common invalidating disorder that can affect the musculoskeletal apparatus in both younger and older ages. The chief component of the intervertebral disc is the highly organized extracellular matrix; maintenance of its organization is essential for correct spinal mechanics. The matrix components, mainly proteoglycans and collagens, undergo a slow and continuous cell-mediated turnover process that enables disc cells to adapt their environment to external stimuli. Cellular senescence and a history of chronic abnormal loading can upset this balance, leading to progressive tissue failure that results in disc degeneration. Although biological treatment approaches to disc repair are still far to come, advances in our understanding of disc biochemistry and in defining the role of genetic inheritance have provided a starting point for developing new concepts in the diagnosis, therapy and prevention of disc degeneration.     

Biological impact of the fibroblast growth factor family on articular cartilage and intervertebral disc homeostasis

Two members of the fibroblast growth factor (FGF) family, basic FGF (bFGF) and FGF-18, have been implicated in the regulation of articular and intervertebral disc (IVD) cartilage homeostasis. Studies on bFGF from a variety of species have yielded contradictory results with regards to its precise role in cartilage matrix synthesis and degradation. In contrast, FGF-18 is a well-known anabolic growth factor involved in chondrogenesis and articular cartilage repair. In this review, we examined the biological actions of bFGF and FGF-18 in articular and IVD cartilage, the specific cell surface receptors bound by each factor, and the unique signaling cascades and molecular pathways utilized to exert their biological effects. Evidence suggests that bFGF selectively activates FGF receptor 1 (FGFR1) to exert degradative effects in both human articular chondrocytes and IVD tissue via upregulation of matrix-degrading enzyme activity, inhibition of matrix production, and increased cell proliferation resulting in clustering of cells seen in arthritic states. FGF-18, on the other hand, most likely exerts anabolic effects in human articular chondrocytes by activating FGFR3, increasing matrix formation and cell differentiation while inhibiting cell proliferation, leading to dispersed cells surrounded by abundant matrix. The results from in vitro and in vivo studies suggest the potential usefulness of bFGF and FGFR1 antagonists, as well as FGF-18 and FGFR3 agonists, as potential therapies to prevent cartilage degeneration and/or promote cartilage regeneration and repair in the future.     

Thermal analysis of the human intervertebral disc

Intervertebral disc (IVD) degeneration is one of the most common musculuskeletal disorders affecting western society. Degeneration alters the morphology and the mechanical properties of the discs. According to previous reports, DSC proved to be a suitable method for the demonstration of thermal consequences of local as well as global conformational changes in the structure of the human intervertebral discs. In the present study, a wide spectrum of degenerated IVD was examined by DSC. The results suggest that definitive differences exist between the stages of disc degeneration in calorimetric measures.     

Neutral proteinases of the human intervertebral disc

Disc tissue consisting of pooled annuli fibrosus and nuclei pulposus from the cadaver of an adolescent aged 19 years was extracted with 4.0 M Gu-HCl. Proteins of low buoyant density (p less than or equal to 1.38 g/ml) containing the disc enzymes and inhibitors were separated from proteoglycans of high buoyant density (p greater than or equal to 1.50 g/ml) by density gradient ultracentrifugation. Sephadex G-75F gel chromatography followed by trypsin affinity chromatography was then used to resolve disc proteolytic and trypsin inhibitory activities. The results obtained were strongly suggestive of the presence of a high molecular weight zymogen which upon activation generated a population of smaller molecular weight proteinases. The disc proteinases obtained by this process showed similar properties in terms of: their pH optima (7.4-7.6); their inhibition patterns by class-specific proteinase inhibitors; their variation of activity as a function of NaCl and lysine concentrations; and the hydrodynamic size of their proteoglycan degradation products. The activated disc neutral proteinase demonstrated many characteristics in common with plasmin; however, unlike the latter, the disc proteinases also showed some calcium dependence.     

The proteoglycans of the canine intervertebral disc

The high-buoyant-density proteoglycans of the nucleus pulposus and annulus fibrosus of the beagle intervertebral disc have been isolated by CsCl density gradient ultracentrifugation. The sulphated proteoglycans were labelled in vivo with 35SO4, 24 h and 60 days prior to killing. The hydrodynamic size and aggregation of the 24 h, 60 day and resident (from hexuronic acid and hexosamine analysis) proteoglycan subunit populations were determined by Sepharose CL-2B chromatography in the presence or absence of excess hyaluronic acid. The hydrodynamic size of the keratan sulphate-proteoglycan core protein complexes were also determined by Sepharose CL-2B chromatography after chondroitinase ABC digestion of proteoglycans. When initially synthesised (24 h) or after 60 days, the percentage aggregation and hydrodynamic size of the proteoglycans derived from the annulus fibrosus were larger than those present in the nucleus pulposus. Hexosamine, hexuronic and protein determination of the high-buoyant-density fractions showed that the proteoglycans of the nucleus pulposus were richer in chondroitin sulphate than those in the annulus. However there was no difference in Mr of the chondroitin sulphate and keratan sulphate attached to the proteoglycans of the two disc regions, nor were differences detected by HPLC between the proportions of chondroitin 4-sulphate and chondroitin 6-sulphate present in these high-density fractions. In contrast, the low-buoyant-density (1.54 greater than p greater than 1.45) proteoglycan fractions and tissue residues remaining after 4 M GuHCl extraction were found to contain dermatan sulphate, suggesting the presence of a third proteoglycan species possibly associated with the collagen of the fibrocartilagenous matrix.     

Proteoglycans of the intervertebral disc. Absence of degradation during the isolation of proteoglycans from the intervertebral disc.

Proteoglycans extracted with 4M-guanidinium chloride from pig intervetebral discs, and purified by equilibrium density-gradient centrifugation in CsCl, were of smaller hydrodynamic size than those extracted and purified in the same way from the laryngeal cartilage of the same animal. Whether this difference in size arose from degradation during the extraction and purification of the proteoglycans of the disc was investigated. Purified proteoglycans labelled either in the chondroitin sulphate chains or in the core protein were obtained from laryngeal cartilage by short-term organ culture. These labelled proteoglycans were added at the beginning of the extraction of the disc proteoglycans, and labelled cartilage and unlabelled disc proteoglycans were isolated and purified together. There was no appreciable loss of radioactivity after density-gradient centrifugation nor decrease in hydrodynamic size of the labelled cartilage proteoglycans on chromatography on Sepharose 2B, when these were present during the extraction of disc proteoglycans. It is concluded that disc proteoglycans are intrinsically of smaller size than cartilage proteoglycans and this difference in size does not arise from degradation during the extraction.     

Calibration of hyperelastic material properties of the human lumbar intervertebral disc under fast dynamic compressive loads

Under fast dynamic loading conditions (e.g. high-energy impact), the load rate dependency of the intervertebral disc (IVD) material properties may play a crucial role in the biomechanics of spinal trauma. However, most finite element models (FEM) of dynamic spinal trauma uses material properties derived from quasi-static experiments, thus neglecting this load rate dependency. The aim of this study was to identify hyperelastic material properties that ensure a more biofidelic simulation of the IVD under a fast dynamic compressive load. A hyperelastic material law based on a first-order Mooney-Rivlin formulation was implemented in a detailed FEM of a L2-L3 functional spinal unit (FSU) to represent the mechanical behavior of the IVD. Bony structures were modeled using an elasto-plastic Johnson-Cook material law that simulates bone fracture while ligaments were governed by a viscoelastic material law. To mimic experimental studies performed in fast dynamic compression, a compressive loading velocity of 1 m/s was applied to the superior half of L2, while the inferior half of L3 was fixed. An exploratory technique was used to simulate dynamic compression of the FSU using 34 sets of hyperelastic material constants randomly selected using an optimal Latin hypercube algorithm and a set of material constants derived from quasi-static experiments. Selection or rejection of the sets of material constants was based on compressive stiffness and failure parameters criteria measured experimentally. The two simulations performed with calibrated hyperelastic constants resulted in nonlinear load-displacement curves with compressive stiffness (7335 and 7079 N/mm), load (12,488 and 12,473 N), displacement (1.95 and 2.09 mm) and energy at failure (13.5 and 14.7 J) in agreement with experimental results (6551 ± 2017 N/mm, 12,411 ± 829 N, 2.1 ± 0.2 mm and 13.0 ± 1.5 J respectively). The fracture pattern and location also agreed with experimental results. The simulation performed with constants derived from quasi-static experiments showed a failure energy (13.2 J) and a fracture pattern and location in agreement with experimental results, but a compressive stiffness (1580 N/mm), a failure load (5976 N) and a displacement to failure (4.8 mm) outside the experimental corridors. The proposed method offers an innovative way to calibrate the hyperelastic material properties of the IVD and to offer a more realistic simulation of the FSU in fast dynamic compression.     

Aquaporin expression in the human intervertebral disc

The nucleus pulposus (NP) of the human intervertebral disc (IVD) is a hyperosmotic tissue that is subjected to daily dynamic compressive loads. In order to survive within this environment the resident chondrocyte-like cells must be able to control their cell volume, whilst also controlling the anabolism and catabolism of their extra-cellular matrix. Recent studies have demonstrated expression of a range of bi-directional, transmembrane water and solute transporters, named aquaporins (AQPs), within chondrocytes of articular cartilage. The aim of this study was to use immunohistochemsitry to investigate the expression of aquaporins 1, 2 and 3 within the human IVD. Results demonstrated expression of both AQP-1 and -3 by cells within the NP and inner annulus fibrosus (AF), while outer AF cells lacked expression of AQP-1 and showed very low numbers of AQP-3 immunopositive cells. Cells from all regions were negative for AQP-2. Therefore this study demonstrates similarities in the phenotype of NP cells and articular chondrocytes, which may be due to similarities in tissue osmolarity and mechanobiology. The decrease in expression of AQPs from the NP to the outer AF may signify changes in cellular phenotype in response to differences in mechanbiology, osmolarity and hydration between the gelatinous NP and the fibrous AF.     

A simple model for the function of proteoglycans and collagen in the response to compression of the intervertebral disc.

The nucleus pulposus of the intervertebral disc exerts a pressure which enables it to support axial compression when contained by the annulus fibrosus. The disc was modelled as a thick-walled cylindrical pressure vessel in which the nucleus was contained radially by the annulus. As a result, the stress in the annulus had radial (compressive) as well as tangential (tensile) components. The radial stress at a given point in the annulus was considered to be balanced by the internal pressure which is expected to arise from the attraction of water by proteoglycans. There was a reasonable agreement between the calculated radial stress distribution and published results on the distribution of water within the annulus. As the internal pressure is expected to be isotropic, the annulus was expected to contribute to the axial resistance to compression of the disc; this contribution would be equal, in magnitude, to the radial stress. Predictions of the pressure distribution within the annulus were similar to published experimental measurements made in the radial and axial directions. The tangential stress within the annulus was considered to arise from the restoring stress in its strained collagen fibrils.